j AT' L INST. OF STAND & TECH R.I.C.

A111DS

1 NBSIR 81-2255

A Diffusive Crack Growth Model

for Creep Fracture

Tzs-jer Chuang

practure and Deformation Division

Nationai Measurement Laboratory

Center for Materials Science

U.S. Department of Commerce

National Bureau of Standards

Washington. DC 20234

March 1S81

for 100 tment of Energy

. 1)56

ngton, DC

61-2255

1981

oj CnJ

NATIONAL. BUREAU

OF STANDARD*

LIBRART JUl 2 0 1981

NBSIR 81-2255

#*

A DIFFUSIVE CRACK GROWTH MODEL

FOR CREEP FRACTURE

Tze-jer Chuang

Fracture and Deformation Division

National Measurement Laboratory

Center for Materials Science

U.S. Department of Commerce

National Bureau of Standards

Washington, DC 20234

March 1981

Prpared for Department of Energy

Washington, DC

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

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director
>v : " X, ; • r*3-i
A DIFFUSIVE CRACK GROWTH MODEL FOR CREEP FRACTURE

TZE-JER CHUANG

Fracture and Deformation Division

National Bureau of Standards

Washington, D.C. 20234

r).\ re D J :

'

< Getter—T§SO

ABSTRACT

A grain boundary creep crack growth model is presented here based on the

assumptions that the crack propagates along the grain boundary by a coupled

process of surface and grain-boundary self-diffusion; the adjoining grains

on either side of the boundary do not behave plastically; and steady state

conditions prevail. Under the action of the applied stress atoms on the crack

surfaces are driven by surface diffusion toward the crack tip from where they

are deposited non-uniformly by grain-boundary diffusion along the grain interface so that the grain boundary opens up in a wedge shape ahead of the advancing

tip which in turn produces a misfit residual stress field. The total grain

boundary normal stresses which are the sum of this misfit stress field and

that due to applied stress as well as the boundary opening displacements due

to materials deposition are solved from a singular integro-differential

equation to give the following equation relating K to u:

K/K

min = 5 Ko/>W1/12 * (»/>W'

1/12

]

where K is the mode I crack tip stress intensity factor, K . = 1.69 is

the minimum K below which no crack growth is predicted, being the stress

intensity based on the Griffith theory; u is the stationary crack tip velocity

and u . is the minimum u for which K = K . . In terms of the conventional min min

expression of u « K

n

, the present model predicts the values of n varying

from 12 to infinity. A comparison with a set of creep crack growth data

on Si-Al-O-N at 1400 °C shows good agreement between the theory and experiment.

A detailed analysis of the energy balance for the present model is also

presented which indicates J or (1-v )K /E is indeed the correct energy release

rate during the crack growth as is true in the theory of elastic fracture

mechanics. However, the energy released in the diffusion processes is in the

form of work done by the normal stress rather than in the form of elastic

strain energy of the grains.

1. Introduction

At high temperatures grain boundary cavitation is a general phenomenon

accompanying the process of creep rupture: cavity nuclei at the grain

interfaces grow to coalesce with each other until the remaining ligament

can no longer support the sustained loads and premature failure takes place.

The prediction of time to fracture is thus important for the design of

structural components subject to creep condi tons. For materials such as

ceramics where pores have been introduced at the boundaries in the process

of fabrication prior to service, emphasis is given to the study of in-service

cavity growth since nucleation plays only a minor role in the entire service

life. Indeed, the subject of integranular cavity growth has been under

intensive research in the past few years. The cavity growth at elevated

temperatures is generally believed to be controlled by surface and grain

boundary- self-diffusion since observations freqeuntly revealed that a

majority of the void population appears at the boundaries normal to the

direction of the applied tensile stress, although other possible mechanisms

such as grain sliding, dislocational slip, etc., may be in operation as

well. The first model of cavity growth by diffusion was proposed by Hull

and Rimmer [1] who considered a square array of spherical voids on a grain

boundary slab normal and subject to a tensile load. It was found that the

void growth rate increases linearly with increasing applied stress assuming

the grains on both sides of the boundary are rigid. Raj and Ashby [2]

considered a network of lenticular cavities and obtained a similar conclusion.

Unfortunately, creep experiment data do not always agree with their predictions and optical metallographs often show that the morphology of intergranular voids look thin and crack-like, rather than spherical or lenticular,

indicating that the quasi -equi 1 ibrium conditions at the void surface are
2

not always fulfilled. This case has been considered by Chuang and Rice [3]

who found the near tip void profile during the steady state growth. In an

overview of the subject, Chuang et al. [4] investigated the time-dependent

aspect of the problem and obtained a class of "self-similar" solutions which

indicate that initially cavities grow slowly in a quasi-equilibrium shape

which gradually becomes crack like as the propagation rate increases. A

growth rate parameter was identified which sets the transition period

between these two extreme modes. In general, the crack like growth mode

prevails when the applies stress, the cavity size, and the grain boundary

diffusivity are large as compared to the capillarity stress, the cavity

spacing, and the surface diffusivity. The equilibrium shape is favored

by the opposite situation.

The present paper restricts the discussion to the limiting case of

crack like cavity growth in steady state as opposed to another extreme case

of Hull-Rimmer type cavities. The central issue to be addressed in the

present paper is "what is the sustained load required for a boundary crack

to propagate at a given velocity and temperature for a known material?"

Mathematically this means that a functional relationship between the applied

stress and the crack tip velocity has to be formulated. In the circumstances,

this can be regarded as an extension of work that was originated by Chuang

and Rice [3].

This problem has been well treated in cases where the separation

between the adjacent cracks is so close that the adjoining grains essentially

behave rigidly. However, if the neighboring cracks are remotely apart or

if a main crack is growing in isolation, the grain deformabi 1 ity becomes

important and therefore must be taken into account. Previous attempts to

model the combined effects of elastic deformation and diffusion on crack
3

growth were made by Vitek [5] and Speight et al. [6]. In the former model,

an enforcement that the crack maintains a constant thickness during the

growth regardless of the level of the applied stress is imposed while in

the latter model, a parabolic distribution of the boundary displacements

in a "wedge" shape ahead of the running crack tip is assumed. Further,

the near tip void shape is not analyzed in detail. As a consequence, the

crack tip conditions together with the related physical laws, as will be

discussed later, are either not satisfied or satisfied only in approximation.

In contrast, our work attempts to solve exactly the coupled problem of

elasticity and diffusion so that both grain boundary stresses and displacements

are determined as part of the solution. As will be seen, the end results

regarding the functional dependence of the applied stress on the crack

growth velocity are quite different although some similar features are

exhibited.

The program of the paper is as follows: In section 2 we present the

self-consistent model of steady state growth, beginning with descriptions

of the mathematical model from which control equations and boundary conditions

at the advancing crack tip are derived based on certain justifiable physical

assumptions. A unique solution is pursued and detailed elaborations leading

to the expression of external loading intensity in terms of the crack growth

rate are given in section 2.4. For a growing crack, it is well recognized

that a portion of energy is released in the body during the growth. This

energy provides a driving force causing the crack to grow. Hence in section 3

we present a detailed calculation on the energy release rate associated with

our crack growth model. This promotes the analytical rigorousness of our

model and provides physical insight as to how and where the energy dissipation

takes place. A discussion follows in section 4 where a comparison between
the present theory and a set of creep test data on ceramics and comparisons I

with the other two models previously mentioned are presented. Finally, I

section 5 summarizes and gives conclusions drawn from the discussion presented.

We conclude that whereas the other models proposed thus far possess some

theoretical deficiencies, the current theory is developed in a self-consistent

fashion. The excellent agreement between the theoretical predictions presented

here and recent test data [7] obtained from creep experiments on Si-Al-O-N at

1400 °C points up a bright prospect. Hopefully, future creep tests on

other ceramics will show similar features predicted by the current theory.

2. The Steady State Crack Growth Model

Before we enter into detailed elaborations on the proposed model,

the physical process is given first by which the model is designed to

represent the case of a creep crack growing in steady state at a fixed

velocity, u, along a grain interface normal to an applied stress, a . As

mentioned in section 1, the kinetics of the crack growth is assumed to be

controlled by a coupled process of surface and grain boundary diffusion.

Of course, at the crack surfaces lattice diffusion as well as evaporation

and condensation may also play a role in the mass transport process whereas

in the grain boundary bulk diffusion may drive atoms out of the boundary

zone. However, it has been shown [4] that at least for the crack size (in

the order of microns) and temperature range (between 0.3 to 0.8 Tm, Tm being

the melting temperature in K) we consider here, surface diffusion and grain

boundary diffusion are the dominate mechanisms at the surfaces of the crack

and in the grain boundary, respectively. Hence atoms are driven along

the crack wall by surface diffusion toward and into the crack tip from

where grain boundary diffusion further carries them away along the grain

boundary. The steady state is reached when all related physical parameters
5

become invariant with regard to an observer stationed at the advancing

crack tip. The grains on both sides of the interface are considered

homogeneous, isotropic, and linear elastic free of dislocations even in

the near tip zone. This is a reasonable assumption for strong brittle

materials such as ceramics where evidence indicates that the dislocations

near the crack tip are essentially immobile [8] during the entire creep

life. On the other hand, this assumption may not be valid for ductile

materials such as metals where substantial dis locational creep might take

place inside the grains especially in the vicinity of the crack tip [9].

Hence the current model has direct application to high temperature creep

crack growth in ceramics.

2. 1 Descriptions of the mathematical model

The limiting case of propagation of an isolated long boundary

crack led us to consider an infinite bicrystal containing a semi-infinite

crack colineared with a grain boundary line on the x^-axis subjected to a

uniform tensile stress, applied remotely in the X

2

~direction and to a

uniform temperature T as illustrated in figure 1. Under the action of ct^,

the crack tip is propagating at a constant velocity, u, in the positive

x-j-direction. A separate moving rectangular coordinate system of (x, x^, x^)

with its origin coinciding with the advancing tip is also established so that

x = x^ - ut. The geometry has a long dimension in x^ resulting in a twodimensional problem with plane strain conditions. This is justified since

the geometry of a long thin crack dictates that the principal curvature of

its tip shape in the plane is much larger than that of the crack

front in the x-j-x^ plane. Accordingly, the two-dimensional plane strain

conditions prevail and consideration of a unit length in the x^-direction

is fully representative of the whole problem. Furthermore, the following
6

physical conditions are also assumed: (1) the crack surfaces are clean and

free of traction (pressure) or environmental attack. In other words, the

interior of the void is a vacuum, free of glassy or viscous phases; (2) the

grain boundary is a perfect matter sink, capable of transporting material

and is a flat slab of constant thickness constituting an effective diffusion

zone; (3) the matrix of the bi crystal is a homogeneous, isotropic body

composed of linear elastic material free of defects; (4) the composition remains|

unchanged at the surfaces and in the grain boundary during the entire period

of the crack growth so that single- valued macroscopic chemical potentials

of atoms can be defined in the diffusion process; and (5) the diffusional

properties of material are isotropic, independent of temperature and crystal

orientation.

This model configuration and physical assumptions are set to represent

the limiting case of boundary cavitation in which the cavity has grown into

a long crack-iike shape and senses no effects from the neighboring defects.

2. 2 Boundary conditions at the crack tip

Our goal is to pursue solutions for the distribution of the normal

stress cr(x) and opening displacement, S(x), along the grain boundary and

thereby correlate the applied stress, ct , or the crack tip stress intensity

factor, K, with the crack velocity, u. It will become clear that the

boundary conditions at the moving crack tip are required in order to do so.

Those are derivable from the steady state crack tip shape solution obtained

by Chuang and Rice [3]. The results are summarized in this section.

It is well known in diffusion theory that a non-uniform distribution of chemical potentials incurs an irreversible diffusional flux of

matter. The chemical potential per atom at the surface is

M

s = k ( 1 )
7

in relation to a reference state of flat surface where we arbitrarily set

P

s = 0. Here Q is atomic volume, surface free energy, and k is the mean

curvature of the surface at the point under consideration, assuming a

positive sign for a concave surface. This expression results from the fact

that there is no normal stress and the strain energy contributions associated

with the surface stresses are small even at the crack tip and hence can be

neglected [10,12]. The linearized theory of diffusion, known as Ficks law,

calls for a linear dependence of matter flux on the gradients of the

chemical potential, namely J « - Vp. This phenomenological law appears to

be followed by nature, at least in the neighborhood of an equilibrium

condition. For surface diffusion, the version of this equation is

QJ

g = -(D

s 6s /kT)(dp

s /ds) where ds is an element of arc length along the

surface, D_ the surface diffusivity, and 6 the thickness of the diffusion

s s 1/3 layer. Adopting 6

g = Q and substituting equation (1) one obtains

J

s = (Ds Ys0

1/3/kT)dK/ds (2)

For a given rate of crack growth, the entire crack surface profile is obtained

by solving the simultaneous equations of mass conservation and surface diffusion (Eq. (2)); the results show a constant crack thickness 2w proportional

-1/3

to u developed far away from the crack tip:

2ui = 2 {Z (1 - Yb/2v5

) 1/2

(D

s Vsn

4/3

/ukT)

1/3

(3)

where is the grain boundary free energy. The solution further indicates

that approximately both surface flux and curvature decay exponentially with

increasing S

' J

s <VtiP

|

K

i

it K. . 1 tip

1 j

exp (-|S) (4)
8

where S = arc length along the crack surface measured from the

crack tip; dS = -ds

Z = (D

s Y$

Q

4/3

/ukT)

,5/3

) l/3

u2/3

(5)

= surface curvature at the point immediately adjacent

to the crack tip

( 6 )

It should be noted here that all parameters summarized above including the

crack shape are dependent on the applied stress since they are functions

of u.

At the grain boundary, the chemical potential per atom is

where again we arbitrarily set = 0 for a reference state in which a = 0,

a being the normal tensile stress at the boundary acting in x^. Further,

the strain energy term has been justifiably neglected [10,12].

In analogy to the case of surface diffusion, the version of Fick's

law in grain-boundary diffusion is = -(Dj

3 6^/kT)dpj

3 /dx. After substituting

the expression of from Eq. (7), one obtains

where is grain boundary atomic flux, the grain-boundary diffusivity

and 6^ the effective thickness (assume constant for simplicity) of diffusion

zone in the grain boundary.

At the crack tip where the two crack surfaces join the grain boundary

(7)

b kT dx

( 8 )

the following conditions have to be met:
9

^s^tip ^b^tip ’

2 <Vtip = <Vtip *

(9a)

(9b)

Eq. (9a) follows from the requirement of continuity in chemical potentials

and Eq. (9b) from conservation of mass. By means of these two equations

the expressions of (^

s

) t ^ p

and (k)^. 1n Eqs. (5) and (6) can be transferred

as follows to a more useful form. First we observed that after substitution

of Eqs. (1) and (7) in Eq. (9a), " K t

-jpY s^ = "CTtipfi from whlch we ^ lnd

a

tip = o(0) = V2(l-Yb

/2-y s

)

1/2 (kTY

s 2

/D

sQ

4/3

) ,/3

o

1/3

( 10 )

Second, the grain boundary diffusion equation of Eq. (8)

crack tip (x = 0) is (J

b

) t1p = O^a^/kT where o[

jp =

It follows from Eqs. (9b) and (5) that

evaluated at the

da/dx

x = 0.

°iip = 2 V2(l-Yb/2Ys

) 1/2

[(D

s Ys

) 1/3(kT) 2/3

/0

b

6

b

n

5/7

]u2/3

(11)

Eqs. (10) and (11) show that the normal stress and its derivative in x at

1 /3 2/3

the crack tip are in proportion to u and u , respectively. Both equations

will be seen useful as boundary conditions for future stress solutions.

2. 3 Derivation of the field equations

The objective of this section is to derive the field equations

governing the interaction of elasticity and diffusion from which the

solution of stress can be obtained to predict the functional relationship

between the stress intensity and the crack tip velocity. Specifically, emphasis

will be placed on two parameters: the grain-boundary normal stresses, a(x),

and opening displacements, 5(x).
As atoms on the crack surfaces diffuse into the grain boundary

at the crack tip, the opening displacements of yet to be determined quantities,

6(x), along the g.b. are generated which in turn create a misfit residual

stress field inside the grains and also along the grain boundary. However,

our interests are restricted to the boundary line so that the relationship

between 6(x) and ct(x) can be formulated. We note that in the case of

Griffith crack extension such as considered by Stevens and Dutton [11], no

S(x) is considered to take place and as a consequence the actual stress

field is produced solely by the applied stress.

Steady state conditions require that 6(x) = 6(x^-ut) so that

3S/3t = -u36/3x. In addition, for mass conservation in the boundary,

36/3t = -QSJ^/Sx must be satisfied. Both conditions lead to udS/dx =

QdJ^/dx. But physically both 6 and must decay to zero as x approaches

infinity. Thus

6(x) = ^J

b

(x) (12a)

At the crack tip we find

fi ti P = 6 <°> = u<Vti P = (12b >

according to Eqs. (3), (5), and (9b).

Finally, we find the relation between 6 and a from Eqs. (8) and

(12a)

6(x) =

°b

s

b

fl ukT

da

dx ( 13 )

which means the 5(x) are in direct proportion to ct'(x).

To evaluate the residual stress field generated by the unknown,

a priori distribution of 6(x), we found it useful to employ the concept of

infinitesimal dislocation theory in mathematics.
11

Consider an edge dislocation of Burgers vector, b, located at

x = x' on the grain boundary as shown in figure 2a. The residual misfit

stress, <?m

(x), introduced by this dislocation at an arbitrary distance, x

on the grain boundary (x coordinate line) is [10]

where E is Young's modulus and v Poissons ratio. This equation was derived

based on the physical assumption (3) in section 2.1, and obtained by superposition of the stress field induced by the dislocation at x = x* (figure 2b)

and that created by dislocational image forces acting along the crack lines

(figure 2c). The resultant two-dimensional stress field thus not only satisfies

the equilibrium and compatibility equations throughout the whole body but

also yields the free traction forces at the outer boundaries at infinity as

well as along the crack line.

boundary are due to a continuous distribution of infinitesimal edge dislocations in the grain boundary so that b = -(36/3x)dx being the total Burgers

vector lying between x and x + dx. Then the residual stress induced by the

non-uniform distribution of 6 over the whole grain boundary is given by

integration of Eq. (14). Thus

(14)

Now, imagine that the opening displacements along the grain

00

dx ( 15 )

Here, 6'(x') denotes dS(x')/dx' and the integration is performed in the

sense of Cauchy principal value.
To evaluate the total stress of the grain boundary we first observed

that, in the absence of diffusion, the stress field created by the external

- 1/2 loads has a standard characteristic singularity of r (where r is the

radial distance from the crack tip) as predicted by Irwin-Wi 11 iams fracture

mechanics theory. Then applying the well-known superposition principle in

the theory of linearized elasticity the actual stress at an arbitrary

location x on the grain boundary is

cr(x) =

^2hx 4tt(1

00 r s'Cx 1 ) fx[

-v^)J x'-x IX

dx 1 (16)

where K is the crack tip stress intensity factor. The first term on the

right hand side of Eq. (16) is due to the applied loads and the second term

is that back stress generated by matter previously diffused from the crack

surfaces.

Eqs. (13) and (16) are the field equations derived for the two

unknown variables, 6 and a, based on the linearized theories of elasticity

and diffusion. These simultaneous equations, together with boundary

conditions at the crack tip, Eqs. (10) and (11) are to be solved in the

following section.

2.4 The numerical solution

In this section we present solutions to simultaneous Eqs. (13) and

(16) subjected to initial conditions (10) and (11) derived in sections 2.2

and 2.3. After a careful study, it is found that this system of differential

equations is difficult to solve analytically and a numerical approach has

to be adopted. To begin, we found some rearrangements and simplifications

of the equations are quite useful. First, we note that 6 can be temporarily
13

eliminated, thus reducing two equations to only one. This is done by substi

tuting Eq. (13) in Eq. (16)

00 (17)

where

\ = ED

b

6

bQ/47i(l-v2

)ukT.

This is a one-dimensional singular integro-differential equation of Cauchy

type for cr(x). To simplify, we apply the well known Cauchy inversion formula

oe

[10] and observe that j* [x‘

^

2/(x'-x)] dx' = 0 for x > 0, Eq. (17) reduces

o

to the following simple form

00 lV(x) = J

dx'

0

(18)

where

L =

n

4

ED

b

6

bQ

(l-v2 )ukT

(19)

is a characteristic length depending on crack velocity, temperature, elastic

and diffusional properties. Eq. (18) can be simplified further if we introduce the following non-dimensional variables

x = x/L ; x' = x'/L and a(x) = ^(20)

CT tip

Then Eq. (18) becomes
14

oo

o"(x) =1

a(x-)

a

, Fx7- dx ( 21 )

and the initial conditions, Eqs. (10) and (11) take the following simple

form

a(0) = 1

a' (0) = a

where

or = o'o L/o

q

=^E/(l-v2)D

s 2/V/18/[(D

bab ) 1/2

Y

s 1/3 (ukT)

1/6

]

(22a)

(22b)

(23)

Equation (23) is obtained by substitution of expressions of a

Q

, o'Q, and

L from Eqs. (10), (11), and (19).

Equations (21) and (22) constitute a well-defined initial value

problem readily solved by numerical method. However, at x = x

1 the singularity in the integral creates problems in the course of numerical integration.

It is found that the singularity can be removed by integration by parts twice

for the integral provided that the stress curve over the whole range (0,®)

satisfies the following conditions: (1) a(x) be continuous, (2) a'(x) be

1 /2 bounded, and (3) x a remains stationary as x approaches infinity. It

will be seen that these are reasonable assumptions and the final solutions

do meet these conditions. The final result is

00 «Kx) = f(x) +

I K(x-x‘ )4»(x' )dx‘ (24)

o

where 4j is the second derivative of stress, f is a given function of x, and

K(|) is the kernel of the integral equation defined as follows
*(») = a" (50

f(x) = £nx + ax(£nx-l

)

(25a)

(25b)

K(x-x') = (x-x')Un|x-x|-l] (25c)

Eq. (24) becomes a regular Fredholm- type integral equation, free of singularities although the upper integration limit extends to infinity and the

function f(x) has a logarithmic singularity at x = 0. Detailed procedures

of solving Eq. (24) are given in the Appendix. The end results are presented

here. First of all, it is clear that for a given a there exists a corresponding solution for ip (and hence for a). Figure 3 illustrates two typical

solutions of ip for a - 0 and 20, respectively. The feature of logarithmic

singularity at the crack tip is observed. The stress a can be obtained by

integration of twice subject to the initial conditions in Eqs. (22). Before

we do so, observe that the integral equation (24) dictates a linear dependence

of tp (and hence a) on a, accordingly it is convenient to express a in terms

of the following form

a(x) = ctq

(x) + aAa(x) (26)

where a

Q

denotes solution of a for a = 0 and Act = <j.|-cto represents the

difference between solutions for a = 1 and 0. Figure 4 shows the basic

solutions of ct , ct^, Act, and Act* obtained by integrating figure 3 with the

aid of Eqs. (22) and (26). It can be seen that the maximum value of Act is

at x = 0.9 which implies that the peak stress for any a cannot occur beyond

the region for which x > 0.9L. With the availability of figure 4, the solutions

for 5 and a at any value of a can be obtained by the help of Eq. (26). To

obtain the distributions of 5, we find from Eqs. (13) and (26) that

5 = (ayo) + Act' (26-1)

where 5 = 6/6^.. is the nondimensional grain boundary opening displacement

normalized against 6. . . Using this equation to assemble the curves of 5
from a'Q

- and Act' curves given in figure 4, figure 5 illustrates typical

|] solutions of 5 for a = 1, 2, 5, 10, and ®, respectively. It is shown that the

grain boundary opens up (5 > 0) in the near tip region where x < 1; elsewhere

along the grain boundary, the 5 is negative implying close-up of the boundary

zone.

Similarly from the curves of a and Aa shown in figure 4, the stress

solutions can be assembled for any a according to Eq. (26). Figure 6 gives

some typical curves of a for various values of a. As predicted, the peak

stress approaches asymptotically to the 1 ine of x = 0. 9 as a becomes larger

and larger.

The mode I crack tip stress intensity factor, K, is obtained by taking

the limiting value of iji or a as x approaches infinity. Of course, K is linearly

dependent on a. From the standpoint of numerical evaluation, the value of

K given by solution of ip is more accurate than that given by a in practice.

A fuller discussion is included in the appendix. The final result is

K = 0.75 ct . L 1/2

+ 0.60 a' L3/2

tip tip

After substitution of arj. , L, and ct^.. from Eqs. (11), (19), and (10) we

obtain

(27)

K = Au

1/12

BU

- 1/12 (28a)

where

A = 1.00 (l-Yb/2Ys

) 1/2

[Y

s 2/3

(KT)

1/12

D

s

‘ 1/3Q" 7/36 ][ED

b

6

b

/(l-v 2 )]

1/4 (28b)

B = 1.42 (l-v

b /2ys

) 1/2

[Q

7/36

(D

s ys

) 1/3

(D

b

5

b

)" 1/4 (KT)'

1/12 ][E/(l-v 2

)]

3/4

(28c)

A plot of K vs u for Eqs. (28) exhibits a cut-off point below which no

crack growth is expected. Denoting the values of K and u for this point as

K . and u . , one obtains from Eqs. (28) min min
17

where

and

umin = (B/A)6 = 8 ' 13 W2

^

8 - 1 ' 69 K

G

(29a)

(29b)

K

g =Ve(2VV/0V) (29c)

is the K value predicted by the Griffith theory for propagation of an

atomistically sharp crack in an interface. It asserts that the energy

release rate must be balanced by the net increase of the surface energy

due to an infinitesimal increase in crack length. A more detailed discussion

of the energy release rate is given in section 3. Using the values of

Kq and u - n for normalization Eq. (28a) reduces to the following non-dimensional

form

Figure 7 shows the curve of the K/K« vs u/u . on a log- log scale. It should

u min

be noted here that for u<u

m ^ p

Eq. (30) predicts a decreasing u with an

increasing K. We contend that this configuration is physically inadmissible

since the predicted crack thickness 2uj from Eq. (3) is large enough for u<u .

to render the model's basic assumption of long thin shape invalid. Further,

the time-dependent analysis of Chuang et al. [4], indicates that the crack

growth mode operates only at higher velocities. In the region of slow growth

rates, the cavity is in quasi -equi 1 ibri urn shape and consequently the low

velocity portion of the curve is not valid.

(30)

It has been conventional to express the u-K curves in the form of

u = const*Kn so that straight lines can be drawn on a log-log paper. In
an attempt to plot Eq. (30) in compliance with this form, we found that n

is not a constant; it's values range between 12 and » depending upon the

ratio of u/u . : min

u/u . = 1 10 10

2

10

3

10

4

10

5

10

6

10

7

10

8 *

mi n

n = oo 63 33 23 19 16 15 14 13 12

as can be seen in figure (7). A fuller discussion of this plot as well

as a comparison with experimental data are given in section 4.

3. The energy release rate in crack growth

It is well recognized that during crack propagation a portion

of the energy is trapped inside the body which is not recoverable. This

released energy due to increment in crack length provides a driving "force"

for crack extension. The energy release rate J" , as a basic definition is

therefore the decrease of the total potential energy P in the system due to

infinitesimal increment of crack length da, i.e. , S’ = - dP/da. Further,

since P = Fe-W, & can be expressed in terms of the steady state crack tip

velocity u as follows*

cT = (W - F

e )/u C

where W and Fg are the mechanical work done by the extenral loads and the

total strain energy stored in the interior of the body. The dot stands for

the total derivative with respect to time.( e.y., v = a)

The Griffith crack extension model is well known and assumes

that all energy loss goes to the creation of new crack surfaces, no other

plastic energy dissipation occurs. Thus, if F and F

g denote total and

surface free energy of the body, respectively, then F = F

0 + F

s and

*In transient state crack growth, is not definite since it depends on the

surrounding stress and displacement fields which are changing continuously.

The reader is referred to [12] for more detailed discussion.
19

energy balance requires that W = F = Fg + (2y

s “ Yb

) u so that

^Griffith %

/u = 2y

s " v

b

(31-1)

This represents the absolute minimum energy consumption required

for growing a crack. Any loading with a S’ level below ^Qri*ff1

* th

Wl1 ^ not

cause a crack to propagate and furthermore, at least in theory, healing

(u < 0) would take place.

In contrast to the Griffith model, it is perhaps interesting to

evaluate the energy release rate for our diffusive crack growth model. Since

diffusion is an irreversible process leading to crack growth and since the

grains are assumed to behave elastically, no energy loss occurs in the

interior of the grains during crack growth. The unrecoverable energy

involves only that association with the surface and grain-boundary diffusional

processes in addition to that which has to be spent on creation of the new

crack surfaces. Accordingly, two extra terms associated with diffusion have

to be added to the energy balance equation for the present model*. Thus,

diffusion processes. These two quantities are in direct proportion to

j'jJ‘(-Vp)ds since (-Vp) represents the chemical force exerts on an atom and

J • (~Ym

)

ds is therefore the total energy used to move atoms across ds per

unit time at the location of our interest. In terms of non-equilibrium thermo-

*The kinetic energy is very small as compared to say work done by grain boundary

normal stress on opening the crack tip, hence will be neglected in the subsequent discussions. For example, even at high u (= 1 pm/s) for sialon at 1400 °C,

the kinetic energy is « 10

^

J/mol and the work done by cr^.. on opening 6^

at the crack tip is «1 J per meter of crack front.

giving ^ = (F

s/u) + (Qs /u) + (Qb

/u) (31b)

where Q

s and Qb are the energy dissipation rates for surface and grain-boundary
dynamics these terms represent local entropy production by irreversible

diffusion flow at the free surfaces and interface. Equation (31b) agrees

with the results obtained by Rice and Chuang [12] based on a detailed consideration of energy variations in cavity growth. (Cf. Eq. (16) of Ref. 12.)

It is also in complete accord with the elaborations made by Speight et al. [6].

To evaluate Q

s

, we recall from Eqs. (1) and (4) that both J

g and

P

s decay exponentially along the crack surface from the tip. Thus, by noting

that there are two crack surfaces we have from Eqs. (1), (5), and (6)

Qs = 2 J*J s (-8p

s /3s)ds

^s^K tip^s^tip

= &s

' Vu (32)

Surprisingly, we find that the surface diffusion component of S’ is identical

to the Griffith energy, 2y

s - and is independent of the crack tip velocity,

u.

To proceed with evaluation of we found it useful to introduce

a new parameter, J--a path- independent integral discovered by Rice [13]:

J = Jwdx^ " T • 8jj/3x^ ds (33a)

Here r is an arbitrary integration path surrounding the crack tip starting

from the lower crack surface and proceeding in a counter-clockwise fashion

to the upper one; w is the local strain energy density; and T and jj are

the traction and displacement vectors acting on the path. In our model,
21

it is possible to evaluate J by choosing a special path as indicated in

figure 1 starting from the point immediately below the tip following the

lower boundary surface, r-, to infinity, crossing the boundary there and

returning to the point immediately above the tip by following the upper

boundary surface, r+. In this way, it is evident that only the upper and

lower path integration along g.b., T+ and T- contribute to J since along the

vertical path at x = + », the values of w, T and jj have decayed to zero.

Thus

o

Here the first term of J in Eq. (33a) involving w is dropped due to

interesting to note, in the context of evaluating J, that by reducing both

T+ and T- as shown in figure 1 to a finite length, the situation of the

well-known Dugdale model is obtained. Thus in a sense the Dugdale model

which has a finite cohesive zone ahead of the stationary crack tip can be

regarded as a special case of the present crack growth model in which the

"cohesive forces" cover the whole length of the grain boundary. With the

help of Eq. (33b) we proceed to evaluate Qb

in the following way

QO

0

00

(33b)

cb^ = 0 for both upper and lower paths, and u+

(x) and u (x) denote the

displacements in the direction of X

2

at the upper and lower surfaces of

the grain boundary, respectively, such that 6(x) = U+

(x) - U (x). It is
«b =

j

Qh = I J

b (-3Mb/3x)dx

0

01 I

J^Q(3a/8x)dx

= o^|6(3£r/ax)dx

*^tip a

tip

u " u Jcr( 35/ax)dx

= Ktip *s

U + uJ

= C-2(2vs - Yb

) + J]u (34)

Here, the second equality follows from Eq. (7), the third from Eq. (12a), the

fourth is obtained by integration by parts, the fifth by Eqs. (33b), (12b),

and (10), and the final equality by Eqs. (3) and (6).

The final expression of S’ results in the following simple form

after combining Eqs. (31-1), (32), and (34) into Eq. (31b):

S' = J (35)

This equation shows that J is indeed the correct energy release rate for

our diffusive crack growth model as is true in the theory of elastic

fracture mechanics.

Alternatively, it is straightforward to evaluate g from the

standpoint of mechanics. From the theory of linear elasticity, the rate of

change of strain energy inside the body due to material removal from the
23

crack surfaces and addition to the grain boundary [14-16] is

Fg = 2 J

*6ds + J

w<5ds + W - Jj<5ds

s ‘b 'b

However, the first two integrals involving w are negligible due to low w

since the stress levels developed even in the vicinity of the crack tip are

low (see figure 6). Thus

cT = (W - F e )/u

00

= (1/0) adds h

00

= (I/O) J°[-u(36/3x)] dx

= J

where the steady state condition 6 = -u 36/3x has been used. Hence we see

that the two independent approaches for evaluating S’ , one based on consideration of diffusive energy dissipation; another derived from theory of

elasticity, both lead to the same conclusion that & = J is indeed the correct

energy release rate.

In what follows we attempt to prove that

«r = J = (1-V

2

)K

2

/E (36)

in the present crack growth model as is true in the case of elastic fracture

mechanics.

Using Eq. (13) to eliminate 6(x) and performing integration by

parts, allow Eq. (36) to be transformed into the following form

00

K = (2/ Jn) [a + /<* ) 2 dx] 1/2 (37a)
24

*

where K and a

1 are non-dimensional versions of K and a1 defined as follows

K = K/(c

t1 /L) (37b)

a' = La'/atip

(37c)

Our task is to provide the proof of the equality shown in Eq. (37a)

regardless of the value of a and, thereby, establish the identity:

S’ - (l-v2

)K

2

/E. To do so, we first recall that a* = + ctAct' from Eq. (26)

and K = VItc (0.24 a + 0.30) from Eq. (A15). Accordingly, the following

three equalities have to be simultaneously satisfied if Eq. (37a) holds for

any value of a:

|

(3^)

2

dx = ^(zr x 0.3)

2 = 0.44413 (38a)

|

(Aa‘ ) 2dx = j(ti x 0.24)

2 = 0.28424 (38b)

i.AAua - ^ Aa'dx = j

(jC x 0.3 x 0.24-1) s -0.14469 (38c)

To prove these identities, Eqs. (38a-c), we performed direct numerical

integrations on (a^) , (Aa‘ ) , and (ct^-Act') as given by solutions obtained in

section 2.4 (see figure 4). Table 1 shows the numerical values obtained by

these solutions and comparisons to the values given by Eqs. (38a-c). It is

seen that the difference between the two columns is less than five percent

2 2 error margin. Hence it can be concluded that in the present model (1-v )K /E

is indeed the correct energy release rate which is consumed in creation of
25

new crack surfaces and in completion of surface and grain-boundary diffusion

processes. Alternatively in mechanics term,^ or J has been shown to be

directly related to the work done by the normal stress in opening up the

grain boundary as indicated by expression of J in Eq. (33b).

Equation (36) shows that the total energy release rate is dependent

on the crack velocity. From this equation it is possible to compute g

for the present model. At u = u

mi

- n

, for example, we find J" 2.85JTGriffnh

from Eqs. (29b) and (36), far exceeding 1.0 a^ so ^ ute minimum

level*. This is expected from the energy balance considerations. The

total amount of energy loss can be further split into the following items:

1.0 1S attributed to the creation of new surfaces as in the case of

Griffith cracks; the identical amount of energy is spent on surface diffusion;

and the remaining 0.85 is consumed in grain boundary diffusion.

However, as u increases due to a higher level of applied stress, the former

two items remain fixed and the increased energy loss goes solely to the

last term, since both F

s /u and Qs /u are not function of u whereas Q^/u is

(see Eqs. (31-1), (32), (34)).

4. Discussion

We have presented a steady state solution relating the applied stress

intensity to the crack tip velocity for the grain boundary crack growth model

proposed here, assuming that the crack growth is controlled by stress-assisted

surface and grain-boundary self-diffusion and the adjoining grains behave

elastically. This solution identifies K, the crack tip stress intensity

factor defined in the absence of diffusion, as the major parameter in

controlling the driving force of the creep crack growth although the true

*0ther diffusive crack growth models considered in Ref. 10 all indicate that

Griffith' For examP 1e « at 0 = <Wi'= <rmi n = 4 Griffith for a spring

model and r.. = 6.45 for a double cantilever beam model.

-’min Griffith
26

stress field no longer exhibits stress singularity at the moving crack tip

(Cf . , figure 6). Further, the solution was obtained from a rigorous analysis

of the coupled effects of diffusion and elastic deformabi 1 ity in which diffusion equations and conservation of mass are satisfied at both crack walls and

along the grain boundary including the crack tip, and the associated stress

and displacement fields not only satisfy the equilibrium and compatibility

requirements at any point inside the body but also yield to the appropriate

boundary conditions at the crack plane and the outer boundaries where the

external loads are prescribed.

A similar crack growth model based on the assumptions set out in

section 2.1 was proposed by Vitek [5]. However, as discussed earlier,

two extra assumptions about the crack shape were made in that model. First,

a point called A was introduced somewhere on the crack surface across which

the slopes of the crack shape profile are discontinuous. This leads to

an undefined k (and hence p

g

) at A, making Eq. (2) governing surface

diffusion inapplicable in the neighborhood of A. Physically, surface

"faceting" can occur due to the anisotropic nature of crystallographic

orientations. However, under the assumption of surface isotropy on a

macroscopic scale implicitly invoked in this work and also by Vitek, this

configuration is not in thermodynamic equilibrium. As a consequence, the

surface facet or discontinuous slope, would be "smoothed" out by the

long-term diffusion process. Second, the crack thickness, 2uj, given by

Eq. (3) based on a detailed analysis of the crack shape was not adopted

in that model, instead an arbitrary constant thickness, independent of u,

was assumed. It is difficult to justify using a particular thickness due

to its arbitrary nature although some identifiable physical parameters
27

such as crack opening displacement can be used as suggested by Vitek. His

paper also criticises Eq. (3) because that equation not only predicts

decreasing crack thickness with increasing growth rate, but also because

it predicts physically unattainable velocities for a-iron at 550 °C. In

response to this criticism, we emphasize that Eq. (3) does indeed predict

a constant crack thickness at steady state in which u is stationary. It

is when the crack growth rate changes from one steady state to another,

perhaps due to a change in load or temperature, does Eq. (3) predict a

decreasing 2u> with increasing u. A discussion of the transient behavior

is outside the scope of the present paper although a brief remark is in

order here. According to a recent study [17] of cavity morphology,

resulting from crack growth in transient state, it was shown that when the

applied stress is high as compared to the sintering stress, a "nose" region

develops ahead of the cavity apex and the direction of the surface flux

near the central portion is actually reversed. This causes the crack

thickness to decrease in close accord with Eq. (3). A scanning electron

micrograph of silver showing nose regions on grain boundary cavities was

also provided as evidence to support that analysis. These interesting

results support the validity of Eq. (3). The high crack velocities

calculated for a-Fe using Eq. (3) may indicate that some other mechanisms,

not considered in our model, are controlling crack growth.

Because of the two extra conditions (a discontinues slope and a constant

crack thickness), Vitek's model suffers the following consequences: (1) the

crack tip conditions regarding p

g and Jg are not properly formulated, as

we have done in section 2.2, with the result that the normal stress at the

crack tip remains constant independent of u (and hence independent of the
28

applied stress); and (2) mass conservation at the crack tip, Eq. (9b),

dictating a^.. could not be used. Instead the physical requirement that

ct(x) far away from the tip approach asymptotically to the stress field

induced by the applied stress in the absence of diffusion was imposed (see

Eq. (14) of Ref. 5). It is noted that in our model this physical behavior

was obtained directly from the solutions shown in figure 6. It remains to

be proved that the solutions based on Vitek's boundary conditions will automatically make Eq. (9b) valid.

Speight et al. [6] have proposed a crack growth model similar to

what has been presented here. In contrast to our model (see fig. 5) as well

as Vitek's, the grain boundary displacement distribution, 6(x), was not pursued

in their analysis. Instead, a parabolic function of S(x) was imposed (see

Eq. (1) of Ref 6) such that 5 and 6' vanish at a short distance (x = b)

from the crack tip. A consideration of the energy release rate associated

with crack growth was also incorporated in their formulation. We have shown

2 2

in section 3 that J or (1-v )K / E is the correct energy release rate of

which, at u = u . , thirty-five percent goes to creation of new crack

surfaces, another thirty-five percent is spent in the process of surface

diffusion, and the remaining thirty percent is consumed by grain-boundary

diffusion. As u increases, the former two terms remain fixed and the

excess energy is dissipated in the grain boundary as a form of mechanical

work done by the boundary normal stresses. In contrast to the present

analysis, Speight et al. contend that the total plastic work of 2u>u

by the applied stress a is the total energy released during the growth

period, i.e., & - 2out. This contention implies that the adjoining grains

behave rigidly so that the opening displacement, 5(x) = 2w, are uniformly

distributed along the grain boundary. This is in direct contradiction
29

to their Eq. (1). Further, it has been shown in Ref. 12 that this term

2a

a)u), appears to be minor in the total energy release expression. As a

result of these assertions, the peak stress is predicted to have a value

5 linearly dependent on cr^ at a fixed point x = b, a(b) = ^

ar^. 1° contrast

our solutions show how the values of the peak stress varies depending on

the values of a and L which in turn are functions of u (see figure 6).

Finally, their model is claimed to be applicable to a relatively short

crack but this is not in agreement with the conclusions obtained by

Chuang et al. [4] based on a time-dependent study. There it was shown

that, when the cavity size is small, presumably in the early stage, the

cavity maintains a quasi-equilibrium shape and as the creep test continues,

the cavity gradually becomes long and crack- like as its growth rate increases.

The present, long crack model, which includes the prediction of fast growth,

fits well in this time-dependent analysis.

Attempts have been made to evaluate the theoretical predictions for

K . , u . , and K versus u curve in the case of pure Cu, Ni , aFe, yFe, Zn, min min

and Ag [10]. It was found that of the six metals, only Ni and Cu yield physically

tenable values for u . . It is fair to say that the materials property data

min

employed for this evaluation involved some degree of uncertainty. Nevertheless,

it is likely that for pure metals some mechanisms not considered in our paper

(particularly some dislocation processes) are in operation during creep.

As pointed out before, the theory was developed for strong brittle materials.

Thus it is worthwhile to compare our theory with the data on ceramics.

Recently, Lewis and Karunaratne [7] have performed creep tests on Si-Al-O-N

ceramics at 1400 °C. In their tests three different techniques were used

to measure crack growth rate depending on the range of crack velocities.
30

In this way they were able to produce reliable, continuous curves on a

K vs V plot. It was found in the ceramic specimen C that the sub-critical

crack growth occurs via the advance of a single crack along the grain

boundaries. The growth behavior can be well described by [7]

K = 3.5 x 10

6 u 1/12

+ 7.4 x 10

4

u'

1/12

(39)

1/2 where K is in units of Pa*m ; and u in m/s. This experimental result

is identical to what the model predicts in equation (28a) where

A = 3.5 MN-m‘

19/12

S

1/12 and B = 74 KN*m" 17/12

*S' 1/12

. The coefficients of

Eq. (39) were computed from the data listed in table 2.

The creep test data are also plotted against our theoretical curve

in figure 7. It is seen that the majority of the data points (except the

lowest three data points) fall along the curve, close enough to lead us to

conclude that the theory successfully predicts not only the functional dependence of crack velocity on applied stress intensity factor, but also the

absolute magnitude of crack tip velocity. The deviation in functional dependence in the low velocity range may indicate the flaw shapes no longer preserve

the crack-like shapes.

All relevant parameters for the creep tests are also caculated in

table 3. Examination across the board shows that the values of most

parameters do fall in physically reasonable ranges except 2uj which seems

to be about one order too low. At those data points for which

u/u . sslO9 ~ 10

4

, the crack thickness 2w reduces to the same order as

mi n the atomic spacing (0.2 nm). It is possible that both surface and grain

boundary diffusivities are one order lower. If such is the case, then

all parameters do fall in the appropriate values. Owing to the scarcity
31

of the creep crack growth data it is clear that more measurements are needed,

but qualitatively our theory seems to be well supported by creep tests on

sialons.

5. Summary and Concluding Remarks

From what we have presented on the present crack growth model , the

following concluding remarks may be given.

5.1 If steady state interfacial crack growth is controlled by coupled

surface and grain-boundary diffusion and if the adjoining grains behave

elastically then the crack tip stress intensity factor K in the absence of

diffusion is identified as the correct parameter driving creep crack growth.

The functional relationship between K and u is found to be K/Kg = 0.865 *

[(u/um

. + (u/u. ) ly^]. This equation predicts that a threshold K min min

equal to 1.69 exists, where Kg is the critical K for Griffith cracks, below

which no crack growth can take place. In terms of u = const. K

n on a log- log

scale, this equation predicts the slop n varying between 12 and infinity

depending on the value of u. The features of the sub-critical crack growth

behavior so predicted are also consistent with that obtained from a timedependent analysis [4].

5.2 The path- independent integral J or the plane strain fracture

2 2 mechanics energy release rate (1-v )K /E is shown to be the correct energy

release rate associated with creep crack growth. This energy is dissipated

in the creation of new crack surfaces, as well as in the completion of

surface and grain boundary diffusional processes. However, it was found

that the energy loss is in the form of work done by grain boundary normal

stresses due to deposition of atoms along the grain interface rather than

the release of strain energy in the near-tip region as is the case in

conventional fracture mechanics.
32

5.3 Excellent agreement is obtained when the predictions are compared

with a set of test data on creep crack growth for Si-Al-O-N at 1400 °C. It

is concluded, at least qualitatively, that under appropriate conditions,

sub-critical creep crack growth behavior of ceramics closely follows what

is predicted by the present theory.

Acknowl edgments

A portion of this study (sections 2 and 6) was based on a part of the

author's Ph.D. thesis and was supported by the U. S. Department of Energy

(formerly U. S. Atomic Energy Commission) under contract EY-76-S-02-3084-A003

with Brown University. The remaining portion of the work was supported by

the Department under interagency agreement DE-AI05-800R20679 between DOE and

the National Bureau of Standards.

The author is grateful to his thesis adviser, Professor James R. Rice

for his effective direction, and to his colleagues, Dr. Richard J. Fields and

Dr. Edwin R. Fuller, Jr., for providing the computer plot of figure 7 and

for proofreading the manuscript.
33

Appendix

Numerical Solution Scheme for integral Equation (24)

In this appendix we present a solution scheme to the following ID

Fredholm- type integral equation for the unknown function tjj(x) as appeared

in the text, equation (24):

4>(x) = f (x)

+J

K(x,t) ijj(t) dt (Al)

o

wherein K(x,t) is the kernel and f(x) a given function:

K(x,t) = (x-t)£n[ x-t

| - (x-t) (A2)

f(x) = In x + ax (£n x-1) (A3)

The unxonwn function <|j(x) has been defined in the text as the second

derivative of normal stress with respect to x along the grain boundary

(0 S x < ®) and is presumed to satisfy the following conditions:

(1) «Kx) is integrable;

( 2 ) £im x

2

tjj(x) = 0

X-xo

These two mathematical assumptions are based on the physical arguments

that a unique stress solution does exist and its asymptotic behavior as

x^<», as depicted by linear fracture mechanics theory makes conditions (1)

and (2) valid.

Since closed form solutions exist only for a limited class of integral

equations, we shall pursue the solution of equation (Al) numerically with

the aid of digital computers. To solve (Al) numerically, we employ the

standard method of replacing the integral equation with a system of coupled

oo

linear algebraic equations, assuming the integral /

Kcpdt can be represented

o
34

by a finite sum. In trying to do so, we encounter two difficulties. First

as pointed out in the text, ip(x) has a logarithmic singularity at x = 0

which makes the direct numerical integration from x = 0 impossible; secondly,

the upper limit extends to infinity which also apparently cannot be reached.

To avoid these two areas of difficulty, it is therefore necessary to introduce two cut-off points, say x^o and X

n

^<» so that integration over

[x

1 ,x ] can give an approximate value of the integral. This means that if

we write

J

K(x,t)t^(t)dt

o /

K(x, t)t|i(t)dt Kipdt +

o

(A4)

then the second and third terms on the right hand side are small as compared

to the first teem. However, in order to increase the accuracy of the results,

it is desirable to evaluate the second and third terms as well*. The asymptoti

behaviors of ip allow us to write

xl

j

Ktpdt ~ j

K(x,t)£nt dt, x ^ x-j

and

00 00

*The solution of the system without addition of these two terms has also

been pursued which showed satisfactory results. However, by including these

two terms the results showed faster convergence and therefore improve the

accuracy of the solutions.
35

Carrying out the integration analytically, one obtains

i

L(x,x.j) =1 K(x,t)£nt dt

= (x-x

1 )[(x+x

1 £nx1 )£n(x-x.| j-Cx^nx-j-x^ )]

+ x

2

(£ nx + |) - (x~x^ ) (3x+x-j )/2

X

1 x x2-, 2, + ^ [2£nx-2(—)-(—) ] x

] £nx

]

« x, n x, n x

2 00 (-1)

00 (J:)

2 xl, , ^x ; 2 — u ;

2 - - -

' X / - - 2

n=3 n=3

for x ^ x. (A5.1)

R(x,x

n

)

oo J

K(x,t)t" 5/2dt

= |(X 1 /2

“X

n 1/2 )£n(xn ~x) - fx

1/2£n(^jT + yx

n

)

- ~x

3/2

[(x -x)£n(x -x) - (x -x)];

3 n LV

n

' v nJ v n for x ^ x (A5.2)

We observe that both L(x,xp and R(x,x

n

) are bounded even at x = x.j and x =

x . The interval [x, ,x 1 is divided into (n-1) subintervals with n discrete

n I n points x^ , x^, .... x

n excluding the point x = 0. Denoting i|i . = 4*(x.)> our

objective is then to find the n unknown values of 4k , i = 1 to n, by

assuming that any two adjacent points (x. and X-j + -|) are sufficiently close

so that 4*(x) can be assumed to vary linearly between 4*.j and 4k +

.j. Thus
36

, „ , *i+r*i , . , x i+l^i~ x i^i+K

<j>(x) = (— ~ ) x + ( 3 73 ) vrx

i x i+l"

x

i

for ^ x S x

i+1

To proceed, we observe that the first term on the right hand side of (A4)

can be decomposed into the following sum:

n-1 .1+1

J

Kijidt = E f

x, i=l

K(x, t)4)(t)dt

'1 Denoting

x

i+l

F(x • x

i> x i+P =j tK(x,t)dt

and

xi

x

i+l

G(x > x

,* x

i + i ) 5

j

K(x, t)dt

x.

i

and substituting 4> of (A6) into (A7), the integral equation (Al) finally

reduced to the following simultaneous equations:

([K] - [I]) {*} = -{f}.

where [ K] is a nxn matrix, [I] is a unity matrix and {f} is a vector with

typical element f. = f(x.). The typical element of [K] can be written as

follows for j = 1 to n:

(A7)

(A8)

(A9)

(A10)
37

Kjl x

2

- x

1

x

2

G(x . .x, .x„)-F(x . ,x-j ,x 2

)

(i = 2 to n-1)

K

n 5/2

where L, R, F, G are functions defined in (A5.1), (A5.2), (A8), and (A9)

respectively.

A computer FORTRAN program was developed to solve {<Jj} in equation (A10)

in which the following parameters were chosen: n = 200, x^ = 0.01,

x =58.2 and

n

Integration range: [0.01,1.2] [1.2, 2. 2] [2. 2,8. 2] [8.2,18.2] [18.2,58.2]

Interval Ax : 0.01 0.05 0.2 1.0 2.0

Thus a non-uniform distribution of x. results, having a more dense distribution near the origin where the behavior of is of major interest.

To test the validity of the FORTRAN program, an integral equation of

the form (Al), with the same kernel as given in (A3) was chosen where f(x)

is given by

(All)
and its closed form solution is known as

<Kx) = (x + a)' 5/2 (A12

Equation (All) was input to the program with a = 1 and the solution given by

the output of the program was then compared with the analytical curve of

(A12) which shows a negligible difference between the two solutions. The

reliability of this computer program was hence confirmed.

Using this program with the input of f given by (A2), the solutions

of equation (Al) for ijj were obtained first for or = 0 and 20, respectively.

5/2 The results show that values of x tj>(x) do attain an approximately constant

value of a distance far away from the origin, say for x ^ 8.0, and as

3 K remarked above, this should be set equal to ^

The correctness of

these solutions can be further verified from the initial conditions on

stress at x = 0 which take the following form:

Uk

J

4<(x)dx = -a (Al 3)

I

xt|j(x)dx =1 (A14)

The numerical integrations of the left-hand sides of equations (A13) and

( A 1 4 ) were performed, using Simpson's rule which yield the following results:
39

00

J

x(»jOdx = -0.17651* for a = 0

= -21.71512* for a = 20

/

xt{j(x)dx = 0.99551* for a = 0

0.97582* for a = 20

From these numerical results, it can be said that the convergence of the

solution is fairly rapid and the accuracy of the solution is within five

percent error range.

The structure of the integral equation (Al) indicates that the nondimensional ized stress intensity factor K should be linearly dependent on

a. Indeed, this is verified by the solutions of tjj for a = 1.0, 1.5, 2.0,

2.5. A plot of K/ -{Zii vs a shows that

AK/ {2n = 0.24 a + 0.30

as given by equation (27) in the text.

The stress distribution ct(x) can be obtained by integration of ij> twice

This was done on the two solutions of i|> corresponding to a = 0 and a = 20.

1/2/s The results show that at large distances from the origin, x a(x) does

not attain a stationary value, instead it either rises or falls linearly.

This phenomenon was not surprising because any disturbance caused by errors

unavoidably induced in ct(x) by numerical integration as a + bx contributes

nothing to i|j. Therefore, a correction should be made over the whole range

(Al 5)
40

of x for a according to the guideline that /xa must approach K/ ^2n for

x ^ x , for example. Figure 6 shows the curves of 6(x) vs x for

or = 0, 1, 5, 10, and 20, respectively, after the corrections have been

made.

^Values include analytical integration over [x

n »].
41

References

1. D. Hull and D. E. Rimmer, "The Growth of Grain-boundary Voids Under

Stress," Philos, Mag., 4 [42], 673-687 (1959).

2. R. Raj and M. F. Ashby, "Intergranular Fracture at Elevated Temperature,"

Acta Metal 1

. , 23 [6], 653-666 (1975).

3. T.-J. Chuang and J. R. Rice, "The Shape of Intergranular Creep Cracks

Growing by Surface Diffusion," ibid., 21 [12], 1625-1628 (1973).

4. T.-J. Chuang, K. I. Kagawa, J. R. Rice, and L. B. Sills, "Overview

No. 2: Non-equilibrium Models for Diffusive Cavitation of Grain

Interfaces," ibid., 27 [2], 265-284 (1979).

5. V. Vitek, "A Theory of Diffusion Controlled Intergranular Creep Crack

Growth," ibid., 26 [9], 1345-1356 (1978).

6. M. V. Speight, W. B. Beere, and G. Roberts, "Growth of Intergranular

Cracks by Diffusion," Mater. Sci. Eng., 36 [2], 155-163 (1978).

7. M. H. Lewis and B. S. B. Karunaratne, "Determination of High Temperature

K-j-V Data for Si-Al-O-N Ceramics," submitted to ASTM STP publication.

8. B. R. Lawn, B. J. Hockey, and S. M. Wiederhorn, "Atomically Sharp Cracks

in Brittle Solids: An Electron Microscopy Study," J. Mater. Sci., 15

[ ], 1207-1223 (1980).

9. A. Needleman and J. R. Rice, "Overview No. 9: Plastic Creep Flow

Effects in the Diffusive Cavitation of Grain Boundaries," Acta Metal 1.,

28 [10], 1315-1332 (1980).

10. T.-J. Chuang, "Models of Intergranular Creep Crack Growth by Coupled

Crack Surface and Grain Boundary Diffusion," Univ. Microfilms Int.

(Ann Arbor, Michigan) Order No. 76-15618, 95 pp. ; Diss. Abstr. Int. B,

37 [1] 402 (1976).

11. R. N. Stevens and R. Dutton, "The Propagation of Griffith Cracks at

High Temperatures by Mass Transport Processes," Mater. Sci. Eng., 8

[4] 220-234 (1971).

12. J. R. Rice and T.-J. Chuang, "Energy Variations in Diffusive Cavity

Growth," J. Am. Ceram. Soc. , 64 [1], 46-53 (1981).

13. J. R. Rice, "A Path Independent Integral and the Approximate Analysis

of Strain Concentration by Notches and Cracks," J. Appl. Mech. , 35 [6]

379-386 (1968).

14. J. R. Rice and D. C. Drucker, "Energy Changes in Stressed Bodies Due

to Void and Crack Growth," Int. J. Fract. Mech., 3 [1] 19-27 (1967).
42

15. J. R. Rice; pp. 191-311 in Fracture, Vol. II. Edited by H. Liebowitz,

Academic Press, New York, 1968.

16. J. D. Eshelby; pp. 77-115 in Inelastic Behavior of Solids, Edited by

M. F. Kanninen, W. F. Adler, A. R. Rosenfield, and R. I. Jaffee,

McGraw-Hill, New York, 1970.

17. G. M. Pharr and W. D. Nix, "A Numerical Study of Cavity Growth

Controlled by Surface Diffusion," Acta Metall., 27 [10], 1615-1631

(1979).
43

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Captions

1 Geometry of the grain-boundary crack growth model, (x-j^) is a cartesian coordinate system fixed in space, whereas (x^) is

moving with the crack tip at the velocity of u such that

x = x^ - ut.

2. Schematic sketches of configurations where the stress field of

figure 2a containing a crack due to an edge dislocation can be

computed from superposition of stress field produced by a dislocation in an external stress free body, figure 2b and that

induced by the dislocational image forces applied at the crack

site, figure 2c.

3. Numerical solutions of ^ vs x for typical values of cy.

4. Basic solutions of a and Aa and their first derivatives with

o x from which any solution of a or S with a given a is obtainable

from b = b + aAa, or 5 = (a'/a) + Aa'

.

o o

5. Typical solutions of 6 vs x for different values of a. Note that

the curve for a = ® crosses the 5 = 0 1 ine at x = 0. 9.

. Typical solutions of a vs x for several values of a. Stresses

are seen to peak within the region of 0 < x < 0.9. For sake of

comparison, stress distributions with and without diffusion are plotted for a = 20.

Figure 7. Theoretical plot of normalized K vs u curve with correlated

creep crack growth data for Si-Al-O-N at 1400 °C [7].
44

Table 1

Comparison of Numerical Values of Integrals Given by

Figure 4 and by Equations (38 a-c)

Integral

Numerical Integration

Based on Figure 4

1 Value Given by

Equations (38)

Percent

Error

00

J

(a^)

2

dx 0.43382 0.44413 2.32

0

00

1 (Aa*

)

2

dx 0.28571 0.28424 0.52

J

0

00

S a

1 • Act

1 dx

J

0

-0.13787 -0. 14469 4.71

J

0

^he method of numerical integration follows Sympson's rule, namely,

Ir Ay

9 y(x)dx « -3 (yQ + 4y

] + 2y£ + 4y3 + . . . + yn

) where y. = y(x-).

i = 0 , 1 ,. . . n.
45

Table 2

Summary of Material Property Data for Sialon at 1400 °C

. Used in Obtaining Equation (39)

3 Atomic volume, Q(m ) 3 Grain boundary diffusivity, /s)

Poisson's ratio, v

-2

Young's modulus, E(GNm ) 2 -I Assumed surface diffusivity, D

g (m s )

-2

Assumed surface energy, Ys (Jm ) -2

Assumed grain boundary energy, )

1.06 x 10

6 x 10

0.27

300

2.5 x 10

0.75

0.375

-29

-25

-18
46

Table 3

Calculated Values of Relevant Parameters for Creep

Crack Growth of Si-Al-O-N at 1400 °C

Parameter Unit Eq. No. Value

k

g

MPa* /in 31 0.6

Kmin

MPa* /in 30 1.0

u . min

-1 nm s 29 0.09

^umin nm 19 990.0

(or) . v y umin — 23 1.26

(ct. . ) tip umin MPa 10 717.0

(2uj) • umin nm 3 3.14

+

!!
)

FOR a

=20

0.00 1.00 2.00 3.00 4.00 5.00

X (= X/L)

wo 3

=

o-'')for

a

=
NORMALIZED

BOUNDARY

COORDINATE

X

(=

X/L)
S=

(<Xo/cc) +

Ao- 1

NORMALIZED

BOUNDARY

COORDINATE, X(=

X/L)
0001

SS3U1S aaznvwaoN
NORMALIZED

CRACK

VELOCITY

l)/V

mi

1/2

STRESS INTENSITY FACTOR, K (MPa -m )

0.6 I 2 3 4

NORMALIZED STRESS INTENSITY, K/KQ

K 9 7

CRACK

VELOCITY

V

(

ms 1)

- I(«ev. ». 78 >

: °7'' lk“<& •'/.‘.Vi-'

SOEPT. of comm. LOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO. NBS 81-2255

r : AND SUBTITLE ' effusive Crack Growth Model for Creep Fracture —MP 5. Publication Date . ^ March 1981-^

J iOR(S)

z-jer Chuang -

: V

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UPPLEMENTARY NOTES

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

•

. •-

. ^WmwCf-’ * *. 5 ‘ .

• '"*•

ABSTRACT (A 200-*rord or leas tactual summary of moat significant information. If documant includes a significant bibliography or

[ literature survey, mention it here.) A grain boundary creep crack growth model is presented here based on the

sumptions that the crack propagates along the grain boundary by a coupled process

surface and grain—boundary self-diffusion; the adjoining grains behave', Plastically

;

1 steady state conditions prevail. Under the action of the applied stress, atoms

the crack surfaces are driven by surface diffusion toward the crack tip

^ from where

ey are deposited non—uniformly by grain—boundary diffusion along the grain interface

that the grain boundary opens up in a wedge shape ahead of the advancing tip which

i turn produces a misfit stress field. The total grain boundary stresses and opening

Lsplacements are solved from a system of integro-differential equations to give the

allowing relating K to v: (k/k min.)=%[ (v/v min. : (v/v min. ) ‘r.1/12

j.3 where K

s stress intensity factor and \> the crack tip velocity. In terms of v « kn , the

quation predicts n varying firm 12 to infinity. A comparison with data on Si-Al-O-N

it 1400 °C shows agreement betveen the theory and experiment.

_ t A consideration of the energy balance for the present model is also presented which

indicates J=(l-v2 )K2/E is the correct energy release rate associated with the present

crack growth model as is true in the theory of elastic fracture mechanics . However

,

the energy released in diffusion is in the form of work done by the normal stress along

the grain boundary, not from the elastic strain energy loss of the adjoining grains.

#4

17. KEY WORDS (six to twelve entriee; alphabetical order; capitalize only the first letter of the firat key word unless a proper name;

s separated by semicolons) . Hrack growth model; (Creep cavitation; diffusive crack growth; ,33nergy release rate;

ciigh temperature fracture; J-integral ; /Cialon; ^Singular integral equation

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