Mixed-mode crack growth predictions

Engineering Fracture Mechanics Vol. 28, No. 2, pp. 211-221. Printed in Great Britain.

MIXED-MODE School

of Engineering,

0013-7944/87 @ 1987 Pergamon

1987

$3.00 + .oo Journals Ltd.

CRACK GROWTH PREDICTIONS Democritus

E. E. GDOUTOS University of Thrace,

CR-671

00 Xanthi,

Greece

Abstract-The problem of slow stable growth of an inclined crack in a plate subjected to uniaxial tension is studied by the strain energy density criterion. The stable crack growth process is simulated by predicting a series of crack growth steps corresponding to a piecewise loading increase when material elements along the direction of crack extension absorb a critical amount of elastic strain energy density. Crack instability takes place when the last ligament of crack extension takes a critical value which is a material constant. The critical stress at the onset of crack initiation and unstable crack extension is determined for various crack inclination angles. Three different loading step increments corresponding to three different loading rates are considered and their effect on stable crack growth is analysed. Furthermore, the influence of loading history on the crack growth process for three different loading types is studied. The complete crack growth patterns for all types of load are determined and analysed. It is obtained that the amount of slow crack growth can be increased by lowering the rate of loading. The effect of the loading history on the failure load and the crack paths is established.

1. INTRODUCTION THE

ANGLED crack problem has been given special attention in the past few years by fracture mechanics investigators since it represents the simplest case of a crack in a mixed-mode stress field. Interest was mainly concentrated in the determination of the critical load for crack extension and the value of the initial crack extension angle. The first study of this problem was made by Erdogan and Sih[l] who, using the Griffith concept, postulated that the crack will propagate along the plane normal to the maximum circumferential stress around the crack tip. Propagation of the crack will start when the maximum circumferential stress reaches the critical fracture stress of the material in tension. Some years later Sih[2-41 has proposed a new concept for the prediction of crack growth under mixed-mode loading conditions. He postulated that crack growth is controlled by the strain energy density around the crack tip and assumed that the crack propagates in the direction of the minimum strain energy density. Fast, unstable crack extension takes place when the strain energy density factor reaches a critical value which is assumed to be a material constant. The strain energy density criterion has been used by Gdoutos[5] for the solution of a series of mixed-mode crack growth problems under conditions of rapid, catastrophic fracture. The same author[6] studied the dependence of the initial crack extension angle in the angled crack problem on the specimen’s geometry. The centre- and edge-cracked specimens under tension and bending were examined. The maximum circumferential stress criterion was further used by Williams and Ewing[7] and Finnie and Saith[g] who incorporated in the analysis the second order term of the crack tip stress field. Ewing and Williams[9] and Liu[lO] used the criterion in pure shear, while Eftis and Subramonian[ 1 l] in biaxial loading. All the aforementioned work concerns the situation when crack growth is rapid and unstable. In this case the crack path can be accurately determined from the initial geometry of the crack prior to propagation. Indeed, the crack growth process is so rapid that the material does not have time to redistribute the stresses during crack propagation. However, instantaneous crack instability is a mathematical abstraction which has no physical counterpart. All fracture processes may be regarded as transitions from stable to unstable crack propagation. The fracture process of a body depends on the geometry, the material properties, the rate of loading, the specimen size and the environmental conditions. A specimen from a brittle material may fracture with a substantial amount of crack growth when it is slowly loaded. On the contrary, a structure from a ductile metal may have brittle fracture when the load is applied suddenly or the temperature is very low, Crack paths for conditions of stable crack growth were obtained by Liebowitz et al. [ 121 for a slanted crack under general in-plane loading. They adopted a step-by-step procedure in 211

212

E. E. GDOUTOS

conjunction with the strain energy density criterion to obtain the path of crack extension. An elastic stress field solution was used to outline the procedure which can be extended to the elastic-plastic problem. To study the crack path, however, for conditions of stable crack growth a more rigorous and consistent criterion than used in the previous paper should be adapted. The strain energy density failure criterion introduced by Sih[ 131 to address the general process of material damage seems quite promising. The relatively new approach makes use of the strain energy density function d W/d V and focusses attention on the fluctuation of d W/d V in the solid. The peaks (or maxima) of d W/d V can be associated with the locations of yielding and valleys (or minima) of d W/d V with fracture, because distortion (or shape change) dominates in the former while dilatation (or volume change) dominates in the latter. The proportion of each is not chosen arbitrarily but determined from the stationary values of d W/d V. The process of slow stable growth of a central crack in uniaxial tension with and without the influence of plastic deformation was studied by Sih and Kiefer[l4] by the strain energy density criterion. Gdoutos[lS] addressed the problem of initiation, stable and unstable growth of a central crack in a sheet. Stable crack growth was simulated by predicting a series of crack growth steps corresponding to piecewise loading increments when material elements ahead of the crack tip absorb a critical amount of strain energy density. Gdoutos and Sih[l6] analysed the dependence of crack growth on load time history for a centrally cracked panel. Load history was shown to effect the process of crack growth and the critical failure loads. In the present paper the growth pattern of an inclined crack in a plate subjected to uniaxial tension is studied by the strain energy density theory. The applied stress is increased by steps and the corresponding crack growth increments are determined. The direction and amount of crack growth are dictated by the relative minima of the strain energy density function d W/d V. The critical stress at crack initiation and instability, and the process of stable crack growth are determined. Although this work treats the elastic problem, the results obtained are very useful in outlining a procedure which can be used for the elastic-plastic problem of crack propagation.

2. CRACK

GROWTH

The strain energy density criterion constitutes an effective and powerful tool for the determination of crack growth under mixed-mode loading conditions. This criterion has been used by the author [S] in a host of engineering problems of unstable crack extension in mixed-mode stress fields. The fundamental quantity is the strain energy density function d W/d V which in general can be calculated for any material from the relation lt!

oij d Eij

where mij and Eij are the stress and strain components, respectively. Only in the linear elastic range d W/d V = gijEij/2. Consider a crack in a general mixed-mode stress field and let us examine the extension of the crack from the tip A (Fig. I). A circle of radius r. centered at the crack tip A is drawn and attention is focused on material elements along the circumference of the circle. The circular region of radius r. represents the core region in which the continuum model fails to describe in detail the state of stress and deformation. The values of the strain energy density d W/d V are calculated along the boundary of the core region and attention is concentrated on the stationary values of d W/dV. It is assumed that the direction of the element that initiates fracture corresponds to the minimum value of d W/d V which is associated to excessive dilatation, while the direction of the maximum value of d W/d V corresponds to yielding due to excessive distortion in this direction. Crack initiation starts when (d W/d V)min takes its critical value (d W/d V), which is equal to the area of the true stress-true strain diagram of the material in tension. (d W/d V), is, therefore, a material constant. Based on the above arguments the basic hypotheses of the strain energy density criterion which govern the process of crack initiation and stable crack growth may be stated as: Hypothesis I. The location of fracture coincides with the location of relative minimum

Mixed

mode crack growth

213

predictions

t Fig. 1. Procedure

strain

energy

density,

to find the path of stable crack extension.

(d W/d V)minr and yielding

with relative

(d W/d V),,,. Hypothesis 2. Failure by stable fracture or critical values. (d W/d V),,, reach their respective Hypothesis 3. The amount of incremental growth load is governed by

maximum

yielding

occurs

of unstable by

when

energy

damage

density

(d W/d V)min or

rl, r2, . . , rj, . . . , r, for a stepwise

.=si-Sj_ . ..=

whereby r = r, corresponds to the onset The quantity S in eq. (2) is defined

strain

varying

I,SC

and r = rOto damage

(2)

arrest.

S represents the local energy release for a segment of crack growth r. Equation (3) is independent of the constitutive relations of the material. The condition of unstable damage expressed by relation r = r, is equivalent, by taking into account eq. (3), to the relation S = S,, where S, is a material constant. S, is related to the critical stress intensity factor k,, through the relation s

=

c

(I+ 4(1-24kL 2rrE

(4)

where v is the Poisson’s ratio and E the modulus of elasticity. Consider now the growth of the crack AB from its tip A for a rising applied load in Fig. I. The direction of crack extension AO, is determined from the minimum value of the strain starts when the energy density function d W/d V along the circle of radius rO. Crack initiation value of d W/d V along the direction of AO, and at a distance r. from A becomes equal to the critical value of the strain energy density, (d W/d V),. Thus, the crack initiation load is determined. The crack after its first growth step becomes BAO,. For a rising load which does not produce unstable crack extension the new crack extension direction is determined by seeking the

214

E. E. GDOUTOS

minimum of d W/d V along the circumference of a circle centered at 0,. The crack extension length 0102 corresponding to a loading increment is determined from eq. (2). By continuing this step-by-step procedure the whole stable crack extension process is obtained. This process terminates at the point of unstable crack extension leading to catastrophic failure. For a continuously rising load the successive crack growth steps generally increase until the critical ligament size r, is reached. The lengths of the successive crack growth steps, however, depend on the history of the application of the load. If the value of the crack growth increment becomes smaller than the radius of the core region the crack arrests. Crack reinitiation depends on the value of loading increment.

3. THE

ANGLED

CRACK

PROBLEM

A crack of length 2a. = 2 cm in an infinite plate is subjected to a remote uniaxial stress cr subtending an angle /3 to the crack axis (Fig. 2). For a time-dependent rising load the process of crack initiation, stable and unstable crack growth is studied. Although the problem of stable crack growth requires elastic-plastic considerations in order to outline the procedure an elastic stress analysis will be adopted. The same procedure can be used for the elastic-plastic crack extension problem. The material of the plate is a steel with the following properties: E = 2.0684

X IO”

MPa;

= 1X3.98 MJ/m”;

V = 0.3;

S, = 13.48 kN/m.

The value of r, is calculated as r, = S,/(d W/d V), = 7.33 X 10Y3 cm, 3.05 x lo-” cm is used in the subsequent numerical work. The strain energy density in the vicinity of the crack tip is calculated from singular elastic solution as [2]

Fig. 2. An inclined

crack

in a plate subjected

to uniaxial

tension.

while the near

r. =

tip

Mixed

mode crack

growth

215

predictions

with

16pa12 = [2 cos 8 -

(K

-

l)] sin 8

(6)

where p is the modulus of rigidity, K = 3 - 4~ or (3 - v)/(l + u) for plane strain or generalized plane stress conditions and kl, kZ are the opening-mode and sliding-mode stress intensity factors, respectively. k, and k2 are given by [17]. kl = a&

sin* fl

k2=a&sinpcosp

(7)

Consider a small increment of crack from both tips so that the newly developed crack is (Fig. 3a). Determination of stress intensity factors k,, k2 at the tips A’, B’ of the bent crack B’BOAA’ necessitates complicated and lengthy calculations. In order to overcome this difficulty we approximate the bent crack B’BOAA’ by a straight crack A’B’. The new crack angle is PI = p + Ap and the new half crack length is al = a~+ CA’. From geometrical considerations it follows that (Fig. 3b) B’BOAA’

Aal sin &, a0 + Aal cos e,

(8)

Aal + a0 cos 0, Aa ‘. ao + AaI cos e,

(3

P1=P+ and al=urJ+

/

Actual

Crack

I

Assumed Crack

Ib) Fig. 3. Crack

\

shape after the first increment

of growth

and assumed straight

crack

approximation.

E. E. GDOUTOS

216

Under these considerations the stress intensity factors ki, k2 at the tips A’, B’ of the crack B’BOAA’ are calculated from eqs (7) by replacing a0 and p by al and pi respectively. The same procedure is used to calculate k,, k2 at the tip of the crack after the second increment of growth by replacing the actual bent crack by a straight crack and so on. This allows the determination of the stress intensity factors during the process of stable crack growth. The crack becomes unstable when the energy release at the final increment takes a critical value, or, in other words, when S becomes equal to S,. Numerical results were obtained for various values of the crack angle /3. Crack initiation takes place when the minimum value of the strain energy density along a circle of radius r. centered at the crack tip becomes equal to the critical strain energy density, (d W/d V),. Figure 4(a) presents the variation of the critical stress ci for crack initiation for various values of the crack angle p. Note that gi decreases with /3 and takes its minimum value when the crack is perpendicular to the applied stress. Crack initiation is followed by stable crack growth until global instability is reached. In order to study the process of stable crack growth the applied stress v is increased by constant intervals Au and the corresponding crack increments are obtained. Referring to Fig. 1 for an applied stress v1 = Oi+ Aal the crack grows from its tip A to the point 0, by an increment (AO,) = rl subtending an angle f&, with the initial crack direction BA. If now the stress is increased by Au2 and takes the value uz = oI +Au2 the newly developed bent crack BAO, extends from its tip 0, to the point 02 by an increment (0102) = r2 that subtends an angle 8, with the last increment of growth AO1. This process is continued un:il the last crack increment r becomes equal to the critical size r, which corresponds to global instability. Thus the critical stress at instability, a,, is obtained. Table 1 presents the successive crack increments ri and angles 0i for a stepwise rising stress at intervals A(T = 3.447 MPa for the first and last ten steps of a growing crack of an initial length 2a0 = 5.08 cm and angle p,, = 30”. Furthermore, the values of the xi and yi coordinates in a

A.0

ibi

Fig. 4. Critical

stress at crack

initiation,

13.790

MPa

P

(a) q and global instability, increments.

(b) o, for two different

loading

step

Mixed mode crack

growth

Table 1. Successive crack growth increments b and angles the first and last ten steps of a growing crack of an initial coordinates xi, yi of the advancing crack tip in a Cartesian along the direction of the q (MPa)

N

8, (degrees)

r,

0, for a length system applied

predictions

217

stepwise rising stress at intervals Au = 3.447 MPa for 2% = 5.08 cm and angle /3 = 30”. The values of the centered at the tip of the initial crack with the y-axis stress are also shown

(x 10e3 cm)

xi (X 10M3 cm)

yi (x 10-s cm)

761.9 765.3 768.8 772.2 775.7 779.1 782.6 786.0 789.5 793.0

-62.127 -62.086 -62.045 -62.004 -61.962 -61.920 -61.877 -61.834 -61.790 -61.747

3.073 3.099 3.124 3.150 3.175 3.226 3.251 3.277 3.302 3.327

97 98 99 100

1092.8 1096.3 1099.7 1103.2

-5G97 -56.316 -56.235 -56.153

6:8$3 6.934 6.985 7.036

457.144 464.007 470.924 477.893

-28.095 -28.679 -29.271 -29.873

I01 102 103 I04 105 106

1106.6 1110.1 1113.5 11 17.0 1120.4 1123.9

-56.071 -55.989 -55.906 -55.822 -55.738 -55.654

7.112 7.163 7.214 7.264 7.315 7.366

484.914 49 1.990 499.120 506.306 513.545 520.837

-30.485 -31.102 -31.732 -32.372 -32.020 -33.680

I 2 3 4 5 6 7 8 9 10

.

0

0

3.061 6.154 9.273 12.426 15.606 18.816 22.057 25.331 28.633

.

-0.114 -0.229 -0.345 -0.467 -0.589 -0.7 14 -0.843 -0.970 -1.102

.

Cartesian orthogonal

system centered at the crack tip and the y axis directed along the applied stress (T are shown. The first value of the applied stress in table a, = 761.9 MPa for which r = 3.073 X lop3 cm corresponds to the critical stress for crack initiation, while the last value gC = 1123.9 MPa for which r, = 7.366 x 10-j cm corresponds to the critical stress for global instability. Note that for constant stress increment Au the crack increments r increase until the critical size r, is reached. Results similar to those of Table 1 are presented in Table 2 for the same initial crack length 2~ = 5.08 cm and angle PO= 30” but when the applied stress rises at intervals A(+ = 13.79 MPa. Note that the critical stress for global instability is CT,= I 172.1 MPa which is larger than the previous value, while the crack growth length from initiation to instability is much less. Figure 4(b) presents the variation of the stress CT,vs the initial crack angle p for Au equal to 3.447 and 13.790 MPa. Higher stress increments correspond to higher loading rates. It is Table

2. As in Table

1 with Au = 13.79 MPa

N

oi (MPa)

0 (degrees)

ri (X 10M3 cm)

I 2 3 4 5 6 7 8 9 10

772.2 786.0 799.8 813.6 827.4 841.2 855.0 868.8 882.5 896.3

-62.127 -62.085 -62.042 -61.997 -6 1.950 -61.902 -61.853 -61.801 -61.748 -61.693

3.150 3.251 3.378 3.505 3.632 3.759 3.861 4.013 4.140 4.267

0 3.144 6.406 9.784 13.282 16.901 20.645 24.5 16 28.517 32.647

ld4d.O 1061.8 1075.6 1089.4 1103.2 1117.0 1130.8 1144.6 1158.3 1172.1

-6;3:975 -60.898 -60.820 -60.739 -60.656 -60.573 -60.486 -60.397 -60.306 -60.212

5893 6.045 3.683 6.401 6.553 6.731 6.909 7.087 7.264 7.442

giLi I 93.289 99.339 105.555 111.940 118.493 125.222 132.126 139.207 146.469

xi (X 10m3 cm)

yi (X lo-‘cm) 0 -0.117 -0.241 -0.368 -0.500 -0.640 -0.785 -0.937 -1.095 -1.262

. 21 22 23 24 2s 26 77 28 29 30

-3.630 -3.904 -4.188 -4.488 -4.795 -5.118 -5.451 -5.801 -6. I65 -6.543

218

E. E. GDOUTOS

9.3 x&ml p =30°

(b)

Au =6.695

MPa

0.15

x cm)

p :300 45O

+y(mm)

(Cl

Ao =13.790MPa

Fig. 5. Paths of stable crack extension for (a) Aa = 3.447 MPa, (b) ha = 6.895 MPa and (c) Au = 13.790 MPa.

observed that a, decreases as the stress increment increases. This trend agrees with experimental observation for specimens with cracks perpendicular to the applied stress (p = 90”) in [18]. Crack growth patterns during the process of stable crack growth are presented in Fig. 5 for Au = 3.447 MPa (a), Acr = 6.895 MPa (b) and Acr = 13.790 MPa (c) for various values of the initial crack angle /3. Note that the extent of crack growth decreases as the stress increment step A(T increases. This is a well-known result in fracture testing. Alternatively speaking, the amount of slow crack growth can be increased by lowering the rate of loading. This is why at higher loading rates, materials tend to behave in a more brittle fashion. 4. THE INFLUENCE

OF LOADING HISTORY

As it was previously established the slow stable growth of an inclined crack depends on the value of the loading step increment for a monotonically increasing load. In order to further study the dependence of the crack growth process on the loading history three different loading types were considered. The first type (A, Fig. 6a) consists of a convex load curve after crack initiation consisting of three segments of stress increment at a constant stress step. In the first two segments which extend up, to twenty stress steps the stress step is Aa = 1.38 MPa (0 < NC IO) and Ao = 6.895 MPa (I I s NS 20). These two stress segments are followed by a third segment in which the stress increases at a step Au = 13.790 MPa. The third segment extends up to the point of unstable crack extension. The second loading type (B, Fig. 6b) consists of a concave load curve consisting of three constant stress increment steps at the rates Au = 13.79 MPa, (0 < NS lo), ha = 6.895 MPa (11 s N=s 20) and Acr = 3.447 MPa (Na 21). Finally, the third loading type (C, Fig. 7a) has an oscillatory from. The stress increases in the first twenty steps at a rate AV = 1.38 MPa, then it decreases in the subsequent twenty five steps at a rate Au = 6.895 MPa and eventually it increases at a rate A.(T= 6.895 MPa until global instability is reached. The results for these three loading types are given in Figs 6 and 7. Figures 6(a), (b) and 7(a) present the loading curves which terminate at the point where is unstable crack extension takes place. Thus, the critical stress Us at global instability determined. These results are presented in Fig. 7b which gives the variation of the stress u= vs the crack angle p for the three loading types A, B and C. It is observed that the influence of

Mixed

mode crack growth

predictions

219

20Loading A

00 0

10

20

30

40

N

(ai

20

Loading B P~l5” 1

15I..-

Oo1,,,,0 lb)

N

Fig. 6. Character of applied stress for various crack angles from the onset of crack growth until global instability for (a) a convex and (b) a concave type load curve.

loading type on the critical stress u= is more pronounced at small crack angles /3. Figures 6 and 7 establish the path dependent nature of the stable growth process of inclined cracks. 5. CONCLUSIONS The majority of the failure in service is of the mixed mode type where the crack does not propagate in a self similar manner and the direction of crack propagation is not known as an a priori. This is a result of the complex crack geometry and loading where the direction of crack growth is not obvious. Thus a realistic fracture mechanics analysis of structural components necessitates mixed-mode considerations. Most of the work done in this area concerned with the situation where crack growth is sudden and unstable and crack initiation triggers global instability. The fracture criteria developed for the mixed-mode problem deal with the determination of the initial angle of crack extension and the corresponding critical load. However, in most fractures there is an interval of stable crack growth prior to catastrophic failure. The amount of stable crack growth depends on the geometry of the structure, the loading rate, the material properties and the environmental conditions. In the present communication an attempt is made to study the problem of stable crack growth under in-plane mixed-mode loading conditions. Considered is a crack in a plate subjected to uniaxial tension subtending an angle with the crack axis. The stable crack growth process is simulated by predicting a series of crack growth steps corresponding to a piecewise loading increase when material elements along the direction of crack extension absorb a critical amount of elastic strain energy density, (d WldV),. Although the present work is based on elastic considerations for the determination of the stress field in the cracked plate, the procedure developed can equally be used when crack growth is accompanied by plastic deformation. The history of fracture in the plate from the onset of slow crack growth to the point of global instability is determined. More specifically, the critical stress at crack initiation and unstable

220

E. E. GDOUTOS

Loading C

01

0

20

40

300 (b)

80

60°

7b0

100

N

ial

51 15O

60

45”

sbo

P

Fig. 7. Character of applied stress for various crack angles from the onset of crack growth instability for (a) an oscillatory type load curve and (b) critical stress at global instability, three load types of Figs 6 and 7(a).

until global o,, for the

fracture and the complete crack growth pattern are calculated. The applied load is increased at three different loadings steps which correspond to three different loading rates. Furthermore, the effect of the loading history on the stable crack growth process is determined. From the whole work it is found that crack growth is a path dependent process which causes nonlinearity effects in the response of a structure even without the influence of plastic deformation.

REFERENCES and G. C. Sih, On the crack extension in plates under plane loading and transverse shear. ASME J. bas. Engng 85D, 5 19-527 (1963). G. C. Sih, Strain-energy-density factor applied to mixed mode crack problems. In?. .I. Fracture 10,305-321 (1974). G. C. Sih, Some basic problems in fracture mechanics and new concepts. Engng Fracture Mech. 5,365-377 (1973). G. C. Sih. A special theory of crack propagation. In Mechanics of Fracture. Methods of Analysis and Solution of Crack Problems, Vol. I (Edited by G. C. Sih), pp. XXI-XLV. Noordhoff, Leyden (1973). E. E. Gdoutos, Problems of Mixed-Mode Crack Propagarion, p. 204, Martinus-Nijhoff, The Hague (1984). E. E. Gdoutos, The influence of specimen’s geometry on the crack extension angle. Engng Fracture Mech. 13, 79-84 (1980). J. G. Williams and P. D. Ewing, Fracture under complex stress-the angled crack problem. Inr. J. Fracture Mech. 8, 441-446 (1972). 1. Finnie and A. Saith, A note of the angled crack problem and the directional stability of crack. Inr. J. Fracrure 9, 484-486 (1973). P. D. Ewing and J. G. Williams, The fracture of spherical shells under pressure and circular tubes with angled cracks in torsion. Inr. J. Fracture 10, 537-544 (1974). A. F. Liu, Crack growth and failure of aluminum plate under in-plane shear. AZAA J. 12, 180-185 (1974). J. Eftis and N. Subramonian, The inclined crack under biaxial load. Engng Fracture Mech. 10, 43-67 (1978). H. Liebowitz, J. D. Lee and N. Subramonian, Criteria for predicting crack extension angle and path in plane crack problems. Proc. Inr. Conf. Analytical and Experimental Fracture Mechanics (Edited by G. C. Sih and M. Mirabile, pp. 232-250), Rome (23-27 June 1980). Sijthoff and Noordhoff, The Hague (1981). G. C. Sih, Experimental fracture mechanics: strain energy density criterion. Mechanics of Fracrure Vol. 7 (Edited by G. C. Sih), pp. XXVII-LVI Martinus Nijhoff. The Hague (1981).

[II F. Erdogan

PI PI [41 :z; [71 PI WI [IO] [I I] [ 121

[ 131

Mixed mode crack growth predictions

221

[14] G. C. Sih and B. V. Kiefer, Nonlinear response of solids due to crack growth and plastic deformation. Nonlinear and Dynamic Fracture Mechanics (Edited by N. Perrone and S. N. Atluri), Vol. 35, pp. 136156. The American Society of Mechanical Engineering (1979). [15] E. E. Gdoutos, Stable growth of a central crack. Theor. appi. Fracture Mech. 1,139-144 (1984). [lh] E. E. Gdoutos and G. C. Sih, Crack growth characteristics influenced by load time record. T&or. q$. Fracture Me& 2, 91-103 (1984). [17] G. C. Sih, Handbook of Stress Intensity Factors. Institute of Fracture and Solid Mechanics, Lehigh University (1973). 1181 H. Kolsky and D. Rader, Stress waves and fracture. In Fracfur+An Advanced Treatise (Edited by H. Liebowitz), Vol. I, pp. 533-569 (1968). (Received 14 January 1987)