Developing and comparing optimal and empirical land-use models for the development of an urbanized watershed forest in Taiwan

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PERGAMON International Journal of Solids and Structures 25 "0888# 3598Ð3516

9919!6572:88:, ! see front matter Þ 0888 Elsevier Science Ltd[ All rights reserved PII] S 9 9 1 9 ! 6 5 7 2 " 8 7 # 9 9 1 9 6 ! 7

Transient analysis of a propagating crack with _nite length

subjected to a horizontally polarized shear wave

Yi!Shyong Ing\ Chien!Ching Ma

Department of Mechanical Engineering\ National Taiwan University\ Taipei\ Taiwan 09506\ Republic of China Received 11 November 0886^ in revised form 18 June 0887

Abstract

In this study\ the transient response of a _nite crack subjected to an incident horizontally polarized shear wave and then propagated with a constant speed in an unbounded elastic solid is investigated[ Initially\ the _nite crack with crack length l is stress!free and at rest[ At time t  9\ an incident horizontally polarized shear wave strikes at one of the crack tips and will arrive at the other tip at a later time[ Then\ two crack tips propagate along the crack tip line with di}erent velocities as the corresponding stress intensity factors reach their fracture toughness[ The correspondent con_guration is shown in Fig[ 0[ In analyzing this problem\ di}racted waves generated by two propagating crack tips must be taken into account and it makes the analysis extremely di.cult[ In order to solve this problem\ the transform formula in the Laplace transform domain between moving and stationary coordinates is _rst established[ Complete solutions are determined by superposition of proposed fundamental solutions in the Laplace transform domain[ The fundamental solutions to be used are from the problems of applying exponentially distributed traction and screw dislocation on crack faces and along the crack tip line\ respectively[ The exact transient solutions of dynamic stress intensity factor for the _rst few di}racted waves that arrive at two crack tips are obtained and expressed in compact formulations[ Numerical calculations of dynamic stress intensity factors for both tips are evaluated and the results are discussed in detail[ Þ 0888 Elsevier Science Ltd[ All rights reserved[

0[ Introduction

Recently\ the transient response of a solid medium containing a crack!like ~aw under dynamic loads has received much attention[ Scattering of elastic waves by cracks has attracted attention over the years for its importance towards the nondestructive evaluation of cracked bodies and the dynamic fracture analysis of materials[ The interaction of a stress wave with a crack is a complicated problem and the analysis is mainly restricted to relatively simple problems[ Most of the work\ however\ has been directed towards the solution of problems with a semi!in_nite crack subjected

Fig[ 0[ Con_guration and coordinate systems of a _nite crack in an unbounded medium[

to distributed impact loading on crack faces[ The complete solutions mentioned above can be obtained by integral transform methods in conjunction with direct application of the WienerÐ Hopf technique "Noble\ 0847# and the CagniardÐde Hoop method "de Hoop\ 0847# of Laplace inversion[ If the cracked problem having a characteristic length or loading condition is unsym! metrical\ then the usual procedure using integral transform methods does not apply[

The stress intensity factors of a stationary _nite crack upon di}raction of a time!harmonic wave have been obtained by Loeber and Sih "0857# and Sih and Loeber "0857\ 0858#[ If integral transforms are applied to solve the transient response of a _nite crack subjected to dynamic loading\ a relationship among sectionally analytic function will be obtained which is more complicated than the form of the standard WienerÐHopf equations[ The generalized WienerÐHopf equation can be solved iteratively to obtain the complete transient solution\ and only the _rst step in the iteration process has been carried out[ Thau and Lu "0860#\ following the work of Kostrov "0853# and Flitman "0852#\ treated the analogous transient problem of di}raction of an arbitrary plane dilatational wave by a stationary _nite crack and a stationary _nite rigid ribbon in an in_nite elastic solid from the iteration process[ Their results are exact only at the time interval that the dilatational wave has traveled the length of the crack twice[ Sih and Embley "0861# have studied the near tip solution of a stationary _nite crack under transient in!plane loading[ They reduced the mixed boundary value problem to a standard Fredholm integral equation and subsequently inverted the Laplace transform of the stress components by a combination of numerical means and an application of the Cagniard inversion technique[ A class of problems involving interaction between a stationary _nite crack and other boundaries was considered by Chen "0866\ 0867# and Itou "0879\ 0870#[ With the exception of Loeber and Sih who considered the time!harmonic incident wave\ all of the authors mentioned have simpli_ed their problems by assuming sym! metrically distributed loading conditions\ and _nally used a numerical Laplace inversion technique to obtain the solutions in the physical domain[ Because of the mathematical di.culties\ the closed form analytical solution for the problem of a _nite crack subjected to transient waves is very rare[ The problem of an unbounded medium containing a stationary semi!in_nite crack subjected to a pair of concentrated inplane loadings on the crack faces has been investigated by Freund "0863#[

A straightforward application of the WienerÐHopf method is not successful and the transient solution of stress intensity factor was obtained by Freund "0863# by an indirect approach based on the superposition of moving dislocations[ He proposed a fundamental solution arising from an edge dislocation climbing along the line ahead of the crack tip with a constant speed to overcome the di.culties of the case with a characteristic length[ The solution can be constructed by taking an integration over a climbing dislocation of di}erent moving velocity[ Basing his procedure on this method\ Brock "0871\ 0873#\ Brock et al[ "0874#\ and Ma and Hou "0889\ 0880# have analyzed a series of problems of a semi!in_nite crack subjected to impact loading on crack faces[ A thorough summary of the application of main direct methods of analysis for transient problems in dynamic fracture for elastic or inelastic problems has been given by Freund "0889#[ Freund "0889# has suggested an alternate approach based on the aforementioned moving dislocation solution to examine the same _nite!crack problem that had been solved by Thau and Lu "0860#[ In practice\ however\ the alternate approach provided a solution that is valid for the same time range as before[ Kostrov "0855# and Achenback "0869a\ b# have used the method based on Green|s function to solve the problems of crack propagation for anti!plane deformation[ In their studies\ the region of integration for the integral equation is in a complicated shape\ generally being bounded by a hyperbola and a number of straight lines[ For points ahead of the crack tip\the region of integration reduces to a triangular region and the stress in the plane of the crack can thus be determined without di.culty[ However\ for material points not on the crack tip line\ the region of integration is very complicated and careful analysis is needed[ Scattering of plane harmonic waves by a running crack of _nite length was investigated by Chen and Sih "0864#[ They found the dynamic stress intensity factors and crack opening displacements of the _nite crack[ Exact transient closed form solutions for a stationary semi!in_nite crack subjected to a suddenly applied dynamic body force in an unbounded medium have been obtained by Tsai and Ma "0881# for the in!plane case and by Ma and Chen "0882# for the anti!plane case[ They determined the transient full _eld solutions by superimposing a fundamental solution in the Laplace transform domain[ The fundamental solution used in the problem is an exponentially distributed traction in the Laplace transform domain on the crack faces[ This fundamental solution has also successfully been applied to solve the problems of a half plane containing a semi!in_nite inclined crack by Tsai and Ma "0882# and Ma and Chen "0883# for in!plane and anti!plane problems respectively[ Brock "0864# has studied the transient response for di}raction of an incident horizontally polarized shear wave by a stationary _nite crack[ His results indicated that the peak dynamic stress intensity factors could occur after the arrival of the second wave\ which means that secondary di}ractions may produce even higher peaks than the earlier peaking[ Ing and Ma "0886# also investigated the same problem solved by Brock "0864# for the long time behavior by using superposition of new fundamental solutions in the Laplace transform domain[ Their results\ however\ indicate that the maximum dynamic stress intensity factor in the transient period always occurs at the instance that the second wave arrives at the right or the left crack tip[

In this study\ the transient response of a _nite crack subjected to an incident plane horizontally polarized shear wave and then propagates after some delay time\ is investigated[ The geometrical con_guration is shown in Fig[ 0[ If the stress intensity factor of the stationary crack tip is greater than the fracture toughness of the material\ then it is assumed that the crack tip will start to propagate along the crack tip line with a constant velocity[ Both tips\ however\ can propagate with di}erent velocities[ The propagation of crack with _nite length can simulate dynamic fracture

problem more realistically[ In analyzing this problem\ the waves di}racted between two stationary "and propagating# crack tips will make the analysis extremely di.cult[ It is impossible to solve this complicated problem by direct application of the standard WienerÐHopf technique and some other approach must be followed[ Two useful fundamental problems are proposed and used to overcome these di.culties[ The proposed fundamental problems\ which form a key element in the analysis\ are solved exactly by the WienerÐHopf method[ The proposed fundamental problems are the problems of applying exponentially distributed traction and screw dislocation on crack faces and along the crack tip line\ respectively[ The _rst few waves di}racted by the stationary and propagating crack tips are constructed by superposition of the proposed fundamental solutions[ Since the stress intensity factor is the key parameter in characterizing dynamic crack growth\ we will focus our attention mainly on the determination of the dynamic stress intensity factor[

1[ Proposed fundamental problems and fundamental solutions

Two alternative fundamental problems will be proposed and solved in this section which can then be used to construct the solution for the problem of a _nite crack subjected to plane polarized shear waves[ The solutions of an exponentially distributed traction applied at the propagating crack faces and exponentially distributed screw dislocations generated along the crack tip line in the Laplace transform domain will be referred to as the fundamental solutions[ The di}racted waves scattered from the crack tips can be constructed by superimposing the proposed fundamental solutions in the Laplace transform domain[

Consider a fundamental problem of anti!plane deformation for a semi!in_nite crack propagating in an unbound medium[ The crack propagates along the crack line with a constant velocity n which is less than the shear wave speed of the material[ In analyzing this problem\ it is convenient to express the governing equation of wave motion in the moving coordinates j−y as follows

"0−b1_v1_#1 1_w 1j1¦ 11_w 1y1¦1b 1_n1 1_w 1j1t−b 11 1_w 1t1 9\ "0#

where w is the out!of plane displacement\ and b is the slowness of the shear wave given by

b 0 cs



X

r m[

Here csis the shear wave speed\ m and r are the respective shear modulus and the mass density of

the material[ The coordinate j de_ned by j  x−nt is _xed with respect to the moving crack tip[ The nonvanishing shear stresses are

tyz m

1w

1y\ txz m 1w

1x[ "1#

1[0[ Fundamental solution of distributed loads on crack faces

We consider _rst that exponentially distributed tractions in the Laplace transformation domain are applied on the upper and lower crack faces of a propagating semi!in_nite crack[ Because the

tractions are equal but opposite on the two crack faces\ the problem can be viewed as a half!plane problem with the material occupying the region y − 9\ and subjected to the following mixed boundary conditions in the Laplace transform domain

t¹yz"j\ 9\s#  eshj for − ³ j ³ 9\ "2#

w¹ "j\ 9\s#  9 for 9 ³ j ³ [ "3# The Laplace transform parameter s is taken as a positive number and h is a constant[ The overbar symbol is used for denoting the transform on time t[ This fundamental problem can be solved by the application of the standard integral transform method[ Applying the one!sided Laplace trans! form over time\ the two!sided Laplace transform over j under the restriction of Re"h# × Re"l#\ _nally the WienerÐHopf technique is implemented[ The solutions of stresses and displacement in the Laplace transform domain\ for the boundary conditions "2# and "3#\ can be expressed as follows t¹yz"j\ y\ s#  0 1pi

g

a¦"l#e−s"ay−lj# a¦"h#"h−l# dl\ "4# t¹xz"j\ y\ s#  0 1pi

g

−le−s"ay−lj# a−"l#"h−l#a¦"h# dl\ "5# w¹ "j\ y\ s#  0 1pi

g

−e−s"ay−lj# msa−"l#"h−l#a¦"h# dl\ "6# where a"l#  a¦"l#a−"l#  zb¦l"0−bn#zb¦l"0¦bn#[ "7#

To ensure Re"a# − 9 everywhere in the l!plane\ branch cuts are introduced from b:"0¦bn# to \ and −b:"0−bn# to −[ The corresponding result of the dynamic stress intensity factor expressed in the Laplace transform domain is

K¹ "s#  lim

j:9t¹yz"j\ 9\ s# 

−z1"0−bn# zsa¦"h#

[ "8#

1[1[ Fundamental solution of screw dislocation distributed alon` the crack tip line

Consider a semi!in_nite crack contained in an unbounded medium[ A distributed screw dis! location ahead of the crack tip line yields the following boundary conditions in the Laplace transform domain

w¹ "j\ 9\s#  eshj for 9 ³ j ³ \ "09# t¹yz"j\ 9\s#  9 for − ³ j ³ 9[ "00#

The particular problem posed can be solved by means of the WienerÐHopf method[ The solutions of stresses and the displacement expressed in the Laplace transform domain are

t¹yz"j\ y\s# 

0 1pi

g

msa−"h#a¦"l#e−s"ay−lj#

h−l dl\ "01#

t¹xz"j\ y\s# 

0 1pi

g

−msla−"h#e−s"ay−lj#

a−"l#"h−l#

dl\ "02#

w¹ "j\ y\s#  0 1pi

g

−a−"h#e−s"ay−lj#

a−"l#"h−l#

dl\ "03#

The corresponding result of stress intensity factor expressed in the Laplace transform domain is K¹ "s#  −mz1s"0−bn#a−"h#[ "04#

2[ Coordinate transformation relations in the Laplace transform domain

The superposition method can be applied successfully only if the fundamental solutions and the integral function of superposition are described in the same coordinate system[ If they are not de_ned in the same coordinate system\ they have to transform into the same one[ Consider two moving coordinate systems "j\ y# and "j?\ y# whose extending velocities are nAand nB\ respectively\

i[e[ j  x−nAtand j?  x−nBt[ If a function described in the "j\ y# coordinate system is represented

in the Laplace transform domain as

Q¹ "j\ y\ s#  sn_esa

g

F"l#e−saA"l#y¦sljdl\ "05#

where n is an arbitrary integer\ and

aA"l#  aA¦"l#aA−"l#  zb¦l"0−bnA#zb−l"0¦bnA#[

Then it can be transformed into the "j?\ y# coordinate system with the following form

Q¹ "j?\ y\ s#  −sn esa Ðð0−l"nA−nB#Ł n−0_F

0

−l 0−l"nA−nB#

1

e−sa_B"l#y¦slðj−a"n_A−n_B#Ł dl\ "06# in which aB"l#  aB¦"l#aB−"l#  zb¦l"0−bnB#zb−l"0¦bnB#[

The transformation relations described in eqns "05# and "06# can be proved if one inverses these two equations to time domain[

3[ Dynamic stress intensity factors of two propagating crack tips

The evaluation of the stress intensity factor for a cracked body is a well!established concept in fracture mechanics\ and it represents the cornerstone of linear elastic fracture mechanics[ We will focus our attentions in this study mainly on the evaluation of the dynamic stress intensity factor[

A speci_c geometry to be considered here is an in_nite medium containing a _nite crack of length l as shown in Fig[ 0[ The origins of two stationary coordinate systems "x\ y# and "x?\ y?# are located at crack tips A and B\ respectively[ At time t  9\ an incident horizontally polarized shear wave arrives at the crack tip A\ and then\ two crack tips will propagate along the crack tip line with di}erent velocities as the corresponding stress intensity factors reach its fracture toughness[ The incident plane wave with an incident angle g is represented by the general form

wi

"x\ y\ t#  F"t¦bx cos g−by sin g#\ "07# where

F"t#  H"t#

g

t 9

f"t# dt\ "08#

in which F is identically zero when its argument is negative\ but is otherwise an arbitrary wave form[ Thus\ the medium ahead of the incident plane wave front is undisturbed[ In eqn "08#\ H" # denotes the Heaviside step function and g is the angle of the negative x!axis and the normal of the wavefront[ The position of the wavefront for time t ³ 9 is also shown in Fig[ 0[ Here the angle g is restricted to the range 9 ³ g ¾ p:1[

At time t  9\ the incident plane wavefront strikes the crack tip A and will generate plane re~ected and cylindrical di}racted waves[ Some time later\ i[e[\ t  bl cos g\ the incident plane wave will arrive at the crack tip B and another di}racted wave will be induced[ It is assumed that each crack tip will propagate along the crack tip line if the dynamic stress intensity factor of the tip reaches its fracture toughness Kc[ The di}racted waves induced from one crack tip will propagate

toward the other crack tip at a later time\ and it makes the problem more di.cult to solve because many waves will be generated from both tips[ An e}ective superposition scheme will be proposed in this study to solve this complicated problem[

The incident horizontally polarized shear wave expressed in eqn "07# will give rise to the following shear stress in the in_nite medium]

ti

yz"x\ y\ t#  −mb sin gf "t¦bx cos g−by sin g#H "t¦bx cos g−by sin g#[ "19#

Consider an incident step!stress wave for which

f"t# t9

mb[ "10#

Then the incident stress _led eqn "19# can be represented in the Laplace transform domain as

t¹i yz"x\ y\ s#  0 1pi

g

Gl t9sin g s"l−b cos g#e −slytan g¦slx dl\ "11#

or expressed in the "x?\ y?# coordinate system as

t¹i y?z?"x?\ y?\ s#  0 1pi

g

Gl t9sin g s"l−b cos g#e sly?tan g¦sl"x?¦l# dl[ "12#

as that derived for a stationary semi!in_nite crack lies in the plane y  9 and − ³ x ³ 9\ and is struck by the same incident plane wave[ The incident stress _eld t¹i

yz"x\ 9\ s# at y  9 generated by

the step!stress shear wave is

t¹i yz"x\ 9\ s#  0 1pi

g

_Gl t9sin g s"l−b cos g#e slx_{dl[ "13#}

The applied traction on the crack face\ in order to eliminate the incident wave as indicated in eqn "13#\ has the functional form eslx

[ Since the solutions of applying traction eshx

on stationary crack faces have been solved in Section 1 by setting n  9\ the re~ected and di}racted _elds can be constructed by superimposing the incident wave traction that is equal and opposite to eqn "13#[ When we combine eqns "6# and "13# "by setting n  9#\ the solution of displacement w¹A0dfor A0d wave "the _rst wave di}racted from the stationary crack tip A# in the upper plane can be expressed in the Laplace transform domain as follows

w¹A0d"x\ y\ s# −0 1pi

g

Gh0 t9sin g s"h0−bcos g# 0 1pi

g

Gh1 −e−say¦sh₁x msa¦"h0#"h0−h1#a "h1# dh1dh0 z1t9sin"g:1# mzbs1 0 1pi

g

_Gl e−say¦slx a "l#"l−b cos g#dl[ "14#

The corresponding stress intensity factor expressed in the Laplace transform domain is

K¹A0d "s# −0 1pi

g

Gl t9sin g s"l−b cos g#

6

−z1 zsa¦"l#

7

dl −1t9sin"g:1# s2:1_zb [ "15#

By using the CagniardÐde Hoop method of Laplace inversion\ the dynamic stress intensity factor at the crack tip A induced by the incident wave expressed in time domain will be

KA0d

"t#  −3t9

X

t

pbsin"g:1#H"t# "16# Equation "16# is a well known solution of a stationary semi!in_nite crack subjected to an incident plane wave[ After some delay time tA

f \ the dynamic stress intensity factor of tip A may reach its

fracture toughness Kc\ and the tip will begin to propagate[ The delay time t A

f can be determined

from eqn "16# as follows

b

−3t9

X

tA f

pbsin"g:1#

b

 Kc\ so

tA f  pb

0

Kc 3t9sin"g:1#

1

1 [ "17#

In this study\ it is assumed that the incident plane shear wave will always cause the crack tip A to propagate along the crack tip line[ Consequently\ the fracture toughness Kcmust be less than

the maximum amplitude of dynamic stress intensity factor of tip A\ i[e[\ Kc¾ K A

max"t#[ However\

it was also known from Ing and Ma "0886# that the stress intensity factor of tip A will arrive at its maximum amplitude at time t  bl"0¦cosg#[ So\ we have the condition that

Kc¾ =K A0d max"t#=  1t9

X

1l psin g[ "18# At time t  tA

f \ the dynamic stress intensity factor of crack tip A reaches its critical value and this

tip starts to propagate with a constant velocity nA[ The incident wave written in the Laplace

transform domain for the moving coordinate system "j\ y# will have the following form

t¹i yz"j\ y\ s#  0 1pi

g

_Gl t9sin g"0¦bnAcos g# sð"0¦bnAcos g#l−b cos gŁ e−slytan g¦sl"j−nAtAf#dl\ "29#

where j  x−nA"t−tAf#[ The applied traction on crack faces as expressed in eqn "29#\ has the

functional form eslj

[ The di}racted _eld generated from the propagating crack tip A can be constructed by superimposing the fundamental solution and the stress distribution in eqn "29#[ The result of displacement expressed in the Laplace transform domain will be

w¹A0n_{"j\ y\ s#}−0 1pi

g

_G h0 t9sin g"0¦bnAcos g# e−sh0nAt A f sð"0¦bnAcos g#h0−bcos gŁ 0 1pi

g

_G h1

−e−saAy¦sh1j

msaA¦"h0#"h0−h1#aA−"h1#

dh1dh0 z1t9sin"g:1#"0¦bnAcos g# 2:1 mzbs1 0 1pi

g

Gl e−saAy¦slj

aA−"l#ð0¦bnAcos g#l−b cos gŁ

dl[

"20# The dynamic stress intensity factor for a propagating crack in an in_nite medium can also be constructed by a similar manner[ The result in the Laplace transform domain can be obtained from eqns "09# and "29# and is expressed as follows

K¹A0n_"s#−0

1pi

g

_Gh0

t9sin g"0¦bnAcos g#e−slnAt

A f sð"0¦bnAcos g#l−b cos gŁ

6

−z1"0−bnA# zsaA¦"l#

7

dl  −z1t9sin"g:1#z0¦bnAcos g#"0−bnA# zbs2:1 e −sbnAtAfcos g:0¦bnAcos g_{[ "21#}

The inversion Laplace transform of eqn "21# will have the following form

KA0n_"t#−3t9sin"g:1#z"0¦bnAcos g#"0−bnA#

zpb

X

t−

bnAtAf cos g

0¦bnAcos g

H"t−tA

The Heaviside step function in eqn "22# results from the e}ect that KA0n_{"t# is valid only for time}

t × tA

f [ The dynamic stress intensity factor expressed in eqn "22# is a well!known solution for a

propagating semi!in_nite crack subjected to an incident step!stress wave[ The analogous solution has also been found by Ma and Burgers "0875# using a di}erent method[

Subsequently\ the incident plane wave will propagate toward the crack tip B and will be di}racted at time t  blcosg[ Following the similar procedure that is used for constructing the A0d wave\ the B0d wave "the _rst wave di}racted from the stationary crack tip B# can be constructed in the coordinate system "x?\ y?# by using eqns "12# and "6# "by setting n  9# as follows

w¹B0d"x?\ y?\ s# −0 1pi

g

Gh0 t9sin ge sh0l s"h0¦bcos g# 0 1pi

g

Gh1 −e−say?¦sh1x? msa¦"h0#"h0−h1#a "h1# dh1dh0 z1t9cos"g:1# e −sblcos g mzbs1 −0 1pi

g

Gl e−say?¦slx? a "l#"l¦b cos g#dl[ "23# The corresponding stress intensity factor at the crack tip B induced by the incident plane wave is

K¹B0d

"s# 1zbt9cos"g:1#e

−sblcos g

s2:1_zb [ "24#

The dynamic stress intensity factor at the crack tip B expressed in time domain will be

KB0d

"t#  3t9

X

t−blcos g

pb cos"g:1#H"t−bl cos g#[ "25# The results expressed in eqns "16# and "25# are well!known solutions of dynamic stress intensity factor for the _rst two di}ractions of a step!stress wave by a stationary _nite crack in an unbounded medium[ The same solutions have also been obtained by Achenbach "0869a# and Ing and Ma "0886# using di}erent methods[

Similarly\ after some delay time tB

f\ the crack tip B begins to propagate with a constant velocity

nBas the dynamic stress intensity factor exceeds its fracture toughness Kc[ It is assumed in this

study that the crack tip B starts to propagate before the A0d wave arrived the tip\ i[e[\ bl × tB

f × bl cos g[ The delay time t B

f can be obtained from eqn "25# as follows

3t9

X

tB f−bl cos g pb cos"g:1#  Kc and we have tB f  pb

0

Kc 3t9cos"g:1#

1

1 ¦blcos g[ "26#

Notice that the fracture toughness Kcmust be less than K B0d max"t#\ so we have Kc¾ =K B0d max"t#=  1t9

X

1l psin g[ "27#

The result in eqn "27# is the same as that in eqn "18#[ Consequently\ from eqns "18# and "27#\ we can obtain the maximum fracture toughness that allows both crack tips to propagate as follows

Kc\max1t9

X

1l

psin g[ "28#

Following the same procedure\ we represent the incident _eld in the moving coordinate system "j?\ y?# as t¹i y?z?"j?\ y?\ s#  0 1pi

g

_Gl t9sin g"0−bnBcos g# sð"0−bnBcos g#l¦b cos gŁ esly?tan g¦sl"j?−nBtBf¦l#dl\ "39#

where j?  x?−nB"t−tBf#[ The displacement _eld and the stress intensity factor KB0n"t# after the tip

B starts to propagate can be obtained from eqns "39#\ "6# and "8#\ and the _nal results are

w¹B0n"j?\ y?\ s# z1t9cos"g:1#"0−bnBcos g#

2:1 mzbs1 e sbcos g"nBtBf−l#:0−bnBcos g ×−0 1pi

g

Gl e−saBy?¦slj? aB−"l# ð"0−bnBcos g#l¦b cos gŁ dl[ "30# KB0n_"t#3t9cos"g:1#z"0−bnBcos g""0−bnB# zpb

X

t− bcos g"l−nBtBf# 0−bnBcos g H"t−tB f#[ "31#

When the di}racted B0d or the B0n wave arrives at the right tip of the _nite crack at a later time\ it carries a discontinuous displacement in the z!direction which violates the boundary condition for j × 9[ In order to satisfy the boundary condition where the displacement must be continuous for j × 9\ a distributed screw dislocation is required to close the opening displacement ahead of the propagating crack tip[ The di}racted A1d and A1n waves will be induced when the B0d and B0n waves arrive at the moving crack tip A\ respectively[ We change the formulation in eqns "23# and "30# to "j\ y# coordinate system by using the transformation relations established in Section 2\ then the displacements we must eliminate ahead of the propagating tip A are

w¹B0d"j\ 9\ s#  0 1pi

g

_G l z1t9cos"g:1#e−sblcos gesl"j−nAt A f¦l¦nAblcos g# mzbs1 "0−lnr# 2 "0−l0nB# 2:1_a B−"l0# ð"0−bnBcos g#l0¦bcos gŁ dl\ "32# w¹B0n"j\ 9\ s#  0 1pi

g

Gl z1t9cos"g:1#"0−bnBcos g#2:1e−stBesl"j−nAt A f−nBtBf¦l¦nrtB# mzbs1 "0−lnr# 2_a B−"l0# ð"0−bnBcos g#l0¦bcos gŁ dl\ "33#

where nr nA¦nBis the relative velocity between two moving coordinate systems and

l0 l 0−lnr \ tB bcos g"l−nBtBf# 0−bnBcos g [

− ³ j ³ 9[ The di}racted A1d and A1n waves generated from the propagating crack tip A can be obtained by superimposing the distributed dislocation that equal and opposite to eqns "32# and "33# ahead of the tip j × 9 in the Laplace transform domain\ respectively[ Here we only analyze the corresponding stress intensity factors by using eqns "32#\ "33#\ and "04# as follows

K¹A1d "s# −0 1pi

g

Gl z1t9cos"g:1#e−sblcos gesl"l−nAt A f¦nAblcos g# mzbs1 "0−lnr#2"0−l0nB#2:1aB−"l0# ð"0−bnBcos#gl0¦bcos gŁ "−mz1s"0−bvA#aA−"l## dl −0 1pi

g

Gl

1t9cos"g:1#z0−bnAe−sblcos gaA−"l# esl"l−nAt

A

f¦nAblcos g#

zbs2:1

"0−lnA# 2:1_a

A¦"l#ð"0¦bnAcos g#l−b cos gŁ

dl\ "34#

K¹A1n

"s# −0 1pi

g

Gl

1t9cos"g:1#z0−bnA"0−bnBcos g#2:1e−stBaA−"l# esl"l−nAt

A

f−nBtBf¦nrtB#

zbs2:1

"0−lnr# 2:1_a

A¦"l0#ð"0¦bnAcos g#l−b cos gŁ

"35#

Inversion of the Laplace transform of eqn "34# yields

KA1d_"t#3t9cos"g:1#z0−bnA p2:1_zb

g

t−blcos g b"l−nAtAf¦nAblcos g# 0−bnA ×zt−t−bl cos g zt¦bðl¦nA"t¦bl cos g−t A f #Ł ðl¦nA"t¦bl cos g−t A f #Ł 2:1 × ðl¦nA"bl cos g−t A f #Ł2:1 "t¦b cos gðl¦nA"t¦bl cos g−t A f #Ł# zt−bðl¦nA"t¦bl cos g−t A f #Ł ×dtH"t−tA1d#\ "36#

where tA1dis the arrival time of the B0d wave at the crack tip A and it can be expressed as

tA1d b"l−nAtAf¦nAblcos g# 0−bnA ¦blcos g blcos g¦b"l−nAt A f # 0−bnA [ "37#

The inverse Laplace transform of eqn "35# is

KA1n "t# 3t9cos"g:1#z0−bnA"0−bnBcos g# 2:1 p2:1_zb

g

t−tB b"l−nAtAf−nBtBf¦nrtB 0−bnA × zt−t−tB "l−nAt A f−nBt B f¦nrtB¦nrt# 2:1× z"0¦bnA#t¦b"l−nAt A f −nBt B f¦nrtB# ðt¦b cos g"l¦nAt−nAt A f −nBt B f¦nrtB#Ł × "l−nAt A f−nBtBf¦nrtB#2:1 z"0−bnA#t−b"l−nAtAf−nBtBf¦nrtB# dtH"t−tA1n#\ "38#

where tA1nis the arrival time of the B0n wave at the propagating crack tip A and it can be obtained as tA1n b"l−nAtAf #¦tBf 0−bnA [ "49#

In addition\ eqn "38# is valid only for l−nAtAf −nbtBf × 9[ But this is adaptable for most propagating

speed except for the high speed cases "nA:cs: 0\ nB:cs: 0#[

Similarly\ the B1d and the B1n di}racted waves scattering from the crack tip B will be induced after the A0d and the A0n waves passed the tip[ We change eqns "14# and "20# to "j?\ y?# coordinate system by using the transformation relations\ and the displacement _elds along the crack tip line in the Laplace transform domain will be

w¹A0d"j?\ 9\ s# −0 1pi

g

Gl z1t9sin"g:1#e sl"j?−nBtBf¦l# mzbs1 "0−lnr# 2 "0−l0nA# 2:1_a

A−"l0# ð"0¦bnAcosg#l0−bcos gŁ

dl\ "40# w¹A0n"j?\ 9\ s# −0 1pi

g

Gl z1t9sin"g:1#"0¦bnAcos g#2:1e−stAesl"j?−nAt A f−nBtBf¦l¦nrtA# mzbs1_"0−ln

r#2aA−"l0#ð"0¦bnAcos g#l0−bcos gŁ

dl\ "41# where tA bnAtAf cos g 0¦bnAcos g [ "42#

Using the fundamental solution in eqn "04#\ the stress intensity factors corresponding to the B1d and the B1n waves can be obtained as follows

K¹B1d "s#  0 1pi

g

Gl 1t9sin"g:1#z0−bnBaB−"l#esl"l−nBt B f# zbs2:1 "0−lnB# 2:1_a B¦"l# ð"0−bnBcosg#l¦b cos gŁ dl\ "43# K¹B1n "s#  0 1pi

g

Gl 1t9sin"g:1#z0−bnB"0¦bnAcos g# 2:1 e−stAa B−"l# e sl"l−n_AtA f−nBtBf¦nrtA# zbs2:1 "0−lnr# 2:1_a B¦"l0#ð"0−bnBcos g#l¦b cos gŁ dl[ "44# Applying the inverse Laplace transform to eqns "43# and "44#\ the dynamic stress intensity factors in time domain are

KB1d "t# −3t9sin"g:1#z0−bnB p2:1_zb

g

t b"l−nBtBf# 0−bnB zt−tzt¦bðl¦nB"t−tbf#Ł ðl¦nB"t−tBf#Ł2:1 × "l−nBt B f#2:1 "t−b cos gðl¦nB"t−tBf#Ł#zt−bðl¦nB"t−tBf#Ł dtH"t−tB1d#\ "45#

Fig[ 1[ Wave fronts of the incident and di}racted waves for a short time period[ KB1n "t# −3t9sin"g:1#z0−bnB"0¦bnAcos g# 2:1 p2:1_zb

g

t−t_A b"l−nAtAf−nBtBf¦nrtA# 0−bnB zt−t−tA "l−nAtAf−nBtfB¦nrtA¦nrt#2:1 × z"0¦bnB#t¦b"l−nAt A f −nBtBf¦nrtA# ðt−b cos g"l¦nBt−nAtAf −nBtBf¦nrtA#Ł × "l−nAt A f −nBt B f¦nrtB# 2:1 z"0−bnB#t−b"l−nAtAf −nBtBf¦nrtA# dtH"t−tB1n#\ "46#

where tB1dand tB1nare the arrival times of the A0d and the A0n waves\ respectively\ and

tB1d b"l−nBt B f# 0−bnB \ "47# tB1n b"l−nBtBf#¦tAf 0−bnB [ "48# 4[ Numerical results

In the previous section\ the transient solutions of dynamic stress intensity factors for the _rst few di}ractions of a horizontally polarized shear wave by a propagating _nite crack have been derived[ The induced wave fronts of incident and di}racted waves in a short time period are shown in Fig[ 1[ Figures 2 and 3 show the dimensionless stress intensity factors KA

:Kcand K B

:Kcversus

the dimensionless time t:bl for di}erent values of the incident angle g at crack tips A and B\ respectively[ It indicates in Fig[ 2 that the dynamic stress intensity factors at crack tip A will increase as the incident angles increase[ However\ the stress intensity factors at crack tip B increase as the incident angles decrease after the _rst four waves passed the tip[ That is\ the dynamic

Fig[ 2[ The dynamic stress intensity factor KA_:K

cfor di}erent values of the incident angle g[

Fig[ 3[ The dynamic stress intensity factor KB

:Kcfor di}erent values of the incident angle g[

stress intensity factor at crack tip B is much greater than that at crack tip A for the same incident angle[

Figure 4 shows the dimensionless stress intensity factors KA_:K

cand KB:Kc versus the dimen!

sionless time t:bl for di}erent values of fracture toughness[ It can be seen that the ratios for KA

Fig[ 4[ The dynamic stress intensity factor for di}erent values of fracture toughness[

Fig[ 5[ The dynamic stress intensity factor KA_:K

cfor various crack propagating velocities[

and KB

:Kcboth increase rapidly for smaller Kcafter the crack begins to propagate[ It means that

for larger Kc\ the crack may stop propagating after it has propagated for a period of time[

Moreover\ Fig[ 4 also indicates that the dynamic stress intensity factor at crack tip B is larger than that at crack tip A for the same value of fracture toughness[ Figures 5 and 6 show the dimensionless

Fig[ 6[ The dynamic stress intensity factor KB

:Kcfor various crack propagating velocities[

stress intensity factors KA

:Kc and K B

:Kc versus the dimensionless time t:bl for di}erent crack

propagating velocities\ respectively[ It shows that the in~uence of secondary di}raction on dynamic stress intensity factor for higher velocity is relatively smaller than that for lower velocity[

5[ Conclusions

Most of the problems that have been studied in the development of fracture mechanics are quasi!static[ Numerous problems have existed for which the assumption that the deformation is quasi!static is invalid and the inertia of the material must be taken into account[ Because of the di.culties in mathematical complexity\ analytical solutions for an elastic solid containing a _nite crack subjected to dynamic loading are very rare[ In conventional studies of a semi!in_nite crack in an unbounded medium subjected to dynamic loading\ the complete solution can be obtained by applying direct integral transform methods[ If a cracked body having a characteristic length or the loading condition is unsymmetrical\ then the same procedure can not be applied directly[ In this investigation\ we propose a powerful superposition methodology\ which is performed in the Laplace transform domain\ and successfully applied to solve the transient response of a _nite crack propagating in an unbounded medium[ The _nite crack is stuck by a horizontally polarized shear wave[ After some delay time\ two stationary crack tips will start to propagate along the crack tip line with constant velocity as the stress intensity factor reaches its fracture toughness[ Two useful fundamental solutions and the coordinate transformation relations are proposed to solve this problem[ The _rst few waves di}racted by the stationary and propagating crack tips are obtained and expressed in very compact formulations[

It is interesting to note that the dynamic stress intensity factor during crack propagation of crack tip B is larger than that of crack tip A\ which is the tip that the incident plane wave _rst strikes[ It means that after crack starts to propagate\ crack tip A is easier to arrest than crack tip B[ Furthermore\ it also indicates in this study that the in~uence of secondary di}racted waves on dynamic stress intensity factor for higher propagation velocity is relatively smaller than for lower velocity[

Acknowledgements

The authors gratefully acknowledge the _nancial support of this research by the National Science Council "Republic of China# under Grant NSC 72!9390!E!991!004[

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