Internátioná/ Journá/ ï! Frácture 63: 201-214, 1993.(ò) 1993 K/uwer Acádemic Pub/ishers. Printed in the Nether/ánds. 201

Áç exact method for cracked elastic strips under concentratedloads - time-harmonic response

H.G. GEORGIADlS1 and L.M. BROCK 2Jr 1 Mechánics Division, Cámpus Âï÷ 422, Aristot/e University ï! Thessá/oniki, 54006, Greece

2Depártment ï! Engineering Mechánics, University ï! Kentucky, Lexington, Kentucky 40506-0046, USA

É,

J Received 1 ÁñÞ!1993; accepted 6 Ju!y 1993

Abstract. Áç exact method is presented for the determination of the near-tip stress fie!d arising from the scattering ofSH waves by a !ong crack in a strip-!ike e!astic body. The waves are generated by a concentrated anti-p!ane shear forceacting ïç each face of the crack. Time-harmonic variation of the externa!!oading is assumed. The prob!em has twocharacteristic !engths, i.e. the strip width, and the distance between the point of app!ication of the concentrated forcesand the crack tip. It is we!!-known that the second characteristic !ength introduces a serious difficu!ty in themathematica! ana!ysis of the prob!em: a non-standard Wiener-Hopf (W-H) equation arises, one that contains a forcingterm with unbounded behavior at infinity in the transform p!ane. Éç addition, the presence of the strip's finite width

resu!ts in a comp!icated W-H kernel. Neverthe!ess, a procedure is described here which circumvents the aforementioneddifficu!ties and ho!ds hope for so!ving more comp!icated prob!ems (e.g. the p!ane-stress/strain version of the presentprob!em) having simi!ar features. The method is based ïç integra! transform ana!ysis, an exact kerne! factorization and

usage of certain theorems of ana!ytic function theory. Numerica! resu!ts for the stress-intensity-factor dependence uponthe ratio of characteristic !engths and the externa!!oad frequency are presented.

1. Introduction

Within the realm of linear elastic fracture mechanics, major attention is usually focused õñïçthe determination of the stress intensity factor (SIF) associated with the tips of sharp cracks. Asis well-known, this factor is indispensable for applying the Griffith-Irwin fracture concepts

[1-3].Éç this context, an important class of problems concerns elastic stress-wave diffraction by

cracks. Such problems are encountered when rapidly varying loads are applied to a body thatcontains stress concentrators, such as narrow cuts or slits, and therefore the computation of thefield requires that inertia effects be taken into account. Contrary to similar situations ßç

't acoustics, geodynamics and electromagnetic waves, ßç fracture mechanics most of the attention

is focused ïç the field behavior near the crack tip for a wide range of load frequencies. Typical'. of such work for either time-harmonic or transient response is that done by Sih and co-workers

[4, 5], Achenbach [6], Brock [7-9], Keer et al. [10], Jiang and Knowles [11], and Georgiadis

[12, 13], among others.Éç these studies, it was observed that the ratio of dynamic SIF to corresponding static SIF

departs from unity as the frequency (ßç a time-harmonic steady problem) or the time (ßç atransient IJroblem) varies, exhibiting a pattern of local maxima and minima. Dynamic SIFovershoots are then possible and it is, therefore, useful to determine such effects by an exact

analysis.Éç the present study, we deal with the time-harmonic elastodynamic problem of a stÞÑ-Éßke

body containing a semi-infinite planar crack which is acted õñïç by a pair of concentratedanti-plane shear forces. As depicted ßç Fig. 1, the crack is situated at the mid-width plane of the

202 H.G. Georgiadis and L.M. Brock

Õ.ÉÉ

É bÑ(ßÙt) é .

Ï ÷ '

b

ÉFig. 1. Áç elastic strip containing a semi-infinite crack under the action of a pair of concentrated anti-plane forces.Time-harmonic response is assumed.

strip arid the two equal and opposite forces are applied along the crack faces at a distance Lfrom the crack éßñ. Thus, there are two characteristic lengths ßç the problem, i.e. the stripsemi-width b and the distance L. The surface tractioQ& vary ßç a time-harmonic manner, and ifthese have been operating for a long time, ßé is reasonable Éï assume that a steady state hasbeen reached. Under that assumption, the governing equation for the out-of-plane displacementis the Helmholtz partial differential equation.

lç a brief discussion of some mathematical aspects of the problem, we note that, ßç view ofsymmetry, the Helmholtz equation wil1 be solved ßç the upper (or lower) half of the strip-likedomain, ßç the presence of the characteristic length L and mixed boundary conditions. Whilethe difficulties associated with the finite strip-width can be handled ßç several ways (e.g.Georgiadis et a1. [14-17] and references cited therein), ßé is general1y known that the existenceof the characteristic length L induced by the point-force.boundary condition introduces seriousdifficulties ßç analyses where potential-theory methods are çïÉ applicable (see e.g. Freund [18,19], Brock [8, 9, 20], Georgiadis et al. [17], and Êõï [21, 22]). ÂÕ contrast, ßé is a1so noted thatanalogous boundary conditions with characteristic lengths ßç problems involving Laplace-typeequations have successful1y been treated by Barenblatt and co-workers [2, 23-25], and Sih [26].The latter studies concerned the steady-state wedging of a long crack (the motions of the wedgeand the crack éßñ were assumed Éï occur at constant velocities) ßç an unbounded body, andjorthe elastostatic opening of a crack ßç anti-plane shear by ñïßçÉ forces. However, a similarapproach, i.e. one based ïç the conformal mapping technique and the Poisson, Schwartz orKeldysh-Sedov integral formulas, apparently cannot be applied Éï situations involving Helm-holtz or wave-type governing equations (ßÉ seems also that even biharmonic problems involvingstrip-like domains cannot be treated by such an approach, see e.g. Gladwel1 [27], andGeorgiadis et al. [15-17]).

ÉÉ is wel1-known that, for problems involving Helmholtz or wave equations and exhibiting çïself-similarity, integral transforms ßç conjunction with the Wiener-Hopf technique are conveni-ent solution techniques [19, 28, 29]. Nevertheless, when a characteristic length is present ßç the

Crácked elástic strips 203

forcing function (like the length L ßç the present problem), an exponential term with unboundedbehavior at infinity aÞses ßç the transform plane, which, ßç light of Liouvil1e's theorem, impliesthat the entire function coming from analytic-continuation arguments is an infinite-degreepolynomialo Clearly, there are not enough physical conditions to determine the unknown

coefficients of such a polynomialoÔï deal with this difficulty ßç the class of problems involving Helmholtz or wave equations

. and a characteristic length ßç the forcing function, an exact approach has been proposedpreviously [8, 9, 18-20], which is based ïç superpositiono This procedure may, however, beÉabïÞïus ßç geïmetÞes involving stÞÑ-Éßke domainso As an alternative, a direct approach whichis based ïç the exact solution of an equation through contour integration and Cauchy'stheorem, is employed here. It is hoped that this method, which could be viewed as ageneralization of the classical Wiener-Hopf method, wil1 be effective ßç dealing with the evenmore complicated situations involving plane-stress/strain conditions.

2. Problem statement

Consider an elastic body ßç the form of an infinitely long stÞñ containing a semi-infinite crack,see Fig. 10 Áç Qxyz Cartesian coordinate system is attached to the cracked body with the originat the crack tipo The strip occupies the region (- 00 < ÷ < 00, - b < Õ < b), and is thick enoughßç the z-direction to al1ow a state of anti-plane shearo The crack is situated along the plane(- 00 < ÷ < ï, Õ = Ï) and is sheared by a pair of concentrated anti-plane forces :!:F e×Ñ(ßÙt),independent of the z-coordinate, acting along (÷ = - L, Õ = 0)0 The faces of the crack aretraction free, except for the point of application of the concentrated forceso Because ofanti-symmetry with respect to the plane Õ = Ï, the problem can be viewed as a haÉf-stÞÑproblem with the mateÞaÉ 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. Further,we also assume that after a sufficient time, a time-harmonic steady state has been reached.Therefore, the pertinent equations governing such an elastodynamic deformation are written as

(see e.g. Achenbach [28])

Ux, uY = ï, uz(x, Õ, t) = uz(x, Õ) e×Ñ(ßÙt), (1)

auz auzó÷, óÕ' óÆ, Ô×Õ = ï, Ô×Æ = ì~, ÔÕÆ = ì-a;;' (2)

a2uz fJ2uz 2 Ù-a 2 + -a2 - á uz = Ï, á = -;-, (3)÷ Õ IC

where (ux, UY' uz) and (ó ÷, ,,0, ÔÕÆ) are the components of the displacement vector and stress

tensor, respectively, ì is the shear modulus, c is the shear-wave velocity, Ù is the circular

frequency of the disturbing force, á is the wave number, and ß = (-1)1/2.This discussion and Fig. 1 suggest that the associated boundary value problem must satisfy

the fol1owing conditions:

--

204 H.G. Georgiádis ánd L.M. Brock

(ß) Boundáry conditions

ÔÕÆ(×, b) = Ï, for - 00 < × < 00, (4)

ÔÕÆ(×, Ï) = Ñä(÷ + L), for - 00 < × < Ï, (5)

uz(x, Ï) = Ï, for Ï < × < 00;:1; (6)

where ä(.) is the Dirac delta function. .(ßß) Edge conditions

ÔÕÆ(×, Ï) = o(ljx), for × -+ 0+, (7)

uz(x, Ï) = 0(1), for × -+ Ï-. (8)

The edge conditions merely state that the near-tip stress and displacement fields cannot be sosingular as Éï correspond Éï line sources of radiated energy. Furthermore, ïç the basis offracture mechanics considerations (see e.g. [1-3,19, 30]), ßé can be shown that ÔÕÆ(×, Ï) '" ÷- 1/2for × -+ 0+ and uz(x, Ï) '" ÷1/2 and × -+ Ï-. However, (7) and (8) are still sufficient conditions for

applying Jordan's lemma ßç subsequent steps of our analysis.

(ßßß) Rádiátion condition

!ÔÕÆ(×'Ï)É < Á exp( - Ñô×) for × -+ + ~, (9)

luz(x,O)I<Bexp(pux) for ×-+-ïï, (10)

where Á, Â, Ñô and Ñõ are positive constants. These equations state that the diffraction field atinfinity consists of outgoing waves ïçlÕ.

The objective here is the exact determination of the stress field near Éï the crack éßñ for theproblem defined by (IÇÉ0), and further, the study of the SIF dependence õñïç geometry and

frequency.

3. Analysis

The first step ßç solving the problem described ßç the previous section is the introduction of thetwo-sided Laplace-transform pair

f +oo j*(p,y)= -00 j(x,y)e-PX dx, (11)

j(x,y) = -21.1 j*(p,y)ePXdp, (12)ðé Â.

Crácked elástic strips 205

where Br denotes the Bromwich path ßç the complex p-plane. Éç addition, the half-linetransforms of the unknown functions ÔãÆ(×, Ï), Ï < × < 00, and uz(x, Ï), - 00 < × < Ï

Ô+(Ñ)=fï"'ÔãÆ(÷,ï)e-Ñ×d× for -pr<Re(p), (13)

1 {ÔãÆ(×, Ï) = -2 ' ô+ (ñ) ePX dp for Ï < × < oo~ (14)ðé ÂÃÉ

. U-(P)=f:",Uz(X,O)e-PXdX for Re(p) <Pu, (15)

uz(x,O) = -2 1. { U-(p)ePXdp for -00 < × < Ï, (16)ðé Br2

are defined, where, ßç light of conditions (9) and (10), ô+ (ñ) and U- (ñ) are analytic functions ßç

the right, -Ñô < Re(p), and left, Re(p) < Pu, half-plane, respectively.Application of.(11) Éï the governing equation (3) results ßç an ordinary differential equation

having the general solution

u:(p, Õ) = C(p) exp[ -(1÷2 - ñ2)1/2 Õ] + D(p) exp[(1X2 - p2f/2 Õ], (17)

where C(p) and D(p) are unknown functions. Then, applying (11) Éï the boundary conditions(4)-(6), ßç view of (13) and (15), and eliminating C(p) and D(p) from the resulting system gives the

following equation

, Ô+(ñ) + FeLP = - ìê(ñ)õ-(ñ), (18)

where the kernel Ê(ñ) is given by

Ê(ñ) = (1÷2 - p2f/2 tanh[(1X2 - p2)1/2b], (19)

and, ßç view of (13) and (15), ßé is noted that (18) holds over a common region of analyticitydefined by the strip - Ñô < Re(p) < Pu.

The problem has now been reduced Éï the determination of the unknown functions ô+ (ñ)and U- (ñ) from the single equation (18). This is possible by supplying (18) with results obtainedby using Cauchy's theorem and Jordan's lemma. Áé this ñïßçÉ, ßé should be noted that theconventional Wiener-Hopf technique [28, 29] is çïÉ applicable Éï (18), since the term eLp isunbounded as É pl-+ 00. This behavior leads Éï an unfortunate consequence of Liouville'stheorem which is indispensable ßç applying the W-H technique. ÉÉ must also be noted that anattempt [21, 22] at a formal splitting ofthe term [eLPjK+(p)] could be ßç error, since a non-zerocïntÞbutßïn of an integral over a semi-circular arc at infinity was apparently omitted ßçapplying Cauchy's integral formula (the term ê+ (ñ) results from a standard product factoriz-ation of the kernel). Thus, the usual argument of analytic continuation ßç conjunction withLiouville's theorem (which are the basic ingredients of the W-H method) is problematic ßç this

206 H.G. Georgiádis ánd L.M. Brock

case, and a different approach should be followed. The one employed here makes use of simplecontour integration along with a product kernel factorization, and, thus could be regarded as ageneralization of the classical W-H method.

Ôï proceed further a product-factorization of the kernel Ê(ñ) is required. Áç exact andclosed-form factorization is necessary because, as will be seen later, ê+ (ñ) appears ßç aCauchy-type integrand and, therefore, the influence of all poles of this integrand must be takeninto account. We employ the infinite-product forms of the pertinent hyperbolic trigonometricfunctions (see e.g. Abramowitz and Stegun [31], ñ. 85) and find

Ê(ñ) = b(tX2 - ñ2) Ð [ (2k _;)2(ñ + Ak)(p - Ak)], (20)'k=1 4k (ñ + Bk)(p-Bk)

where

[( kð)2 ]1/2 [( (2k - 1)ð)2 ]1/2 Ak = b + á2 , Bk = 2b + á2 . (21)

,1,)

Á product factïÞÆatßïn follows easily as

Ê(ñ) = Ê+(ñ)Ê-(ñ), (22)

where

Ê+(ñ) = b1/2(tX + ñ) Ð [ (2k - 1)(ñ + Ak)], (23)k= 1 2k(p + Bk)

Ê-(ñ) = Ê+(-ñ), (24)

and the functions ê+ (ñ) and Ê- (ñ) are non-zero and analytic ßç Re(p) > - inf(Ak= 1, Bk= 1) andßç Re(p) < inf(Ak=1, Bk=1), respectively.

Éç view of (22Ç24), Eqn. (18) can now be rewÞtten as

Ô+(ñ) FeLP --ê+(Ñ) + ê+(Ñ) = - ìÊ (p)U (ñ) (25)

ßç the strip - inf(PT,Ak=1,Bk=1) < Re(p) < inf(Pu,Ak=1,Bk=1).It is convenient to change the vaÞabÉe from Ñ to ù and divide both sides of (25) by 2ðß(ù - ñ), .

getting

Ô+(ù) + FeLw = - ìÊ-(ù)õ-(ù). (26)2ðßÊ+(ù)(ù - ñ) 2ðßÊ+(ù)(ù - ñ) 2ðß(ù - ñ)

ÂÕ taking the point Ñ to lie ßç the Þght half-plane Re(ù) > Ï, (26) can be integrated over theimaginary axis ß Ém(ù) to yield the result

1 ÉßÏÏ Ô+ (ù) F ÉßÏÏ eLw

-dù + -dù2ðß -ßïïÊ+(ù)(ù-ñ) 2ðß -ßïïÊ+(ù)(ù-ñ)

Crácked elástic strips 207

ìfiOO Ê-(ù)õ-(ù)

,= - -2 . dù. (27)ðé -icx> ù - Ñ

Éç the last integral ßç (27), the edge condition that uz(x, Ï) ~ ÷É/2 for ÷ -+ 0- ßç conjunctionwith the Abel- Tauber theorem [28, 29, 32] can be utilized to show that Ê- (ù) ~ ùÉ/2 for

ÉùÉ-+ 00, õ-(ù) ~ ù-3/2 for ÉùÉ-+ 00. Moreover the integrand is an analytic function ßç the lefthalf-plane Re(ù) < ï. Therefore, by applying contour integration (Fig. 2) and Jordan's lemma[32] to this integral, one can conclude that it vanishes when Ñ belongs to the right half-planeRe(ù) > ï.. Now, by employing again contour integration and Jordan's lemma ïç the first integral ßç (27),

and by closing the integration path with a large semi-circle at infinity ßç the Þght half-plane,Cauchy's integral formula can be utilized to give

ô+ (ñ) F fiOO eLro .Ê-+ôÑ) = - ~ -úïïÊ+(ù)(ù - Ñ)dù, (ñ éç Re(ù) > Ï). (28)

., , '

The integral ßç (28) can formally be evaluated by closing the integration path with a largesemi-circle at infinity ßç the left half-plane. Jordan's lemma along with the Cauchy's residuetheorem then leads to the following expression for the transformed crack-line stress

Ô+(ñ)= -~K+(P) ([ Ó [~~ [Ii (~~~ )]]] +b j=l ù+é÷ ù-ñ k=l 2k-l ù+.1lk ro=-Aj

[Ðïï ( 2k -é÷ + Bk)] e-L« )+ - (29)k=l 2k - 1 -é÷ + Ak -é÷ - Ñ .

Here, (23) has also been employed, and the Aj above are given by (21) with j replacing k.

ß Im(w )

ÈÑ

Re(w)

Fig. 2. Deformation of the original integration path ßç the complex ù-ÑÉane.

208 H.G. Georgiádis ánd L.M. Brock

Equation (29) may provide, through the Laplace-transform inversion (14), the crack-linestress. However, ßç the next section, ïçlÕ the singular part of this stress and the associatedintensity factor will be obtained.

4. Singular crack-line stress

Based ïç the Abel- Tauber theorem [28, 29, 32], which relates asymptotical1y functions and theirtransforms, the singular part of stress, limx-"o+ ÔÕÆ(×, Ï), from the large-p approximation of (29),limp-..oo T+(p)can be calculated. .

First, the asymptotic expression for the kernel ßç (19) can be found as

lim Ê(ñ) = - ßñ (30)ñ-" 00

where Ñ is taken along the pertinent Bromwich path. Then, ïç rewriting (33) ßç the form

Ê(ïï) = _(å2_ñ2)É/2, withB-+O, (31)

the asymptotic kernel factorization fol1ows easily as

Ê+(ïï) = ñÉ/2, Ê-(ïï) = - (_ñ)É/2. (32)

Furthermore, the terms (ù - ñ)-É and (-á - ñ)-É ßç (29) are approximated by (- ñ)-É.

Éç light of the above considerations, (29) becomes

. + F 1 ([ Óïï [ù + Aj Lw11m Ô (ñ) = -bl/2 -úú2e .ñ-"ïï Ñ j=l ù+á

.[ð (~~~ )]]] +k=l 2k - 1 ù + Ak ù=-ë)

.+[ð (~ -á + Bk)] e-L«), (33)

k=é2k-1-á+Ák .and its inverse fol1ows from (14) as

:'

lim ÔÕÆ(×, Ï) =( b'p)l/2 (...), (34)

÷-..ï+ ð ×

where the expression ßç the brackets is the same as that ßç (33).Final1y, from the definition of the stress intensity factor as

ÊÉÉÉ = lim [(2ð÷)É/2ÔÕÆ(×, Ï)], (35)÷-..ï+

Crácked elástic strips 209

we find

(2)1/2 ([ 00 [ ù+ Aj L Êééé = F b j~1 -;;;-+-;;e ù.

. .[n (~~~~ )JJJ +k=1 2k - 1 ù + Ak w=-Aj

. [ÐÏÏ ( 2k -á+ Bk)J -L« )+ k=1 2k=1 -á + Ak e . (36)

Equation (36) provides a closed-form, exact formula for the SIF as a function of loading,geometry, shear-wave speed, and frequency. The infinite series and products are convergent.Now, by considering a zero frequency, i.e. setting á = Ï ßç (36), the SIF for the analogous static

problem follows as

[ 2J1/2 K11rtic = F b [1 + te-nLlb + ie-2nLlb +He-3nLlb + ...]. (37)

The result ßç (37) is identical Éï the one obtained by Sih [26] ßç treating the static problem bythe conformal mapping technique. This comparison serves as one check ïç the method and the

result (36).

5. Results and brief discussion

Éç this section, representative results from (36) for the SIF behavior are given. Éç obtaining theseresults, the standard computer program MATHCAD was used. Moreover, 3 terms ßç the infiniteseries and 1000 terms ßç the infinite products were retained. The formula resulting from (36),which was used ßç all computations, reads as

. Êééé = F(2/bfI2 {2 exp( - LIX[1 + (ð/ábf]1/2).

. ~/,'-~[1 + (ð/ábf]1/2 + [1 + (ð/2ábf]1/2. .

1 - [1 +(ð/ábf]1/2

. Ð (~ -[1 + (ð/ábf] 1/2 +[1 + «2k - I)Ð/2ábf]1/2)k=2 2k - 1 -[1 + (ð!ábf]1/2 + [1 + (kð/ábf]1/2 +

+ (two next terms) +'. ,

+ e-L«.kQ1(~ (38)

210 H.G. Georgiádis ánd L.M. Brock

With these terms, the series seemed to have a rapid convergence and for all values of (L/b) theinfinite products showed little variation after at least 10 terms were considered. The computa-tion time ïç a PC 386 for getting a SIF value for each combination of characteristic lengths andfrequency was typically 5 sec.

Figures 3, 4 and 5 present the vaÞatßïç of the normalized SIF, [(Re êééé)2 +(1m KIII)2]1/2/K:lrtic, versus the normalized frequency, (ÙL/c), for the cases L/b = 2, 3 and 5,respectively. It is seen from these plots that for small frequencies the normalized SIF takes ;

values close to unity, but for larger frequencies local minima and maxima occur. One canobserve, for instance, dynamic SIF overshoots, i.e. peaks of the dynamic SIF value ßç respect tothe static one, for certain ranges of frequency. Appreciable overshoots ïç the order of 60 or 75percent often appeared ßç the computations. This means, of course, that the possibility for anabrupt catastrophic crack propagation increases ßç the dynamic case, ßç cïmÑaÞsïç to theanalogous static situation under the same extemalloading amplitude. Another observation thatcould be inferred from the results is that, for an increasing ratio L/b, the frequency at whichovershoot occurs is itself increased.

Next, Figs. 6, 7 and 8 show the variation of the normalized SIF, [(Re êééé)2 + (1m Êééé)2] 1/2/F(2jb) 1/2, versus the ratio Ljb, for Ù = 3 ÷ 104 ÇÆ, 12 ÷ 104 ÇÆ and 18 ÷ 104 ÇÆ, respect-ively. The SIF behavior for the case Ù = 3 ÷ 104 ÇÆ is very similar to that ßç the static caseÙ = Ï (which is not included here but can be found ßç Sih [26]), but departures from the lattersituation can be observed with increasing frequency, i.e. for Ù = 12 ÷ 104 ÇÆ and 18 ÷ 104 ÇÆ.

2.00

L/b=2

1.50~-C/)

'¼Q)Í~ 1.00~ -S '

~ï~ ï

0.50

0.000.00 2.00 4.00 6.00

normalized frequencyFig. 3. Variation of the norrnalized SIF, [(Re Êééé)2 + (1m KIII)2]1/2/KIIrtic, versus norrnalized frequency, (ÙL/c), forL/b = 2.

Crácked elástic strips 211

2.00

L/b=3

1.50

~. U)

"t:1. ~ 1.0

---~S~;:)~ 0.5

0.000.00 4.00 8. . Ï 16.00

normalized frequency

Fig. 4. Variation of the norrnalized SIF, [(Re Êééé)2 + (1m Êééé)2]É/2/ê!/é"ÉÉï, versus norrna1ized frequency, (áé/b), for

L/b = 3.

2.00

L/b=5

1.50

~U)

"t:1

.. ~ 1.00

---, ~, dß - Ì!É " ~r ;:) ~!1 0.50

0.000.00 10.00 . 30.00

normalized frequency

Fig. 5. VaÞatßïn of the norrnalized SIF, [(Re Êééé)2 + (1m KIII)2]l/2/K~:ttiC, versus norrnalized frequency, (ÙL/b), forL/b = 5.

-

212 H.G. Georgiadis and L.M. Brock

2.004frequency = 3* 1 Ï ÇÆ

1.50~ "-"J)

"¼~ 1.00 "-~S~ï~ 0.50

0.000.00 2.00 4.00

L/b

Fig.6. Variation of the normalized SIF, [(Re KIIIf + (1m KIII)2]1/2/F(2/bj1I2, versus L/b, for Ù = 3 ÷ 104 ÇÆ.

2.00

frequency = 12*104ÇÆ

1.50

~-"J)1=' ~~ 1.06 .-~ :;,~ .;S ... ",~ .

~~ '"ï -~ 0.5

0.0. . 4.00

L/bFig. 7. Variation of the normalized SIF, [(Re êééé)2 + (1m KIII)2]lI2/F(2/b)lI2, versus L/b, for Ù = 12 ÷ 104 ÇÆ.

Crácked elástic strips 213

2.00(( " ,. frequency = 18*104 ÇÆ

1.50

~-~" 'ó'~ 1.00'é-4-~S~c~ b.56

0.000.00 2.00 4.00

L/b

Fig.8. Variation of the nonnalized SIF, [(Re êééé)2 + (1m Krn)2]1/2fF(2/b)I/2, versus L/b, for Ù = 18 ÷ 104 ÇÆ.

6. Conclusions

Éç this work, an exact method for solving a class of fracture mechanics problems in which thegoverning equation is a two-dimensional Helmholtz or a wave equation supplied with boundaryconditions involving two characteristic lengths was used. Specifically, this method dealt with anelastodynamic problem of a cracked strip under concentrated 10ads. The first characteristiclength was the strip width and the second one was the distance between the point of application

- of the pair of concentrated forces. and the crack tip. The particular case of diffraction of SH

waves by a 10ng crack in a stÞÑ-Éike elastic body was worked out and numeÞcaÉ resultsconcerning the SIF behavior were obtained.

, Éç the present approach, use was made of some standard results of analytic function theory! (contour integration, Cauchy's integral formula, Jordan's lemma) and integral transform theory

(Abel- Tauber theorems) in order to solve a single equation containing two unknown complexfunctions. It is hoped that this method can also be applied to boundary value problemsoccurring in different physical contexts, e.g. the scattering of water waves by two overlappingbarriers (which is a problem of paramount importance in the design of harbors) [33].

Acknowledgement

This research was supported in part by NSF Grant MSM 8917944 to L.M. Brock.

214 H.G. Georgiádis ánd L.M. Brock

References

1. G.R. Irwin, ßç Hándbuck der Physik 6, SÑÞnger Verlag, Berlin (1958) 551-590.2. G.I. Barenblatt, ßç Advánces in App/ied Mechánics 7, Academic Press, New York (1962) 55-129.3. J.R. Rice, ßç Frácture 2, Academic Press, New York (1968) 191-311.4. G.C. Sih, Internátioná/ Journá/ ï! Frácture 4 (1968) 51-68.5. G.C. Sih and Å.Ñ. Chen, ßç E/ástodynámic Cráck Prob/ems (Mechánics ï! Frácture 4), Noordhoof, Leyden (1977)

1-58. ~,6. J.D. Achenbach, ßç Mechánics Todáy 1, Pergamon Press, London (1971) 1-57. .7. L.M. Brock, Internátiona/ Journá/ ï! Engineering Science 13 (1975) 851-859. .8. L.M. Brock, Internátioná/ Journá/ ï! Solids áÌ Structures 18 (1982) 467-477. ~"9. L.M. Brock, Quárter/y ï! Applied Máthemátics 43 (1985) 201-210.

10. L.M. Keer, W. Lin and J. D. Achenbach, Journá/ ï! Applied Mechánics 51 (1984) 65-70. é,é.

11. Q. Jiang and J.K. Knowles, Internátioná/ Journá/ ï! Frácture 41 (1989) 283-288. ' ,

12. H.G. Georgiadis, ßç Dynámic Fái/ure ï! Máteriá/s, Ç.Ñ. Rossmanith and A.J. Rosakis (eds.), Elsevier, London (1991)

296-308.13. H.G. Georgiadis et aI., Internátioná/ Journá/ ï! Engineering Science 29 (1991) 171-177.14. H.G. Georgiadis, Engineering Frácture Mechánics 24 (1986) 727-735; see also H.G. Georgiadis and P.S. ÔheïcaÞs,

Journá/ ï! Applied Máthemátics ánd Physics (ÆÁÌÑ) 36 (1985) 146-165.15. H.G. Georgiadis and G.A. Papadopoulos, Internátioná/ Journá/ ï! Frácture 34 (1987) 57-64.16. H.G. Georgiadis and G.A. Papadopoulos, Journá/ ï! Applied Máthemátics ánd Physics (ÆÁÌÑ) 39 (1988) 573-578.17. H.G. Georgiadis and G.A. Papadopoulos, Journá/ ï! Applied Máthemátics ánd Physics (ÆÁÌÑ) 41 (1990) 889-899.18. L.B. Freund, Internátioná/ Journá/ ï! Engineering Science 12 (1974) 179-189.

19. L.B. Freund, Dynámic Frácture Mechánics, CambÞdge University Press, CambÞdge (1990).20. L.M. Brock, Journá/ ï! E/ásticity 14 (1984) 4.15-424.21. Ì.Ê. Êõï, Internátioná/ Journá/ ï! Engineering Science 30 (1992) 199-211.22. Ì.Ê. Êõï and S.H. Cheng, Internátioná/ Journá/ ï! Solids ánd Structures 28 (1991) 751-768.23. G.I. Barenblatt and G.P. Cherepanov, Journá/ ï! Applied Máthemátics ánd Mechánics (ÑÌÌ) 24 (1960) 993-1014.24. G.I. Barenblatt and R.V. Goldstein, Internátioná/ Journá/ ï! Frácture 8 (1972) 424-434.25. G.I. Barenblatt and G.P. Cherepanov, Journá/ ï! Applied Máthemátics áÌ Mechánics (ÑÌÌ) 25 (1961) 1654-1666.26. G.C. Sih, Journa/ o!the Fránklin Institute 280 (1965) 139-149.27. G.M.L. Gladwell, Contact Prob/ems in the C/assicá/ Theory ï! E/asticity, Sijthoff and Noordhoff, Alphen aan den

Rijn (1980).28. J.D. Achenbach, Wáve Propágátion in E/ástic Solids, North-Holland, Amsterdam (1973).29. Â. Noble, Methods Básed on the Wiener-Hop!Technique, Pergamon Press, New York (1958).30. H.G. Georgiadis and J.R. Barber, Journá/ ï! E/ásticity 31 (1993) 141-161.31. Ì. Abramowitz and É.Á. Stegun (eds.), Hándbook ï! Máthemáticá/ Functions, Dover Publications, New York (1972).32. Â. van der Pol and Ç. Bremmer, Operátioná/ Cá/cu/us Based on the Two-sided Láp/áce Integrá/, CambÞdge

University Press, CambÞdge (1950).33. I.D. Abrahams and G.R. Wickham, Wáve Motion 14 (1991) 145-168.

~..

'é. ;.