A moving crack problem in a viscoelastic solid

Int. J. Engng Sci., 1972, Vol. 10. pp. 309-322.

Pergamon Press.

Printed

inGreatBritain

A MOVING CRACK PROBLEM VISCOELASTIC SOLID C. ATKlNSONt

IN A

and R. D. LISTS

Department of the Theory of Materials, The University of Sheffield, Sheffield, England (Communicated by I. N. SNEDDON)

Abstract-A model is considered where a semi-infinite crack begins to grow with a uniform velocity at time t = 0 in a linear viscoelastic solid. The time variation of the stress-intensity of crack velocity and material properties for two simple viscoelastic solids.

factor is evaluated as a function

1. INTRODUCTION

A MODEL is considered which assumes that at time r = 0 a semi-infinite crack suddenly appears in an infinite viscoelastic body and the tip of the crack starts to move with a constant velocity I/, under the action of a constant loading along the faces of the crack (since the medium is linear this is equivalent to a uniform stress at infinity). The ‘standard linear solid’ (see (2.5) for its constitutive equation) is taken as the viscoelastic model and only states of anti-plane strain (mode III deformation) are considered. It is shown that by the use of Fourier and Laplace transforms the problem can be reduced to a form suitable for the application of the Weiner-Hopf technique. While the stress intensity factor for a Maxwell solid (a = 0) is readily obtained, algebraic difficulties arise for the standard linear solid. However, it is shown that by a modif%zation of the constitutive equation of the standard linear solid, as suggested by Achenbach and Chao [ 11these difficulties can be overcome. The variation of the stress intensity factor with time is shown in Figs. 2, 3 and 4 in section 5. It is found that for the standard linear solid (Achenbach-Chao model) if V/c, where Y is the velocity of the crack tip and c is the shear wave velocity, is less than the ratio a//3 of the viscoelastic constants (see equation (4.1)) the behaviour of the stress intensity factor for large times is similar to that of an elastic solid. However, if V/c > ry/p the stress intensity factor is bounded for large times and the behaviour of the model is similar to that of a Maxwell solid. Recently, Willis[21 has discussed the steady state motion of a semi-infinite crack in a linear viscoelastic solid. In order to ensure a finite result he considered a particular kind of loading on the crack faces which decreased with the distance behind the crack tip. Taking the difference in loading into account it can be shown that the asymptotic results of the model considered here for large times agree qualitatively with the results of [2]. The model we consider here can be thought of as the anti-plane strain version of the viscoelastic analogue of the model of Baker [3]. the application of the methods outlined here to the plane strain version shouldn’t present any major difficulties. Furthermore, one would expect the small and large time behaviour to have similar characteristics to those obtained in this paper. tNow at the Department of Mathematics, Imperial College, London. SOn leave from Department of Mathematics, The University of Western Australia.

309

310

and R. D. LIST

C. ATKINSON

~-*,

Fig. 1. The crack geometry. 2. ANALYSIS

The basic equations to be satisfied in this problem are as follows d2Ui

atTij

-=p-

ax,,

Lj=

at2

1.2,3

(2.1)

(2.2) where oj, ui and Lji give the stress, displacement and strain fields respectively, the density. The boundary conditions to be satisfied are x,-VVt < 0

g23(x,, 0. t) = -T (a constant) UQ(X,:0, t) = 0

x1 -

CF~j(X~*yjat)+O

I/t > 0

t>O

asx:+y:

+ Go.

Taking a ‘standard linear solid’ as the viscoelastic the shear strains can be written as

t>O

model, the constitutive

and p is

(2.3)

(2.4

equation for

where CY,/? are positive constants. For the case of the antiplane strain

and the set of three equations in (2.1) reduces to the single equation aal -+

acr,,= aQ3

ax,

a1

PT’

It is convenient to shift the origin of co-ordinates Fig. 1, by the transformation x=x1--Vt

(2.6)

to the tip of the crack at x, = Vt,

Y = Yl.

(2.7)

A moving crack problem in a viscoelastic solid

311

The stresses o13(x1. yl, t) , uz3 (x,, yl, t) and displacement u3 (x1, yl, r) now become f13(x, y, t), rz3(x, y, t) and w(x, y, 1) respectively. Further the strains &(x1, yl, t) = 4 (W@,) and &(x1, yl, t) = 4 (I%/@~) become eIS(x, Y. t) = 4 (&v/ax) and e&x, y, t) = 4 (awlay) respectively since a/ax, = a/ax and a/ay, = d/ay. The time derivative occurring in (2.5) and (2.6) transforms to

Thus equation (23,

for the shear stress t13,transforms to

(2.8) For fz3we have

-v~+f++&,=p

(

-v-

a% a% axay+atay

+C2

ay ) *

(2.9)

Both these equations can be reduced to ordinary differential equations by the application of the Laplace transform over t and the Fourier transform over x. Let us denote the Laplace transform by a single bar f-f(x,y,p)

= 7 e”Ef(x,y,

t) dr

0

and the Fourier transform by two bars e”zf(x, Y, P) dx. --m

Applying these transforms in order to (2.8) and (2.9) we obtain (isV_tp+j3)71,

= /_L(vS*--iSp--~iscu)~

(i.sv+p+p)~,=

p(Vis+p+a)

Returning to equation (2.6), the transformation

!?!_$+2_ andafter

(

j/z

(2.10)

$#

of co-ordinates

(2.11) gives

a% a2w axat+Z >

52v-

taking Laplace and Fourier transforms, as above, we have -iidr,,+dt,,=p dy

(-V2s2+2V~Sp+p2)~.

(2.12)

Substituting for’&, and fz3, from (2.10) and (2.1 l), gives d2Y7 dy2- y5

= 0

(2.13)

312

C. ATKINSON

and R. D. LIST

where 1 s2(vis+p+a)+~(vis+p)‘(~i~+~+~) y2 = v/is+p+CY {

. I

The general solution of (2.13) is G (s, y, p) = A (s, p) e++

B (8, p) eyv

where the branches of the functions occurring in the expression for y are defined so that Rly > 0. Since the stresses are bounded as y + m, W(X, y. t) and hence also B (s, y, p) must remain bounded as y -+ m. Hence, the arbitrary constant B (s, p) must be equated to zero giving as the solution of (2.13) V(s, y, p) = A (s. p) e+. The mixed boundary system are

conditions

(2.14)

(2.3) and (2.4) in the transformed

&(x, 0, t) = -T

t>O

(2.15)

t > 0.

(2.16)

x-co x>o

w(x, 0, t) = 0

co-ordinate

Taking the Laplace transform of (2.15) gives xc0

&3(x, 0,P) = -5 Th(x)

x>o

where h(x) is an unknown function. As it is physically reasonable to expect the disturbances ahead of the crack to decrease with distance from the crack tip, we will assume h (x) is exponentially bounded i.e. h(x) = 0 (eeklr)

k, > O,asx+

+w.

Assuming that the stress singularity is weaker than x-l, h(x) is integrable over an interval with end point at x = 0. The Fourier transform then gives

Zz,(s,0, p) = -

&+ TH+(s)

(2.17)

where H+(S) = 7 eisth (x) dx. The integral of & over (-to, 0) converges if and only if Ims = 7 e 0. Fu’rther, as h(x) is exponently bounded at infinity the integral of &, over (0, W) converges if T > -kl and vanishes for large S. Thus H+(s) is analytic and goes to zero at IsI = 00in Zms = T > -k,.

313

A moving crack problem in a viscoelastic solid

From(2.11) L(s,

Y, P) = P

(I’is+p+a)

dE

(Vis+p+/3)

dy

(Vis+p+a) CL(ViS+P+pyNw4

=-

e-yv

using the expression (2.14) for r. Combining this with (2.17) gives

(Vis+p+dyA(s,p)

CL

_ T -ips-TH+(s).

(vis+p+p)

(2.18)

Following through the same procedure with the second boundary condition (2.16) for the displacement w gives x
g+, 0, P) = ij(x)

x>o

0

where j(x) is an unknown function. As the displacement is bounded for large x, we will assume j (x) exponentially bounded and j (x) = 0 (ektz) kz > 0 as x + - m. Taking the Fourier transform now gives 0 qs,

0,

p)

=

dx = fJ_(s).

e’““j(x)

;

(2.19)

I

-cc

As j(x) is exponentially bounded F(s, 0, p) is analytic and vanishes at infinity in the lower half plane Zms = r < k2where it is assumed k2 > -k,. Again, using the expression for Z from (2.14), (2.19) becomes A(s,p) =$s).

(2.20)

Combining equations (2.18) and (2.20) (Vis+p+a)y.Z_(s) (Vis+p+jl)

1

=&-H+(S)

(2.21)

we note that the problem has been reduced into a form suitable for the application of the Wiener-Hopf technique. 3. THE MAXWELL

SOLID

For general (Yand 8, y does not readily factorise. Expressions for the roots can be obtained but these are awkward to handle. We discuss here the case of the Maxwell solid, where (Y= 0, returning to the general linear solid in the next section. In this case y* from (2.13) reduces to

IJES Vd. 10 No. 3-G

314

C. ATKINSON

and R. D. LIST

where the quadratic has the complex factors, 1 [

2 1-s (

_ Vi(2p+P) c2

+

J

-5

(2P+P)2-$

(1 -$(p+a,]

>

one positive and one negative for V < c. These expressions 1 2i 2 l_~ _vi(2P+P) (

c2

C2 )[

simplify to (3.1)

+/~].

Hence (s+iX1)‘/2(s-iX2)112

Y=

RlX,.X,

where X, and X2 are given by (3.1) and the branches s-t-a. Thus, for a Maxwell solid, (2.21) can be written as

(S-S)

(s+iX1)112(s-iX2)1’2J_(s)

> 0

(3.2)

are chosen so that y ---* + CCas

=&-H+(S) -k, < Ims < 0.

(3.3)

Rearranging (3.3) into the form

$I$

ML ,-~ (

(s-ix2)112J_(s)

1

_

’

ips (iX,)*/2 =

1 1 H+(s) ips(s+iX,)1’2-ips(iX1)1’2-(s+iX1)1~2

-k, c Ims < 0.

(3.4)

We see that the function on the left hand side of (3.4) is analytic for Ims < 0 and the function on the right hand side of (3.4) is analytic for Ims > -k,. As they are both equal for -k, < Zms < 0 the principle of analytic continuation gives that either side represents the same entire function E(s) , say. Further, as we have previously noted that J_(s) -+ 0

ISI+

O”

lms < 0

H+(s) + 0

ISI+

O3

Ims > -k,.

Liouville’s theorem gives that E (s) = 0 and thus (3.5)

A moving crack problem in a viscoelastic solid

315

and from (2.17)

T*,(s,0,p)

= -;+

TIT+(s) =

-7

T (s+iX1)“* lps (ix,)“* ’

Inverting the transforms W+iTo

c’+i-

f23(&

0,

t)

=

$d&

-&

I

I

c’-ice

e-i8z(s+iX,)“2 s(iX1)“*

ds

-=+iTo

’

Reversing the order of integration f23(~,0, t) =

-&. j

e-‘““F(s) ds

--+ir,

where c’+im

F(s)

=&

ept(s+iX1)“* dp. ps (iX1)“* c’-_im

I

(3.6)

Expanding F (s) in powers of 1IS we have c’+im

ept dp

I

px;,2

+ow3’*)

&-ice

so that the singular part of the stress at the crack tip can be written f23(0+.

0.

t)

=

(3.7)

+= G

where 1 A=%

I c’-_c

ePt mdp.

(3.8)

In fracture mechanics, it is conventional to write fz3(O+.0, r) in the form K/V% where K is the stress intensity factor. In our case K=+fiTA.

l&

(3.9)

Substituting from (3.1) and (3.2) for X1 gives

As a quick check on the analysis,

we can obtain an expression

for the stress

316

C. ATKINSON

and R. D. LIST

intensity factor in the case of the crack, moving with constant velocity, in an elastic medium by substituting p = 0. With /3 = 0 (3.10) reduces to

=-+l$=Tji Tr

and the stress intensity factor, from (3.9), is +2vz

-TT G

which agrees with Kostrov 141. 4. STANDARD

LINEAR

SOLID

The algebraic difficulties for general viscoelastic constants o and p can be overcome by use of the three parameter viscoelastic model, suggested by Achenbach and Chao [ 11.with constitutive equation (4.1) where 01= mlT, /3 = fil T and 0 c m < 1. T is a relaxation time. Achenbach and Chao obtained creep curves for this model for values of m ranging from m = O-25to m = I.0 and found that “these curves are similar to the creep curves of the standard linear solid. In fact they practically coincide with those of the latter solid for values of m larger than O-5.” Replacing the constitutive equation for the standard linear solid (2.5) by (4.1) the previous analysis can be repeated with some important changes. Now we find

and

giving Y‘= (Vis+lp+n)’

(- (is)‘(Vis+p+u)‘+$

(Vis+p)‘(Vis+p+p)‘}.

The term in the { } readily factorises giving four roots 1

S=

21/

* )i[

(2p+P) c 1-F -l E

(

+i

@+a)

1

[32P+B)-(P+cr)]Z+3~-3dp+p)}

(4.2)

317

A moving crack problem in a viscoelastic solid

4i

;(2P+8)+(p+a)]2-~(l-;)p(p+P)

.

(4.3)

As three of these roots lie in the upper half of the s plane and one lies in the lower half s plane the Wiener-Hopf equation (3.4) is not basically changed. The analysis remains correct with minor changes and x1=

l

_v (2Ps-P) - (p+a) 2V 1-E 1 c c> ( +j/$-~

Ep(2a---p) +@I + (p+o)4}.

Thus the singular part of the stress at the crack tip in this case is (4.4) where c+im

eptdp

c+im

1[

c_*m P

:

f2p+S)

-

tP+a)

+{$-F

[p(2ru-8)

+cw/31+ (,+U)2)“2]“*’

(4.5) It can be quickly verified that this integral will reduce to the known elastic expression for (Y= p. 5. TIME

VARIATION

OF THE STRESS

INTENSITY

FACTOR

The time variation of the stress intensity factor K can be calculated from the results of sections 3 and 4 (see equations (3.10) and (4.5)). The methods used for evaluating these integrals in (3.10) and (4.5) are briefly outlined in the appendix. 5.1 The Maxwell solid From (3.9) we have K = t/rzTA where in the case of the Maxwell solid the expression for A is given by equation (3.10). In Fig. 2 we have plotted K* = (KIT) (p/c)“” as a function of t1 = pr for a range of values of V/c. The results shown in the figure were obtained by numerical evaluation of the

318

C. ATKINSON

0

1

1

I

5

IO

15

and R. D. LIST

1 20

1 25

,

1

30

35

I 40

I 45

50

'I

Fig. 2. K* vs. r1 for a Maxwell solid.

integrals in (3.10) (see the Appendix for a discussion of the methods used). However, for small and large times simple asymptotic expressions can be obtained which are useful as a check on the numerical results and give a simple picture of the form of the solution. In fact drawing the small and large time solution curves and joining them by a smooth curve for intermediate times would give a useful approximate solution for the whole time range. For small times one finds K=

7-(c-1/)‘/22

02 1’2

(5.1)

77

For large times one finds l/2

(cz-V*)“*

Vf

0.

(5.2)

These results show that the small time solution has a similar form to the corresponding elastic solution (cf. Baker[3]) but the large time solution says that K tends to a constant as t + m whereas the corresponding elastic solution says K + m like P as t + m. It is of interest to compare the result (5.2) with the results for the steady state problem obtained by Willis[2]. For the loading on the crack face he takes f23(~, 0) = - T ezlL, x < 0, where L is a characteristic length associated with the applied load. In the special case of the Maxwell solid Willis’s equation (4.7) gives, after some simple algebra (5.3) Tf we let L + m in equation (5.3) i.e. consider a constant load on the crack faces we get tne result (5.2).

A moving crack problem in a viscoelastic solid

5.2 The Achenbach-Chao

319

model

In this case we again have K = fi TA but the expression for A is given by equation (4.5). In Figs. 3 and 4 K* = (KIT) (@/c)‘~* is plotted as a fkmction of tl = Pt for a range of values of V/c for fixed CY~@. The dotted lines which represent the elastic case (cy= p) have been included for reference. “I

I

.* Y *O.l c

/’

7-

.’

,’

6-

5-

1

I

0

5

I

IO

/

I5

1

1

/

/

1

20

25

30

35

40

I

45

50

6

Fig. 3. K* vs. tl for the Achenba~h-Chao

model a/F = 0.71.

./WA/ 5

IO

_--+O.Q

__--

__&---

+oe3

j I5

__--

__.M---

_#--

0

$0.7

I

20

24

I

30

I

*

/

35

40

I

45

50

tl

Fig. 4. K* vs. t, for the Achenbach-Chao

model a/P = 0.89.

320

C. ATKINSON The

and R. D. LIST

results for small and large times are now for K = 2T(c_

small 1

1/)1/Z _f 1/z ( >

(5.4)

and for large t there are two distinct cases (a) for:<

$ K = ~c1,*(;_‘)“‘fi

(5.5)

C

and (b) for;

v

o! > P (5.6)

It is a simple matter to obtain these asymptotic formulae one simply looks at the behaviour of the integrand of A for p large to get the c + 03result, the inversion of the integral can be done using the appropriate Abelian theorem (see for example Noble [51). Some care needs to be taken for the case V/c = a//3 for this case the large time result behaves like P4 as t + 03.From the above results it can be seen that for velocities less than (a//3) c the stress intensity factor has a similar form to that of the K value of an elastic material whilst for velocities above this value it is similar to that of a Maxwell solid. REFERENCES

andC. C. CHAO,J. Mech. Phys. Solids 10,245 (1962). J. R. WILL&J. Mech. Phys. Solids 15,229 (1967). B. R. BAKER. J. oppl. Mech. 29,449 (1962). B. V. KOSTR0V.J. appl. Mark Mech. 28,793 (1964). B. NOBLE, The Weiner-Hppf Technique. Pergamon Press (1958). A. H. STROUD and D. SECREST, Gaussian Quadrature Formulas. Prentice Hall (1966).

111 J. D. ACHENBACH

i2] [3] [4] [S] [6]

(Receioed 24 April 197 1)

Appendix EVALUATION

OF THE

INTEGRAL

A

The integral in (3.10) has a simple pole atp = 0 and branch points at p = -p.

and

Taking the branch cuts along the negative real axis the integral can be considered as a contour integral around the path as shown in Fig. 5.

A moving crack problem in a viscoelastic solid

321

Fig. 5. The integration contour to evaluate A for a Maxwell solid.

This gives

(3.11) where

and a, =

y*=G/~

@a:,a2

R*

=

@a$, tl = /3f

=

(2)

+ (y*)’

Hence from (3.9) and (3.10) the stress intensity factor K=+v5q~)(~+-$(r,+l*)].

(3.12)

322

C. ATKINSON

and R. D. LIST

The real integrals in equation (3.11) can be evaluated numerically, or alternatively the complex integral given in equation (3.10) can be evaluated by some numerical method for the inversion of a Laplace transform. Both of these approaches have been used. The integral I,, of equation (3.11) was evaluated by using a repeated Simpson rule until a specified accuracy was attained, while the integral I, was evaluated by using a 15 point Gauss-Laguerre quadrature formula. The inversion of the Laplace transform of equation (3.10) was completed using a quadrature formula given in Stroud [6]. The agreement between these two approaches was quite good. As an additional check on the numerical results asymptotic properties of equation (3.10) were used. The expression for A given in equation (4.5) was evaluated by numerical inversion of the Laplace transform in a similar way.

Resume-On considere un modtle oti une cassure semi-infinie commence a croitre avec une vitesse uniforme au temps t = 0 dans un solide viscoelastique lineaire. La variation dans le temps du facteur d’intensid de la contrainte est 6valuCe en fonction de la vitesse de la cassure et des proprietts du mat&au pour deux solides viscoelastiques simples. Zusammenfassung- Es wird ein Model1 betrachtet

wo ein einseitig-unendlicher Riss mit gleichfiirmiger Geschwindigkeit-zur Zeit t = 0 in einem linearen viskoelastischen Festkorper zu wachsen beginnt. Die Zeitvariation des Spannungsintensit;itsfaktors wird als eine Funktion von Rissgeschwindigkeit und Materialeigenschaften fur zwei einfache viskoelastische Festkorper ausgewertet.

Sommario- Si considera un modello in cui un’incrinatura seminfinita comincia a crescere con una velocitl uniforme a tempo t = 0 in un solid0 viscoelastico lineare. La variazione di tempo del fattore sollecitazioneintensita B valutata come funzione della velocita dell’incrinatura e delle proprieta del materiale per due semplici solidi viscoelastici. A&X~IWTBvMeHH Tena.

PaCCMOT~Ha

t=O

BwMeHHWI

MOJWIb,

B PaBHOMePHOti 3aBHCHMOCTb

OT CXOPOCTH T,,eIQlfHbI

B KOTOpOti

CKOPOCTH

KO3&&iUIiCHTa

H MaTCpHaJIbHbIX

IIOJIy6ecKOHW4H~

ABHXteHHSI

BHYTPH

HHTeHCHBHOCTH

CTBOkTB

NIll

TpUIUiIia JIHH&HOrO

HaWHaeT

HZllIp%K~Hli5l

JtByX IIpOCTbIX

PaCTWTb

OT MOMeHTa

BI13KO-3JIaCTli’iHOI’O OUWIHBaeTCII

Bfl3KO-3JIaCTWiHbIX

TBepAOrO

KZ3X I$)‘HXUSiS

T8epnbIX

TWIOB.