AlllOe 140440

NBSIR 82-2628

On the Energy-Release Rate

Associated With Diffusional Crack

Growth

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Measurement Laboratory

Center for Materials Science

Inorganic Materials Division

Washington, DC 20234

December 1 982

NBS-DOE Contract No. DE-A1 05-800R20679

Prepared for ~ partment of Energy

&TD Fossil Energy Materials Program

i 1(]0 ishington, DC

,1156

! 82-2628

1982

NATIONAL BUREAU

OF STANDARDS

LIBRARY

NBSIR 82-2628

ON THE ENERGY-RELEASE RATE

ASSOCIATED WITH DIFFUSIONAL CRACK

GROWTH

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Measurement Laboratory

Center for Materials Science

Inorganic Materials Division

Washington, DC 20234

December 1982

NBS-DOE Contract No. DE-A1 05-800R20679

Prepared for Department of Energy

AR&TD Fossil Energy Materials Program

Washington, DC

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

2 1983

ON THE ENERGY-RELEASE RATE ASSOCIATED WITH DIFFUSIONAL CRACK GROWTH

Tze-jer Chuang

Center for Materials Science

National Bureau of Standards

Washington, DC 20234

Work supported by U.S. Department of Energy AR&TD Fossil Energy Materials

Program under interagency agreement DE- A105-800R20679 with NBS

.

ABSTRACT

A general expression for the energy release rate (G) that arise during

steady state crack propagation by diffusion is derived from the standpoint of

irreversible thermodynamics. Three contributing components of G are

identified: (i) the Griffith energy (G_ ); (ii) heat generated in the process

br

of surface diffusion; and (iii) grain-boundary diffusion. Further, the total

G is shown to be directly related to the well-known J-integral if formulated

in the framework of finite deformation elasticity. This expression for G is

valid in general even if the response of the material is not linear and the

mass transport kinetics does not follow Fick's law. Quantitative evaluations

of each component are made for the linear case where field solutions are

available. The results show that component (ii) is approximately equal to G

and is independent of the crack velocity (u) whereas component (iii) is a

monotonically increasing function with G starting from 0.85 G^ when U is at

threshold value; and that strain energy contributions can be neglected leading

to G = J = (1 -v 2 )K2 /E. This means that G is not primarily associated with the

release of the strain energy rather, it stems mostly from the negative work

done by the normal stresses on the thickening of the grain boundary due to

non-uniform deposition of matter along it.

1 . Introduction

This paper aims to investigate energy release rates that accompany with

diffusive crack growth. Crack-like cavities at grain boundaries are

frequently observed in creep ruptured specimens of crystalline solids . Their

growth can be attributed to a mechanism involving coupled crack surface and

grain-boundary self-diffusion. Chuang et al. 1 reviewed the subject of

diffusive cavitation along grain interfaces and gave the conditions under

which the growth of crack-like cavities prevails. In general, creep cavities

favor crack-like (slit) shapes when the ratios of applied stress to capillary

stress and grain-boundary diffusivity to surface diffusivity are high (say

>>1) and when the service time approaches the later stages in the growth

phase. Under these circumstances, the crack travels in a steady-state fashion

at a moderate velocity along the grain boundary. It is then appropriate to

treat the crack as semi-infinite, growing at a constant speed in an infinite

elastic bicrystal under plane-strain conditions. This case has been

considered by Chuang2 who solved the coupled problem of diffusion and elastic

deformation leading to a specific kinetic law for subcritical crack growth.

Although the energy release rate in that case was discussed briefly in Ref. 2,

the present paper attempts to extend the analysis more thoroughly from the

standpoint of irreversible thermodynamics. It will be shown that, under

steady state, the energy release rates are related to the total entropy

production and can be expressed in terms of the well known path- independent

J-integral regardless of the form of phenomenological law describing the mass

transport processes. For completeness, we also include the strain energy

contribution in the formulation. Linearized phenomenological equations of

diffusion are then assumed in order to evaluate each individual contribution

1
at crack surfaces and grain boundary. It is shown that whereas the exact

energy rate quantities are not known, the results presented here always exceed

that exact values but no more than 9%.

It seems restrictive to limit the analysis to steady state. However, it

is worthwhile to point out that whenever steady state conditions do not hold,

3-4

no unique energy release rate can be defined (i.e. it becomes

time-dependent). Hence, it is meaningless to discuss the subject of energy

release rate without imposition of steady-state conditions.

2 . Thermodynamic Formulation

The energy release rate is derived in this section from the approach of

non-equilibrium thermodynamics. For simplicity, but without loss of

generality, consider a single component system in diathermal contact with an

environment. Then the entropic fundamental equation has the form s = s (e, n)

where s, e and n are respectively the entropy, internal energy and number of

atoms per unit volume. Writing e = e (s, n) , it can be shown that

ds = ^

de - !j! dn (1)

00 00 where T = ( —)n , p = ( —)s are the temperature and chemical potential

3s 3n

respectively. These intensive parameters are defined strictly under

thermodynamic equilibrium conditions. However, in a nonequilibrium system

such as a solid undergoing diffusional crack growth, T and p are permitted to

2
vary continuously throughout the body under the implicit assumption that the

functions T = T(n,s), p = p(n,s) remain the same as they would in local

equilibrium. It is then logical to define the entropy flux (and hence the

heat flux as well) consistent with Eq. (I) 5:

J

Q

» T J

s = Je

- II J (2)

where Je and J are the energy and matter fluxes respectively.

The conservation law dictates that the internal energy and the mass must

be conserved while allowing entropy sources (or sinks) to be present. This

means that

(3a)

(3b)

(3c)

where s is the local entropy production rate. It is possible to express s in

terms of summation of terms involving individual flux and its corresponding

intensive parameter as follows

s = it

+

-

* -s

^ by

= ||f “ ^

+ V • |

(Je - pJ) [by (1) and (2)]

= Je • V |

- J • V

£

[by 3(a) and 3(b)]

= (Jq + P j) • V |

- p J • V

| |

J • V p [by (2)]

= J

Q

• Y

i + -•(- v p)

0 = If

+ Y

' is

0 =

ft

+

-

' ^

and

s = 3s

at

+ V

Js

3
Hence in general for nonuniform temperature and chemical potential (and hence

stress) fields, the entropy production rate consists of two terms:

first, the scalar product of heat flux and gradient of (—) , and second, the

scalar product of matter flux and negative gradient of

temperature. Note that at steady state V • Je = V • J

- J • V p according to Eq. (2).

M divided by the local

= 0 and V • Jq =

Hence

and for isothermal processes wherein T = constant, the above two equations

reduce to

s = J J • (- V (j) = ^ V • J

T - - T - -Q

We now define W . , the dissipative work as dis

WJ . = T S dis

where

S = /s dV = ^ /J • (- V m) dV

v

1 *' v

is the total entropy production rate over the whole body.

(4)

(5)

We now turn to the first law of thermodynamics which states that the

increase in total internal energy of the body must be equal to the total input

of the mechanical work done to the body less the heat loss from the body to

its surrounding environment through the body's external "skin". This means

that

4
E = W -

( 6 ) /A

' s “

where A represents the collection of external areas in contact with

environment where occurs, and n is the unit normal vector of A directed

outward from the body.

At constant T, Eq. (2) can be integrated over the whole volume to give

the following equation in a global form:

J

r a*?

f J

Q

• 5

dA = T 3^

- TS (7 )

where Eq. (3c) and the Gauss divergence theorem have been used. This equation

assures the conservation of heat inside the body. Elimination of the integral

in Eqs . (6) and (7) results in

F = W - T S ( 8 )

where F = E - TS is the total Helmholtz free energy of the body. Eq. (8)

shows that the total external work done contributes partially to the increase

in the total free energy with the remaining part becoming the dissipative work

• • • since

^ = TS . Alternatively, we may define the effective heating

rate as f J * n dA and this can be shown to be equal to (W-E) from the

A -Q

first law of thermodynamics, Eq. (6).

It will be seen in Sec. 4 that Eqs. (5) and (8) are important in the

formulation of the general expression for the energy release rate in which

explicit expressions for the thermodynamic parameters in our special case are

required. The governing equations leading to those explicit expressions are

discussed first in the following section.

5
3 . Governing Equations in Diffusive Crack Growth

Consider a semi-infinite crack travelling along a grain boundary (x-axis)

at a constant velocity, u in an infinite bicrystal subject to a remote loading

(see Fig. 1). Under the influence of the applied stress, the crack is

supposed to advance by matter fluxes induced by stress-assisted self-diffusion

along the crack surfaces and grain-boundary . As a result, the grain-boundary

is thickened by an amount 6(x), defined by the discontinuity of vertical

+ + — displacements in y, 6(x) = [u] = u (x)-u (x) , at the upper and lower surfaces

of the grain boundary, due to grain-boundary diffusion. This 6(x), in turn,

produces a misfit residual stress field which upon adding to the applied

stress field, yields to the actual stress distribution ct (x) along the grain

boundary. These two functions 5(x) and ct^(x)

,

illustrated schematically in

Fig. 1, have been shown to play an important role in governing the behavior of

creep crack growth.

Let u be the normal recession rate at the crack surface; 0(s) be the n angle of the tangent at any point with the vertical y-axis and s the arc

length along the curved crack surface as measured from the crack tip (see Fig.

1). Then steady-state conditions require

u (s) = u cos 0(s) (9) n

at any point s on the surface. Strictly speaking, the LHS of Eq. (9) should

include a term u • n where u is the velocity vector of the material point

-m - -m and n is the unit normal vector of the surface area directed outward from the

material. This term (u * n) is important in materials where time-dependent

-m creep deformation takes place. For elastic materials considered here, this

6
term can therefore be omitted. The range of 0 in Eq. (9) for all points on

the crack surface is such that

\

> e > e tip

and when 0 = 7i/2 (i.e. at the center of the crack), = 0.

In Fig. 1, we set up a two dimensional Cartesian coordinate system (x,y)

and locate the origin at the moving crack tip. Thus, under steady-state

conditions

everywhere along the grain boundary.

The explicit expressions for the chemical potentials at the crack

surfaces and grain boundary applicable to the case of diffusive cavity growth

3 have been derived by Rice and Chuang using Herring's procedures. The effects

of surface strains were considered and shown to have no influence on the final

expressions for chemical potential. The results are quoted here:

36/8t = - u Bd/Bx ( 10 )

p, (x) = Q w(x) - Q ct (x) b yy

(11a)

and

p

g (s) = Q w ( s ) - k(s) (11b)

7
where and p

g are the chemical potentials at the grain boundary and crack

surfaces respectively; Q the atomic volume in the current deformed state; Y

s

>

surface free energy per unit area of current configuration; K, surface

curvature, (positive for concave surfaces) and w, the strain energy density.

The law of mass conservation asserts that

u (s) = « 3J /8s (12) n s

at the crack surfaces and

86/8t = Q 8J, /8x

o (13)

at the grain boundary. Also, at the crack tip where x = 0, we must have, from

Eqs. (11),

O . = V K . tip s tip

(14a)

n(VtiP2 h u (14b)

and

(J,l. =2 (J 1 . b tip s tip

(14c)

where J, (x) and J (s) are the matter fluxes at the grain boundary and crack

b s surfaces respectively; 2h is the constant crack thickness (see Fig. 1). For

cases where velocity fields are substantial, Chen and Argon6 argued that

8
instead of Eq. (14c), 2 J

g = + 6J where 6J stands for additional surface

flux induced by the velocity field (jacking). However, although the velocity

field does contribute to the change in volume of the void, it can not alter

the balance of matter flux locally at the crack tip. Hence Eq. (14C) is

generaly valid regardless of the field of strain rates.

An additional equation can be derived if we assume no local entropy

production takes place at the crack tip. The dihedral angle must take a value

such that

Sin 9tiP

= V2 (15)

be satisfied. Here 2 0^ = 7t * 4*> being the dihedral angle at the crack

tip and the grain-boundary free energy. It is interesting to note that two

totally independent approaches, one based on the energy transfer that occurs

when matter is removed from the crack surfaces and deposited along the grain

3 4 boundary , and another , based on invariant dissipative work, both lead to Eq.

(15).

Now we can combine Eqs. (10) and (13) to give

6(x) = 5 J, (x) (16)

u b

if we assume both 6 and decay asymptotically to zero as x approaches

infinity (as expected by physical argument). Eq. (16) asserts that the

boundary fluxes are linearly dependent on 6(x) and that at the crack tip

9
( 17 )

according to Eq. (14b).

Finally, the gradient of surface fluxes can be expressed in terms of 0

upon combination of Eqs . (9) and (12):

The governing equations derived in this section are necessary, as will be seen

in the next section, for the investigation of energy release rate in diffusive

crack growth.

4. Energy release rate in diffusive crack growth

It is well known in fracture mechanics that when a crack is propagating

in a solid, there is always an energy-release rate accompanying crack growth.

Indeed, this provides the thermodynamic "force” driving the crack growth. The

conventional energy-release rate G is defined by

for a class of elastic-brittle cracks without diffusion. Here P is the total

potential energy, w = w (x,y) is the strain energy density and the integration

is performed over the whole volume of the body. For a sake of consistency we

extend this definition to the case of diffusive crack growth by writing

3J

s /3s = (^) cos 0 (18)

w dV

10
UG = W- F (19) v

since the time rate of strain energy is identical to that of the Helmholtz

the special case of diffusive crack growth since in elastic-brittle materials

the motion of dislocations is mild and under creep conditions, the

quasi-static processes lead to little phonon emission (i.e. negligible dynamic

effects). Accordingly, the total dissipative work is predominantly generated

by matter fluxes notably along the high diffusivity paths at internal void

surfaces and grain interfaces. Thus the contribution of by bulk

diffusion could also be neglected unless the temperature approaches the

• 3 •q 4 |) materials melting point. Let and be the dissipative work generated

per unit time at the crack surfaces and along the grain boundary respectively,

we can now write

• • 3 *0,1)

W ^ W + W*! and Eq. (20) describing the energy release rate becomes

free energy for isothermal processes. Here F^ represents the total volumetric

free energy; it relates to the total free energy of the body by F = + F

g

where F is the free energy associated with internal surfaces (cavities and

s interfaces). Thus, by employing eq. (8) derived in Sec. 2

UG = F + W . = F + TS s dxs s

( 20 )

wherein according to Eq. (5), = Jy J * (-V|j)dV. This is justified for

u G = F + W®. + WJ ?

,b

s dis dis ( 21 )

where

( 22 )

11
and

« Si«* * f Jb

• (23)

Notice that a factor of 2 appears in eq. (22), reflecting the fact that there

are two identical crack surfaces. The increase in surface free energy F due

s to steady state crack growth along the grain boundary is simply

*. = 0

- V (24)

since by growing a crack of unit length, two unit areas of free surfaces are

created at the expense of one unit area of grain boundary. It should be noted

that in the absence of diffusion, s = 0 throughout the whole body, our case

then reducing to reversible crack growth of the Griffith model. Then

G_ = F /u = 2 y - y, from Eqs. (20) and (24). This means that (2 y - yu ) is

Gr s s b s b

identical to the Griffith energy.

Eq. (21) serves as the starting point for evaluating G. We tackle the

problem by solving for (eqs. (22) and (23)) explicitly with the

assistance of the equations derived in Sec. 3.

First, we solve for W in Eq. , (22) in the following way:

W

dls = 2 /

J

s <- 3^s /3s) ds

•'o [by Eq. (22)]

r

-2 r ^ CJ M ) ds + 2 f M OJ /8s) ds [integration by

' 3s v s^s' Jo

M

s s' parts]

2u

2 h u (y k - w. . ) + pr- f M cos 0 ds ^ by

VJ

s tip tip' Q

r

s and (18) J

oo

12
The integral in the second term of the RHS is

(w Q - Y

S K Q) cos 0 ds [by Eq. (lib)]

[by Eq. (15)]

Thus, the surface dissipative work per unit cracked area results in the

following expression:

W

dis = 2 h Y K . - (2 Y - Yv ) i " 2 h w . . + 2

s tip v J s

J b' tip dy (25)

This means that, in general, the surface dissipative work is a function of

JUA crack geometric parameters (such as 2h and ) , strain energy distribution

along the crack surface and the intrinsic surface and interfacial free

energies

.

Similarly, the rate of grain-boundary dissipative work can be evaluated

in the following way

W

8-b

dis

J, (-3u /8x) dx

b b

J, (8ct /3x) dx

b yy

5 (3a /3x) dx

yy

J (3w/3x) dx

b

5 (3w/3x) dx

[by Eq. (23)]

[by Eq. (11a)]

[by Eq. (16)]

13
Thus, after integration by parts and noting that 6^ = 2h from Eq. (17) and

= Y

s K t ip f rom Eq. (14a), the grain-boundary dissipative work per unit

cracked area takes the form

W

g-b

dis = * 2 h y

r

tip I c

yy

(36/3x)dx + 2 h u

tip

+ ri

w (36/3x) dx

o ( 26 )

The first term on the RHS of Eq. (26) represents the contribution made by

JUA K t £p> the second term is the mechanical work done by q^ on 6, in analogy to

the cohesive zone models; the remaining terms are the contributions from the

strain energy density.

Substituting Eqs. (24), (25) and (26) into Eq. (21), one obtains the

following equation

rh /*» roo G = 2 I wdy+l w (36/3x) dx - I a (36/3x) dx (27)

•'o •'o *'o ^

This is a general expression for Irwin's energy release rate that arises from

diffusional crack growth in an (linear or non-linear) elastic solid. We note

that in the absence of diffusion, 6=0 and G = 2 w dy becomes the strainenergy release rate appearing in conventional fracture mechanics theory. It

can be seen that G is composed of three parts: (1) the loss of strain energy

in connection with matter that is removed from the crack surface; (2) the

It might be mentioned here that the curvature cannot be defined exactly at

the crack tip mathematically. Hence K t^pi- s referred to as the curvature at a

point immediately adjacent to the crack tip.

14
gain of strain energy by stressing the deposited matter at the grain boundary

to a proper level so as to ensure a coherent fit; and (3) the mechanical work

done by normal stresses to allow insertion of the matter which has diffused

away from the crack surfaces. We emphasize here that Eq. (27) is derived

solely from irreversible thermodynamic principles and therefore is independent

of specific constitutive law or phenomenological equations intrinsic to a

given material, as long as the behavior of the material remains elastic.

Extension of the applicability of Eq. (27) to elastic-plastic materials is

possible if one includes also the plastic dissipative work generated by the

motion of dislocations in the vicinity of the crack tip as formulated by

Thomson^

.

It is of special interest to note that an alternative approach from the

8 9 theory of elasticity can be adopted to derive G. Rice and Drucker , Rice and

Elshelby 10 have shown that the reduction in potential energy for a

smooth-surface notch due to removal (or addition) of matter is

where the integral is performed over the volume dV of the matter being removed

(or added) . Noting that potential energy is the strain energy minus the

mechanical work done (both external and internal) , we can rewrite this

equation so that it is applicable to the present case as a time rate

expression:

15
where ]~b

and I" are grain-boundary and crack surface paths respectively; 6 > 0

at r

fe and 6 < 0 at T . Notice that this equation neglects the surface tension

effects; otherwise an extra term representing the surface work must be added

which is seen to be identical to Eq. (27). Hence, two entirely different

derivations, one based on thermodynamic principles and the other on elasticity

theory, lead to exactly the same expression for G.

As is well known in linear elastic fracture mechanics theory the

path-independent contour integral J and Irwin's strain energy release rate G

2 2 can be shown to be identical, and G = J = (1-v /E) where is the mode I

stress intensity factor, E Youngs modulus and v the Poisson's ratio. Hence it

is interesting to connect G and J for diffusive crack growth as considered

here

.

to the LHS

3

.

Now, since along T , - 6 ds = u cos 0 ds = u dy and along T , 6 = - u (36/3x)

,

S D this equation reduces to

I

The definition of J in the context of finite deformation elasticity is 11

o

where is an arbitrary path around the crack tip starting from the

lower crack surface and ending at the upper one in a counterclockwise

' 16
direction: X and Y are material coordinates in an unstressed state; dS is an

o element of arc length, W

q

is strain energy per unit volume in undeformed state

and T

q

is the nominal stress vector. By taking advantage of the pathindependence property, we can shrink T down to the crack lines (i.e. negative

X-axis in Fig. 1) and along the flat grain-boundary surfaces (dashed lines in

Fig. 1). After a finite deformation into the current deformed state, the

portion of T along the negative X-axis becomes dashed line along the curved

o "notch surfaces" and the other part of I” at the grain-boundary surfaces

deform into lines following the ± 6/2 solutions as shown in Fig. 1. Then,

based on the current configuration, the contribution to J of the integration

along the "notch" surfaces is w dy since T vanishes everywhere there and

the contribution made by the terms involving integration along both the upper

00 00 and lower grain-boundary surfaces is w (36/3x) dx - J

q

ct (36/3x) dx

since, for integration paths following the ± 6/2-curves, dy = - (3u/3x) dx and

T-

• (3u/3s) ds = - ct (3u/3x) dx. Hence - yy

J-h

, 36 ,

, or (t~) dx f yy ox K O dx • -h •'o *// o

>on comparison between this equation and Eq. (27), and noting the symmetry of

le upper and lower crack surfaces, we immediately arrive at

G = J (28)

Eq. (28) shows that G in the case of diffusive crack growth is identical to J

as in the conventional fracture mechanics expression if one takes finite

elastic deformation into account. We emphasize that this expression, Eq. (28)

is exact and is applicable to steady-state diffusional crack growth in a

17
brittle-elastic material even if the constitutive and phenomenological

equations are non-linear. This is because G was derived from energy

considerations and while specific forms of stress-strain and diffusion

equations can affect field parameters , they do not have any influence on the

energy balance.

5. Discussion

We have shown from non- equilibrium thermodynamics that the general

expression for the energy release rate G that incurs during diffusional crack

growth processes is composed of three terms, namely (i) surface energy

increase arising from the creation of new crack surfaces as well as

dissipative work (heat generated) by diffusional processes along (ii) the

crack free surfaces and (iii) the grain boundary. This statement is valid so

long as the conditions of steady state and mass and energy conservation are

met and no plastic processes take place inside the body. Derivation of G from

the approach of elasticity theory was also given which showed an identical

result. In addition, it was shown that G is related directly to the

well-known J-integral as given by Eq. (28).

To further investigate each individual component contributing to G the

curvature at the crack tip, stress and strain energy fields have to be

determined (see Eqs . (25) and (26)). These field parameters must be solved

from phenomological equations. Unfortunately, no solutions are available in

the general case of a diffusional grain-boundary crack, obeying non-linear

diffusion equations and growing in a non-linear elastic material. We

therefore limit our discussion to a special linerized case where solutions are

18
readily available. The diffusion equations then follow Fick's law and take

the form

J = (const.) V |J (29)

where J = J

g and |J = (j g at crack surfaces, and J = and p = along the

grain boundary. Also the stress-strain relations follow Hooke's law, so the

strain energy function takes quadratic form in stress (or strain). Further,

the strain energy contribution to p (and hence to G) can be neglected even at

the crack tip (e.g. see Eq. (8) of Ref. 3). This is because the radius of

curvature developed by surface diffusion at the crack tip is normally large

compared to lattice spacing and misfit residual stresses induced by mass

transport along the grain boundary create "wedging" effects, thus alleviating

the stress concentration near and at the crack tip. Actual stress solutions

given by Chuang confirm this assertion. Thus, the version of W

g

, Eq. (25),

ais’ ^9* (26) and G Eqs . (27) and (28) in the linerized theory can be

rewritten as

W

s dis

= 2 h

*s

K tip (2 *s * V (30a)

W

g-b

dis _ , = - 2 h Y

u ° s r f8«. k . - I a (— „ ) dx

tip J yy 3x (30b)

and

G = J = •/’

Jo

, 86 . j

ct (—) dx

yy 9x

(30c)

• g • g . b

In the following we will investigate W and W

6 ' separately. To evaluate

dis dis

W

( dis), as remarked before, K . and 2h must be determined by solving the

txp

u r

19
entire crack tip profile geometry. However, we may determine the order of

magnitude by considering the limiting case of a unit cell in a rigid material

containing a linear array of cracks with uniform center-to-center spacings 2b

rapid grain-boundary diffusion so that the normal stresses as well as the

matter diffused from the crack surfaces are distributed uniformly along the

respectively. Combining these three equations and Eq. (30a) and solving for

k . yields tip

where f = (b-a)/a is a pure number. Substituting this expression for K

into Eq. (30a) results in

Although this equation is derived in a simple-minded fashion, it does reveal

some important features in a general case. First, this expression shows that

surface dissipative work is independent of the crack velocity u. Second, the

absolute quantity is of the same order of magnitude as the Griffith energy.

Indeed, by solving the linear diffusion equation (Fick's law) at the crack

12

surfaces subject to the conditions of mass conservation, Chuang and Rice

gave an approximate closed form solution for the crack tip profile from which

K is obtainable (see Eq. (24) of Ref. 12):

and crack lengths 2a under a remote applied stress a . Taking an extreme of

3

• 2b = ct

• • s 2h • 2 u, and CT^ • 2b * 6 = 2 due to balances of force, mass and energy

(31)

tip

20
(32)

I

’I

I

I

I

Ktip [2 (l-Yb /2 Yg

)

]

1/2 (kT/D

s Ys Q

4/3

) 1/3

u 1/3

E

(1 ‘ V2 V

substituting Eq. (32) in Eq. (30a) leads to f = 1 for this approximation. When

evaluated precisely from a numerical technique, f = 1.1 for the limiting case

in which = 0- In fact, f lies between 1.0 and 1.1 depending upon 0^ for

any material. Setting f = 1 will suffice for all practical purposes. If we

let = 20° as is true for many crystalline materials, then setting f = 1

will result in an error of approximately 5.7%. (The error which would

correspond to the worst case (0^ = 0) is then 9%)

.

We note in passing the interesting situation regarding the validity of

Eq. (30a) in quasi-equilibrium cavities of Hull-Rimmer type not considered in

the present paper. In that case, the version of Eq. (32) is =

(l-y

b /2y )/h from a simple geometric sketch. Insertion of this expression for

into Eq. (30a) leads to W^

g = 0. This result can be verified by Eq.

(22) since quasi-equilibrium shapes imply uniform curvature everywhere

resulting in grad = 0. Hence the applicability of Eq. (30a) for evaluation

• 3

of W is established for a cavity of general shape.

Now, in addition to K . , the CT and 5 field parameters at the grain

tip yy

boundary must be determined according to Eq. (30b) prior to computing the

dissipative work incurred by grain-boundary diffusion. The governing

equations were derived from the requirements that matter be conserved and

Fick's law be satisfied in the grain boundary, and that the stress and strain

fields so generated in the interior satisfy the -equilibrium and compatibility

21
conditions respectively, and also satisfy boundary conditions at the crack

planes and at the outer boundaries. The results are two differential

2 equations with two unknown functions a (x) and S(x) as follows :

CTyy(x) K_

V 2nx

+

E

47X(1-V

2

)

a6/8x'

x' -x

S(x)

D,6,Q da

b b yy

ukT dx

subject to the initial conditions at the crack tip (x=0):

(33a)

(33b)

tfyy (0) = [2 (l-Yb /2Ys )]

1/2

(kT Ys

2

/D

s 0

4/3

) 1/3

u 1/3

(33c)

CJ^y (0) = 2 [2(l-Yb /2y

s

) ] 1/2 [(D

s Y

s

) 1/3

(kT) 2/3 /D

b

6

bQ

5/9 ]u2/3 (33d)

1/3 where K is the applied stress intensity factor; D

g Q and Db

6

b are surface

and grain boundary diffusivities respectively, and kT has its usual meaning.

The first term on the RHS of Eq. (33a) is just the well-known K-field created

by the applied stress and the second term represents the residual stresses

generated by non-uniform matter deposition due to grain boundary diffusion.

Eqs. (33b) and (33d) follow from Fick's law and Eq. (33c) from Eqs . (14a) and

(32). Note that both initial conditions are dependent on the crack velocity.

Typical solutions for o^(x) and S(x) at a fixed crack growth rate are plotted

schematically in Fig. 1. The reader is referred to Ref. 2 for detailed

solutions. It was shown there that a grain boundary parameter L exists, where

1/2

L = [ztED, 6, Q/4(l-v 2 )ukT] has a dimension of length and is a function of

b b

material properties, temperature and crack velocity, such that the size of the

"diffusion zone" is confined to within 4L ahead of the moving crack tip

22
wherein the influence of diffusion is significant. The resulting small

diffusion zone implies that the applied K field is able to control the crack

growth behavior. Mathematically, this is analogous to the small cohesive zone

in the Dugdale-Baranblatt model or small-scale yielding zone in the linear

elastic fracture mechanics.

Hence, there is a one-to-one correlation between K and u for u > u . : min

K/K_ = 0.845

br [ovv ) 1/12 +

, , .- 1 / 12 . mm (34a)

where

u = 8.13 D

4

Q

7/3 [E/(l-u2

) D.S ] 3 /KTy 2

min s b b s (34b)

and G = J “ K

or G/G

Gr = J/J

Gr = (K/KGr ) 2

= 0.714 [(u/u. ) mm

1/12 ^ f

. .- 1 / 12.2

+ (u/umm) ] (35)

where the subscript Gr relates to an ideal Griffith crack.

Eq. (34a) is the resulting K-U kinetic law applicable to steady state

diffusional crack growth in a linear elastic solid obeying Ficks law. A plot

of this equation shows that a threshold value of K equal to 1.69 exists

below which the crack can not propagate. Moreover, since u . is a constant mm

depending on material properties and temperature (see Eq. (34b)), and has a

23
finite value, it is not possible to have a crack growing at a speed slower

than u

^ unless another crack growth mechanism takes over. The consequence

can be explained in terms of energy considerations given below. At this

13 point, it might be mentioned that Speight et al. claimed that their results

could be made to coincide with the kinetic equation given here (see Eq.

(34a)). Unfortunately, their claim only corresponds to the second term of the

RHS of Eq. (34a) which is negligible when u >> u , a range where slit-like

growth mode is expected to be operative.

The main features of the energy release rate that occurs during diffusive

growth of a sharp crack can be unveiled if we plot nondimensional G vs . u from

Eq. (35) as illustrated in Fig. 2 where the three components of G are shown

for an arbitrary velocity. For example, at the threshold point where u = U ^

an amount of energy 0.85 G_ goes to dissipative work in the grain boundary

br

whereas 1.0 G^ each is spent on F

s /u as well as on surface diffusion. For a

higher crack velocity corresponding to a higher applied load, the first part

increases following the curve while the last two parts remain fixed. However,

when U < u . , there is insufficient energy available to move the crack tip min

and, as a result, the crack will cease to grow. The current theory is unable

to predict, under this condition, whether the crack tip will remain stationary

or begin to heal.

Fuentes-Samaniego and Nix asserted that a steady state exists whenever

the energy dissipation rate remains constant with respect to time, namely

d_

dt

+ W

g.b.

dis ) = 0 . (36)

24
• s

T:,g- b Substituting the expression for and from Eq. (5) into Eq. (36) and

then applying Euler's equation derived from variational principles, they

obtained several governing equations from the natural boundary conditions.

Examination of these equations reveals that plays an important role in

the crack growth behavior. This is a feature consistent with the current

treatment (see Eqs. (25) and (26)). Evidently, an invariant dissipative work

must be a necessary condition for the existence of a steady state. However,

it needs to be proved that Eq. (36) is a sufficient condition as well.

Otherwise, it is likely that a "steady state" defined purely by Eq. (36) may

yield a crack velocity as well as a crack tip shape which are continuously

changing with time.

Conclusion

The energy release rates that take place during steady state diffusional

crack growth were derived in general from the principles of irreversible

thermodynamics. For the sake of completeness, the strain energy contribution

is included in the formulation. The resulting expression is valid regardless

of the form of constitutive law or phenomenological equations that a material

must follow, and can be decomposed into three individual components

representing surface Griffith energy and dissipative work necessary to

complete the processes of surface diffusion as well as grain-boundary

diffusion. We then find that G is related directly to the well known

J-integral when it is formulated properly in the framework of finite

deformation elasticity.

25
Quantitative evaluations of each component were made for a special case

based on a linearized theory where solutions of field parameters are readily

available. The results showed that the strain energy contribution is

negligible and that the surface diffusion component has values ranging between

1.0 and 1.1 G^, depending on material and is independent of the crack

velocity. On the other hand, the grain boundary diffusion component is a

function of the crack tip curvature as well as the J-integral, and is a

monotonically increasing function with increasing crack velocities starting

from a minimum value of 0.85 times the Griffith energy at u = u . . The total mm

G is equal to J which then can be interpreted as (negative) mechanical work

done by the grain boundary normal stresses on the thickening of the interface

2 2 by mass transport. Finally, G remains equal to (1-v )K /E which means that,

as in the case of linear elastic fracture mechanics, the applied stress

intensity factor is the key parameter controlling diffusional crack growth in

a linear elastic material.

Acknowledgement

The author is grateful to Professor James R. Rice of Harvard University

for several stimulating discussions on nonequilibrium thermodynamic formalism.

References

1. T.-J. Chuang, K. I. Kagawa, J. R. Rice and L. B. Sills, Acta Metallurgica

27 (1979) 265-284.

2. T.-J. Chuang, Journal of the American Ceramic Society 65 (1982) 93-103.

3. J. R. Rice and T.-J. Chuang, Journal of the American Ceramic Society 64

(1981) 46-53.

26
4. R. Fuentes-Samaniego and W. D. Nix, Philosophical Magazine A 44 (1981)

601-612.

5. H. B. Callen, Thermodynamics, John Wiley & Sons, Inc. New York (1960).

6. I.-W. Chen and A. S. Argon, Acta Metallurgica 29 (1981) 1759-1768.

7. R. M. Thomson, Chapter 23 in Physical Metallurgy, Ed. by R. W. Cahn and

P. Haasen, North-Holland Publishing Co. Amsterdam 1983.

8. J. R. Rice and D. C. Drucker, International Journal of Fracture Mechanics

3 (1967) 19-27.

9. J. R. Rice, in Fracture Vol. II, Academic Press, Inc. New York, (1968)

191-311.

10. J. D. Eshelby, in Inelastic Behavior of Solids, McGraw-Hill, New York

(1970) 77-115.

11. J. R. Rice, Journal of Applied Mechanics 35 (1968) 379-386.

12. T.-J. Chuang and J. R. Rice, Acta Metallurgica 21 (1973) 1625-1628.

13. M. V. Speight, W. B. Beere and G. Roberts, Materials Science and

Engineering 36 (1978) 155-163.

27
_ 4.
Fig.

1.

Diffusive

crack

growth

along

a

boundary.

Normal

stress

a

(x)

and

opening

displacement

6(x)

distributions

are

shown

schematically.

Special

path

for

evaluating

J

is

indicated

in

dashed

line.
001

o

o

00

O

CDO

m-O

CVJo

o

o

o £ O O O

00 • _j • OJ —

CM

J0 9/9 ‘3±VH 3SV3“l3a A9d3N3 Q3ZnVlAiaON

+J 01 s J= 0 4J u

oo n

<0

CM

oo

NORMALIZED

CRACK

VELOCITY,

v/v

m

\
INTERNAL DISTRIBUTION

1. E. R. Fuller, Jr.

2. S. J. Schneider, Jr.

3. R. P. Reed

4. N. J. Tighe

5. J. B. Wachtman, Jr.

6. S. M. Wiederhorn

EXTERNAL DISTRIBUTION

7.

Ames Laboratory, Iowa State University, Ames, Iowa 50011

0. Buck

8-10. Argonne National Laboratory, 9700 S. Cass Avenue, Argonne,

Illinois 60439

W. A. Ellingson

C. R. Kennedy

D. S. Kupperman

11. Battelle Columbus Laboratories, 505 King Avenue, Columbus,

Ohio 43201

I. G. Wright

12. Bureau of Mines, Tuscaloosa Research Center, P.0. Box L,

University, Alabama 35486

H. Heystek

13-14. Cornell University, Materials Sciences and Engineering Department,

Bard Hall, Ithaca, NY 14853

C. -Y. Li

R. Raj

15-17. Electrical Power Research Institute, 3412 Hillview Avenue,

P.0. Box 10412, Palo Alto, CA 94303

W. T. Bakker

R. I. Jaffee

J. Stringer

18. Exxon Research and Engineering Company, P.0. Box 101, Florham Park,

New Jersey 07932

E. M. Anderson

19. Gas Research Institute, 8600 West Bryn Mawr Avenue, Chicago,

Illinois 60631

V. L. Hill

20. IIT Research Institute, 10 West 35th Street, Chicago, Illinois 60616

S. A. Bortz

21. Lawrence Berkeley Laboratory, University of California, Berkeley,

California 94720

A. G. Evans

22. Metal Properties Council, Inc., United Engineering Center, 345 E. 47 St.,

New York, NY 10017

A. 0. Schaefer
23. National Science Foundation, Division of Materials Research,

Washington, DC 20550

B. A. Wilcox

24-27. Oak Ridge National Laboratory, P.0. Box X, Oak Ridge,

Tennessee 37830

P. F. Becher

R. A. Bradley

P. T. Carlson

V. J. Tennery

28. Pennsylvania State University, College of Earth and Mineral Sciences,

University Park, PA 16802

R. C. Bradt

29. Pennsylvania State University, Department of Materials Science and

Engineering, University Park, PA 16802

30. Southwest Research Institute, 6220 Culebra Road, P.0. Drawer 29510, San

Antonio, TX 78284

F. F. Lyle, Jr.

31

-32. Virginia Polytechnic Institute and State University, Blacksburg,

VA 24601

J. J. Brown, Jr.

D. P. H. Hasselman

33-34. DOE, Office of Fossil Energy, 20 Massachusetts Avenue, NW Washington,

DC 20545

S. J. Dapkunas

J. Fairbanks

35. DOE, Office of Basic Energy Sciences, 20 Massachusetts Avenue, NW

Washington, DC 20545

L. Ianniello

36-37. DOE, Oak Ridge Operations Office, P.0. Box E, Oak Ridge, Tennessee

37830

E. E. Hoffman

Assistant Manager, Energy Research and Development

38. Technical Information Center, P.0. Box 62, Oak Ridge, Tennessee

37830
NBS-114A (rev. 2 -ac) U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET (See instructions)

1. PUBLICATION OR

REPORT NO.

NBSIR-82-2628

2. Performing Organ. Report No. 3. Publication Date December 1982

4. TITLE AND SUBTITLE

ON THE ENERGY-RELEASE RATE ASSOCIATED WITH DIFFUSIONAL CRACK GROWTH

5. AUTHOR(S)

Tze-Jer Chuang

6. PERFORMING ORGANIZATION (If joint or other than NBS. see instructions)

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

7. Contract/Grant No. DE-A1 05-800R20679

8. Type of Report & Period Covered

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. State, ZIP)

AR&TD Fossil Energy Materials Program

U.S. Department of Energy

10.

SUPPLEMENTARY NOTES

~] Document describes a computer program; SF-185, FIPS Software Summary, is attached.

11.

ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here)

A general expression for the energy release rate (G) that arise during steady

state crack propagation by diffusion is derived from the standpoint of irreversible

thermodynamics. Three contributing components of G are identified: (i) the

Griffith energy (Gp ); (ii) heat generated in the process of surface diffusion; and

(iii) grain-boundary diffusion. Further, the total G is shown to be directly

related to the well-known J-integral if formulated in the framework of finite

deformation elasticity. This expression for G is valid in general even if the

response of the material is not linear and the mass transport kinetics does not

follow Fick's law. Quantitative evaluations of each component are made for the

linear case where field solutions are available. The results show that component

(ii) is approximately equal to Gp and is independent of the crack velocity (u)

whereas component (iii) is a monoionically increasing function with G starting from

0.85 Gp when u is at threshold value; and that strain energy contributions can be

neglected leading to G = J = (l-v 2 )K 2 /E. This means that G is not primarily

associated with the release of the strain energy rather, it stems mostly from the

negative work done by the normal stresses on the thickening of the grain boundary

due to non-uniform deposition of matter along it.

12.

KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolons)

Creep cavitation; creep fracture; diffusional crack growth; energy release rate;

high temperature fracture; J-integral; non-equilibrium thermodynamics; steady state

crack propagation

13.

AVAILABILITY

C^Unl i mi ted

j | For Official Distribution. Do Not Release to NTIS

j | Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C.

20402.

[X] Order From National Technical Information Service (NTIS), Springfield, VA. 22161

14. NO. OF

PRINTED PAGES

35

15. Price

$7 . 50
'

1