- Email: [email protected]
Dynamic crack propagation in a viscoelastic strip
J. Mech. i’hys. Solids Vol. 28,pp. 79-93 Q Pergamon Press Ltd. 1980.Printed in Great Britain
DYNAMIC
CRACK PROPAGATION V~SCOELASTIC STRIP C.
IN A
H. POPELAR
Department of Engineering Mechanics, The Ohio State University, Columbus, OH 43210, U.S.A.
and C. ATKINSON Department of Mathematics, Imperial College of Science and Technology, London SW7 ZAZ, England
(Received 17 September 1979)
ABSTRACT THE DYNAMICPROPAGATION of a semi-infinite crack in a finite linear viscoelastic strip subjected to Mode I loading is investigated. Through the use of integral transforms the problem is reduced to solving a Wiener-Hopf equation. The asymptotic properties of the transforms are exploited to establish the stress intensity factor. Plane-stress and plane-strain stress intensity factors as a function of crack speed for both fullyclamped and shear-free lateral boundaries are presented for the standard linear viscoelastic solid. Comparisons are made with previously obtained asymptotic stress intensity factors and with stress intensity factors for the equivalent elastic strips.
1. EVEN WITH
INTRODUCTION
simpIifying assumptions, such as the fracture process being confined to a point, the number of solutions for dynamic crack propagation in viscoelastic materials at crack speeds necessitating the retention of inertia terms is small compared to the number of elastic solutions. The complexities introduced by rate effects make the mathematical analysis difficult. The first such analysis was performed by WILLIS (1967) who considered antiplane, steady-state crack propagation in an infinite viscoelastic medium. With the exception of the transient analysis by ATKINSON and LIST (1972), and the model of POPELAR and KANNINEN (1980) for crack arrest in a viscoelastic double cantilever beam fracture specimen, most of the investigations have been for steady-state crack propagation. A specimen that has been used frequently in experimental studies of quasi-static crack growth in viscoelastic materials is the strip. When unstable crack growth appears, the inertia terms become important and must be included in an analysis of the event. Assuming the parameter E = m/h (where v is the crack speed, r is the relaxation time of the medium, and h is the half-height of the strip) to be much less than unity, ATKINSONand COLEMAN(1977) used matched asymptotic expansions to 6 79
80
C. H. POPELAR and C. ATKINKIN
obtain the Mode I stress intensity factor for a steadily advancing semi-infinite crack in a clamped viscoelastic strip. The equivalent elastic problem has been solved by NILSSON (1972) using integral transforms and the Wiener-Hopf method. Atkinson and Coleman found the asymptotic stress intensity factor to be identical to Nilsson’s factor if the elastic moduli in the latter are replaced by the respective long-time relaxation moduli. Additional generalizations of these solutions can be found in ATKINSON (1979). Recently, ATKINSONand POPELAR (1979) reduced the problem of a steadily moving semi-infinite crack in a linear viscoelastic strip subjected to antiplane shear (Mode III) loading to solving a Wiener-Hopf equation. As is usually the case, the successful solution of this equation hinges upon being able to factor it. The factorization was performed by numerically evaluating a Cauchy integral. For E Q 1 the stress intensity factor was found to be in excellent agreement with its counterpart deduced from the matched asymptotic solutions of Atkinson and Coleman. However, for larger values of E the differences become important. The antiplane problem has been a favorite of the analyst venturing into new areas of fracture mechanics because compared to the in-plane problem the mathematical difficulties are minimal. It is commonplace-the analysis of Atkinson and Popelar not excluded-to assume that the crack extension in Mode III is self-similar. While similarities exist between the antiplane and in-plane problems, and while antiplane solutions provide useful qualitative information, their applicability is questionable because as KNAUSS (1970b) has noted it is physically difficult to achieve rectilinear crack growth in Mode III. As a step toward providing a physically more useful analysis, the dynamic propagation of a semi-infinite crack in a finite, linear viscoelastic strip subjected to Mode I loading is treated in the present paper. A rather general formulation of the problem culminating in a Wiener-Hopf equation is presented. The numerical approach employed by Atkinson and Popelar is used in the solution of this equation. Plane-stress and plane-strain stress intensity factors as a function of crack speed for both fully-clamped and shear-free lateral boundaries are determined for the standard linear solid. Since no restriction upon the magnitude of c: is made, these results represent a significant generalization of Atkinson and Coleman’s asymptotic solution. Because of the close analogue between this problem and the elastic one, frequent reference is made to NILSSON’S (1972) work.
2.
ANALYTICAL DEVELOPMENT
Consider the situation depicted in Fig. 1 in which a semi-infinite crack is propagating in a brittle, linear viscoelastic strip. Suppose a constant displacement uZO of the edges of the strip causes the crack to extend with a constant speed 2’relative to the fixed reference frame xi. Primary consideration is given to establishing the dependence of the viscoelastic stress intensity factor K,, upon the crack speed. This task is facilitated by introducing a moving coordinate system xi attached to the crack tip and defined by xi = x; - L& )
(2.1)
81
Dynamic crack propagation in a viscoelastic strip
_- ------_______^_----f“20 FIG. 1. Viscoelastic strip.
where t is time and 6i, is the Kronecker delta. For this piane problem there is no dependence upon x3. If aij and Ui denote, respectively, the components of the stress tensor and displacement vector, then the equations of motion in the absence of body forces are
d2Ui
aaij
8X;
(2.2)
-P-p
where p is the density of the medium and the usual summation convention is implied. The constitutive relations for a homogeneous, isotropic viscoelastic medium can be written as de+)
sij= 2 j G,(t-T)~d~, --a)
okk =
(2.3)
j G2(t-r)yh, -CC
where sij and eii are the deviatoric stress and strain components sij
=
eij
=
I
-+CkkSij, Eij-+&kk&fjr aij
(2.4)
and G, and G, are relaxation functions. The components of the strain tensor are
(2.5) On the upper and lower boundaries of the strip, x2 = +_h, assume that a2 = 3-u&W)
(2.4)
and either (i) ai = 0
or
(ii) er2 = 0,
(2.7) where H(t) is the Heaviside unit-step function. The former condition {i) describes a clamped boundary and the fatter (ii) a shear-free boundary, and this designation is used in the sequel. It is convenient to make use of the symmetry with respect to the
82
C. H. POPELAR and C. ATKINSON
plane x2 = 0 and superpose the solution satisfying (2.2) through (2.7) for the untracked strip. Consequently, the boundary conditions (2.6) and (2.7) for the residual problem assume the form on x2 = h, u2 = 0
(2.8)
and either (i) u1 = 0
or
(ii) cl2 = 0,
lx11 < 00; (2.9)
and on x2 = 0, G,Z(XI, 0, r) = 0,
(2.10)
Ix11< @J,
(2.11) -cc
o
(2.12)
The functions X(x,, t) and A(x,, t) are respectively the unknown traction ahead of the crack tip and the unknown crack opening displacement, and remain to be determined ; whereas a(~,, t) is the normal stress on x2 = 0 obtained from the solution of the untracked strip. Finally, the dilatational wave potential 4 and shear wave potential rj are introduced through
w
u1=ax,+ax,,
a*
a+ a*
UZ=3g--.
(2.13)
With the Laplace transform over t and the Fourier transform over x1 defined, respectively, by f(xr, x2, P) = 7 e-pfS(xI, x2, r) dt,
Re [PI > 0,
0
(2.14)
T(f(s,x2, P) = i eisx13(x1, x2, p) dx,, -a, the transformed counterparts
of (2.2) and (2.3) are d~i2 __-
dxz
isSi, = p(p+i~s)~fi~,
Sij = 2~L2,,
~~~= (3~+ 2~L)~kk,
(2.15) (2.16)
where p = (p + ius)G,(p + ius),
(2.17)
3J.+ 2p = (p + ivs)G,(p + ius).? I Equations (2.16) have the same form as the transformed elastic constitutive relations except that the Lame parameters 1 and p here are complex functions of p+ius. The 7 To change this plane-strain formulation to a plane-stress one replace I by 24/(E. + 2~):
83
Dynamic crack propagation in a viscoelastic strip
complex longitudinal and shear wave speeds are (2.18) The transformed boundary conditions are on x2 = h, ii, = 0
(2.19) and either
(i) ii, = 0
or
(ii) Zi2 = 0; 1
and on x2 = 0, 5 12 -0, _ 622
=
-~+~+(s),
ii,
=
L(s),
(2.20) 1
where
C+(s)= i
eisxl
E(x,, p) dx,,
0
(2.21)
j eisxlA(xl,p)dxl.
Z_(s)=
-a!
The behavior of the stresses and displacements as lx11+ cc are assumed to be such that E+(S) and k(s) are regular in overlapping half-planes of the complex s-plane. Finally, (2.13) is transformed to yield
. ;,=-- d$ 4, dx,
ii, = d6 dx
+
is?.
(2.22)
2
After some algebraic manipulations
the equations of motion assume the form
(2.23)
where yf =
s2+ (p + ius)2/c:,
(2.24)
7; = s2 + (p + ivs)2/c$. 1 Then, with Cl2 = p
(s2+#-2isz
2
[
E22 = P (s2+y:)J+2isz
1 1
d? , d?
(2.25)
,
2
the equations for solution of the boundary-value problem are complete. It is a straightforward matter, albeit lengthy, to determine 4 and 3 that satisfy
C. H. POPELARand C. ATKINSON
84
(2.19) through (2.25). Having done so, one can ultimately write (2.26)
522(O) = -K(s, P)&(O), where, for the clamped boundary (i), K(s, P) = YI(s:_Y;) {4s2Y,Y&2+Y:)-Yy1Y2[4~4+(s2+Y:)2]
cash (Y,h)cosh
[4s2Y:y: + s2(s2 +~f)~] sinh (Yrh) sinh (y,h)} x {Y1y2sinh (y,h)cosh (Y2h)-s2 sinh (Y,h)cosh (yrh)}-‘;
(y,h)
+
(2.27)
whereas, for the shear-free boundary (ii), K&p)= The combination
[4s2y, Y2coth (y,h)P Y1(S2-Yy22)
(s2 +Y;)~ coth (yr h)].
of (2.20) and (2.26) yields the Wiener-Hopf -Z+C+(s)
(2.28)
equation
= -K(s,p)Z_(s).
(2.29)
The solution of (2.29) depends upon being able to determine the factors K, and K _ each of which is analytic and different from zero in its respective half-plane such that (2.30)
K(s, P) = K+(s, P)K-(s, P).
To facilitate the factorization of K(s, p) it is normalized with respect to its asymptotic value as JsJ--) cc to yield
K(s,P) .ABI cc
N(s, p) = ___
K,(s)
pm
(1-B:. co)?
(2.31)
where p: = (l-?/c;)
(2.32)
and Pi, m = lim
Pi,
ISI-+ m
k
(2.33)
= ,$Fr pL,
in which p, is the short-time shear modulus. For boundary conditions (i), G(s) = s{4BI,mB2,m(l +B:,a:)-B~,mB2,mC4+(l+P:,m)21 x cash M&.mh)cash ($2, doh)+r4B:,,P:,,+(1+8:,,)21 x sinh (s/?r, ,h) sinh (sj12,mh)} sinh (sfil,,h)cosh ($,,,h)-sinh (s/&h) cash (sB~,~~)}-‘, XV 1.00 B2.a;
(2.34)
and for boundary conditions (ii), K,(s) = sC4B1,mBz,m coth (sB2,,h)-(l+&m)2
coth (@l,,h)l.
(2.35)
Except for the sign, K,(s) is identical with NiLssoN’s (1972, equation (33)) expression for the equivalent elastic problem if the elastic moduli in the latter are replaced by the short-time moduli.
85
Dynamic crack propagation in a viscoelastic strip
With N(s, p) so defined, N(s, p) + 1 as IsI+ c;o in a strip a < Im [s] < b in the complex s-plane. If, in addition, N(s, p) is regular and nonzero in this strip, then (for example, see NOBLE (1958, p. 16)) Cauchy’s integral theorem can be used to factor N(s, p) as N = N, N _ with
N+b,p)=ew (2.36) N-(s,p)=ew
where a < c < Im [s] < d < b and the branch of the logarithm is chosen so that In N + 0 when N(i, p) + 1 with I[[ -+ cc in the strip. If, for example, N(s, p) is analytic only on Im [s] = 0, the limits c + O- and d + O+ are taken and the contour is indented accordingly at the origin, i.e. .
N,(s, P) = exp
(2.37)
The factors of K(s, p) can be written as
K+(s, P) = Ko+(sP’+(s, P), Ko-(SW(s> P)P,,
K-(syp)=&&,Jl-&) The factorization particular that
.
(2.38)
of K,(s) has been performed by NILSSON (1972) who found in &+ (0) = Ci~o(W+, K,+(s) = K,_(s) _ [R,s]+
(2.39) as IsI-+ co,
where the Rayleigh factor & = 4P,,,82,~-(1+B:,a:)2.
(2.40)
The short-time Rayleigh wave speed cR,% satisfies R, = 0. The Wiener-Hopf method of solution of (2.29) requires it be split such that each side of the equality is analytic in its respective half-plane and has a common overlapping strip. In order to achieve explicitly this separation the expression for 5 is required. For boundary conditions (i) it assumes the form =
o’=-
F(P)
isp and the split considerably Nevertheless propagation.
(2.41)
is readily made. However, for boundary conditions (ii), the expression is more complicated and a straightforward separation is precluded. (2.41) does hold in the limit as t + a~, i.e. for steady-state crack Mindful of this latter limitation, (2.30) and (2.39) can be used to rewrite
C. H. POPELARand C. ATKINFWN
86
(2.29) as
= -K_(s,
p)L(s,
F(P)
p) +
Y---p.
iVK +(0)
(2.42)
The extended form of Liouville’s theorem yields J(s) = 0 which, when (2.38) is employed, implies
>
‘Y_(s, p)
= ~----
F(P)Pl,m(l4,,)
(2.43)
,
w~,&,+(OW+(O, P)&-@N-h
P)
With the aid of (2.38) and (2.39) the asymptotic forms of (2.43) become
The Tauberian theorem (see NOBLE, 1958) for the asymptotic transforms further permit writing X = R,,(p)(2nx,)-*,
inverses of Fourier
xi --+o+,
(2.45) (2.46)
t
where
(2.47)
is the Laplace transform of the viscoelastic stress intensity factor. It is apparent that the problem of determining the viscoelastic stress intensity factor has been reduced to one of evaluating N, (0, p) and taking the inverse Laplace transform of (2.47). In general this is an extremely complicated task, but one which is feasible for steady-state crack propagation.
3.
STEADY-STATE PROPAGATION
In the sequel it will be assumed that the crack propagation has been occurring for a suffi~ently long period (t -+ co) so that a condition of steady-state crack extension has been attained. The well-known asymptotic relation between a function as t -+ cg and its Laplace transform as p -+ 0 is applied to (2.45) through (2.46) to obtain c = z&(27&-*.
x1 -+o+,
(3.1)
87
Dynamic crack propagation in a viscoelastic strip
(3.2) (3.3)
for steady-state crack propagation.
61,
For boundary conditions
F(0) = (;I+ 2&J y, >
and (ii),
(3.4)
F(O) = 4u,(a)y,j
where PO = cl(O),
10 = A(O)
(3.5)
are the long-time Lame parameters. From NILSSON’S (1972) analysis the stress intensity factors for the equivalent elastic problems based upon the short-time moduli would be
(3.6)
The combination
of (3.3), (3.4) and (3.6) yields
&0+&*
0)
L
1,+2/&J .K,
1 = N+(O,O)’ >
.&+A. (ii) k PO Ao+po
20 + 2~0
Ke
1,+2c1’x .K,
1
(3.7)
= N.&JO)
If, as is frequently done, Poisson’s ratio v is taken to be a real constant, then (3.7) assumes for both sets of boundary conditions the simple form CL, L ----_ PO K,
This expression for steady-state, Whether or N+(O, 0) which
1 N+ (090)’
(3.8)
is identical in form to that obtained by ATKINSONand POPELAR (1979) antiplane, dynamic crack propagation in a viscoelastic strip. not v is a real constant, the determination of K,, reduces to evaluating according to (2.37) becomes N + (0,0) = r exp
(3.9)
when the oddness of arg N([, 0) is employed. The method of residues provides the
88
contribution
C. H. POPELAR and C. ATKMON
r to N+(O, 0) due to the indented contour at 5 = 0. The values are
6) (ii)
Jo+2clo +
r=
(- ) &-t-2/1,
’
y _ j10 J1.m 4P:,0-(l+P;JJ2 gLI*i P 1.0 [ 48:,,-fI+pf,X))2
(3.10) t 1 ’ I
whereiLo = &(Ol and 82,0 = j&(O). While the complexity of arg N([, 0) prevents the formal integration in (3.9), the integral can be evaluated numerically, say, by Simpson’s rule and supplemented by asymptotic values near the origin and at infinity. Based upon an argument of the local work at the crack tip, the energy release rate G is (3.11) which yields for (3.1) and (3.2), (3.12) as KOSTBOV and NIKITIN (1970) have predicted. In general, N + (0,O) depends upon the crack speed and therefore so will the energy release rate. For the equivalent elastic problem, the energy release rate of the strip with clamped boundaries is independent of the crack speed as it is also for the antiplane loading. At first appearance it would seem that this should also be true for the shear-free boundary, but this is not the case. The origin of the velocity terms in this elastic problem and its implications are discussed in the Appendix.
4.
NUMERICAL RESULTS AND DISCUSSION
Whereas the previous development has been for arbitrary viscoelastic materials, the numerical evaluation of N + (0,O) necessitates that a specific material model be considered in order to obtain quantitative results. One restriction implicit in the development is that the material must exhibit nonzero, finite long- and short-time mod&i. In the following a three-parameter, linear viscoelastic solid with relaxation moduli G,(r)=~o~l+~erp(-~t)].
G,(t) =
2(1-t-v)
-m
G,(t),
(4.1)
K0 = I”*(1 +f) is selected as a material model. While it is not necessary, a consrant Poisson’s ratio (v = 0.3) has been assumed for the sake of convenience.
Dynamic crack propagation in a viscoelastic strip I
I
I
89
/
f =0.25, rc2 ,/h: Y
0.2
0
06
04
0.8
I.(I
FIG. 2. Dimensionless stress intensity factor vs dimensionless crack speed, for plane strain and clamped boundary.
For the relaxation moduli (4.1) the integral of (3.9) was numerically evaluated using Simpson’s rule. Because K, vanishes at the Rayleigh wave velocity CR,m = 093c,, m, computations were not performed for crack speeds in excess of this value. A plot of dimensionless stress intensity factor against crack speed normalized with respect to the short-time shear wave speed c 2, m appears in Fig. 2. In this case the
1 7
0.8 -
06 -
0
02
0.4 v’c,,
06 00
08
I IO
FIG. 3. Dimensionless stress intensity factor vs dimensionless crack speed, for plane strain and shear-free boundary.
C. H. POPELARand C. ATKINSON
90
I
, f
q
I
0.25,Tc&h= J
06
-
0.2
04
06
IC
08
FIG. 4. Dimensionless stress intensity factor vs dimensionless crack speed, for plane stress and clamped boundary.
strip is assumed to be in a state of plane strain and to have clamped boundaries. Comparable results for a strip with shear-free boundaries are shown in Fig. 3. Companion plots for conditions of plane stress appear in Figs 4 and 5. These curves are very reminiscent of the results that ATKINSON and POPELAR (1979) obtained for
f
&h=
=0.25,rcp ’
J
IC
04
02
C
'0
02
04
06
00
FIG.5. Dimensionless stress intensity factor vs dimensionless crack speed, for plane stress and shear-free boundary.
91
Dynamic crack propagation in a viscoelastic strip
antiplane dynamic crack propagation in a viscoelastic strip. Qualitatively, all plots are similar and the quantitative differences for the most part can be considered small. Consequently, the discussion will focus upon the plane strain, clamped strip with similar conclusions holding for the other cases. As the parameter zcz, ,/h -+ co it can be shown that N(s, 0) + 1 and it follows that N, (0,O) + 1. It is clear that the numerical results for rcz, ,/h = lo3 are in excellent agreement with this limiting case. For zc2, ,Jh ti 1 such that N, (0,O) w 1, the results, or equivalently (3.3), suggest that K,, can be viewed as the stress intensity factor of an elastic strip with elastic moduli equal to the short-time moduli and with a pressure acting on the crack faces associated with the fixed edge displacement and the longtime moduli. Alternatively, for fixed-grip loading and steady-state crack propagation the stress far ahead of the crack tip is established by the long-time moduli whereas the unloading fracture process at the crack tip is governed by the short-time moduli. A similar interpretation was given by POPELARand KANNINEN(1979) of an analysis of rapid fracture and crack arrest in a viscoelastic double cantilever beam fracture specimen. For vanishing small values of the parameter m/h, ATKINSONand COLEMAN(1977) used matched asymptotic expansions to predict K,, for the clamped viscoelastic strip. They found K,, to be the equivalent elastic stress intensity factor with the elastic moduli equal to the long-time moduli: that is, in this instance,
pm Ke -.--= tie K,
481.0Dz.o -(1+/%,o)2 4Fl,‘w&m -(I +&Y
&J(‘-&) Bl,O(I -s:,o,
zc,,,,h<
1
.
(42) .
Extending this asymptotic analysis to the strip with shear-free boundaries one obtains flu, Ke -._= PO K,
*48:,,-(I+&J2 481,OP~,O-(1+8:,o)2 -(l +a;,m)2 48:*0-(1 +8:.o)2 481,J&,,
J&J
j
B1.m ’ zc2. Jh
G 1. (4.3)
These asymptotic expressions appear as the dashed curves in Figs 2 through 5. The present results for rc2, ,/h = 10V3 are in excellent agreement with the asymptotic predictions. Even for larger values of zc,,,/h th e asymptotic predictions provide reasonably good estimates at the slower crack speeds. It is noteworthy that the limiting cases of Tc2,Jh + co and zc2, Jh + 0 are independent of the specific material model. Therefore, the same general behavior is to be expected for other viscoelastic material models. Until the crack speed approaches the long-time Rayleigh wave speed (equal to the crack speed at which the dashed curve intersects the abscissa) several decades of change in rc,,,/h p ro d uce only small differences in K,,. On the other hand, significantly smaller variations of the difference between the short-time and long-time moduli can produce substantial changes in K,,. Elastic analyses have been used to deduce the dynamic fracture toughnesses of strain-rate dependent materials (see, for example, PAX,WNand LUCAS,1973). The present results would seem to indicate that, depending upon the material and geometric parameters of the strip, such an interpretation could be a source of error. For crack speeds at which bifurcation would be likely (u/c2, m w 0.4) this would lead
C. H. POPELAR and C. ATKINSON
92
to an error of approximately 5-10 per cent in the fracture toughness which might be acceptable. However, if the short-time moduli were used for the elastic moduli, then a further overestimate of the toughness by an amount of p,/pO would occur. For PMMA this error may be as large as 40 per cent. This analysis has assumed the fracture of the material KNAUSS (1970a),
for example,
the quasi-static crack growth observed in shortcoming can be overcome by introducing equivalent
to a Dugdale
occurs
as if it were brittle.
has noted that a brittle model is incapable
zone and requiring
of describing
many viscoelastic materials. This at the crack tip a process region
the singularity
at the crack tip to vanish.
In this regard the present work may be viewed as the first step in the solution problem which will be addressed elsewhere.
of such a
ACKNOWLEDGEMENT The authors wish to extend their appreciation to the Instruction and Research Computer Center, The Ohio State University (Columbus) for the use of computing facilities.
REFERENCES ATKINSON,C. ATKINSON,C. and COLEMAN,C. J. ATKINSON,C. and ESHELBY, J. D. ATKINSON,C. and LIST, R. D. ATKINSON,C. and POPELAR,C. H. BERGKVIST, H KNAUSS, W. G. KOSTROV, B. V. and NIKITIN, L. V. NILSSON, F. NOBLE. B.
PAXSON, T. L.and LUCAS, R. A.
1979 1977
Arch. Mech. (Arch. Mech. Stos.) 31. J. Inst. Maths Applies 20, 85.
1968
Int. J. Fract. Mech. 4, 1.
1972
ht. J. Engng Sci. 10, 309.
1979
J. Mech. Phys.
1973 1970a 1970b 1970
J. Mech. Phys. Solids 21, 229. Int. J. Fract. Mech. 6, 7. Ibid. 6, 183. Arch. Mech. (Arch. Mech. Stos.) 22, 749.
1972 1958
Int. J. Fract. Mech. 8, 403. Methods based on the Wiener-Hopf Technique for the Solution of Partial Differential Equations. Pergamon Press, London. Dynamic Crack Propagation (Proceedings of an International Conference, Lehigh University,
1973
Solids 27, 431.
l&12 July 1972), (edited by G. C. Sib), p. 415. Noordhoff, Leyden. POPELAR,C. H. and KANNINEN,M. F.
1980
RICE, J. R. WILLIS, J. R.
1967 1967
Crack Arrest Methodology and Applications (Papers presented at Symposium, Philadelphia, 6-7 November 1978), (edited by G. T. Hahn and M. F. Kanninen). ASTM STP 711. Trans ASME 89, Ser. E, J. appl. Mech. 34, 248. J. Mech. Phys. Solids 15, 229.
93
Dynamic crack propagation in a viscoelastic strip
APPENDIX The equivalent
elastic problem
If the stress intensity factor for the elastic strip with shear-free boundaries substituted into the Freund-Nilsson relation G
=
fal(l
(3.6ii) were
-m ’
~/JR where the subscript co has been dropped, then G
16(~+~)~do
=
(A+ 2/l)2
h
i&(1 -a3
(A.21
[4g - (1 - p:)*]
which depends upon crack speed. On the other hand, if one were to use an argument similar to RICE’S(1967) for the static loading of the strip and BERGKVIST’S (1973) for steady-state crack propagation in a clamped strip, namely, the energy release rate is equal to the strain energy stored in a unit-width element far ahead of the crack tip, then a speed-independent G would follow. The source of this apparent paradox will be examined. For steady-state crack propagation, G for a linear elastic material is (ATKINSONand ESHELBY,1968)
1
au.au. n,-pj-nj au. au. dL ,7ij-+pfi2-axj ax,ax, ax, which is path-independent. The components of the unit normal For the path indicated in Fig. 1 there is no contribution to because n, = g12 = 0 there. Thus, the only contributions are By exploiting the asymptotic properties of the Fourier particular that as xi -+ co, a,1
=
,3,*
=
(A.3)
to the path L are denoted by ni. G from the lateral boundaries from xi = f co. transform it can be shown in
au,jax, = au21ax, = 0,
u+d
u2o
fJ22=4Pg7y4h1
84 ax,=---‘-
L 1+2/l
(A.4)
u20
h’
and as x, +-co, 6,2
=
a22
=
au21ax, = 0, ng:(l+~:)-(n+2~)(1+8~)+4~8:
u20 h’
(A.51
The above are the total stresses and displacement gradients. When (A.4) and (AS) are introduced into (A.3), then (A.2) is obtained. It is now apparent that the crack-speed dependency of G for the shear-free boundary is due to nonzero &il at xi = + cc. By contrast, ei 1 -+ 0 as x, -+ + cc for the clamped strip. It is worth noting that at xi = - cc there is a u, 1 which vanishes, as equilibrium demands, for v = 0 only. It would appear that for dynamic crack propagation the shear-free-boundary strip would not make a very good test specimen because it would be experimentally very difficult to provide the appropriate stress o 1, at xi = - cc as dictated by Nilsson’s analysis. It goes without saying that the equivalent viscoelastic strip suffers from the same difficulties.
DYNAMIC
CRACK PROPAGATION V~SCOELASTIC STRIP C.
IN A
H. POPELAR
Department of Engineering Mechanics, The Ohio State University, Columbus, OH 43210, U.S.A.
and C. ATKINSON Department of Mathematics, Imperial College of Science and Technology, London SW7 ZAZ, England
(Received 17 September 1979)
ABSTRACT THE DYNAMICPROPAGATION of a semi-infinite crack in a finite linear viscoelastic strip subjected to Mode I loading is investigated. Through the use of integral transforms the problem is reduced to solving a Wiener-Hopf equation. The asymptotic properties of the transforms are exploited to establish the stress intensity factor. Plane-stress and plane-strain stress intensity factors as a function of crack speed for both fullyclamped and shear-free lateral boundaries are presented for the standard linear viscoelastic solid. Comparisons are made with previously obtained asymptotic stress intensity factors and with stress intensity factors for the equivalent elastic strips.
1. EVEN WITH
INTRODUCTION
simpIifying assumptions, such as the fracture process being confined to a point, the number of solutions for dynamic crack propagation in viscoelastic materials at crack speeds necessitating the retention of inertia terms is small compared to the number of elastic solutions. The complexities introduced by rate effects make the mathematical analysis difficult. The first such analysis was performed by WILLIS (1967) who considered antiplane, steady-state crack propagation in an infinite viscoelastic medium. With the exception of the transient analysis by ATKINSON and LIST (1972), and the model of POPELAR and KANNINEN (1980) for crack arrest in a viscoelastic double cantilever beam fracture specimen, most of the investigations have been for steady-state crack propagation. A specimen that has been used frequently in experimental studies of quasi-static crack growth in viscoelastic materials is the strip. When unstable crack growth appears, the inertia terms become important and must be included in an analysis of the event. Assuming the parameter E = m/h (where v is the crack speed, r is the relaxation time of the medium, and h is the half-height of the strip) to be much less than unity, ATKINSONand COLEMAN(1977) used matched asymptotic expansions to 6 79
80
C. H. POPELAR and C. ATKINKIN
obtain the Mode I stress intensity factor for a steadily advancing semi-infinite crack in a clamped viscoelastic strip. The equivalent elastic problem has been solved by NILSSON (1972) using integral transforms and the Wiener-Hopf method. Atkinson and Coleman found the asymptotic stress intensity factor to be identical to Nilsson’s factor if the elastic moduli in the latter are replaced by the respective long-time relaxation moduli. Additional generalizations of these solutions can be found in ATKINSON (1979). Recently, ATKINSONand POPELAR (1979) reduced the problem of a steadily moving semi-infinite crack in a linear viscoelastic strip subjected to antiplane shear (Mode III) loading to solving a Wiener-Hopf equation. As is usually the case, the successful solution of this equation hinges upon being able to factor it. The factorization was performed by numerically evaluating a Cauchy integral. For E Q 1 the stress intensity factor was found to be in excellent agreement with its counterpart deduced from the matched asymptotic solutions of Atkinson and Coleman. However, for larger values of E the differences become important. The antiplane problem has been a favorite of the analyst venturing into new areas of fracture mechanics because compared to the in-plane problem the mathematical difficulties are minimal. It is commonplace-the analysis of Atkinson and Popelar not excluded-to assume that the crack extension in Mode III is self-similar. While similarities exist between the antiplane and in-plane problems, and while antiplane solutions provide useful qualitative information, their applicability is questionable because as KNAUSS (1970b) has noted it is physically difficult to achieve rectilinear crack growth in Mode III. As a step toward providing a physically more useful analysis, the dynamic propagation of a semi-infinite crack in a finite, linear viscoelastic strip subjected to Mode I loading is treated in the present paper. A rather general formulation of the problem culminating in a Wiener-Hopf equation is presented. The numerical approach employed by Atkinson and Popelar is used in the solution of this equation. Plane-stress and plane-strain stress intensity factors as a function of crack speed for both fully-clamped and shear-free lateral boundaries are determined for the standard linear solid. Since no restriction upon the magnitude of c: is made, these results represent a significant generalization of Atkinson and Coleman’s asymptotic solution. Because of the close analogue between this problem and the elastic one, frequent reference is made to NILSSON’S (1972) work.
2.
ANALYTICAL DEVELOPMENT
Consider the situation depicted in Fig. 1 in which a semi-infinite crack is propagating in a brittle, linear viscoelastic strip. Suppose a constant displacement uZO of the edges of the strip causes the crack to extend with a constant speed 2’relative to the fixed reference frame xi. Primary consideration is given to establishing the dependence of the viscoelastic stress intensity factor K,, upon the crack speed. This task is facilitated by introducing a moving coordinate system xi attached to the crack tip and defined by xi = x; - L& )
(2.1)
81
Dynamic crack propagation in a viscoelastic strip
_- ------_______^_----f“20 FIG. 1. Viscoelastic strip.
where t is time and 6i, is the Kronecker delta. For this piane problem there is no dependence upon x3. If aij and Ui denote, respectively, the components of the stress tensor and displacement vector, then the equations of motion in the absence of body forces are
d2Ui
aaij
8X;
(2.2)
-P-p
where p is the density of the medium and the usual summation convention is implied. The constitutive relations for a homogeneous, isotropic viscoelastic medium can be written as de+)
sij= 2 j G,(t-T)~d~, --a)
okk =
(2.3)
j G2(t-r)yh, -CC
where sij and eii are the deviatoric stress and strain components sij
=
eij
=
I
-+CkkSij, Eij-+&kk&fjr aij
(2.4)
and G, and G, are relaxation functions. The components of the strain tensor are
(2.5) On the upper and lower boundaries of the strip, x2 = +_h, assume that a2 = 3-u&W)
(2.4)
and either (i) ai = 0
or
(ii) er2 = 0,
(2.7) where H(t) is the Heaviside unit-step function. The former condition {i) describes a clamped boundary and the fatter (ii) a shear-free boundary, and this designation is used in the sequel. It is convenient to make use of the symmetry with respect to the
82
C. H. POPELAR and C. ATKINSON
plane x2 = 0 and superpose the solution satisfying (2.2) through (2.7) for the untracked strip. Consequently, the boundary conditions (2.6) and (2.7) for the residual problem assume the form on x2 = h, u2 = 0
(2.8)
and either (i) u1 = 0
or
(ii) cl2 = 0,
lx11 < 00; (2.9)
and on x2 = 0, G,Z(XI, 0, r) = 0,
(2.10)
Ix11< @J,
(2.11) -cc
o
(2.12)
The functions X(x,, t) and A(x,, t) are respectively the unknown traction ahead of the crack tip and the unknown crack opening displacement, and remain to be determined ; whereas a(~,, t) is the normal stress on x2 = 0 obtained from the solution of the untracked strip. Finally, the dilatational wave potential 4 and shear wave potential rj are introduced through
w
u1=ax,+ax,,
a*
a+ a*
UZ=3g--.
(2.13)
With the Laplace transform over t and the Fourier transform over x1 defined, respectively, by f(xr, x2, P) = 7 e-pfS(xI, x2, r) dt,
Re [PI > 0,
0
(2.14)
T(f(s,x2, P) = i eisx13(x1, x2, p) dx,, -a, the transformed counterparts
of (2.2) and (2.3) are d~i2 __-
dxz
isSi, = p(p+i~s)~fi~,
Sij = 2~L2,,
~~~= (3~+ 2~L)~kk,
(2.15) (2.16)
where p = (p + ius)G,(p + ius),
(2.17)
3J.+ 2p = (p + ivs)G,(p + ius).? I Equations (2.16) have the same form as the transformed elastic constitutive relations except that the Lame parameters 1 and p here are complex functions of p+ius. The 7 To change this plane-strain formulation to a plane-stress one replace I by 24/(E. + 2~):
83
Dynamic crack propagation in a viscoelastic strip
complex longitudinal and shear wave speeds are (2.18) The transformed boundary conditions are on x2 = h, ii, = 0
(2.19) and either
(i) ii, = 0
or
(ii) Zi2 = 0; 1
and on x2 = 0, 5 12 -0, _ 622
=
-~+~+(s),
ii,
=
L(s),
(2.20) 1
where
C+(s)= i
eisxl
E(x,, p) dx,,
0
(2.21)
j eisxlA(xl,p)dxl.
Z_(s)=
-a!
The behavior of the stresses and displacements as lx11+ cc are assumed to be such that E+(S) and k(s) are regular in overlapping half-planes of the complex s-plane. Finally, (2.13) is transformed to yield
. ;,=-- d$ 4, dx,
ii, = d6 dx
+
is?.
(2.22)
2
After some algebraic manipulations
the equations of motion assume the form
(2.23)
where yf =
s2+ (p + ius)2/c:,
(2.24)
7; = s2 + (p + ivs)2/c$. 1 Then, with Cl2 = p
(s2+#-2isz
2
[
E22 = P (s2+y:)J+2isz
1 1
d? , d?
(2.25)
,
2
the equations for solution of the boundary-value problem are complete. It is a straightforward matter, albeit lengthy, to determine 4 and 3 that satisfy
C. H. POPELARand C. ATKINSON
84
(2.19) through (2.25). Having done so, one can ultimately write (2.26)
522(O) = -K(s, P)&(O), where, for the clamped boundary (i), K(s, P) = YI(s:_Y;) {4s2Y,Y&2+Y:)-Yy1Y2[4~4+(s2+Y:)2]
cash (Y,h)cosh
[4s2Y:y: + s2(s2 +~f)~] sinh (Yrh) sinh (y,h)} x {Y1y2sinh (y,h)cosh (Y2h)-s2 sinh (Y,h)cosh (yrh)}-‘;
(y,h)
+
(2.27)
whereas, for the shear-free boundary (ii), K&p)= The combination
[4s2y, Y2coth (y,h)P Y1(S2-Yy22)
(s2 +Y;)~ coth (yr h)].
of (2.20) and (2.26) yields the Wiener-Hopf -Z+C+(s)
(2.28)
equation
= -K(s,p)Z_(s).
(2.29)
The solution of (2.29) depends upon being able to determine the factors K, and K _ each of which is analytic and different from zero in its respective half-plane such that (2.30)
K(s, P) = K+(s, P)K-(s, P).
To facilitate the factorization of K(s, p) it is normalized with respect to its asymptotic value as JsJ--) cc to yield
K(s,P) .ABI cc
N(s, p) = ___
K,(s)
pm
(1-B:. co)?
(2.31)
where p: = (l-?/c;)
(2.32)
and Pi, m = lim
Pi,
ISI-+ m
k
(2.33)
= ,$Fr pL,
in which p, is the short-time shear modulus. For boundary conditions (i), G(s) = s{4BI,mB2,m(l +B:,a:)-B~,mB2,mC4+(l+P:,m)21 x cash M&.mh)cash ($2, doh)+r4B:,,P:,,+(1+8:,,)21 x sinh (s/?r, ,h) sinh (sj12,mh)} sinh (sfil,,h)cosh ($,,,h)-sinh (s/&h) cash (sB~,~~)}-‘, XV 1.00 B2.a;
(2.34)
and for boundary conditions (ii), K,(s) = sC4B1,mBz,m coth (sB2,,h)-(l+&m)2
coth (@l,,h)l.
(2.35)
Except for the sign, K,(s) is identical with NiLssoN’s (1972, equation (33)) expression for the equivalent elastic problem if the elastic moduli in the latter are replaced by the short-time moduli.
85
Dynamic crack propagation in a viscoelastic strip
With N(s, p) so defined, N(s, p) + 1 as IsI+ c;o in a strip a < Im [s] < b in the complex s-plane. If, in addition, N(s, p) is regular and nonzero in this strip, then (for example, see NOBLE (1958, p. 16)) Cauchy’s integral theorem can be used to factor N(s, p) as N = N, N _ with
N+b,p)=ew (2.36) N-(s,p)=ew
where a < c < Im [s] < d < b and the branch of the logarithm is chosen so that In N + 0 when N(i, p) + 1 with I[[ -+ cc in the strip. If, for example, N(s, p) is analytic only on Im [s] = 0, the limits c + O- and d + O+ are taken and the contour is indented accordingly at the origin, i.e. .
N,(s, P) = exp
(2.37)
The factors of K(s, p) can be written as
K+(s, P) = Ko+(sP’+(s, P), Ko-(SW(s> P)P,,
K-(syp)=&&,Jl-&) The factorization particular that
.
(2.38)
of K,(s) has been performed by NILSSON (1972) who found in &+ (0) = Ci~o(W+, K,+(s) = K,_(s) _ [R,s]+
(2.39) as IsI-+ co,
where the Rayleigh factor & = 4P,,,82,~-(1+B:,a:)2.
(2.40)
The short-time Rayleigh wave speed cR,% satisfies R, = 0. The Wiener-Hopf method of solution of (2.29) requires it be split such that each side of the equality is analytic in its respective half-plane and has a common overlapping strip. In order to achieve explicitly this separation the expression for 5 is required. For boundary conditions (i) it assumes the form =
o’=-
F(P)
isp and the split considerably Nevertheless propagation.
(2.41)
is readily made. However, for boundary conditions (ii), the expression is more complicated and a straightforward separation is precluded. (2.41) does hold in the limit as t + a~, i.e. for steady-state crack Mindful of this latter limitation, (2.30) and (2.39) can be used to rewrite
C. H. POPELARand C. ATKINFWN
86
(2.29) as
= -K_(s,
p)L(s,
F(P)
p) +
Y---p.
iVK +(0)
(2.42)
The extended form of Liouville’s theorem yields J(s) = 0 which, when (2.38) is employed, implies
>
‘Y_(s, p)
= ~----
F(P)Pl,m(l4,,)
(2.43)
,
w~,&,+(OW+(O, P)&-@N-h
P)
With the aid of (2.38) and (2.39) the asymptotic forms of (2.43) become
The Tauberian theorem (see NOBLE, 1958) for the asymptotic transforms further permit writing X = R,,(p)(2nx,)-*,
inverses of Fourier
xi --+o+,
(2.45) (2.46)
t
where
(2.47)
is the Laplace transform of the viscoelastic stress intensity factor. It is apparent that the problem of determining the viscoelastic stress intensity factor has been reduced to one of evaluating N, (0, p) and taking the inverse Laplace transform of (2.47). In general this is an extremely complicated task, but one which is feasible for steady-state crack propagation.
3.
STEADY-STATE PROPAGATION
In the sequel it will be assumed that the crack propagation has been occurring for a suffi~ently long period (t -+ co) so that a condition of steady-state crack extension has been attained. The well-known asymptotic relation between a function as t -+ cg and its Laplace transform as p -+ 0 is applied to (2.45) through (2.46) to obtain c = z&(27&-*.
x1 -+o+,
(3.1)
87
Dynamic crack propagation in a viscoelastic strip
(3.2) (3.3)
for steady-state crack propagation.
61,
For boundary conditions
F(0) = (;I+ 2&J y, >
and (ii),
(3.4)
F(O) = 4u,(a)y,j
where PO = cl(O),
10 = A(O)
(3.5)
are the long-time Lame parameters. From NILSSON’S (1972) analysis the stress intensity factors for the equivalent elastic problems based upon the short-time moduli would be
(3.6)
The combination
of (3.3), (3.4) and (3.6) yields
&0+&*
0)
L
1,+2/&J .K,
1 = N+(O,O)’ >
.&+A. (ii) k PO Ao+po
20 + 2~0
Ke
1,+2c1’x .K,
1
(3.7)
= N.&JO)
If, as is frequently done, Poisson’s ratio v is taken to be a real constant, then (3.7) assumes for both sets of boundary conditions the simple form CL, L ----_ PO K,
This expression for steady-state, Whether or N+(O, 0) which
1 N+ (090)’
(3.8)
is identical in form to that obtained by ATKINSONand POPELAR (1979) antiplane, dynamic crack propagation in a viscoelastic strip. not v is a real constant, the determination of K,, reduces to evaluating according to (2.37) becomes N + (0,0) = r exp
(3.9)
when the oddness of arg N([, 0) is employed. The method of residues provides the
88
contribution
C. H. POPELAR and C. ATKMON
r to N+(O, 0) due to the indented contour at 5 = 0. The values are
6) (ii)
Jo+2clo +
r=
(- ) &-t-2/1,
’
y _ j10 J1.m 4P:,0-(l+P;JJ2 gLI*i P 1.0 [ 48:,,-fI+pf,X))2
(3.10) t 1 ’ I
whereiLo = &(Ol and 82,0 = j&(O). While the complexity of arg N([, 0) prevents the formal integration in (3.9), the integral can be evaluated numerically, say, by Simpson’s rule and supplemented by asymptotic values near the origin and at infinity. Based upon an argument of the local work at the crack tip, the energy release rate G is (3.11) which yields for (3.1) and (3.2), (3.12) as KOSTBOV and NIKITIN (1970) have predicted. In general, N + (0,O) depends upon the crack speed and therefore so will the energy release rate. For the equivalent elastic problem, the energy release rate of the strip with clamped boundaries is independent of the crack speed as it is also for the antiplane loading. At first appearance it would seem that this should also be true for the shear-free boundary, but this is not the case. The origin of the velocity terms in this elastic problem and its implications are discussed in the Appendix.
4.
NUMERICAL RESULTS AND DISCUSSION
Whereas the previous development has been for arbitrary viscoelastic materials, the numerical evaluation of N + (0,O) necessitates that a specific material model be considered in order to obtain quantitative results. One restriction implicit in the development is that the material must exhibit nonzero, finite long- and short-time mod&i. In the following a three-parameter, linear viscoelastic solid with relaxation moduli G,(r)=~o~l+~erp(-~t)].
G,(t) =
2(1-t-v)
-m
G,(t),
(4.1)
K0 = I”*(1 +f) is selected as a material model. While it is not necessary, a consrant Poisson’s ratio (v = 0.3) has been assumed for the sake of convenience.
Dynamic crack propagation in a viscoelastic strip I
I
I
89
/
f =0.25, rc2 ,/h: Y
0.2
0
06
04
0.8
I.(I
FIG. 2. Dimensionless stress intensity factor vs dimensionless crack speed, for plane strain and clamped boundary.
For the relaxation moduli (4.1) the integral of (3.9) was numerically evaluated using Simpson’s rule. Because K, vanishes at the Rayleigh wave velocity CR,m = 093c,, m, computations were not performed for crack speeds in excess of this value. A plot of dimensionless stress intensity factor against crack speed normalized with respect to the short-time shear wave speed c 2, m appears in Fig. 2. In this case the
1 7
0.8 -
06 -
0
02
0.4 v’c,,
06 00
08
I IO
FIG. 3. Dimensionless stress intensity factor vs dimensionless crack speed, for plane strain and shear-free boundary.
C. H. POPELARand C. ATKINSON
90
I
, f
q
I
0.25,Tc&h= J
06
-
0.2
04
06
IC
08
FIG. 4. Dimensionless stress intensity factor vs dimensionless crack speed, for plane stress and clamped boundary.
strip is assumed to be in a state of plane strain and to have clamped boundaries. Comparable results for a strip with shear-free boundaries are shown in Fig. 3. Companion plots for conditions of plane stress appear in Figs 4 and 5. These curves are very reminiscent of the results that ATKINSON and POPELAR (1979) obtained for
f
&h=
=0.25,rcp ’
J
IC
04
02
C
'0
02
04
06
00
FIG.5. Dimensionless stress intensity factor vs dimensionless crack speed, for plane stress and shear-free boundary.
91
Dynamic crack propagation in a viscoelastic strip
antiplane dynamic crack propagation in a viscoelastic strip. Qualitatively, all plots are similar and the quantitative differences for the most part can be considered small. Consequently, the discussion will focus upon the plane strain, clamped strip with similar conclusions holding for the other cases. As the parameter zcz, ,/h -+ co it can be shown that N(s, 0) + 1 and it follows that N, (0,O) + 1. It is clear that the numerical results for rcz, ,/h = lo3 are in excellent agreement with this limiting case. For zc2, ,Jh ti 1 such that N, (0,O) w 1, the results, or equivalently (3.3), suggest that K,, can be viewed as the stress intensity factor of an elastic strip with elastic moduli equal to the short-time moduli and with a pressure acting on the crack faces associated with the fixed edge displacement and the longtime moduli. Alternatively, for fixed-grip loading and steady-state crack propagation the stress far ahead of the crack tip is established by the long-time moduli whereas the unloading fracture process at the crack tip is governed by the short-time moduli. A similar interpretation was given by POPELARand KANNINEN(1979) of an analysis of rapid fracture and crack arrest in a viscoelastic double cantilever beam fracture specimen. For vanishing small values of the parameter m/h, ATKINSONand COLEMAN(1977) used matched asymptotic expansions to predict K,, for the clamped viscoelastic strip. They found K,, to be the equivalent elastic stress intensity factor with the elastic moduli equal to the long-time moduli: that is, in this instance,
pm Ke -.--= tie K,
481.0Dz.o -(1+/%,o)2 4Fl,‘w&m -(I +&Y
&J(‘-&) Bl,O(I -s:,o,
zc,,,,h<
1
.
(42) .
Extending this asymptotic analysis to the strip with shear-free boundaries one obtains flu, Ke -._= PO K,
*48:,,-(I+&J2 481,OP~,O-(1+8:,o)2 -(l +a;,m)2 48:*0-(1 +8:.o)2 481,J&,,
J&J
j
B1.m ’ zc2. Jh
G 1. (4.3)
These asymptotic expressions appear as the dashed curves in Figs 2 through 5. The present results for rc2, ,/h = 10V3 are in excellent agreement with the asymptotic predictions. Even for larger values of zc,,,/h th e asymptotic predictions provide reasonably good estimates at the slower crack speeds. It is noteworthy that the limiting cases of Tc2,Jh + co and zc2, Jh + 0 are independent of the specific material model. Therefore, the same general behavior is to be expected for other viscoelastic material models. Until the crack speed approaches the long-time Rayleigh wave speed (equal to the crack speed at which the dashed curve intersects the abscissa) several decades of change in rc,,,/h p ro d uce only small differences in K,,. On the other hand, significantly smaller variations of the difference between the short-time and long-time moduli can produce substantial changes in K,,. Elastic analyses have been used to deduce the dynamic fracture toughnesses of strain-rate dependent materials (see, for example, PAX,WNand LUCAS,1973). The present results would seem to indicate that, depending upon the material and geometric parameters of the strip, such an interpretation could be a source of error. For crack speeds at which bifurcation would be likely (u/c2, m w 0.4) this would lead
C. H. POPELAR and C. ATKINSON
92
to an error of approximately 5-10 per cent in the fracture toughness which might be acceptable. However, if the short-time moduli were used for the elastic moduli, then a further overestimate of the toughness by an amount of p,/pO would occur. For PMMA this error may be as large as 40 per cent. This analysis has assumed the fracture of the material KNAUSS (1970a),
for example,
the quasi-static crack growth observed in shortcoming can be overcome by introducing equivalent
to a Dugdale
occurs
as if it were brittle.
has noted that a brittle model is incapable
zone and requiring
of describing
many viscoelastic materials. This at the crack tip a process region
the singularity
at the crack tip to vanish.
In this regard the present work may be viewed as the first step in the solution problem which will be addressed elsewhere.
of such a
ACKNOWLEDGEMENT The authors wish to extend their appreciation to the Instruction and Research Computer Center, The Ohio State University (Columbus) for the use of computing facilities.
REFERENCES ATKINSON,C. ATKINSON,C. and COLEMAN,C. J. ATKINSON,C. and ESHELBY, J. D. ATKINSON,C. and LIST, R. D. ATKINSON,C. and POPELAR,C. H. BERGKVIST, H KNAUSS, W. G. KOSTROV, B. V. and NIKITIN, L. V. NILSSON, F. NOBLE. B.
PAXSON, T. L.and LUCAS, R. A.
1979 1977
Arch. Mech. (Arch. Mech. Stos.) 31. J. Inst. Maths Applies 20, 85.
1968
Int. J. Fract. Mech. 4, 1.
1972
ht. J. Engng Sci. 10, 309.
1979
J. Mech. Phys.
1973 1970a 1970b 1970
J. Mech. Phys. Solids 21, 229. Int. J. Fract. Mech. 6, 7. Ibid. 6, 183. Arch. Mech. (Arch. Mech. Stos.) 22, 749.
1972 1958
Int. J. Fract. Mech. 8, 403. Methods based on the Wiener-Hopf Technique for the Solution of Partial Differential Equations. Pergamon Press, London. Dynamic Crack Propagation (Proceedings of an International Conference, Lehigh University,
1973
Solids 27, 431.
l&12 July 1972), (edited by G. C. Sib), p. 415. Noordhoff, Leyden. POPELAR,C. H. and KANNINEN,M. F.
1980
RICE, J. R. WILLIS, J. R.
1967 1967
Crack Arrest Methodology and Applications (Papers presented at Symposium, Philadelphia, 6-7 November 1978), (edited by G. T. Hahn and M. F. Kanninen). ASTM STP 711. Trans ASME 89, Ser. E, J. appl. Mech. 34, 248. J. Mech. Phys. Solids 15, 229.
93
Dynamic crack propagation in a viscoelastic strip
APPENDIX The equivalent
elastic problem
If the stress intensity factor for the elastic strip with shear-free boundaries substituted into the Freund-Nilsson relation G
=
fal(l
(3.6ii) were
-m ’
~/JR where the subscript co has been dropped, then G
16(~+~)~do
=
(A+ 2/l)2
h
i&(1 -a3
(A.21
[4g - (1 - p:)*]
which depends upon crack speed. On the other hand, if one were to use an argument similar to RICE’S(1967) for the static loading of the strip and BERGKVIST’S (1973) for steady-state crack propagation in a clamped strip, namely, the energy release rate is equal to the strain energy stored in a unit-width element far ahead of the crack tip, then a speed-independent G would follow. The source of this apparent paradox will be examined. For steady-state crack propagation, G for a linear elastic material is (ATKINSONand ESHELBY,1968)
1
au.au. n,-pj-nj au. au. dL ,7ij-+pfi2-axj ax,ax, ax, which is path-independent. The components of the unit normal For the path indicated in Fig. 1 there is no contribution to because n, = g12 = 0 there. Thus, the only contributions are By exploiting the asymptotic properties of the Fourier particular that as xi -+ co, a,1
=
,3,*
=
(A.3)
to the path L are denoted by ni. G from the lateral boundaries from xi = f co. transform it can be shown in
au,jax, = au21ax, = 0,
u+d
u2o
fJ22=4Pg7y4h1
84 ax,=---‘-
L 1+2/l
(A.4)
u20
h’
and as x, +-co, 6,2
=
a22
=
au21ax, = 0, ng:(l+~:)-(n+2~)(1+8~)+4~8:
u20 h’
(A.51
The above are the total stresses and displacement gradients. When (A.4) and (AS) are introduced into (A.3), then (A.2) is obtained. It is now apparent that the crack-speed dependency of G for the shear-free boundary is due to nonzero &il at xi = + cc. By contrast, ei 1 -+ 0 as x, -+ + cc for the clamped strip. It is worth noting that at xi = - cc there is a u, 1 which vanishes, as equilibrium demands, for v = 0 only. It would appear that for dynamic crack propagation the shear-free-boundary strip would not make a very good test specimen because it would be experimentally very difficult to provide the appropriate stress o 1, at xi = - cc as dictated by Nilsson’s analysis. It goes without saying that the equivalent viscoelastic strip suffers from the same difficulties.