Elastic-plastic mechanics of steady crack growth under anti-plane shear

J. Mech.Phys. Solids, 1971, Vol. 19, pp. 147 to 163. Pergamon Press.

Printed in Great Britain.

ELASTIC-PLASTIC MECHANICS OF STEADY CRACK GROWTH UNDER ANTI-PLANE SHEAR A. D. CHITALEY The B. F. Goodrich Co., Rrecksville, Ohio

F. A. MCCLINTOCK Department

of Mechanical Engineering,

Massachusetts

Institute of Technology

(Received 17th February 1970) SUMM.~RY

FOR a crack with steady growth under anti-plane shear, analysis shows a primary plastic zone included in an angle of & 19.7” ahead of the crack tip, and two very thin secondary (reverse) plastic zones along the crack flanks, each included in an angle of 0.37”. Numerical solutions give the shape of the plastic zones which determine the active and residual plastic strains, and give the crack tip displa~ment, which is approximately 0.07 of that for monotonic loading without growth. The length of the primary plastic zone is almost the same as that without growth, but the thickness is about 31.5 as great. Coupled with ductile fracture criteria, the present results predict initially stable crack growth, whereas analyses based on the simplification of yielding on just one plane predict unstable fracture immediately following initiation. 1. INTRODUCTION THE ELASTIC-PLASTICfield for a crack without growth, under III), was given by HULT and MCCLINTOCK (1956). Crack tip

anti-plane shear (Mode displacements expected from that solution are considerably larger than those observed in growing stress corrosion cracks, for example, indicating the need for solution of a crack with stable growth. IRWIN, KIES and SMITH (1958) found slow crack extension under increasing load to occur prior to instability. MCCLINTOCK (19.58) and MOTE and FRISCW (1962) have observed it in aluminium foil. Inertia is negligible during such slow crack growth. To avoid considering each previous increment of crack growth, a steady state is studied. By assuming the extent of the plastic zone to be independent of residual stress, MCCLINTOCK (1958, 1965) found a solution for cracks with growth. He found that instability conditions after a steady state were very nearly the same as those after the transient caused by monotonic loading. Relative to the crack tip, an element of material sweeps back through positions l-6 of Fig. la. As will be shown, the element is loaded elastically, then plastically, and then unloaded as it moves out of the plastic zone and enters the ‘wake’ region. The unloading could be severe enough to cause the element to reload plastically in the reverse sense (Fig. 1, position 7), resulting in a secondary plastic zone behind the primary plastic zone. The residual plastic strain field in the wake gives rise to an inhomogeneity of the field equations for the crack with growth. 147

A. D. CHITALEY and F. A. MCCLINTOCK

148

ELASTIC

fal

REGION Y

TRAILING

LEADING

BOUNOARY

i3OUNDARY /

\

’ ‘CRACK i SECONDARY PLASTIC

FLANKS PRlMARV PLASTIC

ZONE

YIELD

FIG. 1. (aj Successive positions of element.

ZChE

LOCUS

(b) Stress history of element.

2. SHAPE OF THE PLASTIC ZONE 2.1 Gowrning equations In steady-state the stress and strain fields relative to the crack tip are invariant. In terms of the derivative at a point in the material relative to crack length c, and the .\: partial derivative, (:,Jx,Y+

(&),,.

(1)

= O.

In Mode III deformation (HULT and MCCLINTOCK, 1956) the shear strains obey the compatibility equation

yx and ;‘,,

%x ---Y=O ..- - a? zy ax .

(2)

The non-zero stress components are z, and T,,. Equilibrium through the definition of a stress function 4: 34 - =: -_T3” l?x

is identically

satisfied

W

(3)

& = zx.

Hooke’s law and the partition of strain components denoted by superscripts e and p respectively, gives

into elastic and plastic

pits,

Elastic-plastic

mechanics of steady crack growth under anti-plane shear

149

Using (4) and (3) in (2) we get

Both the von Mises and the Tresca criteria reduce to k = (~z+r$r~~, where k, the yield stress in shear, is constant assuming a non-strainhardening material. The yield condition and the equilibrium condition give equations with straight characteristics (grad 4 lines) to which the stress vectors are normal (FREIBIJRGER, 1962). Application of this fact shows that the 4 surface for a plastic zone ahead of the crack tip must be a cone and for a plastic zone along the flank it must be a plane. Then the stress components in the primary zone (,) and secondary zones (,) of Fig. 2 are (T,), = -k sin 0, (~,)s = k,

(T,),, = k cos 8, (6) (73S

=

0.

Y

SECONDARY,

>

ELASTIC

PRIMARY,

S

P

FK. 2. Notation.

The Prandti-Reuss flow rule (PRAGER and HODGE, 1951) may be rewritten using the steady-state condition (1) for strain increments :

In the elastic region there is no plastic history and (5) becomes

dz,_ fy - v2t$ = 0. ay

ax -

In the wake region there is no further plastic flow and c*yp/?c= 0. Then, equation (5), through the steady-state condition (I). may be rewritten as

In the wake of the primary plastic zone, S(v) equals the value (V2#)r,, at the same y at a point on l-2 just inside the wake. Further, as shown in the Section 2.2, V2 d, II

150

A. D. CHITALEY and F. A. MCCLINTOCK

is continuous across IY2. V’d, can be calculated just inside the plastic zone from (3) and (6) and substituted forf(y) in (9) to give

where Rz is the radius to lY2(Fig. 2). 2.2 Continuity

of V2#

across the trailing boundary of a plastic zone

Consider the mechanics of crack growth by an incremental amount AC (Fig. 3). Let txo, zYObe the stress field when the crack length is c,. After crack growth, the length AC becomes a free surface with 7, = 0, whereas initially the stress over this TRAILING

BOUNDARY

LEADING

I’2 \

BOUNDARY

l-1

x. PREVIOUSLY

PLASTIC

CURRENTLY

ELASTIC

f UNLOADING

f

PREVIOUSLY

PLASTIC

CURRENTLY ( LOADING

\.

PLASTIC

.

PREVIOUSLY

ELASTIC

CURRENTLY

PLASTIC

(LOADING)

I

FIG. 3. Incremental growth of crack.

length was tvIl = k, txo = 0. The final stress field t,, zY can be obtained by solving the problem in which the initial stress field z,,, zyo is given, along with the initial boundary conditions, and a new traction AX,,= -k is applied just over the length AC, causing the stress distribution At,, AZ,,which is subject to a non-homogeneous yield criterion. Applying the principle of superposition in the wake, which is currently elastic, the stress field t,, ty may be expressed as the sum of the fields txo, zyOand AzX, At, and substituted in (5). For a point in the wake on r2, (11) (v2#)=~~ = (V2A~)~~~ +(V%)rz~, where 4, and A# are the appropriate stress functions for the initial and incremental stress fields. Now AZ,.,AzYis an equilibrium stress distribution satisfying compatibility and is below its yield locus in a currently elastic region, such as the wake. Hence,

Substituting this result in (ll), and noting that V24, is the vatue of V’I# at the same point I”2 W while it was last in the plastic region before crack growth by AC, we get (W)r

w = (WI,,P.

(12)

Elastic-plastic

mechanics of steady crack growth under anti-plane shear

151

2.3 Possible plastic zones Definition of boundary conditions requires knowledge, or assumptions, about the regions in which the plastic zones lie. This was the greatest obstacle to a solution. We here present plausibility arguments for the regions, to be verified by the consequent analysis, with an assumption of uniqueness in steady flow. Through the incremental stress distribution due to each increment of crack growth it is shown by CHITALEY (1968)* that there is always a plastic zone ahead of the crack, attached to its tip. The C#Isurface for parts of a plastic zone to the left and to the right of the y axis can be obtained as follows (Fig. 4a). C#Iis constant along the traction-free crack

i

(a) f _ANE \

CONICAL

x hRACK

FLANKS $:O

SECONCJARY

PRIMARY PLASTIC

FIG. 4. (a) Grad q6lines for roof of a plastic zone. (b) Wedge element between a primary and a secondary plastic zone.

flanks (FRAGER and HODGE, 1951). The straight grad &lines, being normal to the constant &line, would be radial in the right part of the plastic zone and normal to the crack flanks in the left part of the plastic zone. Suppose the primary zone extended to the left of the y axis. Then with grad 4 = k = const., the 4 surface for the right part of the plastic zone would be a cone with the apex at the crack tip and for the left part would be a plane, giving V’+ = 0. Thus, through the continuity of V”+ across the trailing boundary of the left part, V’4 would equal zero in its wake. The plane 4 surface could then be analytically extended in its Laplacian wake. Then, the part of a primary plastic zone to the left would extend indefinitely in its wake. First, this * Henceforth denoted as (C).

A. D. CHITALLY and F. A.

152

MCCLINTOCK

seems implausible through an examination of the mechanics of incremental crack growth which indicates unloading behind the crack tip. Further, it can be shown from (9) that the strain derivative ay:‘/ay would be discontinuous across the y axis and then this would result in discontinuous strain y$ across the y axis. Thus, a plastic zone cannot consist of a conical and a plane &surface joined together. Other curved + surfaces behind x = 0 seem also to be excluded by considerations of incremental growth. The primary plastic zone is thus restricted to the right of the y axis. A secondary plastic zone which has been reloaded in the reverse sense could be expected, from the nature of the kernel of the Poisson-type equation (lo), in the wake behind the primary plastic zone, near the crack Hank. A secondary plastic zone with a conical &roof is impossible (Fig. 4b) because [see (c)] equilibrium cannot be satisfied without exceeding yield in the region between the secondary and the primary plastic zones. Thus, the only type of secondary plastic zone could be a reversed one to the left of the y axis, with a plane &roof. Through arguments given earlier this extends throughout its wake. Thus the primary and secondary plastic zones must be as shown in Fig. 2. 2.4 DeJGtion

of the problem

As summarized in Fig. 5, the boundaries TP and I-S are unknown and (8) and (10) hold in the elastic and wake regions respectively, with the following boundary conditions, found from integration along the straight grad &lines (Fig. 2) inside the plastic zones. Along I‘1 and l-2 = I-P, (f, = -kr.

(13)

--..::y&)* _____________-.__-

0

;‘c

;'Js.15,I5 FIG. 5.

x

i l-P_ r2+ IEOS. 13,I4

Equations.

Derivatives to provide the additional boundary conditions along the unknown plastic zone boundaries can be obtained by transforming the components of the normal derivative of 4 through (3) and (6) on I?, -

an

= k sin (o--U),

Elastic-plastic mechanics of steady crack growth under anti-plane shear where w is the angle made by the local tangent

to the boundary

153

with the x axis. On TS, (15)

4 = ky,

3 =k cos w. an Along the x axis, which is a line of symmetry, a$

- = 0. Sn

(17)

The boundary condition for large r is more difficult. For a crack without growth, the stress function at radii large compared to the plastic zone but small compared to the crack length would be obtained by integrating the stress distribution near a crack tip in an elastic medium, as given, for example, by MCCLINTOCK and ARGON (1966): $J = -(2r)1’2 cos (e/Z). Along the flank of a crack with growth, however, the residual stress makes this invalid. Perhaps a better boundary condition at large r would be that the x-gradient of the stress function be the elastic value:

a4

z= - (2r)-

II2 cos (e/2).

As it turns out, such a difference has negligible effect on the primary plastic zone. The limiting case of the second equation was used at the edge of the residual stress calculations of the secondary plastic region, where a@/& = 0, for asymptotic zone. 2.5 Asymptotic solutions Tangents to the plastic zones at the crack tip can be obtained through solution of the Poisson equation (lo), with the boundary conditions (13) through (16) in the wake, infinitesimally close to the crack tip where the tangents OP and OS represent to the first order the boundaries rP and l?S respectively (Fig. 6a). The Poisson equation in the corner POS for r --) 0 is obtained from (10) with Y/sin 8, replacing R, as follows: -ksin8, r sin 0 ’ The boundary

conditions

(18)

become

(4hp = -kr,

(i)8p=0,

(4)n-0s

= kr sin 0,,

($$)._,, = -kr

cm 4.

(19)

(20)

The unknown angles 8, and es are found by first solving (18) with the boundary condition (19) in terms of arbitrary f!Ip and 0, and then simultaneously solving for ep and es with the remaining boundary condition (20). The solution to 4 for r + 0 must be linear in r to satisfy the boundary conditions (19) and the field equation (18) for r+O: 4 = t-F(O).

(21)

154

A. D. CHITALEYand F. A. MCCLINTOCK

(a)

WAKE

(b) +, i

0

i

d;

_---------WAKE d+, dy+k,

FIG. 6. (a) Asymptotic

Use of this expression

r,=k

\

solution at r -+ 0. (b) Limiting thickness of secondary plastic zone.

for C#Jin (18) and multiplication d2P -&F=--.

The solution

to this equation

F --= k sin 0, The constants the boundary

by Y gives

k sin t$

cw

sin 8

is of the type

Asin8+Bcos8+sinOIn(sin@+

A and B are expressed conditions (19);

as functions

A_Ccosf?~+DcosOp sin (6, + 0,)

’

B=

(23) of 0r and 13~through

C sin 0,--D sin

(e,t

application

of

sin OP

ii,)’

(24)

where

The solution simultaneous

to the # function in the corner POS is given by (21), (23) and (24). Two equations for 8, and OS are obtained from satisfying the remaining two

Elastic-plastic

boundary

conditions

mechanics of steady crack growth under anti-plane shear

(20).

Simplification

cot 8,+0,+arcsin Numerical

solution

and elimination

{(sin B,)/exp ( l~~~Pep)}

of this equation

155

of Bs results in -rc = 0.

(25)

gives

8, = 0.36572”. 8, = 19.693”, (26) Application of this technique with the assumption of no secondary plastic zone shows that the stress tends to infinity as the crack tip is approached along the flank. The assumption gives only a small decrease of 0.033” in BP. An upper bound to the thickness h, of IS is obtained as follows. In (lo), express With the boundary 4 = 4,(x, y) + 41(y) where V24, = 0 and V24, = -k/R,(y). condition (&,/dy) = 0 at y = h,, and with the approximation that I’2 follows a straight line (C) this integrates to d4,/dy

Further, @,/dy taking (d4,/&),,

= -k

sin 8, In (y/h,).

(27) decreases with increasing y and is greater than k on IS. Then, = k gives the greatest value of h,. The procedure finally leads to 1

hs< 11, ’

(28)

e,)’

exp (l/sin

which gives h,/h, < 0.0513 for ep = 19.693”. 2.6 Numerical solution for the shape of the plastic zone boundaries A relaxation program developed for the Compatible Time Sharing System at M.I.T. by TILLMAN (1965) was used. The program is suitable for unknown boundary problems since it permits arbitrary boundary and mesh shapes. The problem was solved in a log-polar plane since this magnified the scale near the crack tip where a large accuracy was desired and gave a simple region in which a reasonably well distributed mesh with shapes approaching squares could be obtained. The transformation to the log-polar plane p, 0 is defined after normalizing as follows (Fig. 7):

r* = r/R,

4* = MkR),

p = In (r*/rX),

R = k:/k2,

LOG-RADIUS

FIG. 7. Log-polar

p

plane.

8 = 8,

(29)

156

A. D. CHITALEY and F. A. MCCLINTOCK

where k, is the stress intensity factor in anti-plane shear and k is the yield strength in shear. For a crack of length c in a semi-infinite body, k3 would be z, ctJ2. The constant rz in the transformation decides the closeness of p = 0 to the crack tip. rg was taken as e-l and this gave r* = l/1100 for p = 0. In the elastic region, (8) transforms into

a’4* -+ax=() a$ de2 In the wake, (10) transforms

(30)

into r,*ePYsin 0 2 sine

’

(31

where e2 is the value on r2 for the same height y* as the point p, N. From (13) through (17) the boundary conditions on 4* become (4*)rP = --r,*ep, ($*)rs = r,*eP sin

8,

(4*)p=10 = -(2r,*)“‘epi2

00sy(e/2). I

(32)

Note that the last two equations are inconsistent as 0 + Z. Numerical calculations were also made with the derivative boundary condition discussed after equation (17), and no detectable difference was found. Further, ($*)p=O is obtained from the asymptotic solution of (21), (23) and (24). The boundary conditions on a$*/& become (a4*lait),, = - r,*ep sin p,

(a4*/an),, = - r,*eP cos (e+p), (a4*/an),=, = 0, i

(33)

where /3 is the angle made by the local tangent to the boundary with the p axis. The unknown boundary TP is obtained through iterative perturbation to match the additional conditions (33) where the normal slopes are obtained from solution of (30) and (31) with the boundary conditions (32). A mesh of 21 x 21 was used. The accuracy was affected by the mesh shapes and size relative to the average and also by the large discontinuity in the kernel of (31) across the line y* =hp*. Figure 8 shows an upper bound (@*/an too large) and a lower bound (@*/an too small) to TP, converted to the real plane. The lower bound gives &$*/an very close to the required values. The average deviation of the stress obtained on the lower bound from the required value is 1.5 per cent, whereas a test run on the known solution to the crack without growth (HULT and MCCLINTOCK, 1956) gave an average deviation of 1.4 per cent from the required value. Since the secondary plastic zone was very thin relative to the mesh size, the lYS in the log-polar plane could be taken as a constant O-line corresponding to the analytical solution 8, = 0.37”. Thus, I-P could be obtained independently of the shape of TS. The numerical solution of (30) and (31) gave (@*/an),, less than the required value, and increasing 0, further reduced (@*j&z>,-,, indicating that if anything the solution for TS must lie below es = 0.37”. Thus, though t& = 0.37” served as an upper bound to r,S the solution was not suitable to give the shape of TS because the smallest practical mesh near IYS was coarse compared to O-37”. The shape of IYS was obtained through a technique similar to that used for an upper bound on the thickness of IYS, except for the following difference.

Elastic-plastic

157

mechanics of steady crack growth under anti-plane shear -----UPPirR NO

FIG.

100,000

(101

Primary plastic zone for a Mode III crack with steady growth.

8.

I

I

L

ANALYTICAL h;

>

GROWTH

LIMITING

,

1

,

I

I

TbfiC~NEss

i 0.017

/

*-g l0,000

-

%i 8 1000

-

100

-

z s: f, w” *

COMPUTER

& E s

IO -

5 I al

1 l0,000,000

I I,000,000

1 100,000

10,000 RADIUS,

1,000 RtS

100

x IO’

IO CRACK

I TIP

4

FIG. 9. Secondary plastic zone for a Mode III crack with steady growth.

The condition z,* = 1 = ~#~/~y~+d#~l~y~ on ES gave an expression for (y*)rS in terms of (&#J,*ay*), [see (C)j. The latter was obtained as a function of Y* from a numerical solution of V”q5, = 0. The result for (y*),-s obtained as a function of (T*)~~ is plotted in Fig. 9. 2.7 Strain distributions and the crack tip displacement In the primary plastic zone, the strain field is shown in the Appendix to be k

Y~=(-&-p-~sinOln

R ,

0r

(34)

A. D.

F. A. MCCLINTOCK

CHITALEY and

R

where subscript IP refers to a point on IP at the same 8. In particular, in front of the crack tip (34) and (35) reduce to (rf)e=o = 0, (Y,“>~=~ =

kIn (:)

(36)

[L+iIn($)],

(37)

in agreement with the expression derived for 8 = 0 by MCCLINTOCK (1963). [Note: an earlier expression given by MCCLINTOCK (1958) is incorrect.] Figure 10 plots -y,PG/k and r,“G/k vs rl(k$/k’) along the trailing boundary I2 and along 8 = 0. In the secondary plastic zone, through a procedure similar to that of the Appendix for (42) it can be shown that (38) Then, the plastic strains rf: and r$ are given by the values on IS at the same x and same

0.1 -0.2

0.3

0.4

RADIUS

FIG. 10.

0.5 r’:

0.6

01

0.8

0.9

I

r/lh;/kzl

Plastic strains inside the primary plastic zone along the trailing boundary

and the x axis.

Elastic-plastic mechanics of steady crack growth under anti-plane shear

159

y respectively. The values on I,S are obtained from the values on I2 at the same height y. The crack tip displacement [see (C)] may be obtained through integration of aw/& = ys (Fig. 5) along the plastic zone boundary I-P:

Calculation of this integral using a simplified shape for I-P approximated numerical solution (Fig. 8) gives the crack tip displacement:

[~l~~an,c-to-~,an~ = @WW)k/~)2~ The expression for the flank-to-flank turns out to be

tip displacement

for a crack without

from the

(39) growth

BP 2k -jRdO G 0

since yf: = 0 = y: throughout the boundary of the plastic zone in this case. Then, with the known result R = R, cos 8 (HULT and MCCLINTOCK, 1956), the tip displacement for the crack without growth is

= 2(k/G)(k,/k)2. CWlflank-to-flank

(40)

3. DISCUSSION The existence of stable growth, discussed in Section 1, is confirmed by an examination of Fig. 10, for a material which fractures when a critical strain is reached throughout a critical region in front of the crack. As the applied stress intensity factor k, is increased, the normalized strain increases and the normalized structural distance p* = p,/(k,/k)2 decreases, following the upper curve of Fig. 10. At the ratio of fracture strain to yield strain, say point A, the fracture criterion is satisfied. Instability, which occurs at very nearly the stress intensity factor for steady-state growth, will not yet occur, for the solid steady-state curve lies to the left of the monotonic loading one, indicating that a higher k, is required to produce the critical fracture strain at a given structural distance ps. The intermediate values of k, correspond to stable crack growth, as discussed in detail by MCCLINTOCK (1958, 1965). Models that assume yielding along a single plane [e.g. those of DUGDALE (1960), BARENBLATT (1962) and BILBY, COTTRELL and SWINDEN (1963)] when applied to growth under load, neglect residual plastic strains and do not distinguish between strain increments due to crack growth and due to monotonic loading. Table 1 shows that these models predict immediate instability at initiation whereas the solution allowing the finite width of plastic zone required by yielding [see also HULT and MCCLINTOCK (1956) and FIELD (1965)] predicts an initial stable crack growth. A dislocation model may be useful in visualizing the effect of growth and in extending it to the plane strain opening Mode I. A simple model consists of screw dislocations being emitted along two inclined planes running ahead of the crack tip, and then being left behind as two linear arrays of dislocations in the wake, parallel to

160

A. D.

CHITALEY

and

F. A. MCCLINTOCK

TABLE 1. Fracture predictiomfor

Fracture criterion

Initiation kz,

--.-. __

stress intensitl’,factor

Instability /izc

-

k,

W&c

Diffuse yielding (la)

L---1 2OGlc 3 Wf

Critical crack tip displacement, w,

l/2

Single-plane yielding (lb) Critical 22 w p A f plastic work per unit area of fracture surf;t,ze, w,”

l/2

(2b) Critical crack tip displacement, )$I,

1

the flank. These arrays represent a residual strain which is incompatible in Mode III, producing a residual stress. However, the residual strain in Mode I may produce much less residual stress if the Burgers vectors of the edge dislocations in the arrays are nearly normal to the flank of the crack, as would be emitted from the centered fans of the continuum solution, directly above and below the crack tip. It is interesting to note that the displacement discontinuity does not appear in a rigid-plastic model of steady-state crack growth in a bar under torsion. Likewise, in the fully-plastic tension mode, the strain is finite but the crack opening displacement is zero; it is as yet undetermined what the character of the solution would be for the elastic-plastic tension mode with growth. 4. CONCLUSIONS Several features of the mechanics of cracks through the anti-plane shear (Mode III) case:

with growth

have been examined

161

Elastic-plastic mechanics of steady crack growth under anti-plane shear

(i) A steady-state condition has been formulated to make the solution more practicable and useful. (ii) The inhomogeneity of the field equations in the wake, caused by the residual plastic strain field, has been shown to be equal to the inhomogeneity just inside the Arguments similar to those applied plastic zone on the trailing part of its boundary. to the present case can be used to show the continuity of the biharmonic of the stress function, V”+, across the trailing boundary of a plastic zone for the in-plane deformation cases. (iii) There is an appreciable effect of residual plastic strains in the wake for the Mode III crack with growth. The volume of the primary plastic zone is reduced by a factor of approximately 9/4 and the crack tip displacement is reduced by a factor of approximately 14 as compared to the crack without growth. The length of the primary plastic zone is almost equal to that for the crack without growth. Two thin secondary plastic zones occur along the crack flanks. (iv) The present results for the Mode III crack predict initially stable crack growth. Models in which yielding is assumed to be restricted to a single plane are incapable of indicating the difference between initiation and unstable fracture because elements are subjected to identical plastic strain histories in monotonic loading and growth under load.

AKNOWLEDGMENTS

This work was carried out as part of an Sc.D. Thesis and was supported by the National Science Foundation through Grant NSF GK-1875 X. Numerical work was done on the Time-Sharing System at the Computation Center at M.I.T. Thanks are due to Mr. C. Weissgerber and Mr. C. Tillman for their kind help with the use of the EPS SAVED program. The services of Mr. R. Harrington for numerical calculations are very much appreciated.

REFERENCES BARENBLATT,G. I.

1962

by

BILBY, B. A., COTTRELL,A. H. and SWINDEN, K. H. CHITALEY, A. D.

DUGDALE, D. S. FIELD, F. A. FREIBURGER,W. F.

1963 1968

(edited and VON Academic

Advances in Applied Mechanics

DRYDEN, H. L. BARMAN, TH.) 7, 55. Press, New York. Proc. R. Sot. A272, 304.

1960

Elastic-Plastic Mechanics of Cracks with Growth, Sc.D. Thesis, M.I.T., Cambridge, Mass. J. Mech. Phys. Solids 8, 100.

1965 1962

J. appl. Mech. 32, 197. Handbook of Engineering

Mechanics

(edited by FL~~GGE,W.) pp. 48.348.4. McGraw-Hill, New York. HULT, J. A. H. and MCCLINTOCK, F. A.

1956

Ninth

Int.

Cong.

appl.

Mech.

Free University of Brussels. IRWIN, G.

R., KIES, J. A. and SMITH, H. L.

1958

Trans. ASTM

58, 640.

8, 51.

162

A. D. CHITALEY and F. A. MCCLINTOCK

MCCLINTOCK, F. A.

1958 1965 1963

MCCLINTOCK,F. A. and ARGON, A. S.

1966

MOTE, C. D., Jr. and FRISCH, J.

1962

PRAGER,W. and HODGE, P. G., Jr.

1951

f. appl. Mech. 25, 582. Proc. R. Sot. A285, 58. Fracture of Solids (edited by DRUCKER, D. C. and GILMAN, J. J.) pp. 65102. Metallurgical Society, AIME Conf., Series 20, Interscience, New York. Mechanical Behavior of Materials, Addisonpp. 405407, 535.

Wesley, Reading, Massachusetts. J. Basic Eng. 84, 257. Theory of Perfectly

Plastic

Soli&

pp. 55-81. Wiley, New York. TILLMAN,C. C., Jr.

Usage Conventions for EPS SAVED, Project MAC, Memo. MAC-M284, M.I.T., Cambridge.

1965

APPENDIX Derivation

of strain in primary

plastic zone

From the flow rule at steady state (7) and the stress field (6), eliminating y; and y,, in turn by expressing the total strain as the sum of the elastic and plastic parts, applying compatibility, and expressing y, in terms of its parts gives

(A.1) Substitution of Hooke’s law and the stress field (6) for the elastic strain, and changing to cylindrical coordinates gives @r: Zr

Integrating

at constant

k sin 8

G

(A.2)

r’

8 gives y,” = (yp)rr; e + r sin 8 In

r

(A.3)

0R

(the first term is zero along TIP but non-zero along I2P). The rate of change with respect to x is found by noting that ar/ax = cos 8, &Y/ax = -(sin Q/r: &%p sin 8 ayl: ax = -dO_~r.-G_ Application

k sin 8 cos 8

I~_.

ax

Integration (see Fig.

k sin’ 8 dR

In R-1 r

+G-,-R-dO.

( > of the flow rule (7) and the stress field (6) gives

k ~0s~ 0 111i - $COS~ ar:_ _-~0s 0 ~Y%P+ -Ci. r

dt9

of (A.5) at constant 2) gives 0

with

e- gE r

y in terms of the variable

de,

cos 0 sin 0 g.

(A.4)

(AS)

0, that goes from 0, to 0

Elastic-plastic

Next,

(i) partial

mechanics of steady crack growth under anti-plane shear

differentiation

with respect W&P

83

cos8

ay

r

ay

1

cose1+

--=

8Y all together

ary” _

of

y = R(B,) sin 01, with respect

to y and solving,

$2 ($)r,,l>

give

602

e

8 d(Y:)rp

r sin8 al, - -~-___--

Now, (AS)

’

of sin 0, = y/R(O,)

801

x, (ii) substitution

= (YyP)rIP= 09

e = sine

’

and, (iii) by differentiation

to y at constant

163

de

k Gy

s

~0s~8 -;a

de-

81

with r sin fI = R(B,) sin 8,, into

and (A.7) combine,

ay;ax ayyPayayp a$ =zav+---=~~O~~+-~ine ar ay ar a,

w ay

e

= --

k Gr

s

01

k = --cOsO-Crln Gr Integration

along

8 then gives (35).

k ’

(A.81