Christos C. Chamis and Louis M. Handler

Glenn Research Center, Cleveland, Ohio

Approximate Micromechanics Treatise of

Composite Impact

NASA/TM—2005-210756

September 2005
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Christos C. Chamis and Louis M. Handler

Glenn Research Center, Cleveland, Ohio

Approximate Micromechanics Treatise of

Composite Impact

NASA/TM—2005-210756

September 2005

National Aeronautics and

Space Administration

Glenn Research Center
Available from

NASA Center for Aerospace Information

7121 Standard Drive

Hanover, MD 21076

National Technical Information Service

5285 Port Royal Road

Springfield, VA 22100

Available electronically at http://gltrs.grc.nasa.gov
NASA/TM—2005-210756 1

Approximate Micromechanics Treatise of Composite Impact

Christos C. Chamis and Louis M. Handler

National Aeronautics and Space Administration

Glenn Research Center

Cleveland, Ohio 44135

Summary

A formalism is described for micromechanic impact of composites. The formalism consists of numerous equations which describe all aspects of impact from impactor and composite conditions to impact 

contact, damage progression, and penetration or containment. The formalism is based on through-thethickness displacement increments simulation which makes it convenient to track local damage in terms 

of microfailure modes and their respective characteristics. A fl ow chart is provided to cast the formalism 

(numerous equations) into a computer code for embedment in composite mechanic codes and/or fi nite element composite structural analysis.

Introduction

Composites in aerospace vehicles in general and composites in engine structures in particular are 

subjected to inadvertent and design-for impact loads. References [1-3] describe various types of impacts, 

approximate methods, while refs. [4-6] describe different experimental techniques to evaluate composite impact. References [7,8] describe impact damage in composite structures by classical formulations 

while refs. [9,10] describe impact damage in composite engine structures. Damage tolerance and dynamic 

loading in composites are extensively described in two recent books [11,12]. From those studies as well 

as others and continuing research on three-dimensional fi nite element simulations, it is fair to note that 

impact damage in composites is very complex and is mostly evaluated by testing [11-13].

The complexity arises from the multitude of composite fracture modes that are caused during impact 

and the attendant failure mechanisms associated with those fracture modes. Impact damage for specifi c 

designs can be evaluated by fi nite element simulations. However, local tracking of damage during impact and identifi cation of local fracture modes is a very tedious and often an elusive process. Because the 

fi nite element model for composite impact is usually very large, it requires substantial personnel expertise 

and effort, and long computer times. Therefore it is cost prohibitive if not impossible to use fi nite element 

simulations in preliminary designs of composite structures for impact resistance. It is also fair to note that 

an inclusive micromechanics treatise of composite impact has not been developed or even proposed as 

yet [14-18]. The work reported in [14-15] is a step in that direction.

It is timely then to develop a micromechanics based generic and inclusive treatise of composite impact. The objective of this report is to describe the development of the formalism for a micromechanics 

treatise of composite impact. The treatise covers all aspects of impact damage from contact to penetration and includes cratering and spallation. It describes local damage as it occurs and the various fracture 

modes that are induced during the damage process. Strain rate effects are neglected because of the in situ 

three-dimensional constraints in the matrix. Only point impact is considered since it is assumed that this 

will be coupled with structural analysis for global response. Equations are derived for penetration time, 

residual velocity, rebound velocity and backface splitting. Equations are also derived for impact forcetime trace and impact containment. A fl ow chart is outlined which cast the formalism into a computer 

module computational algorithm which can be made part of available composite mechanics codes and 
NASA/TM—2005-210756 2

general purpose fi nite element for composite structural analysis. Because of its inclusiveness and

generality the treatise is approximate relative to mathematical rigor but consistent in the approximations 

with mechanics of materials approach.

Nomenclature

A area

D projector diameter

E energy, modulus, E.E. elastic energy

F force

G shear modulus

g gravitation constant

h crater height

K K.E. kinetic energy

M moisture, mass

R projectile radius, transformation array

r contact area radius

S strength

T time, tension, temperature

t thickness, time, tl ply thickness

Vind volume of indentation

v velocity

zp projectile penetration depth

x,y,z coordinate axis

α thermal expansion coeffi cient

β moisture expansion coeffi cient

ΔT temperature change

ε limit strain

∈ strain

θ angle contact area periphery (fi gure 1), ply orientation angle (fi gure 6)

l laminate property

ν Poisson's ratio

ρ material density

Σ summation

σ stress

{}, [] vector, array, respectively

Subscripts

C compression

c laminate, contact crushing

crz crz composite, rz direction

F bending (fl exural) fi ber

i indentation

o initial property

p projectile related, penetration

S strength

s shear 

x,y,z coordinate direction (fi gure 13)

1,2,3 material axes (1-along the fi ber)
NASA/TM—2005-210756 3

Some Geometric Relationships

Express the radius, r, and the angle, θ, of the contact area as a function of z the local indentation of an 

impactor with a spherical tip of radius Rp (fi gure 1).

 

Rz r R

R Rz z r R

r

p p

pp p

( ) − + ( ) =

− ++=

=

2 2 2

2 22 2

2

2

2

(2.1)

Rz z z R z p p −= − 2 ( ) 2

r zR z = − ( ) p

⎡

⎣ ⎤⎦ 2

1 2/

(2.1a)

θ =

⎛

⎝

⎜

⎜

⎞

⎠

⎟

⎟ =

⎡ ( ) − ⎣ ⎤

⎦

⎧

⎨ − − sin sin

/

1 1

1 2

2 r

R

zR z

p R

p

p

⎪

⎩

⎪

⎫

⎬

⎪

⎭

⎪

(.) 2 2

The volume of the spherical cap with radius r and depth z is derived as follows:

dV r dz

V dV r dz

R z z dz

Rz z

ind

ind ind

V

o

z

p

z

p o

z

ind

=

= =

= − [ ]

= − ⎡

⎣

⎢ ⎤

⎦

⎥

∫ ∫

∫

π

π

π

π

2

0

2

2

0

2 3

2 3

2

1

3

(.)

V Rz z z R z ind p p = − ⎡

⎣

⎢ ⎤

⎦

⎥ π = − [ ] 2 3 π 2 1

3 3

3 24 (.)

Check for the volume of a hemisphere: 

πDp

3

12

π

π

π

π

π π

D DD D

D D

D D

D D

O K

p pp p

p p

p p

p p

3 23

3 3

3 3

3 3

12 2 4

1

3 8

8 24 24

3

12 12

= ⎛

⎝

⎜

⎞

⎠

⎟ −

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

= − ⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

= − [ ]

= . .
NASA/TM—2005-210756 4

Penetration Description

Penetration is a time dependent process. Here we consider very fast penetration that is the structure 

responds only locally by material damage (stress waves do not reach the boundaries).

The material getting indented is equal to the volume of the spherical cap in the indentation (fi gure 2). 

The energy required to crush this volume of material is given by

E VS c ind c C = 1

2 33 (.) 3 1

Where Ec denotes the energy of indentation, Vind is the crushed volume, and Sc33C is the fl at-wise laminate strength. If we further assume that this equals the kinetic energy absorbed (relatively stiff indentor 

compared to composite fl at-wise stiffness) a projectile with mass Mp, and an initial velocity vo, then for 

energy conservations the kinetic energy of the projectile equals the energy expanded Ec to crush the

material:

E Mv V S

Mv V S

v

V S

M

c po

p

ind c C

p o ind c C

o ind c C

p

= ⎛

⎝

⎞

⎠ =

=

=

⎛

⎝

⎜

⎞

⎠

⎟

1

2

1

2

3 2

2

33

2

33

33

1 2

(.)

/

It is noted that vo is the projectile velocity prior to contacting the target. Check units:

v

in lb in

in lb in in

in lb in

lb in

v in in

o

o

= ⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥ =

( )

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

= ( ) =

. .

. . . sec

. .

sec

. sec . sec

/ /

3 2

33 2

1 2 3 2

2

1 2

2 2 1 2

/ /

/

Therefore, the units check.

The next step is to assume that the projectile has fi rst to crush the impacted surface up to its

equator and beyond until the crushing force exceeds the remaining thickness shear-stress-resistance to 

push it through as a plug.

V D zR ind p p = ≤ ( ) 1

12

3 3 3 π when ( . )

V

D

ind RzR z R p =+ − pp p ( ) ( > ) π

π

3

2

12 when ( . ) 3 4

Equation (3.3) is half the spherical volume for a spherical projectile and equation (3.4) is that plus the 

volume of the shear plug fi gure 3. Then the crushing velocity required at z > Rp and at z ≤ Rp is given 

respectively by
NASA/TM—2005-210756 5

v

D RzR S

D c z R

p p p cC

p p

= p

+ − ( ) ⎡

⎣

⎢ ⎤

⎦

⎥

⎡

⎣

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

( > )

1

12

1

6

3 5

3 2

33

3

1 2

π π

π ρ / g

/

(. ) a

= ( − ) ⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥ ( ≤ ) 2 3

4

3 5

2

33

3

1 2

gz R z S

R

z

p cC

p p

Rp ρ

(. ) b

Additional energy will be required to push the material ahead as a shear plug for penetration. That energy 

(Es) is given by assuming the friction against the projectile is part of the shear resistance, fi gure 3 and the 

projectile velocity is vs and the time t

s

 required during the shearing plug:

E S Dt R z s = crzS ( ps p p ) ≥ 1

2

3 6 2 π (.)

1

2

1

2

3 7

2 2

2 1 2

Mv S Dt

v

DtS

M

p s crzS p s

s p s crzS

p

= ( )

=

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

π

π

/

(.)

Another way to think is vs is the projectile velocity the instant the shear block starts moving. For a 

spherical projectile of diameter Dp and for Mp ≥ than the mass of the plug:

M

D

v t gS

D

in in lb in

in lb in

in in

p

p

s s crzS

p p

=

=

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥ = ( )

( )

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥ =

⎛

⎝

⎜

⎞

⎠

⎟ =

π

ρ

ρ

3

2

2

1 2 2 23

2 3

1 2

2

2

1 2

6

6

( )

sec

sec sec 

/ / /

/ g

O.K.

the units check.

The total velocity required for penetration Ep is calculated from the sum of the two energies

E EE pcs = +

and the residual velocity is

v

gz RS

R

gt S

R p

pcC

p p

crzS

p p

= ( − ) +

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

3

4

3

2

3 8 33 2

1

12

ρ ρ

(.)

It is noted that vp is velocity that the projectile must have to crush and then start pushing the plug out. 

The residual velocity at the instant of vp is Er which is given by:
NASA/TM—2005-210756 6

EEE EEE r opocs =− =−−

E Mv Mv S Dt

M

v v v Dt S

M

r p o p c crzS p s

p

r o c ps

crzS

p

=−−

⎫

⎬

⎪

⎭

⎪

= −−

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

1

2

1

2

1

2

3 9

22 2

22 2

1 2

π

π

Dividing by the 

equation for the 

residual velocity is 

/

(.)

vr is the residual velocity that drives the shear plug through, where vo is the projectile initial velocity, vc is 

the crushing velocity and the last term in the equation is the shear velocity. 

Several points are worth observing:

 1. vo = [v2

c + v2s

]

1/2: vr

 = 0

 2. vo > [v2

c + v2s

]

1/2: vr = [v2

o – (v2c + v2s

)]1/2

 3. vo < [v2

c + v2s

]1/2: Only partial penetration if at all.

 4. For any velocity vo and a given laminate with known Sc33C and ScrzS, a composite thickness t

c 

 can be found for projectile containment.

 5. Equivalently for a given t

c and Sc33C, an ScrzS can be selected for containment.

The equivalent force at the instant which the plug begins to move will be equal to the normal stress 

times the projected indentation surface area. For a spherical projectile at z = Rp the surface area is:

A

D

cs p = π 2

4

and the force

F AS c cs c C = 33 (. ) 3 10

F DS c p cC = 1

4

3 11 2 π 33 (. )

This force or ratio thereof can then be used to perform evaluations for delaminations, splits and other 

fracture modes in the remaining laminate.

It is important to note that ScrzS very likely includes contributions from fi ber fractures either in shear 

or in local bending. This implies that penetration will be dependent on the laminate confi guration. It is 

suspected that delaminations and/or fi ber crushing will occur when:

F D S D zS

F DS S t

c p c C p crzS

c p c C crF

= >

⎧

⎨

⎩

⎫

⎬

⎭

= >

⎧

⎨

⎩

⎫

⎬

⎭

⎫

⎬

⎪

⎪

⎭

⎪

⎪

1

4

1

4

1

3

3 11

2

33

2

33

2

π π

π

interply strength

at that interply

Ply bending strength

at that intraply

a

l

(. )

If fi bers break in local bending the crushing force is 
NASA/TM—2005-210756 7

F St c crF = 1

3

3 11 2

l (. ) b

Where ScrF is the fl exural strength and t

l is the thickness of that ply, respectively.

Time to Penetration

 Time to penetration (Tp) is the time required for the projectile to penetrate the laminate. That time 

can be estimated by assuming that the following relationship holds in an average sense:

1

2

2

3 12

vT t

T

t

v

pp p

p

p

p

=

= (. )

where vp is given in eq. (3.8) and tp is the penetration thickness (t

p = z). Since both vo and vp can be 

measured in a high-velocity-impact test, suitable tests can be conducted to verify eqs. (3.8) and (3.12) and 

indirectly eqs. (3.10), to (3.11b). It is noted here that equation (3.12) is only a rough estimate.

Cratering

Cratering occurs at the periphery of the contact as the material is pushed out due to high in-plane 

compressive stresses which are induced by the projectile ingress during crushing. The cratering is estimated by the following relationship (fi gure 4):

ε νε

ε ν

czz crz crr

czz crz

c C

c

S

E

= −

= − 33

33

(. ) 3 13

The crater height is determined from

h t crater p czz = ε (. ) 3 14

where t

p, vcrz, Sc33C, and εc33 are evaluated at the depth of crushing at that time. Implicit in eq. (3.14) is 

that no cratering will occur once shearing started. The corresponding time is approximated:

1

2

2

3 15

vT t

T

t

v

cc p

c p

c

=

= (. )
NASA/TM—2005-210756 8

where vc is given by eq. (3.5). We can also determine whether spallation will occur as a result of cratering 

since we know the strain εczz at any (t

p = z).

 A condition for estimating spallation is given by

σc cT 33 33 > S

where sc33T is given by

εν ν c c crz c C c

c

E S crz c C c T

E

E 33 33 33 S S 33

33

=+ = > 33 33 (. ) 3 16

Delamination and Back Surface Splitting

These two conditions will occur if the penetration time is suffi ciently long for shear waves to propagate in a region beyond projectile diameter. The shear stress waves velocities (vcrz) are given by

v

G

crz

crz

c

= ⎛

⎝

⎜ ⎞

⎠

⎟ ρ

1 2

3 17

/

(. )

v

G

cxy

cxy

c

= ⎛

⎝

⎜ ⎞

⎠

⎟ ρ

1 2

3 18

/

(. )

where Gcrz and ρc are the corresponding laminate shear modulus and density, respectively. The distance 

these waves will travel at Tc are estimated by

rvT

xyvT

T t

crz c

cxy c

c cc

=

= =

=

⎫

⎬

⎪

⎭

⎪ 2

3 19

/ v

(. )

Recall that t

p is the crushed thickness and vc the corresponding velocity. Using the last equation in the 

above two equations and also eqs. (3.17) and (3.18) and by assuming average wall stress

where

r

G t

v

xyvT

v

gD R z R

D

S g t S

D

crz

c

c

c

cxy c

p

pp p

p p

c C

s crzS

p p

= ⎛

⎝

⎜ ⎞

⎠

⎟ ⎛

⎝

⎜ ⎞

⎠

⎟

= =

= ( + − ( )) +

⎛

⎝

⎜

⎞

⎠

⎟

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

⎫

⎬

⎪

⎪

⎪

⎪

⎭

⎪

⎪

⎪

⎪

ρ

ρ ρ

1 2

3 2

3 33

2

2

1 2

2

3

2

6

3 20

/

/

(. )
NASA/TM—2005-210756 9

Delamination will occur if:

r D /2 > p

>

⎫

⎬

⎪

⎭

⎪ and D S t S p c C s crzS

1

4

3 21

33

(. )

and back surface splitting will occur if in addition to eq. (3.21) the following conditions hold:

1

4

1

3

1

4

1

3

3 22

2

33

2

22

2

33

2

11

π

π

DS tS

DS tS

pc C c T

pc C c F

>

⎛

⎝

⎜

⎜

⎞

⎠

⎟

⎟

>

⎛

⎝

⎜

⎜

⎞

⎠

⎟

⎟

l

l

tranverse

ply

splitting

longitudinal

fiber

breakage

(. )

Note that the r in eq. (3.21) indicates an average radial stress wave velocity propagation. Values for rx and 

ry are calculated from

r x

G T

r y

G T

x c r

c c

y c r

c c

= = ⎛

⎝

⎜ ⎞

⎠

⎟

= = ⎛

⎝

⎜ ⎞

⎠

⎟

⎫

⎬

⎪

⎪

⎭

⎪

⎪

θ

θ

ρ

θ

ρ

θ

1 2

1 2 3 23

/

/

cos

sin

(. )

where the x and y are the in-plane coordinates. These equations will give an elliptical wave propagation 

front, in general.

Thickness Growth

This growth is due to Poisson's effect. The ply in-plane unrestrained strains along the material axes 

are

{ } εl = ε{ } l/ m + ∆T{ } αl + M{ } βm (4.1)

Those in the in situ laminate are

{ } εl = [ ] Rl/ ε { } ε c (4.2)

The ply in-plane restrained strains are the difference between eqs. (4.2) and (4.1)

εl

R { } = [ ] Rl/ ε { } ε c − ε{ }l (4.3)
NASA/TM—2005-210756 10

The thickness growth is obtained by multiplying eq. (4.3) by the appropriate ply Poisson's ratios

ε νε izz j

R = [ ] l l 3 { } j = 1 2 44 (.) and 

The composite or laminate thickness strain then is obtained from

ε ε czz c i izz

i

N

t = (t ) =

∑

1 4 5

1

l

l

(.)

and the laminate change in thickness is obtained from

∆t t c i izz

i

N

= ( ) =

∑ l

l

ε

1

(.) 4 6

It is noted that these strains are in addition to those produced by impact. It should also be noted the 

ply in-plane strains are already known from laminate analysis. To obtain the thickness strains and thickness changes use the ply strains from laminate analysis in eq. (4.4) and proceed with eqs. (4.5) and (4.6).

Elastic Indentation Stresses

For an elastic indentation of depth Z, we assume continuous contact between projectile surface and 

target indentation. For this case, the in-plane and thickness-size local strains are uniform and the stresses 

will be proportional the local stiffness. Since the strains are uniform and the deformation conforms to the 

projectile surface, then at a depth Z it follows that the in-plane (εcrr and εcθθ) and thickness-wise (εczz) 

strains are:

ε

ε

ε

θθ

crr

c

czz

r

r

r

r

z

z

=

= + ⎛

⎝

⎞

⎠

=

⎫

⎬

⎪

⎪

⎪

⎭

⎪

⎪

⎪

∆

∆

∆

1 50 (.)

Note eq. (5.0) denotes completely recoverable indentation. Note also that for a spherical indentor at any z, 

r = [z(2Rp – z)]1/2, eq. (2.1a) and q = sin–1 r/Rp. Then, the in-plane ply strains along the material axis are 

given by (see fi gure 5)

ε ε { }l = ε θ ⎡

⎣ ⎤ R ⎦{ }cr (. ) 5 1a
NASA/TM—2005-210756 11

ε

ε

ε

θθ θ

11

22

12

2 2 1

2

2

⎧

⎨

⎪

⎩

⎪

⎫

⎬

⎪

⎭

⎪

=

cos sin sin

sin2 2 1

2

2

22 2

θθ θ

θθ θ

cos sin

−sin sin cos

⎡

⎣

⎢

⎢

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

⎥

⎥

⎧

⎨

⎩

⎫

⎬

⎭

εcrr

0 (. ) 5 1b

The corresponding stresses are given by

σ ε { }l ll = [ ] E { } (. ) 5 1c

where El is defi ned by:

σ

σ

σ

ν ν

ν

ν ν

ν

νν νν

ε

ε

ε

l

l

l

l

l l

l l

l l

l l

l l

l

l l

l

l

l

l

11

22

12

11

12 21

21 22

12 21

12 11

12 21

22

12 21

12

11

22

12

1 1

0

1 1

0

0 0

5

⎧

⎨

⎪

⎩

⎪

⎫

⎬

⎪

⎭

⎪

=

−

−

−

− − −

⎡

⎣

⎢

⎢

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

⎥

⎥

⎧

⎨

⎪

⎩

⎪

⎫

⎬

⎪

⎭

⎪

E E

E E

G

( .1d)

Concisely the in-plane strains are given by

{ } ε cl = [ ] Rε θ l{ } εcr (5.2)

and the fl at-wise

ε ε z z cz l Rε l { } = [ ] { } (.) 5 3

A more direct approach will be to obtain the three-dimensional strains from the three-dimensional 

transformation (see appendix A).

ε ε zl Rε θ r cz z l { } = [ ] , { } , (.) 5 4

The last equation will provide all 6-strains from which the corresponding stresses can be obtained 

from the three-dimensional stress/strain relationships:

σ ε z z Ec l l l { } = [ ] { } (.) 5 5

Equations (5.4) and (5.5) can be evaluated by incrementing eq. (5.3) by a single fi ber cell 

s

k f

=

⎛

⎝

⎜

⎞

⎠

⎟ π

4

and continuously checking fracture modes. Here s is the edge of a fi ber square array, kf

 is the fi ber volume 

ration and df is the fi ber diameter. The strains fracture modes are obtained from
NASA/TM—2005-210756 12

ε ε { } z  ≤ { } (.) 5 6

and the stresses fracture modes from

{ } σzl l ≤ { } S (.) 5 7

If eqs. (5.6) or (5.7) are violated failure will occur by that fracture mode and permanent damage will be 

induced in the composite target at that location. Crushing will proceed as described in Penetration Description section.

The elastic energy stored down to depth z will be given by

EE r z z

z

z

z . . (.) = ⎣ ⎦{ } =

∑1

2

5 8

0

2 ε σπ l l

∆

∆

where Δz is the thickness increment and r is evaluated at z from eq. (2.1a), in the section titled Some 

Geometric Relationship.

The loss in kinetic energy then is

KE EE loss ( . . . . (.) ) = 5 9

when

{ } σzl l ≤ S (. ) 5 10

No damage will occur and the projectile will bounce (rebound) with velocity equal to its initial velocity 

assuming no losses due to local heat or vibrations. To assure no heat and no vibration the contact time will 

have to be relatively small so that the change in local temperature is negligible:

∆T ≈ 0 5 11 (. )

where the impact surface temperature and the shear stress waves have not propagated beyond the a

distance t

p (crushed thickness). The contact time for no damage can be evaluated from the rebound

condition

1

2 (v Tz o c − 0 5 12 ) = (. )

T z

v c o

= 2 (. ) 5 13

Assuming an initial velocity of 30.5 m/sec (100 ft/sec) and an elastic indentation of z ≈ 0.013 cm 

(0.005 in.) without damage in any fracture mode

T s c = = ≈× ≈

2 0 005 −

1200

0 010

1200

1 0

1 2

10 8 33 (. ) . . 5

. sec . μ
NASA/TM—2005-210756 13

a very small time indeed.

Also the contact time for onset of damage can be calculated from eq. (5.8) by setting

E E dz z z S z

z

z

z . . (. ) ( ) σ ε σ l l ≥ = ∑⎣ ⎦ l l { }

∆

5 14

These simplifi ed equations can be readily checked by using available experimental data [3-5, 12 and 

13]. Note that the aforementioned derivations are for hard object projectile. They need to be modifi ed for 

soft impact. In that case the z increment will be on the projectile and r on the projectile fl attening.

The other important point to note is that the energy for elastic deformation must be added to that for 

crushing and penetration if it is rather signifi cant, say, more than 3 percent of their sum.

Force-Time Trace

The force time trace can be determined for both elastic impact and penetration impact. Elastic Impact—elastic indentation the force at any local indentation z is obtained from (fi gure 6).

F r z zz ≈π σ2

l (.) 6 1

where r is given by

r zR z = − [ ] p ( ) (.)

/

2 62

1 2

and σczzl is given by

σl l zz c

z

t = E 33 (.) 6 3

and the corresponding time is given by

T z

v v

z

o z

= +

2 (.) 6 4

where vo is the projectile velocity at initial contact and vz is determined from (K.E.—E.E.) is satisfi ed or 

equivalently from the equation:

1

2

1

2

6 5 2 2 M v M v EE z pz po = − . .( ) ( . )

Unloading will be obtained by deincrementing z and the contact time will be obtained from z max 

which occurs when vz = 0 or from eq. (6.5):

EE z M vp o .. . ( max ) = 1

2

2
NASA/TM—2005-210756 14

Impact Damage Response

Impact damage response is evaluated by accounting for the energy expanded as impact damage

progresses and accumulates (fi gure 7). The impact force is

F r z czz

( ) (.) 1 2 =π σ 7 1

F rS z cC

( ) (.) 2 2

33 = π 7 2

F RtS z p c crzS

( ) (.) 3 = π2 73

Fz

( ) (.) 4 = 0 74

The corresponding times are given by:

T zv v z oz

( ) (.) 1 = + 2 75 ( )

where eq. (5.8), 1

2

1

2

2 2 M v M v EE pz po = − ⎡

⎣

⎢ ⎤

⎦

⎥ .. u (ntilσ z C   = S 33 ); (.) 7 6

T zv v z oz

( ) (.) 2 = + 2 77 ( )

where eq. (3.2) 1

2

1

2

2 2 Mv Mv E E pz po = − 2 2

⎡

⎣

⎢ ⎤

⎦

⎥ , , shearing takes place ( ∋ ); (.) 7 8

from the moment that crushing begins until the moment of transition from crushing to shear plug movement, or from the moment that σ22l = sl33 to the moment F F z z

(3 2 ) ( ) ≤ ; where zc is the penetration depth at 

the moment crushing ends; and vz is the projectile's velocity at that same moment.

T z vv z z

v v

z c oz c

z z

( ) / (.) 3 2

2

= + 7 9 ( ) + ( − )

+

where due to shear 1

2

1

2

2 2 Mv Mv E E pz po s s = − ⎡

⎣

⎢ ⎤

⎦

⎥ ( ∋ ); (. ) 7 10

T z

v v

tRz

v v z c

o z

c pc

z e

( ) (. ) 4 2 2 = 7 11

+

+ ( ) + −

+

where t

c is the composite target thickness and where ve is the exit velocity obtained from equation (7.10).
NASA/TM—2005-210756 15

Computation Simulation

The penetration process can be simulated as described below. We can use the previous simple equations to computationally simulate the penetration process: The block diagram in fi gure 8 is one way to 

implement the impact process described, by the equations in the previous sections.

Typical Results

Typical results were obtained by using the computer module generated from the block diagram 

(fl owchart) in fi gure 8 in conjunction with a composite mechanics computer-programmed [19] and 

reprogrammed, respectively in JAVA computer language code. These results are from assuming a hybrid 

[0/90/0]s

 laminate where the 0°-plies are AS graphite-fi ber epoxy-matrix and the 90°-intraply hybrid plies 

are AS graphite-fi ber epoxy-matrix// S-glass-fi bers/epoxy-matrix composites. The constituent properties 

used are typical for these composites. The results are for three different initial velocities 36.6 m/sec 

(120 ft/sec), 366 m/sec (1200 ft/sec), and 3660 m/sec (12000 ft/sec). The projectile is a 0.64 cm (0.25 in.) 

diameter steel ball.

 The force-time traces developed during impact and predicted by eqs. (7.1) through (7.4) are 

shown in fi gure 9. Three points are well worth observing: (1) the impact duration for each velocity, 

(2) projectile containment, and (3) laminate penetration. The locations for the projectile progression 

predicted by eq. (7.5) through (7.11) are shown in fi gure 10. Note containment for the two low velocities 

and early penetration for the highest. The velocity trace predicted by eq. (7.9) is shown in fi gure 11. Note 

the containment velocity is zero for the two low velocity impacts and very high residual (escape) velocity 

(about 90 percent retention) for the high velocity. The interply and intraply stresses develop during impact 

predicted by eqs. (3.11a) and (6.3) are shown in fi gure 12. Note that the lowest velocity reached only 

69 Mpa (10 ksi) while both the others reached about 110 Mpa (16 ksi) denoting most likely interply and 

intraply delaminations for both of these. It is worth noting that the discontinuities in stresses shown in fi gure 12 are from the assumption that the in-plane strains are continuous prior to damage. Suffi ce it to say 

that this type of the laminate local detail can only be obtained by a micromechanics impact treatise such 

as described herein.

Conclusions

 An investigation was performed to develop an approximate formalism for describing composite 

damage due to impact. The formalism consists of numerous composite micromechanic equations describing all aspects of impact from initial conditions (projectile, target) to containment and/or penetration. 

These equations describe damage initiation, damage progression and accumulation in terms of microfracture and respective fracture modes. Equations are also included for impact-force/time history and time to 

penetration as well as conditions for back surface splitting. A fl ow chart is outlined to cast the formalism 

in a computer code module for embedment in composite mechanics codes. The formalism in essence 

constitutes an approximate treatise of micromechanics for composite impact. Results obtained from the 

computer module of this fl ow chart illustrate laminate local detail information that can be generated. The 

three velocities investigated may be characterized as follows: crushing initiation, extensive crushing, and 

hyper velocity impact.

NASA/TM—2005-210756 17

Appendix A

 The three-dimensional strain transformations are given by the equations below where l, m, n are 

direction defi ned by the table below (fi gure A1).

εε εε

εε εε

εε ε ε

εε εε

εε ε

θθ

θθ

θθ

θθ

θθ

cxx crr c czz

cyy crr c czz

czz crr c czz

cxy crr c czz

cyz crr c

m n

m n

m n

mm nn

mm nn

=+ +

=+ +

=+ +

=+ +

=+ +

l

l

l

l l

l l

1

2

1

2

1

2

2

2

2

2

2

2

3

2

3

2

2

2

12 1 2 12

23 2 3 23

22 2

22 2 ε

εε εε θθ

czz

czx crr =+ + mm nn c czz

⎫

⎬

⎪

⎪

⎪

⎪

⎭

⎪

⎪

⎪

⎪ 22 2

1

31 3 1 31 l l

( ) A

r, q, and z are attached to the projectile tangential plane and x, y, z are the laminate global axis. The 

undamaged ply stresses in the laminate at that point are obtained by: fi rst, calculating the ply strains el11, 

elzz, and el23 along the material axis and second, by using the three-dimensional stress/strain relationships. The fl at-wise (normal) ply stress slzz is obtained by multiplying eczz by El33 and the fl at-wise ply 

shear stress from

σ ε lzz l

i

czz i E ( ) = ( ) ( ) 33 A2

are obtained from

σ ε

σ ε

l l

l l

yz cyz yz

zx ezx zx

G

G

=

=

⎫

⎬

⎭

( ) A3

The interlaminar stresses are obtained from (assuming isotropic interply layer behavior)

σ ε

σ ε

σ ε

izz

j

czz m j

iyz

j

cyz m j

izx

j

czx m j

E

E

E

( )

( )

( )

( )

= ( )

= ( )

= ( )

⎫

⎬

⎪

⎪⎪

⎭

⎪

⎪

⎪

A4

In eq. (A4) i denotes interlaminar and j denotes the jth interlaminar layer.

Equations (A1) to (A4) are then compared to their corresponding strengths to identify local fractures 

and respective fracture modes. It is noted that eqs. (A1) to (A4) are general and can treat any material 

including isotropic hybrids and combinations. Thermal, moisture and residual stresses can be included 

as is done in the conventional laminate analysis. For local damage and progressive fracture the projectile 

induced strains are obtained by the incremental growth during the current time increment.
NASA/TM—2005-210756 18

References

 1. Anon., "Foreign Object Impact Damage to Composites," ASTM STP 568, American Society for 

Testing and Materials, 1975.

 2. Greszczuk, L.B. and Chao, H., "Impact Damage in Graphite-Fiber-Reinforced Composites," Composite Materials: Testing and Design (Fourth Conference), ASTM STP 617, American Society for Testing 

and Materials, 1977, pp. 389–408.

 3. Sun, C.T., "An Analytical Method for Evaluation of Impact Damage Energy of Laminated Composites," Composite Materials: Testing and Design (Fourth Conference), ASTM STP 617, American 

Society for Testing and Materials, 1977, 

pp. 427–440.

 4. Lagace, P.A. and Wolf, E., "Impact Damage Resistance of Several Laminated Material Systems," 

AIAA Journal, vol. 33, no. 6, June 1995, pp. 1106–1113.

 5. Habib, S.S., "Impact Response of Glass Fibre Reinforced Composite Plates," Journal of Reinforced 

Plastics and Composites, vol. 14, August 1995, pp. 799–803.

 6. Gross, X.E., "Characterization of Low Energy Impact Damages in Composites," Journal of Reinforced Plastics and Composites, vol. 15, March 1996, pp. 267–282.

 7. Moon, F.C., "Impact of Composite Plates, Analysis of Stresses and Forces," NASA CR–134999, 

Princeton, University, 1976.

 8. Sih, G., "Normal and Radial Impact of Composites with Embedded Penny-Shaped Cracks," NASA 

CR–159538, Lehigh University, 1979.

 9. Alexander, A. and Cornell, R.W., "Interactive Multi-Mode Blade Impact Analysis," NASA CR–

159462, Hamilton Standard, 1978.

10. Lee, C.H., "CELF Coupled Eulerian-Lagrangian Finite Element Program for High Velocity Impact, 

Part I Theory and Formulations," NASA CR–159395, Lockheed Missiles and Space Company, 1978.

11. Scaproni, C. et al., "Reduction of Tensile Strength in Angle-Ply Composite Laminates Due to Low 

Velocity Impact," Journal of Reinforced Plastics and Composites, vol. 18, no. 01, 1999.

12. Cantwell, W.J., and Morton, J., "Comparison of the Low and High Velocity Impact Response of 

CFRP," Composites, vol. 20, pp. 545–551, 1989.

13. Morton, J., and Godwin, E.W., "Impact Response of Tough Carbon Fibre Composites," Composite 

Structures, vol. 13, pp. 1–19, 1989.

14. Sierakowski, R.L., and Newaz, G.M., "Damage Tolerance in Advanced Composites," Technomic Publishing Co., Lancaster, PA, 1995.

15. Sierakowski, R.L., and Chaturvedi, S.K., "Dynamic Loading and Characterization of Fiber-Reinforced Composites," John Wiley and Sons, New York, NY, 1997.

16. Langlie, S., and Cheng, W., "Numerical Simulation of High Velocity Impact on Fiber-Reinforced 

Composites," ASME Pressure Vessels and Piping Division (Publication) PVP., ASME, vol. 159,

pp. 51–64, 1989.

17. Langlie, S., and Cheng, W., "A High Velocity Impact Penetration Model for Thick FiberReinforced Composites," ASME Pressure Vessels and Piping Division (Publication) PVP., ASME, 

vol. 174, pp. 151–158, 1989.

18. Anon., "Impact Damage Tolerance Modelling of Composite Materials and Structures," Euromech Colloquium 400, Imperial College, London, United Kingdom, September 27–29, 1999.

19. Murthy, P.L.N., Ginty, C.A., and Sanfeliz, J.G., "Second Generation Integrated Composite Analyzer 

(ICAN) Computer Code," NASA TP–3290, 1993.
NASA/TM—2005-210756 19

Figure 1.—Spherical indenter-portion diagram.

Rp

Rp

Projectile center

r



z

z

Rp

Rp

Initial surface

Figure 2.—Indentation diagram.

r

, M

2
NASA/TM—2005-210756 20

Figure 3.—Penetration diagram.

Crater

hcrater

tp = z

Figure 4.—Cratering diagram.



2

r

2

1

1

22

11

~12

Figure 5.—Ply-stress transformation diagram.
NASA/TM—2005-210756 21

Fz

TR tcontact

Figure 6.—Impact force history diagram.

T Tc

2 3

4

1 Elastic

Force

Crushing Shear Exit

Figure 7.—Impact damage response.
NASA/TM—2005-210756 22

23) If Ez < 0, set

Ez = 0

36) If Ez < 0, set

Ez = 0

Figure 8.—Computational simulation flow chart.

Print out

summary

40) If v > 0 and z <

thickness + Rp

loop back to 32

39)

41)

Print out

per user specs

42)

T = T' + dT

35) Ez = Ez'-Fz dz

34) Fz = FzX (thickness

+ Rp-z)/(thickness +

Rp-z chg)

32) FzX =  Fz (friction

force)

38) dT = dz/v

Completed initial crushing in

shape of projectile. Continue

crushing with cylindrical walls.

1) Input:

Targe F.E. Model

Impactor F.E. Model

and initial cond

15) Test for failure

modes

14) Integrate crater

height: h crater =

h crater' + epslzz

dz

13) Integrate time:

T = T' + dt

12) dT = dz/v

11) Velocity:

v = sqrt [2 Ez/Mp]

2) Select impact

region and

model as shown

3) Use composite

Mech to synthesize

laminate properties

6) Increment z

7) Compute crushing

force Fz 3.11

9) If Ez < 0, set

Ez = 0

10) Calculate

Vind (Eq. 2.4)

8) Integrate to get

remaining energy

Ez = Ez'-Fz dz

4) Initialize at contact

T = 0, z = 0

Ez = ra 2/2

h = 0

5) Calculate

dz = min {tc, Rp}/

n increments

40

back to 6

31) If Ez < 0, Ez = 0,

If v = 0, skip to

print 

27) Test for failure

modes

26) T = T' + dT

25) dT = dz/v

24) v = sqrt [2 Ez/Mp]

22) Ez = Ez'-Fz dz

21) If shear force <

crushing force

start push

19) Compute crushing

force Fz

If v is zero skip

to print at end

17)

29)

Ez = Ez'-EzDrop

Where EzDrop =

Fz (thickness + Rpz chg) (? see 3.6)

30)

20) Compute shear

force Fb to push

out as a plug

16) If v = > 0 

and

z < Rp loop

18) Increment z

28) If v > 0 and z <

thickness + Rp

loop back to 18

40

Fz = Fb

33) Increment z

37) v = sqrt [2 Ez/(Mp +

 Dp

2

4

tc c)]

z chg = z

If v is zero z =

thickness + Rp

skip to print at

end

If v is zero

skip to

print at end or

z = thickness + Rp
NASA/TM—2005-210756 23

0

4.0010–5 0.00 1.00 2.00 3.00

Time, sec

Figure 9.—Impact force developed in a [0/90/0]s

 interply hybrid laminate (90°-plies intraply hybrid

 graphite/S-glass). (1 lb = 4.45 N, 1 ft = 0.305 m). Force, lbs

300

400

200

100

120

1200

1.200104

Initial

velocity,

ft/sec

–0.2

4.0010–5 0.00 1.00 2.00 3.00

Time, sec

Figure 10 – Projective location during impact [0/90/0]s

 interply hybrid laminate. (90°-plies intraply hybrid

 graphite/S-glass). (1 in = 2.5 cm; 1 ft = 0.305 m). Location, z, ft

0.1

0.2

0.0

–0.1

120

1200

1.200104

Initial

velocity,

ft/sec
NASA/TM—2005-210756 24

0.0

4.0010–5 0.00 1.00 2.00 3.00

Time, sec

Figure 11.—Projectile velocity in a [0/90/0]s interply

 hybrid laminate during impact. (90°-plies intraply

 hybrid graphite/S-glass). (1 ft = 0.305 m). Velocity, ft/sec

1.6104

1.2

0.8

0.4

120

1200

1.200104

Initial

velocity,

ft/sec

0.0

–0.2 –0.1 0.0 0.1 0.2

Location, z, ft

Figure 12.—Interlaminar and intralaminar shear

stress developed in [0/90/0]s interply hybrid

laminate during impact (90°-plies intraply hybrid

graphite/S-glass). (1 psi = 6.9 Pa).

Stress, psi

1.6104

1.2

0.8

0.4

120

1200

1.200104

Initial

velocity,

ft/sec
NASA/TM—2005-210756 25

Figure A1.—Coordinate transformation

 diagram.

z

z

y

x

r

(r, , z)




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NASA TM—2005-210756

E–12680–1

WBS–22–104–08–01

30

Approximate Micromechanics Treatise of Composite Impact

Christos C. Chamis and Louis M. Handler

Penetration; Escape velocity cratering; Coustituents; Failure mechanisms

Unclassified -Unlimited

Subject Categories: 24 and 39

 Responsible person, Christos C. Chamis, organization code RO, 216–433–3252.

A formalism is described for micromechanic impact of composites. The formalism consists of numerous equations

which describe all aspects of impact from impactor and composite conditions to impact contact, damage progression,

and penetration or containment. The formalism is based on through-the-thickness displacement increments simulation

which makes it convenient to track local damage in terms of microfailure modes and their respective characteristics. A

flow chart is provided to cast the formalism (numerous equations) into a computer code for embedment in composite

mechanic codes and/or finite element composite structural analysis.