Transient stress intensity factors for a cracked plane strip under anti-plane point forces

Vol. 30, No. 2, pp. 199-211. Printed in Great Britain. All rights reserved

ht. 1. Engng Sci.

0020-7225/92

1992

$5.00 + 0.00

Copyright @ 1992Pergamon Press plc

TRANSIENT STRESS INTENSITY FACTORS FOR A CRACKED PLANE STRIP UNDER ANTI-PLANE POINT FORCES M. K. KU0 Institute of Applied Mechanics, National Taiwan University, Taipei, Taiwan 106, P.R. China Abstract-The transient elastodynamic stress intensity factors of a semi-infinite crack in an elastic infinite strip of finite width are analyzed. The crack is subjected to a pair of suddenly applied anti-plane concentrated point loadings on its faces at a distance I away from the crack-tip. The crucial steps in the analysis are the direct application of integral transforms together with the Wiener-Hopf technique. The functions of exponential type, which are introduced by the fixed characteristic lengths in geometry and in loading, must be split explicitly into the form of product and sum, respectively, of regular functions in the Wiener-Hopf equation. Instead of performing the quotient splitting directly, however, the corresponding term is expanded first in a series. The solution is then constructed as a series accordingly. Each term in the solution series can be interpreted as the contribution of waves that have reflected at the strip surfaces different times. Exact expressions are obtained for the resulting mode-III stress intensity factors as functions of time. For illustration, the first three terms in the series for the stress intensity factor history are then computed. The results are exact for the time interval from initial loading until the first wave scattered at the crack tip is reflected three times at the strip surface and returns to the crack tip. Numerical results show that the maximum of the transient stress intensity factors of symmetric strips are larger than the ones for asymmetric strips. Moreover the smaller strip height is, the larger is the dynamic overshot on the stress intensity factors for the cases of the symmetric strips.

1. INTRODUCTION An infinite strip of finite width containing a semi-infinite crack has been investigated by several authors. Owing to the strip geometry, there is an inherent difficulty on performing the needed quotient splitting of a kernel function in the Wiener-Hopf equation. Most of the analyses are then either static [I, 21 or steady-state in nature [3-61 such that the complete analysis of above mentioned quotient splitting can either be performed or be eluded in some sense. Knauss [l] studied a mode-1 static crack in an isotropic linearly elastic strip, whereas the orthotropic version of [l] was treated by Georgiadis et al. [2]. The steady-state analyses [3,4] assumed that disturbances have been produced in the strip for a sufficiently long time where a steady pattern has been established in the strip with respect to an observer situated in a frame of reference rigidly attached to the moving crack tip. Under this argument Nilsson [3] obtained the steady-state stress intensify factor of a mode-1 propagating crack in an isotropic linearly elastic strip. Georgiadis [4] considered a mode-III steady-state crack propagation in an orthotropic linearly elastic strip. Atkinson and Popelar [5] formulated the transient mode-III propagating crack problem for a viscoelastic strip, and constructed the formal expression for the Laplace transform of the stress intensity factor in the form of a Cauchy type integral. They then only restricted attention to numerically approximate the Cauchy integral for the steady-state limit case. The mode-1 version of the one in [5] was addressed by Popelar and Atkinson [6]. As with the corresponding mode-III case, the integral was only studied numerically in the limiting special case of steady-state crack propagation. It is also worthwhile to mention the work by Nilson [7] in which a mode-III transient case was analyzed. Owing to the nature of boundary conditions considered, only some specific values of quotient-split functions are needed to perform the sum-splitting. For a very good review on early works of these topics we refer to Nilsson’s paper [8]. In this paper we are interested in the transient anti-plane stress intensity factors of a semi-infinite stationary crack in an elastic strip. The crack faces are suddenly loaded by a pair of concentrated anti-plane point forces which are located a distance 1 away from the crack tip. In some respects, the present problem can be regarded as an infinite strip version but a mode-III analogue of the type considered by Freund [9], where mixed mode-I-II crack in ES 30:2-F

199

200

M.

K. KU0

homogeneous isotropic unboudned media was considered. It was long believed that the Wiener-Hopf technique could not directly appiy to the problem with fixed characteristic lengths in loading, even for the crack in unbounded medium [9, lo]. Clever superpositions were then proposed by Freund [9]. In his analysis, the original problem was considered as :1 superposition of two of Lamb’s problems and a superposition problem. The superposition problem concerned an unbounded solid containing a semi-infinite crack. Ahead of the crack tip, the solid was subjected to displacement discontinuities which were equal in ~nagrlitu~ic hut opposite in sign to the relative displacements of the two half-plane surfaces arising from Lamb’s problems. Two of Lamb’s problems and the superposition problem possessed no fixed characteristic length and were then solved by the Wiener-Hopf technique. A similar superposition scheme has also been applied to various problems by Brock 111f and Lee and Freund [12]. Recently, Kuo 1131 successfully applied the Wiener-Hopf technique dtrcctly to deal w&h elastodynamic mixed mode-I-II crack problems with fixed characteristic length in loading. As an illustration of the solution procedure, the problem considered by Freund I’?‘] was reconsidered as a special case. The crucial steps in the analysis were integral transforms together with the direct application of Wiener-Hopf technique and C’agniard-de Hoop method, as if there were no characteristic length. The summary of the Wiener--~opf technique and Cagniard-de Hoop method can be found in the book by Achenbach [14]. It was observed in [13] that the characteristic length in loading introduced an exponential term in the Wiener-Hopf equation. The sum splitting of the function, which exhibited exponential behavior. was one of the most important steps in the analysis. The solution proocdurc has then been applied to a mode-III interface crack problent by Kuo and Cheng 1IS]. where the complexities of the problem are increased by the presence of anisotropic blmaterial. lt is fair to say that whilst the superposition argument proposed in [9] offers better physical interpretation to the solution. the solution procedure presented in [ 131 is much more elegant. For the strip geometry with point impact loading. the fixed characteristic lengths arise not only in loading but also in geometry. The crucial steps in the anaiysis arc again the integral transforms together with the direct application of Wiener-Hopf techmquc and Cagniard-de Hoop method. In contrast to the work of [7], the functions of exponential type. which are introduced by the fixed characteristic lengths in loading and in geometry 1 must he split of regular functions in the explicitly into the form of product and sum, respectively, Wiener-Hopf equation. Instead of performing the quotient splitting directly, however, the corresponding term is expanded first in a series. The solution is then c~~nstr~Icted in series form accordingly. Each term in the solution series can be interpreted as the contribution of waves that have reflected against the strip faces different times. The first three terms in the series for the stress intensity factor history are then computed. The numerical results are hence exact for the time interval from initial loading until the first wave diffracted at the ct-:+ck lip is reflected three times at the strip surface and returns to the crack tip. The results for anisotropic cracked strip are also discussed by a transformation of relevant coordinates and parameters into the corresponding isotropic strip as discussed in [ 15, 161. The anisotropic cracked strip is assumed to possess certain material symmetry and the plane coincides with OIW of the planes of materiai symmetry, so that the in-plane and the anti-plane motions are not coupled. It is also of intrrest to mention the work by Thau and Lu [ 171 who used the idea of successive approximation to solve the finite crack problem in unbounded medium.

2. PROBLEM

STATEMENT

Consider anti-plane deformations of a semi-in~nite stationary crack m a linear, isotropic infinite strip which is characterized by the eiastic shear modulus p, and mass density p,_ A Cartesian coordinate system (x, y) is defined in such a way that the only nonzero displacement

201

Transient stress intensity factors Y

4

a

FH (0

0

-X

c9--

Fig. 1. Loading and cracked strip geometry.

is normal to the xy-plane, the strip locates from y = -b to y = (I, and the crack lies in the line y=O, x
* d2W dt2

(1)

is the slowness of the shear wave. The relevant stress component

is

aw (2)

Oyz= “ay’

The elastic strip is initially at rest, for time t < 0. For time t 2 0, a pair of concentrated anti-plane shear forces in the z-direction of magnitudes fF are suddently acting on the crack faces y = O* at x = -1. Thus the crack-face boundary condition is o,,(x, y, t) = -FH(t)G(x

x CO,

+ I),

y=Of

where H( ) and 6( ) are the Heaviside unit-step and Dirac delta functions, respectively, conditions of zero tractions on the upper and lower faces of the strip are a, t) = 0,

--co
o,,yr(x, -b, t) = 0,

--co
oJx,

(3) The (4)

(5) When shear forces F are applied, a pattern of cylindrical wave radiates first from the points y = O*, x = -I into the strip. This pattern defines only the wave motion in the strip for t < to, to = min(sa, sb, sl). For t > to, when the waves reach the strip surfaces or the crack-tip, they are reflected or diffracted, respectively. As time goes, the waves continue to reflect at the strip surfaces and crack faces and to diffract at the crack tip. The conditions which correspond to the continuity of displacement and traction along the crack line and ahead of the crack tip are w(x, o+, t) = w(x, o-, t), fJy*(x, o+, t) = q&

o-, t),

x>o

(6)

x > 0.

(7) Let us further denote the not-yet-determined displacement and stress along the crack line and ahead of the crack-tip as w+(x, t) and 0+(x, t), respectively. We will be particularly interested in the elastodynamic mode-III stress intensity factor which is defined as K&t)

= Xl~~+[21Gx]“*u~,(x,0, t).

Notice that the fields are anti-symmetric

with respect to y = 0 only if a = b.

(8)

202

M. K. KU0

3. METHODS

OF

SOLUTION

Apply one-sided Laplace transform over t, with kernel exp(-pt), transform over X, with kernel exp(-p&), to the governing equation the transformed domain, which satisfy the conditions of traction-free (4) and (5), can be written as

S(E,Y, PI = {

A cosh[py(y- a)], B coshbvty + b)l>

and two-sided Laplace (1). General solutions in on the faces of the strip

Osy
(9)

where y = (s2 - &!2)1’2,and A and B are arbitrary functions of 6 and p. The transformed function of one-sided and two-sided Laplace transforms are denoted by superposed bar and tilde, respectively. From (3), the transformed shear stresses along the whole crack line, y = 0, are

~yzt5,0, p) = 3, - f exp(piY)

(10)

where 5, is the double integral transform of the unknown stress field, u+(x, t). Because of the anticipated circular wavefront emitting from the point of applied force and the reflected and diffracted waves, the wave fields along y = 0 are zero beyond IX- II = t/s. By virtue of the theory of Laplace transform, both 5+ and $+ are analytic and go to zero as IQ* +m in the half complex g-plane Re(Q > -s, where r?+ is the double integral transformed unknown displacement w+(x, t). Of particular interest, the one-sided Laplace transformed stress intensity factor can be obtained from B+(& 0, p) by using the Abelian theorem (see e.g. [18], p. 36) which concerns functions and their two-sided Laplace transforms as

&I(P) = 21’2Dliim (PE)"2~+t& From (9) the double integral transformed are expressed in terms of A and B as G- + G+ = =

o+ - % exp(p&) =

displacement

0, PI. and shear stress along the crack line

A cosh(pyu),

y = Of

B cosh(pyb),

y =

-ppyA

PPYB

sinh(pw),

sinhtmWl

o-

( 12)

y=o+ y=o-'

(13)

In (12) i;_(& O+, p) and a-(& O-, p) are the double integral transformed unknown crack-facedisplacements [w-(x, O+, t) and w-(x, OK, t)] of the upper and lower crack faces, respectively. Since w_ vanishes for x > 0, moreover, because of anticipated wavefronts it is zero for x < -tJs - 1, y = o*. By virtue of the thoery of Laplace transform, $_ is analytic and goes to zero as lgl+ m in the half complex g-planes Re(f)
PR-

= 8,

- $ exp(PW

(14)

where

L(E, PI

=

tanh(pya)tanh(pyb) tanh(pya) + tanh(pyb)

and fi_ 5 i;_(E, O+, p) - i;_(g, O-, p), which denotes the double integral transformed crack opening displacement. Notice that L(c$, p) in (14) depends only on the geometry of the cracked strip and the boundary conditions on the strip surfaces, while the last term in the right hand side of (14) depends on other loading conditions. By (14) the problem has been reduced into a form suitable for the application of the Wiener-Hopf technique. The solution of (14) then

203

Transient stress intensity factors

depends upon being able to write these related respectively, of regular functions.

4.

WIENER-HOPF

functions

as the product

and the sum,

TECHNIQUE

Since the function I,(& p) has infinite numbers of zeros and poles in the complex g-plane, the direct application of the formula of sum splitting [18] to the logarithm of L(& p) can then only be evaluated numerically in the complex c-plane. On the other hand, it is possible to introduce a new function which has only simple poles and zeros but no branch cuts in the E-plane. It can be then expressed as an infinite product ([18], p. 15) and the quotient splitting is immediately followed. Unfortunately, there then is some difficulty to perform the sum-splitting in this case. Moreover, the convergence of the infinite product expression is very slow, as pointed out in [l]. The numerical computation shows that the first 5000 term in a typical infinite product has at least 2% error. Instead of finding the solution by performing the quotient splitting directly to L as is discussed in [7], it is sometimes convenient to find only the early time solution. Expressing the hyperbolic tangent functions in terms of exponential functions, and using the binomial expansion to the denominator of L, the function L can then be written as L

-_ k [I_

e-2~~

_

e_Wb

+

2e--2pYfafb)

-

e--2gY@+b)

_

e--2pY(ff+a)

+

, . ,] .

06)

The idea is similar to the Bromwich expansion method (see e.g. [19], pp. 195-196) in the solution of one dimensional wave propagation problem for a bounded region. The branch cuts of y have been taken as from g-+ --03 to --s and from 5 = s to E+ 03, such that Re(y) 10 in the entire cut complex E-plane, where “Re” denotes the real part. Let us construct series for the crack opening displacement @- and the crack-line stress ii, of general forms I?_(& p) = I@?(& p) + $‘i”(& p) + I@?(& p) + - * * E+(5;p) = V(5,

p) + ii’:‘($, P) + @‘(5, p) + - * ’

(17) (18)

with $2) and ;iy’ being solutions of the following problems - (0)

-;ppy_w=~-;D(9) ppy_ @(_o) . (e-2PW + e-W+)

(20) (21)

where D(5) =exp(p&)/y+. It is easy to verify that the infinite series of (17) and (18) constructed in this mmaner certainty satisfies (14). It becomes more clear later that 8:) and I?!!) correspond to the solutions that experience j-times of reflection at faces of the strip. By using the definition of KIII given as (B), it is apparent that the series corresponding to (17) and (18) is

204

M. K. KU0 5. EARLY

TIME

SOLUTION

Equation (19) is exactly the corresponding Wiener-Hopf equation for a cracked infinite plane with concentrated point forces acting on the crack faces. It, as one would expect. implies that in the short time approximation the Mode-III elastodynamic stress intensity factors for a cracked strip can be computed for a crack in an infinite plane, provided that the crack face tractions are the ones corresponding to the original strip problem. The infinite plane containing a semi-infinite crack is only a special case of the work considered by Kuo and Cheng [15] where semi-infinite cracks along the interface of an anisotropic bimaterial were analyzed. Here we outline only the brief results. As discussed in [15], since exp(p@) is an entire function in the complex c-plane, D(E) is analytic in the right half E-plane Re(E) > --s. The left-hand and the right-hand sides of (19) are then analytic in the overlapping half planes Re(E) -s, respectively. Hence both sides represent one and the same entire function and it might appear that our problem has been solved. Unfortunately, D(E) is not bounded as Re(g)+ m in the right half E-plane owing to the presence of the exponential term exp(ptl). That prevents one from determining the entire function. The sum splitting of D(E) into two functions, D, and D-, which are analytic and bounded at each point of the right and the left half planes, respectively, is therefore necessary. Since D(t) is analytic and goes to zero as (51- 00, at least, in the strip --s < Re(g)
exp(-pvf) dr] = D(g) + 5)

(rj -sy2(q

-D-(E).

(23)

Notice that D+(E) has branch cut along 6 = --s to --oo, like D(g), while D_(E) is not bounded as Re( 5) --, 03. The expression for D_(E) can be simplified further. Considering first the case of Re(g) > -s, the expression of D+(E) in (23) can then be regarded as a function of I, say f(1), for a fixed 5. Notice that the integration can be evaluated analytically when 1 = 0, yields f(0) = (s + g)-‘“, moreover f

[f(Oexp(-z-%01 =-

The use of (23) and the integration

D-(E) = (p/n)

(~)“2e~p~-AE +sY1

of (24) with respect to 1 yield expl-p(c

+ s)ql dq.

Since D_ is regular in the half complex plane Re(g)
(25) be

(26) The left-hand and right-hand sides of (26) are clearly regular in the overlapping half-planes Re(g) -3, respectively. By analytic continuation, therefore, each side of (26) is the unique analytic continuation of the other into the complementary half plane, and both sides represent one and the same entire function, say Z. Liouville’s theorem for bounded entire functions allows the conclusion Z = constant. The magnitude of the constant can be obtained from order conditions on Z as ]E]+ 00, which in turn are obtained from order conditions on the dependent field variables in the vicinity of x = 0. As usual, the near-tip stress fields and the crack opening displacement exhibit inverse square root behaviors and square root behavior,

Transient stress intensity factors

respectively,

205

with the result that Z vanishes identically, hence, bT’(& 0, p) = pE Y+D+(E) &y-@c”)(&

(27)

o,p)=gD_(Q.

(28)

The zeroth order stress intensity factor can now be concluded. By inverting the two-sided Laplace transform to (27), changing the order of integration between the integral over the Hoop transformed variable E and the integral in D+(g), and then using the Cagniard-de method, one has KjY&t) = ($)l’*FH(r It is of interest that (29) is exactly the appropriate

- sl).

(29)

static result for crack in the infinite plane.

5.2 6’1(_1) and by’ From (20) and (28), the equation governing @!!) and S!+!)is obtained as -~~~Y_l??)=~-~[G(h.

(30)

2a) + G(g, 26)]

where GE

LY= 2a, 2h.

m) = exp(-pY@-(6%

(31)

The solution for the problem described by (30) can be obtained again by the Wiener-Hopf technique. Careful study of the function G(& a) show that the function has branch cuts along w, and is not bounded as Re(g) + ~0. Since G(& a) is analytic and -oc,
G+tEi,a) = (PIJG)“*GT(~, 4 = G(5, a) - G-(5, a)

(32)

where (33) As in the last section, conclude

the uses of the analytic continuation

and the Liouville’s

E

ii!“(& 0, p) = p~+.[G+(5,2a)+G+(5,2b)l ~~~r-f?‘(& The double

WW,

integral

transform

0, p) =g[G_(&

2a) + G_(& 26)].

of (34) can now be inverted.

theorem

(34)

(35)

Notice

that 1G+(& a)[ -

as El +-ccin the right half plane. From Abelian theorem (11) concerning asymptotic

relations between functions and their Laplace transforms, KMp)

= ($)“‘~[Kl(p,

one has

2a) + K,(p, 2b)]

(36)

where K&,, a) = i’4-112( &

I,r

e-~[(“+~)9+(~2-uz)‘n,-url

du}

dq.

(37)

Finally, by letting (U + s)~ + (s2 - ~*)~‘*a,- uf = t and using of the Cagniard-de Hoop method, the one-sided Laplace transform in (37) can be inverted by inspection. The first order

206

correction

M. K. KU0

to the stress intensity factor is 7

K$(t)

= i$)’ _F[K,(t, 2a) + K,(t, 2b)l

(38)

where (Y H(t - fl) K,(t, (u) = Jc[2s(t-sz)]“7

Yh I() q yqa

t-sq - q)l’2[(q

- I)’ + (Y2]dq

- sl)]. The integral in (39)

and cl(m) = s(Z* + LYE)“‘,qh = min(Z, qa), q, = [t’ - s2(I’ + a’j]/[2,s(t can be integrated analytically as K,(Z, a) =

(39)

1 Ke (t - %)N(q”) Jr[2S(f - sf)]“’ L%Y2(qa- 4”P2 1

(40)

where q. = f + ia, the expression of N,(q) is defined for s(Z’ + &Ii2 < t s(Z + a) as N,(q) = i log

-{2]4Z(% - 4)(&l - W + (qa - 2q)Z + qnq) (Z - 4 )40

(41)

It is of interest that for the case of s(Z’ + (Y’)“~< t cs(f + (u) the result of (40) with N, = n can be obtained easily by evaluating (39) as a contour integration in the complex q-plane, moreover, after some manipulations (42) where q = n - tan-‘(all). Equation (42) is exactly the appropriate static and also the transient (see e.g. Cheng and Kuo [20]) results of KI,, for semi-infinite cracks in an infinite plane loaded by a concentrated point force of magnitude 2F at (x, y) = (-1, a) for time t > 0 with the crack at y = 0, x < 0. It in turn is the transient KI,, for a subsurface semi-infinite crack (at y = 0, x < 0) in a half plane without counting the interaction effects between the crack and the surface of the half plane. The crack is parallel to and located (Yaway from the surface of the half plane as in Fig. 2(a). The cracked half plane, which is defined in y < (Y, is loaded by a surface load of magnitude F at (--I, cu). In the method of image, the solution of a cracked strip is the same as for an unbounded medium with infinite number of parallel cracks as shown in Fig. 2(b), provided that each crack face is subjected to “image forces” that correspond to those of the original crack. Equation (42) is then the effect of the image force at (-I, (u) from the stand point of the method of images. From the stand point of wave propagation, it is the contribution of cylindrical waves emitted from the point force at (-I, 0) and reflected once at the strip surface of y = a/2. On the other hand, the arrival time suggests that the Z(t, (u) with the expression of N1 defined as (41) is composed of two contributions: one corresponds to the crack-tip diffracted waves which reflect at the strip surface of y = a/2 once and return to the crack tip again, the other corresponds to waves which reflect first at the strip surface of y = ru/2 and the crack face and then are diffracted at the crack-tip.

From (21), (28) and (35), we obtain for &!!) and SC:’ -&&=~-;

-G(~,4~)+G(j,46)-~~~~~~Q(~, L .

where

Q(5, w PI = exp(-m$W+(Et

a).

.

a, /3 = 2a, 2b.

ff, P) ]

(43) (44)

In (43), the relation G-(6, o) = G(E, (u) - G+(& (u) has been used to arrange G-(5, cy)exp( pyP) as G(5, cy + P) - G+(& a)exp(-py6). C areful studies of the function Q(g, cr, p) reveal that the function has only branch cuts along --OO< g < --s and s < 5 < M. Since Q(g, (Y,p) is

Transient stress intensity factors

(b)

(a)

Y

FH 0) I

A

Gl b-l--

* t

a

Fig. 2. (a) The loading and cracked half-plane geometry corresponding to the early part of contribution in the second term of the solution series. (b) The cracks, loadings and the geometry of the corresponding infinite plane from the stand point of method of images.

analytic and goes to zero as I& + 00in the strip --s < Re(c) < s, it can be expressed as the sum of two regular functions Q+( 5, (Y,/I) and Q-(5, (Y,/3) by the direct application of the formula of sum splitting. As in the last section, the uses of the analytic continuation and the Liouville’s theorem conclude a!“)( 5, 0, p) = E y+ * [ G+(& da) + G+(f> 46) P

Notice that IQ+<& a, @I - O(l5l-‘1, Abelian theorem (ll), @3(p)

c

a,/3=2a,Zb

Q+(E, a; 8)].

(4%

as l,!j-* 00 in the right half plane. From (45) and

one has = (f)lnF[

&(P,

&I + &(P,

46) -

2

Kz(P,

a,

a,j3=2a,Zb

P,]

(46)

where K2(pj a, B) = &

I,, G(Q

+qC-p(~2

-

rl’Y”B1drl

(47)

and K,(p, a) and Gf(q, LY)are defined as (37) and (33), repectively. The Laplace inversion of x2 is performed as follows. First of all, by letting (s2 - q2)lR@ = v/3, the integral in (47) is rearranged as an integration of v along the positive real v-axis from v = s to v + 03. Then, by letting v/? + (s’ - ~‘)~~a, - ul + (u + s)q = t and using of the Cagniard-de Hoop method, the one-sided Laplace transformed K2(p, cu, /I) is easily inverted by inspection (48) where iW2= i H(t - td)Im td = /Iv + sq +

- au+iat [ iv+ -u+ iv+ + u+ du+ldt

1

(49)

[a?+(1- w)2]“2s, ?,7+(v)= (v” - s2yn and “Im” stands for the imaginary part.

M. K. KU0

208

The u+ is a branch

of the Cagniard

u+ where

path in the complex

+-(I-q)T+i(T‘-

R* = (Y’ + (I - q)2, and T = t - /!h - .rq. After

u-plane

detined

as

.s2R’)“2a)

(SO)

some manipulation,

one has

(52)

(53)

qc = qd = min(l,

(t -

/sv)’ - s’(F i 2) (54)

2s(t - pv - .FI) -

qc) and t,(cu, p) = sp + s(l’ + d)“‘.

Notice

tor of J are polynomials in q of the fourth degree real coefficients. J can then be expressed by partial

that the denominator

and of the third fraction as

degree.

and the numera-. respectively,

with

(55) where Aj(V) = lim,,,,(q - qj)J(V, q), and qi(j = 1, . . . , 4) are the zeros of the denominator of J. Moreover, since the denominator of J is in the form of a sum of complete squares, its zeros (namely qj’s) are generally in the form of two pairs of complex conjugates, q1 and q2 being those two with positive imaginary parts. Hence NZ can analytically

and expressed

Let us denote be integrated

as

(56)

11# s where

the expression

of N,(q)

is defined

as JG for qc < f and as (41) for q<.:z 1 with y(, being

replaced by qc. Notice that, the only degenerate case is when t’ = s. In that case. the zeros of the denominator of J are q = t/s - p f a. They are real and double roots. Since they are also zeros of the numerator of J as u = s, they behave as simple poles of J in the complex w-plane. Consequently,

N*(u) at u = s is still finite,

its value

does not affect the result

of the integral

in

(51).

6. ANISOTROPIC

CRACKED

STRIP

Consider a semi-infinite stationary crack in an anisotropic strip which is characterized by three elastic moduli (e,,), i, k = 4, 5 (rather than one for the isotropic solid), and mass density p. A Cartesian coordinate system (a, 9) is defined in such a way that the only nonzero The strip locates from jj = -6 to E = ci, and the displacement, G’, is normal to the @-plane. crack lies in the line jj = 0, ,f < 0. The material has been assumed to possess certain material symmetry and @-plane coincides with one of the planes of material symmetry, such that the in-plane and the anti-plane motions are not coupled. In the Cartesian coordinate system, two dimensional anti-plane wave motions of homogeneous, anisotropic, linearly solids are governed by

Transient stress intensity factors

209

hat indicate that the quantities are associated with the anisotropic relevant stress component is

The superposed

solid. The

The anisotropic strip is initially at rest. For time t 2 0, a pair of concentrated anti-plane shear forces in the i-direction of magnitudes fF are suddently acting on the crack faces jj = Of at f = -1. Thus the crack-face boundary condition and the conditions of zero tractions on the faces of the strip are &&, y^,t) = -FH(t)G(f

+ I),

fP&, 8, t) = 0, a&f,

As discussed in [15], the coordinate

-6,

f co, --co
t) = 0,

--co
jj=O’

(59) (60) (61)

transformation x = 2 - (e45/e44)yl

(62)

Y=

(63)

we44)Y

where ~1= (E&55 - ?i5)1’2, reduces (57) to the standard wave equation. The positiveness of is guaranteed by the positive definite of strain energy. It can be shown that a &4~55 - Ei5) transformation of relevant parameters will deduce the solution for an anisotropic strip, formulated by (57)-(61), from a corresponding solution for an isotropic strip. The corresponding isotropic problem in the xy-plane concerns a semi-infinite stationary crack along the line y = 0, x < 0, in an elastic strip which is characterized by shear modulus y = [c144?s5 - ~?i~]l’~and mass density p = (@Jo). The strip locates from y = -b to y = a, where b = &/Z, and a = M/C,. The anti-plane wave motions of the corresponding isotropic strip are then governed by (l)-(5). We will be particularly interested in elastodynamic Mode-III stress intensity factor which is related to the one for the corresponding isotropic problem by

Here &I1 is defined as Qt)

= lim [2n$]lni?&?, P-o+

7.

0, t).

(65)

RESULTS

The elastodynamic Mode-III stress intensity factors of a semi-infinite crack in the strip, as in (22), is the sum of (29), (38) and (46). Figure 3 shows the dimensionless stress intensity factor KJK$, for cases of symmetric strips (i.e. crack along the central line of the strip, a = b) vs the dimensionless time t/[(~l)~ + s’(~u)~]‘~ for various values of I/u, where K& is the appropriate static value which is defined as [21] K& = F f ln[l - exp( -d/u)]-‘n. 0

(66)

The KII, actually jumps first from zero to K&= [2/( nl)] ‘“F, which corresponds to appropriate static stress intensity factor of a cracked infinite plane, at time t = sl, as (29), any values of f/u and then retains the constant value until t = s(12 + 4~2’)‘~. The discrepancies the magnitudes and the positions of the first discontinuities for various of l/u in the figure the consequences of the normalization. At time t =s(12+ 4a2)l”, the KII, jumps again

the for in

are to

210

M. K. KU0

0

0.5

1.0

t/[(d)* + (6sa)2 ]1’2 Fig. 3. Mode-III stress intensity factors for symmetric strips (a = h).

another constant values with the amount of discontinuities 2(2/~c)“*(1~ + 4a*)-“*F sin(q/2) until t = s(l + 2a), where W = at - tan’(2all). It shows that the elastodynamic overshot of the stress intensity factors are around 1.2 - 1.4 for l/a = 0. 1 - 2. The larger l/a (i.e. the smaller strip height) is, the larger is the dynamic overshot. Moreover, the maximum of the stress intensity factors, for smaller l/a values (e.g. l/i2 < 0.2) seem to occur at s(P+ 4a*)‘“< t< s(l + 2a), which corresponds to the time before the tip-diffracted waves reflect twice at the strip surfaces and return again to the crack tip. The results for asymmetric strips (b # a) are shown in Figs 4 and 5. For l/a = 0.1, Fig. 4 shows KJK& as a function of t/(sf) for various values of b/u. It shows that the maximum of the elastodynamic transient stress intensity factors of the symmetric strips (b = a) are larger than the ones for the asymmetric strips. Moreover. the maximum of the stress intensity factors are larger for the cases with the value of b/u closer to 1 (i.e. for the strip closer to the symmetric one). These due to the fact that the larger b is, the smaller are the effects from the strip face at y = -b. In addition, the larger b is, the latter the effects of the strip face at y = -b enter the stress intensity factors and hence the lesser are of the effects of strip face y = a left at that moment. It is of interest to point out that for the cases of b/u 2 3, the strip surface of y = -b has no effect on the stress intensity factors for the time interval t < s[l’ + (6~)‘]~“. Hence the results in that time interval for b/u = 3 should be the same as the ones for the subsurface crack in the half plane. Similarly, for the time interval t < s[1’ + (1.4~)~]“*, there is no difference on the stress intensity factors for cases b/a 1: 1.4. The analogous curves for 2.5 I--

2 K Ill

K Ill

KlPP

K,:

,

1 .o

0

20

40

60

t/s1 Fig. 4. Mode-III stress intensity factors for Z/a = 0.1.

0

4

2 t/s1

Fig. 5. Mode-III stress intensity factors for i/a =

6

i

Transient stress intensity factors

211

I/a = 1 are shown in Fig. 5. The variation of the history of A&, is very similar to that for the case of I/a = 0.1.

8.

CONCLUSION

In this paper the mode-III transient elastodynamic stress intensity factors of a stationary semi-infinite crack in an infinite strip has been determined for a particular case of a pair of anti-plane point crack-face loadings. The characteristic lengths of the problem arise not only from the loading but also from the geometry. The exact solution has been obtained by integral transforms together with the direct application of the Wiener-Hopf technique and the Cagniard-de Hoop method. Instead of performing the quotient splitting directly in the resulting Wiener-Hopf equation, the corresponding term has been expanded first into a series. The solution has then constructed in a series form accordingly. The first term in the solution series is exactly the solution of the crack in an infinite plane and hence the waves have not yet reflected against the strip surfaces. With each further term in the solution series, it corresponds to the solution that the waves has reflected more times against the strip surfaces. Each term in the solution series can be interpreted as the contribution that experience different times of reflection at the strip surfaces. The first three terms in the series for the stress intensity factor history have been computed numerically. The results are exact for the time interval from initial loading until the first wave scattered at the crack tip is reflected three times at the strip surface and returns to the crack tip. The numerical results are presented only for cases of isotropic strips. The results for anisotropic strips can be deduced from the results of corresponding isotropic strips by the transformation of relevant parameters. It has been shown that the maximum of the transient stress intensity factors of the symmetric strips (b = a) are larger than the ones for the asymmetric strips (b #a). It has also been shown that the elastodynamic overshot of the stress intensity factors for symmetric strips are around 1.2 - 1.4 for L/a = 0.1- 2. Moreover; the smaller strip height is, the larger is the dynamic overshot.

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