INDENTATION CONTACTMECHANICS− Micro/Nano Physics of Materials −(REVISED EDITION)

Mototsugu Sakai

Preface
This textbook is intended to make considerations on the continuum mechanics of indentation contact
in micro/nano-scales, experimental details for measuring/analyzing mechanical properties (elastic,
plastic, elastoplastic, and viscoelastic properties), instrumented indentation apparatuses, and on thematerials physics in micro/nano-regimes that makes a great contribution to developing various typesof engineering materials and composites through designing their microstructures.
The contact hardness has been conventionally defined as a mean contact pressure calculated fromthe applied indentation load. The most widely utilized and well-known hardness number is the Vickers
hardness; this hardness number is determined by pressing a Vickers indenter (a tetrahedral pyramidal
diamond tip) on a material, and then by calculating the apparent contact pressure from the indentationload divided by the total contact area of the residual impression. Besides the Vickers indenter,depending on the tip-geometry of indenter, various types of indenters including Berkovich (trigonal
pyramid), Brinell (sphere), and Rockwell (cone with a rounded-tip) indenters have also been widely
acknowledged.
The history of the concept, test procedure, and the physics of contact hardness number date back to
the mid-19th century, in which the hardness number was defined as the indentation load divided by the
total contact area of residual impression. However, on the basis of the experimental results forspherical indentation on various metallic alloys along with the considerations on the geometricalsimilarity of indentation contact, in 1908, E. Meyer proposed the concept of “contact hardness, theMeyer hardness H M ” as a mean contact pressure defined by the applied load divided by the projectedcontact area of residual impression. The Meyer hardness of very ductile materials is closely related to
the yield strength Y via the relation of H M = cY with the constraint factor c . In 1920s, C.
Prandtle examined the constraint factor in relation to the plastic flow beneath a two-dimensional flatpunch, and then R. Hill extended his considerations to what is now known as the slip-line field theory,leading to the c -value of 2.57. Based on their extensive as well as intensive research activities, theMeyer hardness H M has long been recognized as a “measure of the yield strength Y ” of ductile
metals.

Since the mid-20th, the engineering of organic polymers as well as of ceramic materials has

significantly been progressed. Due to a significant elastic deformation of these engineering materials,
their Meyer hardness numbers are by no means the measure of yield strength; the Meyer hardness isgiven by a function of the elastic modulus (Young’s modulus E ) and the yield strength Y . Thoroughunderstanding of the materials physics of Meyer hardness will be given in this textbook for thematerials including perfectly elastic, elastoplastic, fully plastic, as well as time-dependent viscoelastic
materials.
In 1881, H. Hertz, 24 years old at that time, published a classic paper, On the contact of elastic
solids, in which he made theoretical considerations on the contact pressure distributions of sphericalisolids in order to understand the optical interference fringe patterns at contact. This paper founded thepresent contact mechanics as the “Hertzian contact theory”. Four years after his paper, in 1885, J.
Boussinesq made a classical approach to finding the elastic stresses and their spatial distributions
induced by arbitrary surface tractions on the basis of potential theory.

Though the Boussinesq’s

theoretical framework included the Hertzian contact problem as a specific case, its analytical solution
was first derived by A.E.H. Love in 1930s, and then by I.N. Sneddon in 1960s by applying the Hankel
transforms to the Airy’s/Love’s stress functions, having led to the basis of the present elastic theory
for conical/pyramidal indentation contact problems.
The classic book, “Contact Mechanics”, published by K.L Johnson in 1985, encompasses various
types of contact problems in a systematic manner, including the Hertzian contact problems on the basisof continuum mechanics, but not deeply discussing the materials physics. Accordingly, none ofconsiderations were made on the materials physics of elastoplastic/viscoelastic deformation and flow
induced by indentation contact.

On the other hand, D. Tabor, through his book “Hardness of Metals”

published in 1951, discussed in an intensive manner the correlation between the elastoplastic
characteristics of metals and their indentation contact behavior, while no discussions were made forthe contact behavior of organic polymers and engineering ceramic materials in micro/nano-regimes,
due to the significant development of these engineering materials that has been made after the
publication of his textbook.
In 1980’s, several types of conventional instrumented indentation apparatuses came onto the market.
These apparatuses can measure the indentation load P and the associated penetration depth h in
micro/nano-scales but cannot determine the indentation contact-area under the applied load.

This

incapability of the contact-area determination in experiments is fatal, since none of material
characteristics (elastic modulus, yield strength, any of viscoelastic functions, etc.) cannot be obtainedin a quantitative manner without the experimental information on the in-situ contact-area.

We haveto make several undesirable approximations/assumptions prior to estimating the contact-area in theseconventional instrumented indentation apparatuses in order to determine the material characteristics.
Due to the axisymmetric nature of indentation, we need the information not only on the deformation
along the loading direction, i.e., the penetration depth, but also on the deformation perpendicular tothe loading axis, i.e., the contact area, in order to describe the indentation-induced surface deformationand flow. Accordingly, the quantitative information on the indentation contact-area is very essential indetermining any of material characteristics. To overcome these difficulties included in the present
conventional instrumented indentation systems, an instrumented indentation microscope, as a newgeneration of instrumented indentation apparatuses, was design and fabricated by T. Miyajima and M.
Sakai in 2003. This indentation microscope can determine in experiment the indentation contact-areaalong with the load and the depth of penetration. Furthermore, once we utilized the indentation
microscope, we can readily determine the elastic/elastoplastic material characteristics, as well asii

quantitatively analyze any-types of viscoelastic functions by conducting computer-assisted indentation
tests for the applied load, depth of penetration, as well as the applied contact area as functions of time.
Up to the present date, have not been published none of the appropriate textbooks that deal with themicro-nano materials physics based on the continuum mechanics of indentation contact. Throughout
this textbook, therefore, the author intends to discuss the materials physics as well as the detailed
considerations on the testing apparatuses, quantitative analyses of experimental data, etc. on the basisof continuum mechanics of indentation contact.

In the first several chapters, the continuum

mechanics of indentation contact will be given, and then in the subsequent chapters, discussed will be
the micro-nano materials characteristics (elastic, elastoplastic, viscoelastic properties) of several typesof engineering materials by the use of the experimental data obtained with the conventional indentation
apparatus as well as the instrumented indentation microscope. Intensive considerations on the finite
element analyses will also be given that are definitely invaluable for the considerations on the materials
physics of engineering materials with complicated microstructures such as film/substrate laminates.
The author acknowledges the collaborations of his research stuffs and students in The Toyohashi
University of Technology. Without their contributions, the author cannot complete this textbook. In
particular, the author appreciates the great contribution of Dr. T. Miyajima (National Institute ofAdvanced Industrial Science and Technology (AIST)) who successfully designed and fabricated theworld’s first instrumented indentation microscope.

As will be repeatedly emphasized in this book,the quantitative determination of the contact area by the use of the instrumented indentation
microscope is very essential in experiments for determining the micro-nano material characteristics
without making any of undesirable assumption and approximation. The potentials of the instrumented
indentation microscope will play an extremely significant role in establishing the theoretical as well
as experimental frameworks of indentation contact mechanics.

M. Sakai(Revised in January, 2020)

iii

iv

CONTENTS
PREFACE
1 General Concept of Deformation -Strain and Stress -1

2 Continuum Mechanics of Perfectly Elastic Body
2.1 Constitutive Equation

7

2.2 Plane-Strain Problems

9

2.3 Axisymmetric Problems

11

2.4 Solution of Bi-Harmonic Equation in Axisymmetric Problems- Application of Hankel Transform -15

3 Indentation Contact Mechanics of Perfectly Elastic Body

19

3.1 Flat-Ended Cylindrical Punch

21

3.2 Spherical Indenter (Hertzian Contact)

22

3.3 Conical Indenter24

4 Onset of Plastic Yield and Indentation Contact Problems

28

4.1 Maximum Shear Stress Criterion -Tresca Criterion-29

4.2 Maximum Shear Strain Energy Criterion -von Mises Criterion-30

4.3 Onset of Plastic Yield under Spherical Indentation Contact

32

5 Elastoplastic Indentation Contact Mechanics
5.1 Materials Physics of the Meyer Hardness

34(1) Cavity Model

38(2) Unified Theory of the Meyer Hardness:
The Additivity Principle of Excluded Volume of Indentation

39(3) Elastoplastic Deformation and Flow of Cone/Pyramid indentation

42

5.2 The P –h Loading/Unloading Hysteresis Curve of Conical Indentation

45(1) Work-of-Indentation, WOI47(2) Correlation between the Loop Energy Ur and the Yield Stress cY

48

6 Linear Viscoelastic Formulation
6.1 Glass Transition Behavior and Viscoelastic Response(1) Glass-Transition Behavior

49(2) Viscoelastic Models: Maxwell Liquid and Zener Solid50

6.2 Differential Form of Viscoelastic Constitutive Equation -Wiechert Model-

55

6.3 Integral Form of Viscoelastic Constitutive Equation-Boltzmann’s Hereditary Integral-57

6.4 Time-Dependent Poisson’s Ratio in Viscoelastic Deformation

59

v

7

Viscoelastic Indentation Contact Mechanics
7.1 Mathematical Formulation

63

7.2 Viscoelastic Meyer Hardness

67

7.3 Indentation Contact Mechanics of the Viscoelastic Models

68

8 Indentation Contact Mechanics of Soft Matters with Surface Adhesion
8.1 Elastic Bodies with Surface Adhesion -The JKR Theory-72

8.2 Elastoplastic Bodies with Surface Adhesion -FEA-Based Numerical Analyses-

81

8.3 Viscoelastic Bodies with Surface Adhesion(1) Load Relaxation (Cone/Pyramid Indentation)

84(2) Creep Deformation (Spherical Indentation)

85

9 Indentation Contact Mechanics of Coating/Substrate Composite
9.1 Elastic Composites

88

9.2 Viscoelastic Composites

98(1) Stress Relaxation Behavior of an Elastic Film Coated on a Viscoelastic Substrate

99(2) Stress Relaxation Behavior of a Viscoelastic Film Coated on an Elastic Substrate 100
9.3 Elastoplastic Composites

102

10 Numerical Indentation Contact Mechanics

106

10.1 Outline of Finite Element Analysis in Indentation Contact Mechanics

107

10.2 Elastic Analysis

107

10.3 Elastoplastic Analysis

109

10.4 Viscoelastic Analysis

115

10.5 Application to Laminate Composites

117

11 Experimental Apparatus and Data Analysis
11.1 Tip-geometry of the Conventional Indenters11911.2 Instrumented Indentation Test System

121(1) Experimental Determining of the Frame Compliance

122(2) Experimental Determination of Elastoplastic Characteristics

125(3) Experimental Determination of Viscoelastic Functions

128

11.3 Instrumented Indentation Microscope

131(1) Basic Configuration of the Apparatus and the Principle ofMeasuring the Contact Area

132

(2) Determination of Elastoplastic Characteristics

133(3) Determination of Viscoelastic Functions

134

vi

12 Materials Physics Based on the Experimental Data of Instrumented Indentation Testing
12.1 The Effect of Tip-Geometry of Pyramidal Indentation on
the Contact Behavior

139

12.2 Indentation Contact Mechanics of Engineering Materials(1) Loading/Unloading P-h Hysteresis Curves and the Elastoplastic Characteristics

141(2) Indentation-Induced Elastoplastic Behavior of Carbon Materials

143(3) Viscoelastic Indentation Contact Mechanics of Inorganic Glasses

147(4) Instrumented Indentation Microscope Applied to
The Viscoelastic Studies of Amorphous Selenium (a-Se)

150(5) Instrumented Indentation Microscope Applied to
The Viscoelastic/Plastic Studies of Polycarbonate Resin in the Glassy State

151(6) Instrumented Indentation Microscope Applied to
The Elastoplastic Studies of Coating/Substrate Composites

152(7) Instrumented Indentation Microscope Applied to
The Soft Matter with Surface Adhesion

156

APPENDIX

A Kinematic Equations in Cylindrical Coordinate

161

APPENDIX

B Bessel Function

167

APPENDIXC Laplace Transforms

169

APPENDIX

D Relaxation Time Spectrum and Retardation Time Spectrum

172

APPENDIXE History of Indentation Contact Mechanics

174

Index

181

vii

List of Symbols
A

Contact area of indentation

Aproj

Projected area of residual impression

Ae

Elastic contact area

HTT

Hear-treatment-temperature

Ap

Plastic contact area

Relaxation time spectrum

Ave (t)Viscoelastic contact area

H (τ )HM

aH
a
B

Contact radius of homogeneous bodyh , hmax , h (t )gdefined byIndentation contact radiusIndenter’s shape factor

Frame compliance of indentation
apparatus

Cg ; Ce

Meyer hardnesshc

Contact depth of penetrationhr

Depth of residual impression

Jn ( x)Bessel function of the first kind of order n

J creep ( t ) Creep function in shear

Glass compliance; equilibrium compliance

C creep (t ) , C 'creep (t ) , C '(t )Creep compliance functions

C *( p )A = gh 2

Penetration depths of indentation

defined in Eq. (3.34)

Cf

Area index of cone/pyramid indenter

K

Bulk moduluskCritical yield stress in shear (Eq. (4.2))

kh

Indentation loading coefficientdefined in Eq. (3.33)

Carson transform of
creep compliance defined byke , kp , keppC ( p )c

Indenter’s shape factorcConstraint factor defined in Eq. (5.7)

E ; E'

Elastic modulus (Young’s modulus)

Loading coefficients (slopes of

P - h 2 linear plots) of cone/pyramid

defined in Eq. (3.34)

Eg ; Ee

Glass modulus, equilibrium modulus

EM

Elastic modulus of Maxwell’s springindentation

k1

Slope of P - h 2 loading linear plotk2

Slope of P - h 2(loading coefficient)unloading linear plot

(unloading coefficient)EV
Elastic modulus of Voigt’s spring
Erelax ( t ) , E 'relax ( t ) , E ( t )Laplace transform

Unloading stiffness

Distribution of

Carson transformof relaxation modulus;

E 'effMp(r )of relaxation modulus

E *( p)Retardation time spectrum

P , P ( t ) Indentation loadPI
Plastic strain (plastic index)
defined by ε I E ' cY

Relaxation moduli

E ( p)L (τ )

indentation contact pressurepE ( p )pC , pF , pS

Effective elastic modulus

Coefficients of contact pressureof laminate compositee

Bulk strain defined in Eq. (2.16)

G

Shear modulus;

distribution define in Eqs. (8.1) - (8.3)

pm

Mean contact pressureof indentation； P π ap0elastic modulus in shear

G relax ( t ) Relaxation modulus in shear

Maximum contact pressure(of spherical indentation； 3 2

FEA, FEM Finite Element Analysis,
Finite Element Method

viii

2

R

Radius of spherical indenter

Sobs

Unloading stiffness

) pm

Tg

Glass transition temperature

T
t, t'
tf
U

Absolute temperature
Time
Film thickness of laminate
Density of strain energy definedin Eq. (4.6)

UT , Ue , Ur
Energies defined in Eqs. (5.19) - (5.21)
for characterizing P - h hysteresis curveu (t )

Heaviside step function

V , Ve , Vp
Indentation-induced excluded volumes
defined in Eq. (5.12)

WOI

Work-of-Indentation defined in Eq. (5.24)

Y

Yield stress (Yield strength)ix

βInclined face-angleof cone/pyramid indenterβcχ (t )

Inclined face-angle of cone indenter

Δ

(3.4)

γ , γ EP

Surface energy

Relative contact-depth of

Laplace operator (Laplacian) defined

penetration defined byin Eq. (2.3)

ΔG
ΔH

Change in Gibbs free energyΔSδ (t )

ηeChange in entropy

Shear viscosity in steady-state

Dirac’s delta functionκBulk compliance, 1 K

λ , λ EP Adhesion toughnessν , ν ( t ) Poisson’s ratios

Representative strainsof work-hardening materials

εI
εIY

Indentation strain

Non-dimensional radius;ΣSum of normal stressesσiσ (t )

Normal stress

Normal strain

ε (t )ε ( p)ϕ ( r, z )r adefine in Eq. (2.3)

plastic deformation

Time-dependent stressof viscoelastic body

Viscoelastic strain

σ ( p)
τ
τM
τZ
τ ij
ξ

Laplace transform of ε ( t )Airy’s stress function;
Love’s stress functionHρ

Threshold indentation strain for

εi

hc h

η -value of elastic bodyη -value of fully plastic bodyηpηChange in enthalpy

εR , ( ε p ) Rϕn( a aH )hγ ij
Shear strain
η , η c , ηc ( t )Sneddon’s function definedin Eq.

γ(ζ , z )Hankel transform of ϕ ( r , z )defined in Eq. (2.23)

Laplace transform of σ ( t )Relaxation time
Relaxation time of Maxwell model
Relaxation time of Zener model
Shear stressNon-dimensional thicknessof coating defined by

ξr

tf a

Non-dimensional depth of residual
impression defined by

hr hmax or hr hx

GENERAL CONCEPT OF DEFORMATION−STRAIN AND STRESS−CHAPTER

1

Indentation contact mechanics is a field of the science and engineeringof continuum mechanics that focuses on several types of contact
problems. Very complicated is the mechanical field of the contacting areathat is formed between two of the continuum bodies with different
mechanical properties. Accordingly, we are sometimes facing to numbersof mathematical difficulties in describing the mechanical field of thiscontacting area. In applying the general stress/strain analysis to theindentation contact mechanics, therefore, it will be appropriate to makemathematical formulations for describing the mechanical equilibrium andthe constitutive equations. This is the objective of the present chapter[1.1-1.3].
The definition and the quantitative expression for the deformation of asolid body are given in this chapter. If all points within the body
experience the same displacement, the body moves as a rigid translation,uB(=uA+δ)

dx
Α

Β

A'

uA

being not stretched or deformed internally. Under the applied load, theyelongationbody must experience different displacements for strain to occur.

D

A and B initially separated by a small distance of dx . These points areexperiencing translational motion along the x -axis.

The displacements

at point A and B are uA and uB ( = uA + δ ) , respectively.

Accordingly,

D'

γ

dy

Suppose two points

xuD

Consider the deformations shown in Fig. 1.1, where the deformations are
depicted in two-dimensional plane, for simplicity.

B'

C

uC

C'

δ

shear deformation

Figure 1.1 Elongation (incremental

by the use of Taylor expansion around the point A, the differential motion

deformation) and shear deformation in(the net displacement; i.e., elongation) δ is then

two-dimensional spaceδ = uB − uA
∂u 
∂u

=  uA + dx  − uA = dx
x
∂
∂x


The x -component of strain as the differential displacement per unit
length, i.e., the strain for elongation is then defined by

εx =

δ
dx

=

∂u
∂x

(1.1)

Hence, the strain is a displacement gradient. Applying similar arguments

1[1.1] S.P. Timoshenko, J.N. Goodier,“Theory of Elasticity”, McGraw-Hill(1970)[1.2] I.H. Shames, F.A. Cozzarelli, “Elastic
and Inelastic Stress Analysis”,
Prentice Hall (1992)[1.3] D. Roylance, “Mechanics of Materials”,
John Wiley & Sons (1996)

to the differential motions in the y - and z - directions, the y - and z- components of normal strains are given by

εy =

∂v
∂y

εz =

∂w
∂z

(1.2)

where v and w are, respectively, the displacements in y - and z directions.
Elongation or compression is always associated with the change in thenormal separation of atomic planes of the body deformed, while shear
distortion results in relative sliding of these atomic planes, but no changes
occur in their normal direction. The vertical line element CD having thelength ofdy in y-direction is experienced with the shear displacementof δ to the horizontal direction (x-direction), and tilted to the line
element C’D’ with the shear angle γ , as depicted in Fig. 1.1;

δ = u D − uC =  uC +


∂u 
∂u
dy  − uC =
dy
∂y 
∂y

The shear strain in y -direction is thus defined by the displacement
gradient of shear, ∂u ∂y ( = tan γ ) ;γ ( ≈ tan γ ) =

δ
dy

=

∂u
∂y

In addition to the shear displacement of u along the x -direction, there
exists the coupled shear displacement of v to the y-direction. This is arequisite for prohibiting the rotation of this two-dimensional xy -plane inmechanical equilibrium, resulting in the shear strain of ∂v/∂x.
Accordingly, a general formula for shear strain in the xy-plane is givenbyγ xy =

∂u ∂v
+
∂y ∂x

(1.3)

In a similar way, the shear strains in yz- and zx-planes are described,
respectively, by

γ yz =

∂v ∂w
+
∂z ∂y

(1.4)

γ zx =

∂w ∂u
+
∂x ∂z

(1.5)

Since these six strains [the normal strains of ( ε x , ε y , ε z ), and the shear
strains of ( γ xy , γ yz , γ zx )] are written in terms of only three displacements2( u , v , w ), they cannot be specified arbitrarily; there must exist somecorrelations (compatibility equations) among these six strains. These
compatibility relations stand for the physics that any adjacent two points
in the continuum are neither overlapped nor cracked in between during
the deformation. Equations (1.1) - (1.3) are rewritten with∂ 2ε x∂ 3u=∂y 2 ∂x∂y 2∂ 2ε y∂x=2∂ 2γ xy∂3v∂x 2 ∂y

∂x∂y

=

∂ 3u
∂ 3v
+
,
∂x∂y 2 ∂x 2 ∂yresulting in22∂ 2ε x ∂ ε y ∂ γ xy
+ 2 =
∂y 2
∂x
∂x∂y

Similarly, using the following four equations∂γ yz∂ 2ε x∂ 3u=∂y∂z ∂x∂y∂z

∂x

∂γ zx
∂u
∂ w
=
+
∂y
∂y∂z ∂x∂y
2

2

=

∂2v
∂2 w
+
∂x∂z ∂x∂y

∂γ xy
∂z

=

∂ 2u
∂2v
+
∂y∂z ∂x∂z

we have2∂ 2ε x∂  ∂γ yz ∂γ zx ∂γ xy 
= −
+
+

∂y∂z ∂x  ∂x∂y∂z Applying a similar differential operation to the other strain components,
the following Saint Venant’s compatibility equations are finally obtained:22∂ 2ε x ∂ ε y ∂ γ xy
+
=
∂y 2
∂x 2
∂x∂y

∂ 2ε y22∂ 2ε z ∂ γ yz
+ 2 =
∂z 2
∂y
∂y∂z

2

∂ ε z ∂ ε x ∂ γ zx
+ 2 =
∂x 2
∂z
∂x∂z

2

2

2

2

∂ 2ε x∂  ∂γ yz ∂γ zx ∂γ xy 
= −
+
+

∂y∂z ∂x  ∂x∂y∂z 
∂ 2ε y∂x∂z=∂  ∂γ yz ∂γ zx ∂γ xy −+∂y  ∂x∂y∂z ∂ 2ε z∂  ∂γ yz ∂γ zx ∂γ xy 
= 
+
−

∂x∂y ∂z  ∂x
∂y
∂z 

(1.6)
These compatibility equations along with the equations for mechanicalequilibrium, the details of which are given below, play an important rolein deriving the bi-harmonic equation that describes the mechanicalresponses of the continuum body under a given boundary condition.
Suppose a continuum body in mechanical equilibrium under several
arbitrary tractions and displacement constraints acting on the boundary
surfaces. These boundary tractions and constraints induce internal strains
and forces (stresses) in the continuum body as the requisites formechanical equilibrium. Consider now an infinitesimal rectangular3parallelepiped embedded in this continuum. Choose the x-, y-, and z-axesyso as to be parallel to the edges of this rectangular parallelepiped, as

σyshown in Fig. 1.2. The normal stresses σx, σy, and σz are defined as thedznormal forces acting on the x-, y-, and z-unit planes with their normal

τzy

dy

vectors directed to x-, y-, and z-coordinates, respectively. The shear

τyx

τyz

τzx

σz

stresses ( τ xy , τ xz ), ( τ yx , τ yz ), and ( τ zx , τ zy ) are the shear forces acting

τxy
τxz

σx

x

dx

in these x-, y-, and z-unit planes; the first subscript identifies thezrespective unit planes in which the stress acts, while the second subscript

Figure 1.2 Stress components in

identifies the coordinate direction of the stress itself; τ xy is the shear

three dimensions acting on/in the

stress acting in the x-plane and directing to the y-coordinate, by way ofrespective surfaces of a rectangularexample.

parallelepiped in a continuum

The parallelepiped shown in Fig. 1.2 neither rotate nor translate,because it is embedded in a continuum with mechanical equilibrium.
This fact implies that there exist momentum balances around the x-, y-,and z-axes, resulting in the following symmetric relations of shear
stresses;

τ xy = τ yx
τ yz = τ zy

(1.7)

τ zx = τ xz
Furthermore, the following force-balance must be satisfied along the xaxis∂σ x  σ x + ∂x dx  dydz − σ x dydz
∂τ yx dy  dzdx − τ yx dzdx
+  τ yx +
∂y


∂τ


+  τ zx + zx dz  dxdy − τ zx dxdy∂z
+ Xdxdydz = 0,

resulting in the following equilibrium equation along the x-axis:∂σ x ∂τ xy ∂τ xz+++ X =0dxdydz(1.8a)

where X stands for the x-directed body force (gravitational force,
centrifugal force, etc.) acting on the continuum body. In a similar way to
the x-axis, we have the following mechanical equilibrium equations forthe y-, and z-axes, as well:4∂τ yx∂σ y∂τ zx ∂τ zy ∂σ z
+
+
+Z =0
dx
dy
dz

(1.8c)

dz

This

z

τzr
τrz

σθ

respectively.
The mechanical field induced by indentation loading/unloading

τzθ

τθz

where Y and Z are the body forces directing to the y- and the z-coordinates,

processes is, in general, axisymmetric along the penetration axis.

σz

dr

dz

(1.8b)

dy

+

∂τ yz

+Y = 0

dx

+

τθr

θ

τrθ

σr

dθ

r

fact implies that the axisymmetric cylindrical coordinates (r, θ, z)Figure 1.3

depicted in Fig. 1.3 might be much more appropriate than the Cartesianstresses in the cylindrical coordinate(x, y, z)-coordinate for representing the indentation-induced contact field,

systemcompatibility relations, and the mechanical equilibrium equations.

In

Fig. 1.3, the r-, θ -, and the z-axes represent, respectively, the radial,azimuthal, and the vertical coordinates. In the indentation contact
mechanics, the vertical coordinate (the z-axis) is to the direction ofindentation penetration. The respective components of the displacementin this coordinate system are denoted by ur , uθ , and u z .

The three-dimensional strains in the cylindrical coordinate system are;∂ur∂r
1 ∂uθ ur
εθ =
+
r ∂θ
r
∂u z
εz =
∂z
∂u
1 ∂ur uθγ rθ = θ +−r∂r r ∂θ∂u
1 ∂u zγθ z = θ +∂z r ∂θ∂u ∂uγ zr = r + z
∂z
∂r

εr =

(1.9)

The Saint Venant’s compatibility equations, i.e., Eq. (1.6) in (x, y, z)Cartesian coordinate system, are rewritten with the following expressionsin (r, θ, z)-cylindrical coordinate system:

5

Three dimensional1 ∂ 2ε r ∂  ∂εθ ∂  ∂γ rθ γ rθ + r
− ( ε r − εθ )  =+r ∂θ 2 ∂r  ∂rr  ∂θ  ∂r
1 ∂ 2ε z
2

+
2

2

2

r ∂θ
∂ εz
∂r

2

+

∂ 2εθ
∂z

∂ εr
∂z

2

2

=

+

1 ∂ε z 1 ∂  ∂γ θ z
=+ γ rz r ∂r r ∂z  ∂θ∂ 2γ rz∂r ∂z

2 ∂ 2ε r 1 ∂  ∂γ rθ ∂  1 ∂γ rz ∂γ θ z  1 ∂γ rθ γ θ zr=− γθ z  + 
−
+
+ 2

r ∂θ∂z r ∂r  ∂z∂r  r ∂zr
 ∂r  r ∂θ2

∂  ∂εθ ε r − εθ  1 ∂  ∂γ θ z 1 ∂γ rz ∂γ rθ  1 ∂γ θ z
−
=
−
+
+
r  r ∂θ  ∂rr ∂θ
∂z  ∂r∂z  r 2 ∂θ

2 ∂  ∂ε z ε z  ∂  ∂γ θ z 1 ∂γ rz ∂γ rθ γ θ z − = 
+−−r ∂θ  ∂rr  ∂z  ∂r
r ∂θ
r 
∂z

(1.10)The mechanical equilibrium equations in the cylindrical coordinate
system, that are corresponding to Eqs. (1.8a) - (1.8c) in the Cartesiansystem, are given in Eq. (1.11) in terms of the R-, Θ-, and Z-body forces;

σ r − σθ∂σ r 1 ∂τ θ r ∂τ zr
+
+
+R=0
∂r
∂z
r
r ∂θ
2τ θ r ∂τ θ r 1 ∂σ θ ∂τ zθ
+
+
+
+Θ=0
∂r
∂z
r
r ∂θ
τ zr ∂τ zr 1 ∂τ zθ ∂σ z
+
+
+
+Z =0
∂r
∂z
r
r ∂θ
+

(1.11)The details for the derivation of Eqs. (1.9) - (1.11) are given in
APPENDIX A.

6

CONTINUUM MECHANICS OFPERFECTLY ELASTIC BODY
2.1 CONSTITUTIVE EQUATIONIn the second-half of 17th century, Robert Hook made measuring smallchanges in the length of a long wire under various loads. He observed thefact that the load P and its resulting elongation δ were related linearly as
long as the loads were sufficiently small. Furthermore, he observed thefact that the wire recovered to the original length after unloading.

This

relation has been generally known as the Hook’s law, and can be writtenbyP = kδ

where the constant of proportionality k is referred to as the stiffness or thespring constant having the unit of N/m. He also recognized that k is nota material characteristic parameter, depending on the specimen shape aswell as on the dimension.
A century afterword from the experiment of R. Hook, Thomas Young
introduced a useful way to adjusting the stiffness so as to be acharacteristic material property by introducing the concept of tensilestress σ defined as the load per unit cross-sectional area combined to theconcept of tensile strain ε as the deformation per unit length of the test
specimen. In terms of the stress and the strain, the Hook’s law is writtenby
σ = EεThe constant of proportionality E in this linear law is referred to as theYoung’s modulus or the modulus of elasticity, one of the most importantmechanical characteristics of an elastic material.

It has the same unit as

stress, Pa.
In general, a somewhat subtle contraction in the radial direction isobserved, when a load is applied to an elastic rod in its axial direction,
just as stretching a rubber band to make it longer in on direction makes itthinner in the other direction. This lateral contraction accompanying alongitudinal extension is called the Poisson’s effect, after the Frenchmathematician, Simeon Denis Poisson. The Poisson’s effect is resulted
from the natural law of isometric deformation, i.e., no changes in volume
after deformation. The Poisson’s ratio is a material characteristic property

7

CHAPTER2

defined asν=−ε radial

ε axial,

where the minus sign accounts for the sign change between the radial andaxial strains; due to the Poisson’s effect, ε axial > 0 results in ε radial < 0 .
The three-dimensional expression for the constitutive equations of anelastic body is given in Eq. (2.1) in terms of the Young’s modulus E andthe Poisson’s ratio ν ;1
σ x − ν (σ y + σ z ) E1
ε y = σ y − ν (σ z + σ x ) E1
ε z = σ z − ν (σ x + σ y ) Eεx =γ xy =

τ xy

γ yz =

Gτ yz
Gγ zx =

(2.1)

τ zx
G

where the shear modulus G as the elastic modulus for shear deformationis related to the Young’s modulus E via the Poisson’s ratio ν in thefollowing equation
G=E
2 (1 + ν )(2.2)

Combining the Saint Venant’s compatibility (Eq. (1.6)), mechanicalequilibrium equation (Eq. (1.8)), and the constitutive equation (Eq. (2.1))
for elastic body leads to the following Beltrami-Michell compatibility
relations:(1 + ν ) Δσ x +(1 + ν ) Δσ y +(1 + ν ) Δσ z +

∂2Σ
∂x

=0
2

∂ 2Σ
∂y

=0
2

∂ 2Σ
∂z

=0
2

(1 + ν ) Δτ xy +

∂2Σ
=0
∂x∂y

(1 + ν ) Δτ yz +

∂ 2Σ
=0
∂y∂z

(1 + ν ) Δτ zx +

∂ 2Σ
=0
∂z∂x
(2.3)

In Eq. (2.3), Δ is the Laplace operator (so called Laplacian) that isdefined by Δ = ∂ 2 ∂x 2 + ∂ 2 ∂y 2 + ∂ 2 ∂z 2 , and Σ stands for the sumof normal stresses, Σ = σ x + σ y + σ z . In general, mathematically very
cumbersome procedures are required in solving the Beltrami-Michell
equations under a given boundary condition. However, as the details

8will be discussed in the following section, the analytical solution of theBeltrami-Michell equations can be rather easily obtained in the states ofplane-strain and of axial symmetry.

In Sec. 2.2, the elastic theory of

two-dimensional plane-strain problems is given, and then followed by theconsiderations on the elastic indentation contact mechanics in Sec. 2.3 asaxisymmetric problems.

2.2 PLANE-STRAIN PROBLEMS
Let us consider the plane strain problems where the displacement w
in the z-direction is zero, i.e., w = 0 , and the displacements u and v
in the x- and y-directions are respectively independent of the z-coordinate[1.1-1.3].

Substituting these plane-strain conditions into Eqs. (1.2),(1.4), and (1.5) leads to the following expressions for the normal andshear strains:

ε z = γ zx = γ zy = 0
εx =

∂u
∂x

εy =

∂v
∂y

γ xy =

∂u ∂v
+
∂y ∂x

(2.4)

Furthermore, by substituting Eq. (2.4) into the Saint Venant’s
compatibility relation of Eq. (1.6), we have the following simple
expression that holds among the normal and shear strains:22∂ 2ε x ∂ ε y ∂ γ xy
+ 2 =
∂y 2
∂x
∂x∂y

(2.5)

In plane-strain problems, the equilibrium equations, Eqs. (1.8a) and(1.8b), reduce to∂σ x ∂τ xy
+
=0
∂x
∂y
∂τ y x∂x+∂σ y

τ xy = τ yx∂y

(2.6a)=0(2.6 b)τ yz = 0 = τ zx

(2.6 c)where the body forces of X and Y are assumed to be zero for simplicity.
On the other hand, by substituting the relation of ε z = 0 into Eq. (2.1),
the constitutive equation of an elastic body in plane-strain is given as

9

1(σ x − ν ' σ y )E'1
ε y = (σ y − ν ' σ x )E'εz = 0εx =γ xy =

τ xy
G

(2.7)

γ yz = 0 = γ zx

In Eq. (2.7), the elastic modulus E ' and the Poisson’s ratio ν ' in“plane-strain” are defined by(

E ' = E 1 −ν 2) and ν ' = ν (1 −ν ) ,respectively.
The Saint Venant equation (Eq. (2.5)) in plane-strain state represents
the compatibility among the strains, while the compatibility among thestresses, i.e., the Beltrami-Michell compatibility of elastic body, can begiven by substituting Eq. (2.7) (the constitutive relations of perfectly
elastic body in plane-strain) into Eq. (2.5);∂ 2τ xy
∂2
∂2
'
'
2
1
'
σ
−
ν
σ
+
σ
−
ν
σ
=
+
ν
(
)
( x
( y
y)
x)
∂y 2
∂x 2
∂x∂y
(2.8)
Differentiating the equilibrium equations, Eqs. (2.6a) and (2.6b), with x
and y results in the following relation∂ 2τ xy2∂ 2σ x ∂ σ y=− 2 −
2
∂x∂y
∂x
∂y 2

and then substituting it into the Beltrami-Michell compatibility (Eq.
(2.8)) finally leads to the following Laplace equation

 ∂2
∂2 
 2 + 2  (σ x + σ y ) = 0
∂y 
 ∂x

(2.9a)

Eq. (2.9a) is alternatively written by
Δ (σ x + σ y ) = 0(2.9b)

using the Laplace operator (Laplacian)
Δ=

∂2
∂2
+ 2.
2
∂x
∂y

The solution of Laplace equation is referred to as the harmonic function.
Equation (2.9) encompasses all the requisites included in thecompatibility relation (Eq. (2.5)), equilibrium equation (Eq. (2.6), as wellas the elastic constitutive equation (Eq. (2.7)).

Accordingly, under the

given boundary conditions applied to Eq. (2.9), we can solve any of10plane-strain problems of elastic bodies.
G.B. Airy showed that if one introduces a stress function ϕ ( x, y)such that

σx =

∂ 2ϕ
∂y 2

σy =

∂ 2ϕ
∂x 2

τ xy = −∂ 2ϕ,∂ x ∂y(2.10)

the equilibrium equations (Eqs. (2.6a) and (2.6b)) are automatically andreciprocally satisfied.

We have the following forth-order differential

equation of the stress function ϕ ( x, y) by substituting Eq. (2.10) into
Eq. (2.9);

∂ 4ϕ
∂ 4ϕ
∂ 4ϕ
+
+
2
∂x 4
∂x 2 ∂y 2 ∂y 4
= Δ ⋅ Δϕ(2.11)=0
The stress function ϕ is, therefore, also called the bi-harmonic function
since Eq. (2.11) is the Laplace equation of Δϕ .

Accordingly, thestresses and strains in elastic problems under any boundary conditions in
two-dimensional plane-strain state can be found in a quantitative mannerby substituting the solution ϕ of Eq. (2.11) into Eq. (2.10), and theninto Eq. (2.7).

2.3 AXISYMMETRIC PROBLEMS
The axisymmetric problems along the penetration axis (z-axis) ofindentation are independent of the azimuthal coordinate θ , and areappropriately represented as the three-dimensional problems incylindrical coordinate.

Equation (1.9) in axisymmetric problems is,therefore, recast into
∂ur
∂r

εθ =

γ rθ = 0γθ z = 0

εr =

ur
r

∂u z
∂z
∂ur ∂u z+
γ zr =∂z∂r

εz =(2.12)

Furthermore, applying Eq. (2.12) to the Saint Venant’s compatibility (Eq.
(1.10)) leads to the following axisymmetric compatibility expressions:11∂  ∂εθ− ( ε r − εθ )  = 0
r∂r  ∂r

∂ 2εθ
∂z

2

1 ∂ε z 1 ∂γ rz=
r ∂r r ∂z+∂ 2ε z(2.13)∂ 2ε r∂ 2γ rz
+
=
∂r ∂z
∂r 2
∂z 2
∂  ∂εθ ε r − εθ −
=0∂z  ∂rr On the other hand, the axisymmetric expressions of the equilibrium
equation (Eq. (1.11)) are given byσ r − σθ
r

τ zr
r

+

+

∂σ r ∂τ zr
+
=0
∂r
∂z

(2.14a)∂τ zr ∂σ z
+
=0
∂r
∂z

τrθ = 0 = τzθ(2.14b),(2.14c)where we assume that the body forces ( R , Θ , Z ) in Eq. (1.11) are all
zero for simplicity.
The constitutive equations for perfectly elastic body ((Eq. (2.1)) in the
Cartesian (x, y, z) coordinate turn to the following equations ofaxisymmetric problems in the cylindrical coordinate;1
σ r − ν (σ θ + σ z ) E1εθ = σ θ − ν (σ z + σ r ) E1
ε z = σ z − ν (σ r + σ θ ) Eεr =γ rθ = 0γθ z = 0(2.15a)τγ zr = zr
G

The conjugate expressions of Eq. (2.15a) in terms of strain anddisplacement are

∂ur
∂r
u
σ θ = λ e + 2Gεθ = λ e + 2G rr∂u
σ z = λ e + 2Gε z = λ e + 2G z∂z

σ r = λ e + 2Gε r = λ e + 2Gτ rθ = 0τθ z = 0(2.15b) ∂ur ∂uz 
+
∂r 
 ∂z

τ zr = Gγ zr = G 

12In Eq. (2.15b), the volumetric strain e and the Lame constant λ arerespectively defined bye = ε r + εθ + ε z=

λ=∂ur ur ∂uz
+ +
∂r
∂z
r

ν

(1 + ν )(1 − 2ν )(2.16)

E

Combining Eqs. (2.14) – (2.16) with Eq. (2.13) results in the following
Beltrami-Michell compatibility in axisymmetric problems;

1 ∂ 2Σ
=0
1 + ν ∂r 2
r
2
1 ∂ΣΔσ θ − 2 (σ θ − σ r ) +=0+ν r ∂r
1r

Δσ r −

2(σ r − σ θ ) +2(2.17)

1 ∂ 2Σ
Δσ z +=0
1 + ν ∂z 2
Δτ rz −

1
r

τ +
2 rz

1 ∂2Σ
=0
1 + ν ∂r ∂zwhere Σ represents the sum of normal stresses, Σ = σ r + σ θ + σ z ,while the Laplace operator Δ in axisymmetric cylindrical coordinate is
written by∂2 1 ∂∂2++∂r 2 r ∂r ∂z 2
1 ∂  ∂  ∂2=
r  +r ∂r  ∂r  ∂z 2

Δ=

(2.18)We have introduced the Airy’s stress function in Eq. (2.10) that enables
the quantitative description of two-dimensional plane-strain problems via
the bi-harmonic equation (Eq. (2.11)). In a similar way to the plane-strain
problem, we introduce the Love’s stress function ϕfor three-dimensional axisymmetric elastic problems to represent the stresscomponents in cylindrical coordinate;

13

σr =

∂ 
∂ 2ϕ 
νΔϕ − 2 
∂z 
∂r 

σθ =

∂ 
1 ∂ϕ 
νΔϕ −
r ∂r 
∂z (2.19)∂ ∂ 2ϕ 
σ z =  (2 − ν )Δϕ − 2 
∂z 
∂z 

τ rz =∂ ∂ 2ϕ 
(1
)
−
Δ
−
ν
ϕ


∂r 
∂z 2 

Furthermore, once we assume that the stress function ϕ is the solutionof the following bi-harmonic equation

Δ ⋅ Δϕ = 0 ,(2.20)

not only Eq. (2.14a) but also Eq. (2.14b) are automatically satisfied. In
addition to these facts, by the use of Eqs. (2.15), (2.16), and (2.19)
combined with the relations of ur = rεθ , andu z =  ε z dz , thedisplacements ur and uz are easily written in terms of the stressfunction ϕ , as follows;

2Gur = −

∂ 2ϕ
∂r ∂z

(2.21)∂ 2ϕ
2Gu z = 2(1 − ν )Δϕ − 2∂zOne can therefore quantitatively describe the stress-strain fields of theaxisymmetric problem of any perfectly elastic body in cylindricalcoordinate by substituting the stress function ϕ (the solution of the
biharmonic equation, Eq. (2.20)) into Eq. (2.19) combined with Eq.
(2.15a).

14

2.4 SOLUTION OF BI-HARMONIC EQUATIONIN AXISYMMETRIC PROBLEMS- APPLICATION OF HANKEL TRANSFORM As mentioned in the preceding section, in order to find the stressfunction ϕ for elastic problems, we need to solve the bi-harmonicequation, Eq. (2.11) or Eq. (2.20), under a given boundary condition. In
the second half of 19th century to the beginning of 20th century, some ofphysicists and mathematicians have contended with solving the biharmonic equation under the specific boundary conditions by the uses ofpolynomial equations and/or Fourier series. In general, the theories ofcomplex functions have been widely applied to solving the bi-harmonicequation in the Cartesian ( x , y ) coordinate. On the other hand, in thecylindrical coordinate that is more appropriate for describing theindentation contact problems, it has been well-known that the use of theBessel function-based Hankel transform is effective and useful forsolving the harmonic/bi-harmonic equations. We will discuss in thissection the application of Hankel transform to the bi-harmonic equation(Eq. (2.20)) for axisymmetric indentation contact problems of elastic
body [2.1-2.2].
In cylindrical coordinate system, the application of the method ofseparation variables to the axisymmetric Laplace equation Δϕ ( r , z ) = 0
leads to the ordinary differential equation with the variable r or z, andthen the resultant equation with the variable r forms the first-kind Bessel
differential equation. On the other hand, the Hankel-transform integral
includes the Bessel function as its integral kernel. This is the reason why
we can easily find the solution of the harmonic or the bi-harmonicequation via the Hankel transform. The details of Bessel function are
given in Appendix B at the back of this book.
The Bessel function J n ( x) of the first kind of order n is defined as
the solution of the following differential equation;d 2 J n ( x) 1 dJ n ( x)  n2 ++ 1 − 2  J n ( x) = 0
dx 2
x dx
 x 

(2.22)The Hankel transform of the stress function ϕ (r, z ) is defined in Eq.
(2.23) using the Bessel function of order n as the kernel of the following
transform integral;
15[2.1] I.N. Sneddon, “Fourier Transforms”,
McGraw-Hill (1951)[2.2] D. Maugis, “Contact, Adhesion and
Rupture of Elastic Solids”, Springer(2000)

∞ϕnH (ξ , z ) =  rϕ (r , z ) J n (ξ r )dr ,(2.23)

0while its inverse is written by∞

ϕ (r , z ) =  ξϕ nH (ξ , z ) J n (ξ r )d ξ

(2.24)0

Let us now consider the zero-order Hankel transform of Δ ϕ ;∞I =  r Δϕ ( r , z ) J 0 (ξ r )dr
0

∞

∂2 ∞
 ∂  ∂ϕ 
=
+
r
J
(
ξ
r
)
dr
rϕ J 0 (ξ r ) dr 0∂z 2 0 0 ∂r  ∂r 

where Δ is the Laplacian operator (refer to Eq. (2.18)), and ϕ means theLove’s stress function. Applying the partial integration twice to the firstterm of the right-hand side of the above equation, and then using the zeroorder Bessel differential equation ( n = 0 in Eq. (2.22)), we haved 2  J 0 (ξ r ) d (ξ r )2+

1 d  J 0 (ξ r ) 
+ J 0 (ξ r ) = 0 ,ξ r d (ξ r )and then finally obtain the following result of the integral I ;∞I =  r Δϕ ( r , z ) J 0 (ξ r )dr
0

 d2
 ∞
=  2 − ξ 2   rϕ J 0 (ξ r ) dr
 dz
 0

(2.25) d2
=  2 − ξ 2  ϕ 0H (ξ , z ) dz
where use has been made of the mechanical requirement that the stressfunction ϕ and its spatial gradient d ϕ / dr always diminish withr→∞.

Replacing the stress function ϕ with ϕ ≡ Δ ϕ in Eq. (2.25),

and repeating the same mathematical operations used in deriving Eq.
(2.25), we finally have the Hankel transform of the bi-harmonic equation,
as follows;2 d2
2  d2 H
 2 − ξ  2 − ξ  ϕ0 (ξ , z ) = 0
 dz
 dz


(2.26)

Accordingly, one can get the solution ϕ0H (ξ , z ) of bi-harmonic equation
by solving the fourth-order ordinary differential equation, Eq. (2.26), andthen the application of inverse Hankel transform to ϕ0H (ξ , z ) finallyresults in the stress function ϕ ( r , z ) of the bi-harmonic equation (Eq.
16

(2.20)). In other words, we can say that the Hankel transform of biharmonic

function

satisfies

ϕ (r , z )thefourth-order

ordinarydifferential equation, Eq. (2.26). Accordingly, once we notice the factthat the characteristic (auxiliary) fourth-order polynomial equation (Eq.
(2.26)) has the multiple roots of ±ξ , it is rather easy to solve Eq. (2.26),
resulting in the following general solution;

ϕ0H (ξ , z) = ( A + Bξ z)e−ξ z + (C + Dξ z)eξ z ,(2.27)

in which the integral constants A, B, C and D will be finally fixed under
the given boundary conditions.
Let us now derive the analytical expressions for the stresses and thestrains in terms of ϕ0H (ξ , z ) . These expressions will play some ofimportant roles in the elastic theory of indentation contact mechanics, thedetails of which will soon be given in the following sections. The orderzero Hankel transform of Δ ϕ is given in Eq. (2.25), and the first order
Hankel transform of the derivative d ϕ / dr is reduced to the order-zero
Hankel transform of ϕ , as follows (refer to Appendix B),∞∞

 r ( dϕ / dr )J (ξ r ) dr = −ξ  rϕ J (ξ r )dr
1

0

0

0

Using these facts combined with the Hankel transform of Eq. (2.21) and
its inverse, we finally describe the displacements, ur and u z , in thereal space through using the stress function ϕ0H (ξ , z ) in the Hankelspace;∞dϕ2Gur =  ξ 2 0 J1 (ξ r ) d ξdz
0H(2.28)∞ 
d 2ϕ 0H
2Gu z =  ξ (1 − 2ν )− 2 (1 − ν ) ξ 2ϕ 0H  J 0 (ξ r ) d ξ
2
dz
0 


(2.29)In a similar manner to the above considerations, by applying the Hankeltransform operations to Eq. (2.19), we can uniquely express the stresscomponents, σ r , σθ , σ z , and τ rz by the use of ϕ0H (ξ , z ) , as follows;

17

∞

H
  d 3ϕ0H
2 dϕ0 
+
−
(1
ν
)
ξ
σ r =  ξ νJ 0 (ξ r ) d ξ3dz 0  dz(2.30)∞dϕ H
1
−  ξ 2 0 J1 ( ξ r ) d ξ
r 0
dz
∞

  d 3ϕ H
dϕ H σ θ = ν  ξ  30 − ξ 2 0 J 0 (ξ r ) d ξdz 0  dz(2.31)∞dϕ H
1
+  ξ 2 0 J1 ( ξ r ) d ξ
r 0
dz
∞

 
d 3ϕ Hdϕ H 
σ z =  ξ (1 − ν ) 30 − (2 − ν )ξ 2 0 J 0 (ξ r ) d ξ
dz
dz 
0 


∞

 d 2ϕ 0H
+ (1 − ν )ξ 2ϕ 0H  J1 (ξ r ) d ξ
2
dz



τ rz =  ξ 2 ν0(2.32)(2.33)

Equations (2.28) - (2.33) will play an important role in describing theindentation load P vs. the penetration depth h relation ( P - hrelation) for

axisymmetric indentation problems (cylindrical punch,spherical/conical indentation), the details of which will be given in thesubsequent chapter for perfectly elastic body.

18

INDENTATION CONTACT MECHANICS
OF PERFECTLY ELASTIC BODY

CHAPTER3

The tip-geometry of the conventional indenter includes sphere (Brinell
indenter), cone with rounded tip (Rockwell indenter), and pyramid(Vickers tetrahedral indenter; Berkovich trihedral indenter).

These

indenters have been utilized in experiments for determining not only thecontact hardness, but also other mechanical properties.

One canestimate/determine the mechanical characteristics of the material tested
through analyzing the indentation load P vs. penetration depth h relation,i.e., P-h relation. It will be readily expected that the P-h relation is highly
dependent on the tip-geometry of indenter.

We will discuss in thischapter the elastic indentation contact mechanics of the axisymmetricindenter with an arbitrary tip-geometry [2-2].
A schematic contact view is depicted in Fig. 3.1 for an axisymmetricindenter with arbitrary tip-geometry that is pressed onto an elastic halfspace with the indentation load of P (the contact friction between the
indenter and the surface of elastic body is assumed to be zero forsimplicity). The geometrical parameters of indentation are: thepenetration depth h, contact depth hc , contact radius a, and the surface
displacement u z ( r , 0) of contact plane at the location r from thepenetration-axis (z-axis). The boundary conditions at the free surface(z=0) are given byuz ( ρ ,0 ) = h − f ( ρ ) ; 0 ≤ ρ ≤ 1

σ z ( ρ ,0 ) = 0;

ρ >1

(3.1)

τ rz ( r ,0 ) = 0

P

where ρ = r a is the non-dimensional radius normalized with thecontact radius a, and f ( ρ ) is the shape-function of the axisymmetrichmeans the free surface with no stresses along the z-direction outside thecontact area ( ρ > 1 ), and the third relation represents the frictionlesscontact.
All of the stresses at the locations far away of the contact area ( r → ∞ )must be zero. This fact implies that the Hankel transform of the bi19hc

the surface displacement within the contact area ( 0 ≤ ρ ≤ 1 ), the secondrrf(r/a)

hs

indenter defined with f (0) = 0 . The first relation in Eq. (3.1) describesuz(r,0)0azFigure3.1 The surface deformationf ( r a ) of elastic half-space for an
axisymmetric indentation contact.

harmonic stress function ϕ0H (ξ , z ) given in Eq. (2.27) must be finite in
the region of ξ → ∞ , leading to the following formula;ϕ0H (ξ , z) = ( A + Bξ z)e−ξ z(2.27a)Substituting the frictionless boundary condition τ rz ( r ,0 ) = 0 into Eq.
(2.33) results in d 2ϕ0H =0,2  dz  z =0

(1 −ν ) ξ 2ϕ0H (ξ ,0) + ν and then combining Eq. (2.27a) with the above relation leads to
A = 2ν B . Accordingly, the Hankel transform of the bi-harmonic stressfunction ϕ0H (ξ , z ) is represented with

ϕ0H (ξ , z ) = B ( 2ν + ξ z ) e−ξ z(3.2)

Furthermore, Eqs. (2.29) and (2.32) combined with the boundary
conditions of Eq. (3.1) finally fix the unknown integral constant B as
follows;
B=−

E 'a 1
χ ( t ) cos ( aξ t ) dt
2ξ 3 0(3.3)

in which E ' denotes the plane-strain elastic modulus, E ' = E (1 − ν 2 ) .
The Sneddon’s function χ ( t ) in Eq. (3.3) is defined byt
2
 f '( x )dx χ (t ) =  h − t π 0 t 2 − x 2


(3.4)

in terms of the shape-function of indenter f ( r ) and the penetration
depth h .

We, therefore, finally obtain the analytical expression of

ϕ0H (ξ , z ) ;ϕ0H (ξ , z ) = − ( 2ν + ξ z ) e−ξ z

E 'a 1χ (t ) cos(aξ t )dt
2ξ 3 0(3.5)

by substituting Eq. (3.3) into Eq. (3.2). Equation (3.5) makes it possible
for us to quantitatively describe the surface deformation and the stressesof elastic half-space for an arbitrary axisymmetric indentation contact(see Fig. 3.1) through the uses of Eqs. (2.28) - (2.33).
Let us start discussing the indentation load P vs. the penetration depth
h relationship, i.e., P - h relationship, in terms of ϕ0 (ξ , z ) (Eq. (3.5)).
H

Substituting Eqs. (3.4) and (3.5) into Eq. (2.32) and using the fact that the20

aindentation load P is described by P = −2π  rσ z ( r , 0 )dr in terms of the0

contact pressure σ z ( r,0) within the contact area ( r ≤ a ) , we finallyhave the following P - h relation;1
xf ( x)
P = 2aE ' h − dx   1 − x2 
0



(3.6)

Furthermore, combining Eq. (3.5) with Eq. (2.32) results in the contactpressure σ z ( ρ , 0) within the contact area ( ρ < 1 ) as follows;

σ z ( ρ , 0) = −1
 χ '(t )dt 
E '  χ (1)−
2a  1 − ρ 2  ρ t 2 − ρ 2 



(3.7)

On the other hand, by substituting Eq. (3.5) into Eq. (2.29), thedisplacement uz (ρ ,0) along the z-direction, i.e., the displacement along
the loading direction of the free-surface outside the contact zone ( ρ > 1 )is written by1 χ (t )dt
u z ( ρ , 0) = 
0 ρ 2 − t 2= χ (1)sin

−1

1

ρ

1

−  χ '(t )sin
0

−1

t

ρ

(3.8)
dt

Based on the preceding considerations on the indentation contact
mechanics of elastic body for an axisymmetric indenter with arbitrary tipgeometry of f ( r ) , let us discuss the elastic contact mechanics for the flatpunch, sphere, as well as the cone indentation in what follows.

3.1 FLAT-ENDED CYLINDRICAL PUNCHPIn Fig 3.2, depicted is the deformation of the indented elastic half-space
for a flat-ended cylindrical punch with the radius of a0 . The indenter’s
tip-geometry is denoted by f ( ρ ) ;f (ρ ) = 0 ;0 ≤ ρ ≤1

. r

0

radius a0 . Equation (3.6) combined with this tip-geometry function
reduces to the following simple P - h relation, since the integral in theright-hand side of Eq. (3.6) turns to zero;rhwhere ρ (= r / a0 ) is the non-dimensional radius normalized with theuz(r,0)a0zFigure 3.2 The surface-displacement

P = 2a0 E ' h(3.9)

Equation (3.9) means that the indentation load P linearly increases with

21of an elastic half-space indented by aflat-ended cylindrical punch

the increase in the penetration depth h , and the slope of this linear
relation gives the Young’s modulus E ' of the elastic body indented. In
addition to this fact, noticing the expression of χ ( t ) = 2h π in Eq. (3.4),
the contact stresses σ z ( ρ ,0) within the contact zone ( 0 ≤ ρ ≤ 1 ) is
written as

σ z ( ρ , 0)
pm

=−

1
1
; 0 ≤ ρ <1
2 1− ρ2(3.10)

by the uses of Eqs. (3.7) and (3.9) combined with the mean contactpressurepm ( = P π a0 2 ) . Equation (3.10) implies that the contactpressure p ( r )( ≡ −σ z ( ρ ,0) ) has its minimum value at the center ofcontact ( ρ = 0 ), and then increases monotonically with the increase inρ to infinity at the edge of the flat-punch, i.e., at ρ = 1 .
On the other hand, the displacement u z ( r , 0) of the free surface
outside the contact zone ( ρ ≥ 1 ), namely the sink-in of the free-surface,is described byu z ( ρ , 0) =
=

2h

π

sin −11

ρ(3.11)1 P
1sin −1
π E ' a0ρ

where use has been made of the relations χ '(t ) = 0 and χ (1) = 2 h / π
in Eq. (3.8).

3.2 SPHERICAL INDENTER (HERTZIAN CONTACT)The displacement of the surface of an elastic half-space contacted witha spherical indenter (radius R) is shown in Fig. 3.3, where the shapefunction f ( ρ ) of spherical indenter is given byf (ρ ) = R − R2 − a2 ρ 2
1a 2
ρ ; aR
2 R

0uz(r,0)rrh

We will assume a  R in the following considerations for simplicity.

R

hs

≈

P

(3.12)

2hc

In fact, it is well known that the approximation given in Eq. (3.12) issufficient enough for describing the experimental results in the region of(

2 2
a ≤ 0.4 R . Hence, by applying the approximation f ( ρ ) ≈ a ρ 2 R

to Eq. (3.4), we have the following Sneddon’s function χ ( t ) ;

22

)

a
z

Figure 3.3 The surface-displacementof an elastic half-space indented by asphere (Hertzian contact)χ (t ) =2a2 2 
h − t R π(3.13)

Furthermore, the substitutions of Eq. (3.12) into Eq. (3.6) and of Eq.
(3.13) into Eq. (3.7) lead to the following P - h relation along with thecontact stress distribution σ z ( ρ , 0) of spherical indentation;

1 a2 
P = 2aE '  h −

3 R


(3.14)

E '  χ (1)4a 2
1− ρ2 ; 0 ≤ ρ ≤1+
2a  1 − ρ 2 π R(3.15)σ z ( ρ , 0) = −As has been mentioned in the preceding section, the contact stressσ z ( ρ , 0) of cylindrical flat-punch changes in a discontinuous manner at
the contact periphery, i.e. σ z ( ρ , 0 ) ↑ −∞ at ρ = 1 − ε ( ε ↓ 0 ) andσ z ( ρ,0) ≡ 0 at ρ = 1 + ε ( ε ↓ 0 ) . This discontinuity in the contact

stress is resulted from the discontinuous flexion of the contact surface at

ρ = 1 , as depicted in Fig. 3.2, while the free surface of sphere-indented
elastic body changes continuously from the indented area to the outsideof the contact, as shown in Fig. 3.3. In addition to this fact, when we
notice the fact that the surface free-energy, i.e. the surface tension, is
negligibly small in the indentation contact of an elastic body having
higher elastic modulus (refer to Chapter 8), the contact stress σ z ( ρ ,0)at the contact boundary [ ρ = 1 − ε ( ε ↓ 0 )] is required to coincide withthe stress σ z ( ρ ,0 ) = 0 ( ρ ≥ 1) of the free surface outside the contact
zone, leading toχ (1) = 0in Eq. (3.15).

(3.16)

The contact radius a and the contact load P in

Hertzian contact problems, therefore, are expressed in terms of thepenetration depth h , as follows

a = Rh1 2P=(3.17)

4
E ' Rh3 23where uses have been made of Eqs. (3.13) and (3.14).

(3.18)
Equation (3.18)

means that the indentation load P increases linearly with the increasein the penetration depth h3 2 , and the slope of this linear line gives theelastic modulus E ' of the elastic body indented. Furthermore, combining
23

Eqs. (3.15), (3.17), and (3.18), the contact stress distribution σ z ( ρ , 0)is written by

σ z ( ρ , 0)
pm

=−

3
1− ρ2 ; 0 ≤ ρ ≤1
2

(3.19a)orσ z ( ρ , 0)p0

= − 1− ρ2 ; 0 ≤ ρ ≤1

(3.19b)in which pm = P (π a 2 ) and p0 = (3 / 2) pm are the mean and the
maximum contact pressures, respectively.
Equation (3.8) combined with Eq. (3.13) describes the sink-in profileuz (ρ ,0) of the free-surface outside the contacted area ( ρ > 1 );u z ( ρ ,0) =a2 ( 2 − ρ 2 ) sin −1 ρ1 + ρ 2 − 1
π R ρ 2 



h 
1
= 2 ( 2 − ρ 2 ) sin −1 + ρ 2 − 1  ; ρ ≥ 1πρ ρThe sink-in depth hs(3.20)

at the contact periphery (see Fig. 3.3) is

hs ( = u z (1, 0 ) ) = h 2 that is obtained by substituting ρ = 1 into Eq.

(3.20), resulting in the contact depth hc = h / 2 . The relative contact
depth ηc ( = hc h ) for Hertzian contact is, therefore,
 h 
 h

ηc  ≡ c  =1
2(3.21)

that is independent of the penetration depth h.

3.3 CONICAL INDENTER
The surface deformation of an elastic half-space indented by a conewith its inclined face-angle of βis shown in Fig. 3.4. The shapefunction of the indenter is described byf ( ρ ) = a ρ tan β(3.22)P

The Sneddon’s function χ ( t ) in Eq. (3.4) combined with Eq. (3.22) is
0

hs hc
h

given byuz(r,0)

t

2
1

dx 
 h − ta tan β π 0 t 2 − x 2 
2h
= − ta tan β

r

r
β

a

χ (t ) =(3.23)z

Figure 3.4 The surface displacementπof an elastic half-space contacted

When we notice the fact that the boundary value ofwith a conical indenter24

must satisfy the physical requirement ofχ (1) = 2 h / π − a tan βχ (1) = 0 at the contact periphery like as Eq. (3.16) for the Hertziancontact, we finally obtain the following relationship between the contact
radius a and the depth of penetration h;a=2cot βπh(3.24)

Furthermore, conducting similar mathematical operations that we madein the Hertzian contact problem, we have the following key results forconical indentation contact;

2 cot βπσ z ( ρ , 0)pm

u z ( ρ ,0) =E ' h2= − cosh −1(3.25)1ρ

; 0 < ρ ≤1

(3.26)

2h  −1 1sin− ρ + ρ 2 − 1 ; ρ ≥ 1

π 
ρ


(3.27)

Equation (3.25) means that the indentation load P increases linearlywith the increase in the penetration depth h 2 , and the slope of this linear
line gives the elastic modulus E ' of the body indented. Substitution ofNormalized contact pressureσz(ρ,0)/pm

P=ρ = 1 into Eq. (3.27) leads to the contact depth hc  ≡ u z (1, 0 )  , and thenyields the relative contact depth ηc ( = hc h ) as follows;ηc  ≡
hc  2=h  π: cylindrical flat-punch: spherical indenter: cone indenter

3

2

1

0

0

0.2

0.4

0.6

0.8

1

Normalized radius, ρ(=r/a)

Figure 3.5 Normalized contact(3.28)

stress distributions for flat-ended
cylindrical punch, spherical, andAs in the case of spherical indentation contact, for the conical indentation

conical indenters

the total penetration depth h . We show in Figs. 3.5 and 3.6 the tipgeometry dependence of the normalized contact stress distribution

σ z ( ρ ,0) / pm ; ρ ≤ 1
u z ( ρ , 0) / h; ρ ≥ 1

and

the

normalized

surfacedisplacement

plotted against the normalized contact radiusρ ( ≡ r a ) for the flat-punch, sphere, and the cone indentations,
respectively.
In the preceding considerations, the experimentally observable
relations, i.e. the indentation load P vs. the penetration depth h relations,have been given in Eqs. (3.9), (3.18), and (3.25) for the respective
indenter’s tip-geometries.

These P - h relations are essential in25

Normalized
surface displacement, u z/hcontact as well, the relative contact depth ηc is always independent of
0
0.2
0.4
: cylindrical flat-punch: spherical indenter: cone indenter

0.6
0.8
1
1

2

3

4

5

6

7

8

9

10

Normalized radius, ρ(=r/a)

Figure 3.6 Normalized surface
deformations (sink-in profiles) ofelastic half-space indented by flatended cylindrical punch, spherical,
and conical indentersexperimentally determining the plane-strain elastic modulus E ' of thetested body. It must be noticed in this context thatboth Eq. (3.18)( P ∝ h3 2 ) and Eq. (3.25) ( P ∝ h2 ) of spherical/conical indentations do
not satisfy the linear P - h relationship although the body tested isperfectly elastic.

However, once we introduce the following indentation

strain ε I , we have the generalized Hook’s law of pm = E ' ε I throughusing the mean contact pressure pm ( = P π a 2 ) , where the indentation
strain ε I is defined as follows;
Flat-ended cylindrical punch:εI =2 h  
π  a0 εI =4  a  4
  =
3π  R   3π

εI =

tan β
2

(3.29)

Spherical indenter：

h

R 

(3.30)

Conical indenter:(3.31)

It must be noticed in these equations that the indentation strain ε I of
conical indenter is independent of the penetration depth h , not like theflat-punch and the spherical indenter.

This fact is resulted from thegeometrical similarity of the conical indenter, i.e., the contact radius a
linearly changes with the penetration depth h as shown in Eq. (3.24).
The geometrical similarity of cone/pyramid indenters is, therefore, very
essential in quantitatively as well as easily determining the material
characteristics not only in elastic but also in elastoplastic/viscoelastic
regimes. This is the reason why the cone/pyramid indenters have long
been conventionally utilized in the history of indentation contact
mechanics.
As mentioned in the preceding considerations, we have the linear
relation between the mean contact pressure pm ( = P π a 2 ) and theindentation strain ε I ;pm ( = P π a 2 ) = E ' ε I(3.32)

This simple linear relation implies the fact that the indentation contact26

2area A(= π a ) plays a very essential role in the indentation contact

mechanics not only in the elastic but also in the elastoplastic/ viscoelastic
regimes, as well. We will discuss/emphasize in the following chaptersthat the in-situ measurement of the contact area A along with theindentation load P is a very essential requisite for quantitatively
determining the material characteristics in the elastic/elastoplastic/
viscoelastic indentation tests.
Among the indentation contact responses for various types of tipgeometry of the indenters discussed in Sec. 3.1-3.3, the most essential
ones are the P − h , a − h , and the a − P relations;P = kh E ' h n(a ≡(3.33))

A π = Bh n −1

(3.34)= cP ( n −1)/ n

The parameters denoted in these equations are listed in Table 3.1.

Table 3.1 List of indentation contact parametersflat-ended
平坦円柱圧子cylindricalpunchspherical
indenter
球形圧子

conicalindenter
円錐圧子

f (ρ )

0( a 2R) ρaρ tan βσσz z((ρρ,0,0))/ ppmm

−1 2 1 − ρ 2−3 1 − ρ2 2

− cosh−1(1/ ρ)

n

1

3/2

2

kh

2a0

4 R3

2cot β πB

a0R

2cot2 β π

Cca0

(34 E' ) R

2cot βhcc / h

1

1/2

2π
/2
π

εI

2h π a0( 4 3π ) h R

tan β 2

2

1/3

2

1/3

* ρ = r/a

27

πE '

ONSET OF PLASTIC YIELD AND
INDENTATION CONTACT PROBLEMSCHAPTER

4

Engineering materials including metals and ceramics deform in elastic
manners under minute strain. The mechanical work for inducing elastic
deformation of these materials leads to the increase in the Gibbs freeenergy Δ G through the increase in the enthalpy ΔH via the reversible
deformation of the inter-atomic separation of these materials. On theother hand, the external work applied to organic polymers/rubbers isstored by the increase in Δ G ( ≈ − T Δ S ) through the decrease in theentropyΔ S ( < 0)of polymeric chains. Accordingly, the elastic

deformation of the former materials is referred to as the enthalpic
elasticity, while the latter is named as the entropic elasticity.

Once theloads or the constrained displacements of these elastic bodies are taken
away, the deformation instantly as well as reversibly disappears, and the
body recovers to its original shape and size. [1.2, 1.3]
Elasticity is, therefore, the characteristic material property of
immediate recovery of deformation upon unloading, while there exists
the elastic limit, at which the material experiences a permanent
deformation (plastic flow) that is not lost on unloading. This is the
beginning of plastic yielding process, in which the shear slip of atomic ormolecular planes causes atoms/molecules to move to their new
equilibrium positions.

The stress-strain curve (S-S curve) of a rod

g

E

E

u nl ostress (or yielding stress, yielding strength), and defines the elastic limit,

ad i nThis specific stress Y is referred to as the yield

i ng

becomes nonlinear.

Y

load

depicted in Fig. 4.1. There is a specific stress Y at which the S-S curvestress, σundertaken with a simple tension or compression is schematicallybeyond which the deformation becomes elastoplastic, meaning that theεYεpdeformation includes not only elastic deformation but also plastic flow as

0

well;

strain for tension/compression
σ ≥Y

ε

(4.1)

Equation (4.1) expresses the criterion for the onset of plastic yield fortension/compression. A complete unload from an elastoplastic S-S state

Figure 4.1 Stress-strain curve of arod undertaken with a simple tension
or compression, indicating the onset

results in a finite residual strain ε p as shown in Fig. 4.1. The residualof plastic yielding at the yield stressstrain ε p represents the plastic flow that is induced during loading in theY28plastic region exceeding the elastic limit.
Let us now scrutinize the microscopic processes for the onset of plastic
yield of a metallic single crystal as a representative of elastoplastic body.
Not only normal but also shear deformations are imposed in the threedimensional crystal under externally applied loads, as mentioned in Chap.
1.

Among these deformations, shear deformation contributes to therelative slips of crystalline planes. If the shear deformation is smallenough, the relative slips are mechanically reversible, leading to theelastic shear deformation that is dictated by the shear modulus G .
However, once the imposed shear stressτ

crosses over thecharacteristic value of k termed the yield stress in shear, these shear
slips become irreversible, and lead to the onset of plastic yield. The
criterion for the onset of plastic yield in shear, therefore, is written by
τ ≥k

(4.2)

The criterion for the onset of plastic yield under multi-axial S-S states israther complicated not like the simple tension/compression or the shear
deformation given in Eq. (4.1) or Eq. (4.2); the details will be given in
the following considerations.

4.1 MAXIMUM SHEAR STRESS CRITERION-TRESCA CRITERIONIn a simple tension test of a rod with its axial stress σ , the maximum
resolved shear stress τ max appears on the plane tilted at 45° to theapplied axial stress, leading to the correlation of τ max = σ 2 .
Accordingly, the maximum shear stress criterion, i.e., the Tresca criterion,
leads to the following formula;max {σ } ≥ 2k = Y(4.3)

In more general three-dimensional S-S state, the Tresca criterion can be
written as follows;

max { σ1 − σ 2 , σ 2 − σ 3 , σ 3 − σ1 } ≥ 2k = Y
where σ1 , σ 2 , and σ 3 stand for the principal stresses.

(4.4)
In threedimensions, as shown in Figs. 1.2 and 1.3, not only normal stresses σibut also shear stresses τ ij exist on an arbitrary plane. However, as

29

shown in Fig. 4.2, we can find a specific orientation of the coordinate2

system such that no shear stresses appear, leaving only normal stresses in

σ2

its three orthogonal directions. The 1-, 2-, and 3-coordinate axes are
termed the principal axes, and the planes normal to these axes are named
as the principal planes.

τNo shear stresses exist on these principal

planes except the normal stresses that are termed the principal stresses of

σ3

σ1

1

σ1 , σ 2 , and σ 3 , respectively. In Fig. 4.2, the shear stress τ on theplane encompassed by the white broken lines is induced by the combined

3

normal stresses of σ1 and σ 2 , the maximum shear stress τ max of

Figure 4.2 Principal axes andwhich is written by τ max = σ1 − σ 2 2 through a similar consideration

principal stresses

made in Eq. (4.3). The Tresca criterion for the onset of plastic yield is,therefore, given bymax { σ1 − σ 2 } ≥ 2k = Y(4.5)

Equation (4.4) is the more generalized extension of Eq. (4.5) in the threedimensional S-S state.

4.2 MAXIMUM SHEAR STRAIN ENERGY CRITERION-VON MISES CRITERIONThe Tresca criterion is convenient for using it in practical plastic
analyses, but a somewhat better fit to experimental data of ductile metals
can often be obtained from the von Mises criterion, on which the onset ofplastic yield is dictated with shear strain energy.
energy density

U

The elastic strain

stored in a unit cube (see Fig. 1.2 withdx = dy = dz = 1 ) under a three-dimensional S-S state is given by

U = (1 / 2)(σ xε x + σ y ε y + σ z ε z+ τ xy γ xy + τ yz γ yz + τ zxγ zx )having the physical dimension of J/m3. Applying the elastic constitutive
equation, Eq. (2.1), to the above formula of strain energy density, we
finally have the following expressions of U in terms of only the stresscomponents:30

1
σ x 2 + σ y 2 + σ z 2 − 2ν (σ xσ y + σ yσ z + σ zσ x )2E
1 2+(τ xy + τ yz2 + τ zx2 )2G22
1 2 1 +ν2=σm +σ x − σ y ) + ( σ y − σ z ) + (σ z − σ x )
(
2K
6E
1 2
+
τ xy + τ yz2 + τ zx2 )
(
2G

{

U=

}

{

}

(4.6)

In Eq. (4.6), K stands for the bulk modulus that is related to the Young’s
modulus with K = E 3 (1 − 2ν )  , and the mean normal stress σ m is(defined by σ m = σ x + σ y + σ z) 3 . The first term of the right-hand sideof the second equation in Eq. (4.6) means the strain energy density
associated with volumetric change, and the second and the third terms
represent the shear strain energy density U S .

As mentioned in thepreceding considerations and in Fig. 4.2, when we choose the x -、 y-,andz -axes as their principal axes, the shear stress components τ xy , τ yz ,and τ zx all diminish, then the shear strain energy density U S can be
described by the use of the principal stresses of σ1 , σ 2 , and σ 3 , asfollows;
US =

1 +ν
6E

{(σ − σ ) + (σ − σ ) + (σ − σ ) }
2

2

1

2

2

2

3

3

1

(4.7)

As an example, when a rod yields at σ 1 = Y and σ 2 = 0 = σ 3 in a
simple tension test, Eq. (4.7) gives US (Tension) = (1 +ν ) Y 2 3E .

On

the other hand, in a pure shear test, the onset of plastic yield occurs at

σ 1 = −σ 2 = k,

and

US (PureShear) = (1 +ν ) k 2 E

σ3 = 0 ,

then

Eq.

(4.7)

results

in(notice that the mean normal stressσ m = (σ1 + σ 2 + σ 3 ) 3 turns to zero in pure shear, leading to the presenceof only shear deformations in the body tested).

At the onset of plastic

yield, the critical strain energy densities both in the simple tension and
the

pure

shear

tests

must

agree

with

each

other,i.e.,

US ( Tension ) = US ( PureShear ) , resulting in the relationship of Y = 3k

for the von Mises criterion. These critical strain energy densities are also
supposed to be equal to Eq. (4.7) in the von Mises criterion, and thus we
finally have the following general expression for the onset of plastic

31

yield;

{

}

1
2
2
2
(σ 1 − σ 2 ) + (σ 2 − σ 3 ) + (σ 3 − σ 1 ) ≥ Y 2 = 3k 2
2
Note that this result of k = Y(4.8)

3 is somewhat different from the Tresca

criterion with k = Y 2 .

4.3 ONSET OF PLASTIC YIELDUNDER SHERICAL INDENTATION CONTACT
Let us consider the plastic yield induced beneath the contact surface of
spherical indentation [4.1]. The onset of plastic yield will appear along
the penetration axis. i.e., z-axis ( ρ = 0 ), since the principal stresses

σ1 ( = σ r ) , σ 2 ( = σθ ) , and σ 3 ( = σ z ) have their maximum values on
the z-axis for spherical indentation contact in an elastic half-space (seeSec. 3.2). These principal stresses are given in the following equations
by the uses of Eqs. (2.30) - (2.32), (3.5), and (3.13);

σ r  σθ 1
−1 1  ≡  = −(1 +ν ) 1 − ς tan+ς  2 (1 + ς 2 )p0  p0 


σz
p0

=−

(4.9)

1
1+ ς 2

Normalized
maximum shear stress, τmax /p0(4.10)

0

0.4

0.3

where p0 is the maximum contact pressure defined in Eq. (3.19b), andcontact

radiusa.

Furthermore,themaximumshear

stressτ max (= σ r − σ z / 2) (see Eq. (4.5)) is written by
τ max
p0

=

σr −σz
2 p0


1
1
3
= −(1 + ν ) 1 − ς tan −1  +ς  2 (1 + ς 2 )2
(4.11)

The analytical result of τ max / p0 vs. ς ( = z / a ) relation is plotted in
Figure 4.3

actually shows that the shear stress attains its maximum value ofτ max = 0.31p0 at

z = 0.48 a along the z-axis.

Accordingly, the
32

0

1

0.31

2

3

4

5

in which uses have been made of the relations of Eqs. (4.9) and (4.10).

Fig. 4.3 for the Poisson’s ratio of ν = 0.3 as an example.

0.1

0.48

Normalized z-coordinate, z/aς = z a is the non-dimensional z-coordinate normalized with the

0.2

Figure 4.3 Distribution of the
maximum shear stress (Eq. (4.11))
along the penetration axis[4.1] K.L. Johnson, “Contact Mechanics”,
Cambridge University Press (1985)

maximum contact pressure p0 ( ≡ 3 pm 2) as the threshold for the onsetof plastic yield is

p0 ≥ 3.2k = 1.6Y ,(4.12)()

for the Tresca criterion ( 2τ max ≡ max { σ 1 − σ 2 } ≥ 2k = Y ),andp0 ≥ 2.8k = 1.6Y

(4.13)()

for the Von Mises criterion ( 2τ max ≡ max { σ 1 − σ 2 } ≥ 3k = Y ).
The preceding considerations combined with the relations of

p0 = ( 3 2 ) pm , pm = P π a 2 , Eq. (3.17) and Eq. (3.18), therefore, finallygive the threshold Hertzian contact load PY for the onset of plastic

PY

yield;

PY = (1.6 )3

π 3 R2 Y 3
6

E '2

,

(4.14)ameaning that the resistance to the plastic flow increases with the increaseinY 3 E '2

0.48a

for spherical indentation contact (Hertzian contact).

Namely, Eq. (4.14) implies that the elastoplastic body with a larger yieldstress and a smaller elastic modulus resists to the onset of plastic flow
under Hertzian indentation contact.
Figure 4.4 (the numerical result of finite element analysis (FEA), thedetails of which will be given in Chap. 10) demonstrates the plastic core
beneath the contact surface induced at the threshold Hertzian contact load

PY . As actually predicted in Eq. (4.11) and in Fig. 4.3, the plastic coreappears at the location of z=0.48a that is clearly seen in Fig. 4.4.

33

Figure 4.4 A plastic core appears at
the location of z=0.48a beneath thecontact

surfaceunderspherical

indentation (FEA-based numericalresult with von Mises criterion)

ELASTOPLASTIC
INDENTATION CONTACT MECHANICS

CHAPTER5

5.1 MATERIALS PHYSICSOF THE MEYER HARDNESSThe science and engineering of contact plasticity significantly evolved
with the invention of steam locomotives in the second half of 19th century,
where the plastic contact deformation induced between the rail and thewheel was very critical. In the beginning of 20th century, the concept andthe test methods of indentation hardness were proposed for characterizing
the plasticity of metallic materials. The well-known test methods ofindentation hardness include “Brinell test” (Johan August Brinell, 1900)
and “Vickers test” (Vickers Ltd., 1925). In these tests, the hardness H
is defined as the indentation load P divided by the total contact areaSof the residual impression;

H=

P
S

The details for various types of indentation hardness test are given in theJapanese Industrial Standards (JIS) [5.1].
The physics of indentation contact hardness as a material characteristic
parameter was first examined by E. Meyer in 1908. He defined the Meyer
hardness HM as the indentation load P divided by the projectedcontact area Aproj of residual impression;
HM =

P
Aproj

(5.1a)

He emphasized the importance of projected contact area in determining
the contact hardness through the theoretical considerations on themechanical processes of spherical/conical indentation. In ductile metals,
the projected contact area

Aat the maximum indentation load

approximately equals its projected contact area of the residual impression
Aproj , i.e. A ≅ Aproj , because the elastic recovery in unloading is
negligibly small. However, in ceramics and organic polymers, due to asignificant elastic recovery of contact impression during indentation
unloading, there always exists the relation of A > Aproj . Accordingly,
when we want not only to measure the plasticity, but also to characterize
the elasticity in an indentation contact test, the use of contact area Aproj
34[5.1] Examples;
JIS Z2243, “Brinell Hardness Test” (2008)
JIS Z2244, “Vickers Hardness Test” (2009)
JIS Z2245, “Rockwell Hardness Test” (2011)
JIS Z2246, “Shore Hardness Test” (2000)

of the residual impression we get it after a complete elastic recovery is
inappropriate.

In-situ measurement of the projected contact area A

at the indentation load P

is very essential in characterizing theelasticity/plasticity of the material indented. In this text, therefore, we
redefine the Meyer hardness H M by the use of the projected contact
area A at the indentation load P , as follows;

HM =

P
A

(5.1b)We use Eq. (5.1b) as the definition of the Meyer hardness, since theprojected contact area A under load is very essential for understanding
the materials physics of indentation contact mechanics, the details of
which will be discussed in the following chapters. However, upon using
the conventional instrumented indentation apparatuses, it is impossible inexperiments for us to determine the projected contact area A underload, though these apparatuses are capable of determining the penetration
depth h in relation to the associated indentation load P . This fact
implies that we cannot determine in a quantitative manner not only the
Meyer hardness, but also the material characteristics (elastic modulus,
yield stress, viscoelastic functions, etc.) on the conventional indentation
apparatuses. We have to circumvent this difficulty through making anappropriate approximation/assumption to estimate

Afrom theobserved P - h relation prior to determining the material characteristics.
The instrumented indentation microscope has been designed to overcome
this essential difficulty in conventional instrumented indentation testing.
The indentation microscope enables in-situ measurement of the projectedcontact area A in relation to the applied indentation load P .

Thedetails (the concept/principle of measurement, the mechanical structureof the apparatus, experimental procedure for determining material
characteristics, etc.) of the indentation microscope will be given in Sec.
11.3.
Let us first consider the Meyer hardness of a perfectly elastic body. The
Meyer hardness of an elastic body is related to its elastic modulus E 'via Eq. (5.2) combined with Eq. (3.32);

 P
HM  =  = εI E ' Ae 

(5.2)

where Ae stands for the projected contact area created at the indentation35

load P(the subject index “e” means “elastic”), and ε I is theindentation strain that has been defined in Eqs. (3.29) - (3.31).

Asemphasized in Chap. 3, the indentation strain ε I depends not only on
the tip-geometry of the indenter, but also on the depth of penetration forflat-ended cylinder and sphere. Once we apply Eqs. (3.29) - (3.31) to Eq.
(5.2), the H M vs. E ' relations for the respective tip-geometries ofindentation are given as follows;
Flat-ended cylindrical punch:
HM =Spherical indenter:HM =Conical indenter:

HM =

2

π A0

h⋅E'

4
3π 3/ 2 R

(5.3)

Ae ⋅ E '

(5.4)

tan β
⋅E'
2

(5.5)

The Meyer hardness of flat-ended cylindrical punch exhibits a linear
increase with the increase in the penetration depth h , i.e., H M ∝ h , due
to the invariance of the contact area A0 . In spherical indentation, on theother hand, the Meyer hardness depends on the square root of the
projected contact area or the square root of the penetration, i.e.,H M ∝ Ae R ∝ h R .

In contrast to the tip-geometries of theseindenters, the Meyer hardness of conical indentation, however, is
independent of the depth of penetration, resulting in a constant value ofthe hardness that is proportional to the elastic nodulus E ' .

This fact isvery essential in characterizing the mechanical characteristics from the
Meyer hardness observed in experiments. The geometrical similarity of
the cone indentation (the linear relation between the penetration depth
and the contact radius; a ∝ h as shown in Eq. (3.24)) actually results in
the Myer hardness that is independent of the depth of indentation
penetration. In the history of indentation contact mechanics, thisgeometrical similarity of indentation yields the reason why the cone
indenter (Rockwell indenter) and the pyramid indenters (Vickers indenter,
Berkovich indenter) have long been utilized as the standard test methods
for determining the contact hardness numbers.

36

Let us now consider the elastic limit and the onset of plastic yield thatappears when the indentation strain ε I increases to its threshold valueof ε IY (see Fig. 4.1). The indentation strain ε I increases with theincrease in the penetration depth h of a flat-ended cylindrical punch ora spherical indenter.

It is also possible for us to increase ε I by the useof a spherical indenter with smaller radius R , or a conical indenter withlarger inclined face angle β . When ε I reaches to it threshold valueof ε IY , the plastic core appears beneath the contact surface, as shown in
Fig. 4.4 for a spherical indentation by way of example. The Meyer
hardness at the threshold strain ε IY is given by

H M ( = ε IY E ') ≈ Y ,(5.6)

as has been suggested both in the Tresca and in the von Mises criteria (seeEqs. (4.12) and (4.13)). In other words, the plastic yield is resulted inwhen the elastic Meyer hardness ε I E ' (Eq. (5.2)) increases up to theyield stress Y .
Most of ductile materials fall into the category of perfectly plastic body(fully plastic body), once the plasticity overwhelms the elasticity in the
indentation-induced deformation. The Meyer hardness of a fully plastic
body is independent of the indentation strain ε I , and described by

 P
= cY
HM  =
 A 
p 


(5.7)

in which Ap means the projected contact area of indentation (the subject
index “p” stands for plasticity). The major of plastic deformation induced
beneath the indentation contact surface is accommodated in the elastic
field surrounding the plastic core of indentation. However, when theplastic deformation becomes so significant that the surrounding elastic
field becomes insufficient for accommodating this plastic deformation,
the excess of the in-surface plastic deformation flows out to the freesurface, resulting in the indentation pile-up. On the other hand, thesurface energy (i.e., the surface tension) of the free surface constrains thisplastic flow-out. The constraint factor c in Eq. (5.7) thus describes theconstraint resistance to the plastic flow-out. The constraint factor c is
not only dependent on the tip-geometry of the indenter used, but also on
the contact friction. The experimental results along with the FEM-based
numerical analyses suggest that the constraint factor c ranges from 2.5

37

to 3.5, i.e., 2.5 ≤ c ≤ 3.5 [4.1].
The elastoplastic deformation induced with indentation contact cantherefore be divided into the following three regions [4.1]:

HM < Y ：elastic region(5.8a)Y ≤ H M < cY :elastoplastic region(5.8b)

H M ≈ cY :fully plastic region(5.8c)

Since the Meyer hardness is described by H M = ε I E ' (see Eq. (5.2)) in
the elastic region ( H M Y < 1 ), Eq. (5.8a) can be rewritten with

εIE '
Y

<1

(5.9)

εI =

2 h
；flat-ended cylindrical punch
π a0

εI =

4 a
；spherical indenter
3π R

εI =

tan β
；conical/pyramidal indenter
2

Namely, the deformation of an elastoplastic body having the elastic
modulus of E ' and the yield stress of Y is always perfectly elastic
when the indentation strain ε I applied to the body satisfies ε I E ' Y < 1P(elastic region). Accordingly, in the elastic region, the Meyer hardness isba material characteristic parameter for representing the elastic modulus•a

E ' , i.e., H M = ε I E ' . On the other hand, the Meyer hardness representspm

H M = cYin the fully plastic region

plastic( ε I E ' Y > 20 ), while in the elastoplastic region ( 1 < ε I E ' Y < 20 ), it iselastic

the yield stress Y , i.e.,

an elastoplastic parameter given as a function of E ' and Y , the detailsof which will be given in the followings.

Figure 5.1 Cavity model of an(1) Cavity model [4.1]elastoplastic indentation contact by a

Figure 5.1 shows the cavity model of the Meyer hardness, where a cone

cone; the model comprises a hemi-

indenter with the inclined face angle β (see Fig. 3.4) is pressed onto

spherical cavity of radius a withthe free surface of an elastoplastic half-space with the elastic modulus

internal hydrostatic contact pressure

E ' and the yield stress Y .

pm ( = P π a 2 ) . The cavity is

The cavity model comprises (i) a hemi-

spherical cavity of radius a with the internal hydrostatic contact pressure,surrounded by a plastic core withpm ( = P / π a 2 ) , (ii) a plastic core ( a < r ≤ b ) with radius b surrounding

radius b embedded in an elastic half-

the cavity, which is embedded in (iii) an elastic half-space ( r > b ). The

space

38

volume of the cavity ( ≡ ( 2π 3) a 3 ) stands for the excluded volume ofindentation. The materials (atoms and molecules) thus excluded fromthe cavity is assumed to be accommodated in the elastic half-space
through the plastic core. In the cavity model, therefore, the indentation
contact mechanics is simply reduced to the elastoplastic continuum
mechanics of a semi-infinite elastic half-space with a surface cavity ofradius a that is surrounded by a plastic core of radius b, where the cavity
is subjected to the hydrostatic pressure of pm . The cavity model finallygives the following formula for the Meyer hardness,
H M ( ≡ pm ) = P π a 2 [4.1]:

 (1 + ν ) E ' 2(1 − 2ν )  
HM 2 εI +=  2 + ln 

Y
Y
3
3(1 − ν )  
 3(5.10a)Equation (5.10a) is simply rewritten by

HM 2 
 1 E ' 
=  2 + ln  ε I  Y3 2 Y (5.10b)for incompressible elastoplastic body with ν = 1 2 . It must be noticed
that the cavity model can describe the indentation contact behavior onlyin the elastoplastic region (Eq. (5.8b)).

The model neither can beapplied to the elastic region (Eq. (5.8a)), nor to the fully plastic region(Eq. (5.8c)).

In the subsequent section, we will introduce a unified

theory that is fully capable of describing the Meyer hardness from theperfectly elastic to the fully plastic regions.
(2) Unified theory of the Meyer hardness:
The additivity principle of the excluded volume of indentation

The theory describes the Meyer hardness H M in terms of the elastic
modulus E ' and the yield stress Y as a function of the tip-geometryof indentation in a unified manner that encompasses the elastoplastic
Meyer hardness ranging from perfectly elastic to fully plastic indentation
contacts. Since the theory is based on a phenomenological principle, i.e.,a thermodynamic principle, any of elastoplastic models can be adopted.
In this textbook, we apply an energy-based consideration to the preceding
cavity model. Let us consider the virtual increment of the Gibbs free39

energy δ G that is associated with the virtual increment of the excluded
volume δ V under the virtual increment of the contact pressure frompm ( ≡ P / π a 2 ) to p m + δ p acting on the internal surface of the cavity.

The increment of the free energy, i.e., the externally applied work

δ W ( ≡ δ G ) is, therefore, given byδ W ( ≡ pmδ V ) = pmδ Ve + pmδ Vp(5.11)

where the higher-order term of δ p⋅δV is assumed to be negligibly small.
In Eq. (5.11), δ Ve and δ Vp are the volumetric increments associatedwith the elastic and plastic indentation deformations, respectively. The
first term ( pmδ Ve ) in the right-hand side of Eq. (5.11) stands for theelastic stored energy, and the second term ( pmδ Vp ) is the energy
dissipation associated with plastic flow.
Equation (5.11) is thus rewritten by δ V = δ Ve + δ Vp , and then finally
we have the additivity principle of excluded volume of indentation as
follows;
V = Ve + Vp

(5.12)Let us consider the alternative expression of this additivity principle interms of the contact area A. Since the excluded volume V is related to the()

contact area A via V = tan β 3 π ⋅ A3 2 for cone/pyramid indentation(inclined face-angle of β ), and to V = (1 6π R ) ⋅ A2 ( a R  1 ) for
spherical indentation (radius of R), there exists the relation of V ∝ A m( m = 3 2 for cone/pyramid indenter; m = 2 for spherical indenter).
The additivity principle of the excluded volume, therefore, finally resultsin the following formula of the contact area A ;Am = Aem + Apm(5.13a)or
1/ m

 A /A m pe=A 1 + A / A m 
p
e



Ap

(

(

)

(5.13b))In these equations, A is the contact area of an elastoplastic body (elastic
modulus E ' and yield stress Y ) generated by the indentation load of
P . Ae and Ap are, respectively, the contact areas of a perfectly elastic

40

body with the elastic modulus E ' and of a fully plastic body with theyield stress of Y ; both are created by the same indentation load P and
the same tip-geometry of the indenter applied to the elastoplastic body.
Substituting A = P H M (Eq. (5.1b)), Ae = P ε I E ' (Eq. (5.2)), and
Ap = P cY (Eq. (5.7)) into Eq. (5.13b), we can express the elastoplastic
Meyer hardness H M in terms of the elastic modulus E ' and the yieldstress Y , as follows;1m  ε E ' m IH M   cY  =
cY   ε I E ' m 
1 +    cY  
m=3 2;
m=2;(5.14)

conical indenter
spherical indenter

It should be emphasized that Eq. (5.13b) is equivalent to Eq. (5.14),
though these equations are expressed in somewhat different ways.
Equation (5.13b) plays an important role in determining the elastoplastic

experiments by the use of the “instrumented indentation microscope”(refer to the details in Chap. 11).

The elastoplastic parameterε I E ' cY ( ≡ Ap Ae ) in Eq. (5.14) represents the relative ratio of plastic
to elastic deformation, namely, describes the amount of plastic
deformation relative to the elastic deformation.

In this text book,

Normalized hardness
HM/cY(= Ap/A)

material characteristics, once we directly measure the P - A relation in
1 Eqs. (5.13b), (5.14): spherical indenter: cone indenter
0.8
0.6
0.4
0.2
0

: Johnson's Cavity model(Eq. (5.10b))
10-1

100

101

Plastic index, εIE'/cY(= Ap/Ae)

therefore, we define it as the plastic index, PI ; PI = ε I E ' cY .
The comparison between the cavity model (Eqs. (5.10a), (5.10b)) and

Figure 5.2 The relation between thethe unified theory of volumetric additivity (Eqs. (5.13b), (5.14)) is shownnormalized hardness and the plastic

in Fig. 5.2. As emphasized in the preceding considerations, the cavity

index.

model is applicable only in the elastoplastic region, while the additivity

volumetric additivity (Eqs. (5.13b)

principle analytically predicts the elastic/plastic indentation contact

and (5.14)) is given by the solid

encompassing the all ranges from perfectly elastic to fully plastic(sphericaldeformation.

As recognized in Fig. 5.2 and in Eq. (5.14), thebroken (conical indentation) lines.

elastoplastic contact behavior depends not only on the elastoplastic

The Johnson’s cavity model (Eq.

characteristics E ' cY , but also on the tip-geometry of the indenter used

(5.10)) is given by the dashed-dotted(spherical, conical indenters, etc.) via ε I . The elastoplastic contact

line.

41

The unified theory ofindentation)

and

the

behavior is, therefore, classified in terms of the plastic index PI, as
perfectly elastic region:

PI <0.2

sphere; coneelastoplastic region:

0.4 ≤ PI <8sphere0.4 ≤ PI <20

cone

PI ≥ 8spherePI ≥ 20

cone

fully plastic region:

Normalized hardness
HM/cY(= Ap/A)

follows;

1
0.8
0.6

spherical indenter: Numerical FEA-results: Volumetric additivity
principle

0.4
0.2
0 -1
10

Figureconical indenter: Numerical FEA-results: Volumetric additivity principle
100

101

102

Normalized

Meyer

Plastic index, εIE'/cY(= Ap/Ae)

5.3

The validity and the reliability are scrutinized in Fig. 5.3 for the

hardness vs. plastic index plots inadditivity principle of the excluded volume of indentation, where thespherical and conical indentation

analytical predictions of the volumetric additivity principle (the solid andcontacts: The solid and the broken

the broken lines) are compared with the finite-element-based numericallines are the analytical predictions

results (the triangles and circles).

As clearly seen in Fig. 5.3, the(Eqs. (5.13b) and (5.14)) of theanalytical results based on the principle of volumetric additivity faithfully

volumetric additivity principle; theas well as precisely predict the FEA-based numerical results not only forsymbols (triangles△ and circles ● )spherical but also for conical indentation contacts.

are the finite-element-based
numerical results(3) Elastoplastic deformation and flow of cone/pyramid indentation

As mentioned in the preceding sections, due to the geometricalsimilarity of cone/pyramid indentation, the Meyer hardness is2

2

P = H M ⋅ A ∝ a ∝ h . Accordingly, it will be easily expected incone/pyramid indentation that the indentation load P is proportional to
the square of penetration depth h2 , i.e., the indentation load linearly2

increases with the increase in h . Figure 5.4 shows the numerical resultsof finite element analysis (FEA); the P - h2 hysteresis relations aredemonstrated for the perfectly elastic (PI (= ε I E ' cY )=0.05) ,elastoplastic (PI=1.0), and the fully plastic (PI=20) bodies in theirloading/unloading cycles of the Vickers/Berkovich equivalent cone
indentation.

In all ranges of the perfectly elastic to the fully plastic

indentation contact, the loading P - h2 relations (the solid lines) arelinear. It should be noticed, as shown in Fig. 5.4, that the subsequentunloading P - h2 relations (the broken lines) are also linear; the loading
linear P - h2 line coincides with the subsequent unloading
for the perfectly elastic body with PI=0.05.

linear lines

The slopes of42

Indentation load, P(N)independent of the penetration depth, leading to the relation of

6 E=100 GPa

εIE'/cY=0.05
1.0

4

2

20
0
0

10

20

3

Penetration depth, h2 (μm)

Figure 5.4 P - h2 hysteresis
relations of the elastic (PI=0.05),
elastoplastic (PI=1.0), and the fully
plastic bodies (PI=20) in theirloading/unloading cycles of theVickers/Berkovich equivalent cone
(FEA-based numerical results). The
solid and the broken lines indicate theloading and the subsequentunloading P - h 2 relations,
respectively.

material characteristic parameters of the Meyer hardness H M , elastic
modulus E ' , and the yield stress Y , the details of which will be givenin the subsequent chapters.
Outside the contact area of indentation, the free-surface of an elastic
half space always sinks-in along with the penetration, i.e.,ηc ( ≡ hc h ) < 1 .

As discussed in Chap. 3, ηc ( ≡ hc h ) = 1 2 forspherical indentation, and ηc ( ≡ hc h ) = 2 π for conical indentation

Relative contact depth 、hc/hloading/unloading P - h2 linear relations are intimately related to the

1.2

pile-up1sink-in
0.8

0.6
0

10

20

Plastic index、εIE'/cY

Figure 5.5 Relative contact depth
hc h under load is plotted against(refer to Figs. 3.3 and 3.4; Eqs. (3.21) and (3.28), and Table 3.1). On

the plastic index PI (FEA-based

the other hand, due to a significant plastic deformation of elastoplastic

numerical results). The profile of theindentation contact (PI>0.4), the indentation-displaced molecules/atoms

free surface outside the contact area

beneath the contact area flow out to the free-surface, resulting in a pile-undergoesup outside the contact area, i.e., ηc ( ≡ hc h ) > 1 .

sinking-in ( hc h < 1 ) to piling-up

The FEA-based

numerical results of the relative contact depth hc h under load areplotted against the plastic index PI ( ≡ ε I E ' cY ) in Fig. 5.5 for the
Vickers/Berkovich equivalent cone, where the transition from sinking-inthetransitionfrom( hc h > 1 ) with the increase in theplastic index (FEA-based numericalresults)

to piling-up occurs at about PI=6. Figure 5.6 shows the side views of
the surface profiles outside the contact area of the equivalent cone at theand the profiles of residual impression after unloading (the left-hand halfin Fig. 5.6); the profiles are of PI=0.318, 3.18, and 31.8, respectively. As

uzz (r)( r) (μm)( μm)penetration depth of h=5μm under load (the right-hand half in Fig. 5.6),

2
0
-2
-4
-6
40

●: PI = 0.318
○:

3.18

□:

31.8

30

20

10

0

( μm)
r r(μm)

10

20

30

40

well recognized in Fig. 5.6, in the elasticity-dominant region of PI<3, theFigure 5.6 The side views of theelastic recovery along the penetration axis is very significant in unloading.

surface profiles outside the contact

Furthermore, it should be noticed that the threshold value of plastic index

area

PI at the sink-in/pile-up crossover is about PI=6 under load in Fig. 5.5,

equivalent cone for the elastoplastic

while it is about PI=3 after unload shown in the left-hand half of Fig. 5.6;

bodies with three different values ofthis significant discrepancy is resulted from the enhanced pile-up

the plastic index PI = ε I E '/ cY . The

of

the

Vickers/Berkovich

right-hand half is the side views at

associated with the elastic recovery in unloading.

the penetration depth h=5μm, and theleft-hand half is the profiles ofresidual impressions after unload43

***** Tip-Geometries ofVickers/Berkovich Indenters *****The Vickers indenter was designed to circumvent thea(= sin β ) = 0.375Rβ

β = 22.0°Rβainconvenience of the Brinell’s spherical indenter, the hardness ofwhich is always dependent on the penetration depth. As shown in
Fig. 5.7, the inclined face-angle of the Vickers indenter is designed
to be β = 22.0 , being consistent to the including face-angle ofsphere at a R = 0.375 of the standard Brinell hardness test
condition.

On the other hand, the geometry of the trigonal

Berkovich indenter is designed with the inclined face angle ofβ

β = 24.7 , resulting in the same excluded volume of the Vickers

Figure 5.7 Tip-geometry of theindentation. A conical indenter having the inclined face-angle ofVickers indenter designed to be

β = 19.7 is called as the Vickers/Berkovich equivalent cone, theexcluded

volumeofwhich

Vickers/Berkovich indenters.

coincides

with

that

of

The details of the equivalent cone

will be given in Chap. 11

44

consistent with the Brinell indenter5.2 THE P-h LOADING/UNLOADING
HYSTERESIS CURVE OF CONICAL INDENTATIONThe P - h relationship of a perfectly elastic body ( PI < 0.2 ) for
conical indentation contact is given in the following equation by2

substituting the elastic contact area Ae = π (ηe cot β ) h 2 into Eq. (5.2);

P ( ≡ ε I E ' Ae ) = ke h2((5.15))

ke = πηe2 cot 2 β ε I E 'in which ηe ( ≡ hc h ) = 2 π is the relative contact depth of the elastic
body.
the following P - h relationship;()

P ≡ cYAp = kp h 2((5.16))

k p = πη p 2 cot 2 β cY

In Eq. (5.16), ηp denotes the relative contact depth hc h of a fully

Indentation load， P (N)For a fully plastic body ( PI ≥ 20 ), on the other hand, Eq. (5.7) leads to10plastic body, the value of which is ηp ≈ 1.2 as shown in Fig. 5.5. The
P - h relation of an elastoplastic body with its plastic index PI ranging
from 0.2 to 20 is given in Eq. (5.17) in terms of the Meyer hardness H M ;P ( ≡ H M A) = kep h2((5.17))

kep = πηep 2 cot 2 β H M

In Eq. (5.17), the subscript “ep” stands for “elastoplastic”, and the relative

Berkovich indenter

20

10
20
Penetration depth, h (μ m)

Figure 5.8 The P - h loadingunloading hysteresis curves of three
engineering materials with different
elastoplastic behavior (Berkovich
indentation)

contact depth ηep changes in the range of 2 π < ηep < 1.2 , depending(Berkovich indentation) of silicon nitride ceramic (Si3N4), soda-lime
glass (Glass), and metallic copper (Cu) as the representatives ofengineering materials exhibiting various elastoplastic behaviors, i.e.,having the different PI -values.
The loading-unloadingP - h linear hysteresis is plotted in Fig. 5.9of the Si3N4 ceramic that has been demonstrated in Fig. 5.8. The
considerations on Fig. 5.4 combined with Eqs. (5.15) – (5.17) lead to therelation of P ∝ h2 orP ∝ h both in the loading and in the unloading

processes due to the geometrical similarity of cone/pyramid indentation.

45

1/2
1/2

Figure 5.8 shows the loading-unloading P - h hysteresis curves

Indentation load, P P (N )on the plastic index PI (refer to Fig. 5.4).
Si 3N 4 ceramic

4
3
2
1
0

kep
0

2

hr

ke

4

6

Penetration depth, h (μm)

Figure 5.9 The loading-unloadingP - h linear plot of Si3N4 ceramic,
the P - h hysteresis of which isplotted in Fig. 5.8

The indentation loading process is described by Eq. (5.17), and thesubsequent unloading process is well approximated by
P ( ≡ ε I E ' Ae ) = ke ( h − hr )(2

(5.18))

ke = πηe 2 cot 2 β ε I E 'like as Eq. 5.15 for the P - h relation of perfectly elastic body since theunloading process of any elastoplastic body is essentially resulted fromthe elastic recovery of the indentation contact impression. In Eq. (5.18)
and in Fig. 5.9, hr denotes the residual depths of contact impression
formed after a complete unload.
The subsequent unloading P - h2 linear line of a perfectly elastic
body overlaps with the preceding loading P - h2 line (refer to the P h2 loading-unloading relations for PI =0.05 in Fig. 5.5), resulting innone of P - h hysteresis phenomenon. This fact implies that theexternal work applied to the system is completely released during theelastic recovery in unloading. The change of Gibbs free energy Δ G of
the system, therefore, is zero in this loading-unloading mechanical cycle.
In contrast to the elastic indentation process, the hysteresis loop areas
enclosed with the loading and the unloading curves demonstrated in Figs.
5.5, 5.8, and 5.9 stand for the plastic energy that dissipates to the outsideof the system as a heat flux during indentation loading-unloadingmechanical cycle (note: the hysteresis loop energy may partly include theelastic strain energy stored in the field surrounding the residual
impression).
The energy diagram associated with the indentation loading- unloading
processes of an elastoplastic body is depicted in Fig. 5.10. The external

Pmax

work U T applied to the system in the loading process up to its

UT=Ur+Ue

maximum indentation load Pmax is described by

UT = Ur + Ue

(5.19)

as a sum of the loop energy (the plastic energy dissipation) U r and theelastic strain energy U e .

Using Eqs. (5.17) and (5.18), we finallyUr

have the following expressions for the energies of U T and U e ;UT = 

hmax

0

Ue = 

hmax

hr

Pdh =

kep
3

hmax 3hr(5.20)

hr

Ue
he

hmax

Figure 5.10 Energy diagram fork3
Pdh = e hmax 3 (1 − ξ r )3(5.21)

the indentation loading- unloading
process.

46

where ξ r means the relative residual depth defined by ξ r = hr hmax .
The plastic energy dissipation U r is, therefore, related to the externallyapplied work U T as follows;

Ur ( ≡ UT − Ue ) = ξrUT

(5.22)In the derivation of Eq. (5.22), we have utilized the compatibility relationof kep hmax 2 = ke ( hmax − hr ) , or kep ke = (1 − ξr ) , due to the fact that22

the loading curve crosses the unloading curve at the coordinate of ( hmax ,Pmax ).
(1) Work-of-Indentation, WOI

The work-of- indentation WOI is defined as the externally applied
work to create a unit volume of residual impression;
WOI =

Ur
Vr

(5.23)

where Vr means the volume of residual impression. Substituting Eqs.
(5.17), (5.20), and (5.22) into Eq. (5.23) and noticing the approximations()

Vmax ≈ π cot 2 β 3 hmax 3

andVr ≈ ξ rVmax

finally give thefollowing important conclusion that the work-of-indentation WOI isequivalent to the Meyer hardness:
WOI ( ≡ U r Vr ≡ U T Vmax )(5.24)= HM

In other words, the Meyer hardness H M that has been historicallydefined as the indentation load divided by the contact area H M = P A

Work-of-Indentation, WOI (GPa)of(see Eq. (5.1b)) is re-defined in this context as the plastic energy

2

1

0
0

1

2

Meyer hardness, HM (GPa)dissipation to create a unit volume of residual impression U r Vr , or theexternally applied work to create an indentation-excluded unit volume

Figure 5.11 One-to-one correlation

U T Vmax .

betweenHM

and WOI of the

The one-to-one correlation between the Meyer hardness H M and theelastoplastic materials having thework-of-indentation WOI is shown in Fig. 5.11 for the elastoplastic

elastic modulus of E = 10 GPa andmaterials having their material characteristics of the elastic modulus

theE = 10 GPa, and the yield stresses ranging 0.1GPa ≤ Y ≤ 3GPa , all

0.1GPa ≤ Y ≤ 3GPabeing the numerical results examined in a finite element analysis (FEA).

numerical results)

47

yield

stressranging(FEA-based

(2) Correlation between the loop energy U r
and the yield stress cY

Pmax

As discussed in the preceding section, the loop energy U r of anelastoplastic body defined by the area enclosed by the indentation
loading-unloading hysteresis curve stands for the heat dissipation
associated with the plastic flow in the indentation contact process.
Accordingly, the elastoplastic loop energy U r will nearly equal theplastic energy dissipation U p of a fully plastic body having the same

Ur
Pmax

hr

h

hp

h

yield stress Y as that of the elastoplastic body, where the plastic energy
dissipation U p can be given through integrating the P - h relation of Eq.
(5.16);() k3 hhp

U p ≡  Pdh =
0

p

p

3

(5.25)The elastoplastic and the fully plastic loop energies U r and U p areUpdepicted in Fig. 5.12 for comparison. As shown in the figure, the relations

Figure 5.12 The loop energy U r ofof U r ≈ U p and hp ≈ hr exist under the same maximum indentation

an elastoplastic body and the plastic

load, since both of the elastoplastic and the fully plastic bodies have theenergy dissipation U psame yield stress Y , resulting in the following unique expression thatplastic body, both having the same

correlates the elastoplastic loop energy U r with the yield stress Y

value of the yield stress Y .

through the following linear relation of U r vs. P

(

P

3/2

tan β P

) 3 k = 3η π

Ur ≈ Up =

p

p

3/2

;

exist the approximations

3/2(5.26)

cYof a fully

There

Ur ≈ Upand hp ≈ hr under the same maximum
indentation load of Pmax

We can, therefore, determine the yield stress cY from the slope of theobserved linear U r vs. P3/2 plot as well demonstrated in Fig. 5.13 for

Elastoplastic loop energy, U r (mJ)the several engineering materials (Vickers indentation) [5.2].

[5.2] M. Sakai, Acta Metall.
Mater., 41, 1751 (1993)

Vickers indentation

3
Al
Cu
MgO
Si3N4
SiC
Glassy carbon

2

Figure 5.13 U r - P3/2 linear plots (Vickers indentation
test results). These linear plots are, respectively, formetallic aluminum, metallic copper, magnesium oxide,1

silicon nitride, silicon carbide, and for glass-like carbon,
0

0

200

400

600

800
3/2

Indentation load, P

3/2

(N )

1000

from the top to the bottom. The slope of the respective
linear plot gives the yield stress cY (see Eq. 5.26)

48

LINEAR VISCOELASTIC FORMULATION

6CHAPTER

6.1 GLASS TRANSITION BEHAVIOR AND
VISCOELASTIC RESPONCE(1) Glass Transition Behavior [6.1, 6.2]
The mechanical responses of elastic and elastoplastic bodies are
independent of the time as well as of the rate of deformation/load applied
to the system. These mechanical responses instantaneously onset once thestress/strain is applied to the system. Furthermore, the system does notshow the time-dependent relaxation phenomena and the creep
deformation against a stepwise application of stress/strain. On the other
hand, amorphous materials including organic polymers, inorganic glasses,
etc., exhibit time/rate-dependent mechanical responses at temperatures
above their glass transition point.

These materials fall into the group ofviscoelastic materials.
The volumetric change of an amorphous body is shown in Fig. 6.1

super
glass
cooled liquidtransiition

glass



discrete change in their volumes at their freezing point (crystalline point)

Volume, V

against the temperature in comparison to that of a polycrystalline bodysuch as a metallic body. Most of liquids including molten metals show aassociated with cooling, and turn to the crystals, as shown in Fig. 6.1.
Since this volumetric change goes through a thermodynamicallyliquid

crystalequilibrium process, the resultant crystal reversibly turns to liquid at theTLmelting point (Tm) with a discrete volumetric dilation on heating. On theother hand, molten organic polymers and inorganic glasses change toglass-like amorphous solids in their cooling processes through the glasstransition region without any discrete volumetric changes, as shown in
Fig. 6.1, where the glass-transition temperature (glass-transition point) Tg
as a material characteristic temperature is defined at the intercepting
temperature between the liquid and the solid lines with the associated

Tg

TU

Tm

Temperature, T

Figure 6.1 Volumetric changes ofcrystalline and amorphous bodies in
the cooling/heating cycle, and theircharacteristic temperatures of TL, Tg,
TU, and Tm.

upper- and lower-temperatures, TU and TL, respectively. The glasstransition behavior is thermodynamically irreversible, resulting in thesecharacteristic temperatures of Tg, TU, and TL dependent on thecooling/heating rate; these temperatures shift to lower for larger thecooling rate, and vice versa.

49[6.1] E.J. Donth, “The Glass Transition:
Relaxation Dynamics in Liquids andDistorted Materials”, Springer (2001)[6.2] G.W. Scherer, “Relaxation in Glassand Composites”, John Wiley (1986)

(2) Viscoelastic Models: Maxwell Liquid and Zener Solid

Maxwell model°

amorphous bodies are time/rate-dependent viscoelastic. The MaxwellηEMmodel and the Zener model shown in Fig. 6.2 are the most common for
describing in a phenomenological manner the viscoelastic behaviors ofamorphous bodies in its glass-transition region.

°

As mentioned in the preceding section, the mechanical responses of

Zener model I

The Maxwell modelη

comprises a Hookean spring (the elastic modulus EM) and a Newtonian
dashpot (the viscosity η) connected in series.

The spring visualizes theelastic deformation, while the dashpot represents the viscous flow. The

Zener model II

Maxwell model is the simplest for describing a viscoelastic liquid. On theother hand, the simplest models for representing viscoelastic solids

ηinclude the Zener models of I and II. The Zener model I comprises a Voigt
model placed in series with a Hookean spring (the elastic modulus Eg),
where the Voigt model is given by a parallel combination of a spring (theelastic modulus EV) and a dashpot (the viscosity η).

The parallel

combination of a Maxwell model and a Hookean spring (the elastic
modulus Ee) leads to the Zener model II [1.3].

σ = EεIn a dashpot, the stress produces not a strain ε but a strain rate d ε dt
as follows:dεdt

In many of the following considerations, it will be convenient tointroduce the ratio of viscosity η (Pa･s) to stiffness E (Pa);

τ =

η
E

The unit of τ is time (s), and it will be seen that this characteristic timeplays an important role in describing the material’s viscoelastic response;τ is referred to as the relaxation time in stress relaxation phenomena,
and as the retardation time in creep deformation.
The mechanical responses of the viscoelastic liquid/solid models
shown in Fig. 6.2 are generally described with the following linear firstorder differential equation;

a0ε + a1dεdσ
= b0σ + b1
dt
dt

(6.1)

50

EM

Ee
Figure 6.2 The Maxwell model torepresent a viscoelastic liquid and the
Zenermodelsviscoelastic solids

The mechanical response of a Hookean spring is given by

σ =η

Eg

EV

for

describing

in which the coefficients are

a0 = 0a1 = 1(6.2a)

b0 = 1 η ( = 1 τ M EM )b1 = 1 EM
for the Maxwell model,

a0 = EV η ( = 1 τ V )a1 = 1(6.2b)

b0 = ( Eg EV + 1) τ V Eg
b1 = 1 Eg
for the Zener I model, anda0 = Ee τ M ( Ee + EM ) 
a1 = 1(6.2c)

b0 = 1 τ M ( Ee + EM ) 
b1 = 1 ( Ee + EM )for the Zener II model. The Zener I model with EV = 0 and Eg = EM ,or the Zener II model with Ee = 0 is reduced to the Maxwell model.
In Eqs. (6.1) - (6.2c), the subscript “g” of Eg , and “e” of Ee stand forthe “glassy modulus” and the “equilibrium modulus”, respectively.
The Laplace transform is very effective in solving linear differential
equations like as Eq. (6.1), the mathematical details of which are given
in APPENDIX C. The Laplace transform of a function f ( t ) in the time-()

space of t ≥ 0 is denoted here as L f (t ) ≡ f ( p) , and is defined by∞

L f ( t ) = f ( p ) =  f ( t ) e− pt dt
0where p is a nonnegative real parameter with the unit of inverse time (1/s)
termed the transform parameter. Applying Laplace transform to bothsides of Eq. (6.1) turns the differential equation to the following algebraic
equation;σ ( p) = E * ( p)ε ( p)

(6.3a)

where E * ( p ) is referred to as the pseudo elastic modulus define by

E * ( p) =a0 + a1 p
b0 + b1 p

(6.4)

51

By way of example, the pseudo elastic modulus E * ( p) of theE * ( p) = p ⋅EM
p+(6.5a)

1

τM

Relaxation modulus
Erelax(t) (GPa)viscoelastic Maxwell liquid is written by

1

0.6
0.4
0.2
0

through substituting Eq. (6.2a) into Eq. (6.4). In a similar mathematical



 1
1
p 

E * ( p) = Eg +1 
τV p + 1p+τZ
τ Z 


(6.5b)

3

Eq. (6.9a)

2
1
0

0

100

200

300

400

500

Time, t (s)τ Z = ( Ee Eg )τ V ; 1 Ee = 1 Eg + 1 EV

Figure 6.3 Relaxation modulus andfor the Zener I viscoelastic solid, andcreep compliance (Maxwell model):

(



E E
1
p 
e
g

E * ( p ) = Eg +
1  τM p + 1
p+τMτ M 
Eg = Ee + EM)

EM ≡ Eg = 1 GPa, τ M = 200 s,(6.5c)

for the Zener II viscoelastic solid.
As readily seen in Eqs. (6.5b) and (6.5c), their mathematicalexpressions are formally the same, though the respective coefficients
describing their E * ( p ) -values are different. As a matter of fact, both ofthe models I and II coincide with each other, and their viscoelastic
responses become equivalent when we substitute the relations ofEe Eg = τ M τ V and τ M = τ Z into the Model II.
The mathematically conjugated formula of Eq. (6.3a)

ε ( p ) = C * ( p )σ ( p )(6.3b)

defines the pseudo compliance C * ( p) that is related to E * ( p ) withC * ( p) = 1 E * ( p ) . As shown in Eqs. (6.3a) and (6.3b), it will be worthyof note that the formulas of linear viscoelastic constitutive equations

σ ( p ) = E * ( p )ε ( p ) and ε ( p ) = C * ( p )σ ( p ) in the Laplace spaceare equivalent to those of linear elastic constitutive equations σ = Eε
and ε = Cσ in the real space. Furthermore, there exists one-to-one
correspondence of C = 1 E and4

Creep compliance
Ccreep(t) (1/GPa)

substitution of Eq. (6.2b) or Eq. (6.2c) into Eq. (6.4), we finally have

Eq. (6.8a)

0.8

C * ( p ) = 1 E * ( p ) in the real andthe Laplace spaces. This one-to-one correspondence is named as “TheElastic-Viscoelastic Correspondence Principle” [1.2, 1.3], playing an52

η ( ≡ τ M ⋅ EM ) = 200 GPa･sessential role in formulating the viscoelastic indentation contact
mechanics, the details of which will be discussed in the later chapters.
Let us now consider; (i) the stress relaxation under stepwise strain and(ii) the creep deformation for stepwise loading in order to make further
understanding the viscoelastic responses of the Maxwell liquid and the
Zener II solid.
(i) Stress relaxation under stepwise strain
The stepwise application of a constant value of strain ε 0 to thesystem can be mathematically made byε (t ) = ε 0u (t ),0,u (t ) = 
1,

t<0
t≥0

(6.6)

using the Heaviside step function u (t ) . Accordingly, substituting the
Laplace transform of Eq. (6.6), i.e., ε ( p) = ε 0 p (refer to APPENDIXC) into Eq. (6.3a) leads to the following viscoelastic constitutive equationin the Laplace space;σ ( p ) =  E * ( p ) p  ε 0(6.7)= E relax ( p ) ⋅ ε 0Erelax ( p) represents the relaxation modulus Erelax (t ) [ ≡ σ (t ) / ε 0 ] inthe Laplace space, relating to the pseudo modulus E * ( p ) with
Erelax ( p ) = E * ( p ) p , and then through the inverse of Laplacetransforms of Eq. (6.5a) (Maxwell model) and Eq. (6.5c) (Zener II model)(refer to APPENDIX C), we finally have the following relaxation moduli
in the real space;[Maxwell liquid] −t σ (t )= Erelax (t ) = EM exp  
ε0
τM 

(6.8a)

Eg = EM[Zener II solid] −t σ (t )= Erelax (t ) = Ee + ( Eg − Ee ) exp 

ε0
τM 
E g = Ee + E M

53(6.8b)

(ii) Creep deformation for stepwise loading

stepwise application of a constant stress σ 0 to the system withσ (t ) = σ 0u (t ) , and then the substitution of its Laplace transform ofσ ( p ) = σ 0 p into Eq. (6.3b) combined with the creep function of
Ccreep (t ) [ ≡ ε (t ) σ 0 ]results

inthepseudocomplianceofC * ( p )  ≡ 1 E * ( p )  = pCcreep ( p ) with Ccreep ( p ) ( ≡ ε ( p ) σ 0 ) . The
creep functions Ccreep (t ) of the Maxwell and the Zener II models intheir real spaces are, therefore via the inverse Laplace transform, finally

0.8

0.6
Ee=0.5 GPa
0.4
1/Ee=2.0 GPa-1

2

Eq. (6.9b)

1.5

1

0.5

written with

0

100

200

300

400

500

Time, t (s)[Maxwell liquid]

Figure 6.4 Relaxation modulus andε (t )
1t= Ccreep (t ) =
+
σ0
EM η

(6.9a)

Eg ( = EM + Ee ) = 1.0 GPa

ε (t )= Ccreep (t )σ0

τ M = 200 s、
τ Z ( ≡ Egτ M Ee ) = 400s

 −t  
1  1
1 
=
+
−
1 − exp   Eg  Ee Eg   τ Z  

(6.9b)

Eg = Ee + EM

τ Z = (1 + EM Ee )τ M
It must be noticed in the Zener viscoelastic solid that the relaxation timeτ M in Eq. (6.8b) for stress relaxation is always smaller than the
time

τZ

in

creep compliance (Zener II model):EM = 0.5 GPa、 Ee = 0.5 GPa、

[Zener II solid]

retardation

Eq. (6.8b)

Creep compliance
Ccreep(t) (1/GPa)Using the Heaviside step function u (t ) , we readily describe theRelaxation modulus
Erelax(t) (GPa)1

Eq.

(6.9b)

for

creep

deformation;τ Z  ≡ (1 + EM Ee )τ M  > τ M . This fact indicates that the creep
deformation always proceeds in a more sluggish manner than the stressrelaxation (see Fig. 6.4).
The specific discrepancies in their viscoelastic responses observed in
the Maxwell liquid and the Zener solid are graphically compared in Figs.
6.3 and 6.4. The viscoelastic liquid exhibits a complete stress relaxationin time ( Erelax (t ) ↓ 0 ), while the induced stress of the viscoelastic solid
relaxes to a finite value of its equilibrium modulus ( Erelax (t ) ↓ Ee ).

54

6.2 DIFFERENTIAL FORM OF
VISCOELASTIC CONSTITUTIVE EQUATION-WIECHERT MODELWe cannot describe the time-dependent viscoelastic responses with asingle value of the relaxation/retardation time τfor most ofσ (t )......
η1

η2

ηi
• • •

Ee

E1

E2

the multiple characteristic times or the relaxation/retardation time

6.5

with

specific relaxation time τ i ( ≡ ηi Ei )( i = 1, 2,3) and an elastic spring

relaxation times

the model is, therefore, the sum of the stress σ i ( t ) on the i-th Maxwell
element and on the equilibrium spring σ e ( t ) , as follows;σ ( t ) = σ e ( t ) +  σ i (t )(6.10)

i

Due to the parallel combination of each element, the strain ε (t ) of themodel equals to the strain of each element, that is, ε (t ) = ε i ( t ) , leading
to the following constitutive relations;

σ e (t ) = Eeε (t )d ε (t ) 1 dσ i (t ) σ i (t )
=
+
dt
Ei dt
ηi

(6.11)The Laplace transforms of Eqs. (6.10) and (6.11) result in Eq. (6.3a),where the pseudo modulus E * ( p ) is written withE * ( p) = pErelax ( p)


E

E
i

= p e + 
1
 p
i
p
+

τ i 


(6.12)The relaxation modulus in the Laplace space is, therefore, given by
Erelax ( p ) =

Ee
Ei
+
1
p
i
p+

τi

Applying the inverse Laplace transform to Eq. (6.12), the stress relaxation
modulus Erelax (t ) of the Wiechert model as the generalized expression for

55

Wiechert

model

comprising the Maxwell models

model; a parallel combination of the multiple Maxwell models having thewith the equilibrium elastic modulus of Ee . The stress σ ( t ) applied to......

......
σ (t )

FigureThe Wiechert model is shown in Fig. 6.5 that is an extended Zener II

• • •

Ei

engineering materials including organic polymers, being required to use
spectrum.

......

variousvaluesoftheτ i ( ≡ ηi Ei ) ( i = 1, 2,3 ⋅⋅⋅)and the equilibrium spring Ee in aparallel combination.

Eqs. (6.8a) and (6.8b) is given by −t 
Erelax (t ) = Ee +  Ei exp  i τi 

(6.13)

in which the model with Ee = 0 stands for the viscoelastic liquid as a
specific of the Wiechert model.

Since the viscoelastic constitutive

equations that we have considered in this section are based on the
differential equations of Eqs. (6.1) and (6.11), these equations are referred
to as the “differential type of viscoelastic constitutive equation”.
As has been discussed in this section, the generalized linear
viscoelastic constitutive equations in the Laplace space are written byσ ( p) = pErelax ( p)ε ( p)
= E * ⋅ ε ( p)

(6.14)The inverse Laplace transform of Eq. (6.14), thus, leads to the following
viscoelastic constitutive equation in its real space (see APENDIX C);t

σ (t ) =  Erelax (t − t ')0d ε (t ')
dt '
dt '

(6.15)In deriving Eq. (6.15), we have used the fundamental formulas of Laplacetransform;

L [ d ε (t ) / dt ] = pε ( p) − ε (0− )with ε (0 ) = 0 , and−L   f (t − t ') g (t ')dt ' = f ( p) g ( p)
t

 0



The fact that none of external strains are applied to the system in pasttimes ( ε (t ') = 0 ; t ' < 0 ) gives the initial condition of ε (0− ) = 0 . In asimilar way, by the use of the creep function Ccreep ( t ) instead of using
the relaxation modulus Erelax (t ) , the following linear viscoelastic
constitutive equation is also given;tε (t ) =  Ccreep (t − t ')0dσ (t ')
dt 'dt '(6.16)

Accordingly, when we compare the Laplace transform of Eq. (6.16)

ε ( p) = p ⋅ Ccreep ( p)σ ( p)

(6.17)= C *( p)σ ( p)with Eq. (6.14), we can find the following essential relationship between

56the relaxation modulus Erelax ( t ) and the creep compliance Ccreep ( t )in the Laplace space as follows;p 2 Erelax ( p)Ccreep ( p) =(6.18)= E *relax ( p ) ⋅ C *creep ( p ) = 1

Furthermore, the inverse Laplace transform of Eq. (6.18) leads to thefollowing relation in the real space;
t

E
0

relax

(t − t ')Ccreep (t ')dt ' = t(6.19)

Due to the integral expressions of the constitutive equations given in Eq.
(6.15) and (6.16), they are referred to as the integral type of viscoelastic
constitutive equation. The differential type and the integral type of thelinear viscoelastic constitutive equations we discussed in this section aremathematically equivalent. However, the integral equation is more
universal and more extensive in applying it to describing the viscoelastic
behavior of engineering materials, since the integral constitutive equation
is described in terms of the generic viscoelastic function of Erelax (t ) or
Ccreep (t ) instead of using the specific mechanical combination of
Hookean springs and Newtonian dashpots in the differential equation.

6.3. INTEGRAL FORM OF
VISCOELASTIC CONSTITUTIVE EQUATION– BOLTZMANN’S HEREDITARY INTEGRAL –An elastic body responds in an instantaneous manner to the mechanicalstimulus such as the stress/strain externally applied, as repeatedly
mentioned in the preceding sections. In other word, it recovers in an
instantaneous manner to the original state once the applied stimulus istaken away.

On the other hand, as seen in the time-dependent

phenomena of stress relaxation and creep deformation of a viscoelastic
body, the mechanical history in past time t ' has an affect on the
stress/strain state that we observe at the present time t . Due to the factthat the past memory remains in the present mechanical state ofviscoelastic body, it is referred to as the memory material; the rheological
functions such as the relaxation modulus Erelax (t ) and the creep function
Ccreep (t ) are, therefore, often called as the memory functions.
The Boltzmann’s convolution integral (the Boltzmann’s hereditary57

integral) forms the basis of the linear viscoelastic theory [1.2, 1.3].
Suppose a strain ε (t ') at the past time t ' as a mechanical stimulus,
and the induced stress σ (t ') as its linear response. In such a way, we
denote the induced stresses σ (t1 ) and σ (t2 ) for the applied strains ofε(t)Δε (t ')ε (t1 ) and ε (t2 ) at the past times of t ' = t1 and t ' = t2 ( t 1 < t2 < t ),t'respectively. The Boltzmann’s convolution principle supports theassumption that the stress σ (t ) observed at the present time t ' = t is
described with the convolution of the respective stresses σ (t1 ) and

σ(t)

•

σ (t2 ) ;t1 t2 t3

σ (t ) = σ (t − t1 ) + σ (t − t2 )

σ (t)

⋅⋅⋅ t

t'

for the total strain ε (t ) = ε (t1 ) + ε (t2 ) having been applied to the system

Figure 6.6 Boltzmann’s hereditaryat the present time. Namely, the Boltzmann’s principle indicates that the

principle. The stress σ (ti ) induced

stress (or strain) observed at the present time can be described with theby a stepwise strain at the past time

convolution (i.e., the sum) of the respective stresses (or strains) induced

t ' = ti dictates the stress σ ( t ) at

at past times.

A schematic of the Boltzmann’s hereditary principle isdepicted in Fig. 6.6 for an arbitrary strain history.

As demonstrated in

the figure, the stress σ (t ) we are observing at the present time t isgiven byσ (t ) =  Erelax ( t − ti ) Δε (ti )(6.20)iin terms of the stepwise strain Δε (ti ) applied to the system at the pasttime t i via the stress relaxation modulus Erelax (t ) as its memory
function. Equation (6.20) turns to its integral form;tσ (t ) =  Erelax (t − t ')0byreplacingthed ε (t ')
dt '
dt '

stepwisestrain(6.15)
inEq.

(6.20)withΔε (ti ) = [ dε (t ') dt '] dt ' . By the use of the creep function Ccreep (t ) as
the memory function in the Boltzmann’s hereditary integral, in a similarway, we derive Eq. (6.16);t

ε (t ) =  Ccreep (t − t ')0dσ (t ')
dt 'dt '(6.16)In the mathematical framework of Laplace transform, the so-called
Carson transform is defined by;∞f * ( p) = pf ( p) = p  f (t )e− pt dt(6.21)

058

the present time

t in a hereditary

as well as a convolutional manneras a specific type of Laplace transform. Accordingly, as mentioned above,
the pseudo elastic modulus
C * ( p)E * ( p)and the pseudo complianceare, respectively, the Carson transforms of Erelax (t )andCcreep (t ) ;E *( p) = pErelax ( p)

(6.22)C *( p) = pCcreep ( p)with the simple interrelation of E * ( p ) ⋅ C * ( p ) = 1 that is the one-toone corresponding relation of E ⋅ C = 1 for a perfectly elastic body, as
discussed in the preceding section (see Eq. (6.18)). In other word, theCarson transform of a viscoelastic constitutive equation yields thecorresponding perfectly elastic constitutive equation. This one-to-one
correspondence leads to the most essential principle, i.e., the elasticviscoelastic correspondence principle; the very complicated timedependent viscoelastic problems are simply reduced to the perfectly
elastic problems through Carson transform; an example will bedemonstrated in the following section.

6.4 TIME-DEPENDENT POISSON’S RATIOIN VISCOELASTIC DEFORMATIONThe Poisson’s ratio is also time-dependent in a viscoelastic regime,
leading to somewhat complicated mechanical analysis [1.3]. To
circumvent this difficulty, a simple shear test has widely been utilized in
the conventional rheological tests, due to the iso-volumetricdeformation/flow under shear, i.e., no need of the considerations on the
Poisson’s effect. This is the reason why we have long been making use of
the relaxation shear modulus Grelax (t ) and the shear creep compliancefunction J creep ( t ) in most of rheological tests and analyses. However,
the mechanical processes of indentation contact are always as well as
inevitably associated with the volumetric change of the material indented.
The time-dependent viscoelastic Poisson’s ratio ν ( t ) , therefore,
significantly affects the indentation contact mechanics through its effecton the elastic moduli Erelax ( t ) and E 'relax ( t ) . As an appropriate exampleof the application of Carson transform combined with the elasticviscoelastic corresponding principle to the viscoelastic deformation and59flow, we will discuss in this section the time-dependent Poisson’s effecton the viscoelastic functions.
The Young’s modulus E , shear modulus G , bulk modulus K and
the Poisson’s ratio ν of a perfectly elastic body are correlated to eachother through the following equations [1.2];

ν=

3K − 2G 3 − 2κ G=
6 K + 2G 6 + 2κ G

(6.23a)E = 2 (1 +ν ) G(6.24a)E  2G=
E ' ≡2 
 1 −ν  1 −ν

(6.25a)In Eq. (6.23a), the bulk modulus K is defined by p = K ( ΔV V ) interms of the volumetric strain ΔV V under hydrostatic pressure p ,and the bulk compliance κ is defined as the inverse of bulk modulus,i.e., κ = 1 K . Since the bulk compliance of incompressible body is zero,i.e., κ = 0 , Eqs. (6.23a), (6.24a), and (6.25a) lead to the well-known
relations of ν = 1 2 , E = 3G , and E ' = 4 G .
In linear viscoelastic bodies, once we apply the elastic-viscoelastic
correspondence principle combined with the Carson transform to theirtime dependent viscoelastic moduli of E ( t ) , G ( t ) , and ν ( t ) throughthe formulas of Eqs. (6.23a), (6.24a), and (6.25a), we have the following
relations in the Laplace space;ν* =3K * −2G * 3 − 2κ * G *=6 K * +2G * 6 + 2κ * G *(6.23b)

E* = 2 (1 +ν *) G *(6.24b)E *  2G *
=E '*  ≡2 
 1 −ν *  1 −ν *(6.25b)Suppose a model viscoelastic solid having the following shear modulus
with a single relaxation time τ 0 ; t 
G (t ) = Ge + ( Gg − Ge ) exp  −  τ0 

(6.26)

where Gg ( = G (0) ) is the glassy modulus (instantaneous modulus) andGe ( = G(∞) ) stands for the equilibrium modulus; substituting Ge = 0
in Eq. (6.26) turns the model to a viscoelastic liquid, so-called the60Maxwell model. Since the time-dependent nature of the bulk complianceκ (t ) is very minor when we compare it to other viscoelastic functions[1.3], the bulk compliance is, in the present context, assumed to be timeindependent, i.e., κ (t ) = κ 0 (a constant value), making the subsequent
mathematical operations/procedures rather simple and compact.
Substituting the Carson transform G * of Eq. (6.26) into Eqs. (6.23b)– (6.25b), and then making their inverse Laplace transform, we finallyhave the following viscoelastic functions in the real space;

3R+
2 (1 + ν g ) (1 + γ R )
1  1 − 2ν g  E (t )
2  2   2 (1 + ν g ) Gg ν (t ) = − (6.27)(6.28)

andE '(t )
1+ γ R= 2 (1 − ν g )R
2Gg 1 − ν g
1 + 4γ R
+

(

2

+

)





τ 0 

(1 + ν ) 1 − R exp  −t
g

2

1.5
Ge/Gg=0.0
1.4
0.2
1.3
0.4
1.2

0.6

1.1

0.8
1.0

1
0

1 + 4γ R(6.29)

0.2

0.3

0.4

0.5

6.7

The

characteristic

relaxation time τ 1 (Eq. (6.30)) that
dictates the time-dependent naturesofτ 2 

0.1

Glassy Poisson's ratio, νgFigure(1 − ν ) 1 − R exp  −tgNormalized relaxation time, τ1/τ0E (t )
3R
 exp  − t 
= 1 −
τ1 
2 (1 + ν g ) Gg  2 (1 + ν g ) (1 + γ R ) E (t )and ν ( t ) . Notice that

there exists the relation of τ 1 ≥ τ 0

The characteristic relaxation times τ 1 and τ 2 in these equations aredescribed in terms of the shear relaxation time τ 0 through the following

τ1 =

τ2 =

3
1
τ
+
γR 0
1
2 (1 + ν g )(6.30)

1 + 4γτ0
1 + 4γ R(6.31)

in which ν g means the glassy (instantaneous) Poisson’s ratio defined asν g = ν (0) ; R and γ

(

)

(

are, respectively, defined by R = Ge Gg ,)viscoelastic liquid (G e/Gg=0.0)

Normalized relaxation time
τ1(t)/τ0, τ2(t)/τ0

equations;

τ1(t)/τ0
τ2(t)/τ0

3

2

1
0

0.1

0.2

0.3

0.4

0.5

Glassy Poisson's ratio, νgFigure 6.8 Characteristic relaxation

γ = 1 − 2ν g 2 1 + ν g  ; the R-value, therefore, becomes zero inviscoelastic liquid ( Ge = 0 ). The characteristic relaxation time τ 1times of τ 1 and τ 2 plotted against

dictates the relaxation behavior of the present viscoelastic model throughto Eqs. (6.30) and (6.31))

61

the glassy Poison’s ratio ν g (referAs well demonstrated in Fig. 6.7, it should be noticed that the relaxation
time τ 1 is always larger than the shear relaxation modulus τ 0 , i.e.,τ 1 ≥ τ 0 , leading to a more sluggish relaxation of E ( t ) than the shear
relaxation G ( t ) .

On the other hand, the relaxation behavior of E ' ( t )(see Eq. (6.29)) is described via the twin characteristic times of τ 0 andτ 2 with τ 2 ≥ τ 0 ; Figure 6.8 shows the interrelation between thesecharacteristic relaxation times and the glassy Poisson’s ratio ν g ofviscoelastic liquids, indicating the relations of τ 2 > τ 1 for arbitraryvalues of 0 ≤ ν g < 1 2 . Furthermore, it should be noticed that thesecharacteristic relaxation times agree with each other, i.e., τ1 = τ0 = τ 2 ,for the incompressible viscoelastic body with

νg = 1 2 .

Therelaxation behaviors of E ( t ) and E ' ( t ) are plotted in Fig. 6.9 in

Normalized relaxation moduli
G(t)/G(0), E(t)/E(0), E'(t)/E'(0)E ( t ) (see Eq. (6.27)) and the time-dependent ν ( t ) (see Eq. (6.28)).

τ0=200 s; νg=0.3

1

G(t)/G(0)0.9

E(t)/E(0)0.8

E'(t)/E'(0)

0.7
0.6
0.5
0.4

0

100

20

Time, t (s)Figure 6.9 The stress relaxation
moduliG ( t ) , E ( t ) , and E ' ( t ) .

Due to the fact of τ 0 ≤ τ 1 ≤ τ 2 , theirrelaxations become progressively
more sluggish in the order of G ( t ) ,E ( t ) , and E ' ( t )comparison to the shear relaxation of G ( t ) for the viscoelastic liquid( ν g = 0.3 , τ 0 = 200 s). Due to the fact of τ 0 ≤ τ 1 ≤ τ 2 , the relaxation
behaviors are progressively more sluggish in the order of G ( t ) , E ( t ) ,and then E ' ( t ) . The time-dependent nature of the viscoelastic Poisson’sτ 0 = 200 s) with various values of Ge Gg .

As readily seen in Eq.

(6.28), the time-dependent nature of ν ( t ) stems from the relaxation
modulus E ( t ) , resulting in its monotonic increase with time: Figure
6.10 demonstrates the time-dependent Poisson’s ratio ν ( t ) of theviscoelastic bodies, where the Poisson’s ratio of viscoelastic liquid( Ge Gg = 0 ) converges to ν (t ) = 1 2 with time, i.e., lim ν ( t ) = 1 2 ,t →∞implying that the liquid turns to an incompressible fluid in time.

62

Poisson's ratio, ν(t)ratio ν ( t ) is shown in Fig. 6.10 for the viscoelastic bodies (ν g = 0.1 ,

0.6
0.5

Ge/Gg=0.0

τ0=200 sνg=0.1

0.4

0.3

0.3

0.6

0.2
1.0

0.1
101

102

103

Time, t (s)Figure 6.10 Time-dependent
viscoelastic Poisson’s ratio

VISCOELASTIC INDENTATIONCONTACT MECHANICS

CHAPTER7

7.1 MATHEMATICAL FORMULATION
We can readily make a mathematical formulation of viscoelastic
indentation contact mechanics throughthe“elastic-viscoelastic

correspondence principle” applied to the elastic indentation contact
mechanics. Namely, the elastic theory of indentation contact mechanics
we developed in Chapter 3 is readily extended to the corresponding
viscoelastic theory via the Laplace transform and its inversion (seeChapter 6) [7.1-7.5].

The constitutive equation of the elastic indentation

contact given in Eq. (3.32) is rewritten in terms of the indentation load
P and the contact area A(= π a2 ) , as follows;

P = ( E 'ε I ) A(3.32’)

Once we apply the elastic-viscoelastic correspondence principle to Eq.
(3.32’), the following constitutive equations of viscoelastic indentation
contact are obtained;d [ε I (t ') ⋅ A(t ') ]P (t ) =  E 'relax (t − t ')
dt '
dt '
0
t

(7.1)

d [ P (t ') ]dt 'ε I (t ) ⋅ A(t ) =  C 'creep (t − t ')
dt '
0
t

In Eq. (7.1), E 'relax (t ) stands for the plane-strain relaxation modulus
defined as E 'relax (t ) = Erelax (t ) / 1 −ν 2 (t )  . There is also a similar relation
to Eq. (6.19) for the plane-strain creep function C 'creep (t )
t

 E'
0

relax

(t − t ')C 'creep (t ')dt ' = t(7.2)

As readily expected from Eqs. (3.29) - (3.31), it must be noticed thatthe indentation strains ε I (t ) of flat-ended punch and spherical indenter
are time-dependent in viscoelastic regime. In Tab. 7.1 listed are theviscoelastic constitutive equations for different indenter’s tip geometries.
As clearly seen in the table, due to the geometrical similarity ofcone/pyramid indenter (including Vickers/Berkovich indenters), theindentation strain ε I is dependent only on the inclined face-angle β ,but not on the penetration depth, leading to the time-independent
indentation strain, i.e., ε I (t ') = tan β 2 , and making the constitutive
equations very simple and compact;
63[7.1] J.R.M. Radock, Q. Appl. Math., 15,
198 (1957)[7.2] S.C. Hunter, J. Mech. Phys. Solids,
8, 219 (1960)[7.3] M. Sakai, Phil. Mag, A, 82[10], 1841
(2002)[7.4] M. Sakai, S. Shimizu, J.
Non-Crystal. Solids, 282, 236 (2001)[7.5] M. Sakai, J, Soc. Rheol. Japan,
39(1･2), 7 (2011)P (t ) =tan β  tdA(t ')dt ' E 'relax (t − t ')
2 0
dt '

(7.3)

t

dP (t ')A(t ) = 2 cot β dt ' C 'creep (t − t ')0
dt 'In the following sections, using Eq. (7.3), we will discuss the viscoelastic
indentation contact behaviors in the various types of tests ofcone/pyramid indentation.
(i) Step-wise penetration tests
Let us discuss the indentation load relaxation for the stepwise
application of a constant contact-area A0 ;A(t ) = A0 ⋅ u (t ) ,(7.4)

and the creep deformation of contact-area under the stepwise applicationof a constant indentation load P0 ;P(t ) = P0 ⋅ u (t )(7.5)

The applications of Eqs. (7.4) and (7.5) to the right-hand side of Eqs. (7.3)
combined with the relation of du ( t ) dt = δ ( t ) ( δ ( t ) ; Dirac’s delta
function) result intan β
A0 ⋅ E 'relax (t )2A(t ) = 2cot β P0 ⋅ C 'creep (t )P(t ) =

(7.6)

Accordingly, we can determine the relaxation modulus E 'relax (t ) and/or
the creep function C 'creep (t ) through measuring the indentation load
relaxation P(t ) and/or the creep deformation of the contact area A(t ) .
However, in the conventional instrumented indentation apparatus, wecannot measure the indentation contact areas not only A0 but also A(t ) ,although it is possible for us to measure the indentation load relaxationP(t ) and the time-dependent penetration depth h(t ) . We need,
therefore, to use somewhat undesirable approximation/assumption toestimate A(t ) from the h(t ) observed. To circumvent this difficulty,
we have to use the instrumented indentation microscope, the details of
which will be given in the later chapter (Chapter 11).

64

(ii) Constant-rate of penetration tests
In the indentation tests with a constant-rate k A of contact-area
incrementA(t ') = k A ⋅ t ' ,(7.7)

and with a constant-rate of loading k P

P(t ') = kP ⋅ t ' ,(7.8)

the resultant viscoelastic responses of P(t ) for the former, and of A(t )for the latter are, respectively, described by the use of Eq. (7.3) combinedwith Eqs. (7.7) and (7.8), as follows;

P (t ) =k A tan β t
0 E 'relax (t − t ')dt '
2

(7.9)

t

A(t ) = 2k P cot β  C 'creep (t − t ')dt '0

We can, therefore, readily determine the relaxation function E 'relax (t )and/or the creep function C 'creep (t ) via using the time-derivative ofP(t ) and/or A(t ) observed in experiments;
E 'relax (t ) =2 cot β dP (t )kA
dt

(7.10)

tan β dA(t )C 'creep (t ) =2k P dt

In these indentation tests, in order to measure A(t ) , the instrumented
indentation microscope plays a very essential role (see Sec. 11.3). The
details of the viscoelastic constitutive relations are listed in Tables 7.1 –7.3 for the flat-ended cylindrical punch, spherical indenter, and theconical indenter.

Table 7.1 Integral-type viscoelastic constitutive equations for the axisymmetric indenters
Flat-ended cylindrical punch

Spherical indentertP(t ) =1 t
dP(t ')dt ' C 'creep (t − t ')2a0 0dt 'A3/2 (t ) =h(t ) =

a0 : radius of the cylindert

3/2
 E ' (t − t ') dA (t ') dt ' relax
3π R 0dt 'dh(t ')P(t ) = 2a0 dt ' E 'relax (t − t ')
0
dt '

4

Conical indenter

3/2

3π 3/2 R  tdP(t ')dt ' C 'creep (t − t ')
4 0
dt '

R : radius of the sphere65

P( t ) =tan β  tdA(t ')dt ' E 'relax (t − t ')
2 0
dt '
t

dP(t ')A(t ) = 2cot β dt ' C ' creep (t − t ')0dt 'β : inclined face-angle of the cone

Table 7.2 Viscoelastic responses for the step-wise indentation
Stepwise penetration to a
constant contact-area A0

Stepwise penetration to a
constant load P0

A(t)

P(t)

A0
t

P(t)

t

A(t)

t

0

flat-ended
cylinderP0t0P(t) = 2a0h0E 'relax (t)h (t ) =P0
C 'creep (t )2a0spherical
indenter

4  A0 
3π 3/2 RP0
3/2
P(t ) =
C 'creep (t )  E 'relax (t ) A (t ) =3R  π 
4conicalindenterP(t ) =

3/2

tan β
A0 E 'relax (t )2A(t) = 2P0 cot β ⋅ C 'creep (t)* Use the penetration depths of h ( t ) and h0 instead of the contact areas ofA(t ) and A0 for the flat-ended cylindrical punch

Table 7.3 Viscoelastic responses for the constant-rate indentation

Constant-rate of contact
area increment

Constant-rate of loading

A(t)

P(t)

kP

kA
t

P(t)

0

flat-ended
cylindertdP(t )= 2a0kh E 'relax (t )dtspherical
indenterP ( p) =conicalindenterdP(t ) k A= tan β ⋅ E ' relax (t)dt2

k A3/2 E 'relax ( p)
πR
p3/2

t

A(t)

0

t

dh(t ) k P=
C 'creep (t )dt
2a0
dA3/2 (t ) 3π 3/2 Rk P=
C 'creep (t )dt
4dA(t )= 2kP cot β ⋅ C 'creep (t)dt* Use the penetration depth of h (t ) ( = k h ⋅ t ) instead of the A(t ) ( = k A ⋅ t ) for
the flat-ended cylindrical punch667.2 VISCOELASTIC MEYER HARDNESSThe Meyer hardness H M of elastic/elastoplastic bodies is time/rateindependent, and defined by the ratio of the applied indentation load P
and the resultant contact area A as H M = P A (see Eq. (5.1b)).
Accordingly, on the basis of the elastic-viscoelastic corresponding
principle, when we apply the Laplace transform inversion to thetransformed Meyer hardness P = H *M ⋅ A , we have the following

formula of the viscoelastic Meyer hardness H M ( t ) ;tdA(t ')P (t ) = 
dt ' H M ( t − t ')0
dt '(7.11)The viscoelastic Meyer hardness H M (t ) is, therefore, by no means
given by the indentation load P ( t ) divided by the contact area A(t )at the respective time t ; H M (t ) ≠ P(t ) A(t ) .
In order to understand the physics of viscoelastic Meyer hardness,further considerations will be made in this context for the stepwise
penetration, as an example.

Substituting A(t ') = A0 ⋅ u (t ') into Eq.

(7.11) and ε I (t ') ⋅ A(t ') = ε I0 A0 ⋅ u (t ') into Eq. (7.1), and comparing
these resultant relations, we finally find the fact that the viscoelastic

H M (t ) is equivalent to the relaxation modulus

Meyer hardness

E 'relax (t ) ;H M (t ) = ε I0 E 'relax (t )(7.12)

in which the indentation strain ε I0 (refer to Eqs. (3.29) - (3.31)) is;flat-ended cylindrical punch:2h 

ε I0 =  0 
π  a0 (7.13a)spherical indenter：ε I0 =

4  a0 
3π  R 

(7.13b)conical indenter:ε I0 =In Eq. (7.13a),

tan β
2

(7.13c)h0 is the penetration depth of the cylindrical punch (theradius of a0 ). In Eq. (7.13b), a0 is the contact radius

induced by the

stepwise penetration in spherical indentation. Due to the geometricalsimilarity of cone indentation, its indentation strain ε I0 is independentof the penetration depth applied, and described by ε I0 = tan β 2 in67

terms of the characteristic inclined face-angle β of the cone. This factmeans that Eq. (7.13c) is always applicable to arbitrary depths ofpenetration.

The equivalence of the viscoelastic Meyer hardness andthe relaxation modulus is also well recognized from the fact that theviscoelastic Meyer hardness is given by H M (t ) = P (t ) A0 in terms of
the indentation load relaxation P(t ) observed in the indentation test ofa stepwise penetration to a constant contact-area A0 .

7.3 INDENTATION CONTACT MECHANICS OF
THE VISCOELASTIC MODELSThe numerical indentation contact mechanics will be discussed in thissection for the Maxwell liquid and the Zener II solid ( EM = Ee isassumed

in

Fig.

6.2

for

simplicity)

in

the

constant-rate

penetration/loading tests (Eqs. (7.7) and (7.8); Table 7.3) of a conicalindenter with the inclined face-angle β .
[Constant-rate penetration test]
In the constant-rate penetration test ( A(t ') = k A ⋅ t ' ), the viscoelastic
indentation load P(t ) ) is given by the first formula of Eq. (7.9);P(t ) =k A tan β t
0 E 'relax (t − t ')dt '2(7.9a)

Through applying the relaxation moduli E 'relax ( t ) of the Maxwellliquids (Eq. (6.8a)) and of the Zener II solid (Eq. (6.8b) withformulas of the time-dependent indentation load P(t ) :(i)

Viscoelastic Maxwell liquidP(t ) =
(ii)

E 'M tan β( k Aτ M ) 1 − exp ( −t τ M )
2(7.14)

Viscoelastic Zener solid

E ' tan βP(t ) = M
( k Aτ M ) t τ M + {1 − exp ( −t τ M )} (7.15)
2

Normalized indentation load
PN(t/τM)

E 'M = E 'e = E 'g 2 for simplicity) to Eq. (7.9a), we have the following

5
4

constant-rate of penetration
A(t)=kAt

3

Zener model; Eq. (7.15)

2
Maxwell model; Eq. (7.14)

1
0
0

1

2

3

4

5

Normalized time, t/τM

Figure7.1

The

normalizedindentation load in the constant-rate

The normalized indentation loadof penetration test ( A(t ') = k A ⋅ t ' ):PN ( t τ M )  ≡ ( 2 P(t ) cot β ) ( k Aτ M ⋅ E 'M ) of Eqs. (7.14) and (7.15) are plotted in Fig. 7.1 against the normalized

The solid line represents the Maxwell

time t τ M . Figure 7.1 shows that the indentation load of the Maxwellliquid, and the dashed line indicatesliquid converges to PN (t τ M ) → 1.0 or P(t ) → ( kA tan β 2)η in thethe Zener solid.

68long-time region of t  τ M . This fact indicates that the Maxwell liquid
in its steady state behaves as the Newtonian liquid with the shear
viscosity η . On the other hand, as shown in Fig. 7.1 for the Zener solid,
there exist a linear relation between PN (t τ M ) and t τ M in the longtime region of t  τ M , i.e.,

PN ( t τ M ) ∝ t τ M ∝ A ( t ) k Aτ M
or

P ( t ) ∝ ( E 'e tan β 2) A ( t )This linear relation indicates that the Zener model behaves as theperfectly elastic body with the equilibrium elastic modulus of E 'e inthe long-time region.
[Constant-rate of indentation loading]
The time-dependent viscoelastic contact area A ( t ) for constant-rateof indentation loading P ( t ') = k p ⋅ t ' is given by the second formula ofEq. (7.9);tA(t ) = 2kP cot β  C 'creep (t − t ')dt '(7.9b)

0

When we apply the creep functions of the Maxwell liquids (Eq. 6.9a) andof the Zener II solid (Eq. 6.9b) to Eq. (7.9b), we have the following
formulas of the time-dependent contact area A ( t ) :Viscoelastic Maxwell liquidA(t ) =
(ii)

2cot β1( kPτ M ) t τ M + ( t τ M )2 
E 'M
2



(7.16)

Viscoelastic Zener solid

2cot βA(t ) =( kPτ M ) t τ M − {1 − exp ( −t τ Z )}E 'M

(7.17)

τ Z = (1 + EM Ee )τ M
The Zener’s relaxation time τ Z is given by τ z = 2τ M because of theNormalized contact area, A N(t)(i)

constant-rate of loading, P(t)=kPt

10

Maxwell model (Eq. (7.16))
8
6
4
Zener model (Eq. (7.17))

2
0

0

2

4

6

8

10

Normalized time, t/τM

relation EM = Ee we assumed for simplicity. The normalized contact

Figure 7.2 The normalized contact

areas AN ( t τ M ) [ ≡ A(t ) E 'M tan β 2kPτ M ] of Eqs. (7.16) and (7.17) arearea in the constant-rate of loadingplotted in Fig. 7.2 against the normalized time t τ M . In the long-time

test ( P(t ') = kP ⋅ t ' ):region of t  τ M , the AN (t τ M ) vs. t τ M relation of the Zener solid

represents the Maxwell liquid, andbecomeslinear

with

the

slope

of

1.0,

i.e.,

AN ( t τ M ) ∝ t τ M = P ( t ) kPτ M , thus leading to the linear relation
69

The solid line

the dashed line indicates the Zenersolid.

between the indentation load P ( t ) and the resultant contact area A ( t ) ,behaves as a perfectly elastic body having the elastic modulus of E 'e
in the long-time region, like as the steady-state deformation observed in
the constant-rate penetration test (refer to the consideration on Eq. (7.15)

viscoelastic Maxwell liquid

Normalized hardness[HM(t)]N, [P(t)/A(t)]N

i.e., P ( t ) ∝ E 'e ⋅ A ( t ) . This fact implies that the viscoelastic Zener solid

We emphasized in Sec. 7.2 and in Eq. (7.11) that the viscoelastic Meyer
hardness H M ( t ) is by no means given by H M ( t ) = P ( t ) A ( t ) at all, butThesignificant discrepancies existing between the viscoelastic Meyer
hardness H M ( t ) (see Eq. (7.11)) and the ratio of P ( t ) A ( t ) calculated
via Eqs. (7.14) – (7.17) are well demonstrated in Figs. 7.3 and 7.4 fortheir normalized plots; the hardness numbers of P ( t ) A ( t )  N

 H M ( t )  N

normalized with E 'g tan β 2 . The relaxation behaviors of the hardness
number P ( t ) A ( t ) (indicated by the symbols of ○ and ● in Figs. 7.3
and 7.4)

are always more sluggish than those of the Meyer hardness

H M ( t ) (indicated by the solid lines), leading to always larger values ofP ( t ) A ( t ) than those of H M ( t ) . The hardness numbers both P ( t ) A ( t )and H M ( t ) of the Maxwell liquid (Fig. 7.3) progressively diminish withtime, while those values of the Zener solid (Fig. 7.4) converge to a finite
value of E 'e E 'g = 0.5 in the long-time region of t τ → ∞ (notice that
we assumed E 'M = E 'e = E 'g 2 for the model examined).

10 P ( t ) A ( t )  N

plottedagainstthenormalized time of t τ M (viscoelastic
Maxwell liquid)
1.2
viscoelastic Zener solid1: [P(t)/A(t)]N; constant-rate of loading: [P(t)/A(t)]N; constant-rate ofcontact area increment: Meyer hardness, [HM(t)]N

0.8
0.6
0.4

0

10

20

Normalized time, t/τM

Figure 7.4 Normalized viscoelastic
Meyer hardness
 H M ( t )  N , and P ( t ) A ( t )  N

plottedagainstthenormalized time of t τ M (viscoelastic
Zener solid)

70

20

Figure 7.3 Normalized viscoelastic
Meyer hardness
 H M ( t )  N , andandare, respectively, defined as H M ( t ) and P ( t ) A ( t )0Normalized time, t/τM

Normalized hardness[HM(t)]N, [P(t)/A(t)]NH M (t ) ≠ P (t ) A( t ) .

: [P(t)/A(t)]N; constant-rate of loading: [P(t)/A(t)]N; constant-rate ofcontact area increment: Meyer hardness, [HM(t)]N

0.50and in Fig. 7.1).

there exists an inequality relation of1

INDENTATION CONTACT MECHANICS
OF SOFT MATTERS
WITH SURFACE ADHESIONCHAPTER

8

In the preceding sections, we have discussed the elastic/elastoplastic
indentation contact mechanics of engineering materials including metals
and ceramics so called the hard materials having rather large elastic
modulus of E ' ≥ 100 GPa.

Due to the large elastic modulus of such

hard materials, the surface adhesion, even if there exists, makes a rather
minor effect on their contact behavior. In most cases, therefore, we canneglect the surface adhesion in their indentation contact mechanics. On
the other hand, the elastic moduli of soft materials including organic
polymers, biomaterials, and microbiological mediums are rather small,and fall in the range of E ' ≈ 1 kPa - 100MPa, where the surface adhesion
plays an essential role in their mechanical characteristics and functions.
It must be also noticed that the mechanical characteristics of most of thesesoft matters are time dependent viscoelastic and/or elastoplastic.
The indentation contact mechanics of perfectly elastic bodies withsurface adhesion has well been appreciated through the JKR-theory(Johnson-Kendall-Roberts theory) ([8.1], [8.2], [8.3]), the details of
which will be given in Section 8.1. While on the other hand, neither
theoretical considerations nor well-conducted experiments have been
conducted for elastoplastic/viscoelastic bodies with surface adhesion. To
overcome these difficulties and issues, the considerations on FEA-based
numerical indentation contact mechanics will be made in Section 8.2 forelastoplastic bodies. As to the indentation contact mechanics of timedependent viscoelastic bodies with surface adhesion, the “elasticviscoelastic correspondence principle” (refer to Chapters 6 and 7) will beapplied to the JKR-theory in Section 8.3. The details of experimental
studies of elastoplastic/viscoelastic soft matters with surface adhesion
will be given in Chapter 12.

71[8.1] K.L. Johnson, “Contact Mechanics”,
Cambridge University Press (1985)[8.2] K.L. Johnson, K. Kendall, A.D. Roberts,“Surface energy and the contact of elastic
solids”, Proc. Roy. Soc. London A, 324:3018.7(1971)[8.3] D. Maugis, “Contact, Adhesion and
Rupture of Elastic Solid”, Springer (2000)

8.1 ELASTIC BODIES WITH SURFACE ADHESION- THE JKR THEORYThe JKR theory focuses only on the contact mechanics of sphere. In
this section, however, we will also discuss the JKR theory extended tothe conical/pyramidal indentation contact, since the Vickers/Berkovich
indentation is very conventional in the practical indentation tests.
Suppose an axisymmetric indenter with an arbitrarily shape indented
onto an elastic half-space, resulting in the contact radius a and thepenetration depth h under the indentation load P . The indenter will
be withdrawn to the elastic body having surface adhesion, i.e., the surface
adhesion induces negative contact pressure. This fact implies that theindentation load P at the contact radius a will be smaller than that of
the elastic body without surface adhesion. The JKR theory models thesurface adhesion as the negative contact pressure acting on thecylindrical flat punch with the radius a .
Using Eq. (3.10), the contact pressure p ( r ) of a cylindrical flatpunch having the radius a is given by  a 2 
pF (r ) = pF 1 −     r  

−1 2

; 0≤r <a(8.1)

where use has been made of the relation p (r ) = −σ z (r ) . In Eq. (8.1), thesubscript F stands for Flat punch. The coefficient of the contact pressure

pF

must be a negative value ( pF < 0 ) when it describes surface

adhesion.

In a similar way, using Eqs. (3.19) and (3.26), the contactpressure is12  a 2 
pS (r ) = pS 1 −    ; 0 ≤ r < a  r  

(8.2)

for spherical indentation, andpC ( r ) = pC cosh −1 ( a r ) ; 0 ≤ r < afor conical indentation.

(8.3)

In these expressions, the subscripts S and C

indicate Sphere and Cone, respectively.

In the JKR theory, therefore,

the contact pressure distribution beneath the indenter of an elastic bodywith surface adhesion is described by superposing Eq. (8.1) on Eq. (8.2)
or on Eq. (8.3) as follows;72

Spherical indentation12  a 2 
p(r ) = pS 1 −     r  

  a 2 
+ pF 1 −     r  

−1 2

(8.4)

Conical indentation  a 2 
p(r ) = pC cosh ( a r ) + pF 1 −     r  

−1 2

−1

(8.5)

On the other hand, using the geometrical formula of Eq. (3.1), the contact
surface profile beneath the indenter ( 0 ≤ r ≤ a ) is described by
Spherical indentationu z (r ) = h −

r2
2R

(8.6)

Conical indentationu z (r ) = h − r tan β(8.7)

Appling the analytical procedures in Sec. 3.1 combined with Eqs. (8.1) –(8.5) to Eq. (8.6) or to Eq. (8.7), we finally have the following relations;【Spherical indentation】

π pS

2

π ap
r
2a − r ) +
=h−
(
E'
4aE '
2R
2

2

F

(8.8)

Notice the algebraic identity of Eq. (8.8) as to the variable

r , then we have the following relations;
pS =
h=

2aE '
πR

πa
2E '

(8.9)

( pS + 2 pF )(8.10)aFurthermore, applying P =  p (r )2π rdr to Eq. (8.4),0

the indentation load is given by2
P =  pS + 2 pF  π a 2
3
(8.11)【Conical indentation】
A similar mathematics in the preceding spherical indentation contact
leads to the following relations;

π apC 2 r  π apF= h − r tan β1 −+E'  π aE'(8.8’)

73

pC =
h=

E'
tan β
2

πa
E'

(8.9’)( pS + pF )(8.10’)

P = ( pC + 2 pF ) π a 2

P (a)
P1

A
•

(8.11’)

The coefficients of contact pressure distribution pS and pC in Eqs.
(8.9) and (8.9’) are, therefore, successfully related to the elastic modulus
E ' . It must be noticed that these coefficients of contact pressure
distribution are identical to those we have already derived in Secs. 3.2
and 3.3 for the perfectly elastic body without surface adhesion.
Unfortunately, however, we have failed in describing the pressure

U1
B
•
h1

0•

P
P1

(b)

h

A
•

distribution coefficient pF of cylindrical flat punch in terms of thesurface adhesion in the preceding context. To overcome this difficulty, let
us make an energy-based consideration on the indentation contact
processes, since the surface adhesion γ (N/m) is equivalent to thesurface energy γ

P2

C•
D•

0•

(J/m2).

In the first step of indentation contact process, suppose anaxisymmetric indenter pressed onto an elastic body without surface

P (c)
P1

adhesion to the indentation load P1 as depicted in Fig. 8.1(a), at which

-U2
B

h1

h2

h

•A

UE=U1+U2

the penetration depth h1 and the contact radius a1 are assumed to be(h1

induced. The elastic strain energy U1 =  Pdh
0) stored in the elastic

body associated with this indentation contact process is

P2

Spherical indentation

U1 =

2

a13

2π
pS12
15 E '(8.12)

π 2 a133E '(8.13)The elastic stored energy U1 is represented by the area OABO in Fig.
In Eqs. (8.12) and (8.13), pS1 and pC1

are, respectively, the contact pressure coefficients of spherical andconical indentations at the contact radius a1 ; pS1 = 2a1 E ' π R and

pC1 = ( E ' 2) tan β . It must be noticed that pC1 is independent of thecontact radius

a1

h1

h

Figure 8.1 Energy-based consideraprocesses of an elastic body with

pC12 .

8.1(a) of the P - h diagram.

B

•D
h2

tions on the indentation contact

Conical indentation

U1 =UE • C0•

due to the geometrical similarity of conicalsurface adhesion: (a) the indentation
loading path of an elastic bodywithout surface adhesion (the line
OA),

(b)

associatedtheunloadingwith the74

incremental

increase of surface adhesion (the lineAC), (c) the elastic stored energy

U E of the elastic body with surface
adhesion at the point A

indentation.

path

In the subsequent second step of indentation contact process, at the
point A ( P1 , h1 , a1 ) in Fig. 8.1(b), let us consider the variation of thestrain energy associated with the incremental surface adhesion from 0 toγ

on the contact surface maintaining its contact area at π a1 . This2

mechanical process implies that the indenter is progressively pulled to thecontact surface, resulting in unloading, as shown in Fig. 8.1(b), along thelinear line AC, since the contact area maintains constant. The indentation
contact state at the point C ( P2 , h2 , a1 ) is the equilibrium state of theelastic body having the surface energy of γ . This mechanical process
along the line AC will be equivalent to the unloading process of a flatpunch with the radius a1 . The total energy released through thisunloading process along the line AC is denoted by U2 ( < 0) , and givenby the area ABDCA (＝ −U 2 ) in Fig. 8.1(b). The P - h unloading pathalong the line AC is, therefore, described by

P = P1 + 2π a12 pF1= 2a1 E ' ( h − h1 ) + P1
h = h1 +π a1
E'(8.14)

pF1

where use has been made of Eq. (3.9) for the indentation of a flat-punch.
In Eq. (8.14), we must notice the fact that the coefficient of contactpressure distribution pF1 (see Eq. (8.1)) of the cylindrical flat-punchwith the radius a1 is negative due to surface adhesion. The indentation()

load P1 at the point A is given by P1 = 2π a12 3 pS1 for sphericalindentation, and P1 = π a p

2
1 C1

for conical indentation. On the other hand,

the released energy U2 ( < 0) associated with the incremental surface
adhesion is finally given by
Spherical indentation

U2 =π 2 a13  2

2
 pS1 pF1 + pF1 
E'  3(8.15)

Conical indentation

U2 =π 2 a13(p p + p )
E'
C1 F1

2
F1

(8.16)

75

h

P

h1

P1

through using the formula of U 2 =  2 Pdh =  2 P

dP
P 2 − P12.
= 2
2 a1 E '4 a1 E 'Accordingly, when we apply an indentation load to an elastic half-spacewith surface adhesion to the contact radius of a , using Eqs. (8.12) (8.16), the associated elastic strain energy U E stored in the body (thearea of ACDOA in Fig. 8.1(c)) is;
Spherical indentation
U E ( ≡ U1 + U 2 ) =π 2 a3  2

222
 pS + pS pF + pF 
E '  153

(8.17a)or Eqs. (8.9) and (8.10) combined with Eq. (8.17a) result in
U E ( ≡ U1 + U 2 )π 2 a 3  1  aE '  2 E '2
 E '  2 
h
=
+
 
 −

 h 
E '  5  π R 
3 π 2R
πa 

2

2

(8.17b)
Conical indentation
U E ( ≡ U1 + U 2 ) =π 2 a3  1

22
 pC + pC pF + pF E' 3(8.18a)or Eqs. (8.9’) and (8.10’) combined with Eq. (8.18a) lead toU E ( ≡ U1 + U 2 )=π 2 a 3  1  E ' tan β  
E '  3 2

2
E '2 tan β E '  2 
h+
 −
 h 
2π a

πa 

2

(8.18b)As mentioned in the preceding considerations, in the present
indentation contact problem, the adhesive surface force introduces thesurface energy U S which decreases when the surfaces come into
intimately contact and increases when they separate. Therefore, we canwrite
U S = −2γπ a 2(8.19)The total energy (the Gibbs free energy) U T of the present mechanicalsystem, therefore, is given by

U T = U E + US

(8.20)

76

In the mechanical equilibrium under a fixed depth of penetration meaning
none of external works, the variation of total energy associated withincremental contact radius δ a results in ∂U T  ∂a  = 0h(8.21)

Substituting Eqs. (8.17) - (8.20) into Eq. (8.21) and using Eq. (8.17b) or
Eq. (8.18b), we have the relation of ( ∂U E ∂a )h = (π 2 a 2 E ') pF 2 bothfor spherical and conical indentations. We can, therefore, finally relate
the pressure distribution coefficient pF to the adhesive energy γ

asfollows;

4γ E '
πapF = −(8.22)

Once we get the analytical formulas of pS (Eq. (8.9)), pC (Eq.
(8.9’)), and pF ((8.22)), substituting these coefficients into Eqs. (8.10)
and (8.11), or into Eqs. (8.10’) and (8.11’), we can finally describe thepenetration depth h and the indentation load P in terms of thecontact radius a :Spherical indentationh=a2
πγ 1 2−2
aRE'(8.23)

P=

4E ' 3a − 4 πγ E 'a3 2
3R(8.24)

Conical indentationh=π tan β2a−2

πγ 1 2E'a(8.23’) E'
P =  tan β  π a 2 − 4 πγ E 'a3 2
 2


(8.24’)γE' E'
=  tan β  A − 4 1 2 A3 4
π
 2


(

in which A = π a 2) stands for the contact area. Substitution of γ = 0

into the above formulas naturally reduces them to those of purely elastic
body that we have already derived in Secs. 3.2 and 3.3. The use of the
instrumented indentation microscope (see the details in Sec. 11.3) makes
us possible for determining not only the indentation load P and thepenetration depth h , but also for measuring the contact radius a and77the contact area A in experiment. We, therefore, readily determine inγ

without using any

undesirable approximation and assumption. As readily seen in Eqs. (8.23)- (8.24’), however, the P - h relationship can only be obtained in an

E'=5kPa10.5

γ=0 N/m

2

5

P(N)

experiment the quantitative values of E ' and

(a)

10

3

0

implicit manner via the contact radius a as its intermediate variable.

4-5

In order to understand the role of surface adhesion in the indentation
contact mechanics, Figs 8.2(a) and 8.2(b) demonstrate the P - A

surface adhesion. In Fig. 8.2(a), the linear P - A relation

P = ( E ' tan β 2 ) A

indicatesof the perfectly elastic body without surface

adhesion ( γ = 0 N/m). As clearly seen in Fig. 8.2, with the increase in theadhesive energy γ

and with the decrease in the elastic modulus E ' ,the effect of surface adhesion on the contact behavior becomes
progressively more significant, leading to the enhanced nonlinearity intheir P - A relations.

Indentation load, P(N)relations (Eq. (8.24’)) of conical indentation for the elastic bodies with

10

0

0.01

A(m 2)

0.02

0.03

γ=1N/m(b)E'=10kPa

5

5

3
2

0

1
0

0.01

0.02

0.03

Contact area, A(m 2)

Figure 8.2

The P - A relationsof conical indentation for the elastic
bodies with surface adhesion:

The normalized expressions of Eqs. (8.23) – (8.24’) will lead us to(a) E ' =5kPa; γ =0N/m〜4N/m

deeper and more comprehensive understanding for the role of surface(b) γ = 1 N/m; E ' = 1kPa〜10kPa

adhesion in the indentation contact mechanics. As readily seen in Fig.
8.2, since the adhesive force pulls the surface into contact, the indentationload has a minimum value of Pc at the critical contact radius ac , i.e.,( dP da )a = 0 . In this textbook, we adopt ac , Pc , and the critical
penetration depth hc ( = h ( ac ) ) as the critical parameters in normalizingcEqs. (8.23) – (8.24’) (notice that we will use the absolute values of Pc
andhc , whenever these parameters are negative). These criticalparameters thus defined are summarized as follows;
Spherical indentation13 9πγ R 2 
ac = 

 4E ' 
Pc = −3πγ R(8.25)
13 3π 2γ 2 R 
hc = − 2  16 E ' 78Conical indentation
ac =

36γ

π E ' tan 2 β3 6  γ2
Pc = −  π tan β  E '6γ
hc =E ' tan β(8.25’)(a)

The normalized P - a and h - a relations thus derived in terms of
4these critical parameters are given by
PN

Spherical indentation
PN = aN 3 − 2aN 3/ 2

(8.26)

hN = 3aN 2 − 4aN1 2

2

0: conical indentation: spherical indentation

0

5

10

hN

Conical indentation
4

PN = 3aN 2 − 4aN 3/2: conical indentation: spherical indentation(8.26’)

3aN

hN = 3aN − 2aN1 2(b)2

where aN , PN , and hN indicate, respectively, the normalized contact
parameters of aN = a ac , PN = P Pc , and hN = h hc
indentation) or hN = h hc (conical indentation).

1(spherical0

On the other hand,05

the PN - hN relation is numerically given in an implicit manner via

3(c): conical indentation: spherical indentation

the contact radius aN as its intermediate variable via Eq. (8.26) or Eq.
(8.26’). These normalized expressions thus obtained are plotted in Fig.

(conical indentation), while the open circles ( ○ ) indicate Eq. (8.26)(spherical indentation).

Furthermore in Fig. 8.3, the solid lines

represent the elastic body with surface adhesion, and the broken lines are
for without surface adhesion.

Due to the face-contact of sphericalindentation while the point-contact of conical indentation in their initial2aN

8.3; the closed circles (●) are the normalized relations of Eq. (8.26’)

10

PN

1

0

-2

0

2

4

hN

Figure 8.3 Normalized indentation
contact diagrams; the solid lines
indicate the adhesive contact (JKR

stages of indentation contact, the surface adhesion makes moretheory), while the broken lines are

significant effect on the spherical indentation contact than that of conicalthose of the elastic body without

indentation, as well recognized in Figs. 8.3(a) and 8.3(c).

surface adhesion

79

[FEA-Based Numerical Studies]

The numerical results of Finite Element Analysis (FEA) of theloading/unloading P - A relations for the perfectly elastic bodies areshown in Fig. 8.4 (Vickers/Berkovich-equivalent cone indentation for the
penetration depth up to hmax = 30 μm). Both the loading and theunloading P - A relations of the elastic body without surface adhesion( E ' = 20 kPa, γ = 0.0 mJ/m2; ● symbol) are linear and coincide witheach other, where the dashed linear line is the analytical solution of

P = ( E ' tan β 2 ) A . On the other hand for the elastic body with surface
adhesion ( E ' = 20 kPa, γ = 5.0 mJ/m2; 〇symbol), however, asignificant hysteresis is observed in its P - A loading/unloading
relation, although the body is perfectly elastic. It must be noticed in theirloading P - A relations that the indentation load of the elastic body withsurface adhesion is always as well as significantly smaller than thatwithout surface adhesion. This is resulted from the fact that the surface
adhesion pulls the tip of indenter downward to the contact surface, i.e., a

It will be also worthy of note that the contact area at the maximum depthof indentation ( hmax = 30 μm) is about Amax ≈ 100 (x 100μm2) for theelastic body without surface adhesion, while it is about Amax ≈ 180(x100μm2) for the elastic body with surface adhesion, i.e., about 1.8 times

E=20kPa, ν=0.49; hmax=30μm

Indentation load, P (μN)

negative force is superimposed on the externally applied indentation load.

50

0

-50
0

larger in its contact area under surface adhesion: The JKR-theory predicts
this effect of surface adhesion on the contact radius as readily seen in Fig
8.3(c), although the JKR-theory cannot predict the unloading P - A
path.
As readily seen in Fig. 8.4, the FEA-based numerical results (thesymbols 〇) well signify the JKR-theory (the dashed P - A path: Eq.
(8.24’) );2

γ=5.0 mJ/m2

γ=0.0 J/m

50

100

150

Contact area, A (x100μm2)

Figure 8.4 The

effect of surface

adhesion on the P - A loading/unloadingrelations(FEA-based

numerical results for the Vickers/
Berkovich equivalent cone). The
symbols 〇 and ● indicate theresults of the elastic bodies ( E ' = 20

 E ' tan β 34P= A − λE A ,2(8.27)kPa) with ( γ = 5.0 mJ/m2) andwithout ( γ = 0.0 ｍ J/m2) surface

where λ E is defined withadhesion, respectively. The dashedγE'
λE = 4 1 2 ,π(8.28)

lines indicate the JKR-theory (Eq.
8.24’)

80

being referred to as the adhesion toughness (the subscript E indicates
Elastic) that stands for the fracture toughness of interfacial delamination
between the tip-of-indenter and the material indented. The physicaldimension of λE is [Pa・m1/2] that is the same as the mode-I fracture(

hmax=30μm60)(a) Elastic body （E=20kPa, ν=0.49）

toughness, K Ic ≡ 2γ E ' .

: γ=0.0 mJ/m22: γ=5.0 mJ/m

40

8.2 ELASTOPLASTIC BODIESWITH SURFACE ADHESION- FEA-Based Numerical AnalysesThe plastic deformation and flow will lead to the mechanical processes
that reduce the effect of surface adhesion on the elastoplastic contact
mechanics. Suppose an elastic body with the elastic modulus E ' and anelastoplastic body with the same elastic modulus E ' along with theyield stress Y , and then inspect the effect of surface adhesion on thenormalized indentation load  P ( γ ) − P ( γ = 0 ) P ( γ )  of these twoAelastic and elastoplastic bodies;
the normalized indentation load isdefined by the indentation loads of P ( γ ) and P ( γ = 0) with andIndentation load, P (μN)

20
0
-20
60
(b) Elastoplastic body
（E=20kPa, Y=2kPa, ν=0.49）: γ=0.0 mJ/m2: γ=5.0 mJ/m2

40
20
0
-20
-40

0

50

100

2

Figure 8.5

P - A loading/ unloading relations(FEA-based numerical results) of(a) perfectly elastic bod

without surface adhesion at a given contact area A . Due to the reducing

E ' = 20.0 kPa, andsurface adhesion under plastic flow of the contact surface, the normalized(b) elastoplastic bodyindentation load  P ( γ ) − P ( γ = 0 ) P ( γ )  of the elastoplastic bodyAwill always be smaller than that of the perfectly elastic body. To confirm
this plastic effect on the adhesion toughness, the FEA-based numericalresults are plotted in Fig. 8.5, indicating that the plastic flow reduces theeffect of adhesion on the P - A loading/unloading relation.
Based on these FEA-based numerical studies, the JKR-theory (Eq.
(8.27)) of perfectly elastic body can be modified into the following
formula through including the plastic contribution to the indentation
contact of elastoplastic body;

P = H M A − λEP A3 4

(8.29)

in which H M stands for the Meyer hardness, and the elastoplastic
adhesion toughness λEP is defined by

λEP = 4γ EP E 'π1 2(8.30)

81

150

Contact area, A (x100μm )E ' = 20.0 kPa， Y = 2.0 kPa

The open circles (○) and the closed
circles ( ● ) indicate the P - A
relationswith ( γ = 5.0 mJ/m2) andwithout ( γ = 0.0 mJ/m2)
surface adhesion, respectively

In Eq. (8.30), γ EP

represents the elastoplastic surface adhesion(adhesion energy), although we have none of analytical formulas tocorelate it to the yield stress Y . To circumvent this difficulty, therefore,
we have to rely on the FEA-based approaches.
The FEA-based numerical results of elastoplastic bodies ( E = 20 kPa;
1.0 kPa ≤ Y ≤ 5.0 kPa) are shown in Fig. 8.6, indicating that theelastoplastic adhesion energy γ EP monotonically decreases with thedecrease in the yield stress Y , i.e., with enhancing plastic flow; γ EP
Normalized adhesion energy, γEP / γ

approaches to γ , i.e., γ EP → γ with the increase in Y , while itdiminishes to zero, i.e., γ EP → 0 , with the decrease in Y . The
correlations

( (

of

λEP E ' ≡ 162

theelastoplastic

adhesion

toughness) ) and the surface energy γ of elastoplastic

π γ EP

bodies with various value of E and Y are shown in Fig. 8.7.

As

readily seen in Figs. 8.6 and 8.7, the FEA-based numerical studies for the1
E=20 kPa
0.8
0.6
0.4
0.2
0

0

1

2
3
Yield stress, Y (kPa)45adhesive elastoplastic indentation contact problems lead to the following

Figure 8.6 Correlation between the

conclusions;(i) both the elastoplastic adhesion toughness λEP and the adhesive

surface energy γ EP decrease with the decrease in the yield stressY , i.e., with enhancing plastic flows,γ EPelastoplastic surface energy
and the yield stressof theY

elastoplastic bodies ( E = 20 kPa)
(FEA-based numerical results)

2

increase in the yield stress Y , leading to the perfectly elastic JKR-

2

of perfectly elastic body, i.e., γ EP → γ and λEP → λE , with theAdhesion toughness, λEP /E' (mJ/m )(ii) these elastoplastic adhesion parameters increase to the values

theory.
2
Figure 8.8 shows the ( λEP E ' vs aγ γ )-master curve that is made by2
horizontally shifting the respective λEP E ' vs γ

curves having

60
50
40
30

: E=20kPa, Y=1kPa: E=20kPa, Y=2kPa: E=20kPa, Y=3kPa: E=20kPa, Y=5kPa: E=100kPa, Y=12kPa: E=100kPa, Y=18kPa: E=20kPa; Elastic: JKR; λE2/E'=(16/π1/2)γ

20
10
0

0.5

1

5

2

10

Adhesive surface energy, γ (mJ/m )various values of E and Y in Fig. 8.7 along the log γ -axis to

Figure 8.7 Correlations between

superimpose on the perfectly elastic JKR-curve. The superposition isthe adhesion toughness λ EP 2 E 'satisfactory as readily seen in Fig. 8.8, where the shift factor a γ stands

and adhesive surface energy γ

for the amount of horizontal shit in superposition, and is related to theof elastoplastic bodies (FEA-

elastoplastic adhesion energy γ EP as follows;

basedaγ = γ EP γ

(8.31)

numerical

results). The

dashed line indicates the JKRtheory of perfectly elastic body82

There may exist a well-defined relationship between the shift factor a γ
and the plastic index PI ( ≡ ε I E ' cY ) (refer to Chap. 5), being expectedthataγ → 1 ( γ EP → γ ) for PI ↓ 0 (perfectly elastic)aγ → 0 ( γ EP → 0 ) for PI ↑ ∞ (fully plastic)
The relationships between aγ and PI ( ≡ ε I E ' cY ) are shown in Fig.
8.9 (the FEA-based numerical results); the best-fitted empirical formulaof this relation is given byaγ = 1 ( 3.5PI ) ; PI ≥ 0.286(8.32): E=20kPa, Y=1kPa, aY=0.17: E=20kPa, Y=2kPa, aY=0.3: E=20kPa, Y=3kPa, aY=0.53: E=20kPa, Y=5kPa, aY=0.8: E=100kPa, Y=12kPa, aY=0.38: E=100kPa, Y=18kPa, aY=0.55: E=20kPa; Elastic
2

Shift factor, aγ

40

2

2

λEP /E' (mJ/m )

1

60

1/2

: JKR; λE /E'=(16/π )γ20: E=20kPa: E=100kPa

0.8: aY=1/(3.5PI)

0.6
0.4
0.2

0
0.1

0.5

1

2

γEP(=aγγ) (mJ/m )

5

0

10

100

101

PI[=(E'tanβ/2)/cY]

Figure 8.8 λEP 2 E ' vs γ EP masterFigure 8.9

curve made by horizontally shifting theplastic index PI ( ≡ ε I E ' cY ) and therespective λEP E ' vs γ

shift factor2

curves inCorrelation between theaγ ( ≡ γ EP γ ) utilized inFig. 8.7 and superimposing onto theFig. 8.7 to make the master curve shown

perfectly elastic JKR-curve (the dashedin Fig. 8.8 (FEA-based numericalline)

result). The dashed line indicates thebest-fitted

empirical

relationaγ = 1 ( 3.5PI ) (Eq. (8.32))

83

of8.3 VISCOELASTIC BODIESWITH SURFACE ADHESION(1) Load relaxation (cone/pyramid indentation)

Applying the “elastic-viscoelastic corresponding principle” to the
JKR-theory leads to the following constitutive equation of viscoelastic
bodies with surface adhesion in Laplace-space; E '* ( p) tan β 34
P ( p) =  A( p) − λVE * ( p) A( p)2(8.33)

where P( p) ，A( p) ，E '* ( p) ，and λVE * ( p) are, respectively, definedwith∞P ( p) =  P(t )e− pt dt
0∞A( p) =  A(t )e− pt dt
0∞

E '*( p) = p  Erelax '(t )e− pt dt0∞

λVE * ( p ) = p  λVE ( t ) e − pt dt
0

=4

∞
γ
p  E 'relax (t )e − pt dt
120πThe invers Laplace transform of Eq. (8.33) for viscoelastic bodies withsurface adhesion, therefore, results in the following constitutive equationin real space;dA(t ')
 tan β  tP(t ) = dt '
 0 E 'relax (t − t ')
dt '
 2 

(8.34)γ tdA(t')3 4
dt '− 4 1 2  E 'relax (t − t ')
dt '
π 0

Let us suppose, as an example, the indentation load relaxation test
under the stepwise application of contact area to A0 ;A(t ) = u (t ) A0 , u (t ) ; Heaviside step-function(8.35)and then inspect the effect of surface adhesion on the load relaxation.
Substituting Eq. (8.35) into Eq. (8.34) along with noticing the relation ofdu ( t ) dt = δ ( t ) （ δ ( t ) : Dirac’s delta function）, we finally have thefollowing expression for the indentation load relaxation of cone/pyramidindentation;84

−4(8.36)γA 3 4 E 'relax (t )π1 2 0In order to simply the subsequent numerical analyses, by way of example,

0.4
0.2
0

-0.2

100

200

300

400

500

600

Time, t (s)(8.37)

Figure 8.10 Effect of surface

and the Zener viscoelastic solid.

adhesion on the indentation load tE ' ( t ) = E 'e + E 'g − E 'e exp  −  τ

)(8.38)

are applied to Eq. (8.36) for examining the effect of surface adhesion on
the load relaxation behavior of P (t ) vs t ; the results are demonstrated

relaxation curve of the Maxwell
viscoelastic liquid(stepwise application of the contact
area to A0 = 2 ×10−7 m2)

in Fig. 8.10 for the Maxwell liquids ( E 'g = 20 kPa， τ = 50 s) and in Fig.

without surface adhesion for Vickers/Berkovich-equivalent cone
indentation ( β = 19.7 , A0 = 2 × 10−7 mm2). It will be worthwhile
noticing in Figs. 8.10 and 8.11 that the surface adhesion yields thenegative indentation load, i.e., P < 0 , due to the adhesive force at thecontact interface, and the time to the complete load relaxation or to thesteady-state , i.e., the time to P → 0 or dP dt → 0 , is shifted tolonger side with the increase in the surface adhesion γ .

Zener solid

Indentation load, P(t) (mN)

8.11 for the Zener solid ( E 'g = 20 kPa, E 'e = 5 kPa, τ = 50 s) with and: γ = 0.0 mJ/m2
:
10
: 100

0.5

0

-0.5
0

100

200

300

400

500

600

Time, t (s)Figure 8.11 Effect of surface
adhesion on the indentation load
relaxation(2) Creep deformation (spherical indentation)

The P vs a relation of perfectly elastic bodies for spherical

curve

of

the

Zener

viscoelastic solid(stepwise application of the contact

indentation (JKR-theory; see Eq. (8.24))
4E ' 3a − 4 πγ E 'a 3 2
3R(8.42)

can be recast into the following quadratic equation in terms of the variable(: γ= 0.0 mJ/m2
: 10
: 100

0.6

-0.6
0

 t
E 'relax (t ) = E 'g exp  − 
 τ

P=

Maxwell liquid

0.8

-0.4

the Maxwell viscoelastic liquid,(

Indentation load, P(t) (mN)

 tan β P(t ) = 
A0  E 'relax (t )
 2


)

x ≡ a3 2 ;4E ' 2x − 4 πγE ' x − P = 0 ,3R(8.43)and then be solved as

85

area to A0 = 2 ×10−7 m2)

{ (

4a 3
= C 'P 1+ 2 k + k + k2
3R)}

(8.44)

in which

C ' =1 E'
k =Γ P(8.45)

Γ = 3π R ⋅ γ＊ E ' : elastic modulus， C ' : complianceΓ : adhesion force
γ : surface adhesion (adhesive energy)
R : radius of spherical indenter
The application of “elastic-viscoelastic correspondence principle” to Eq.
(8.44) under stepwise indentation load P(t ) = P0u(t ) ( u (t ) : Heaviside
step-function) along with the Laplace transform and its inversion finallyresults in the following constitutive equation for creep deformation andflow; 4a ( t )3 
0
 = C ' ( t ) 1 + 2 k 0 + k0 + k0 2

3RP0 { (α0 (t )  ≡)} (8.46)

where C ' ( t ) stands for the creep function and the subscript “0”indicates the stepwise indentation loading to P0 . Substituting k 0 = 0 ,i.e., γ = 0 , in Eq. (8.46) yields
3a0 ( t ) =3RC ' ( t ) ⋅ P0
4

(8.47)

that describes the creep deformation and flow of linear viscoelastic
bodies without surface adhesion; there exists a linear relation ofa3 ( t ) ∝ P under stepwise spherical indentation. As readily seen in Eq.
(8.46), however, the surface adhesion violates this linear relationshipeven if the indented body is linear viscoelastic. It must be also noticed
that Eq. (8.46) can be rewritten witha3 ( t ) = 3RC ' ( t ) ⋅ Γ(8.48)

when the adhesion force Γ is large enough and/or the applied load P
is small enough, i.e., k  1 .

In other words, due to the surface

adhesion that pulls the tip-of-indenter toward the material’s surface, thefinite creep deformation/flow will be observed even under the dead loadof P ≡ 0 .

86

In order for further examining the effect of surface adhesion, let makenumerical considerations on the creep deformation and flow of thefollowing Zener viscoelastic solid; t
C ' ( t ) = C 'e + ( C 'g − C 'e ) exp  −  τ

(8.49)

with C 'g = 0.01 mPa-1 ， C 'e = 0.035 mPa-1, and τ = 200 s. The
numerical results of the creep curve a3 ( t ) vs. t are shown in Figs.
8.12 and 8.13, leading to the following important conclusions as to theeffect of surface adhesion on the creep deformation/flow;(1)

a3 ( t ) -creep curve is not linearly proportional to the applied
indentation load P(2) creep deformation is enhanced with the increase in thesurface adhesion

P=0.10 mN
Contact radius, a3(t) (10-3 mm3)

γ=10mN/m

600
500
400

3

-3

3

Contact radius, a (t) (10 mm ))(3) significant creep is observed even under the dead-load of P ≡ 0

300: P=0.4 mN: 0.2 mN
: 0.1 mN
: 0.0 mN

200
100
0
0

200

400
600
Time, t (s)

800

1000

200

γ=4.0 mN/m

2.0

100

1.0
0.0

0
0

200

400
600
Time, t (s)

800

1000

Figure 8.12 Creep curves of the

Figure 8.13 Creep curves of theviscoelastic Zener solid with surface

viscoelastic Zener solid with surface

adhesion ( γ = 10 mN/m) underadhesion under stepwise loading ofstepwisespherical indentation ( R =3.0mm;

loadingofspherical

indentation ( R =3.0mm). The curves

P =0.1mN). The curves are γ =4.0,

are P =0.4, 0.2, 0.1, and 0.0mN from

2.0, 1.0, and 0.0mN/m from the top

the top to the bottom. It must be

to the bottom

noticed that creep deformation is
induced even under the dead-load of
P ≡ 0 (the dashed-dotted line)

87INDENTATION CONTACT MECHANICS
OF COATING/SUBSTRATE COMPOSITE

CHAPTER

9

Japanese craftworks “SHIKKI” (Japanese lacquered crafts), potteries(glazed ceramics), plated metals, etc. are all the typical examples ofcoated materials/arts that have long been making our daily lives enriching
and comfortable.

Painted walls and adhesive tapes, by way of example,

are also classified as coating/substrate composites; the paint (coating) not
only protects the walls (substrate) from hostile environments, but also
make the wall more tactile; the adhesive coating makes the substrate more(a) homogeneous half-spacePffunctional. The thickness of these coating films is in the range fromnanometer to sub-millimeter.

This fact makes the experimentalβhdetermination of film-only mechanical characteristics critically difficult,2af

thus remaining much works/problems/ issues for use to overcome.
On the other hand, as readily expected from the considerations we

Ef, υfmade in the preceding chapters, the science of indentation contact
mechanics combined with the instrumented indentation apparatuses will(b) film/substrate composite

Pprovide a powerful tool for experimentally determining the mechanicalcharacteristics of coating/substrate composites in their micro/ nanoregions. The theoretical formulation of the indentation contact mechanics
of coating/substrate composites will be given in this chapter, though its
present status is still immature [9.1-9.5]. The details of experimental
techniques/analyses

fordetermining

mechanicalproperties

ofcoating/substrate composite will be given in Chap. 11.

β

h
2a

Ef, υfEs, υsFigure 9.1 Geometrical details of aconical indenter pressed onto(a) homogeneous half-space

9.1 ELASTIC COMPOSITES
The mathematical formulation of indentation contact mechanics isfully established for the laminate of an elastic film coated on an elastic
substrate.

tf(i.e., film-only half-space), and(b) film/substrate composite

Figure 9.1 depicts the geometrical details of a conicalindenter pressed onto (a) an elastic half-space (file-only half-space), and(b) a film having the thickness tf
extending to a half-space.

coated on an elastic substrate

The elastic moduli and the Poisson’s ratiosof the film and the substrate are denoted by ( Ef , ν f ) and ( Es , ν s ),respectively. At a given penetration depth h , the induced indentationloads and the associated contact radii are described by ( Pf , af ) and( P , a ) for the film-only half-space and for the laminate composite,

88[9.1] C.H. Hsueh, P. Miranda, J. Mater.
Res., 19[1], 94 (2004)[9.2] C.H. Hsueh, P. Miranda, J. Mater.
Res., 19[9], 2774 (2004)[9.3] H.Y. Yu, S.C. Sanday, B.B. Rath, J.
Mech. Phys. Solids, 38[6], 745 (1990)[9.4] F. Yang, Mater. Sci. Eng., A358, 226(2003)[9.5] M. Sakai, J. Mater. Res., 24[3], 831 (2009)

respectively.

When the thickness of the film is much larger than thepenetration depth, i.e., tf >> h , we can neglect the substrate effect on thecontact behavior, leading to the perfectly elastic indentation contact
mechanics of a homogeneous half-space of the film, the details of which
are already given in Eqs. (3.33) and (3.34) (see Chapter 3, Tab. 3.1);

Pf = kh Ef ' h n
af ( ≡ aH ) = Bh

(9.1)
n −1

(9.2)

= cP( n −1) n

Due to the fact that the frontal factor B in Eq. (9.2) of a perfectly elastic
half-space is independent of the elastic modulus (see Tab. 3.1), it must be
noticed that the indentation contact radius af of the elastic film can be
described in terms ofthe contact radius aH of a homogeneous elastic

body having arbitrary elastic modulus (the subscript H stands for“homogeneous”).
On the other hand, in the cases of tf ≈ h and tf < h , as shown in Fig.
9.2, the indentation contact behavior is significantly affected by thesubstrate. Figure 9.2 demonstrates the substrate-effect on the indentation
contact behavior for the coating/substrate composite with Ef << Es , as
an example. We must keep in mind the fact that there is a significant
discrepancy in their contact behaviors of coating/substrate composite
under a fixed depth h of penetration and under a fixed load P of(a) under a fixed depth of penetration, hindentation. As a matter of fact, as shown in Fig. 9.2, the substrate has an

PfP

affect in an entirely different manner on the indentation contact behaviorof ( P Pf )h and ( a af )h  ≡ ( a aH )h  observed under a fixed depth hof penetration, or of ( h hf ) P and ( a af ) P  ≡ ( a aH ) P  under a fixedloadPof indentation.

2af

understanding the effective elastic modulus of laminate composites that
we discuss in the following considerations.

2aEs

Ef<<Es(b) under a fixed load of indentation, P
PPIt should be noticed, in particular,( a aH )h ≠ ( a aH )P ; this inequality will play an essential role in

Ef

h
Ef

hf
Ef

h

2af

Ef

2a

Es

Ef<<EsFigure 9.2 Substrate-effect on thelaminate composite with Ef << Es

In order to discuss the indentation contact behavior of elastic

for the indentations (a) under a fixedcoating/substrate composites in a phenomenological manner, let us now

indentation depth h , and (b) under aintroduce the concept of effective elastic modulus E 'eff ( tf a ) to

fixed indentation load P

89

Eq. (9.1) as follows;
P = kh E 'eff ( tf a ) h n(9.3)

where the effective elastic modulus E 'eff ( tf a ) satisfies the following
extremes;lim E 'eff ( tf a ) = E 'f

tf a →∞

lim E 'eff ( tf a ) = E 's

tf a → 0

Accordingly, through Eqs. (9.1) and (9.3), the normalized values ofindentation load

( P Pf )hand penetration depth( h hf ) P

are,respectively, given by( P Pf )h = E 'eff E 'f

( h hf ) P = 1

(9.4)
(9.5)

( E 'eff Ef )1n

In our next step of mathematical formulations through the Boussinesq
Green function, we will make an analytical derivation of the effective
modulus E 'eff ( tf a ) that has been phenomenologically defined in Eq.
(9.3). The contact problem of a concentrated point load P applied onan elastic half-space having the elastic modulus E and the Poisson’s
ratio ν was first solved by J. Boussinesq in 1885 [1.1, 9.6]. The problem
is reduced to the elastic contact of a flat-ended cylindrical punch withP

radius a0 (see Sec. 3.1) in its extreme of a0 → 0 , leading to theuz ( r,0)

following result of the displacement gradient ∂u z ∂z ;∂u z P (1 + ν )  3r 2 zz=− ( 3 − 2ν )
3 
2
2
2
∂z
2π E  r 2 + z 2 5 2)( r + z ) 
 (

h

tf

E f, υf
E s, υ s(9.6)

Integrating both sides of Eq. (9.6) along the z -axis of the elastic

Figure 9.3 The Boussinesq problem

composite shown in Fig. 9.3, we finally have the contact surface-ofdisplacement u z (r , z = 0) induced by the concentrated force P [9.1,composite under a concentrated load

9.2];

P

an

elastic

coating/substrate[9.6] J. Boussinesq, “Application des potentials
á l’etude de l’équilibre et du mouvement dessolides élastiques”, Gauthier-Villars (1885)

90

t

0

f
 ∂u z ∂u zu z (r , z = 0) ( ≡ G ( r , z; Ei ,ν i ) ) = 
dz + 
dz
∞ ∂z stf ∂z f

P (1 − ν f 2 ) 1

=

π Ef

(9.7)

r





r2
1 + ν s  ( 3 − 2ν s ) −


1
3 
 E  22
2s
( r 2 + tf 2 ) 2  P 
 ( r + tf )
+


2π 


2
3 − 2ν f )r − 1 +ν f  (
−
1
3 


2
2
2
2
2
2 
Ef
( r + tf )  
 ( r + tf )Equation (9.7) is the so-called Boussinesq Green function G ( r , z; Ei ,ν i )of the coating/substrate composite, providing a basic equation in order tomake the mathematical formulation of spherical/conical indentation
contact mechanics, the details of which will be given in the following
considerations.
Combining Eq. (9.7) with the indentation contact pressure distributionp ( r ) ( ≡ −σ z ( r ,0 ) ) ( r ≤ a ; contact radius a ) (refer to Chap. 3 (Fig. 3.5,
Tab. 3.1)) for the conventional axisymmetric indenter’s geometry, thepenetration depth h of the elastic laminate can be described in terms ofp (r )

asthe

integration

of

the

contact

displacementuz (r , z = 0) ( ≡ G ( r , z; Ei ,ν i ) ) along the z -axis;ah =  p(r )G ( r , z; Ei ,ν i ) 2π rdr

(9.8)

0

Accordingly, substituting Eq. (9.7) into Eq. (9.8), and furthermore using
Eq. (9.4), we finally have the effective elastic modulus E 'eff ( tf a ) as afunction of the normalized film thickness ξ ( = tf a ) along with thematerial characteristics ( E 'f , ν f ) of the film and ( E 's , ν s ) of thesubstrate;−11 + ν s( 3 − 2ν s ) I1 (ξ ) − I 2 (ξ )  
E 'eff   P    E 'f  Es≡    = 1 + (9.9)
E 'f   Pf h   2 I 0  1 + ν f−{( 3 − 2ν f ) I1 (ξ ) − I 2 (ξ )}


E


f

where I0 , I1 (ξ ) , and I2 (ξ ) are respectively defined by91

1

I 0 =  pN ( ρ ) d ρ0ρ

1I1 ( ξ ) =  p N ( ρ )0(ρ + ξ )
2

0

1

2

ρ3

1

I 2 (ξ ) =  pN ( ρ )23(ρ2 + ξ 2 ) 2(9.10)dρdρin terms of the normalized contact pressure distribution pN ( ρ ) ：pN ( ρ ) = p (r ) pm ;pm = P π a 2 ; ρ = r a

In the extreme of thick film, i.e., tf  a (ξ ↑ ∞ ) ,since there exists the3and the coating film dictates the indentation contact behavior of thelaminate composite. On the other hand, in the extreme of thin film, i.e.,tf  a (ξ ↓ 0 ) , we have the relation of I1 (ξ ) = I 2 (ξ ) = I 0 , resulting inE 'eff = E 's ; the substrate characteristics dominate the contact behavior of
the composite. The analytical formulas of the functions of p N ( ρ ) ,I1 (ξ ) , and I 2 (ξ ) of the axisymmetric conventional indenters (flat-ended
cylindrical punch, sphere, and cone) are listed in Tabs. 9.1 and 9.2.
The analytical expression for the contact pressure distribution pN ( ρ )is very essential prior to conducting the integral in Eq. (9.10), though wehave no information onpN ( ρ ) of the substrate-affected laminate

composite in an analytical manner. Accordingly, in Tab. 9.1,

I1 (ξ )and I 2 (ξ ) are approximately calculated by the use of p N ( ρ ) for the
homogeneous half-space (see Chap. 3; Tab. 3.1) neglecting the substrateeffect.

Due to the fact of I 0 = I1 (0) = I 2 (0) , the integral I 0 is readily

obtained from I1 (ξ ) or I 2 (ξ ) with ξ = 0 . On the other hand, for athin-film ( tf a << 1 ) coated on a rigid substrate ( E 's ↑ ∞ ) as a specificcase of laminate composites, since there exists the analytical expressionofpN ( ρ ) [9.4], the analytical solutions of I1 (ξ ) and I 2 (ξ ) isreadily obtained, the results of which are also listed in Tab. 9.2. In Fig.
9.4, the contact pressure distributions pN ( ρ ) of the homogeneous halfspace (the solid lines) and of the thin-film coated on a rigid substrate (thebroken lines) are respectively plotted for comparison.
The elastic contact mechanics based on the preceding Boussinesq
Green function G ( r , z; Ei ,ν i ) has the merit of providing the analytical
92

Normalized contact pressure distribution, pN[=p(r)/pm]

relation of I1 (ξ ) = 0 = I 2 (ξ ) , Eq. (9.9) naturally leads to E 'eff = E 'f ,pm=P/πa2flat-ended
cylindrical
punch

2

1

0

spherical
indenter

1.5
1
0.5

: homogeneous half-space: thin film on rigid substrate0

conicalindenter

3
2
1
0
-1

-0.5

0

0.5

1

Normalized radius, r/a

Figure9.4

distributions

Contact

pressurepN ( ρ )ofaxi-

symmetric indenters;
the solid line:
homogeneous half-space
the broken line:
a thin-film coated on a rigid
substrate [9.4]solution and its extension to the viscoelastic indentation contact
mechanics via the elastic-to-viscoelastic correspondence principle.
However, as mentioned above, there is a critical difficulty in obtaining
the analytical solution for the contact pressure distribution pN ( ρ ) ofsubstrate-affected laminate composite.
In contrast to the analytical solution based on the Boussinesq Greenfunction given in the preceding considerations, on the other hand, we canget the numerical solution of the elastic contact mechanics ofcoating/substrate composites in a quantitative manner

through theFredholm integral equation; we first express the stresses/strains of thecoating film and the substrate using the characteristic harmonic function(Papkovich-Neuber function) followed by the Hankel transform (refer toChap. 3), and then we finally derive the following Fredholm integral
equation of the second-kind [9.3];

H (ρ ) −11

 K ( y + ρ ) + K ( y − ρ ) H ( y )dy = F ( ρ )π 0In Eq. (9.11),

H (ρ)

stands for the non-dimensional function

expressing the indentation contact pressure distribution.

K ( y)(9.11)The kernelof the integral includes the information of the material

characteristics ( Ef , ν f ), ( Es , ν s ), and the normalized film thicknessξ ( = tf a ) . The function F ( ρ ) describes the geometry of the indenter;flat-ended cylinder

F (ρ ) = 1 ；F ( ρ ) = 1 − ( γρ ) ；sphereF ( ρ ) = 1 − γρ ；

cone

2

in which γ stands for the relative contact radius ( a aH )h at a given
depth of penetration h as shown in Fig 9.2(a), i.e., γ = ( a aH )h( ≡ ( a af )h ).

The solution H ( ρ ) of Eq. (9.11) is related to thenormalized contact load ( P Pf )h , as follows;1 P   E 'eff   ≡
 = c 0 H ( ρ )d ρ Pf h  E 'f 

(9.12)

The frontal factor c in Eq. (9.12) is the indenter’s tip-geometry
dependent coefficient; c = 1 , 3/2, and 2 for the flat-ended cylindrical
punch, spherical indenter, and the conical indenter, respectively. As
readily seen in Eq. (9.11), the integral equation includes the unknown

93

parameters of γ = ( a aH )h , H ( ρ ) , and ( P Pf )h of the film/ substratecomposite that we are interested in.
In the first step of solving the integral equation (Eq. (9.11)), we
transform the integral equation into the linear simultaneous algebraic
equations by expressing the unknown function H ( ρ ) in terms of the
Chebyshev series expansion, and then we substitute F ( ρ ) with γ = 1
as its initial value into this algebraic equations, leading to the zerothorder-solution H ( ρ ) . In the next step, we repeat the preceding
numerical procedures by systematically changing the γ -value until we(meaning that the contact pressure diminishes outside the contact area( ρ ≥ 1 )). The normalized elastic modulus of the laminate composite
E 'eff E 'f  ≡ ( P Pf )h  thus obtained by substituting the solution H ( ρ )into Eq. (9.12) is plotted in Fig. 9.5 against the normalized thickness

tf a of the coating film for various values of the modulus-ratio

E 's E 'f . We also plot in Fig. 9.5 the analytical solution (Eq. (9.9)) (thesolid lines) derived from the Boussinesq Green function with the contactpressure distributions pN ( ρ ) for the homogeneous half-space (see Tab.
9.1, Fig. 9.4), excepting the results of the composite E 's E 'f = 100 for
which we utilize pN ( ρ ) of the thin-film coated on rigid substrate (seeTab. 9.2, Fig. 9.4). As readily seen in Fig. 9.5, the analytical solution (Eq.
(9.9); the solid lines) with the approximated pN ( ρ ) rather faithfully101

Normalied effective elastic modulus, E'eff /E'f [(=P/Pf)h]finally get the solution H ( ρ ) having the boundary value of H (1) = 0

100
10-1
spherical indenter10

10-1
conical indenter
10

demonstrates that the effective elastic modulus

E 'effnaturally

converges to the modulus of the substrate E 's , i.e., E 'eff → E 's , in theextreme of thin film or major penetration ( tf a < 0.1 ), while it converges

1

100
10-1
10-1

100

101

102

Normalized film-thickness, t f /a(Eq. (9.11); the respective symbols with broken line). In particular for

numerical and the analytical results is satisfactory. Figure 9.5 furthermore

1100

realizes the precise numerical solution of the Fredholm integral equation

spherical indentation contact, the coincidence between both of the: E's/E'f=100
5
2
1
0.5
0.1

flat-punch

Figure9.5

Normalized

effective

elastic modulus of coating/substratecomposites

plottednormalizedfilmagainstthe

thickness.

Thesymbols with broken line are thenumerical results of the Fredholm

to the modulus of the coating film E 'f , i.e., E 'eff → E 'f , in the extreme

integral equation (Eq. (9.11)), and theof thick film or minor penetration ( tf a > 50 ). As has been given insolid lines are the analytical results of( h hf ) P = 1 ( E 'eff E 'f ) , we canreadily describe the normalized penetration depth ( h hf ) P under a fixed

the Boussinesq Green function (Eq.

1nEq. (9.5), noticing the relation of

94(9.9))

indentation load P in terms of the modulus ratio E 'eff E 'f via its
numerical or analytical solution given in Eqs. (9.9) and (9.12), and in Fig.
9.5.
As mentioned in Chap. 3 for the elastic indentation contact mechanics
of a homogeneous body, its free surface outside the indentation contact
zone always exhibits sinking-in along with the penetration of the indenter,
resulting in the relation of hc h < 1 (see Tab. 3.1) irrespective of the tipgeometry of the indenter we used. On the other hand, forcoating/substrate composites, their surface profiles are significantly
affected by the modulus ration E 's E 'f . Figure 9.6 shows the details of
the contact profiles (sink-in/pile-up) of laminate composites having
various values of E 's E 'f , in which the normalized contact radiusγ  ≡ ( a aH )h 

is numerically determined through the iteration

procedure in solving Eq. (9.11) as mentioned in the preceding

Normalized contact radius, (a/a H)hconsiderations [9.3].

2

spherical indenter

conical indenter: E 's/E 'f=100
:
5
:
2
:
1
:
0.5
:
0.1

1.5

1

0.5
0

2

4

6

8

0

2

4

6

8

Normalized film thickness, t f /a

Figure 9.6 Effect of substrate on the normalized contact radius γ  ≡ ( a aH )h (numerical solutions of the Fredholm integral equation)

95

10

As shown in Fig. 9.6, the relation of( a aH )h vs. tf a for thecomposites with E 's E 'f > 1 is convex upward, while it is concave
downward for E 's E 'f < 1 at around tf a ≈ 1 ; in the former, the majorof penetration-induced deformation at the fixed depth of penetration his accommodated in the coating film due to the stiffer substrate, resulting
in suppressing the sink-in or in inducing the pile-up of the free-surface
outside the contact zone, while in the latter, due to the more compliant
substrate, the major of penetration-induced deformation at the fixed depthof penetration h is accommodated in the substrate, resulting in theenhanced sink-in of the free-surface. In the conical indentation as an
example, noticing the fact that the relative contact depth hc h of ahomogeneous elastic half-space is given by 2 π(see Tab. 3.1), thepreceding considerations lead to the following expression for thecoating/substrate composite;hc 2  a =  
h π  aH hAccordingly, using the results shown in Fig. 9.6 for the composite with

E 's E 'f = 5 , by way of example, the relative contact depth hc h at thepenetration of tf a ≈ 1 is given by hc h ≈ ( 2 π ) × 1.28 ≈ 1.0 , implying
none of sink-in nor pile up. Furthermore, for the composite having anearly rigid substrate ( E 's E 'f = 100 in Fig. 9.6), the relative contact
depth becomes larger than 1.0, i.e., hc h > 1 , leading to an enhanced
pile-up of the free-surface, like as the ductile plastic body that we havediscussed in Chap.5 (see Figs. 5.6 and 5.7), although the composite isperfectly elastic. On the other hand for the laminate composite with very
compliant substrate such as E 's E 'f = 0.1 , the normalized contact
radius at

tf a ≈ 1is( a aH )h ≈ 0.5(see Fig. 9.6), and thenhc h ≈ ( 2 π ) × 0.5 ≈ 0.3 , meaning an enhanced sink-in more than that ofa homogeneous elastic body ( hc h = 2 π ( = 0.637 ) ).

96

Table 9.1 The contact pressure distributions pN ( ρ ) of the axisymmetric indenters pressed onto a
homogeneous elastic half-space, based on which the integrals I1 (ξ ) and I 2 (ξ ) are determined (see Eq.
(9.10))

pN(r;a)1
Flat-ended
cylinder 2 1 − (r / a) 2
Sphere3
1 − ( r / a) 2
2

Conecosh −1 ( a / r )

I1(ξ)

I2(ξ)

 ξ 
1 π
 − sin −1 
 1 + ξ 2 
22




 ξ 
1 π
ξ 
−
 − sin −1 
 1+ξ 2  1+ξ 2 
22





π
 ξ  3(1 + ξ 2 )  − sin −1  −ξ 
 1 + ξ 2  
2
2


 




π
 ξ 
3(1+ 3ξ 2 )  − sin −1 

3
ξ
−


 1+ ξ 2 
2
2









π

π

ξ2
− tan ξ + ξ ln
2
1+ξ 2

ξ

ξ2
− tan ξ + ln
2
2 1+ξ 2
−1

−1

Table 9.2 The contact pressure distributions pN ( ρ ) of the axisymmetric indenters pressed onto a thin elastic
film ( tf a << 1 ) coated on a rigid substrate, based on which the integrals I1 (ξ ) and I 2 (ξ ) are determined(see Eq. (9.10))

pN(r;a)

I1(ξ)

I2(ξ)

Flat-ended
cylinder1

1+ ξ 2 −ξ2( 1 + ξ 2 − ξ ) −Sphere2 1 − (r / a)2 Cone

3[1 − (r / a) ]1
1+ ξ 22[(2 / 3)(1 + ξ 2 ) 1 + ξ 2

4[(1/ 3)(1 + 4ξ 2 ) 1 + ξ 2− ξ − (2 / 3)ξ 3 ]− ξ − (4 / 3)ξ 3 ]3[(1/ 2) 1 + ξ 2 − ξ{}

3[(1/ 2) 1 + ξ 2 − 2ξ{}−(1/ 2)ξ 2 ln ξ /(1 + 1 + ξ 2 ) ] −(3/ 2)ξ 2 ln ξ /(1 + 1 + ξ 2 ) ]97

9.2 VISCOELASTIC COMPOSITES
Based on the “elastic-to-viscoelastic correspondence principle” (refer
to Chap. 7), i.e., the linear viscoelastic constitutive equations in the
Laplace-space coincide with those of perfectly elastic body in the realspace, combined with the preceding considerations on the elastic
laminate composites, we can rather easily extend the indentation contact
mechanics of elastic laminate composites to that of viscoelastic
composites [9.7].
The indentation load relaxation of a viscoelastic laminate composite
under a stepwise penetration to the contact area A0 , i.e., A(t ) = A0 ⋅ u (t ) ,is given byP(t ) =tan β
A0 ⋅ E '(t )2(9.13)

like as Eq. (7.6). In Eq. (9.13), we denote the relaxation modulus

E 'relax (t ) by E ' ( t ) for simplicity. Once we apply the “elastic-toviscoelastic correspondence principle” to Eq. (9.9), the relaxation
modulus E '(t ) ( ≡ E 'eff (t ) ) of the viscoelastic laminate composite is
described by1 + ν s( 3 − 2ν s ) I1 (ξ ) − I 2 (ξ )  E '(t )   P(t )    E 'f (t )  Es (t )

≡ 


  = 1+
E 'f (t )   Pf (t )  A  
2I0  1 + ν f−{( 3 − 2ν f ) I1 (ξ ) − I 2 (ξ )}


E
t
(
)
 f



−1

(9.14)
where E 'f (t )  ≡ Ef (t ) / (1 − ν f 2 )  and E 's (t )  ≡ Es (t ) / (1 − ν s 2 )  are therelaxation moduli of the coating film and of the substrate, respectively.
Due to the assumption of stepwise penetration A(t ) = A0 ⋅ u (t ) , thenormalized film thickness ξ ( ≡ tf a0 )( a0 = A0 π ) is a time-

independent constant. As emphasized in Sec. 6.4, the Poisson’s ratio νof viscoelastic body is time-dependent; by way of example for a
viscoelastic liquid such as the Zener II model with Ee = 0 in Eq. (6.8b),
the Poisson’s ratio ν (0) ( ≈ 0.1) at the onset of stress relaxation
monotonically increases with time to the value of ν ( ∞ ) ↑ 0.5 in its
steady-state, while its time-dependence is very minor for a viscoelastic
solid ( Ee ≠ 0 ) (see Fig 6.10).

In this chapter, therefore for simplicity,

we assume that the Poisson’s ratios of coating/substrate composites are98[9.7] M. Sakai, Phil. Mag., 86[33-35], 5607 (2006)

time-independent having the values of ν f = ν s = 0.3 in Eq. (9.14). This
assumption by no means results in any critical issues in discussing theviscoelastic contact mechanics of laminate composites.
Let us suppose the Zener II model (refer to Chap. 6, Fig. 6.2, and Eq.
(6.8b)) with a single relaxation time of τ = η E 'M ,
 −t 
E '(t ) = E 'e + ( E 'g − E 'e ) exp  τ 
E 'g = E 'e + E 'M

(9.15)in order to describe the relaxation moduli of Ef (t ) and Es (t) in Eq.
(9.14). The Zener II model with Ee = 0 is readily reduced to theMaxwell model.

We will discuss in what follows the viscoelastic(a)contact mechanics of laminate composites subjected to a stepwiseof a cone indentation, i.e.,A(t ) = A0 ⋅ u (t ) , or to the contact radiusa(t ) = a0 ⋅ u (t )witha0 = A0 π .
(1) Stress relaxation behavior of an elastic film coated on aviscoelastic substrate
The indentation load relaxation P ( t ) of a laminate for stepwise
penetration of a cone indenter is well described in terms of the relaxation
modulus E ' ( t ) through the relation given in Eq. (9.13). The relaxation
modulus

E '(t )is shown in Fig. 9.7(a) for the laminate with

E 'f = E 's (0) = 10 GPa as an example, i.e., for the laminate composite
with the elastic modulus E 'f of the coating film matching to the glassy
modulus E 's ( 0 ) of the viscoelastic substrate.

On the other hand in

Fig. 9.7(b), shown is the relaxation behavior of the laminate with the film
modulus smaller than that of the glassy modulus and larger than the

Relaxation modulus, E(t) (GPa)A0

Es(0)=10 GPa, Ee=1 GPa, τs =10 s

10film (Ef=10 GPa)
tf/a=20

8

10

6

5

4

1

2

substrate

0
0

10

0
20
30
Time, t (s)

40

50

(b)
Relaxation modulus, E'(t) (GPa)penetration to the contact area

1212Es(0)=10 GPa, Ee=1 GPa, τs =10 s

108film (Ef=5 GPa)64

tf/a=102

substrate
0

0
0

10

20

1

30

5

40

50

Time, t (s)E 's (0) > E 'f > E 'e ( ≡ Es (∞ ) ) ( E 'f = 5 GPa; E 's (0) = 10 GPa、 E 'e = 1

E '(t )

9.7 Relaxation moduli
of the laminate composites

with

the

GPa).

( E 's (0) = 10 GPa,equilibrium

modulus

E 'e

of

the

viscoelastic

substrate,i.e.,

In Fig. 9.7, the relaxation moduli of the laminate composites are

Figureviscoelastic

substrate

E 'e = 1 GPa,plotted for the various values of the normalized film thickness tf a in

τ s = 10 s) on which the elastic film

order to demonstrate how the film-thickness or the depth of penetration((a)

dictates the relaxation behavior.

GPa) is coated99

E 'f = 10 GPa; (b)E 'f = 5elastic substrate
The contact mechanics of the composite comprising a viscoelastic film
coated on an elastic substrate is scrutinized for the stepwise penetrationof conical indentation, in contrast to the composite with an elastic film
coated on a viscoelastic substrate that we discussed above. We utilize
the viscoelastic Zener II solid as the coating film;

 −t 
E 'f (t ) = E 'e + ( E 'g − E 'e ) exp  , τf 
τ f = 10s E 'g = 10GPa E 'e = 1GPa(a)Relaxation modulus, E'(t) (GPa)(2) Stress relaxation behavior of a viscoelastic film coated on an15
tf/a=1

E's=20 GPa
10 10
5
5

0
0

2
1
10

20

30

40

50

Time, t (s)while we suppose the perfectly elastic substrates having various valuesof the elastic modulus E 's in order to examine the effect of substratemodulus on the viscoelastic behavior of laminate composite.
Figure 9.8(a) demonstrates the effect of elastic modulus E 's of thesubstrate on the relaxation behavior of the laminate composite for theNormalized
relaxation modulus, E'(t)/E'(0)(b)
1

E's=0.5 GPa

0.8

1

0.6

2
5

0.4

10

0.2
0

tf/a=1

20
0

10

20

30

40

50

Time, t (s)conical indentation at a fixed depth of penetration, tf a = 1 . As readily

Figure 9.8 Effect of the elastic

seen in Fig. 9.8(a), the substrate-effect is predominant in the initial stage

modulus E 's of the substrate on theof stress relaxation. In order to further examine the substrate-effect, therelaxation modulus E ' ( t ) of therelaxation modulus normalized by its initial value, i.e., E '(t ) E '(0) is

laminatecomposites.

All

plotted against time in Fig. 9.8(b); the stress relaxation of the composites

relationsarestepwisebecomes progressively more sluggish along with the elastic substrate thatpenetration ( tf a = 1 ) of conicalbecomes progressively more compliant, namely the elastic modulus ofindentation;

the substrate becomes progressively smaller. In other word, the relaxation
time τ of the laminate composite shifts to the longer side by decreasing
the elastic modulus of the substrate.

It must be noticed that when theelastic modulus of the elastic substrate is larger than the initial modulusof the viscoelastic film, i.e., Es > Ef ( 0 ) (=10 GPa), the relaxation timeof the composite shifts to the smaller side, and vice versa. In Fig. 9.8, wedemonstrate the substrate-effect at the fixed stepwise penetration of

tf a = 1 , although the substrate effect must be more enhanced with thedecrease in tf a -value, or the increase in a tf . To confirm thissubstrate-effect in a quantitative manner, the relaxation time of thelaminate composite is plotted against the normalized contact radius a tf
in Fig. 9.9 for various values of the elastic modulus E 's of the substrate.
In Fig. 9.9, the relaxation time τ of the composite is defined as the timeat which the normalized modulus[ E '(t ) − E '(∞)] [ E '(0) − E '(∞) ]
100

for

the

the

(a) viscoelastic film coated onelastic substrate, and(b) the normalized moduli of thecomposites shown in (a).

relaxes to the value of 1 / e .

In the region of smaller penetrationa tf  1 , due to no substrate-effect, the relaxation time τof thecomposite naturally coincides with that of the viscoelastic coating filmτ f ( τ = τ f = 10 s) itself. However, along with increasing the penetration,i.e., along with increasing a tf , the relaxation time τ progressively
increases due to the enhanced substrate-effect. Furthermore, thissubstrate-effect is more significant for the substrate having a smaller
value of the elastic modulus, as clearly seen in Fig. 9.8. The above
considerations on the substrate effect thus lead to an important conclusionthat we can control the viscoelastic behavior of the laminate composite
by only adjusting the elastic modulus of the substrate without any
modification of the viscoelastic coating film.

Relaxation
time,τ τ(s)
(s)
緩和時間,

40

E 's = 0.1 GPa

0.5

Figure 9.9 Effect of the elastic

1.0
5.0

30

10

modulus E 's of the substrate on therelaxation time of the laminate
composite. The symbols in the figure

50

20

100

10
10-2

10-1

100

10 1

規格化接触半径,a/tf a/tf
Normalized
contact radius,

indicate the elastic modulus E 's of
the respective substrates;○(0.1 GPa)●(0.5 GPa)△(1 GPa)▲(5.0 GPa),□(10 GPa)■(50 GPa),▽(100 GPa)

101

P

9.3 ELASTOPLASTIC COMPOSITES
The elastoplastic Meyer hardness H M of a homogeneous body is
uniquely described in terms of the elastic modulus E ' and the yieldhE'f, Yf

stress Y , the details of which have been well mentioned in Chap. 5. On

E's, Ys
2athe other hand for film/substrate composites, its mechanical field induced
by indentation contact continuously changes from the field controlled by
the coating film having the elastoplastic characteristic ( E 'f 、 Yf ) to thatof the substrate ( E 's 、 Ys ) along with the progressive increase in thePpenetration depth. In other words, as depicted in Fig. 9.10, we canquantitatively make modeling the laminate composite as a homogeneous
elastoplastic half-space with spatially graded mechanical characteristicshE'(tf/a)Y(tf/a)E ' ( tf a ) and Y ( tf a ) along the axis of indentation penetration. [9.5]2aWe introduced in Chap. 5 the additivity principle of the excluded
volume of indentation in order to quantitatively describe the Meyer
hardness of a homogeneous elastoplastic body (see Eq. (5.14)).
Accordingly, by simply replacing E ' with E ' ( tf a ) and Y withY ( tf a ) in Eq. (5.14), we can readily extend this additivity principle to

Figure 9.10 Mechanical equivalence
between the elastoplastic laminate
composite

and

the

the elastoplastic homogeneous body having a spatially gradedelastoplastic

characteristics, resulting in the following analytical expression for the

anisotropic mechanical characteristics

Meyer hardness H M ( tf a ) :m
H M ( tf a )  {ε I E ' ( tf a ) / cY ( tf a )} =cY ( tf a ) 1 + {ε I E ' ( tf a ) / cY ( tf a )}m (9.16)In the extreme of infinite yield stress, Y ( tf a ) ↑ ∞ , namely in theextreme of both of the coating film and the substrate exhibiting none ofplastic yielding, the laminate composite is reduced to the perfectly elastic
laminate that we discussed in Sec. 9.1, and then the Meyer hardness
H M ( tf a ) in Eq. (9.16) is rewritten into the following simple formula(notice the similarity to the elastic Meyer hardness given in Eq. (3.32));(9.17)

where the details of indentation strain ε I have already been given in
Chaps. 3 and 5.

The indentation contact load P of the present

laminate model (i.e., a homogeneous body having spatially graded elastic
modulus E ' ( tf a ) ), therefore, is readily related to its contact area A
through the following relation;
P = ε I E ' ( tf a ) ⋅ A(9.18)

102

withspatiallyof E ' ( tf a ) and Y ( tf a ) (refer toFigs. 9.11 and 9.12)1m

H M ( tf a ) = ε I E ' ( tf a )bodyhomogeneousOn the other hand, we defined the effective elastic modulus E 'eff ( tf a )in Sec. 9.1

that relates the indentation load P to the penetration depth

h as follows;
P = kh E 'eff ( tf a ) h n(9.3)

Since Eq. (9.18) in terms of the contact area A must be equivalent to
Eq. (9.3) in terms of the penetration depth h , we finally relate

homogeneous half-space at a fixed depth of penetration h as defined inSec. 9.1.

Equation (9.19), therefore, means that the substrate-effectboth on the contact area A (or on the contact radius a ) and on theelastic modulus E ' ( tf a ) dictates the effective modulus E 'eff ( tf a )in a synergetic manner. The essential discrepancies of the substrate-effectE ' (t a) E 'the contact radius of an axisymmetric indenter pressed onto aff /a)/E'
規格化弾性率,E'(tff

In Eq. (9.19), ( AH )h(9.19)2 a =   E ' ( tf a ) aH  hand ( aH )h are, respectively, the contact area andNormalized elastic modulus

 A 
E 'eff ( tf a ) =  E ' ( tf a ) AH hNormalized effective modulus
E 'eff ( tf a ) E 'f
規格化弾性率,E'eff (tf /a)/E'f

E 'eff ( tf a ) to E ' ( tf a ) through the following relation;
101

(a)

100

10-1
0
101

10

20

規格化膜厚, tf /a: (E'eff/E' f)/(a/a H)

(b)

2

100

10-1
0

1

2

3

4

5

on E 'eff ( tf a ) and on E ' ( tf a ) are plotted in Fig 9.11 for conical

規格化膜厚,
tf/a
tf a
Normalized
film-thickness,

indentation. As shown in Fig. 9.11(a), the substrate-effect on E 'eff ( tf a )becomes significant when the contact radius a exceeds 0.1tf , i.e.,

Figure 9.11 Substrate-effect on (a)E 'eff ( tf a ) and on (b) E ' ( tf a )a ≥ 0.1tf ( tf a ≤ 10 ), while it eventually becomes significant in the region

forof a ≥ 0.5tf ( tf a ≤ 2 ) for E ' ( tf a ) as demonstrated in Fig. 9.11(b).

symbols

conicalindentation.

indicatetheThe

numerical

In contrast to the elastic extreme that we discussed above, suppose theresults of the finite element analysis.

plastic extreme with infinite elastic modulus E ' ( tf a ) ↑ ∞ in which

The solid lines in (a) are theboth the coating film and the substrate are subjected to fully plastic

numerical solutions E 'eff ( tf a ) of

yielding. For such a fully plastic laminate composite, the Meyer hardness

the Fredholm integral equation (Eq.

H M ( tf a ) is given by substituting E ' ( tf a ) ↑ ∞ into Eq. (9.16),(9.11); refer to Fig. 9.5). The solid

resulting inlines in (b) are the numerical solution
H M ( tf a ) = cY ( tf a )(9.20)of E ' ( tf a ) in Eq. (9.19) combined

Equation (9.20) represents the Meyer hardness of the fully plastic

with the E 'eff ( tf a ) -values given inhomogeneous model having spatially graded yield stress along the(a). The solid lines and the symbols

penetration axis, corresponding to Eq. (5.7) for a fully plasticin (a) and (b) are the results for the

homogeneous body we discussed in Chap. 5.

Unfortunately, however,modulus ratio E 's E 'f of =10, 5, 2,

we have so far no analytical theories for describing Y ( tf a ) as a1, 0.5, 0.2, and 0.1 from the top to thefunction of tf a , not like the case of E ' ( tf a ) . To circumvent thisbottom103

Y ( tf a ) by mapping the FEA-derived Meyer hardness H M ( tf a ) asa function of

tf afor laminate composites having various

combinations of elastoplastic characteristics of the coating film and of thesubstrate; we first determine the Meyer hardness H M ( tf a ) by the useof FEA-based numerical results, and then substitute this H M ( tf a ) value along with the E ' ( tf a ) -value into Eq. (9.16) to calculate
Y ( tf a ) as a function of tf a , some of the examples thus obtained areshown in Fig. 9.12.
Once we get the detailed information on E ' ( tf a ) (Fig. 9.11) and on
Y ( tf a ) (Fig. 9.12) as the functions of tf a in a numerical manner

via FEA, we inversely predict in an analytical manner the Meyer
hardness H M ( tf a ) of the elastoplastic laminate composite using Eq.

Normalized yield stress, Y(tf/a)/Yf

difficulty, we can quantitatively/successfully describe the yield stress

E(=Ef =Es )=10 GPa
Yf =1.0 GPa

100

10-1
0

on elastic substrate). In these figures, the symbols are the FEA-based
numerical results, and the solid/broken lines are the analyticalpredictions via Eq. (9.16). As clearly seen in these figures, the analyticalpredictions of Eq. (9.16) faithfully realize the elastoplastic behaviors oflaminate composites having wide varieties of their elastoplastic
characteristics. This fact implies that we can quantitatively determine inexperiments the elastic modulus ( Ef , Es ) and the yield stress ( Yf , Ys )from the FEM-based numerical analysis combined with the Meyer
hardness

H M ( tf a ) observed on the instrumented indentation

microscope; the details of experimental procedures will be given in Sec.
11.3.

104

2

3

4

5

Figure 9.12 Normalized yield stressof the homogeneous
Y ( tf a ) Yf
anisotropic elastoplastic bodies having
variousYs Yf-values(FEA-based

numerical results)

Meyer hardness, H(a/t f) (GPa)substrate, elastoplastic film coated on ductile substrate, and ductile film

1Normalized film-thickness, tf/a

(9.16). In Figs. 9.13-9.15, shown are the Meyer hardness H M ( tf a ) ofvarious types of laminate composites (elastoplastic film coated on elastic: Ys /Yf =10
:
5
:
2
:
1 (film-only)
:
0.5
:
0.2
:
0.1

10

elastoplastic film coated on elastic substrate
Ef=10 GPa, Yf=1.0 GPa

8Es=
100 GPa

6

50

4

20

2
0
0

5

10
5
21

10

15

Normalized contact radius, a/tf

Figure 9.13 Meyer hardness of anelastoplastic film coated on elastic
substrates ( 2 ≤ Es ≤ 100 GPa). The
symbols are the results of FEA, andthesolid/broken

lines

are

analytical predictions (Eq. (9.16))theelastoplastic film coated on ductile substrate

Meyer hardness, H(a/t f) (GPa)Meyer hardness, H(a/t f) (GPa)10

Ef=50 GPa, Yf=5 GPa; Ys=1 GPa

8
6
Es= 500 GPa

4
2

20

50 100

5

0
0

5

10

15

3

2

ductile film/elastoplastic substrate(Ef=100 GPa, Yf=1 GPa;
Es=10 GPa, Ys=10 GPa)1elastoplastic film/ductile substrate(Ef=10 GPa, Yf=10 GPa;
Es=50 GPa, Ys=1 GPa)

0
0

5

10

Normalized contact radius, a/tf

Normalized contact radius, a/tf

Figure 9.14 Meyer hardness of an elastoplastic film

Figure 9.15 Meyer hardness of an elastoplastic film

coated on ductile substrates. The symbols are thecoated on a ductile substrate, and a ductile film on

results of FEA, and the solid/broken lines are thean elastoplastic substrate. The symbols are theanalytical predictions (Eq. (9.16))

results of FEA, and the solid lines are the analyticalpredictions (Eq. (9.16))

105

NUMERICAL INDENTATIONCONTACT MECHANICS

CHAPTER

10

None of analytical theories have been established for the indentation
contact mechanics except the elastic contact due to the extremelycomplicated mechanical processes induced beneath the tip of indenter, as
repeatedly emphasized in the preceding chapters. When we overview
the long history of the indentation contact mechanics since the end of the
18th century, we readily notice that we will not be able to overcome these
difficulties within the present century.

On the other hand, in order tonumerically solve the problems of deformation and fracture ofmechanically complicated structures, the Finite Element Method (FEM)
has been proposed over the period 1950-1962 by the pioneers of The
Boeing Co. Ltd and The University of Washington. The prominent
development of electronics-based computer science and engineering
since the mid-20th century along with the mathematical contributions of
numerical analyses has enhanced and boosted the application of FEM to
the numerical analyses for solving the problems of structural engineering.
The FEM, therefore, has long been provided powerful tools to the specificengineering fields of the contact mechanics, nonlinear fracture mechanics,
and the hydrodynamic science/engineering, etc., in which we cannot
analytically solve their problems [10.1]. Currently available commercial
FEM software packages include (1) NASTRAN developed in 1968 by
NASA (United States National Aeronautics and Space Administration),(2) ABAQUS that has been intensively applied to the field of numericalindentation contact mechanics since 1980s, and (3) ANSYS that has been
designed and developed for the users of interactive personal computer(PC) through Graphical User Interface (GUI). In this chapter, based on
the Finite Element Analysis (FEA), we will extensively discuss thereliability and the usefulness of the numerical indentation contact
mechanics applied to the axisymmetric indentation contacts problems(spherical and conical indentation).

106[10.1] O.C. Zienkiewicz and R.L. Taylor, andJ.Z. Zhu, The finite element method: its basis andfundamentals, Seventh Ed., Elsevier, 2013;
David V. Hutton, Fundamentals of finite element
analysis, McGraw-Hill, 2004; J.N. Reddy, An
introduction to the finite element method, thirdEd., McGraw-Hill, 2006.

10.1 OUTLINE OF FINITE ELEMENT ANALYSISIN INDENTATION CONTACT MECHANICS
In this section, we will focus on the FEM applied to the indentation
contact mechanics (the mathematical theory, concept of finite elements,
modeling the mechanical structures, boundary and initial conditions, etc.
of FEM are published in the literatures [10.1]). Most of the commercially

Figure 10.1 An example of theavailable FEM-packages equip an interactive user interface, throughaxisymmetric finite element model

which we readily make modeling the geometry of the solid structures wefor a spherical indenter pressed onto

are interested in, defining the element type and the materiala homogeneous half-spacecharacteristics/boundary conditions, meshing to create the finite element
model, solving the problems, and then plotting/viewing the solution.
An example of the axisymmetric finite element model is demonstrated
in Fig. 10.1 of a spherical indentation on an infinite half-space. We needto make finer meshing beneath the contact area that is requisite for
improving the reliability/accuracy of the numerical solution due to thevery steep spatial gradient of stresses/strains. Figure 10.2 shows anexample of FEA for the surface-profile (sink-in profile) and the
distribution of indentation-induced in-surface displacements along theaxis of penetration at h = 3 μm (Vickers/ Berkovich-equivalent cone) fora perfectly elastic body. As clearly seen in Fig. 10.2, one of theadvantages of FEA is the visualization of in-surface mechanicalinformation (stress-strain details) that is very essential for us to get theintuitive as well as deep insight into the physics of indentation contact(some FEA-based numerical results have already been given in Figs. 4.4,
5.3, 5.5, 5.7, 5.11, 8.4-8.9, and 9.11-9.15).

10.2 ELASTIC ANALYSIS
As well described in Chap. 3, there exist the analytical solutions in aclosed form for the axisymmetric indentation contact onto a
homogeneous elastic half-space. We will scrutinize in this section thereliability and the accuracy of the numerical FEA-results for the
spherical/conical indentation contact by comparing them with theanalytical solutions. Furthermore, we will make the FEA-based
numerical approach to the Poisson’s effect on the indentation contact
behavior since the analytical theories have been implicitly based on anassumption of incompressible elastic body, i.e. ν = 0.5 .

107Figure 10.2 The surface-profile andthein-surface

distributionofdeformation at the penetration depth

h = 3 μm for the Vickers/Berkovich
equivalent cone pressed onto a
perfectly elastic bodyAs discussed in Chap. 3, the indentation contact load P is related tothe induced penetration depth h through the following relation;indentation load, P (mN)

4
P = E ' Rh3 2
3

3000(3.18)

for spherical indentation (the Hertzian theory), andP=

2 cot βπE ' h2(3.25)

ratio of most of engineering materials is ν ≈ 0.3 , suggesting thedifficulty in applying the above analytical formulae to the indentation
contact behavior of engineering materials. Equations (3.18) and (3.25)
are described in terms of the plane-strain elastic modulusE'( = E 1 − ν 2 ) that includes the Poisson’s ratio, suggesting that thetheoretical framework of the above equations seems to include the
Poisson’s effect.

However, it must be noticed the fact that these theories

ν=0.5

E=100 GPa; R=10μm
2000

;
;

FEM numerical results
Hertziananalytical solutions
ν=0.3

1000

0
0

1

2

3

penetration depth, h (μm)

for conical indentation (the Sneddon’s theory). These analytical formulae
implicitly assume the Poisson’s ratio of ν = 0.5 , though the Poisson’s

spherical indenter

Figure 10.3 The P - hloading

curves of spherical indentation (thetip-radius R = 10 μm). The solid line( ν = 0.5 ) and the broken line( ν = 0.3 ) indicate the Hertziananalytical solutions. The symbols ○( ν = 0.5 ) and ● ( ν = 0.3 )) are the
FEM-numerical results

make a priori assumption of incompressible elastic body. On the other
hand, we can introduce any value of the Poisson’s ratio ν to the FEAbased model material, leading to the resultant numerical solution that is
applicable to any of engineering materials. The FEA-results with the
Poisson’s effect on the P - h loading relations (the plots with thesymbols of ○ and ●) are shown in Figs. 10.3 and 10.4, where the solid( = E 1 − ν ) for ν = 0.5 and ν = 0.3 . As clearly seen in these figures,2

the analytical solutions for ν = 0.5 (the solid lines) well realize the
FEA-numerical results, while the analytical solutions for ν = 0.3 (thebroken lines) always underestimate the FEA-numerical results due to thea priori assumption of the incompressible elastic body included in the
Hertz/Sneddon theories.
perfectly elastic body is described by

=

2000

E=100 GPa, β=19.7°,
FEM-numerical results,
Sneddon's analytical solutions

ν=0.3

1000

0
0

ν=0.5

1

2

3

penetration depth, h(μm)

Figure 10.4 The P - h loading

The profile of the free-surface outside the contact region ( r > a ) of au z ( ρ , 0) =indentation load, P(mN)lines and the broken lines indicate the analytical results with E 'conical indenter

curves of conical indentation (theinclined face-angle β = 19.7 o ). The
solid line ( ν = 0.5 ) and the brokena2 ( 2 − ρ 2 ) sin −1 ρ1 + ρ 2 − 1 
π R 


(3.20)h 
12 − ρ 2 ) sin −1 + ρ 2 − 1  ; ρ ≥ 1
2 (
πρ 
ρ


line (ν = 0.3 ) indicate the Sneddon’sanalytical solutions. The symbols ○(ν = 0.5 ) and ● (ν = 0.3 ) are thethat is indented by a sphere with the radius R , andFEM-numerical results1082h  −1 1sin− ρ + ρ 2 − 1 ; ρ ≥ 1

π 
ρ


0

(3.27)

for conical indentation (refer to Chap.3). The comparison of theseanalytical solutions with the FEA-based numerical results (ν = 0.5 ) isshown in Fig. 10.5; the excellent agreement between the analytical andNormalized surface
displacement, uz(r)/hu z ( ρ , 0) =0.1

conical indentation

0.2

spherical indentation

0.3
0.4

: FEA numerical results: analytical solutions

0.5
0.6
0.7
1

the numerical results ensures the accuracy/reliability of the FEA.

2

3

4

5

Normalized contact radius, r/a

Figure 10.5 The elastic contact

10.3 ELASTOPLASTIC ANALYSIS

profiles (ν = 0.5 ) of the free-surface(1) P - h Hysteresis Curveoutside the contact zone of sphericalAs emphasized in Chap.5, we have no analytical solutions for theor conical indentation contact: theelastoplastic indentation contact mechanics, excepting the unified theory

symbols (●, ○）are the FEA-based

based on the additivity principle of excluded volume of indentation (seenumerical

Sec. 5.1(2)). The FEM-based numerical analysis, therefore, exclusively

solid/broken lines are the analyticalplays an essential role in the elastoplastic indentation contact mechanics.

solutions (Eqs. (3.20) and (3.27))

Due to the irreversible plastic flow, we observe a hysteresis in the

results

andthe(refer to Fig. 3.6)

indentation loading-unloading P - h curve (see Figs. 5.5, 5.8 - 5.10, and5.12). This fact implies that the observed P - h hysteresis curve
combined with FEA makes it possible for us to quantitatively determine
the elastoplastic characteristics, i.e., the elastic modulus E ' and theyield stress Yof the material tested.

Figure 10.6 depicts theloading/unloading P - h2 linear plot along with the contact geometriesof an elastoplastic half-space for pyramid/cone indentation. Asemphasized in the following considerations, the contact depth hc isPessential for calculating the contact area A , and the residual depth hrplays an important role in determining the plastic flow; both are the

key

k1

parameters for determining the elastoplastic characteristics. The
geometrical similarity of the pyramid indenter (Vickers/Berkovich
indenter) leads to the penetration-depth-independent indentation strain

0

k2

hr 2

h2

P

ε I (= tan β 2 ), depending only on the inclined face-angle β (refer tohs

Chaps. 3 and 5, and Tab. 3.1). This fact of penetration-depth- independent

β

β

hr
hc h

βr

he

strain is very essential for quantitatively determining the elastoplastic
the loading and the unloading P − h2

relations shown in Fig. 5.5 for

2ar

2ac

characteristics through P - h hysteresis curve. The linear plots both for

Figure

10.6

Loading/unloading

the wide range of plastic index ( 0.05 ≤ PI ≤ 20 ) are actually resulted

P − h2 linear plot and the pyramid/from the penetration-depth-independent ε I . In Fig. 10.6, we define thecone indentation contact geometries

contact depth hc at the applied load P as well as the depth hr and theof an elastoplastic body109

inclined face-angle βr of residual impression after unload. Since there
pyramid/cone indentation, we can describe the loading/unloading process
in terms of the loading coefficient k1 and the unloading coefficient k2 ;P = k1h 2(10.1)
2

2

P = k 2 ( h − hr )100

Normalized loading
coefficient. k1/Y

exist the linear P - h2 relations both in loading/unloading processes for

80
60
40
20

(10.2)

010-2Furthermore, due to the fact that the loading line crosses the unloading

10-1

100

101

Plastic index, PI(=εIE'/cY)

102

line at the maximum penetration depth h , there is the interrelation

Figure 10.7 Normalized loading

between k1 and k2 ;coefficient k1 Y vs. plastic index

k1 = k2 (1 − ξ r 2 )(10.3)PI (FEA-based numerical results for
the Vickers/Berkovich equivalent

in which ξr is the normalized residual depth hr h . Accordingly, thecone)loading coefficient k1 is related to the unloading coefficient k2 via ξr .
loading coefficient k1 and the normalized residual depth ξr are
intimately related to the plastic index PI, while the unloading coefficient

k2 is closely related to the elastic modulus E ' . The FEA-based

Normalized unloading
coefficient, k2/E'

On the other hand, as given in the following considerations, both the

3.5

3

2.5

2
10-2

numerical results of the normalized elastoplastic characteristics, k1 Y ,k2 E ' , and hr h are, respectively, plotted against the plastic index PI

10-1

100

101

Plastic index, PI(=εIE'/cY)

102(= ε I E ' cY ) in Figs. 10.7 – 10.9. These figures show that the normalized

Figure 10.8 Normalized unloadingparameters of k1 Y , k2 E ' , and hr h are quantitatively related in acoefficient k2 E ' vs. plastic index

unique manner to the plastic index PI ( = ε I E ' cY ) in the wide ranges

PI (FEA-based numerical results forof 0.5 GPa ≤ E ' ≤ 500 GPa and 0.5 GPa ≤ Y ≤ 100 GPa.

the Vickers/Berkovich equivalentconsiderations imply that we can quantitatively determine the elastic
modulus E ' and the yield stress Y through Figs. 10.7 – 10.9

after

we determine in experiment the loading coefficient k1 , unloading
coefficient k2 , and/or the residual depth of impression hr by the use of
the conventional instrumented indentation apparatus; the detailed
experimental procedures are given as follows;cone)
Normalized residual depth, h r/h

The preceding1cone indentation

0.8
0.6
0.4
0.2
0

10-1

100

101

102

Plastic index, εIE'/cY

Figure 10.9 Normalized residual
depth hr h ( ≡ ξ r ) vs. plastic index
PI (FEA-based numerical results for
the Vickers/Berkovich equivalent
cone)
110

(i)

Apply the experimentally observed hr h -value to the ordinate ofFig. 10.9 to estimate the plastic index PI ( = ε I E ' cY ) from thecorresponding abscissa, followed by the application of this PIvalue to the abscissa of Fig. 10.7 in order to determine the k1 Y value from the corresponding ordinate. This

k1 Y -value

combined with the experimentally observed loading coefficient k1
and the PI-value are then utilized to determine the elastoplastic
characteristics ofthe elastic modulus E 'and the yield stressY , or(ii) Assume the initial value Y0 of the yield stress, and then calculate

k1 Y0 using the experimentally observed k1 . Apply this k1 Y0 value to the ordinate of Fig. 10.7 to estimate the corresponding
plastic index

ε I E ' cY0 from the abscissa, leading to the initial

estimate of the elastic modulus E0 ' . Substitute these initial valuesof Y0 and E0 ' into the abscissa (the plastic index) of Fig. 10.8
to make the first-order approximation of the elastic modulus E1

1.4from the ordinate combined with the experimentally observed k2

1.2-value. And then apply the ε I E1 ' cY0 -value to the abscissa of Fig.

1

stress Y1 from the ordinate. Repeat these procedures until theconvergences of En and Yn are obtained, finally resulting in theelastoplastic characteristics of E ' and Y .
Among these elastoplastic parameters, the most essential one is thecontact depth hc at the indentation load P (see Fig. 10.6), though it
cannot be determined in experiment on the conventional instrumented: FEA results: Oliver-Pharr approximation

Normalized contact depth, h c/h10.7 for determining the first-order approximation of the yieldconical indenter

0.8

2/π

0.6

1.4 spherical indenter: FEA results
1.2: Field-Swain approximation
1
0.8
0.6
0.4
0

1/2
0.2

0.4

0.6

0.8

1

Normalized residual depth, hr/hindentation apparatus. On the other hand, as shown in Fig. 5.6 in Chap.
5, using the FEA-based numerical results, the normalized contact depth

Figure 10.10 Correlation between thehc h is intimately related to the plastic index ε I E ' cY . As a matternormalized contact depth hc h andof fat, combining Fig. 5.6 and Fig. 10.9 results in Fig. 10.10. By the use

the normalized residual depth hr h .

of Fig. 10.10, therefore, we can readily estimate the contact depth hc

The broken lines are, respectively, thefrom the experimentally observable residual depth hr . Once weOliver-Pharr (conical indenter) anddetermine the contact depth hc , we can successfully calculate the contact

the Field-Swain (spherical indenter)radius ac , and then the contact area A that is requisite for determining

approximations. The symbols (●, ▲)the elastoplastic characteristics, the details of which have already been

indicate the FEA-based numericalgiven in Chaps. 3 and 5.

results
111

The considerations on the conical indentation in Fig. 10.10, by way of
example, make clear that (1) the normalized contact depth hc hconverges to the Sneddon’s solution of 2 π
extreme ofin the perfectly elastic

hr h → 0 , (2) the normalized contact depth

hc hmonotonically increases with the increase in the residual depth hr hdue to the enhanced plastic flow-out to the free-surface outside thecontact zone, implying the depression of the sink-in profile of the surface,(3) none of sink-in nor pile-up of the free-surface, i.e., hc h = 1 , isobserved at hr h ≈ 0.85 , (4) a significant pile-up of the free-surface is
induced in the perfectly plastic extreme of hr h → 1 , resulting in theincrease of the contact depth up to hc h ≈ 1.1 . The FEA-visualized
contact profiles of the free-surface have already been given in Figs. 5.6
and 5.7 in relation to the plastic index PI ( = ε I E ' cY ) . Figure 10.10 alsosuggests that the elastoplastic contact behavior of spherical indentation issimilar to that of conical indentation; the normalized contact depth hc hconverges to the Hertzian value of 1/2 in the elastic extreme of hr h → 0 ,and goes up to about 1.4 in the plastic extreme of hr h → 1 .
(2) The Oliver-Pharr/Field-Swain Approximation [10.2-10.4]
In the preceding section, using the FEA-based numerical results, we
correlate the normalized residual depth hr h to the normalized contact
depth hc h in a quantitative manner. In this section, we will discuss the
Oliver-Pharr approximation for conical indentation and the Field-Swain
approximation for spherical indentation; both of the approximations
stand on the same elastic assumption prior to analytically corelating

hr h to hc h . In elastoplastic indentation contact, however, this elastic
assumption is very critical in estimating the contact penetration depth hc ,and then for determining the elastoplastic characteristics in theconventional instrumented indentation apparatus. There exist thefollowing geometrical relations in the elastoplastic indentation contact(see the details in Fig. 10.6);h = hs + hc = hr + he(10.4)The Oliver-Pharr/Field-Swain approximation assumes that the sink-in
depth hs and the depth of elastic recovery he of the elastoplastic bodywith the elastic modulus E ' are the same as those of the perfectly elastic
body with the same elastic modulus E ' when both of the bodies are112[10.2] W.C. Oliver, G.M. Pharr, J. Mater. Res.,7, 1567 (1992)[10.3] J.S. Field, M.V. Swain, J. Mater. Res., 8,297 (1993)[10.4] M. Sakai, “Principle and Application ofIndentation”, in Micro and Nano
Mechanical Testing of Materials andDevices, Edited by F. Yang, J.C.M. Li,
1-47, Springer (2008)

indented at the same load of P .

The elastic indentation contact

mechanics (refer to Eqs. (3.21) and (3.28)) leads to the following
relations between he and hs ;hs 1=
(spherical indentation)
he 2(10.5)hs2=1−(conical indentation)πhe(10.6)

Substituting these elastic relations into the elastoplastic relations of Eq.
(10.4), we finally have the following approximations that relate hr h to

hc h ;
Field-Swain approximation:hc 1  hr 
= 1 +  (spherical indentation)h 2 
h(10.7)

Oliver-Pharr approximation:(10.8)

YR

strainhardeningYnonhardeningσhc 2   π  hr 
= 1 − 1 −   (conical indentation)h π  2 hThe broken lines in Fig. 10.10 represent the Oliver-Pharr/Field-Swain
approximations. Since both of Eqs. (10.7) and (10.8) assume the elastic
relations of Eqs. (10.5) and (10.6), these approximations converge to

,
s
s
e
r
s
l
a
i
x
a
i
n
U

E

hc h → 1 in the plastic extreme of hr h → 1 , being impossible toE0describe the plastic-flow-induced surface pile-up ( hc h > 1.0 ), as well

( εp ) RεR

Uniaxial strain, εrecognized in Fig. 10.10.

(3) Strain Hardening and the Representative Stress/Strain

Figure 10.11

The discussions in Chap.5 and the preceding considerations have only

Stress σ vs. strain ε curve (S-S

been applicable to elastic-perfectly-plastic solids having a constant yield

curve) of elastoplastic material. The

stress Y in the region exceeding the elastic limit at which the plasticity

S-S

index PI ( ≡ ε I E ' cY ) is about 0.2. However, in most of engineering

materialmaterials, such as mild steel, aluminum, copper, etc., ever-increasing

exceeding the elastic limit. The

actual stress is required for continued deformation beyond the yield point,

representative flow stress YR

isi.e., strain hardening (work hardening), as shown in Fig. 10.11. The

definedtheexperimental results of D. Tabor as well as the FEA-based numericalrepresentative plastic strain

studies have shown that the conclusions obtained in the indentation

or at the representative total strain

contact mechanics of non-hardening elastoplastic solids may be applied

εR

113

curve

of

strainincreasesasthehardening

with

value

strain

at

(ε )

p R

with a good approximation to strain hardening solids if Y is simply
replaced with the representative flow stress YR . This fact indicates thatthe preceding elastoplastic analyses in terms of PI that we made, by
way of examples, in Figs. 10.7–10.9 still survive by simply replacing PI
with( PI )R ( = ε I E ' cYR ) [4.1, 10.5, 10.6]. The representative flow

stress YR is defined by the representative plastic strain

(ε )

p R

or therepresentative total strain ε R as sown in Fig. 10.11. The FEM-based
numerical result, on the other hand, indicates that

(ε )

p R

is related tothe indentation strain ε I as follows [10.6];(ε ) = 0.44ε
p R(10.9)

I

Accordingly, substituting the indentation strain ε I (see Eqs. (3.30) and(3.31)) into Eq. (10.9), the representative plastic strain of conical indenterwith the inclined face-angel β is written by(ε ) = 0.22 tan β ,(10.10)p R

and of spherical indenter with the radius R is given by(ε ) = 0.19 Ra ,(10.11)p R

We can, therefore, readily determine the respective flow stresses YR in
Fig. 10.11 by the use of these ( ε p ) -values.
R(4) The Effects of Indenter’s Tip Geometry
and of Contact Friction on the Constraint Factor c
The role of contact friction at the interface between the indenter andthe material tested is not well understood even for the perfectly elastic
contact mechanics. Furthermore, the indenter-tip geometry and thecontact friction will have a significant effect on the sink-in/pile-up of anelastoplastic body (see Figs. 5.6 and 5.7), as well as on the constraint
factor c (refer to Chap. 5 and Eq. (5.7)), since the constraint factor
dictates the in-surface plastic flow out to the free surface outside thecontact region. The contact friction may resist this plastic flow-out along
the indenter’s face, leading to a larger c -value. Furthermore, when the
inclined face-angle of conical/pyramidal indenter becomes larger, i.e., thetip of indenter becomes sharper, it will be expected that the plastic flow
114

[10.5] D. Tabor, “Hardness of Metals”,
Clarendon (1951)[10.6] M. Sakai, T. Akatsu, S. Numata,K. Matsuda, J. Mater. Res., 18[9],2087 (2003)

c -value.
As discussed in Chap.5, the experimental results indicate that theconstraint factor of ductile metals falls in the range of 2.5 ≤ c ≤ 3.5
depending on the contact friction and the indenter’s geometry. On theConstraint factor, cout to the free surface will be enhanced, leading to progressively smaller

other hand, the FEM-based numerical analyses enable us to determine in

3

2.8

2.6 -310

a quantitative manner the constraint factor of an arbitrary tip-geometry ofindenter in a wide range of its contact friction.

3.2

10-2

10-1

Contact friction, μ100

Figure 10.12 Effect of contact

The FEA-based numerical results of the constraint factor c are

friction on the constraint factor for

plotted against the contact friction μ in Fig. 10.12 (for Vickers/

Vickers/Berkovich equivalent cone

Berkovich-equivalent cone), and against the inclined face angle β (forindentation (FEA-based numericalconical indenters) in Fig. 10.13. The constraint factor of the equivalent

results)

cone significantly increases from the initial value of c ≈ 2.65 to theplateau of c ≈ 3.2 with the increase in the contact friction ( 0 ≤ μ ≤ 0.5 ),materials against diamond falls in the range of 0.1 ≤ μ ≤ 0.5 , the FEAbased preceding conclusions quite well satisfy the experimentallyobserved c -values of 3.0 ≤ c ≤ 3.2 for Vickers diamond indentation([10.5], [10.6]).

3.5

Constraint factor, cas shown in Fig. 10.12. Since the contact friction of most of engineering

μ=0.2
3

μ=0.0

2.5

As well seen in Fig. 10.13, the constraint factor cdecreases with the increase in the inclined face-angle β . This fact
implies the enhanced in-surface plastic flow out to the free surface when
the conical indenter becomes sharper, i.e., the inclined face-angle β
becomes larger.

10

20

30

Inclined face-angle, β (degree)

Figure 10.13 Effect of the inclined
face-angle of conical indentation on
the constraint factor (FEA-based
numerical

10.4 VISCOELASTIC ANALYSIS
In contrast to the preceding elastoplastic analysis, we can readilyextend the elastic contact mechanics to the viscoelastic contact
mechanics by the use of “the elastic-to-viscoelastic corresponding
principle” (refer to Chaps. 6 and 7). However, the time-dependent
Poisson’s effect (see Figs. 6.7 - 6.10) associated with indentation contact
makes the analysis rather difficult. As a matter of fact, when we indent a
Vickers/Berkovich equivalent cone onto a viscoelastic body in a stepwisemanner to a constant penetration depth of h0 , due to the increase in the
Poisson’s ratio associated with stress relaxation, we observe theprogressively increasing contact area induced by creeping-up of the freesurface along the face of indenter. An example of the FEA-based

115

results;

thecontact

frictions of μ = 0.0 and μ = 0.2 )ν g = 0.1 ). This result indicates that we cannot determine the stressrelaxation modulus E 'relax (t ) from the observed load relaxation P(t )using Eq. (7.6) because of the time-dependent contact area A ( t ) . We,
therefore, need to use the following integral equation2

Gg = 36.4 GPa, relaxation time τ = 200 s, and the glassy Poisson’s ratio

A(t) (μm )relaxation shear modulus G(t ) = Gg exp ( −t τ) with the glassy modulus

Maxwell liquid ( νg=0.1)
5
4 penetration depth
3210cntact area
115
110
105

P(t) (mN)contact area under a stepwise penetration onto a Maxwell liquid (theh(t) (μm)

numerical result is shown in Fig. 10.14 for the creeping-up of indentation

200 load relaxation10000tan β  tdA(t ')P(t ) =dt '
 E 'relax (t − t ')
2 0
dt '

100

(7.3)

200

300

400

500

Time, t (s)Figure 10.14

Load relaxation andin order to determine the relaxation modulus E 'relax (t ) . We have to,contact-area creep under a stepwise

therefore, measure in experiment the time-dependent contact area A ( t )penetration of a conical indenter

along with the indentation load relaxation P(t ) , though it is impossible(FEA-based numerical results of afor us to measure the time-dependent contact areaA ( t ) in theMaxwell liquid)

conventional instrumented indentation apparatus (the details will begiven in Sec. 11.3 of the instrumented indentation microscope thatenables to measure A ( t ) as a function of time). In the FEM-based
numerical analyses, on the other hand, it is easy for us to numericallydetermine the plane-strain relaxation modulus E 'relax (t ) , since we caneasily determine the time-dependent contact area A ( t ) in a quantitative
manner.
Comparison between the FEA-based numerical result and theanalytical solution of a viscoelastic liquid having a time-dependent

Poisson's ratio, νg=0.3

Gg = 3.85 GPa, Ge = 0 GPa, τ 0 = 200 s, and ν g = 0.3 ,is shown in Fig. 10.15, where the relaxation modulus E 'relax (t ) obtainedby substituting the FEA-derived P(t ) and A ( t ) into Eq. (7.3) isplotted along with the analytical solution (Eq. (6.29)), demonstrating that
the FEA-based numerical result faithfully realizes the analytical solution.

Relaxation modulus
E'relax(t) (GPa)Poisson’s effect with shingle relaxation time (Eq. (6.26));

10

FEA-based result: analytical solution

5

0
0

100

200

300

400

500

Time, t(s)This excellent agreement between the analytical solution and the FEA-

Figure 10.15 Relaxation modulus of abased numerical result indicates the usefulness of the FEA–based

viscoelastic liquid; the numerical result

numerical indentation contact mechanics in the studies of engineering(○) via Eq. (7.3) with FEA-derived

materials having rather complicated mechanical characteristics such as

P(t ) - A ( t ) relation, and the analyticalthe viscoelastic laminate composite discussed in the following section.

solution (Eq. (6.29); the solid line)11610.5 APPLICATION TO LAMINATE COMPOSITESAs mentioned in Secs. 9.1 and 9.2,
there exist the analytical solutions for

5μm

elastoplastic film

elastoplastic film

the indentation contact mechanics ofcoating/substrate composites in elastic

1μm

viscoelastic
bonding layer

as well as viscoelastic regimes, thoughelastic substrate

we inevitably face the extremely

elastic substratecomplicated mathematical procedures
such as solving the Fredholm integral
equation (Eq. (9.11)) and/or integrating

Bilaminar composite

the Boussinesq Green function (Eqs.
(9.7) and (9.8)).

Even in these

Trilaminar composite

Figure 10.16 Bilaminar and trilaminar composites used in the FEA

analytical approaches, however, wecannot control the mechanical consistency at the interface between thecoating film and the substrate. To circumvent this difficulty, the FEA-

quantitative manner, as well as deal with the indentation contact
mechanics of multi-laminar composites.
As an example, let us make the FEA-based numerical analysis of atrilaminar composite depicted in Fig. 10.16; an elastoplastic coating film
( Ef = 100 GPa, Yf = 10 GPa, ν f = 0.3 ) with the thickness of 5 μm is
bonded to an elastic substrate ( Es = 100 GPa, ν s = 0.3 ) through aninterfacial

viscoelastic

bondinglayer

Erelax ( t ) = Eg exp ( −t τ ) , Eg =10GPa, τ =200s)(Maxwellliquid;having the thicknessof 1 μm. In order to scrutinize the role of this viscoelastic interface in
the indentation contact behavior, the bilaminar composite without
viscoelastic bonding layer is also examined for comparison (see Fig.
10.16).
Figure 10.17 shows the FEA-based numerical results of the load P ( t )plotted against time t for the bilaminar and the trilaminar composites
indented by a Vickers-equivalent cone under a constant-rate ofpenetration to the depth of h = 3.0 μm in 400s, and then the depth is kept
to t = 1000 s (see the top view of Fig. 10.17).

The indentation load

relaxation is observed for the trilaminar composite due to the presence of

117

Indentation load, P(t) (N)we can take into account the mechanical consistency at the interface in a

Penetration depth, h(t) (μm)

based numerical indentation contact mechanics plays an essential role;

3

2

1

0
2 bilaminar composite

1
trilaminar composite
with viscoelastic interface
0
0

200

400

600

800

1000

Time, t (s)Figure 10.17 Time-dependent loads
for the bilaminar and the trilaminar
composites indented by a Vickersequivalent cone under a constant-rateof penetration to the depth of
h = 3.0 μm in 400s, and then thedepth is kept to t = 1000 s (FEAbased numerical results)

is very sluggish while the relaxation time of the viscoelastic interface isτ =200s, being resulted from the geometrical constraints of the timeindependent elastoplastic coating film as well as of the elastic substrate.
In Sec. 9.2, we emphasized in the indentation contact mechanics ofbilaminar composites that we can control and design the viscoelastic

1

Normalized load relaxation
P(t)/P(400)viscoelastic interface. It must be noticed the fact that this load relaxation

0.95

Es=10 GPa

0.9

100 GPa

0.85

500 GPa

0.8
400

500

600

700

800

900

1000

Time, t(s)behavior of the laminate composite by only adjusting the elastic modulus

Figure 10.18 Effect of the elasticof the elastic substrate without any modification of the viscoelastic

modulus Es of the substrate on thecoating film (refer to Fig. 9.9). We can also control the viscoelastic

relaxation behavior of the trilaminar

behavior of the trilaminar composites by only adjusting the elastic

composite

modulus of the substrate. In Fig. 10.18 plotted are the FEA-based

bonding layer (the right-hand view innumerical results of the trilaminar composites shown in Fig. 10.16 with

Fig 10.16). A Vickers-equivalent

the substrate having its elastic modulus ranging 10GPa ≤ Es ≤ 500GPa .

cone

In Fig. 10.18, the normalized load relaxation P ( t ) P ( 400 ) is plottedagainst the time t (≥ 400s) , where the penetration depth is kept constant

with

under

a

a

viscoelastic

constant-rateofpenetration to the depth of h = 3.0

μm in 400s, and then the depth is kept

at h = 3.0 μm (see the top view of Fig. 10.17). As shown in Fig. 10.18,

constant to t = 1000 s (see the top

the load relaxation becomes progressively significant, i.e., the relaxation

view of Fig. 10.17).

time of the laminate shifts to the shorter side, with the increase in theelastic modulus Es of the substrate.

The normalized load relaxation
P ( t ) P ( 400 ) is plotted against thetime t ( ≥ 400s)(FEA-based numerical results)

118

EXPERIMENTAL APPARATUS ANDDATA ANALYSISCHAPTER11

11.1 TIP-GEOMETRY OF
THE CONVENTIONAL INDENTERS
Figure 11.1 depicts the tip-geometries of the conventional Vickers andthe Berkovich indenters. In order to add the geometrical consistency to
the Brinell hardness test, the inclined face-angle β of the Vickers
tetrahedral pyramid is 22.0 , and thus the apex angle, i.e., the diagonal
face-to-face angle 2θ is designed to be 136 , since this is the angle
subtended by the tangent of a Brinell sphere when the ratio of the contact
diameter 2a to the diameter 2R of Brinell sphere is 0.375, being themost recommended ratio in the Brinell hardness test (refer to Fig. 5.4 in(a) Vickers tetrahedral indenter
2ψ

Chap. 5). The diagonal edge-to-edge angle 2ψ is, therefore, 148.1 . On

2athe other hand, the tip-geometry of the Berkovich trihedral pyramidindenter is designed in order that its excluded volume of the material
beneath the indenter at a penetration depth of h is the same as that of
the Vickers pyramid, resulting in β = 24.7 and ψ = 77.1 . The

β

h
2θ

projected contact area A of cone/pyramid indenter is, due to its
A = gh 2

geometrical similarity, described byprojected areag

( g = 4 cot 2 βusing the index ofβψ
2θ
Vickers 22.0° 74.05° 136.0°

for the Vickers indenter, andg = 3 3 cot 2 β for the Berkovich indenter). Accordingly, the g -value(b) Berkovich trihedral indenterof the Vickers coincides with that of the Berkovich, resulting in g = 24.5 ,and then giving the same projected contact area A = gh 2 , as well as the()

same excluded volume of V ≡ gh3 3 = 8.17h3 at a given penetrationadepth h .
Upon machining a pyramidal tip on a tiny diamond single crystal, it is
impossible to make an ideally sharp-tip, i.e., an atomic tip, always
resulting in a rounded or a truncated tip. Due to the trihedral tip-geometry,machining a sharp tip on the Berkovich indenter is rather easier than thetetrahedral Vickers indenter. This is the reason why the Berkovich
indenter is rather preferable than the Vickers indenter in nanoindentation
testing. Commercially available Berkovich indenter, in general, has the
119

β

ψ

h
β

Berkovich 24.7°

ψ
77.1°

Figure 11.1 Tip-geometries of theconventional pyramid indenters:(a) Vickers indenter,(b) Berkovich indenterrounded tip-radius of about 100 nm. This rounded tip may yield crucialdifficulties in nanoindentation testing when the penetration depth isseveral tenth of nanometers or less.

The effect of a rounded tip on the
Rt

indentation contact behavior will be discussed in what follows for thehcone

Vickers/Berkovich equivalent conical indenter ( β = 19.7 ) via the FEA-

hsphereβbased numerical results.
The tip-geometry of Vickers/Berkovich equivalent cone with arounded tip (radius of curvature, Rt ) is shown in Fig. 11.2. Suppose theapex radius of Rt = 100 nm, by way of example, then the lost-tip-depth
is to be Δht ( = Rt (sin β tan β + cos β − 1) ) = 6.22 nm, the spherical depth
is hsphere ( = Rt (1 − cos β ) ) = 5.85 nm, and the region of conical depth isΔht

Figure 11.2 Tip-geometry of theVickers/Berkovich equivalent cone
( β = 19.7 ) having a rounded tip
with the radius of curvature, Rt

given by hcone ≥ 5.85 nm. Once we press this indenter onto a flat halfspace, we will observe the spherical contact response in the initial stageof penetration followed by the conical indentation contact.
Figure 11.3 shows the FEA-based numerical result of the P - h plot
for this rounded-tip cone indented on a perfectly elastic body with theclearly seen in Fig. 11.3, it must be noticed in the region of h ≤ 150 nm
that there exists a finite deviation from the conical indentation in its P -hrelation.

This numerical study suggests that we have to pay anattention in nanoindentation tests upon using the conventional
Vickers/Berkovich pyramid. In other word, we have to make an

Indentation load, P(mN)

elastic modulus E = 100 GPa and the Poisson’s ratio ν = 0.5 . As

elastic contact(E=100 GPa, ν=0.5)
conical indenter (Eq. (3.25))

101

100 spherical indenter(Eq. (3.18))
10-1

10-2

101

102

Penetration depth, h (nm)experimental examination/correction of the tip-radius in a quantitative

Figure 11.3 P - hmanner prior to nanoindentation testing.

However, on the conventional

perfectly

elastic

instrumented indentation apparatuses unlike the indentation microscope

modulusE = 100 GPa, Poisson’s(refer to the subsequent Sec. 11.3 for the details), we can neither measureratio

the contact depth hc nor determine the contact area Ac in experiment;

Vickers/Berkovich equivalent cone

it is only possible for us to determine the indentation load P and thewith a rounded tip (radius; Rt = 100

associated penetration depth h . This fact implies that, even after wenm). The broken lines indicate thecompleted the quantitative correction for the tip-radius, we have to makeanalytical relations, and the symbol

approximations/assumptions to estimate hc and/or Ac , before we(○) are the FEA-based numericalexperimentally determine the material characteristics such as the elastic

results

modulus and the yield stress of the material tested.

120

body

plot of a(elastic

ν = 0.5 ) indented by a

11.2 INSTRUMENTED INDENTATION TEST SYSTEM
The studies on instrumented indentation testing have been started in
1980s in order to make a quantitative measurement/determination of thematerial characteristics in micro/nano region [11.1-11.3].

The

schematic of instrumented micro/nano indentation test system is given in
Fig. 11.4. A piezo or an electromagnetic actuator is widely utilized as theload/displacement driving unit. Load sensor with the precision of about±0.5μN and displacement sensor with the precision of ±0.1nm arecommercially available. As shown in Fig. 11.4, depending on the spatialconfiguration between the displacement sensor and the test specimen, thetest system is grouped into (a) the apparatus-frame-referenced and (b) the
specimen-holder-referenced test systems [9.4, 11.4-11.6]. In the former
test system (a), since the displacement sensor can be fixed away from theload/displacement
actuator

specimen tested, we can easily make indentation testing at elevatedload sensor

temperatures, in a hostile environment, etc. by setting a specimen

load trainchamber, while the elastic deformations of the frame, load-train, and of
displacement sensor

the specimen-holder are inevitably included as the frame compliance inindenter
specimenthe penetration depth observed, leading to the fatal defect of this test
system. The frame-compliance-associated problem becomes moresignificant in micro/nano indentation testing. Accordingly, we have toa priori determine the frame compliance of the test system in aquantitative manner (refer to the details in the subsequent section). In thelatter test system (b), on the other hand, due to detecting the displacementof the indenter-tip relative to the specimen-surface, the undesirable effectof frame compliance is insignificant. Accordingly, we can determine themechanical characteristics such as the Meyer hardness, elastic modulus,
and the yield stress in a quantitative manner without any corrections of
the frame compliance, while it is rather difficult for us to conductindentation testing at elevated temperatures that is contrary to the former
test system (a). In both of the test systems, we must pay attention totightly fixing the test specimen on the holder in order to eliminate
undesirable errors in measuring the penetration depth.

121

frame
(a) relative displacement
sensing

specimenholderframe(b) absolute displacement
sensing

Figure 11.4 Instrumented micro/
nano indentation test systems:(a) sensing the relative depth ofpenetration, and(b) sensing the absolute depth ofpenetration[11.1] M.K. Shorshorov, S.I. Bulychev,
V.A. Alekhim, Sov. Phys. Doki., 26, 769(1981)[11.2] D. Newey, M.A. Wilkins,
H.M. Pollock, J. Phys. E15, 119 (1982)[11.3] J.B. Pethica, R. Hutchings, W.C.
Oliver, Phil. Mag., A48, 593 (1983)[11.4] G.M. Pharr, R.F. Cook, J. Mater.
Res., 5, 847 (1990)[11.5] M. Sakai, Acta. Metall. Mater., 41,
1751 (1993)[11.6] M. Sakai, S. Shimizu, T. Ishikawa,
J. Mater. Res., 14[4], 1471 (1999)

(1) Experimental determination of the frame compliance
Aw well discussed in the preceding section, the elastic deformation of
the frame, i.e., the frame compliance, has always undesirable effect on
determining the mechanical characteristics such as the elastic modulus,
viscoelastic functions, etc. in micro/nano indentation testing.
Suppose a perfectly elastic indentation contact and define the observed
penetration depth as hobs , the actual depth of penetration as h , and theindentation load as P , then the following relations are given;hobs = Cobs P(11.1)

hobs = h + hf

where hf represents the frame-deformation-induced penetration, andCobs is defined as the elastic compliance observed. Once we define theelastic compliance of test specimen C by h = CP , and the frame
compliance Cf by hf = Cf P , then the observed compliance Cobs isrelated to the specimen’s compliance C as follows;

Cobs = C + Cf

(11.2)

via the frame compliance Cf . In most of the commercially available
conventional instrumented indentation test systems, their frame
compliance is to be Cf ≈ 200 nm/N.
The load vs. penetration depth hysteresis curve (P–h loading/the unloading curve is resulted from the elastic recoveries of the test
specimen and of the apparatus frame, the initial slope of the unloading
P–h curve at the maximum penetration depth hmax , being referred to as(

the unloading stiffness Sobs ≡ ( dP dh )hmax) , is related to the observed

π
2E '

1
Ac

Pmax

Indentation load, P

unloading curve) of an elastoplastic body is depicted in Fig. 11.5. Since

Cobs ( = 1 / Sobs ) = Cf +

Sobs

hr

hmax

elastic compliance Cobs through the inverse relation of Sobs = 1 Cobs .

Penetration depth, hOn the other hand, when we substitute the relation of C = dh dP into

Figure 11.5 Loading/unloading P-hEqs. (3.33) and (3.34), for arbitrary axisymmetric indenters including

hysteresis curve of an elastoplastic

spherical, conical, and pyramidal indenters, the unloading compliance isbody. The initial slope Sobs of therelated to the elastic modulus E ' of the specimen tested [9.4];

Cobs ( ≡ 1 Sobs ) = Cf +π

1

2 E ' Ac

unloading curve is referred to as theunloading stiffness; there is an,(11.3)

where Ac stands for the contact area at the maximum indentation load

Pmax . The contact area Ac is related to the contact depth of penetration

122

inverserelation

betweentheobserved compliance Cobs and theunloading

Sobs = 1 Cobs

stiffness Sobs

,

i.e.,

hc via the formula of Ac = ghc 2 in terms of the area index g (theVickers/Berkovich indenters; g = 24.5 ) for deep penetration, where we
can neglect the effect of indenter’s rounded-tip on the indentation contact
behavior. Furthermore, there exists a quantitative correlation of

hc = ηc hmax between the contact depth hc and the penetration depth
hmax at the maximum indentation load, where the relative penetration
depth ηc is given by the function of plastic index

ε I E ' cY (refer toFig. 5.6). Substituting the preceding formulas into Eq. (11.3), both ofthe experimentally observable parameters Cobs ( ≡ 1 / Sobs ) and hmax
are correlated in the following equation via the frame compliance Cf ;Cobs ( ≡ 1 / Sobs ) = Cf +π 1
12 E ' gηc hmax

(11.4)An example of the Cobs - 1 hmax plot is shown in Fig. 11.6 in order toscrutinize the preceding procedure for determining the frame compliance

Cf , where all the data are of the FEA-created model elastoplastic body
3000

Poisson’s ratio ν = 0.3 ) indented by a Vickers/Berkovich equivalentcone with the truncated tip-radius Rt = 100 nm (see Fig. 11.2). The frame
compliance is a priori set to be Cf = 200 nm/N in this FEA-based

Cobs (nm/N)(the elastic modulus E = 100 GPa, yield stress Y = 10 GPa, and the

2000

1000

modeling. In Fig. 11.6, we can readily confirm that the frame compliance

Cf determined from the intercept of the y-axis (the axis of ordinate), andthe elastic modulus E ' calculated from the slope of Cobs - 1 hmax
linear plot coincide well with those a priori fixed in the FEA-model. In
other words, Fig. 11.6 indicates that we can successfully determine inexperiment the frame compliance Cf from the Cobs - 1 hmax linear
plot obtained in the conventional instrumented indentation apparatus.

123

π 1
2 E ' g ηc00

Cf=200 nm/N

0.001
1/hmax (1/nm)

Figure 11.6 Cobs vs. 1 hmax plot
for determining the frame compliance

Cf in experiment(FEA-based numerical results)

Experimental procedure for determining the frame compliance of
instrumented indentation apparatus(1) Experimental determination of

Cobs ( ≡ 1 Sobs )

vs.

hmax

relations at several discrete values of hmax ( ≥ 500nm ) by the use of ahigher-modulus material such as a brittle ceramic; a higher-modulus
material enhances the deformation of the frame, thus leading todetermining the frame-compliance Cf with a higher precision.
(2) Determination of Cf from the intercept of the Cobs vs. 1 hmax
linear plot (see Fig. 11.6) obtained in the procedure (1).

* Once we use a test material having the known elastic modulus E 'such as a fused silica, we can readily determine ηc from the slope(= π( 2E ' gη ) ) of the linear Ccobs

vs. 1 hmax plot, then we cancorrelate hmax to the contact depth hc = ηc hmax as well as to the(contact area Ac = ghc 2)

numerical results is plotted in Fig. 11.7 against the penetration depth

hmax in a double-logarithmic way in order to evaluate the tip-radius Rtof the indenter used, since there exist the relations ofC ≡ ( dh dP ) ∝ 1/ hmax1/2 → − (1 2) log hmax for spherical, andC ∝ 1/ hmax ( → − log hmax )Unloading compliance, C(nm/N)

The unloading compliance C ( ≡ Cobs − Cf ) of these FEA-based
106

conical region

1/2

5

10

104
spherical region

1103
102 -1
10

100

101

10 2

103

Penetration depth, hmax (nm)Figure 11.7 Double-logarithmic plot

for conical indentation (refer to Eqs. (3.18) and (3.25)). As a matter ofoffact, Fig. 11.7 demonstrates the linear log-log plots with the slope ofC ( ≡ Cobs − Cf ) vs. the penetration-1 in the region of conical indentation and with the slope of -1/2 in thedepth

region of spherical indentation. There exists, therefore, the spherical-to-elastoplastic body with the elastic

conical transition in the region of 10nm ≤ hmax ≤ 60nm that is marked

modulusby the hatched zone in Fig. 11.7. This transition is actually resulted fromratio ν = 0.3 and the yield stress

the tip-geometry of the indenter (Fig. 11.2) modeled in the present FEA

Y = 10 GPa)

124

the

unloadinghmax

compliance(FEA-results for an

E = 100 GPa, Poisson’shaving the tip-radius Rt = 100 nm and hsphere [ = Rt (1 − cos β ) ] = 5.85 nm.
As well demonstrated in the preceding FEA-results, once we determine
the frame compliance Cf in experiment, we can readily determine the
tip-radius of the commercially available pyramidal indenter through the
double-logarithmic plot of the unloading compliance C ( ≡ Cobs − Cf )vs. hmax as shown in Fig. 11.7. Along with this compliance method,
making direct observation for the tip of indenter through a scanning probe
microscope refines the experimental determination of the indenter’s tipradius Rt .
(2) Experimental determination of elastoplastic characteristics
The mechanical characteristics such as the elastic modulus and theyield stress can, therefore, be determined in indentation tests throughanalyzing the indentation loadPvs. the penetration depthhhysteresis, i.e., the loading/unloading P - h hysteresis relation, once wehave completed an a priori evaluation of the frame compliance and the
tip-geometry of the indenter used. As mentioned in the preceding sections,however, due to the incapability of determining the indentation contact
area A

in the conventional instrumented indentation test systems, wehave to reckon the approximation and assumption and/or the FEA-based
calibration to estimate the contact area, leading always to somewhat
uncertainty and inaccuracy in experimentally determining the mechanicalcharacteristics from the P - h hysteresis relation observed.

125

【Procedure for determining the elastoplastic characteristics
from the P - h hysteresis observed in experiment】
The flow chart for determining the elastoplastic characteristics from the observed P - hhysteresis of pyramidal indentation is given in Fig. 11.8: Figures 1 and 2 in the flow chart
exhibit the key parameters; the maximum penetration depth hmax , depth of residual
impression hr that is attributed to the plastic deformation, loading coefficient k1
including the elastoplastic information, and the unloading coefficient k2 that is resulted
from the elastic recovery of indentation-induced impression.

(1) Apply the observed dimensionless residual depth hr hmax to the y-axis of Fig. 3 (FEAderived plot; see Fig. 10.9), and then determine the plastic index PI ( = ε I E ' cY ) from thex-axis(2) In a similar way, apply the observed hr hmax to the x-axis of Fig 4 (see Fig. 10.10) in
order to determine the dimensionless contact depth hc hmax from the y-axis. Instead ofusing this graphical procedure, the Oliver-Pharr elastic approximation (Eq. (10.8)) is also
applicable to determining hc hmax(3) Determine the contact radius ac through the relation ac =hc cotβ by the use of hc
obtained in the preceding procedure (2), then calculate the contact area using the relation of

(

) (

)

Ac = π ac 2 = π cot 2 β hc 2 , leading to the determination of the Meyer hardness HM
from Ac combined with the maximum indentation load Pmax ;(H M ( ≡ Pmax Ac ) = Pmax tan 2 β

) πh

c

2

(11.5)(4) Apply the plastic index PI determined in the preceding procedure (1) to the x-axis of Fig.
5 (see Fig. 10.7) and of Fig. 6 (see Fig. 10.8), then determine k1 Y and k2 E ' from therespective y-axes.

Since the values of k1 and k2 have already been known in theobserved P - h2 plot in Fig. 2, we are ready to determine the elastic modulus E ' and theyield stress Y of the elastoplastic body(5) Apply the plastic index PI obtained in Fig. 3 to the x-axis of the FEA-derived Fig. 5.3,
then determine the constraint factor c from the corresponding y-axis of the normalized
hardness of H M cY , because the values of H M and Y have already been determined in
the preceding procedures of (3) and (4)126P Fig. 1

P Fig. 2

k2

k1

2

h

h

hmax

hr

hr

Fig. 4

•

•

PI =
PI

εI E '
cY•••

Fig. 5•hr/hmax••

Fig. 6

•

k2/E'

k1/Y

•

2

hmax

hc/hmax

hr/hmax

Fig. 3

2

•

•

PI

PI

Figure 11.8 Flow chart for determining the elastoplastic characteristics from the P - h hysteresis
obtained in experiment127

(3) Experimental determination of viscoelastic functions
As discussed in Chap. 7, without measuring the indentation contact
area A ( t ) in a quantitative manner, it is impossible for us to obtain anyof viscoelastic functions such as the stress relaxation modulus E 'relax (t )and the creep compliance function C 'creep (t ) . In the conventional
instrumented indenters, however, we have to estimate the time-dependentcontact area A ( t ) from the penetration depth h ( t ) observed inexperiment through the assumption/approximation, resulting in the very
qualitative viscoelastic functions, not the quantitative material
characteristics.
As emphasized in the preceding sections, due to the luck ofgeometrical similarity, the indentation strain ε I of spherical or flatended cylindrical indenter is dependent on the penetration depth or thecontact area. This fact results in undesirable complexity for analyzing thetime-dependent viscoelastic characteristics. To circumvent this difficulty,
we make the viscoelastic analysis only for conical/pyramidal indentation
contact in this section like as the considerations we made in Chap. 7; theviscoelastic constitutive equations for conical indentation given in Chap.
7 (see Eq. (7.3)) are

P (t ) =tan β  tdA(t ')dt ' E 'relax (t − t ')
dt '
2 0

(11.6)tdP(t ')A(t ) = 2 cot β dt ' C 'creep (t − t ')0
dt 'Equation (11.6) alternatively represented in terms of the penetration
depth h ( t ) is given as follows;
P (t ) =π cot β 2t

 E 'relax (t − t ')0d {ηc 2 (t ')h 2 (t ')}
dt '

dt '

2 tan β  t
dP (t ')
η c 2 (t ) h 2 (t ) =dt ' C 'creep (t − t ')
π 0
dt '

(11.7)

in which the relative contact depth of penetration ηc ( t ) is defined byηc ( t ) = hc ( t ) h ( t ) . The ηc ( t ) -value of a perfectly elastic body isalways time-independent having the value of ηc = 2 π (refer to Eq.
(3.28)). Furthermore, ηc for elastoplastic indentation is also timeindependent material characteristic as shown in Eq. (10.8) and in Fig.
10.10. On the other hand, as seen in Eq. (11.7) even for conical128

indentation contact, we have to take into account the time-dependent
for viscoelastic indentation contact. The time-dependent ηc ( t ) -values are shown in Fig. 11.9, by way of example,
under the step-wise penetration of h ( t ) ≡ 3 μm ( t ≥ 0 ) for the Vickers/
Berkovich-equivalent cone indented on the viscoelastic Maxwell liquids
having various values of their Poisson’s ratios (ν ( 0 ) = 0.1, 0.3, and 0.5)
(FEA-based numerical results). The ηc ( t ) -value of the incompressibleν ( 0 ) = 0.5

viscoelastic body withis always time-independent( ηc = 2 π ) like as a perfectly elastic body, while for the viscoelastic
liquids with ν ( 0 ) < 0.5 , ηc ( t ) monotonically increases along with theload relaxation, though the applied penetration is kept constant ofh ( t ) ≡ 3 μm in a step-wise manner. This fact implies that the free-surface
outside the contact area creeps-up during load relaxation.
We can neither measure the time-dependent hc ( t ) nor determineηc ( t ) in experiments on the conventional instrumented indenter. We
have therefore no choice but to simply assume the time-independent
elastic value of ηc ( t ) ≡ 2 / π in Eq. (11.7), leading to the following
approximated viscoelastic constitutive equations;d {h (t ')}2 cot β 
E 'relax (t − t ')
dt '

dt '
π 0
t

P (t ) =2h (t ) =2(11.8)

π tan β  t
2dP (t ')dt '
 C 'creep (t − t ')0dt '(i) Step-wise penetration test
In Eq. (11.8), the indentation load relaxation P ( t ) observed under
the step-wise penetration to h0h(t ) = h0 u (t ) ,(11.9)

is related to the relaxation modulus E 'relax (t ) as follows,

P(t ) =

2h02 cot βπ

E 'relax (t ) ,(11.10)while under the step-wise loading to P0

P(t ) = P0u (t ) ,(11.11)the resultant creeping penetration depth h(t ) is related to the creep
compliance function C 'creep (t ) in the following equation;
129

Relative contact depth, ηc(t)nature of ηc ( t )Maxwell liquid (E g=80 GPa, τ=200 s)

0.67
0.66
Poisson's ratio: ν(0)=0.5
:
0.3
:
0.1

0.65
0.64
0.63
0

100

200

300

400

500

Time, t (s)Figure 11.9 Poisson effect on thetime-dependent relative contact depthηc ( t ) for the Maxwell liquid with therelaxation modulus:
Erelx (t ) = Eg exp ( −t τ )( Eg =80GPa, τ =200s) under a stepwise penetration (numerical FEAresults)

h 2 (t ) =π P0 tan β2(11.12)

C 'creep (t )We can, therefore, determine the relaxation modulus E 'relax (t ) from therelaxation load P ( t ) and the creep function C 'creep (t ) from thecreeping penetration h 2 (t ) in these step-wise penetration tests. In
addition to neglecting the time-dependent nature of ηc ( t ) in deriving
Eq. (11.8), it must be noticed in this context that the frame complianceCfof the instrumented apparatus inevitably induces the creeping

penetration h ( t )during the load relaxation

P ( t ) even for the

stepwise penetration to h0 ;h(t ) = h0 − Cf P(t )(11.13)This frame-compliance effect implies that the relaxation time τ

isalways significantly overestimated, resulting in an undesirable estimateof the relaxation modulus E 'relax (t ) in Eq. (11.10). On the other hand,for step-wise loading of P(t ) = P0u (t ) as well as for constant-rate ofpenetration h ( t ) = k h ⋅ t or P ( t ) = k P ⋅ t , as will be discussed in thesubsequent section, we do not need to take care of the complexity and the
difficulty associated with the frame compliance due to their indentationP(t ) ≥ P (0) .

tests without load relaxation, i.e.,

Even for theviscoelastic deformations without load relaxation, however, it must be
noticed the fact that there exist crucial difficulties in the reliability andthe accuracy of the viscoelastic functions estimated through using Eq.
(11.8), because it has been derived by making a fatal assumption of thetime-independent ηc ( t )( ≡ 2 π ) .
(ii) Constant-rate of penetration test
The application of Eq. (11.8), i.e., Eq. (11.7) with ηc ( t )( ≡ 2 π ) , tothe constant-rate of penetration testh(t ') = kh t '(11.14)and to the constant loading test

P (t ') = k P t '(11.15)leads to the following viscoelastic constitutive equations, respectively;
P (t ) =2h (t ) =4 k h 2 cot βππ k P tan β
2

t

 E'
0

relax

(t − t ')t ' dt '

t

 C'
0

creep

(11.16)(t − t ') dt '130

Once we apply the Laplace transform and its inversion to Eq. (11.16), we
finally have the following formulae of the relaxation modulus and thecreep compliance function;
E 'relax ( t ) =C 'creep ( t ) =π tan β d 2 P (t )
4kh 2

dt 2

{

(11.17)}2
2 cot β d h ( t )

π kP

dt

We can, therefore, readily determine these viscoelastic functions in termsof the time-dependent indentation load P ( t ) and the penetration depthh 2 ( t ) observed in experiments.

11.3 INSTRUMENTED INDENTATION MICROSCOPE
We have repeatedly emphasized the key roles of the indentation contact
mechanics in the micro/nano materials science for studying the materials
characteristics (elastic, elastoplastic, viscoelastic characteristics). We
have also emphasized the importance of the indentation contact area A
induced by the applied indentation load P , or the interrelation of theapplied contact area A to the resultant indentation load P (see in
Chaps. 4 ~ 9). As described in Sec. 11.2, however, it is impossible for
us to measure the contact area A in the conventional instrumented
indentation apparatuses, although we can measure the penetration depth
h in a quantitative manner. In these conventional test apparatuses,

therefore, we have to determine the mechanical characteristics throughmaking appropriate assumptions/approximations to estimate the contact
area A from the penetration depth h observed in experiments. On theother hand, upon using the instrumented indentation microscope, we caneasily measure the contact area A as a function of the indentation load
P and/or of the penetration depth h , then readily determine themechanical characteristics without relying on any of undesirable
assumptions/approximations.

131

(1) Basic configuration of the apparatus and

CCD-camera

the principle of measuring the contact area [11.7 - 11.9]

penetration
depth sensor

optical microscope

The basic configuration of the instrumented indentation microscope isindenter

depicted in Fig. 11.10. The optical image of the contact area A

specimenmeasured at the contact interface is stored into a personal computer as thedigital data through an optical microscope combined to a CCD camera.

load
sensordisplacement actuator

In Fig. 11.11, shown are the schematic details of (a) the dark field and (b)Piezo
actuator
Z stage
Z-stagethe bright field microscopies that are utilized in the indentation
microscope. As shown in these figures, not only the inlet but also the

Figure 11.10 Basic configuration ofoutlet beams must be transferred through the indenter, indicating that thethe instrumented indentation micro-

indenter must be transparent for the beams. In this context, thescopeconventional diamond indenter is always applicable to the indentation
test in air, i.e., in oxidative environments under 300oC, while an indenter(a) dark field nicroscopy

made of sapphire is the most appropriate at temperatures in
300oC~1200oC.

The test specimen shown in Fig. 11.10 is lifted up to

the indenter fixed to the test frame. The indentation load P is measured

microscopeby the load cell that is arranged in series beneath the specimen, and thepenetration depth h is monitored by a displacement sensor (a linear
transducer or a capacitance sensor).

All of the mechanicaloutletbeam

inlet beamcharacteristics can be readily determined in a quantitative manner once
we get the contact area A as a function of the applied load P on thisspecimen

instrumented indentation microscope, implying no need of monitoring
the penetration depth h , the details of which will be discussed in thefollowing section.

(b) bright field microscopy

The penetration depth sensor equipped in the

indentation microscope, however, is required in order to confirm theconsistency of theP - hrelation to that of the conventional

instrumented indentation apparatuses.

microscope

It must be noticed furthermorethat we do not need to take into account the effect of the frame
compliance Cf(refer to Sec. 11.2(1)), since all the mechanicalinlet/outlet
beamcharacteristics are quantitatively determined from the P - A relation
directly observed on the instrumented indentation microscope.

specimen

Figure 11.11 The inlet/outlet beams＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊[11.7] T. Miyajima, M. Sakai, Phil. Mag, 86[33-35], 5729 (2006)[11.8] M. Sakai, N. Hakiri, T. Miyajima, J. Mater. Res., 21[9], 2298 (2006)[11.9] N. Hakiri, A. Matsuda, M. Sakai, J. Mater. Res., 24[6], 1950(2009)

to monitor the indentation contact
images: (a) dark field microscopy, (b)bright field microscopy

132

The optical image of contact area A is digitized and stored in a

optical image

binary image

personal computer as the binary image in relations to the scanning time,
indentation load and the penetration depth. An example of the opticalimage (bright-field image) and its digitized binary image is shown in Fig.
11.12 for Berkovich indentation onto a polycarbonate resin. In the

80μm

instrumented indentation microscope, it is also possible for us to control
the contact area A applied to the test specimen as a function of
scanning time by feeding these digital date back to the displacement
actuator; in time-dependent viscoelastic indentation tests, by way of

Figure 11.12 Optical and its binary
images of the contact area of a
Berkovich indenter pressed onto a
polycarbonate resin

constant-rate contact area to the specimen ( A ( t ) = k A ⋅ t ), but also for astep-wise penetration to a constant contact area ( A ( t ) = A0u ( t ) ), leading
to the quantitative determination of the stress relaxation modulus and thecreep compliance function, the details of which have already been given
in Chap. 7 .

Indentation load, P (N)example, we can conduct the indentation tests not only for applying a(2) Determination of elastoplastic characteristics
as the Meyer hardness H M , elastic modulus E ' and the yield stress Y
in the instrumented indentation microscope through analyzing theloading-unloading hysteresis relation of the indentation load P and thecontact area A , i.e., the P - A hysteresis relation without making any
assumptions and approximations. The loading-unloading

P - A

hysteresis relations are shown in Fig 11.13(a) for the Vickers indentation
test results of silica glass and silicon nitride ceramic, and in Fig. 11.13(b)for the FEA-based numerical result of an elastoplastic body indented by
the Vickers/Berkovich-equivalent cone.

As clearly shown in thesefigures, not only the loading but also the unloading P - A relations arelinear. The slope of the loading line gives the Meyer hardness itself (seeEq. (5.1b)), while the unloading slope M ( ≡ ε I E ') (the unloading
modulus) is directly related to the elastic modulus E ' ;

tan β
M=
E'
2

4
3

HM

2

M

1
0

0

4

100

200

300

Contact area, A (μm2)

400

(b)
FEA-numerical result
E=100 GPa, Y=5 GPa, ν=0.3

3
2

M

HM

1
0
0

100

200

300

400

Contact area, A (μm2)

500Figure 11.13(a) Loading-unloading P - A
linear relations of silica glassand silicon nitride ceramic
observed on the instrumented
indentation microscope(Vickers indentation)(b) FEA-based numerical result(11.18)where use has been made of the indentation strain of ε I = tan β 2 forpyramid/cone indentation. We can, therefore, readily determine the133silica glass
silicon nitride ceramic

5

5

Indentation load, P (N)We can quantitatively determine the elastoplastic characteristics such(a)6of an elastoplastic body
( E ' = 100 GPa, Y = 5 GPa)for Vickers/ Berkovich
equivalent con indentation

elastic modulus E ' from the unloading modulus M observed on the
indentation microscope. Furthermore, “the additivity principle of the
excluded volume of indentation” (see Eqs. (5.13a) and (5.14) in Chap. 5),
leads to the following formula of cone/pyramid indentation;
1( cY )

3/ 2

=

1
HM

3/ 2

1

−

(11.19)M 3/ 2

Substituting the Meyer hardness H M and the unloading modulus M
thus directly determined from the loading/unloading slopes of the linear
P - A relations into the right-hand side of Eq. (11.19), we finally obtainthe yield stress cY , and then the plastic index PI as a measure ofplasticity (refer to Chap. 5)
PI ( ≡ ε I E ' cY ) = M cY

(11.20)is readily fixed in terms of M and cY in experiment by the use of the
instrumented indentation microscope.
(3) Determination of viscoelastic functions [7.5]The viscoelastic constitutive equation in terms of the contact area forcone/pyramid indentation is described by

Ave ( t )32

= A(t )

32

32

− Ap ( t )(11.21)through applying the elastic-to-viscoelastic corresponding principle (seeChap. 6) to Eq. (5.13a). Equation (11.21) implies, therefore, that we haveto eliminate the plastic contribution of Ap from the contact area A( t )observed on the instrumented indentation microscope prior to

Ave ( t ) , and then thedetermining the viscoelastic contact areaviscoelastic functions, as follows:(1) Experimental determination of cY through the procedures given in the preceding section (2) using theelastoplastic P - A loading/unloading linear relations in the higher-rate penetration test (i.e., thetime/rate-independent indentation test with invisible viscoelastic flow) , followed by(2) Determination of the plastic contact area Ap ( t ) by substituting this cY -value into Ap ( t ) = P ( t ) cY ,and then(3) Determination of the viscoelastic contact area Ave ( t ) by the uses of P ( t ) and A ( t ) both observed on
the instrumented indentation microscope via the following relation derived from Eq. (11.21);{}

3/2 2/3

Ave (t ) = A(t )3/2 − [ P(t ) cY ](11.22)134

(i) Step-wise penetration tests
Using the function of feed-back controlling of the instrumented
indentation microscope, we can conduct the test of step-wise penetration
to a constant contact area A0 , i.e., A ( t ) = A0u ( t ) , and then measure theresultant indentation load relaxation P ( t ) . The viscoelastic indentation
contact area Ave ( t ) in this test is, therefore, given by}

3/2 2/3(11.23)

via Eq. (11.22). It must be noticed in Eq. (11.23) that Ave ( t ) ( ≡ Ave ) is
time-independent, because the plastic contact area(

Creep compliancelog C'creep(t) (Pa -1){

Ave (t ) ( ≡ Ave ) = A03/2 − [ P(0) cY ]o

T=30.0 C

Ave (t)

-8.5

A(t)

-9

-9.5

)

Ap (t )  ≡ Ap (0) = P (0) cY 

01

induced at the onset of the load relaxation P ( 0 ) is time-independent
constant due to the irreversible nature of plastic flow, although theviscose flow still results in the load relaxation under this step-wise
penetration of A ( t ) = A0u ( t ) . Accordingly, substituting the Ave ( t ) value calculated in Eq. (11.23) combined with the observed load
relaxation P ( t ) into Eq. (7.6), we readily determined the stressrelaxation modulus, as follows;(11.24)3

Figure 11.14 Creep compliancefunction of an amorphous selenium at

T = 30.0 C for Berkovich indentation in step-wise loading to a constantload P0 , i.e., P ( t ) = P0 u ( t ) .
To demonstrate the effect of theplastic

P(t )E 'relax (t ) = 2cot β
Ave

2Time, log t (sec)deformation,compliance

the

creep

functions determinedfrom the viscoelastic contact area

In step-wise loading to a constant indentation load P0 , i.e.,Ave ( t )withplastic

correction

P ( t ) = P0 u ( t ) , the indentation contact area A ( t ) observed on the( Ap ( t ) = P0 cY ) (the closed circles,

indentation microscope increases with time in a monotonic manner●) and from the observed contact

resulting in a creep deformation. Accordingly, the creep compliancearea

function C 'creep ( t ) is determined in experiment by substituting thecorrection (the open triangles, △) are{

Eq. (7.6);

C 'creep (t ) =}3/2 2/3

viscoelastic contact area Ave (t ) = A(t )3/2 − [ P0 cY ]into

A (t )

without

plastic

plotted against time. Both of the
creep

functionsrepresent

anasymptotic coincidence in the long-tan β Ave (t )2
P0

(11.25)The creep compliance C 'creep ( t ) of an amorphous selenium (Se) at
its glass-transition temperature ( Tg = 30.0 C)) determined on thetime

regionsince

theplastic

deformation induced at the onset ofthestep-wise

relatively

loading

insignificant

becomeswiththeindentation microscope is plotted against time in Fig. 11.14 for aincrease in the viscoelastic creep

Berkovich indentation, where demonstrated are both the complianceunder viscus flow135

functions determined from the viscoelastic contact area Ave ( t ) withplastic correction (the closed circles) and from the observed contact areaA ( t ) without plastic correction (the open triangles) for comparison.

(ii) Constant-rate of loading test
In constant-rate of loading test, the contact area A( t ) combined withthe applied indentation load P ( t ) = kP ⋅ t gives the viscoelastic contact
area Ave ( t ) via the relation of{

Ave (t ) = A(t )3/2 − ( kP ⋅ t cY )for cone/pyramid indentation.

}

3/2 2/3

Substituting this viscoelastic contact

area into the second formula of Eq. (11.6) leads to the creep compliancefunction, as follows;
C 'creep (t ) =tan β dAve (t )2k P
dt

(11.26)Equation (11.26) indicates that we can readily determine the creep
compliance function through the time-derivative of the viscoelastic
contact area Ave ( t ) that has been determined by eliminating the plastic
contribution from the contact area A( t ) observed on the indentation
microscope.
(iii) General-purpose indentation tests
The “step-wise penetration to a constant contact area” and/or the“constant-rate of loading” are the most appropriate and wellrecommended in the conventional viscoelastic indentation tests, since we
can quantitatively eliminate the undesirable plastic contribution to theviscoelastic deformation and flow, leading to readily determine theviscoelastic functions without any approximations/assumptions, asmentioned in the preceding sections (i) and (ii).

On the other hand, in

more general-purpose indentation test, we face on several mathematicaldifficulties in applying the observed data to Eq. (11.6) for determining
the viscoelastic functions in an analytical manner. However, we cancircumvent these difficulties through the numerical approach via using
personal computers (PCs) that have made a remarkable breakthroughsince the end of 20th century.

We can easily determine the viscoelastic

functions in numerical manners by the use of PCs even for rather
136

complicated viscoelastic/viscoplastic stimulus-response phenomena inindentation contact problems.
The basic protocols for determining the viscoelastic functions are
given as follows through the numerical analyses of the observed P ( t )vs. A ( t ) relation:
First of all, the application of Laplace transform to the viscoelastic
constitutive equation (7.3) leads to the following viscoelastic functions in
the Laplace space

E 'relax ( p) = 2cot β

P ( p)
pAve ( p)(11.27)tan β Ave ( p)C 'creep ( p) =2 pP ( p)In advance of conducting these Laplace transforms, we have to calculate
the viscoelastic contact area Ave ( t ) by applying the observed P ( t )and A ( t ) into Eq. (11.22), and then represent these P ( t ) and Ave ( t )in their Prony series expansions (refer to the table given below)
respectively, followed by their Laplace transforms ofP ( p) andAve ( p ) , leading to E 'relax ( p ) and C 'creep ( p ) in Eq. (11.27). The

viscoelastic functions

E 'relax ( p )andC 'creep ( p )are, therefore,

finally given by making their Laplace transform inversions.

Thesenumerical procedures for determining the viscoelastic functions are
summarized in the following table, where we focus only on the stressrelaxation modulus E 'relax ( t ) , as an example.

Similar procedures arealso applicable to deriving the creep compliance function C 'creep ( t ) . It
must be noticed, furthermore, that we can also derive the creep function
from the stress relaxation modulus by the use of the convolution integral(Eq. (7.2)) in its Laplace space;(11.28)pC 'creep ( p ) = 1 pE 'relax ( p )137

【Numerical analyses for determining the viscoelastic functions】(1) Determination of the yield stress cY in Eq. (11.19) in terms of the Meyer hardness H M and theunloading modulus M using the slopes of the loading/unloading P ‐ A linear plot (see Fig. 11.13)
in the time-independent elastoplastic regime through conducting a higher-rate indentation test.
(2) Determination of Ave ( t ) by substituting the observed P ( t ) ‐ A ( t ) relation combined with the cY value into Eq. (11.22), where P ( t ) must be fixed to P ( 0 ) in the load-relaxation test whenever theload P ( t ) at time t is smaller than that at the onset of the relaxation, i.e., P ( t ) ≤ P ( 0 ) (refer to thedetails given in Sec. (i) ).
(3) Description of P ( t ) and Ave ( t ) in their Prony series expansions using the collocation method;nP(t ) = Pe +  Pi exp ( −t / λ1i )i =1(11.29)nAve (t ) = Ae +  Ai exp ( −t / λ2i )i =1(4) Substitution of the Laplace transform of Eq. (11.29),
P ( p) =

Pe n
Pi
+
p i =1 p + 1 λ1inAAi
Ave ( p) = e + 
p i =1 p + 1 λ2i,(11.30)

into Eq. (11.27) in order to describe E 'relax ( p ) in the following formula by the use of collocation
method;E 'relax ( p) =

(5)

n
Ee
Ei
+
p i =1 p + 1 τ i(11.31)Laplace transform inversion of Eq. (11.31) finally results in the stress relaxation modulus;n

E 'relax (t ) = Ee +  Ei exp ( −t / τ i )(11.32)i =1

138

MATERIALS PHYSICS BASED ON THE
EXPERIMENTAL DATA OF INSTRUMENTEDINDENTATION TESTING

CHAPTER

12.1 THE EFFECT OF TIP-GEOMETRY OF

2ψb

12
β

2b

PYRAMIDAL INDENTATIONON THE CONTACT BEHAVIOR [12.1]

2a

The most favorable feature of pyramidal/conical indentation is in its
2ψageometrical similarity, resulting in the penetration-depth-independent
mechanical characteristics such as the Meyer hardness, elastic modulus,

Figure 12.1 Tip-geometry of theand the viscoelastic functions observed in instrumented indentation

Knoop indenter:testing, unlike the penetration-depth-dependent characteristics of Ratio of

spherical indentation. On the other hand, however, there exists a minor

major/minor axes; a / b = 7.11

demerit resulted from the geometrical discontinuity at the pyramidal

 Inclined face-angle; β = 25.2oedges that leads to the mathematical complexity in analyzing theindentation contact mechanics.

 Diagonal apex angles;
2ψ a = 172.5o and 2ψ b = 130o

This is the reason why we havefor major/minor axes

exclusively adopted in the preceding chapters the Vickers/Berkovich
equivalent cone indenter in order to circumvent the pyramidal edgeIn this section, based on the experimental examinations, we discuss theeffect of the tip-geometry of the conventional pyramidal indenters on the
indentation contact behavior of the several engineering materials; theindenters examined include the conventional pyramid indenters (thetetrahedral Vickers, trihedral Berkovich (see Fig. 11.1), tetrahedral

Indentation load, P (N)related difficulties.

Si3N4 ceramic

40

Vickers
Berkovich
Knoop

30
20
10
0

0

2

4

6

8

10

Penetration depth, h (μm)

pyramid indenters with various values of the inclined face angle β , andFigure 12.2 Effect of the tip-

the Knoop indenter shown in Fig. 12.1. The Knoop indenter has widelygeometry

been utilized in the fields of crystallography and mineralogy since its

pyramidal indenters on the P − hanisotropic

loading/unloading

tip-geometryisappropriate

forexamining

the

of

the

conventional
hysteresis

silicon nitride ceramic;

crystallographic orientation.
Figure 12.2 shows the P − h loading/unloading hysteresis curves of●：Vickers indentera silicon nitride ceramic for the Vickers/Berkovich/Knoop indentations.

○：Berkovich indenter

As readily seen in Fig. 12.2, the loading/unloading hysteresis curve of the▲：Knoop indenter

Berkovich indentation coincides well with that of the Vickers indentation,
because the index of projected area g of the Berkovich indenter

139[12.1] J. Zhang, M. Sakai, Mater. Sci. Eng.
A:381, 62 (2004)of( g = 3 3 cot 2 β = 24.5 ) is design to coincide with the value ofthe2Vickers indenter ( g = 4 cot β =24.5), resulting in the same values of
their induced contact area ( A = gh2 ) and the excluded volume( V = gh3 3 ) for a given penetration depth h , the details of which havealready been mentioned in Sec. 11.1.

The good coincidence of theVickers/Berkovich indentations shown in Fig. 12.2 means that theindentation

contact

responses

areequivalent

both

for

the

Vickers/Berkovich indentations, implying the insignificant “edge-effect”
of the trigonal and the tetragonal pyramids.

On the other hand for

Knoop indentation, its g -value (= 2 tanψ a ⋅ tanψ b = 65.4) is 2.7
times larger than the value of the Vickers/Berkovich indentation,
resulting in its P − h loading/unloading hysteresis curve much steeper
than that of Vickers/ Berkovich indentation, as demonstrated in Fig. 12.2.
Furthermore, the loop energy U r (refer to Sec. 5.2) of the P − hloading/unloading hysteresis curve of the Knoop indentation and its
normalized residual depth ξ r ( = hr hmax ) are significantly smaller thanthose of Vickers/ Berkovich indentations, indicating that the elastoplastic
response of Knoop indentation is more elastic than those of Vickers/
Berkovich indentations.
Since the inclined face-angel βc of the Vickers/Berkovich equivalentcone is 19.7o (see Chap. 5), while the face-angle of the equivalent cone ofKnoop indenter is β c = 12.4o (the subscript c of the face-angle β inthese context indicates the equivalent cone). This fact leads to moreelastic Knoop contact than that of Vickers/Berkovich, the details of which
The face-angle-dependence of the elastoplastic responses (the P − hloading/unloading hysteresis curves) of a silicon nitride ceramic is shownin Fig. 12.3 for the tetragonal indenters with various values of their faceangle ( β = 10 , β = 22 (Vickers indenter), and β = 40 ); the g 2

values ( = 4 cot β ) of these indenters are, respectively, 129, 24.5

Indentation load, P (N)is given in the following considerations.

tetragonal pyramid indenters
20

o

β=10

o

β=22
(Vickers indenter)

10
β=40o

0
0

5

10

Penetration depth, h (μm)(Vickers indenter), and 5.68, and then the corresponding face-angles of

Figuretheir equivalent cone are, respectively, βc = 8.87 , 19.7 , and 36.6°. As

P − h hysteresis curves of a siliconwell recognized in Fig. 12.3, the indentation contact responses become

nitride ceramic for the tetragonal

more plastic along with the increase in the inclined-face angle, i.e., along

pyramid indenters with their inclinedwith the pyramid-tip becoming shaper. This fact is resulted from theface-angles; β = 10 (●), β = 22

increase in the plastic index ( PI = ε I E ' cY ; ε I = tan β 2 ) with the(Vickers)(○), and β = 40 (▲)

o

o

140

12.3

Loading/unloading

The Meyer hardness H M of a silicon nitride ceramic determined inexperiments for the pyramid indenters with various values of their
incline-face angle is plotted in Fig. 12.4 against their equivalent coneangle βc (these indenters have their tip geometry of regular tetrahedral
pyramid except the Knoop indenter). As shown in Fig. 12.4, the Meyer

Meyer hardness, H M (GPa)increase in the face angle β (refer to Sec. 5.1).

20
18
Vickers indenter
161412Knoop indenter
10

hardness is uniquely represented by the use of the equivalent-cone angle

10

20

30

40

Inclined face-angle, βc (degree)βc even for the Knoop indenter having its highly anisotropic tip-

Figure 12.4 Meyer hardness of ageometry.

silicon nitride ceramic determined inexperiments for tetrahedral pyramid

12.2 INDENTATION CONTACT MECHANICS OF
ENGINEERING MATERIALSIn this section, we discuss the instrumented indentation testing and theindenters. The solid line is theanalytical prediction (Eq. (5.14) with
m=3/2)

indentation contact mechanics/physics of engineering materials ranging
obtained on the conventional instrumented indenter and/or the
instrumented indentation microscope. The mechanical characteristics ofthe engineering materials demonstrated in this section will play importantroles not only in designing the microstructures of materials, but also in
scrutinizing the reliability/accuracy of the instrumented indentation

Indentation load, P(N)from ductile metals to brittle ceramics by the uses of the test results
20

SiC MgO SL-Glass

GLC

HMV100
Al

10

PMMA
00

200

400

600

800

Penetration depth, h2 (μm2)

apparatuses.

Figure 12.5

P − h2hysteresis(1) Loading/unloading P − h hysteresis curve and the elastoplastic

curves for the Vickers loading/characteristics [12.2]

unloading indentation:

The loading/unloading P − h hysteresis curves of the engineering
materials indented by Vickers pyramid are plotted in Fig. 12.5 (theP−h2plots discussed in Secs. 5.2, 10.3, and 11.2). It is readily seen that

SiC: silicon carbide
MgO: magnesia
SL-Glass: soda-lime glass

the hysteresis loop energy U r of the ductile metal (aluminum, Al) is

GLC: glassy carbon (GL-200H)very significant comparing to that of the brittle ceramic such as silicon

HMV100: copper/zinc alloy

nitride ceramic (SiC). Since the loop energy U r stands for the energy(Vickers hardness standard;

dissipation associated with indentation-induced plastic flow (refer to Sec.

HV=1GPa)5.2), it plays an important role in characterizing the elasticity/plasticity

Al: metallic aluminum (99% pure)of engineering materials, as clearly seen in Fig. 12.5.

PMMA: methyl methacrylate resin

The normalized residual depth of indentation-induced impressionξ r ( = hr hmax ) is also an important elastoplastic characteristic (there
exists one-to-one correlation between

ξ r and the loop energy U r ;141[12.2] M. Sakai, Y. Nakano, J. Mater. Res.,
17[8], 2161 (2002)refer to Secs. 5.2 and 10.3), and easy to be determined in experiment
without any undesirable complications that are required in calculating thearea integral for determining U r . The ξ r -values are shown in Fig. 12.6
for the various types of engineering materials. Table 12.1 summarizes theelastoplastic characteristics of the engineering materials, in which theξr ( = hr hmax )materials demonstrated in Figs. 12.5 and 12.6 are also included.

Figure 12.6 Normalized residual depth ξ r ( = hr hmax )of indentation-induced impression ofengineering materials; the loading and the unloading paths of a perfectly elastic material coincide witheach other, resulting in ξ r = 0 , while the perfectly plastic body yields ξ r = 1 due to none of elastic
recovery in its unloading process. The details of the elastoplastic characteristics of these engineering
materials are summarized in Table 12.1.

142

Table 12.1 Material characteristics of engineering materials

It must be noticed in Figs. 12.5, 12.6, and Tab. 12.1 that the mechanicalcharacteristics of the glassy carbon and the magnesia are very unique.
The glassy carbon looks like a perfectly elastic body; the elastic recovery
is very significant in its unloading process, leading to none of residual
impression, ξ r ≈ 0 .

This very unique indentation contact behavior isresulted from its microstructure, the details of which will be discussed in
the subsequent section. On the other hand, the contact behavior ofmagnesia is very ductile with ξ r = 0.86 like a ductile metal as
demonstrated in Fig. 12.6, although it is a ceramic. The isotropic cubic
crystallography of magnesia results in such a very ductile nature. In
general, however, the very anisotropic crystallography of most ofceramic materials leads to their brittleness, resulting in their ξ r -values
less than 0.7.
(2) Indentation-induced elastoplastic behavior of carbon materials
Most of organic materials are carbonized by heat-treatment at elevatedtemperatures, yielding various types of carbons with their uniquemicrostructures depending on the chemical structures of these organic
143

precursors. Carbon materials are categorized into the graphitizablecarbon and the non-graphitizable carbon: The former is easily graphitizedat the heat-treatment-temperatures (HTT) exceeding 1500oC, yielding
extensively developed graphite crystals that lead to significant plastic
deformation resulted from the prominent slippages along the graphite
basal planes, while the latter is rather brittle even heat-treated attemperatures exceeding 2000oC due to the spatially suppressed growth ofgraphite crystals in nano-regions.
(a) Glassy carbon [12.3]
The significant brittleness of glassy carbon (Glass-Like Carbon; GLC)
leads to engineering difficulties in its machining such as cutting/grinding.
Due to the conchoidal appearance of its fracture surface like that ofinorganic glasses, it is also referred to as “glassy” carbon, although theindentation contact behavior of GLC is totally different from that ofinorganic glasses. Details of the indentation contact mechanics of GLC
will be given in this section along with its nano-structure.
The

specificbrittleness

of

GLC

is

resulted

from

the

crystallographically disordered carbons that are made by the pyrolysisof thermosetting polymers (phenol/ epoxy/cellulose resins) heat-treatedat elevated temperatures (1000oC~2500oC). The microstructure of GLC
contains a significant amount of “closed nano pores”; the porosity is
surface area of the nanopores is of the order of several hundred m2/cm3.
All of the nanopores are essentially inaccessible to nitrogen gas; therespective nanopores are surrounded by the partition walls of hexagonal
networks of carbon atoms.
In Fig. 12.7, the P − h loading/unloading hysteresis curve of a GLC
is given in comparison to that of a soda-lime glass (SLG); it should benoticed for the GLC that the unloading path goes closely back along its
loading path to the origin of the P − h curve with insignificant amountof energy dissipation (i.e., very small hysteresis loop energy, U r ) andwithout any residual indentation impression after complete unloading,unlike the SLG that leaves behind a well-defined residual indentation
impression. The purely elastic deformation of nano-size partition walls
surrounding the nanopores of GLC results in such a unique P − h144

Indentation load, P (N)25~30 vol%, the pore diameter ranges from 0.5 to 5 nm, and the specific200

soda-lime glass
100

glassy carbon (GLC)
0

0

20

40

60

Penetration depth. h (μm)

Figure 12.7
P − h loading/unloading hysteresis

curves of soda-lime glass (SLG) andglassy carbon (GLC)[12.3] M. Sakai, H. Hanyu, M. Inagaki,
J. Am. Ceram. Soc., 78[4], 1006 (1995)

loading/unloading hysteresis.
Scanning electron micrographs (SEM) of the residual indentation(a)impressions for GLC and SLG made by Vickers indentation (P=98.1N)
are shown in Fig. 12.8. As well illustrated in Fig 12.8 (c) of the SLG, theVickers indentation usually results in two sets of median/radial cracks inthe form of mutually perpendicular cracks in directions parallel to the(b)pyramidal indentation diagonals. On the other hand, the most importantfeature of the indentation-induced surface damage patterns of the GLC is
its concentric ring cracks (Fig. 12.8 (a)), which are very anomalous inbrittle materials under a sharp indenter like the Vickers pyramid indenter.
Furthermore, as clearly seen in Fig. 12.8 (b), a well-defined cone crack(c)develops from one of the surface ring cracks. Such a well-developed cone
crack induced by a sharp indenter has never been reported in other brittle
ceramics, although a partially developed cone crack by Vickers
indentation in fused silica glass has only been reported. These anomalous
cone cracks of GLC and fused silica make an angle of about 30o with thespecimen surface, being significantly different from the Hertzian cone

100 μmangle (about 22o) usually made by a spherical indenter on brittle materials.

Figure 12.8 Scanning electron

These anomalous ring/cone cracks induced by a pyramid indenter are

micrographs

resulted from the microscopically open structures of GLC and fusedindentation impressions of GLC andsilica; the nano-porous structure in the former and the three-

SLG

dimensionally developed silica (SiO2) chain networks in the latter.

indentation (P=98.1N); (a) glassy

induced

of

the
by

a

residual
Vickers

carbon (GLC), (b) the in-surface
ring/cone cracks of GLC, and(b) Polycrystalline graphite [12.4]
As mentioned above, due to its non-graphitizable nature, GLC is very(c) soda-lime glass (SLG)

brittle and hardly machining. On the other hand, polycrystalline graphites,that are referred to as the graphitizable carbon, are highly plastic/ductile
and easily machining with high precision. They are extensively utilized
not only as the graphite blocks of nuclear power reactors, but also as
various types of engineering tools in semiconductor industries. Most ofpolycrystalline graphites are produced by baking molded green bodies[coal/petroleum-derived coke grains (several microns or less) bonded by
pitch binder] at about 1000oC, and then followed by graphitization attemperatures of 2000oC~3000oC.
Figure 12.9 shows the P − h loading/unloading hysteresis curves of
the isotropic carbon/graphite materials heat-treated at three different

145[12.4] M. Sakai, Y. Nakano, S. Shimizu,

J. Am. Ceram. Soc., 86[6], 1522 (2002)

880oC) has its P − h hysteresis curve similar to that of brittle ceramic,
while the P − h hysteresis is progressively enhanced with the increasein HTT, resulting in highly significant loop energy U r , i.e., leading tosignificant amount of plastic energy dissipation in the loading/unloading
processes. Since the HTTs exceeding 2000oC make highly developed
graphite crystals, the very unique as well as the significant elastic

Indentation load, P (N)temperatures. The carbon with HTT880 (Heat-Treatment-Temperature of
HTT88020

HTT1550

HTT2600

10

0
0

20

40

60

80

recovery is observed in their indentation unloading paths due to thePenetration depth, h (μm)

microscopic slippages along the hexagonal graphite basal planes. As

Figure 12.9 P − h loading/actually demonstrated in Figs. 12.9 and 12.10, the residual depth of

unloading hysteresis curves of theindentation impression ξ r ( = hr hmax ) progressively reduces to zeroisotropic carbons: their HTTs arewith the increase in HTT; the carbon material with well-developed

880oC,

graphite structure leaves no residual indentation impression after

respectively

complete unloading although the prominentP−h1550oC,and

2600oC,hysteresis iswith HTT that is resulted from the enhanced plastic deformation
associated with the development of graphite structures.
The well-defined residual indentation impression left behind on an
ordinary elastoplastic body means that the dislocation-derived plastic
flow is irreversible. In other words, the forwarded slip of dislocation in
the indentation loading process is not fully recovered or only partially
recovered in its unloading process, i.e., the slip of dislocation networks
is always irreversible in the ordinary elastoplastic bodies. For the wellgraphitized carbons as demonstrated in Figs. 12.9 and 12.10, on the other
hand, the slip of dislocation networks on graphitic basal planes ispartially or fully reversible, resulting in the disappearance of residual

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

1000

1500

2000

2500

Heat-treatment temperature, HTT(℃)

Figure 12.10 Normalized residual
depth ξr ( = hr hmax )and Meyer

hardness H M of the polycrystalline
carbon and graphite materials plotted

impression, i.e., ξ r ↓ 0 after complete unloading, though the significant

against the heat-treatment

plastic energy dissipation U r is observed in its loading/unloading process.

temperature (HTT)

It may be easy for us to understand this reversible dislocation slippages
once we notice the van der Waals forces acting on graphite basal planes.
In order to further appreciate these reversible dislocation slippages
observed in the well-graphitized carbon material, its P − h2 hysteresis
curves observed in the sequential “loading → unloading → reloading”
indentation processes are shown in Fig. 12.11 along with those of asilicon nitride ceramic for comparison. It is worthwhile noticing for the1460

Meyer hardness, H M (GPa)12.10, furthermore, the Meyer hardness H M progressively decreases

Normalized residual depth, ξr(=hr/hmax)

observed in its indentation loading/unloading process. As shown in Fig.

demonstrated in Fig. 12.11. These very unique indentation contact
behaviors of well graphitized carbons are resulted from the reversible
dislocation slippages along the basal planes of graphite crystals.

1000

d
lo a

10

0

g
din

lo
a
3000

siliconSi 3N4セラミック
nitride ceramic

0

in g

20

2

40

60

2)

圧入深さ,
h (μmh2 (μm2)
Penetration
depth,

Figure 12.11(3) Viscoelastic indentation contact mechanics

2000

g

paths like as that of the silicon nitride ceramic

in g

d in

unloading P − h

2

d
lo a

2
2)
圧入深さ,
h (μmh2 (μm2)
Penetration
depth,

lo ameans observed in the ordinary elastoplastic materials that have linear

20

re

re

unloading P − h2 path of the graphite material is very unique, by no

Indentation
load,
圧入荷重,
P (N)P (N)to-reloading paths. It must be also noticed that the significant concaving

g

un

0
0

d in

in

recognized as well as a very unique hysteresis loop along the unloading-

lo a

ad

however, for the well graphitized carbon (HTT2300), there exists a well-

10lo

behavior of the ordinary elastoplastic materials. On the other hand,

HTT2300
HTT2300g

behaviors is elastic, being a typical indentation unloading/reloading

20

ununloading path, implying that the unloading and the subsequent loading

圧入荷重,load,
P (N)P (N)Indentation

silicon nitride ceramic that the reloading path coincides with its preceding

P − h2hysteresis

curves (Vickers indentation)of inorganic glasses [7.4]
The detailed considerations on the glass transition behavior,observed in the sequential“loading→unloading→reloading”

viscoelastic theories and the indentation contact mechanic of amorphousindentation processes of the well-bodies have already been made in Chaps. 6 and 7.

graphitized carbon (HTT2300) andIn this section, theviscoelastic indentation contact mechanics at elevated temperatures will
be given of the inorganic glasses including silica glass.
Silica glass (vitreous pure silica; SG) is a covalent-bonded amorphousmaterial comprising three-dimensionally extended continuous random
silica networks, resulting in rather open microstructures as inferred fromits density (d=2.200 g/cm3) and molar volume (27.3 cm3/mol); the former
is the smallest and the latter is the largest among other silicate glasses
includingsoda-lime-silicate

temperature Tg

glass(SLG).

The

glass-transitionof SG is extremely high ( Tg =1050oC). Three-dimensionally extended silica networks can be modified by adding lowmolecular-weight oxides (the network modifiers) such as Na2O, CaO, andB2O3; these modifiers fragment the three-dimensional silica networks
leadingtodensification

of

the

microstructures

and

then

controlling/designing the material characteristics. By way of example,
SLG (so called soda-lime float glass or window glass) is manufactured
by adding a given amount of Na2O, CaO, and MgO into SG. The glasstransition temperature is Tg =540 oC and the density is d=2.500 g/cm3;147

of a silicon nitride ceramicthis significant decrease in Tg with an amount of about 500 oC iswhich will be given in what follows.
The softening curves of SLG and SG are shown in Fig. 12.12 measured
by a Berkovich indenter made of sapphire, since it is incapable of using
the conventional diamond indenter due to the oxidation in ambient air attemperatures exceeding 400 oC. The softening behavior is represented as
the penetration depth

hs (T )

vs. the scanning temperature T

relationship, in which a constant indentation load P0 = 0.98 N is applied
to the specimen at a temperature beneath the glass-transition point, andthen the monotonically increasing penetration is measured that is
associated with a constant-rate of rising temperature( T = qt ;q = 5.0 C min ). Due to the time/rate-dependent viscoelastic nature, thesoftening temperature Ts of glassy material is always affected by thescanning rate q , resulting in the lager Ts for the higher scanning rateof q . The softening temperatures Ts for the scanning rate ofq = 5.0 C/min demonstrated in Fig. 12.12 nearly coincide with the glasstransition temperatures Ts of SLG and of SG, respectively. It is worthyof note in Fig. 12.12 that SLG exhibits a sharply rising penetration depth

Penetration depth, h s (μm)

significant influences on its viscoelastic characteristics, the details of

Penetration depth, h s (μm)

attained by adding these network modifiers to SG, resulting in veryorete of temperature scanning: q=5.0

C/min

60
soda-lime silicate glass

50
40
30
20
10

0
500
520
540
60
silica glass
50

560

580

600

40
30
20
10
0
1000

1100

1200

1300

o

Scanning temperature, T ( C)

Figure 12.12 Softening curves ofsoda-lime silicate glass (SLG) andsilica glass (SG); penetration depth
hs (T )

vs. scanning temperature

T curves for temperature scanningq = 5.0 C minwith the rate ofunder a constant load ( P0 = 0.98 N)
of Berkovich indentation

curve against the scanning temperature. In fact, the scanning temperatureof about 60oC (corresponding to the scanning time of about 12min) is
enough for SLG to result in the penetration of 60μm, while SG requires
about 250oC (the scanning time of about 50min) to attain the same
penetration depth. This significant difference in the softening behavior isresulted

from

their

time/rate-dependent

viscoelastic

characteristics (i.e., relaxation-time spectrum, retardation-time spectrum;refer to APPENDIX D for the details) of these glasses having
significantly different their three-dimensional silica network structures;the spatially fragmented rather small silica-network clusters of SLG andthe spatially well-extended large clusters of SG.

10

Indentation load, P(t) (N)directly

500oC

520oC

soda-lime silicate glass
Berkovich indentation

o

540 C

kh=0.045 μm/s

560oC

8

570oC

6
4

580oC

2
0

590oC
0

20

40

60

80

600oC
100

Penetration depth, h(t) (μm)

Figure 12.13 shows the experimental results of SLG in the constant-Figure 12.13 Constant-rate ofrate penetration test ( h ( t ) = kh ⋅ t ; kh = 0.045 μm/s), where thepenetration test results of SLG

indentation load P ( t ) is plotted against the penetration depth h ( t ) .

( kh = 0.045 μm/s;

The plot of SLG at 570 oC combined with Eq. (11.17) leads to the stress148

Berkovich indentation)and the creep compliance function

C 'creep ( t ) shown in Fig. 12.14 and in Fig. 12.15, respectively. Furthermore in Fig. 12.16, the retardation time spectrum L (τ ) of SLG derived
from C 'creep ( t ) is given in comparison to that of SG.
As emphasized in Chap. 11, the accuracy and the reliability of theseviscoelastic functions given in Figs. 12.14 to 12.16 are by no means

Stress relaxation modulus
log E'relax(t) (Pa)relaxation modulus E 'relax ( t )Time, log t (s)satisfactory, since all of the experimental data are obtained on the

Figure 12.14 Stress relaxation
modulus E 'relax ( t ) of SLG at 570oC

conventional instrumented indentation apparatus that is incapable ofmeasuring the indentation contact area.

The details of moreaccurate/reliable determination for the viscoelastic functions will begiven in the subsequent section on the basis of the indentation contact
area directly observed via the instrumented indentation microscope.
The detailed considerations on the concept, rheological meanings andthe experimental determination of relaxation-time/retardation-time

-8

-10

-12
-2

0

2

4

6

SG

2

-11

-1

Pa )

T=Tg

L(τ) (10soda-lime silicate glass(T=570 ℃)-1

log Ccreep(t) (Pa )-6

Retardation time spectrum

spectra are given in Appendix D.

1

0

SLG

-2

0

2

4

Retardation time, log τ (s)Time, log t (s)Figure 12.15 Creep compliance

Figure 12.16 Retardation-time spectra of SLG and SG at thefunction of SLG at 570 C. The

respective glass-transition temperatures; Tg = 1050 oC: silicaodashedline

conversion

is

the

numerical

from

the

relaxation

glass (SG), Tg = 540 oC: soda-lime silicate glass (SLG)

modulus E 'relax ( t ) shown in Fig.
12.14 (refer to Eqs. (6.18) and (6.19))

149

(4) Instrumented indentation microscope applied to
the viscoelastic studies of amorphous selenium (a-Se) [7.5]
In Chap. 11 briefly mentioned was the creep compliance of a-Se as an
example for the application of instrumented indentation microscope toviscoelastic studies (refer to Fig. 11.14).

We will make a detailed

consideration on the stress relaxation function of a-Se in this section.
Selenium (Se) is a non-metallic element classified in the 16th group (theoxygen group; chalcogens) of the periodic table. Se forms a hexagonal ora monoclinic crystal as well as an amorphous state like as those of sulfurthat belongs in the same periodic group. Amorphous selenium (a-Se)
comprises entangled random chain networks resulting in viscoelastic
behavior at temperatures (25oC~40oC) around its glass-transition point of
Tg = 30.0 oC.
The instrumented indentation microscope is capable of not only
measuringindentation

load

and

penetration

depth,

but

also

programming/determining the in-situ indentation contact area as afunction of time. By way of example, it is possible to measure theindentation load relaxation P ( t ) under a stepwise application of a fixedcontact area of A ( t ) = A0 ⋅ u ( t ) ( u ( t ) : Heaviside step function) to thetest specimen, readily leading to the stress relaxation modulus E 'relax ( t )by substituting the observed P ( t ) into Eq. (11.24). The stress relaxation
moduliE 'relax ( t )of a-Se thus determined on the instrumented

indentation microscope (Berkovich indentation) are plotted against timein Fig. 12.17 for various temperatures of measurement including the
In the subsequent considerations, it will bedemonstrated that the respective relaxation curves shown in Fig. 12.17
satisfy the so-called “time-temperature superposition principle”.
First of all in the procedure of time-temperature superposition, therelaxation curve is chosen at a specific temperature, i.e., at the standard
temperature T0 , and then the other relaxation curves beside the standard
relaxation curve at T0 are shifted along the logarithmic time axis inorder to superimpose to this standard curve for making a single mastercurve; the curves at temperatures exceeding the standard temperature areshifted to the longer-time region, and vice versa for the curves at the150

Stress relaxation modulus
log E'(t) (Pa)glass-transition point.

108
temperature ( oC)
15252830(=Tg)
4
32
35
38
2
0
1
6

2

3

Time, log t (s)Figure 12.17 Stress relaxation
moduli of a-Se at various
temperatures of measurement

by applying this superposition procedure to the relaxation curves given
in Fig. 12.7 (the standard temperature T0 ( = Tg ) = 30.0 oC)).

In Fig.

12.18, the solid line is the master curve of a-Se obtained in the uniaxial
compression test (the macro test) of a rectangle test specimen with thedimensions of 1 x 1 x 3 mm3 for comparison. As well recognized in Fig.
12.8, the stress relaxation of the indentation test (the micro test) is moresignificant in the shorter time region of 0.1s ≤ t ≤ 100s than therelaxation of the macro test. This is resulted from the two-dimensional
unconstrained molecular motions on the free surface beneath the indenter,
whereas the stress relaxation proceeds with three-dimensionallyconstrained molecular motions in the macro test. The shift factor aT
utilized in the time-temperature superposition procedure (see Fig 12.18)
is related to the shear viscosityηthrough the relation of

Relaxation modulus, log E'(t) (Pa)lower temperatures. Figure 12.18 shows the master curve thus obtained10oT0=30 C

8 temperature of o642measurement( C)
15
25
28
30
32
35
38
E'(t) in uniaxial compression test
-2

0

2

4

Time, log (t/aT) (s)Figure 12.18 Master curves of thestress relaxation modulus of a-Se at()

T0 = Tg =30.0 oC in terms of theshift factor aT . The solid line is therelaxation master curve observed in
the uniaxial compression test (macro
test)

viscosity at the standard temperature). The temperature dependence ofthe shift factor aT utilized in Fig. 12.18 is shown in Fig. 12.19,
indicating that the shift factor aT of a-Se is well described with the
Arrhenius plot, aT = A exp ( ΔH T ) . It is worthwhile noticing in Fig.
12.19 that there exists a finite difference in the activation enthalpy ΔH
at the temperatures bounded above/below the glass-transition point Tg .

Shift factor, aT [ = η(T ) η(T0 )]aT = η (T ) η (T0 ) ( η (T ) : the viscosity at temperature T , η (T0 ) : the(Tg = 30.0o C

) ○

○

●
○
○

Temperature, 1 T (10−3 K −1 )This fact indicates that the rheological deformation/flow processes of the

Figure 12.19 Arrhenius plot of theviscoelastic liquid ( T > Tg ) are somewhat different from those of theshift factor aT utilized in Fig. 11.18

viscoelastic solid ( T < Tg ).
(5) Instrumented indentation microscope
applied to the viscoelastic/plastic studies ofpolycarbonate resin in the glassy state [12.5]
The glass-transition temperature Tg of polycarbonate resin (PCR) is
about 145 oC, meaning that it is a glassy solid at room temperature, themechanical characteristics of which are, therefore, time/rate- independentat room temperature in the conventional macro tests. Due to theunconstrained molecular motions at the free-surface of glassy solids
including most of organic polymers as well as a-Se mentioned above, on
151[12.5] C.G.N. Pelletier, J.M.J. Den Toonder,
L.E. Govaert, N. Hakiri, and M. Sakai,Phil. Mag. 88[9], 1291 (2008)

the other hand, their rheological characteristics are time/rate-dependent
well below the glass-transition point.
The indentation loading/unloading hysteresis curve of PCR at room
temperature is shown in Fig. 12.20; a spherical diamond indenter (tipradius 100μm) is penetrated with the rate of 1μm/s to the depth of 13μm

Indentation load, P (N)in the indentation contact tests (micro/nano tests) even at temperatures

Hertzian contact
1 * loading/unloading rate: 1 μm/s** holding time at thepeak penetration: 100s

0.8
0.6
0.4
0.2
0

(i.e., to the peak indentation load of 1.0N), and then held for 100s at this

510

Penetration depth, h (μm)

peak load followed by unloading with the rate of 1μm/s. As readily seen

Figure 12.20

in Fig. 12.20, the well-defined load relaxation is observed during the

P-h loading/unloading hysteresis

holding time of 100s at the peak depth of penetration. This fact implies

curve of polycarbonate resin underthat PCR behaves as a viscoelastic solid in the indentation test (i.e., in thespherical

micro test) even at room temperature, whereas PCR is a glassy solid attemperature. The dashed line is theroom temperature. The dashed line in Fig. 12.20 indicates the elastic

analyticalP - h analytical relation of the Hertzian contact, P = ( 4 3) E ' Rh3 2

contactindentation
solution

at

room

of

Hertzian60

80(refer to Eq. (3.18)); the purely elastic response dominates the initial stage
indentation

loading,

followedbytheviscoelastic/plastic

deformation/flow that becomes significant with the increase in thepenetration. The indentation load relaxation and the associated creep of
the contact area Ac proceeding at the fixed depth of h ( = h0 ) = 13 μm

1

Indentation load, P (N)

of

0.95

0.85

fixed penetration depth. This fact indicates the viscoelastic creeping-upof the free-surface of PCR along the spherical indenter’s side-surface dueto the time-dependent Poisson’s effect, the details of which have alreadybeen discussed in Sec. 10.4 through the FEA-based numerical results (seeFig. 10.14).

-3

2

Contact area, Acx10 (μm )are plotted in Fig. 12.21 against the holding time. It is interesting to notethat the contact area Ac monotonically increases with time under the

0.9

6.6
6.4
6.2
6
5.8

0

20

40

100

Holding timeat the peak depth of penetration, t (s)Figure 12.21 Indentation load
relaxation and the associated creep of(6) Instrumented indentation microscope applied tothe contact area of polycarbonate

the elastoplastic studies of coating/substrate composite [12.6]

resin observed at room temperature

The elastoplastic indentation contact mechanics will be given in thisunder the fixed penetration depth ofsection of the sol-gel-derived film/substrate laminates (methyl-h0 = 13 μm (the test result given insilsesquioxian (MeSiO3/2) film coated on various types of engineering

Fig. 12.20)

materials). In the conventional instrumented indentation testing forlaminate composites, due to the incremental increase of the substrateeffect with the increase in the indentation penetration, it has been well152

[12.6] H. Hakiri, A. Matsuda, and M.Sakai
J. Mater. Res., 24[6], 1950 (2009)

known that there exist crucial difficulties not only in estimating themechanical characteristics of the coating film and of the substrate
respectively, but also in measuring the elastic modulus and the Meyer
hardness of the composites as a function of the penetration depth (refer tothe details in Chap. 9).

On the other hand, once we use the instrumented

indentation microscope, we can determine in experiments the respective
mechanical characteristics (elastic modulus, yield stress, viscoelastic
functions, etc.) of the film and of the substrate in a precise manner (refer
to Chaps. 9 and 10).
MeSiO3/2 films (elastic modulus;

E 'f = 3.9 GPa) with various

thicknesses ( tf = 3 ~15μm) are coated on the five engineering substrates;
poly carbonate resin PCR ( E 's =3.5GPa； E 's E 'f =0.95), polyacrylate
resin PAR (4.5GPa；1.2) , polyphenol resin PPR (7.1GPa；1.9), glasslike carbon GLC (28.7GPa ； 7.7), and soda-lime silicate glass SLG(80.5GPa；22). Using these laminate composites having a rather wide

examine the substrate-effect on the indentation contact mechanics. The
indentation load P vs. contact area A hysteresis curves in theloading/unloading processes of Berkovich indentation are plotted in Fig.

Indentation load, P (N)range of film/substrate modulus ratio 0.95 ≤ E 's E 'f ≤ 22 , let us

0.4

tions are both linear for cone/pyramid indentation; the slope of theloading linear line gives the Meyer hardness H M and that of the

Indentation load, P (N)homogeneous bodies, the elastoplastic loading/unloading P - A rela-

In contrast to

0.2
0.1
1000

MeSiO3/2

0.2

500
1000
Indentation contact area, A (μm2)

Figure 12.22 Indentation load P

semi-infinite homogeneous body, on the other hand, the loading/vs.

unloading P - A hysteresis relations of the laminate composites are notrelationslinear due to the substrate-effect, as shown in Fig. 12.22, where the P A hysteresis of MeSiO3/2/PCR laminate is somewhat linear due to its
film/substrate modulus ratio ( E 's E 'f = 0.95 ) being nearly 1.0, i.e.,
film/substrate modulus matching.

On the other hand, the nonlinear

P - A hysteresis of the MeSiO3/2/SLG laminate composite is very

153

2000

MeSiO3/2 / SLG laminate(film thickness, tf=9.5 μm)
0.40unloading line, i.e., the unloading modulus M provides the elastic
modulus E ' through the relation of M = ( tan β 2 ) E ' .

MeSiO3/2

012.22 for MeSiO3/2/PCR and MeSiO3/2/SLG laminate composites by way
of example. As clearly demonstrated in Fig. 11.13 for semi-infinite

0.3

MeSiO3/2 / PCR laminate(film thickness, tf=10.0 μm)

contactofarea

A hysteresis

MeSiO3/2/PCR

andMeSiO3/2/SLG laminate composites
in the multi-step loading/unloading
cycle tests (Berkovich indentation).
The dashed line is the

P - Aloading linear relation of the semiinfinite homogeneous MeSiO3/2

significant

that

is

( E 's E 'f = 22 ).

resultedfrom

The slope of thethemodulusmismatching

P - A loading path of

MeSiO3/2/PCR laminate slightly decreases with the increase in the
indentation penetration due to the little bit lower modulus of the substrate,i.e., E 's E 'f = 0.95 . On the other hand, for MeSiO3/2/SLG laminate,reflecting its larger modulus mismatching ( E 's E 'f  1 ), the slope of the
P - A loading line progressively and significantly increases withpenetration, approaching to the slope of semi-infinite homogeneous SLG.
Based on the detailed considerations made in Sec. 11.3 for the
instrumented indentation microscope, we can make the quantitative
assessment of the elastoplastic characteristics of laminate composites. As
mentioned above, the P - A relation of laminate composite is not linear
due to the substrate-effect, whereas it is possible for us to estimate theMeyer hardness H M ( A ) from the tangential slope of the P - A
loading line at a given value of the contact area A :H M ( A) = dP dA(12.1)

In other words, due to the substrate-effect, the Meyer hardness oflaminate composite is dependent on the contact area A : the Meyer
hardness is approaching to the value of the coating film by extrapolatingA ↓ 0 , and to the value of the substrate in the extreme of A ↑ ∞ . On
the other hand, as shown in Fig. 12.22, the unloading modulus
M ( A ) ( ≡ ( dP dA ) ) (the initial slope of the unloading curve at the peakindentation load for the respective multi-step loading/unloading cycles)
combined with the following relation (see Eq. (11.18))(12.2)

E '( A) = 2 cot β ⋅ M ( A)gives the elastic modulus E '( A) of the laminate at a given value of thecontact area A . Like as the Meyer hardness, the elastic modulus E '( A)approaches to E 'f

for A ↓ 0 , and is extrapolated to E 's withA ↑ ∞ . We can also determine the yield stress Y ( A) of the laminate by

substituting the values of H M ( A) and M ( A) thus determined in theabove procedures into Eq. (11.19).

154

The Meyer hardness H M ( a tf ) and the effective elastic modulus

E 'eff ( a tf ) of the MeSiO3/2-coated laminates are plotted in Figs. 12.23
and 12.24, respectively, against the normalized contact radius a tf .

E ' ( A) andNotice that the interrelation between

E 'eff ( A)hasNormalized effective modulus
E'eff (a/tf)/E'f

Meyer
hardness, HM(a/tf) (GPa)Meyer硬度,

already been given in Sec. 9.3 with Eq. (9.19).

E' s/E'f=22

0.5

7.7

0.4
1.9

0.3
1.2

0.2
0.1

0.95
film only

0

1

2

3

4

5

Normalized
contact radius, a/tf
無次元圧子接触半径,E's/E'f=22
10

7.7

5
1.9
1.2

1

0.95

0.5
0

0.5

1

1.5

Normalized contact radius, 1/(a/tf)

Figure 12.23 Meyer hardness is plotted

Figure 12.24 Normalized effective modulus

against the normalized contact radius a tf ofE 'eff ( a tf ) E 'f vs. normalized contact radius

the MeSiO3/2-coated laminate composites

1 ( a tf ) relations of the MeSiO3/2- coateddetermined on the indentation microscope. The

laminate composites (refer to Fig. 12.23 for thesymbols from the top to the bottom indicate therespective

materials used for the respective substrates;

symbols)

determined

on

the

●: soda-lime silicate glass (SLG)

indentation microscope. The solid lines are the△ : glass-like carbon (GLC)

numerical solution of the Fredholm integral■: polyphenol resin (PPR)

equation given in Eq. (9.11)▽ : polyacrylate resin (PAR)◆ : polycarbonate resin (PCR)

155

(7) Instrumented indentation microscope applied tothe soft matters with surface adhesion
The technology of cell culture media has been advancing in these few
decades. It is critically essential in the studies of cell culture to find agood medium, since the optimized mechanical characteristics of the geladhesion (surface energy) play essentially important roles in the studies
of stem cell such as the induced pluripotent stem cell (iPS cell).

Indentation load, P (mN)medium such as the contact hardness, elastic modulus and the surface

aloe-gel (Berkovich indentation)

0

-0.1

-0.2

0

(i) Elastoplastic indentation contact characteristics of aloe-gel

12[×105]

Contact area, A ( μm)

The contact test results of an aloe-gel as a model bio medium are

Figure 12.25 P - A loading-

given in this section through utilizing the indentation microscope forunloading hysteresis (Berkovich

determining the adhesive surface energyγalong with theelastoplastic characteristics of E ' and Y ).

indentation) of aloe-gel. The
indentation

An example of the P - A loading-unloading hysteresis (Berkovich
indentation) is shown in Fig. 12.25. In general, both the P - A loading

load

is

alwaysnegative ( P < 0 ) due to surface
adhesionand the unloading relations of cone/pyramid indentation are linear as welldemonstrated in Fig. 11.13 for elastoplastic bodies without surface
adhesion. However, as readily seen in Fig. 12.25, the surface adhesion of
Furthermore, it must be noticed the fact that the observed indentationload always turns to negative, implying that the tip-of-indenter is pulled
to the contact surface due to surface adhesion (refer to Eq. (8.29) in Chap.

Indentation load, P (mN)

the tested aloe-gel leads to a significant nonlinear P - A hysteresis.

aloe-gel (Berkovich indentation): Observed P-A plot
: JKR-corrected P-A plot

0.4

: P+λA3/4(=HMA)

3/2: P=HM-λA

0

-0.2
0

8 for the modified JKR-theory);
P = H M A − λEP A3 4

(8.29)

HM

0.2

1
2
Contact area, A(μm2) [×105]

Figure 12.26 P - A loading

The non-linear P - A loading relation shown in Fig. 12.25, therefore,

relations of aloe-gel (Berkovich

stems from the elastoplastic adhesion toughness λEP . The linear P vs.

indentation). The closed symbols

A loading line, i.e., P = H M A of the aloe-gel is plotted in Fig. 12.26( ● ) and the dashed curvethat is derived by applying Eq. (8.29) to the observed P - A non-linear
loading line in Fig. 12.25 and then making the numerical correction for34
the non-linear adhesion term, λEP A . The Meyer hardness H M andthe adhesion toughness as well as the surface energy γ can thus be

indicatetheexperimental

observation obtained on the
indentation

microscope.

The

open symbols (◯) and the solidlinear line stand for the relation

successfully determined in experiments through these linearization

with the JKR-correction for theprocedures.

adhesion toughness λ ;

One

can

determineinexperimentstheelastic

156

modulusP + λEP A3 4 = H M A

E '  = ( tan β 2 ) M  in a quantitative manner via the unloading modulus
M as the slope of unloading linear P - A line (refer to Fig. 11.13)
whenever the surface adhesion of the specimen is negligibly smallenough. However, as shown in Fig. 12.25, the surface adhesion not only
leads to the nonlinear loading P - A relation, but also results in a very
steep unloading slope that leads to a significant overestimate of themodulus M , yielding the fatal error in determining the elastic modulusE'.

As a matter of fact, the unloading modulus estimated from the M-value turns to infinity when the surface adhesion of the specimen tested
is large enough, since the tip-of-indenter sticks to the contact surface andthe contact area

A ≡ Amax

A keeps its maximum value

duringunloading. In such a specific unloading contact process proceeding witha constant contact area, the unloading stiffness S defined by theunloading slope of the indentation load P vs. the penetration depth his quantitatively related to the elastic modulus E ' as follows (refer toChap. 11 and Eq. (11.3)) for the axisymmetric indenter with any geometry
including flat-ended cylinder, sphere, cone/pyramid, etc.;

E'=π
2 AmaxS(12.3)The P - h loading/unloading hysteresis of the aloe-gel is shown in
Fig. 12.27, indicating the unloading stiffness S as the initial slope of
penetration depth h as well as the unloading stiffness S throughutilizing the instrumented indentation microscope, the application ofthese experimental data to Eqs. (8.29) and (12.3) yields the Meyer

Indentation load, P (mN)the unloading path. Since one can measure the contact area Amax ,-0.1

Chap. 5) as well as the adhesive surface energy γ of the test specimen,
i.e., aloe-gel;

S

-0.2

hardness H M , elastic modulus E ' , yield stress Y (refer to the detailsof the additivity principle of the excluded volume of indentation given inaloe-gel (Berkovich indentation)

0

0

10

20

30

40

50

Penetration depth, h (μm)

Figure12.27
Loading /unloadingP - hhysteresis of aloe-gel (Berkovich

Elastic modulus

E ' =25.5kPa

Meyer hardness

H M =2.40kPa

Yield stress

Y =1.25kPa

Adhesive surface energy

γ =15.6mJ/m2

157

indentation).

The

unloading

stiffness S combined with Eq.
(12.3) yields the elastic modulusE'(ii) Viscoelastic indentation contact characteristics ofpolyurethane-gel
The creep test results (spherical indentation) on the instrumented
indentation microscope are given in this section of a commerciallyavailable sticky polyurethane gel mat that is to be utilized as vibration
dumping mediums.
3

The creeping contact radius a0 ( t )of a spherical indenter combinedwith Eq. (8.46) in Chap.8 is capable of determining in experiment not
only the creep function C ' ( t ) but also the surface energy γ (and/or
the adhesive surface force Γ ) ; these material characteristics can be3

obtained by solving the simultaneous equations of a0 ( t )and a1 ( t )3

in Eq. (8.46) for the stepwise indentation loads of P0 and of P1 ,respectively.

An example of the creep curve a0 ( t ) under theindentation load P0 = 0.11 mN of the spherical indenter (tip-radius,R = 3 mm) is shown in Fig. 12.28 with the indentation contact images in

time.

3

Figure 12.29 shows the creep curves of a0 ( t )and a1 ( t )3

forP0 = 0.11 mN and P1 = 0.05 mN, respectively. It must be noticed in Fig.
3

0.4

a(t)
P0

0.3

0.1

0.2
0.1
0
0

200

400

0
600

Contact radius, a(t) 3 x10-7 (μm3)

0.2

Indentation load, P 0 (mN)Contact radius, a(t)x10 -3 (μm)

12.29 that the a ( t ) -creep curve is not linearly proportional to theTime, t (s)Figure 12.28
a (t )

Creep curve of the contact radius

P0=0.11mN

6

4

P1=0.05mN

2

0
0

200

400

600

Time, t (s)Figure 12.29

Creep curves of the contact3

of polyurethane-gel for stepwise loading

radius a ( t ) of polyurethane-gel for stepwise（ P0 = 0.11 mN）(spherical indentation with the tip-

loading of P0 =0.11mN and P1 =0.050mN

radius of R = 3 mm). The contact images at the(spherical indentation with the tip-radius ofrespective creeping times of t = 0, 200, 400, andR = 3 mm)600s are shown on top of the graph158

applied load due to the significant effect of surface adhesion on the creep
deformation, the details of which have already been discussed in Chap. 8.
In this creep test, a significant increase in the contact radius a ( t ≈ 0)
is observed even right after the tip-of-indenter contacts to the specimen’ssurface and just before the load is applied to the indenter ( P ≈ 0 ). Thisobservation confirms that the adhesion force of the polyurethane gel mat
plays an essential role in its creeping behavior.
The creep function C ' ( t ) thus obtained by applying Eq. (8.46) to theobserved creep curves shown in Fig. 12.29 is plotted in Fig. 12.30 alongwith the relaxation modulus E ' ( t ) in Fig, 12.31 that is converted fromC ' ( t ) through Laplace transform and its inversion (refer to Eqs. (6.18)
and (6.19) in Chap. 6 for the details). Once we determine the creep
function C ' ( t ) in the preceding procedures, by applying it to Eq. (8.46),
we successfully estimate the surface energy (adhesive energy) of thepolyurethane gel (26.5 ℃)

polyurethane gel (26.5℃)

Relaxation modulus, E'(t) (kPa)

Creep function, C'(t)x10 3 (mPa-1)

polyurethane gel mat as γ =6.61mN/m (=6.61mJ/m2).

30

20

10

0

0

200

400

600

Time, t (s)

150

100

50

0
0

100

200

300

400

500

600

Time, t (s)Figure 12.31 Relaxation modulus E ' ( t )Figure 12.30 Creep function C ' ( t ) of
polyurethane-gel

converted from the creep function C ' ( t )(the temperature of measurement: 26.5℃)

shown in Fig. 12.30 through Laplacetransform and its inversion (the temperature ofmeasurement: 26.5℃)

159

KINEMATIC EQUATIONSIN CYLINDRICAL COOEDINATEA

APPENDIX(1) Displacement and Strain
Suppose a point A is displaced to the point A’ in a continuum body bystretching and torsion, as shown in Fig. A1. This displacement (u, v ) in
the Cartesian coordinate is related to that in the cylindrical coordinate(ur , uθ ) ;u = ur cos θ − uθ sin θ
v = ur sin θ + uθ cos θ(A1)

On the other hand, there exist the following interrelations between the
Cartesian and the cylindrical coordinates;r 2 = x 2 + y2
x = r cos θ
y = r sin θtan θ = y x(A2)

Substituting Eq. (A1) into Eqs. (1.2) ~ (1.5), the strains in the Cartesian
coordinate, ε x , ε y , ε z , γ xy , γ yz , and γ zx can be described interms of ur , uθ , and θ

through the following mathematicaloperations.
For the normal strain in the x-direction, as an example, is represented
by

εx =

∂u ∂u ∂θ ∂u ∂r=+∂x ∂θ ∂x ∂r ∂x(A3)

Furthermore, Eq. (A1) combined with Eq. (A2) gives the following
expressions;y∂u∂u ∂urcos θ − ur sin θ − θ sin θ − uθ cos θ
=
∂θ
∂θ
∂θ
∂uθ
∂u ∂urcos θ −
sin θ
=
∂r
∂r
∂r
sin θ
∂θ
=−
∂x
r
∂r
= cos θ
∂x

●

ur

uθ

(A4)

Substituting Eq. (A4) into Eq. (A3), therefore, we finally have

r

●

θ

A

θ

r

u

x

Figure A1 Displacement of a point A
to the point A’ in a continuum bodyby stretching/torsion in Cartesian( x, y )andcoordinates

161

A'

v

cylindrical( r ,θ ) ∂ur sin θ∂ucos θ − ur sin θ − θ sin θ − uθ cos θ 
 ∂θ
 r
∂θ


 ∂ur

∂uθ
cos θ −sin θ  cos θ
+
 ∂r

∂r



εx = − (A5)Once we notice ε x → ε r as θ → 0 in Fig. A1, we obtain(

εr = ε x) = ∂∂ur

r

θ →0

(A6)

via θ → 0 in Eq. (A5). Analogously, noticing ε x → εθ as θ → π 2 ,we haveεθ  = ε x 1 ∂uθ ur=+
θ →π 2 r
 r ∂θ

(A7)

After applying a similar procedure conducted above to the shear(

strain γ xy = ∂u ∂y + ∂v ∂x) in the xy -plane (see Eq. (1.3)), we

have the following expression; ∂ur cos θ∂ucos θ − ur sin θ − θ sin θ − uθ cos θ 
 ∂θ
 r
∂θ


 ∂ur

∂uθ
+
cos θ −sin θ  sin θ
 ∂r

∂r


 ∂ur
  sin θ ∂uθ+sin θ + ur cos θ +cos θ − uθ cos θ   −



r 
∂θ
 ∂θ

 ∂u

∂u
+  r sin θ + θ cos θ  cos θ
 ∂r

∂r



γ xy = (A8)
and then noticing γ xy → γ rθ as θ → 0 , Eq. (A8）leads to(γ rθ ≡ γ xy

) = ∂∂ur + r1 ∂∂uθ − ur
θ

θ →0

r

θ

(A9)

that has already been given in Chap. 1 (Eq. (1.9)).
The expressions given in Eqs. (A6), (A7), and (A9) for the strains in
the cylindrical coordinate are derived from the strains in the Cartesian
coordinate by the use of coordinate-conversion from the (x , y ) -space tothe (r, θ ) -space (Eqs. (A1) and (A2)), in mathematical manner, notderived from the actual displacements and deformations of continuum
body. In order to make clear the physics of these strains of ε r , ε θ , and162

γ rθ associated with the deformation of continuum body, therefore, let us
derive again these strains ε r , ε θ , and γ rθthrough using thedisplacements/deformations depicted in Figs. A2 ~ A5.
The normal deformations are depicted in Figs. A2 and A3; thequadrangle abcd in Fig. A2 is deformed to the quadrangle a’b’c’d’
through the normal displacement ur from bc (length dr ) to b’c’(length δ r ) along the radial direction, and from ab (length rdθ ) to a’b’(length(r + ur ) dθ ) along the azimuthal direction. Accordingly, theradial strain ε r is given by

εr =

δr

=

( ∂u ∂r ) dr = ∂u
r

r

(A6)

∂r
dr
dr
and the azimuthal strain εθ 1 is described byεθ 1 =(r + ur ) dθ − rdθ = ur

(A10)
rdθ
r
that is associated with the radial displacement of ur . In addition to theazimuthal strain εθ 1 induced by ur , there also exists the strain εθ 2induced by the azimuthal displacement of uθ as shown in Fig. A3;

εθ 2 =( ∂u ∂θ ) dθ = 1 ∂uθrdθ

The azimuthal strain ε θ

θ
(A11)
r ∂θ
is, therefore, given by the sum of εθ 1 andεθ 2 , as follows;εθ ( = εθ 1 + ε θ 2 ) =

1 ∂uθ

r ∂θ

+

Figure A2 Normal deformation ofthequadrangle

abcd

to

the

quadrangle a’b’c’d’ through theradial displacement ur

ur

(A7)

r

The shear deformations are depicted in Figs. A4 and A5; thequadrangle abcd in Fig. A4 is deformed to the quadrangle a’bcd’ throughthe shear displacement dur along the radial direction, and the same
quadrangle abcd in Fig. A5 to the quadrangle a’b’c’d’ through the shear
displacement uθ along the azimuthal direction. As shown in Fig. A4,
the shear strain γ r associated with the displacement dur along theradial direction is given by1  ∂u 

γr =  r ,
r  ∂θ (A12)

On the other hand, as shown in Fig. A5, the shear strain γ θ through the

Figure A3 Normal deformation of

azimuthal displacement uθ is described withthequadrangle

abcd

to

the

quadrangle a’b’c’d’ through theazimuthal displacement uθ163γθ = β − α =

∂uθ
∂r

−

uθ

(A13)

r

where β indicates the relative shear angle between the lines cd and abof the quadrangle abcd that is induced by the azimuthal displacement uθ ,while α is the angle of rigid rotation of the quadrangle abcd associatedwith the azimuthal displacement uθ , leading to the resultant shear angleγ θ ( ≡ β − α ) as given in Eq. (A13). Accordingly, the total shear strain

γ rθ in the (r, θ ) -plane associated with the displacements along the r and θ - directions is finally given byγ rθ = γ r + γ θ =1 ∂ur ∂uθ uθ
+
−
r ∂θ
r
∂r

(A9)

Figure A4 Shear deformation of the

Figure A5 Shear deformation of thequadrangle abcd (the solid lines) toquadrangle abcd (the solid lines) tothe quadrangle a’bcd’ (the broken

the quadrangle a’b’c’d’ (the brokenlines) associated with the radiallines) associated with the azimuthal

displacement dur

displacement uθ164

(2) Equilibrium Equations
The mechanical equilibrium shown in Fig. A6 along the radial
direction results in



∂σ
 σ r + r dr  r + dr dθ − σ r rdθ 

∂r





∂σ
dθ
dθ 
−  σ θ + θ dθ  sin dr + σ θ sin dr 

∂θ
2
2






∂τ
dθ
dθ
+ τ θ r + θ r dθ  cos dr − τ θ r cos dr 


∂θ
2
2




(

)

(A14)

+Rrdθ dr = 0

and then,σ r − σθ
r

+

∂σ r
∂r

+

1 ∂τθ r ∂τ zr++R = 0
∂z
r ∂θ

(A15)

by noticing dθ  1 , sin ( dθ 2 ) ≈ dθ 2 , cos ( dθ 2 ) ≈ 1 , and by letting
2

the higher-order variables ( ( dr ) , drdθ ) to be negligible, although theshear stress τ zr on the z-plane (the plane with its normal vector directed
to z-axis) is not depicted in Fig. A6 for simplicity (Eq. (A15) has alreadybeen given in Chap. 1 (see Eq. (1.11)).
The equilibrium equation for azimuthal direction ( θ -direction) is also
given by




∂σ
θ
 σ θ + θ dθ  dr − σ θ dr  cos


2
∂θ






∂τ
+ τ rθ + rθ dθ  r + dr dθ − τ r θ rdθ 

∂r






∂τ
dθ
+ τ θ r + θ r dθ  dr + τθ r dr  sin


∂θ
2




(

)

(A16)

+Θrdθdr = 0

After a similar consideration we made above on the radial direction, we
finally have the following equilibrium equation for azimuthal direction;

2τθ r
r

∂τθ r

1 ∂σ θ ∂τ zθ
+
+
+
+Θ=0
∂r
∂z
r ∂θ

through taking into account ofdθ  1 ,(A17)
sin ( dθ 2 ) ≈ dθ 2 ,cos ( dθ 2 ) ≈ 1 , the contribution of shear stress τ zθ on the z-plane, as

165

Figure A6 Stress components ( σ r ,σ θ , τ θ r , τ rθ ) acting on theinfinitesimal quadrangle abcd withunit thickness under the external
body forces ( R , Θ )well as the symmetric nature of the stress tensor, τ rθ = τ θ r .
(A17) has already been given in Chap. 1 (see Eq. (1.11))

166

Equation

BESSEL FUNCTION

APPENDIX

Daniel Bernoulli first introduced the concept of Bessel functions in1732 to solve the problem of an oscillating chain suspended at one end,
although Bessel functions are named for Friedrich W. Bessel (1784-1846).
In 1824, F.W. Bessel incorporated Bessel functions in the Kepler’s
perturbation problems for determining the motion of three planets
moving under mutual gravitation. In 1878, Lord Rayleigh (John W.
Strutt) employed Bessel functions in the analysis of a membrane
stretched within a cylinder, that has been the historical origin of theapplication of Bessel functions to solving the bi-harmonic equation
through Hankel transformation as demonstrated in Chap. 2 of thistextbook.
Bessel equation (see Eq. (2.22)) is defined by the second orderdifferential equation given as

d 2 y 1 y  n2 ++ 1 −  y = 0
dx 2 x dx  x 2 (B1)

The solution of Eq. (B1) is described in terms of Bessel functions of thefirst and the second kind, J n ( x) and Yn ( x) of the n-th order, as
follows;

y = AJ n ( x) + BYn ( x)with arbitrary integral constant A(B2)
and B . The second kind Bessel

function Yn ( x) is sometimes referred to as the Weber function or the
Neumann function, and related to the first kind Bessel function in thefollowing formula;Yn ( x) =J n ( x) cos(nx) − J − n ( x)sin(nx)(B3)

The function J n ( x) is always finite for all values of n , while thefunction Yn ( x) becomes singular, i.e., becomes infinite at x = 0 , asreadily seen in Eq. (B3). In order to describe the problems ofphysics/mechanics, therefore, we need to set the integral constant B as

B = 0 in Eq. (B2). In other words, only the first kind Bessel function
J n ( x) is essential in analyzing any of physical/mechanical phenomena
and problems without singularity.
The Bessel function of the first kind of order n can be determined using
an infinite power series expansion as follows;

167

B

∞

J n ( x) = (−1)k ( x 2 )

k =0

n+ 2k

(B4)

k !(n + k )!

Equation (B4) leads to the following recurrence formulas, i.e., Bessel
functions of higher order are expressed by Bessel function of lower order
for all real values of n;d  n
x J n ( x)  = x n J n −1 ( x)dx d  −nx J n ( x)  = − x − n J n +1 ( x)
dx 

(B5)

The equivalent alternatives of Eq. (B5) are2n
J n ( x) − J n −1 ( x)x
dJ n +1 ( x) 1
= [ J n −1 ( x) − J n +1 ( x)]
dx
2

(B6)

dJ n ( x)n= J n −1 ( x) − J n ( x)dxxdJ n ( x) n
= J n ( x) − J n +1 ( x)
dx
x

(B7)

J n +1 ( x) =or

Using these recurrence formulas, it will be easy to prove that the firstorder

Hankeltransformofthederivativedϕ / dr

,

i.e.,

∞

 r ( dϕ / dr )J (ξ r ) dr , is expressible in terms of the zeroth-order∞
Hankel transform  rϕ J 0 (ξ r ) dr of the function ϕ , as follows0 r ( dϕ / dr )J (ξ r ) dr = −ξ  rϕ J (ξ r )dr
0

1

∞

0

∞

1

0

0

This formula has already been utilized in Chap. 2 for describing thedisplacements, ur and u z , in cylindrical coordinate. As emphasized in
Chapt.2, Hankel transform plays an important role in solving theharmonic/bi-harmonic equations in cylindrical coordinate.

168

LAPLACE TRANSFORMS

APPENDIX

The Laplace transform, like the Fourier series, give us a method oftackling differential/integral equation problems we couldn’t otherwise
solve. In the linear response theory such as the linear viscoelastic theory,
the Laplace transform is a very powerful tool that enables us to solve
complicated viscoelastic problems, as demonstrated in Chap. 6.
Linear differential/integral equations in real space are transformed tothe algebraic equations in Laplace space, and then the solutions of these
algebraic equations are inversely transformed to the real space resulting
in the solutions of these original differential/integral equations.

(1) Definition of Laplace TransformAn arbitrary function f ( t )with the variable t in real space istransformed to f ( p ) with the variable p in Laplace space throughusing the Laplace operator L as follows;∞

L f (t ) = f ( p) =  f (t )e− pt dt

(C1)

0

The variable p in Laplace space has the inverse physical dimension of
the variable t in real space; p has its physical dimension of [1/s],
provided that t is referred to as time [s]. The variable pt in the
integrand in Eq. (C1) is therefore dimensionless.

(2) Fundamental Properties
The most fundamental properties of Laplace transform are listed
below;(i)

L c1 f1 ( t ) + c2 f 2 ( t )  = c1 f1 ( p ) + c2 f 2 ( p )(C2)where c1 and c2 are constant(ii) df (t ) = pf ( p ) − f ( 0− ) dt 

L

( ) ε

(C3)

with f 0− = lim f ( 0 − ε )→0df ( t ) d 2 f (t ) −2
 = p f ( p ) − pf ( 0 ) −2dt
 dt 

L
(iii)

t = 0−L   f ( t − t ') g ( t ') dt ' = f ( p ) ⋅ g ( p )
t

 0

(C4)
(C5)



169

C

t

in which the integral  f ( t − t ')g ( t ') dt ' is the0

so-called convolution integral; the Boltzmann’s hereditary
integral (Eqs. (6.15) and (6.20) in Chap. 6.3) typifies the
convolution integral.

(3) Some Useful Transform Pairs

Table C1 Laplace Transform Pairs useful in The Linear Response Theoryf (t )f ( p)1u (t − a) ; Heaviside step function

e−ap p2

δ ( t )( ≡ du (t ) dt ) ; Dirac function

1

3

1

1 p

4

t n ; n = 1, 2,3 ⋅ ⋅ ⋅n! pn+15e at

1 ( p − a)6(e − e ) (b − a ) ; a ≠ b(be − ae ) ( b − a ) ; a ≠ b

1 ( p − a )( p − b ) 

7

bt

bt

at

p ( p − a )( p − b ) at

(p +a )p (p +a )8

sin at9

cos at10

J 0 ( at ) ; zeroth-order Bessel functiona1(4) An Example of the Application of Laplace Transformto Linear Differential Equation
Let us solve the following linear differential equation (see Eq. (6.1)) asan example:
a0 ε ( t ) + a1

d ε (t )dt= b0σ ( t ) + b1

dσ (t )

(C6)dt

Laplace transform of Eq. (C6) results in the following algebraic equation;

170

2

2

2

2

p2 + a2( )a0ε ( p ) + a1  pε ( p ) − ε 0− = b0σ ( p ) + b1  pσ ( p ) − σ 0− 



(C7)

( )

Since none of external loads and displacements are applied to the body inthe past ( t < 0 ), Eq. (C7) with the initial condition of( )( ) leads to the following algebraic equation in the

ε 0− = 0 = σ 0−
Laplace space;σ ( p) = E * ( p)ε ( p)(C8)a +a pE * ( p) = 0 1
b0 + b1 p

In Eq. (C8), E * ( p ) stands for the pseudo elastic modulus (refer to Eq.
(6.4)), and the constants a0 , a1 , b0 , and b1 are related to theviscoelastic characteristics such as the elastic modulus, viscosity and therelaxation time (see Eqs. (6.2a) ~ (6.2c)) of the viscoelastic model(Maxwell/Zener models; refer to Eqs. (6.5a) ~ (6.5c)).
Suppose a step-wise strain ε ( t ) = ε 0 u (t ) in Eq. (C6), and then notice
its Laplace transform ε ( p) = ε 0 p via Table C1#1. Equation (C8) is
therefore rewritten bya0a1
σ ( p)=+
ε0b0 + b1 p p ( b0 + b1 p )(C9)The inverse Laplace transform of Eq. (C9) via Table C1 finally results inthe solution of the linear differential equation (Eq. (C6)) as follows;σ ( t ) a0  a1 a0  t=
+  −  exp  − 
ε0b0  b1 b0 
 τ

(C10)

b
τ= 1
b0

Equations (6.2a) ~ (6.2c) applied to the constants a0 , a1 , b0 , and b1
in Eq. (C10), therefore, finally give the constitutive expressions for the
stress relaxation behaviors of the respective viscoelastic models we
discussed in Chap. 6.

171

RELAXATION TIME SPECTRUM AND
RETARDATION TIME SPECTRUM

APPENDIXD

It is incapable of quantitatively describing the viscoelastic behaviors
of engineering materials in terms of a simple viscoelastic model like as
the Maxwell/Zener model discussed in Chaps. 6 and 7. We need,
therefore, to adopt a more general viscoelastic model such as the
Wiechert model (see Fig. 6.5 in Chap. 6) with multiple relaxation timethrough introducing the concept of “time spectrum” to the constitutive
equations, Eqs. (6.8) ~ (6.13) in Chap. 6, for expressing the relaxation
modulus and the creep compliance [A1].
The Wiechert model (refer to Chap. 6) well describes the generalized
relaxation modulus Erelax ( t ) in terms of a discrete distribution ofrelaxation times;

Erelax ( t ) = Ee +  Ei e−t τ i ,(6.13)iwhile its corresponding expression with a continuous time distribution is
given by∞

Erelax ( t ) = Ee +  M (τ )e − t τ dτ

(D1)

0

in Eq. (D1), the function M (τ ) is referred to as the “relaxation time
distribution” that stands for the elastic modulus Ei ( ≡ ηi τ i ) with therelaxation time τ i in the discrete model (Eq. (6.13)). The distributionfunction M (τ ) is a kind of the memory function having the physicaldimension of [Pa/s], being not the dimension of elastic modulus [Pa],though the function M (τ ) in its physical meaning stands for the elastic
modulus itself.

To circumvent such a confusing physical meaning, Eq.

(D1) is rewritten to∞

Erelax ( t ) = Ee +  H (τ )e − t τ d ln τ

(D2)

−∞

by introducing the relaxation time spectrum H (τ )  ≡ τ M (τ )  with thephysical dimension of [Pa]. By the use of the relaxation time spectrum
H (τ ) instead of using the time distribution M (τ ) , we can, therefore,

successfully express the elastic modulus of the springs having theirrelaxation times ranging from ln τ to ln τ + d ln τ .
In a similar manner to the relaxation modulus, we can describe thecreep compliance as follows,
172[A1] J.D. Ferry, Viscoelastic Properties ofPolymers, 3rd Ed. Wiley (1980)

Ccreep (t ) = Cg ++∞t
+  1 − e−t /τ L(τ )d ln τ
2 (1 + ν )η −∞

(

)

(D3)

in terms of the retardation time spectrum L (τ ) for the springs having
the retardation times ranging from ln τ to ln τ + d ln τ .

In Eq. (D3),

Cg is the glass compliance, and η is the steady-state viscosity. The
relaxation modulus

Erelax ( t )is related to the creep compliance

Ccreep (t ) via the convolution integral (refer to Chap. 6);t E ( t − t ')C ( t ') dt ' = t
0

relax

(6.19)

creep

In this context, therefore, the relaxation time spectrum H (τ ) iscorrelated to the retardation time spectrum L (τ ) as follows [A2-A4];

L (τ ) =H (τ )∞ H (1 s )
ds π H (τ )  +  Eg − 
0 s −1 τ
L (τ )

2

2

H (τ ) =(D4)∞ L (1 s )


τ
ds 
π L (τ )  + Cg + + 0η
s −1 τ 2

2

The retardation time spectra L (τ ) in Fig. 12.16 are determined via
the Schwarzl-Staverman approximation [A3-A4]
 dCcreep (t ) d 2Ccreep (t ) L(τ ) = −2 d ln td ( ln t )  t = 2τ(D5)

applied to the experimental data of creep compliance curves Ccreep (t ) in
Fig. 12.15. In a similar way, the relaxation time spectrum H (τ ) is also
determined from the experimentally observed relaxation modulus
Erelax ( t ) by the use of the following relation;

 dE(t ) d 2 Erelax (t ) 
H (τ ) =  − relax +2d ln td ( ln t )  t = 2τ(D6)[A2] B. Gross, Mathematical Structureof the Theories of Viscoelasticity,
Hermann (1953)[A3] F. Schwarzl, A.J. Staverman,
Physica, 18, 791 (1952)[A4] F. Schwarzl, A.J. Staverman,
Appl. Sci. Res., A4, 127 (1953)

173

HYSTORY OF INDENTATION CONTACT MECHANICS

APPENDIXE(1) Thermodynamics and Continuum Mechanics
in The Era of Industrial Revolution
In the early 19th century in Great Britain, the industrial revolution
began in earnest with the technological innovations associated withtextile industry, development of steam locomotives/ships, etc., followedby the development of iron/steel industry with the engineering innovationof coal-to-coke producing processes. The transportation of coal in South
Wales to Glasgow/Manchester, the center of the industrial revolution,
significantly improved Britain’s transport infrastructure with railway

Thomas Young

networks, canal/waterway networks, etc.

(1773-1829)

It was the historical necessities in the era of the industrial revolutionthat there were numbers of the “superbrains” for making great
contributions to founding the thermodynamics [N.L.S. Carnot (17961832), B.P.E. Clapeyron (1799-1864), J.P. Joule (1818-1889), H.L.F. vonHelmholtz (1821-1894), R.J.E. Clausius (1822-1888), W. Thomson(Load Kelvin) (1824-1907), J. W. Gibbs (1839-1903), L.E. Boltzmann(1844-1906), W.H. Nernst (1864-1941)], and to founding the continuum
mechanics [T. Young (1773-1829), S.D. Poisson (1781-1840), C-L.
Navier (1785-1836), A. Caucy (1789-1857), G. Green (1793-1841), G.
Lame (1795-1870), B.S. Saint-Venant (1797-1886), F. Neumann (1798-

Simeon Denis Poisson(1781-1840)

1895), G.G. Stokes (1819-1903), J.C. Maxwell (1831-1879), J.
Bauschinger (1834-1893), J.V. Boussinesq (1842-1929), W. Voigt (18501919), H. Hertz (1857-1894)].

(2) The Role of Measuring the Indentation Hardness
in The Era of Industrial Revolution
Establishment of the test methods for determining the mechanicalcharacteristics (elasticity, strength, plasticity, etc.) of iron/steel-products
were the fatal issues in The Industrial Revolution. The test methods forJames Clerk Maxwell

characterizing “elasticity” have already been established in the Era with

(1831-1879)

the pioneering works made by R. Hook (1635-1703) and T. Young. On
the other hand, as to the test method and the physics of “strength” in The
Mid-Term Industrial Revolution (the mid-19th century), there were noneof systematic studies until the Weibull statistics (W. Weibull (1887-174

1979)) in the end of 19th century and the Griffith’s deterministic strength
theory (A.A. Griffith (1893-1963)) in the beginning of 20th century. The
science and engineering for the study of “plasticity” of iron/steel was in
the most earnest with the practical issues of rolling/sliding problems ofwheel/rail contact along with the developments of steam locomotives,
although the concept, physics, as well as the testing techniques ofplasticity were still in predawn darkness. In 1886, however, J.
Bauschinger examined the elastic limit and the plasticity-related
irreversible deformation that has been widely recognized as the
In these historical ambience,Bauschinger-Ewing extensometer

indentation hardness testing played an important role in practically(precision of strain measurement:characterizing the plasticity of iron/steel products, though it was requiredε ≈ 2.5 × 10−6 )“Bauschinger effect” in the 20th century.

one more century for us to get a deeper understanding of the physics ofindentation hardness.

(3) Test Techniques for Measuring Indentation Hardness
and the History of Elastoplastic Contact Mechanics
F. Mohs (1773-1839) introduced the “Mohs hardness” in order to rank
so-called “scratch hardness”, not the indentation hardness. It is based on
the relative scratch resistance; talc (Mg3Si4O10(OH)2) as the softest
mineral is assigned a value of 1, and diamond (C) as the hardest mineralis assigned a value of 10. Most of minerals are relatively ranked between
1 and 10. There is a tight correlation between the Mohs hardness andthe indentation hardness, while we cannot use the Mohs hardness as aquantitative material characteristic, not like the indentation hardness. The
Brinell indentation test was proposed by J.A. Brinell (1849-1925) in 1890
to quantitatively characterize the hardness of steel. In its standard test
procedure, the Brinell hardness HB is defined as the indentation load
P (=3ton) divided by the total surface area

A of the residual

impression, i.e., HB = P A , by the use a spherical steel indenter with thediameter of 10mm, having been widely utilized up to the present date. In
1898, A. Martens developed a breakthrough indenter that may be apioneering instrument of the present instrumented indentation apparatus.
The Martens indenter is capable of measuring not only the indentationload P , but also the penetration depth h , namely measuring the P - h

175

Vickers hardness, H V

the hardness (scratch resistance) of minerals. The Mohs hardness is the

104

103

102

101

2

4

6

8

10

Mohs hardness

Correlation between the Vickers
hardness and the Mohs hardness

loading curve. He proposed the Martens hardness HM in terms of the
maximum indentation load Pmax and the in-situ penetration depth

hmax at the maximum load. The Martens hardness is widely in use
nowadays as the Universal Hardness HU . The concept of hardness via
in-situ determination under load was actually very pioneering in the endof 19th to the beginning of 20th centuries because it has been usual todetermine the indentation hardness such as the Brinell hardness and theVickers hardness by the use of the residual indentation impression.

Martens indenter
The indentation test techniques with a spherical-tip indenter such as theBrinell hardness test became inappropriate for characterizing the contact
hardness of high-tech hard materials developed in 20th century, leading
to the diamond conical/pyramidal-tip indenters. R. Smith and G.
Sandland at Vickers Ltd. proposed in 1923 the indentation test techniqueusing a diamond tetrahedral pyramid, and then commercialized in 1925
the “Vickers hardness tester”. As we made detailed considerations in
Chap. 5, the Vickers indenter has a specific inclined face-angle (refer toFig. 5.4) in order to keep a consistency with the Brinell hardness testing.
This design concept has then been transferred to the trigonal pyramidindenter proposed by E.S. Berkovich in 1950.
The Brinell hardness HB and the Vickers hardness HV are definedas the indentation load divided by the total contact area of the resultant
residual impression. On the other had in Martens hardness testing, theprojected contact area under load is utilized in order to calculate theMartens hardness HM that means the mean contact pressure underload. The significance of the indentation hardness defined as the mean

176

contact pressure was first emphasized by E. Meyer in 1908 on the basisof materials physics. He demonstrated that the indentation-induced
normal stresses on the contact surface solely as well as exclusively
contribute to the applied indentation load, and then concluded theimportance of the projected contact area not the total contact area incalculating the indentation hardness. The indentation contact hardness interms of the indentation load divided by the projected contact area is now
referred to as the Meyer hardness H M .
The Meyer hardness H M of spherical indentation is dependent on thepenetration depth, i.e., H M increases with the increase in penetration
depth due to the geometrically non-similar tip-configuration unlike theE. Meyer, Zeit. Verein. Deutsch. Ing.

cone/pyramid indenters with geometrical self-similarity (see the details

52,740(1908)

given in Chap. 5).

On the basis of experimental observations, Meyer

proposed the Meyer’s law; the Meyer hardness of spherical indentation
is expressed with H M = f ( d D ) in terms of the diameter D of thespherical indenter and the diameter d of the projected circle of residual
impression (refer to Eqs. (3.30), (5.4) and (5.9)). In other words, theMeyer hardness is independent of the diameter D of the sphericalindenter once we control the penetration depth in order to make the ratioof d D constant (the Meyer’s law of geometrical similarity).
In the beginning of the 20th century, it has been well recognized thatthe contact hardness H M is a “measure of plasticity”, although there
existed none of materials physics to correlate the contact hardness to theyield stress Y . In the 1920s, R. Hill and then C. Prandtle derived theformula of H M = c ⋅ Y (the constraint factor c = 2.57 ) by applying thetwo-dimensional slip-line field theory to a perfectly plastic body indented
by a flat punch. The extensive as well as the intensive indentation testing
and the finite-element-based numerical analyses were conducted in thedecade from 1960 to 1970 by D. Tabor, et al. for evaluating the constraint
factor c . They proved that the constraint factor ranges in 2.5 ≤ c ≤ 3.2
depending on the tip-geometry of the indenter used as well as the contact
friction of the material indented.
In 1961, N.A. Stilwell and D. Tabor emphasized the importance ofmeasuring the indentation loading/unloading P − h hysteresis curve
and then demonstrated that it is capable of determining the elastic
modulus from the slope of the unloading P − h line. Based on these177experimental results and analyses, D. Newey, M.A. Wilkins, and H.M.
Pollock designed a pioneering instrumented indentation apparatus in
1982, having been the historical origin of the present conventional
micro/nano instrumented indentation apparatus.
The Meyer hardness H M of a fully plastic body is a quantitative
measure for the yield stress Y as mentioned above. On the other hand,
not only the plastic flow but also the elastic deformation significantly
affects the Meyer hardness of elastoplastic body. R. Hill (1950) first
applied the cavity model to describe the Meyer hardness in terms of theelastic modulus E ' and the yield stress Y of an elastoplastic bodyindented by a sphere, and then followed by K.L. Johnson (1970) for
conical indentation. Based on these pioneering works, we can now fully
understand the contact physics of the Meyer hardness of elastoplastic
body as a function of the plastic index PI ( = ε I E ' cY ) in a unified
manner based on the additivity principle of the excluded volume ofindentation, the details of which have already been given in Sec. 5.1(2)of this textbook.

(3) Theoretical Foundation of
Elastic/Viscoelastic Contact Mechanics
The classic paper of H.R. Hertz (1882),“ Über die Berührung fester elastischer Körper (On thecontact of elastic solids)”, J. reine und angewandte

Heinrich Rudolf Hertz (1857-1894)

Mathematik, 92, 156-171,
is the historical origin of the indentation contact mechanics. He
successfully solved an elastic contact problem; the analytical derivationof the contact pressure distribution p ( r ) as a function of the contact
displacement u z ( r ) for an elastic sphere (radius R1 , elastic modulus

E1 ) pressed into contact with another sphere ( R2 , E2 ) by an external
force P . Only three years afterward from the Hertz theory, J.V.
Boussinesq (1885) published a paper based on the potential theory for the
more generalized elastic contact problems including the Herzian contact
problem. Due to the mathematical difficulties in applying the Boussinesq
theory to various types of axisymmetric contact problems, its analyticalsolution was first made by A.E.H. Love in the 1930s, and then by I.N.

Joseph Valentin Boussinesq

Sneddon in the 1960s through using the Hankel transformation, the(1842-1929)178

details of which have already been demonstrated in Chap.3.
We have had a significant upsurge of the science and engineering oforganic polymers and plastic materials since 1940s. In these periods, thegreat scientists including P.J. Flory, J.D. Ferry, et al. made a sound basisof the theoretical as well as experimental frameworks of the polymer
rheology and the molecular theories of viscoelastic deformation and flow.
The theoretical extension of indentation contact mechanics to the timedependent viscoelastic studies was first made by J.R.M. Radock(1957)
through applying the “elastic-to-viscoelastic correspondence principle”
to the elastic Hertz theory, and then followed by S.C. Hunter, M. Sakai,
W.H. Yang, et al. through extending the Radock’s study for sphericalindentation to the various types of axisymmetric indentation contact
problems and experiments.

179【INDEX】

Contact【A】
Acrylate resin 141, 153, 155

contact area 19, 34, 40-47, 63, 122, 131

Amorphous 49, 135, 147, 150

contact depth, relative c. d.
19, 25, 45, 96, 19, 126

Amorphous selenium 135, 150

contact friction 1, 37, 114, 177

Adhesion toughness 81, 156

contact pressure
21, 25, 32, 40, 72, 91-97, 176【B】

contact radius 19, 23-36, 67, 72-80, 155

Beltrami-Michell equation 8-10, 13
Bessel function 15, 167

Continuum mechanics 1, 7, 39, 174

Body force 4-6, 9, 12, 165

Creepcreep compliance

Boltzmann’s convolution integral 57-58, 170

52, 59, 128-137, 149, 172

Boussinesq contact problems 90-94, 117

creep deformation 49-64, 85-87, 135, 159

Bright-field method 133

creep function 54-86, 130, 158→dark-field method

Creeping-up 115, 152
Cylindrical coordinate 5, 11-15, 161, 168【C】
Carbon; graphite
glass-like carbon; glassy carbon 48, 144【D】

non-graphitizable/ graphitizable) 144-145

Dark-field method 132→bright-field method

polycrystalline graphite 144-145

Deformation 1

Carson transform 58-61
Cavity model 38-41, 178

normal; elongational 1

Chebyshev polynomials 94

shear 1-8, 29, 164
Displacement gradient 1, 2, 90

Compatibility equation 3, 5, 8-13, 47
Compliance 52, 57, 60, 86, 121, 131, 149, 172
frame compliance

121-125, 130, 132【E】
Elastic modulus 8, 20, 25, 33-59,

Constitutive equation

71, 91, 100, 171→viscoelastic
bulk modulus 31, 60

Constraint factor 37, 114, 126, 177

effective elastic modulus 89, 103, 155
equilibrium modulus 51, 60, 99
glassy modulus 51, 60, 99, 116
pseudo modulus 53, 55
shear modulus 8, 29, 59, 116
Young’s modulus 7, 22, 31, 60, 72

181

Elastic body 7-45, 57-82, 96, 107, 112, 120, 143【H】
Hankel transform 15-20, 93, 167

linear elastic 52

Hardness

Elasticity
elastic limit 28, 37, 113, 175

Brinell 19, 34, 44, 119, 175

elastic recovery 34, 43, 126, 142 146

Meyer 34-47, 67-70, 81, 102, 121, 133, 153

Elastic-viscoelastic corresponding principle

Vickers/Berkovich 42- 44, 63, 72, 107, 115
Harmonic equation 15

59, 67, 84, 115, 134
Elastoplastic 26, 34, 44-48, 71, 81, 102, 109-114

bi-harmonic e. 3, 11-20, 167

Enthalpy 28, 151

harmonic function 10, 93
Hertz theory

Entropy 28

Hertzian contact 22-25, 33, 152

Equilibrium equation 4, 8, 10, 12, 165

Hookean spring 50, 57

mechanical equilibrium equation 8

Hydrostatic pressure 38, 60

Excluded volume 39-44, 53, 102, 119, 134, 140

Hysteresis curve 42-48, 80, 109, 122-127,

additive principle of excluded volume

133, 139-147, 152-157, 177

39-42, 102, 109, 134, 157
【F】

【I】

Field-Swain approximation 111-113

Inclined face-angle 24, 37-44,
63, 108-119, 139-141, 176→Oliver-Pharr approximation

Incompressible

Film, coat 88-105, 117, 152
thickness of 88-104, 117, 153

elastic body 60, 107Film/substrate composite 88, 117, 152elastoplastic body 39

Finite element method/analysis 43, 98, 105

viscoelastic body 62, 129
Indentation

FEM, FEA 33, 42, 37, 71, 104, 106-116
Fredholm integral equation 93, 103, 117, 155

instrumented 35, 64, 88, 110, 121, 139, 141

Free energy

microscope 41, 65, 77, 104, 114,
131-135, 150-158

Gibbs free energy 28, 39, 46, 76

Indentation contact mechanics

Freezing point; crystallization point 49

elastic 19

Fused quartz; silica glass 124, 133, 145-155

elastoplastic 34【G】

numerical 106

Geometrical similarity 26, 36, 42, 63, 74, 109,

viscoelastic 63
Indentation load 18-27, 34, 72

119, 128, 139, 177

Indentation strain 26, 36-38, 63, 102, 114, 128

Glass, glassy solids 49, 151
Glass transition 49, 135, 147-152
Graphite 144-147

182

Loop energy 46, 48, 140-146

Indenter
Berkovich 44, 119
Brinell (spherical) 22, 44【M】

cone 24

Magnesia, magnesium oxide 141, 143

cylindrical 21

Master curve 82, 150

diamond 115, 132, 148, 152

Maxwell model→ viscoelastic

Knoop 139-141
Rockwell 19, 34, 36

Melting point 49

sapphire 132, 148

Memory function 57, 172

Vickers 44, 119

Metalsaluminum 48, 113, 141

Vickers/Berkovich equivalent cone

copper 45, 48, 113, 141

53-55, 113, 122, 139
Index of projected area 119, 139

Methyl methacrylate resin 141

Instrumented indentation microscope

Methyl-silsesquioxian (MeSiO3/2) 152→ indentation

Meyer hardness

Instrumented indenter→ hardness→ indentation
Internal pressure 38【N】
Network modifier 147【J】
JKR theory 71, 79-85, 156【O】
Oliver-Pharr approximation 111-113, 126

【K】
【P】
【L】

Penetration depth 18-27, 35, 42-45, 63-66

Lame constant 13

Perfectly elastic bodyLaminate composites 88-104, 117, 152-155→elastic body

Laplace equation 10, 15

Perfectly plastic body 37, 112, 142, 177

Laplacian operator 8, 10, 16

Pile-up 37, 43, 95, 112-114

Laplace transform→sink-in

51-61, 84, 131, 137, 159, 169

Plane strain 9-13, 20, 63, 108, 116

Load, force 1

Plane-strain elastic modulus 20, 26, 108

Loading-unloading curve

Plasticity 34-37, 113, 134, 141

hysteresis curve 42-48, 80, 109, 122-127,

plastic energy dissipation

133, 139-147, 152-157, 177

40, 46-48, 141-146

Loading/unloading-coefficient 110, 126

183

Soda-lime glass 45, 141, 144, 145, 147-155

Plastic index, PI 41-45, 83, 109-112,

Sol-gel method 152

123, 126, 134, 140

Soft matters 71, 156

Poisson’s effect, Poisson’s ratio 7, 10, 32, 60, 88,

Spectrum 55, 148, 172

90, 98, 107, 115, 120, 129, 152
Polyacrylate resin 153, 155

relaxation-time spectrum

Polycarbonate 153, 155,

retardation-time spectrum
Stimulus-response theory

Polymer rheology→ viscoelastic→ rheology

StrainPrincipal axis, principal stress 29-32

elongation 1
normal strain 2, 161【Q】

strain-/work-hardening 113【R】

strain energy 30, 46, 74-76

Relaxation modulus

shear strain 2, 30, 162
volumetric strain 13, 60

relax. shear modulus 52-58, 62, 98-100,

Stress 1-11, 165

116, 128-138, 149-151, 172

shear s.

Relaxation time 50, 54, 60-62, 100, 116,

normal s.

130, 148, 172

Stress function

Representative yield stress, strain 113, 114
Residual impression 44, 54, 125, 143, 173

Airy (Airy’s s. f.) 11, 13

Retardation time

Love (Love’s s. f.) 13, 16
Substrate→Relaxation time

→Laminate composites

Rheology 179
polymer r. 179

Supercooled liquid 49

Rigid substrate 92-97

Surface adhesion 71-87
surface tension, surface energy 37, 71-83→ JKR theory【S】
Saint Venant equation 3-11
Scanning probe microscope 125【T】

Shear angle 2, 164

Taylor expansion 1

Shift factor 82, 151

Time-temperature superposition rule 150→ shift factor

Silicon carbide 48, 141

Tresca criterion 29

Silicon nitride 45, 48, 133, 139, 140, 147→ Von Mises criterion

Sink-in 22-25, 43, 95, 107, 112→pile-up
Sneddon’s function 20-24

184

【X】

【U】
Unloading stiffness 122, 157【Y】Yield stress/strength 38-60, 100-112
Young’s modulus【V】→ elastic modulus

Van der Waals force 146
Viscoelastic【Z】Zener model

linear viscoelastic 49-58, 86, 98, 169→ viscoelastic

Maxwell model 50-53, 61, 99
Meyer hardness 67-70
stimulus-response theory 57, 137
viscoelastic function 35, 60, 122, 128-138
Voigt model 50
Wiechert model 55, 172
Zener model 50-55, 68-70, 85, 98-100, 171
Viscoelastic constitutive equation→Viscoelastic
Viscosity 50, 69, 151, 171
Voigt model→ viscoelastic
Volumetric additivity 39-41→ excluded volume
Von Mises criterion 30-33, 37→ Tresca criterion【W】
Wiechert model→ viscoelastic
Work-hardening→strain-/work-hardening 113
Work-of-indentation, WOI 47

185

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ⓒ Mototsugu Sakai 2020186