A model for creep cracking by diffusion-controlled void growth

Materials Science and Engineering, 49 (1981) 31 - 39

31

A Model for Creep Cracking by Diffusion-controlled Void Growth D. S. WILKINSON

Department of Metallurgy and Materials Science, McMaster University, Hamilton, Ontario L8S 4M1 (Canada) (Received November 24, 1980)

SUMMARY

Grain boundary cracks at elevated temperatures propagate by the nucleation and growth o f cavities ahead o f the crack tip. A model is presented for crack propagation controlled by the diffusional growth o f an interdependent array o f cavities. In the model a variety o f possible cavity shapes and the different mechanisms controlling the nucleation o f cavities are treated. The dependence o f the crack growth rate on such parameters as the cavity spacing and the number o f cavities which grow simultaneously is predicted. In addition the crack growth rate varies as a power o f the stress intensity factor K, the power varying between 1 and 4 for different situations.

1. INTRODUCTION

The process of creep crack growth has received considerable theoretical attention over the past few years, essentially for two reasons. The first is the importance o f creep cracking to the utilization o f high strength alloys at elevated temperatures. Many of these materials show a susceptibility to creep cracking along grain boundaries in and near welded joints, which post-weld heat treatments are not always able to control [ 1, 2 ]. A second reason is the importance o f creep crack growth to the development of models for creep failure in materials w i t h o u t prior stress concentrators. Models [ 3 - 6 ] in which the growth of a single cavity (or a uniform array of cavities) is assumed to be the rate
throughout the material. Some coalescence of cavities into microcracks occurs well before final failure. It may be the propagation of these microcracks across the material that controls the fracture process. Many models have been developed to predict the rate at which cracks grow at high temperatures. In most of these models [ 7 - 19 ] the region ahead of the crack is considered as a homogeneous solid (i.e. without cavities). Damage is assumed to accumulate in the region preceding the crack, and crack growth occurs continuously at the crack tip. Both plasticity [7 - 13] and diffusion~ontrolled growth [14 - 18] have been considered in this way, b y setting up appropriate b o u n d a r y value problems. Most stable cracks, however, do not grow continuously. Rather they propagate b y the nucleation and growth of voids which eventually link up with the main crack. This has several important consequences. First, the crack growth is discontinuous, occurring in steps as each cavity links up with the crack. Secondly, because the cavities are separated from the crack, t h e y see a stress field which is different from that at the crack tip. Finally, and most important, the growth rate of the crack must depend on microstructural variables such as the spacing o f cavities and their ease of nucleation. Continuous growth models cannot account for these effects. Only a few models have been developed in which the growth of cavities ahead of a crack is considered. The first of these, due to Nix et al. [ 2 0 ] , treats crack growth in a power law creeping solid (i.e. plasticity-controlled cavitation). The cavities are assumed to grow in the elastic stress field of the main crack, unaltered b y plastic relaxation or the presence of the cavities. Nix et al. calculated the void growth rate for a spherical shell surrounding © Elsevier Sequoia/Printed in The Netherlands

32 the cavity in a manner analogous to the sintering of cavities under pressure [21 ]. This leads to a crack growth velocity of the form K~ cn/2 -- 1

where K is the stress intensity factor, c the cavity spacing and n the stress sensitivity exponent for power law creep. Models for diffusion~controUed cavity growth ahead of a crack have also been developed [22 - 2 5 ] . For this case it is clear that t w o different types of steady state are possible depending on the relative rate of cavity growth and the spacing o f the cavities. They are distinguished b y the way in which atoms are deposited ahead of the cavity, as shown in Fig. 1 and outlined below. A cavity grows b y emitting atoms from the cavitygrain b o u n d a r y junction. These atoms diffuse d o w n the boundary and plate o u t onto either side of the boundary. Thus a wedge of material is deposited ahead of the crack, compressing the crystal. This relaxes the elastic stress concentration created b y the presence o f the cavity. If the cavity grows quickly or if the cavities are far apart, then the length ~d of this wedge of deposited material will be relatively short with respect to the cavity spacing c. Under these conditions the cavities grow independently of one another. A stress field of the t y p e shown in Fig. l(a) develops, in which the stress is a maximum at a b o u t the P(x)

I-

C

4

(a)

~

_--_ __--_- _

-

-

-

~

_

~

(b) Fig. 1. Two different types of steady state cavity growth are possible and the stress field is altered as a result: (a) if cavities grow rapidly, they behave like independent microcTacks; (b) if cavities grow slowly, matter is deposited uniformly over the grain boundary between them.

in which the stress is a maximum at a b o u t the distance ~d from the cavity. A steady state can be achieved in which the shape of the deposited wedge is constant and move~with the same velocity as the cavity growth velocity [ 1 7 ] . It is found [23, 26] that, for crack-like cavities at least, ~d is given approximately b y

v/

where ~bDb~G

kT

is a temperature
33

equation [15] for cavity growth and derived a crack growth rate equation which is similar to it b u t which contains a stress intensification factor and a grain size dependence o f D 1/2. In the model presented here the elastic stress distribution ahead of the crack is used to c o m p u t e cavity growth rates. As will be seen, the cavities nearest the crack tip grow much more rapidly than those further away. Thus the Miller and Pilkington t h e o r y , while giving a qualitative picture of the effect of grain size of crack growth, cannot usefully predict crack growth rates. A second model, due to Raj and Baik [ 2 4 ] , is similar in several respects to the one presented here. In b o t h models, diffusional stress relaxation from cavities ahead of a crack is treated. However, Raj and Baik used a numerical m e t h o d to solve the problem that t h e y posed rather than the analytical approach that is used here. Moreover, in the present model, cavities growing with different shapes controlled b y surface or grain b o u n d a r y diffusion are treated. Although Raj and Baik developed a more exact picture of the stress ahead of the crack, the assumptions used here are felt to be justified in order to retain an analytical solution. This allows a wider range o f situations to be analysed and greater physical insight to be gained from the solutions.

2. THE MODEL

We wish to consider the situation outlined in Fig. 2 in which a plane strain crack of halflength a, in an infinite solid, is placed under a tensile load e. The crack propagates b y the growth of cavities ahead of it. Let us assume that N cavities (where N is to be determined) spaced a distance c apart grow simultaneously. N o w let us consider a state in which the

number N o f growing cavities remains constant as the crack propagates and the size of each cavity is determined solely b y its distance from the main crack. (A different approach in which N cavities are assumed to nucleate simultaneously is considered later.) Thus, when the cavity nearest the crack grows to a size necessary to link up with it, a new cavity is nucleated. The crack therefore grows in jumps of distance c, each jump requiring a time interval t. During this time, each cavity grows from a length 2ln to the length 2ln-1 of its neighbour nearer the crack (Fig. 3). This results in a set

v.-,...-~--Vn

Fig. 3. Each cavity grows towards its neighbour at a velocity v n where n is the number of cavities separating it from the crack tip.

o f N simultaneous equations for the time interval t: z~-1 dl

t = (" l~

©

C)

o-

o-

Fig. 2. A plane strain crack loaded in tension propagates by the growth of cavities ahead of it.

n = 1, .... N

(1)

c -

t

(2)

The cavity half-length l, of course, is bounded. The cavity nearest the crack grows from half-length11 to that for linkage l~ in the time interval t. Thus (3)

The N t h cavity begins with a length equal to its nucleation length, i.e. l N = lc

1

-v~

where v. is the velocity o f the cavity into the nth ligament (see Fig. 3). We assume that the velocities for both cavities growing into a given ligament are the same. Thus each cavity grows faster towards the crack than away from it, as expected. Except for very small values o f N, the effective movement of the cavity centres towards the crack has a negligible effect. The average crack growth rate is given simply b y

l0 = l~

C)

Vn--0--.. v..,

(4)

With these t w o conditions the set of equations (1) becomes well defined if either t or N is unknown. We can thus determine a relation-

34 ship between the steady state growth rate and the number of cavities which grow ahead of the crack.

1.5

1

I

|

I

1,0

2.1. The growth of a cavity in the stress field of a crack We need to k n o w the growth velocity v, of each cavity as a function of its size and distance from the crack tip, given that a steady state of the Hull and Rimmer t y p e {i.e. uniform deposition of matter} has been achieved for each ligament. (It should be noted that this does not mean that the deposition rate is the same for every ligament, which would imply that each cavity grows at the same rate.) This t y p e of boundary value problem has been solved often for different situations (e.g. see ref. 6). However, it is necessary to consider the boundary conditions, of which there are three for the present case. The first t w o conditions arise from the imposition of a known stress at the junction between the grain boundary and void surface owing to the curvature of the surface. The third condition is that o f mechanical equilibrium. For crack growth this means that the integrated load ahead of the crack must equal that due to a pure elastic response. Let us c o n s i d e r a single void ahead of a crack, to which a tensile stress o is applied. The elastic stress field due to this situation has recently been calculated [26, 2 8 ] . From it the total load on the ligament between the crack and the cavity can be found. When the cavity is very small we expect a load W G(2aS) 1/2 on the ligament where s is the crack-cavity separation. As the cavity grows, however, some of this load must be transferred ahead of the cavity, this transfer being complete once linkage occurs. As Fig. 4 illustrates, this transfer of load is delayed until I is very large. Thus we can assume that W = a(2as) 1/2 to an error o f less than 15% for l < 0.8s. This gives the condition for mechanical equilibrium. If more than one cavity is present, those further from the crack are smaller and thus the error is less. In the following we assume that the mechanical equilibrium condition can be applied to each ligament in turn, i.e. it is due to the elastic load on each ligament in the absence o f a cavity: =

W n = a ( 2 a c ) 112 (rt 1/2

--

(rt

--

1 ) 112 )

(5)

W crq-d'~ 0.5

o

0

i

I

I

I

0.2

0.4

0,6

0.8

1.0

J~/s

Fig. 4. A plot of the total load W o n the ligament separating the crack tip and a single cavity whose centre is located a distance s from the crack tip.

This is not strictly correct in that stresses can be redistributed over distances larger than c. However, it does provide an equilibrium stress field and therefore ensures an upper b o u n d solution for crack growth. The boundary value problem can n o w be solved subject to several assumptions. First, cavitation is assumed to be b o u n d a r y diffusion controlled and not lattice diffusion controlled. The relative contribution of the t w o mechanisms is of order 5Db/cDv and it is easily shown that at the temperatures of interest this ratio is much greater than unity. (Chuang et al. [29] have recently produced a derivation in terms of characteristic relaxation times.) It is also assumed that the grain b o u n d a r y acts as a perfect source and sink for vacancies. The solution of the boundary value problem is n o w straightforward, the result being

6)~ I W, v,,- (e ---'2l.)2 d.G

%(c-- 21,) I d---nGr,

(6}

where v, is the growth velocity into the nth ligament from the crack t i p , r , is the radius of curvature of cavity n while d, is determined b y the geometry of the cavity such that a cavity has an area A , = d n l n . If 0 is the equilibrium dihedral angle at the cavity-grain boundary junction and the cavity maintains a constant curvature, then

d, = 21,

O -- sin0 cos0 sin20

If the cavity is crack like and maintains a constant width d, then d = 2d. In the work which follows, the surface curvature term in eqn. (6)

35

is not used. However, it provides a lower b o u n d on the size of cavity required for a positive growth velocity.

O.i

2.2. The crack growth velocity We now want to solve the set of equations (1) using our expression for the crack velocity, eqn. (6). This is achieved by first assuming t h a t surface diffusion is rapid and the cavities maintain an equilibrium shape. Thus d = lg(O )

, ~

f(~)

O.5 0.2~ 0

0.;'5

0.50 "e/

(7)

where g(O ) = 2(0 -- sin0 cos0 )/sin20. It is convenient to normalize all lengths by c. Thus we define l, 7n --

,

c/2

075

Fig. 5. Plots o f the functions f(7), f ' ( 7 ) and f " ( 7 ) defined by eqns. (11), (16) and (21) in the range 0<7<1.

(8)

OE

(9)

04 17

Equation (1) then becomes

g(O ) G c 4 tin 24~W, n f

t-

- 1

7(1--72)2d7

The integral is straightforward and when combined with the b o u n d a r y conditions, eqns. (3) and (4), gives

02

50

7o = 1

f ( 7 . - z )--

f(Tn)

= t * ( n 112 - -

(n

--

1 ) 1/2}

n = 1, ..., N - - 1

(10)

21c 7N

=7c

=

---

K)O

n

150

200

2'30

Fig. 6. Plots of the normalized cavity size 7 against the n u m b e r n of cavities f r o m the crack tip for different values of the c o n s t a n t t*. The n u m b e r N o f cavities which m u s t grow for a given value o f t* can be read directly from the n axis for any value o f 7c-

c

where /(7) = 672 -- 8,7 s + 374

(11)

and t* = 288

~o(2ac) 1/2 Gg(O)c 4 t

(12)

The function f(7) is shown in Fig. 5. It ranges from 0 to 1 as the cavity grows to full size. A relationship between N and t* can then be f o u n d , assuming that 7~ is zero, by summing eqns. (10) over n between 0 and N. This gives N

1 -

(t*) 2

(13)

To determine how the cavity size changes as a function of distance from the crack, eqns. (10) were solved numerically. The results are shown in Fig. 6. Cavity growth is most rapid near the crack tip. The slope is also quite

steep for small values o f 7. It thus appears that the choice o f 7c is n o t critical and little is lost by using 7c = 0. Substituting eqns. (12) and (13) into eqn. (2) we arrive at an expression for the crack velocity: a = 288

n ~

g(O) c 2 Gc 112

(14)

The relationship between t* and N given by eqn. (13) is a rather general one, in the sense t h a t it holds for any function f(n) which varies continuously from 0 to 1 over the range 7 between 0 and 1. For example, let us suppose t h a t the cavities are forced to maintain a crack-like shape of constant width d owing to surface diffusion. The equation for v, is altered since d is now given by d = 2d. This changes the analysis slightly in t h a t f(7) is replaced by

36 f'(,/) = 3 , / - - 3'/2 + '/3

(15)

and t* is replaced b y

Xo(2ac) 1/2 t' = 18

Gdc3

(16)

However, eqn. (15) is still valid, i.e. N = 1/t '2, and the modified equation for crack velocity is

(2N)l/2 X K d = 18 - - ~ dc GC1/2

(17)

A different situation can arise as the cavity growth velocity continues to increase. Chuang and Rice [15] have pointed o u t that the cavity width can become dependent on the microcrack velocity..The maximum velocity which can be maintained b y surface diffusion for a given crack width [17] is k,

(18)

V ---- {8(1 - - COS0 )}312 d--~

where ks = 6~Ds~2%/kT. Above this velocity the crack becomes narrower as it accelerates. Furthermore, the radius of curvature at the cavity tip decreases. Thus the surface curvature term in eqn. (6) m a y no longer be negligible. Chuang and Rice have shown that rn = d , / 2 ( 1 -- cos0). By substitution into eqn. (6), d, and r, can be eliminated as independent variables. Thus v, =

8XsG(c--21,)

Thus the dependence of growth rate on the stress o can vary between (/3/2 and o 3 depending on the relative magnitudes of the surface and grain b o u n d a r y diffusivities. The previous analysis for determining average crack growth rates can be used b u t only for the two limiting situations. When curvature is neglected, eqn. (1) is replaced b y f" (?7 n -

1)

--

f" ('/n ) = tl" { (n -- 1 )3/4

__

n3/4}

where f"('/) = 4 ' / - - 6 ' / 2 + 4'/3 _ ' / 4 and 3

3

1]2

tl" =

l q(2ac)l[213/2 t X Gc 2 ~ c The number of growing cavities is equal to N -

1 tl "4/3

(20)

and the resulting crack velocity is

= [{2(1 3

- coso)}'/2j K

]3 ( x 11/2 X \~/

t 3/2

When surface curvature is important and eqn. (19b) is used, a similar analysis shows that

X

1

N = t2,,----~

Xks -3 i--co-~! + 1 }1/2 -- 1] 3

a

k \G/

(22)

where

(19)

This equation has two limiting forms depending on the importance of surface curvature. If the curvature is small, then I 3 . 3 , ~, ,1/2

ks I °(2ac) 1/2 t3 t t2" = 16(2) 1/2-~- (1 -- COS0)i/27sC C Thus d = 16(2) 1/2 ~

7r(1 -- eosO)

Ge 1/2

(23) X

w,, 1~/~ G(c --2/n) 2

(19a)

If the curvature is large, then

v. = 2(2) 1/2

In either case, eqns. (21) and (23) show a coupled dependence o f crack growth on the grain b o u n d a r y and surface diffusivities through the ratio X/Xs.

2.3. Cavity nucleation

~ (1 -- cosO )~/2~,s(c -- 21~ i

(19b)

From the model that has just been developed it is predicted that the crack growth rate depends on the number of cavities which grow

37

ahead of the crack. An upper limit on N is provided b y the condition that the load on the cavity is large enough to overcome the sintering force due to surface tension. This can be found b y equating VN to zero in eqn. (6), giving _

Nmax

C

(leK] 2

2~ \ 7-7 /

(24)

This condition is satisfied b y a mechanism for nucleation b y vacancy condensation at grain b o u n d a r y particles, proposed b y Raj and Ashby [6, 3 0 ] . However, an incubation time is required for this process to occur. If the crack growth is fast enough, such that the time interval t per step is small compared with the incubation time, then nucleation will occur closer to the crack. In this case, nucleation may be due to the fracture o f grain b o u n d a r y particles or the decohesion o f the particle-matrix interface. A mechanism dependent on grain b o u n d a r y sliding [31, 32] or dislocation pile-ups [33] is possible. Thus nucleation m a y occur when the elastic stress on the b o u n d a r y reaches a critical value ac. (ac is not the local stress required for particle fracture to occur, which will be much higher. Rather it is the average elastic stress on the particle (a¢ = K/{2nNc) u2) at which a sufficiently high local stress is present for nucleation.} In this case K2 N (25) 2~Cac 2

The effect of b o t h eqn. (24) and eqn. (25) is to alter the stress dependence or the K dependence o f crack growth to b e t w e e n d ~ K 2 and d ~ K 4 depending on the governing equation for d. The conditions also lead to a threshold stress intensity factor Kth, below which no cavity growth is possible. Since N is an integer, its lowest possible value is unity. Thus eqn. (25) gives Kth =

tion can be envisaged in which all the cavities are nucleated simultaneously on a facet oriented normal to the tensile axis, b y sliding on an adjacent grain b o u n d a r y oriented more closely to the tensile axis (Fig. 7). N is then equal to the number of cavities across each facet, i.e. N~

1

D

(26)

31/2 C

where D is the grain diameter. In this case, steady state crack growth in which N remains constant as the crack grows is no longer appropriate. Rather, we need to consider the growth of the crack across a facet assuming that all the cavities nucleate at once. This has been done using the present model b y integrating the time required for each cavity to grow to a length l~ as the crack approaches. If a >> D, then K remains constant as the crack grows across the facet. The results show that for N > 5 the time tN required to grow across the facet is related to the time Nt required b y the steady state model to grow b y the same length b y only a numerical constant, i.e.

tN = 1.3Nt

(27)

For example, combining with eqns. (12), (13) and (26) gives the average crack growth rate for this process (using equilibrium-~haped cavities): D ( 2 ) 1 / 2 ~, i (D)1/2 K d - ~N --252 g(0--)-~ C C-~ (28) It should be noted that for this situation nucleation depends on grain size rather than on stress, so that the crack growth is linearly dependent on K.

ac(27rC)1/2

This can be included empirically in the equations for growth rate b y replacing N b y N -- 1; this has a negligible effect except near the threshold. The dependence on cavity spacing is also affected b y the nucleation process. It may be that neither of these nucleation models is correct. It is c o m m o n l y observed that grain b o u n d a r y fracture is accompanied b y extensive grain b o u n d a r y sliding. A situa-

0"

Fig. 7. A schematic drawing of the nucleation of grain boundary cavities by boundary sliding.

38 3. DISCUSSION A model has been developed in which the rate of creep crack growth applicable at low applied stress intensity factor is predicted. The crack grows b y cavitation ahead of the crack tip. At low velocities we expect matter to diffuse uniformly along the grain b o u n d a r y between cavities (Fig. l ( b ) ) . This model complements previous work [22, 23, 27] in which models applicable at higher applied K values were developed. In those models, matter diffuses along the grain boundary only to a distance

away from the cavity (Pig. l(a)). The approximate range of application o f each model can be found b y equating ~'a to the cavity separation e. This defines a critical velocity below which uniform deposition should occur: 4k 5DbI2G v~ c2 4 kTc2 Using data for a-Fe (as in ref. 17) this velocity varies b e t w e e n a b o u t 10 -11 and 10 -9 m s -1 as c varies between 1 and 10 pm. These velocities are close to or below the lower limit of crack velocities measured in laboratory tests. However, t h e y are not below velocities o f technological interest. If we wish to limit cracks to a length of 10 mm over a life of say 10 years, the average crack growth rate must be less than 3 × 10 -11 m s -1. For much of this time the crack velocity will be well below this average value. It is important to note that in the crack-like cavity models a dependence o f crack growth rate on K, of K s or greater powers, is predicted while in the uniform deposition models a weaker K dependence is predicted. Thus laboratory data extrapolated from above the transition velocity will yield non¢onservative estimates for the crack growth rate. The models presented here indicate that, as the cavity growth velocity increases, so does the K dependence. For very slow growth, cavities maintain an equilibrium shape and the growth velocity varies linearly with stress. As vn increases above a threshold value given approximately [22, 34] b y sin40 vc = Xs lc30 2( 0 _ s i n 0 c o s 0 )

the cavity is no longer able to maintain its equilibrium shape. It grows instead with constant width equal to that which it has at vc, i.e. d = dc = lc

0 -- sin0 cos0 sin20

The growth velocity still varies linearly with stress. Further increases in velocity lead eventually to a situation where cavities can no longer maintain a constant width; rather d itself depends on velocity, becoming smaller as the velocity increases (eqn. (18)). Initially this results in an increase o f the stress dependence to a p o w e r of 3/2 (eqn. (19a)). However, as the cavity (or microcrack) b e comes sharper, the curvature stresses at the tip become significant, increasing the stress dependence towards a cube power (eqn. (19b)). The actual crack growth rates will also follow this trend o f increasing K dependence as c} increases. This is difficult to analyse precisely, since at any time a number of cavities are growing with different velocities. However, we would expect a continuously increasing K dependence. Two important assumptions were made in producing the models presented here. First, the possible effect of stress relaxation b y plasticity was n o t considered. Thus the model is most applicable to creep brittle materials. Secondly, an equilibrium stress field was chosen, providing an upper b o u n d solution for the crack growth rate [ 3 5 ] . This stress field is based on the assumption that the load per ligament remains equal to the initial elastic load in the absence o f cavities. Raj and Baik [24] have produced a model from which the long-range redistribution of the elastic stress caused b y material deposited in the ligaments near the crack tip can be calculated. They used a numerical m e t h o d which is not described in their paper and a direct comparison is not possible. However, the effect should be to increase the number of cavities which grow ahead of a crack and so to slow the rate of cavitation near the crack tip. A similar t y p e o f model has been constructed b y Nix et aI. [20] for a plastically deforming solid, exhibiting a constitutive equation for flow of the power law t y p e , g = A o n. They allowed an infinite number of voids to grow and calculated the steady state growth rate for the crack. As t h e y noted in

39

their paper, the solution is only convergent if n > 2. Since this is always satisfied for p o w e r law creep, the restriction is n o t serious. However, in the diffusion case (n = 1) this is n o t so. A finite steady state crack growth rate can only be reached if the number of growing cavities is finite. Thus a nucleation parameter is required b o t h physically and mathematically in the present model. Two types of nucleation parameter were presented. A critical stress criterion increases the effective stress dependence o f crack growth rate and introduces a threshold stress into the solution. Alternatively, nucleation m a y be limited b y grain b o u n d a r y sliding in which case the crack growth rate will depend on the grain size of the material. This helps to explain the c o m m o n observation that a large grain size promotes creep brittleness, although effects due to plasticity m a y also be important.

4. S U M M A R I Z I N G R E M A R K S

A model was presented for diffusioncontrolled creep crack growth b y cavitation, in which matter is assumed to deposit uniformly on the ligaments between cavities. From the model it is predicted that the crack growth rate will depend sensitively on such parameters as the spacing between cavities and the distance from the crack tip at which the cavities are nucleated. Depending on the shape with which the cavities grow and the mechanism which controls nucleation the growth rate dependence on K varies b e t w e e n K 1 and K 4 and increases as the crack grows faster. Furthermore, a threshold K value is predicted below which no cavities are nucleated and the growth rate is zero. ACKNOWLEDGMENTS

Several stimulating discussions with Professor V. Vitek are gratefully acknowledged. Funding for this work was received from Imperial Oil Ltd. and the Natural Sciences and Engineering Research Council, Canada. REFERENCES 1 D . J . G o o c h , B. L. King a n d H. D. Briers, Mater. Sci. Eng., 32 ( 1 9 7 7 ) 81 - 91,

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