lnternátionál Journál ÏÉ Frácture 65: 49-61, 1994.(!:;) 1994 Kluwer Acádemic Publishers. Printed in the Netherlánds. 49

Áç analyticaljnumerical approach for cracked elastic strips underconcentrated loads - transient response

H.G. GEORGIADIS and Í. CHARALAMBAKIS:; Mechanics Division. Cámpus Âï÷ 422. School ÏÉ Technology. Aristotle University ÏÉ Thessáloniki. 54006. Greece

Received 1 Ju1y 1993; accepted ßç final form 7 December 1993

Abstract. Áç aça1ÕtßcaÉjçumeÞcal approach is presented for the determination of the near-tip stress field arising fromthe scattering of SH waves by a long crack ßç a strip-like elastic body. The waves are generated by a concentratedanti-plane shear force acting sudden1y ïç each face of the crack. The problem has two characteristic lengths, i.e. thestrip width, and the distance between the point of application of the concentrated forces and the crack tip. It iswell-known that the second characteristic length introduces a seÞïus difficulty ßç the mathematica1 ana1ysis of theproblem. Éç particular, a non-standard Wiener-Hopf (W-H) equation aÞses, that contains a forcing term withunbounded behaviour at infinity ßç the transform plane. Éç addition, the presence of the strip's finite width results ßç acomplicated W-H kernel introducing, therefore, further difficulties. Nevertheless, a procedure is described here whichcircumvents the aforementioned difficulties and holds hope for solving more complicated problems (e.g. the plane-stressjstrain version of the present prob!em) having simi!ar features. Our method is based ïç integra! transform ana!ysis,an exact kerne! factoiization, usage of certain theorems of ana!ytic function theory, and numerica! Lap!ace-transforminversion. Numerica! resu!ts for the stress-intensity-factor dependence õñïç the ratio of characteristic lengths arepresented.

1. Introduction

Within the realm of linear elastic fracture mechanics, the main emphasis is usually placed ïçdetermining the stress intensity factor (SIF) at the tip of sharp cracks existing ßç the body. Asis well-known the determination of this factor is indispensable for applying the Griffith-Irwinfracture concepts [1-3].

Éç this context, an important class of problems concerns elastic stress-wave diffraction bycracks having the form of narrow cuts or slits. These problems are encountered ~hen rapidlyvarying loads are applied to a body that contains such stress concentrators, and therefore thecomputation of the field requires that inertia effects be taken into account. Contrary to similarsituations ßç acoustics, geodynamics and electromagnetic waves, ßç fracture mechanics most ofthe attention is focused ïç the field behaviour near the crack tip during the time intervalimmediately after the application of the external loads. Relative work for analyzing transientresponse has been done by Sih and co-workers [4, 5], Achenbach [6], Brock [7-9], Jiang andKnowles [10], and Georgiadis [11, 12], among others.

Éç the above-mentioned studies, it was observed that the ratio of dynamic SIF to correspond-ing static SIF departs from unity as the time varies, exhibiting a pattern of local maxima and

~ minima. Dynamic SIF overshoots are then possible and it is interesting to determine these

effects by an exact analysis.. Éç the present study, we deal with the transient elastodynamic problem of a strip-like body

containing a semi-infinite planar crack which is acted õñïç by a pair of concentrated anti-planeshear forces. As depicted ßç Fig. 1, the crack is situated at the mid-width plane of the strip andthe two equal and opposite forces are applied along the crack faces at a distance L from the

)""" ;\\;c,"" º ..".,50 H.G. Georgiádis ánd Í. Chárálámbákis .\

b

Õ+ééÉé

F H(t) é. é -.. "

Ï ÷b

..

ÉFig. 1. Áç elastic strip containing a semi-infinite crack under the action of a pair of concentrated anti-plane shear forces.The forces are applied suddenly and a transient response is considered.

crack tip. Thus, there are two characteristic lengths ßç the problem, i.e. the stÞñ semi-width band the distance L. The surface tractions are applied suddenly (containing a Heaviside stepfunction oftime). Thus, the governing equation for the out-of-plane displacement is the 2-Dwave partial differential equation.

Coming now to discuss briefly some mathematical aspects of the problem, we note that the2-D wave equation has to be solved ßç the upper (or 10wer) half of the stÞÑ-Éßke domain, dueto symmetry, ßç the presence of the characteristic length L and mixed boundary conditions (thetraction is prescribed along the crack face, whereas the displacement is zero along the crack-lineprolongation). While the difficulties associated with the finite stÞÑ-wßdth can be removed byseveral ways (see e.g. Georgiadis et al. [13-16] and references cited therein), it is widelyrecognized that the existence of the characteristic length L induced by the point-force boundarycondition introduces serious difficulties ßç analyses where potential-theory methods are notapplicable (see e.g. Freund [17, 18], Brock [8,9,19], Georgiadis et al. [16], and Êõï [20,21]).It is noticeable that analogous boundary conditions and existence of point-force or wedgingcharacteristic lengths ßç problems involving Laplace-type equations have successfully beentreated by Barenblatt and co-workers [2, 22-24] and Sih [25]. The latter studies concerned thesteady-state wedging of a 10ng crack (the motions of the wedge and the crack tip were assumedto occur at constant velocities) ßç an unbounded body and/or the elastostatic opening of a crackby anti-plane shear point forces. However, an approach similar to that utilized ßç theaforementioned works, i.e. one based ïç the conformal mapping technique and the Poisson,Schwartz or Keldysh-Sedov integral formulas, cannot be applied to situations involvingHelmholtz or wave-type governing equations (it seems also that even biharmonic problemsinvolving strip-like domains cannot be treated by such an approach, see e.g. Gladwell [26], andGeorgiadis et al. [14-16]). "

It is well-known that ßç such a class of problems involving Helmholtz or wave equations andexhibiting çï self-similarity, integral transforms ßç conjunction with the Wiener-Hopf techniqueconstitute powerful tools ßç obtaining analytical solutions [18, 27, 28]. Nevertheless, when a "

characteristic length is present ßç the forcing function (like the length L ßç the present problem),an inconvenient exponential term (having unbounded behaviour at infinity) arises ßç thetransform plane, which, ßç turn, implies by Liouville's theorem that the entire function coming

Crácked elástic strips 51

from analytic-continuation arguments is an infinite-degree polynomial. Clearly, one cannotdispose so many physical conditions to determine the unknown coefficients of such a poly-nomial, and thus usage of the Wiener-Hopf technique is inhibited.

Ôï deal with this difficulty ßç the latter class of problems (involving Helmholtz or waveequations and a characteristic length ßç the forcing function), an exact approach has beenproposed previously [8, 9, 17-19], which is based ïç superposition considerations. Thisprocedure may, however, be laborious ßç geïmetÞes involving strip-like domains (as ßç the

" present case). As an alternative, Georgiadis and Brock [29] have introduced recently a direct

approach which is based ïç the exact solution of an equation containing two unknown;; functions through contour integration and Cauchy's theorem. This method could be viewed as

a generalization of the classical Wiener-Hopf method. lç the latter study, the time-harmonicsteady-state version of the present problem was treated and an exact closed-from solution wasobtained. Here, however, such a solution cannot be obtained and we had to resort to numericstoo. lç particular, the same analytical tools as ßç [29] were utilized to get an exact solution ßçthe Laplace-transform domain, but then a numerical procedure was followed ßç order to invertthe Laplace transformed SIF back to the time domain. As will be seen later, the latter taskcannot be accomplished analytical1y., due to the very complicated form of the transformed SIF.Áç advanced technique for çumeÞcal1Õ inverting Laplace transforms was utilized as ßç otherrecent works by Georgiadis [11, 12]. This technique is based ïç FïuÞer series and pertinentdiscretizations of real integrals [30, 31] being proved particularly suitable for elastodynamic

problems.It is hoped that the present approach wil1 be effective ßç dealing with even more complicated

situations involving plane-stress/strain conditions.

2. Problem statement

Consider an elastic body ßç the form of an infinitely long strip containing a semi-infinite crack,see Fig. 1. Áç Qxyz Cartesian coordinate system is attached to the cracked body with the ïÞgiçat the crack tip. The stÞñ occupies the region (- 00 < ÷ < 00, -b < Õ < b) being thick enoughßç the z-direction that a state of anti-plane shear is achieved. The crack is situated along theplane ( - 00 < ÷ < ï, Õ = Ï) and is sheared by a pair of concentrated anti-plane forces :t F H(t),independent of the z-coordinate, acting along (÷ = - L, Õ = Ï). The faces of the crack aretraction free, except for the point of application of the concentrated forces. Because ofanti-symmetry with respect to the plane Õ = Ï, the problem can be viewed as a haÉf-stÞÑproblem with the material occupying the region (- 00 < ÷ < 00, Ï < Õ < b).

Under the above conditions, it is reasonable to assume that a state of anti-plane shearprevails, i.e. the ïçlÕ nonzero component of the displacement vector is the one ßç the z-directionand that the problem is two-dimensional depending ïçlÕ ïç the coordinates ÷ and Õ. Therefore,

- the pertinent equations governing such an elastodynamic deformation are wÞtteç as (see e.g.Achenbach [27])

'" Ux, uY = ï, uz(x, Õ, t) # Ï, (1)

ouz ouzó÷, óÕ' óÆ, Ô×Õ = ï, Ô×Æ = ìfu' ÔÕÆ = ìÂÕ' (2)

52 H.G. Georgiadis and Í. Charálambakis

~ ~- 2~- 0 -! (3)~ 2 + ~ 2 É× ~2 - , É×- ,õ× uy ut c

where (ux, uy, uz) and (ó÷, ..., ÔÕÆ) are the components ofthe displacement vector and stress tensor,respectively, ì is the shear modulus, c is the shear-wave velocity, and é÷ is the wave slowness.

The boundary value problem can now be supplied with the following conditions:

(ß) Boundáry ánd initial conditions: -

ÔÕÆ(×, b, t) =0 for - 00 < × < 00, (4.É)

ÔÕÆ(×, Ï, t) = Fä(÷ + L)H(t) for - 00 < × < Ï, (4.2)

uz(x, Ï, é) = Ï for Ï < × < 00, (4.3)

uz(x, Õ, Ï) = iJuz(x, Õ, O)jiJt = Ï, (4.4)

where ä(.) and Ç(.) is the Dirac delta and the Heaviside step function, respectively.

(ßß) Edge conditions:

ÔÕÆ(×, ï, t) = ï(éé÷) for × -+ 0+, (5.É)

uz(x, Ï, é) = ï(É) for × -+ Ï-, (5.2)

which state that the near-tip stress and displacement fields cannot be so singular as tocorrespond to line sources of radiated energy. Furthermore, ïç the basis of fracturemechanics considerations (see e.g. [É-3, É8, 32]), it can be shown that ÔÕÆ(×, Ï, é) ,.., ÷-É/2 for× -+ 0+ and uz(x, Ï, é) ,.., ÷É/2 for × -+ Ï-. However, (5) are still sufficient conditions forapplying Jordan's lemma ßç subsequent steps of our analysis.

(ßßß) Finiteness conditions át remote regions:

ÉÔÕÆ(×, ï, s)1 < Aexp(-prx) for ÷-+ + 00, (6.É)

ÉßßÆ(÷, ï, s)1 <  exp(puX) for × -+ + 00. (6.2)

where Á, Â, Ñô and Ñõ are positive constants, an overbar denotes the Laplace transform of afunction (as defined below), and s is the Laplace transform variable. These equations statethat the diffraction field at infinity consists of outgoing waves only.

Our objective here is the determination of the stress field near to the crack tip for theproblem defined by (ÉÇ6). ,

3. Analysis "

The first step ßç solving the problem described ßç the previous section is the introduction ofthe fol1owing one- and two-sided Laplace transform pairs. These are utilized to suppress t-

Crácked elástic strips 53

and x-dependence ßç the stress and displacement field and ßç the partial differential equation

(3)

J(x, Õ, s) = foOO !(×, Õ, t)e-Stdt, (7.1)

. J(x, Õ, t) = -21. ß J(x, Õ, s)est ds, (7.2)ÐÉ JBr

" J*(p, Õ, s) = I~oo J(x, Õ, s)e-SPX dx, (8.1)

J(x, Õ, s) = -2s . ß J*(p, Õ, s)eSPX dp, (8.2)

ðéJÂr

where Br denotes .the Bromwich path ßç pertinent complex planes. Éç addition, we define

half-line transforms of the unknown functions ÔÕÆ(×, Ï, s), Ï < × < 00, and uz(x, Ï, s),

-ïï<÷<Ï

ô+ (p,s) = foOOfyz(x,o,s)e-SPXdx for -(pT/s) < Re(p), (9.1)

ÔÕÆ(×, Ï, s) = ~ ß ô+ (ñ, s)eSPX dp for Ï < × < 00, (9.2)2ðé JBrl

u- (ñ, s) = I~ 00 Uz\X, ï, s)e-SPX dx for Re(p) < (Pu/s), (10.1)

uz(x, Ï, s) = ~ ß U- (ñ, s)eSPX dp for - 00 < × < Ï, (10.2)2ðé J Br2

where, clearly, ßç light of conditions (6), ô+ (ñ, s) and U- (ñ, s) are analytic functions ßç the right,

-(pT/s) < Re( ñ), and left, Re( ñ) < (Pu/s), half-plane, respectively.Further, applying (7.1) and (8.1) to the governing equation (3) results ßn an ordinary

differential equation having the general solution

u:(p,y,s) = C(p, s) exp[ -S(cx2 - ñ2)1/2 Õ] + D(p, s) exp[s(cx2 - ñ2)1/2 Õ], (11)

"where C(p, s) and D(p, s) are unknown functions. Then, applying (7.1) and (8.1) to the boundaryconditions (4), considering (9.1) and (10.1), and eliminating C(p, s) and D(p, s) from the resulting

" system gives the following functional equation

FeLspT+(p,s) + - = - ìÊ(Ñ,S)U-(Ñ,S), (12)s

54 H.G. Georgiádis ánd Í. Chárálámbákis

where the kernel Ê(ñ, s) is given by

Ê(ñ, s) = s(á2 - ñ2)1/2 tanh[s(á2 - p2)1/2b], (13)

and, ßn view of (9.1) and (10.1), we notice that (12) holds over a common region of analyticitydefined as the stÞñ -(PT/s) < Re(p) < (Pu/s).

The main objective ßn the analysis is to determine both the unknown functions ô+ (ñ, s) .and U- (ñ, s) from the single equation (12). This is possible by supplying (12) with resultsobtained by using Cauchy's theorem and Jordan's lemma. At this point, we emphasizethat the conventional Wiener-Hopf technique [27, 28] is not applicable to (12), since the .term eLsp has an inconvenient (unbounded) behaviour as É pl = 00. This, ßn turn, inhibitsthe use of Liouville's theorem which is indispensable ßn applying the W-H technique. Thus,the usual argument of analytic continuation ßn conjunction with Liouville's theorem (whichare the basic ingredients of the W-H method) is useless ßn this case, and a different approachhas to be followed. The one introduced here makes use of simple contour integration alongwith a product kernel factïÞÆatßïn, and, ßn our ïñßnßïn, this approach can be regarded as ageneralization of the classical W-H method.

Ôï proceed further we need to perform a Ñrïduct-factïÞÆatßïn of the kernel Ê(ñ, Î). Wenote that an exact and closed-form factïÞÆatßïn is necessary at this stage because, as will beseen later, ê+ (ñ, s) enters a Cauchy-type integrand and therefore the influence of all poles ofthis integrand should be taken into account. We employ the infinite-product forms of thepertinent hyperbolic trßgïnïmetÞc functions (see e.g. Abramowitz and Stegun [33], ñ. 85) andfind

+ 00 [ (2k - 1)2(ñ + Ak)(p - Ak)JÊ (ñ, s) = bs(á - ñ) kl]l 4k2(p + Bk)(p - Bk) , (14)

where

[( kð)2 J1/2 [( (2k - l)ð)2 Jl/2 Ak = b; + á2 , Bk = 2bs + á2 , (15)

the latter being real and positive quantities for s considered as a real and positive parameter.Then, a product factorization follows easily from (14) as

K(p,s) = K+(p,s)K-(p,s), (16)

where -

K+(p,s) = bÉ/2s(á + ñ) Ð [ (2k -1)(ñ + Ak)J, (17.1)

k=1 2k(p + Bk)

K-(p,s) = K+(-p,s), (17.2)

Crácked elástic strips 55

and the functions ê+ (ñ, s) and Ê- (ñ, s) are non-zero and analytic ßç Re(p) > - inf(Ak= É, Bk= é)

and ßç Re(p) < inf(Ak=l, Bk=l), respectively.Some re-arrangements ßç (12) are now ßç order. More specifically, the kernel factïÞÆatßïç (16),

(17) allows us to write

ô+ (ñ, s) p~op --Ê +( ) + ê +( ) = -ìÊ (p,s)U (p,s), (18)

ñ, s s ñ, s

which holds ßç the strip -inf(pT/s,Ak=l,Bk=l) < Re(p) < inf(Pu/s,Ak=l,Bk=l).Further, we change the vaÞabÉe from Ñ to ù and divide both sides of(18) by 2ðß(ù - ñ), getting

Ô+(ù,s) FeLsro Ê-(ù,s)U-(ù,s)2ðßÊ+ (ù, s)(ù -ñ) + 2~isK+ (ù, s)(ù - Ñ) = - ì 2ðß(ù - Ñ) ,.. (19)

Then, we integrate (19) over the imaginary axis ß Ém(ù), by taking also the point Ñ to lie ïç the

Þght half-plane Re(ù) > Ï only

1 ÉßÏÏ ô+ (ù s) F ÉßÏÏ ~où

- 'dù+-dù=2ðß -ßïïÊ+(ù,s)(ù-Ñ) 2ðßs -ßïïÊ+(ù,s)(ù-Ñ)

= - ~ I ßoo K-(ro,s)U-(ro, S)dù. (20)

2ðé -ßïï ù - Ñ

Now, working ïç the last integral ßç (20), we observe that Ê-(ù,s) ~ ùÉ/2 for ÉùÉ => 00,

U-(ù,s) ~ ù-3/2 for ÉùÉ =>00 (the edge condition that uz(x,O,t) ~ ÷É/2 for ÷=>Ï- ßç conjunc-

tion with the Abel- Tauber theorem [27, 28, 34] was utilized), and also that the integrand is ananalytic function ßç the left half-plane Re(ù) < ï. Therefore, , by applying contour integration(Fig. 2) and Jordan's lemma [34] to this integral, we conclude that it vanishes when Ñ belongs to

the right half-plane Re(ù) > ï.

ß Im(w )

ÈÑ

Re(w)

"

Fig. 2. Defonnation of the ïÞgßna! integration path ßç the comp!ex w-p!ane.

56 H.G. Georgiádis ánd Í. Chárálámbákis

Then, by employing again contour integration and Jordan's lemma ïç the first integral ßç (20),and by closing the integration path with a large semi-circle at infinity ßç the Þght half-plane, weobtain through Cauchy's integral formula

ô+ (ñ s) F Éßé÷> eLs'"+ ' = - --:- +)( ) dù, (ñ ßç Re(ù) > Ï). (21)

Ê (p,s) 2ðés -ßé÷>Ê (ù,s ù-ñ

Finally, the integral ßç (21) can formally be evaluated by closing the integrationpath with a largesemi-circle at infinity ßç the left half-plane. Jordan's lemma along with the Cauchy's residuetheorem lead to the following expression for the transformed crack-line stress .

F ([ é÷> [ ù+Á eLs'"T+(p,s) = -b1iYK+(p,s) j~l ~-ù-=-Ñ'

.[ÃÉ(~ ~~ )]]] + [ÃÉ(~ -É× + Bk)] -~ ) (22)k=é2k-1ù+Ák ",=-Aj k=l2k-1-IX+Ak -é÷-ñ'

where (17.1) was also employed, and Aj above is given ßç (15) with j replacing k.Equation (22) may provide, through the Laplace-transform inversions (9.2) and (7.2) the

crack-line stress. However, ßç the next section, ïçlÕ the singular part of this stress and theassociated intensity factor will be obtained.

4. Singular crack-line stress

Based ïç the Abel- Tauber theorem [27, 28, 34], which relates asymptotically original functionsand their transforms, we can calculate the singular part of stress, limx...o ÔÕÆ(×, Ï, s), from thelarge-p approximation of (22), limp...cx> ô+ (ñ, s).

First, we obtain the asymptotic expression for the kernel ßç (13) as

lim Ê(ñ, s) = - isp, (23)ñ"'é÷>

where Ñ is taken along the pertinent Bromwich path. Then, ïç rewriting (23) under the form

Ê( 00, s) = - S(B2 - ñ2 f/2 with å => Ï, (24)

the asymptotic kernel factorization fol1ows easily as

K+(oo,s) = (Sp)l/2, K-(oo,s) = - s1.f2(_p)l/2. (25) .Furthermore, the terms (ù - ñ)-É and (-É× - ñ)-É ßç (22) are approximated by (-ñ)-É,Éç light of the above considerations, (22) becomes ~

. + F 1 ([Ó é÷>[ ù + Aj Ls",11m Ô (p,s) =-bl/2 (~ e .ñ"'é÷> S sp, =É ù+é÷

Crácked elástic strips 57

.[Ii (~ ~~ )]]] +k=1 2k - 1 ù + Ak ù=-ë}

+[Ii (~ -É× + Bk)] e-LSIl ), (26)

k=1 2k - 1 -É× + Ak

and its inverse follows from (9.2) as

Flim ÔÕÆ(×, Ï, s) = ( b )1/2 (...), (27).. ÷=-Ï+ Ð × s

where the expression inside the brackets is the same as that ßç (26).Finally, from the definition of the stress intensity factor as

Êééé = lim [(2ð÷f/2ÔÕÆ(×' Ï, é)], (28)÷=-Ï+

and employing (27), we find the Laplace transformed SIF

- (2)1/2 1([ 00 [ù + Á.Êééé = F - - Ó JeLsw.

b s j=1ù+é×

.[Ð (~~~ )]]] +k=12k-1w+Ak ù=-ë}

+[Ii (~ -É× + Bk)] e-LSIl ). (29)k=12k-1-é×+Ák

Equation (29) provides a closed-form, exact formula for the transformed SIF as a function ofloading, geometry, shear-wave speed, and the Laplace transform variable s. The infinite seriesand products are convergent, whereas by considering an infinite shear-wave speed (é× = Ï) ands = 1 ßç (29), we get the SIF for the analogous static problem as

K~:ric = p[~]1/2 [1 + !e-ÐL/b + je-2ÐL/b +Çe-3ÐL/b + ...]. (30)

The result ßç (30) is identical to the one obtained by Sih [25] ßç treating the static problem bythe conformal mapping technique. This comparison provides a check ïç our method and thefinal result (29) of the analytical part of the solution.

;. It remains now to invert the transformed SIF ßç (29) by the operation (7.2). However, itappears that an analytical inversion is impossible for this complicated expression and, thus, anumerical technique should be employed. Previous experience with numerical Laplace trans-

~.. form inversions (see e.g. Georgiadis [11, 12] and survey articles by Davies and Martin [35], and1tß Narayanan and Beskos [36]) suggests the DAC technique (Dubner and Abate [30], and CrumpÉ [31]) as a potential candidate among the multitude of such techniques. This technique is briefly

described ßç the next section.

58 H.G. Georgiádis ánd Í. Chárálámbákis

5. Inversion of the one-sided Laplace transform

One may start from (7.2) and notice the following alternative form of the ïÞgßnal function KIII(t)

KIII(t) = (ewt /ð) f ï'" [Re KIII(w + iu). cos(ut) - 1m KIII(w + iu). sin(ut)] du, (31)

where s = w + iu. If the trapezoidal rule for integral over semi-infinite intervals is applied to the ~

latter equation, then the resulting approximate expression for KIII(t) is a Fourier series (Davisand Rabinowitz [37]) ~

KIII(t) ~ (ewt /Ô) [ ! KIII(w) + k~ é [Re KIII(w + ßkð/Ô).

. cïs(kðt/Ô) - 1m KIII(w + ßkð/Ô). sßç(kðt/Ô)] ]. (32)

Crump [31] has presented a systematic analysis of errors ßç the above procedure, fromwhich KIII(t) can be computed to a predetermined accuracy. First, Ô is chosen so that2Ô> tmax and then w is determined by w = q - [lç(ÅÉ)]/2Ô, where q is a number slightly

larger than max[Re(S); S is a pole of KIII(s)] and the relative error is to be çï greater thanÅÉ (say ÅÉ = 10-8). It is possible also to increase the rate of convergence of (32) andthus reduce the truncation error by using a suitable series transformation like the epsilon

aÉgïÞthm

e(m) - e(m+l) +[e (m+l)-e(m) ] -l withe(m) =0 (33)ç+É - ç-É ç ç -É,

B~m) being the mth partial sum of (32), which was utilized here and ßç other recent studies [11, 12].

6. Results and concluding remarks

lç this section, we include representative results for the SIF behaviour both ßç the Laplacetransform domain and ßç the time domain. lç obtaining these results from (29), 4 terms ßç theseries and 1000 terms ßç the infinite products were considered. Ïç the other hand, 100 terms ßçthe series (32) along with the epsilon algorithm were enough to obtain convergence.

Figure 3 shows the vaÞatßïç of the normalized transformed SIF [KIII/F(2/bfI2], versus (c/bs),where s is taken as a real variable. Two different ratios L/b = 1.0 and 3.0 were considered. Forlarge s, íßÆ. small t, one could observe that the SIF is zero, which is expected since çï wavefrontreaches the crack tip up to time t = (L/c). For small s, íßÆ. large t, the SIF tends to reach a ;plateau, which corresponds to the steady-state SIF value ßç the time domain. Ïç the other hand,the monotone and smooth vaÞatßïç of KIII(s) implies that çumeÞcaÉ inversion could be feasible.

Figures 4 and 5 show the variation of the normalized SIF [KIII/F(2/b)lI2] versus time (ßç sec) "

for specific values of the shear-wave velocity (c = 1200 m/sec) and the strip width (b = 0.02 m),for two different ratios L/b = 1.0 and 3.0, respectively. lç both graphs, one could observe astep-type increase of the transient SIF, up to the value taken by the static SIF ßç a body of

Crácked elástic strips 59

1.00

~-,. (/),,F "Ï., " Q)

" Í" . ;.:: 0.50

.. ~10 S

~ï~

0.000.00 2.00 4.00 6.00 8.00

cjbsFig. 3. Variation of the normalized transformed SIF [KIIl/F(2/b)lf2], versus (c/bs), where s is taken as a real variable.

1.60

~-1.20(/)

"ÏQ)Í;.:: 0.80~

S~ï~ 0.40 c = ,200 m,/sec

b = 0.02 mL/b = 1.0

0.00ï 0.00000 0.00004 0.00008 0.000' 2 0.000,6. time (sec)- Fig.4. Variation of the normalized SIF [KIu/F(2/b)lf2], versus time, for L/b = 1.0."

infinite extent containing a semi-infinite crack sheared by point forces, once the initial waíefrontemitted at (÷ = - L, Õ = Ï, t = Ï) reaches the crack tip. The latter íalue of the transient SIFremains steady (or almost steady - due to some small ineíitable inaccuracy of the numerical

60 H.G. Georgiadis and Í. Charalambakis

1.60

é-ïÓ..é-1.20(/)

'¼ ~Q,) -

Í .;.:: 0.80 .éro ~

S~ï~ 0.40 c = 1200 m/sec

b = 0.02 mL/b = 3.0

0.000.000000 0.000040 0.000080 0.000120 0.000160

time (sec)Fig.5. Variation of the normalized SIF [KIlI/F(2/bjlI2], versus time, for L/b = 3.0.

Laplace transform inversion) until the first reflection of the initial SH wavefront, ïç the lateralstrip faces Õ = :t b, reaches the crack tip. Then, a step-type increase of SIF occurs again.

É Afterwards, several reflections of the diffracted field cïçtÞbute more or less severe KIII(t)vaÞatßïçs until a steady-state is reached. For L/b = 1.0 (Fig. 4), dynamic effects are pronounced

É for a time int~rval of about ~60 ìsec, for instanc~. . . . .Our numencal results, WhlCh extend over a wlde range of (L/b)-ratl0s, mdlcate dynamlc SIF

overshoots ïç the order of 25-40 percent ßç respect to the steady-state (static) value. This means,of course, that the possibility for an abrupt catastrophic crack propagation increases ßç thedynamic case, ßç comparison with the analogous static situation under the same external

loading amplitude.Éç conclusion, the present work accompanies a recent attempt for analyzing with an exact

method the time-harmonic problem of a cracked elastic strip subjected to a pair of concen-trated anti-plane shear forces (Georgiadis and Brock [29]). Here, an analytical/numericalprocedure is introduced for analyzing the transient version of the aforementioned problem. Éçthe present approach, use was made of certain results of analytic-function and integral-transform theory ßç order to obtain an exact and closed-form solution ßç the Laplacetransform plane. Áç efficient numerical technique was employed next ßç order to invert thetranformed SIF. ~

AcknowIedgements ~

HGG is grateful to Prof. L.M. Brock (University of Kentucky, USA) for several discussions ïçthis problem and also ïç related crack problems.

Crácked elástic strips 61

References

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