The unsteady motion of a rate-dependent crack model

J. Mech. Phys. Solids, 1971, Vol. 19, pp. 329 to 338. Pergamon Press.

THE UNSTEADY

Printed in Great Britain.

MOTION OF A RATE-DEPENDENT CRACK MODEL By E. B. GLENNIE

Department

of Applied Mathematics and Theoretical Physics, University of Cambridge

(Received

24th May 1971)

.%JMMARy THE UNSTEADY

motion of a finite crack preceded by thin plastic zones is calculated. The yield stress in the plastic zones is linearly dependent on the strain rate. It is shown that the small-scale yielding approximation can be used to predict the motion of a crack, even if there is a considerable plastic flow. The effect of the increased yield stress at high strain-rates on the crack speed is assessed.

I.

INTRODUCTION

IT HAS BEEN suggested

that the speed of ductile cracks in, for example, mild steel, by the increased resistance to plastic flow at high strain-rates (KANNINEN, MUKHERJEE, ROSENFIELD and HAHN, 1969). This possibility has been evaluated quantitatively by these workers and by the present writer (GLENNIE, 1971), using a Dugdale-type crack model. The latter calculated the motion of a semi-infinite crack preceded by a thin plastic zone, in which the yield stress depends linearly on the strain-rate, and concluded that, provided the data of ROSENPIELD and HAHN (1966) for the strain-rate dependence of mild steel are used, the relatively-low speed of ductile fracture propagation can be explained by the increased resistance to flow. However, if the data of CAMPBELL and FERGUSON (1970) are used, the increase in yield stress at high strain-rates does not affect the crack speed significantly. The present paper extends the writer’s previous work [see GLENNIE (1971)] to the more realistic case of a finite crack in an infinite medium, at the price of being able to consider onZy slowly-moving cracks (with speeds up to about 300 m s-l). The effect of strain-hardening is also neglected. KANNINEN ef al. (1969) included However, the neglect of strain-hardening strain-hardening in their calculations. considerably simplifies the calculations. Progress can then be made analytically, and more general predictions made of the effect of strain-rate dependence. To specify the positions of the end of the plastic zone and the end of the crack, it is assumed that the stress at the end of the plastic zone is finite, and that the lateral displacement takes a critical value, half the ‘crack opening displacement’, at the end of the crack. In Section 2, the problem is stated and solved. In Section 3, the motion of a crack in the small-scale yielding case is calculated, and the state of stress near the crack tip is shown to correspond to the case of small-scale yielding near the end of a is limited

23

329

E. B. GLENNIE

330

semi-infinite crack, as would be expected. In Section 4, the motion of a crack with plastic zones of finite size is calculated, and it is shown that the small-scale yielding solution can be used to a good degree of approximation when there is considerable plastic yielding. 2.

STATEMENT AND

SOLUTION OF THE PROBLEM

Consider a finite crack in an infinite (x, y) so that the crack lies along

plane,

y = 0,

and choose

Cartesian

coordinates

1x1 < c.

(2.1)

At each end of the crack there is a thin zone of plastic yielding,

along

c < 1x1< a.

Y = 0,

(2.2) There is a constant stress T at infinity, perpendicular to the crack, and the yield stress in the plastic zone is Y, which may depend on position. The medium behaves elastically, except along the plastic zone, and has Poisson’s ratio v and shear modulus ,u. The displacements in the x-, y-directions are U, u respectively, and U+ is defined by u+(x) = lim v(x, y). (2.3) Y-rO,Y'O

is the purpose of this paper to investigate the unsteady motion of a finite crack. At each instant the moving crack is approximated by a static crack, and the crack motion enters the problem only through the way it affects the yield stress, Y, which is assumed to depend linearly on the strain rate. This ‘quasi-static’ approximation is valid only when It

U/CR 6 1,

(2.4)

u = dcldt

(2.5)

where and cR is the Rayleigh wave speed; for the details, see Appendix 1 and Fig. 1. This means, in practice, that the theory can be expected to apply for crack speeds up to about 300 m s-l. For the purpose of mathematical analysis, the plastic zone is regarded as part of the crack. The static problem outlined here can be solved, using a method of potentials [see MUSKHELISHVILI (1953a) and RICE (1968)]. The solution for &+/ax is { jcdr + j di} y(‘)r:;‘x]. (2.6) [ i T(u2~~~ dz -(1 c -(I In equation (2.6) and throughout this paper the Cauchy principal value of each singular integral is taken. Here, (a2-x2)-3

3-4v Since, by the symmetry

in plane strain,

Ic = 1 (3 - v)/(l f v) of the problem,

in plane stress. >

(2.8)

Y(T) = I;( - z), equation

(2.6) can be written

(2.7)

in the form

l)x(a2-x2)-+

T j (x 0

dz _ { Y(r;!a_Z_--Zr’)t &If c

(2.9)

The unsteady motion of a rate-dependent

331

crack mode

A

-x FIG. 1. The quasi-static

At this stage, a particular that the formula

approximation.

form for Y must be assumed.

It has been suggested (2.10)

Y = Y,+FB,

where F and Y, are constants, represents the dependence of the yield stress on the true-strain rate, 8, at strain rates in excess of 1000 s-l (CAMPBELL and FERGUSON, 1970, and ROSENFIELDand HAHN, 1966). It has been pointed out (LEE and WOLF, 1951, and CONN, 1965) that care must be taken in interpreting the results of experiments to find the strain-rate dependence of the yield stress at high rates, and that a spurious rate-dependence may be introduced. However, in this paper, the linear rate-dependence of (2. lo), with the two different values of F obtained experimentally by CAMPBELLand FERGUSON (1970) and ROSENFIELDand HAHN (1966), will be used. In addition, it is assumed that the true strain E is proportional to the displacement a+ along the plastic zone. Thus, Vf

where d is a constant

=

ds,

(2.11)

length and

y=yo+gag. Now, v+ can be written

as a function

(2.12)

of x and c, where c = C(l).

(2.13)

-=U?c,

(2.14)

Then, &I+

at

ac

and (2.9) becomes

(t!t?!td” .

I

If it is assumed

(2.15)

that ll+

=

w(4c),

(2.16)

332

E.

B. GLENNIE

then

au+ f?X

=

g’fxjc),

av+ - = @(X/C)ac

; g’(x/c),

and (2.15) can be written in the self-similar form

- _Fduf (g(z)--g’(z))

@;--y

dt] ,

(2.19)

where (2.20)

P = a/c, C$= xjc.

(2.21) Two conditions are required to determine the positions of the crack end and plasticzone end. First, the stress at the end of the plastic zone is required to be finite. This is equivalent to the condition lim (au+/&)

< co.

I“X+0

(2.22)

For the second condition, the displacement at the end of the crack is kept constant, equal to ZJ,,half the ‘dynamic crack opening displacement’. If we let 5 + p in (2.19) and apply (2.22) then (2.23) The ‘crack opening displacement’ condition requires that C@(l)= 0,. (2.24) Equations (2.19), (2.23), (2.24) have an entirely self-consistent, and therefore exact, solution for g, and hence for uf. Equation (2.19) can be solved, in principle, for g in terms of 5 and p, with U as a parameter. Equation (2.23) can then be used to determine p as a function of U; finally, (2.24) gives a relation between U and c, and hence determines the motion of the crack. But, since these equations cannot be solved analytically, and numerical solution would be extremely tedious, we make the approximation

au+ -=--ax

do+ dC

(2.25)

for C < .X< a. (2.26) KANNINENet al. (1969) made an equivalent approximation. The self-consistency of this approximation can be checked by comparing the numerical values of each side of (2.25) from the final solution for u*. This is done in Section 4. Using the above approximation, (2.15) can be written in the form (2.27)

The unsteady motion of a rate-dependent

crack model

333

where x =

(2.28)

2,

(2.29) (2.30) (2.31)

The function f is non-singular at x = 0, since &+/ax is zero there. The method of solution of (2.27) has been given elsewhere (MUSKHELISHVILI, 1953a, and GLENNIE, 1971) and will not be repeated here. The solution forfis f(x) = cos (7~0) h(x) cos (7~0)+

dsin (7~0)($$r

1 ($$y

2

dq}.

(2.32)

We now apply (2.22) to obtain 0 h(q)--h(a2) q--a2

dq = 0,

(2.33)

and hence T i

,,2-r2,e ~

0 (u2-rZ)++e

dz = cos (TI~)(Y. - T) 1 (,‘;I--;;;;

dz.

(2.34)

Using (2.33) and (2.31), equation (2.27) can be put in the form

au+

-=dX

&K+l)YoXCOS(ne)

x2 ++‘; (c’-T~)~ (a2) (xZ-c2)0 0 (u2-,2)*+0 c
for

The ‘crack opening displacement’ condition then leads to v, =

-&(K+ l)Y,

“(u~-x)~+~ dx i

cos (7&l) j c2 (X-c2)e

(c2-t2)’

o (a2_Z2)e+*

dz X-t2.

(2.36)

Equations (2.34) and (2.36) determine a and c as functions of 8, which is related to the crack velocity by (2.30). When 0 = 0, the equations reduce to the standard equations for a finite static-crack. In Section 3 some numerical results for the small-scale yielding approximation are presented, and computer calculations for a crack with plastic zones of finite size are given in Section 4. 3. SMALL-SCALE YIELDINGAPPROXIMATION Here, the equations governing crack motion in the case of small-scale yielding are derived. It is shown that they are the same as equations derived previously for the small-scale yielding approximation in a semi-infinite crack configuration (GLENNIE, 1971). The results of calculations showing the dependence of stress-intensity factor on velocity are presented. In order to reduce the number of physical parameters in (2.34) and (2.36), we calculate analytically the integrals in these equations when 0 = 0 and the crack is stationary, and let a = a, and c = c0 in this case. We then

334

E. B.

GLENNIE

substitute into (2.34) and (2.36) for T/( Y,--?‘) and 47c,uu,/{(h-+l)Y,,}, which are independent of velocity and therefore of 0, to obtain

We now let A = c2/a2,

and non-dimensionaIize

(3.3)

the variables x and z by substituting x = c2u, @” cfi

Z=

in (3.1), in (3.2).

(3.4)

Equation (3.1) then becomes

; Bf(l_Ajjd~= 3.zP)e

o

A-’ (D-1)” (~,/a,)] - I} cos (7rO) J1 p+(l -- _qj)“‘”

it 71’ ITarcsin

dp,

(3.5)

which can be rearranged to give lj[arccos (c,ja,)] -2/n

with use of GRADSHTEYN and RYZHIK (1965, eqs. (3.197.3),

(3.197.4),

(8.384.1)).

Here, r is defined by r(z) = ye-$ t”-’ Lit (2 > 1) 0

and F is the hypergeometric Equation (3.2) becomes

2 In (a&,)

function (GRADSHTEYN and RYZHIK, 1965, Section 9. I).

= cos (&I)-

c “-1(l-A&Z)8*+ da J! (1 -p$+ J ~.--

CO 1

@-1)”

4

0 (1 -@P)e+* Ci-P”

(3.7)

Until now, the analysis applies whatever be the scafe of’yieiding.

In the limit as so/co --) 1 for small-scale yielding, the equation C r2(+ - e>l-( 1-t e) - = cos (y$) ~_. ~. 7q - 8) CO can be derived by eliminating /z from (3.6) and (3.7). If the stress-intensity is introduced, so that

K = I’(nc)f,

(3.8) factor

K

(3.9)

it follows that

2 =

- B)l-(--1+ rc;cos (7&Qr-2(4?C(l --0)

e) ’

(3.10

where ~~ is the value of ICwhen 0 is zero. Equation (3.10) has been obtained for semiinfinite cracks also (GLENNIE, 1971), with the right-hand side multiplied by a velocity-

The unsteady motion of a rate-dependent crack model

k/c,

335

I)

FIG. 2. The motion of a crack with small-scale yielding, and critical strain equal to (A) 0.2, (B) 0.4, (C) 0.8, (D) 1.32.

dependent factor. This factor is equal to unity here because of the quasi-static approximation (see Appendix I). To complete the solution it is necessary to specify the relation between 8 and U, using (2.30). The crack half-length for a given velocity can then be obtained from (3.8). The relation between crack length and velocity predicted by these equations is shown in Fig. 2. Calculations were made for the plane-strain case using the data of ROSENFIELD and HAHN (1966), who gave F the value l-73 x 10’ N s m-’ (see (2.10)). It was assumed further that v = O-3 and ,U = 7.69 x 10” N m-’ and that The proportionality constant, d, the shear wave speed, c2, was 3-14x lo3 m s-l. was calculated assuming that fracture takes place at a given displacement, v,, half the critical crack opening displacement (FEARNEHOUGH and NICHOLS, 1968), and at a given strain, E,. Following FEARNEHOUGH and NICHOLS (1968), U, was taken to be 14 x 10e3 in. or 3.56 x 10e4 m. Four values of the critical strain were used, viz. 0.2, 0.4, 0.8, l-32. Then, d = u&,. Equation

(1.30) was rewritten

tan (nU) = and the relation

between

(3.11)

as F(K+ l)c,

u

4dp

cl’

crack length and velocity calculated

(3.12) for

For larger velocities, (2.4) is violated. Figure 2 shows that, for this model, there is an approximately linear relation between crack length and velocity. The experiments and calculations of KANNINEN et al. (1969) concerned thin steel foil, with the crack opening displacement 2v, = 7 x lob4 in. or I.78 x 10d5 m and the strain at fracture E, = 1 *I. Figure 3 shows approximate upper (C) and lower (D) bounds for the crack velocity as a function of crack length obtained experimentally by KANNINEN et al. (1969). Curve (C) was obtained from a simple theoretical formula, and agreed well with a curve, which they computed from a model similar to the present one. Calculations using the present model, with rc taking the value appropriate to plane stress

E. B. GLENNIE

336

[equation (2.7)] yielded curves (A) and (B), using the strain-rate yield-stress data of CAMPBELL and FERGUSON(1970) and ROSENFIELD and HAHN (1966) respectively. According to CAMPBELLand FERCUSON(1970}, the coefficient F takes the value 2-2x IO3 Nsm- ’ for the anti-plane strain situation.

FIG. 3. The veiocity of cracks in steel foil. Curves (A) and (B) are theoretical predictions from the and FERGUSON(1970) and ROSENFIELD and HAHN (1966) respectively. Curves (C) and (D) are upper and lower bounds obtained from the experimental work of KANNINEN et al. (1969).

data of CAMPBELL

Assuming the Tresca yield condition, it can be shown that I? should be multiplied by 3J2 for use in plane stress, and F then takes the value 7.53 x lo3 N s mM2. It will be seen from Fig. 3 that the predictions of the model from the data of ROSENFIELD and HAHN (1966) agree well with the experimental results, whereas the crack velocity predicted by the data of CAMPBELL and FERCUSON(1970) is too high. The discrepancy between the present model and that of KANNINENet al. (1969) is explained by the fact that the present work does not allow for the reduction in foil cross-section in the plastic zone, due to necking, This effect is expected to be significant only for cracks in thin foil. Finally, we note that, although the small-scale yielding approximation, employed here, is not strictly applicable to the experiments of KA~~~~~N et al. (1969, Section 4) shows that the error involved in this approximation is small. We conclude that the increased resistance to plastic ffow at high strain-rates can account for the observed velocity of propagation of ductile cracks in steel. However, since there is still considerable uncertainty about the coefficient, F, of strain-rate dependence, no definite conclusion can yet be reached concerning the importance of this effect. 4.

MASTIC

ZONE OF FINITE SIZE

The motion of a crack was calculated by computer from (3.6) and (3.7) for three cases, with a,/~, = 1*I, l-5, 2 in turn. The calculations were made as follows. With 8 given, d was calculated from (3.6). The value of /z was then substituted into (3.7) and the double integral was computed. Thus, c/c,, was calculated. Finally, a/co was calculated from the identity a/co =

%--t(c/co)

(4.1)

The unsteady motion of a rate-dependent

crack model

337

Equation (2.30) was used to relate 8 to U. Since U does not appear explicitly in (3.6) and (3.7), one set of computer calculations provides the solution for al values of the parameters F and d. It was found that the motion of the crack differed only slightly from the prediction of the small-scale yielding solution, for each of the three values of so/co. In Fig. 4 the motion of the crack for so/co = 2 is compared with the prediction of the smallscale solution (equation (3.8)), with F given the value 1.73 x 10’ N s rnb2, and sc set equal to 0% When aofco = 1*l, l-5 the crack length for a given velocity differs from the small-scale yielding solution by less than 4 per cent. This analysis indicated that using the stress-intensity factor to predict crack growth when considerable plastic yielding occurs will overestimate onl_r slightly the rate of growth.

C/C,

FIG. 4. The motion of a crack (A) with small-scale yielding and (II> with ac/cO = 2.

To check the self-consistency of the approximation (2.25), v+ was computed along the plastic zone for various values of 8, when so/co = l-5, using the integral of (2.35), and then ~~+/~c was calculated at 5 points and compared with -&+/ax; both derivatives were found through use of a finite-difference method. The calculations indicated that the error made in this approximation was smalier than the error introduced in other ways, at least for 0 < 0.3.

Provided the parameters are chosen suitably, it is concluded that the increased resistance to plastic flow at high strain-rates can account for the relatively-slow velocity of cracks in steel. However, at present, no firm conclusions can be drawn. It has been shown that the stress-intensity factor concept can be used to a good degree of approximation in cases when there is considerable plastic yielding. ACKNOWLEDGMENT The writer is indebted to Dr. J. R. Willis for his heIp and encouragement. was done during tenure of a scholarship awarded by the Gas Council.

This work

E. B. GLENNIE

338

REFERENCES CAMPBELL, J.D. and FERGUSON,W. D. CONN, A.F. FEARNEHOUGH, G. D. and NICHOLS,R. W. GLENNIE,E. B. GRADSHTEYN, I. S. and RYZHIK, I. M.

1970 1965

Phil. Mug. 21, 63. J. Mech. Phys. Solids 13, 3 1 I.

1968 1971 1965

ht. J. Fract. Mech. 4, 245. J. Mech. Phys. Solids 19, 255. Tables of Integrals, Series and Products Academic Press, New York.

KANNINEN,M. I;., MUKHERJEE,A. K., ROSENFIELD,A. R. and 1969 HAHN, G. T.

Battelle Memorial Institute,

Report 5668

l-MC. 1951 1953a

J. a&. Mech. 18, 379. Sonre Basic Problems of the Mathematical Theory of Elasticity. Noordhoff,

1953b

Singular

RICE, J. R.

1968

Groningen. Fracture, Vol.

ROSENFIELD,A. R. and HAHN, G. T.

1966

LEE, E. H. and WOLF, H. MUSKHELISHVXLI, N. I.

Groningen. Integral

Equations.

Noordhoff,

2-Mathematical Fundamentals (edited by LIEBOWITZ, H.), ch. 3. Academic Press, New York. Trans. Am. Sot. Metals 59, 962.

APPENDIX I Validity

of quasi-static

~pproximut~on

The calculations given in this paper are made assuming that the ends of the crack are stationary at B and C in Fig. 1, while the ends of the plastic zone are stationary at A and D. In reality, the situation is d_wwnic, and the stress and displacement at a point E, for example, are determined by waves arriving from other points along the crack, which may be regarded as ‘sources’ of elastic waves. Since the boundaries of the medium are taken to be at infinity, the only ‘sources’ are along the crack. If some waves travel along the crack from the point F to the point E with a finite speed, say the Rayleigh wave speed, then the other end of the crack, as viewed by an ‘observer’ at E, will be at G and the plastic zone at F. The situation is made more complicated by the fact that there are two wave speeds. However, the main point remains clear, that the error involved in the quasi-static approximation is of the order of magnitude of the change in crack half-length during the time taken by an elastic wave to travel from one end of the crack to the other, divided by the crack half-length. In Fig. 1, the time interval is about 2a/c,, the change in crack half-length is about 2Ua/cR, and the error is of magnitude 2U/c,.