SOME ELASTODYNAMIC PROBLEMS OF CRACKS G. C. Sill* ABSTRACT Elastodynamie problems of both stationary and moving cracks are discussed. Previous work on the dynamics of fracture has been limited mostly to the model of a semi-infinite crack. In this paper, special emphasis is placed on the configuration of a crack having finite length. By retaining the additional characteristic dimension in the dynamic problem, namely tlaecrack length, severaltypical features of the dynamical stress solution can be ex. hibited. These are the peak values of the crack-tip stress intensity factors occurring at certain wave length for cracks opened by periodic cyclic loadings, and at a given time for cracks opened by rapidly applied tractions. A general solution to the problem of cracks travelling at constant speed is also given, It is shown that the extension of Dugdale's plasticity model to the case of a moving eraek is straightforward. INTRODUCTION From the viewpoint of continuum mechanics, the knowledge of the state of stress and strain near the crack point is one of the key requirements for a fracture strength analysis of structural members weakened by flaws. In the static theory of elasticity, extensive treatment has been given to problems involving the stress distribution around sharp cracks and notches under various loadhag conditions. As an instance of application to fracture mechanics, the two-dimensional elliptical hole solution became the basis of almost all present-day theories of brittle fracture. However, the analogous problem of a finite crack in elastodynamics has yet to receive adequate attention. In general, the literature on elastodynamic problems of cracks is somewhat meager. The main reason for this scarcity is the severe mathematical complexity encountered in finding effective solutions for the crack geometry not covered by the classical method of separation of variables. Of importance in fracture mechanics is the detailed character of the dynamic stresses in the vicinity of the crack point. The far field behavior of the stresses can usually be estimated by means of known techniques with comparative ease. One of the purposes of this paper is to update the contribution on dynamic crack problems and to present certain information that may be interesting and stimulating to the workers in the field. The dynamics of a crack of fixed length travelling at constant velocity in a stressed medium of infinite extent have been discussed by Yoff6(1). Using a criterion of maximum circumferential stress ahead of the crack, she concluded that the crack tends to branch for velocities greater than approximately 0.6 times that of the shear wave velocity. A similar problem was solved later by Craggs(2) who assumes the configuration of a semi-infinite crack extended by tractions applied over a segment of the crack surfaces. Bilby and Bullough(3) obtained the local distribution of longitudinal shear (or anti-plane shear) stress around the tip o f a moving crack. Their work was extended to other crack configurations by McClintock and Sukhatme(4). To unify and simplify the method of solutions used in References 1 4 , it will be shown that the basic problem of moving cracks can be readily reduced to the solution of the Riemann-Hilbert problem(5) in complex function theory. A more realistic model of the moving crack problem has been proposed by Broberg(6). He assumes that the crack tips move in opposite directions with constant velocities and found that the speed of propagation of such a crack cannot exceed the Rayleigh surface wave velocity**. The * Professor of Mechanics, Lehigh University,Bethlehem, Pennsylvania, U.S.A. ** This result has also been obtained by other authors independently.

52

8.C.Sih

consideration of cohesive forces acting near the edge of the crack was due to Barenblatt et al(7). Craggs(8), based on the notion of dynamic similarity, also solved the problem described in Reference 6. The case of a penny-shaped crack growing axisymmetricaUy under tension was investigated by Kostrov(9). Another class of dynamic problems of interest is concerned with the effect of time-dependent loading on a stationary crack. If the applied loads fluctuate periodically in time, the resulting stresses and displacements are propagated through the structure in the form of waves. At an obstacle such as a line crack, these waves are reflected and refracted causing high local stress intensification about the tips of the crack. Previous work on this subject has been restricted to the simple geometry of a half-plane crack. In fact, the problem of the diffraction of polarized harmonic shear waves by a semi-infinite crack is mathematically analogous to the corresponding problem in electromagnetic theory. For the case of harmonic shear waves impinging on a semi-infinite crack lying along the negative x-axis, the anti-plane displacement component* of the incident field can be taken as wi = Wo exp ~-i [a (x cos 13+ y sin 13) + oat] }

(1)

in which Wo is the amplitude of the wave, fl the angle of incidence measured from the x-axis, and co the circular frequency. The wave number is ~ = oa/c2, c2 being the shear wave velocity given by ( ~ / 9 ) 1/2 , where # is the shear modulus and p the mass density of the elastic medium. With a slight change in notation, the dynamic stress-intensity factor for this problem can be found in Noble's book{10): ka = ~

exp [-i (oat + 4) 1 sin-2

(2)

As the wave length ~ = 27r/a tends to infinity, ka reduces to the trivial static solution. It is seen that the static counterpart to this problem fails to exist, and no comparison between the static and dynamic solutions could be made. Hence, it would be more informative to solve the diffraction problem of a double-ended crack by introducing an additional characteristic dimension into the problem. To this end, Loeber and Sih(11) have developed a method for finding the local stresses produced by the interaction of a finite crack with stress waves. Their work will be discussed briefly in this paper. The scattering phenomenon of plane waves due to the presence of a crack has been discussed qualitatively by Papadopoulos(12). In his work, the emphasis is placed on the closure of the crack surfaces rather than obtaining quantitative results which are essential in the formulation of fracture theories. Transient problems in which a semi-infinite crack appears instantaneously in a uniformly stressed medium have been explored by Maue( 1 a) using conical coordinates and by Ang(14) employing the technique of Wiener-Hopf. Baker(15) extended their problem to the case where the crack propagates at constant velocity after it has appeared suddenly in the stretched elastic body. The mathematical formulation of the problems stated in References 13-15 are equivalent to the specification of a uniform impact loading, which can be described by the Heaviside unit step function, on the surfaces of the crack. The salient feature of the local stresses, say oij, in the vicinity of a semi-int~mite crack is that they are directly proportional to the square root of time t in the form The displacement components ui and vi in the x - and y--directions are assumed to vanish in the problem of ~nti--nl~n~ defnrm~tinn.

G.C.Sih oij = c

fij(0) r

In equation (3), C is a constant depending linearly on the load, r the radial distance measured from the crack fronf, and fij(0) the angular distribution. In the limit as t -~ o% the static stresses aij increase without bound. This is to be expected since the-static problem of a uniformly stressed medium containing a semi-infinite cut is improperly formulated. Sih and Rice(16) have pointed this out previously. To obtain a meaningful limit of the transient crack Problem as time increases, Ravera and Sih(17) adopted the model of a finite crack. The surface tractions may be prescribed as an arbitrary function of time. Their method is particularly effective in f'mding dynamic stresses near the singular crack point. Since, in principle, any time-dependent load function on the crack may be generated by superposing a sequence of step functions, the method of Ravera and Sill{17) covers all the elastodynamic problems of stationary cracks. However, from the physical standpoint, it is more clear to discuss the steady-state and transient problems of cracks separately. This will be done in the work to follow. MOVING CRACKS The Riemann-Hilbert problem in static plane elasticity has provided a significant extm,sion of the range of crack problem that can be solved with the aid of the theory of functions of a complex variable. It ~ be shown that the same method can be applied to solve dynamic problems of cracks of constant length moving at a constant speed. Let an elastic medium be stretched in such a way that the cracks move with velocity c along the x-axis, then the stress components of dynamic plane elasticity can be expressed in terms of two functions(18) cbj(zj) (j = I, 2) of the complex variables zj as 1

Ox = - 2 Re {[s~ + 3 ( 1 . s ~ ) ]

1

~I)l(Zl)

"~7(1 "J-S2)

Oy = (1 + s~) Re [ ' 1 ( z l ) + ~:(z~)]

(I +

rxy = 2 Im [sl ~1(zl) + ~

(I)2(Z2)}

(4)

~2(z~)]

4s2

where zj =~ + i s j L j = 1 , 2 with ~ and ~"being the moving coordinates. In equation (4), Sl = 1 -

77

,

s2 = 1 - -

\c=/3

are functions of the compressional (dilatational) wave velocity cl and shear (equivoluminal)wave, velocity c= in an infinitely extended region. These wave velocities are C1 =

~

,

C2

where X and ~ are the Lam4 coefficients. Similarly, the displacement components may be represented by

54

G.C.Sih

1

1

u =---Reu [qh (zl) + 2 ( 1 + s~) ¢2(z2)]

(s) 1

1 (1 +s~] v = - - I m [sl~bx(Zl) + - - ~ ~b2(z2)] /~ 2 kS2 / in which q~j(zj) = f ~j(zj) dzj,

j = 1,2

The primary objective is then to determine the complex functions~j(zj) or Cj(zj) from the boundary conditions.

SOLUTION OF RIEMANN-HILBERT PROBLEM Suppose that the elastic medium occupying the entire xy-plane contains a f'mite number of moving cracks, all collinear. The prescribed normal stresses on the upper and lower edges of each crack are taken to be equal and opposite so that Txy = 0,

for all ~

(6)

is satisfied. As in the plane problem of stationary cracks, it is required to find the sectionally holomorphic functions* ~j(zj) by reducing the boundary condition to the standard form of the Riemann-Hilbert problem. Without going into details**, ~j(zj) are given by

¢,(z,) -

1 -

j

2hi X(z1) -

X+(~)fl(~)d~ ~ Pn(zl.......~)

-

L

~ -- Z1

X(Zl) (7)

J Y+(~)f2(~)d~ Qn(z2.....~) - - 1 2rri Y(z2) ~ - z2 Y(z:) L

where L = LI + L2 + • . . + Ln represents the union of coUinear cracks. The functions

1 +s~ fl(~) (1 +s~) 2-4sis2 =

+ Oy), (8)

f2(~) =

4s i S2 - fl (~) (1 + s~)

are related to the normal tractions oy+and Oy on the upper and lower sides of L, respectively, and they must satisfy the H61der condition on L. The singularities at the end points ajbj (j = 1,2 . . . . . n ) of the cracks are described by the Plemelj functions n X ( z l ) = II

(z 1 _ ~ ) 1 / 2 (z 1 _bj)I/2,

j=l n

Y(z2) = II (Z 2 __~)1/2 (Z 2 _ b j ) l / 2 j=l * See Muskhelishvili(5) for definition of a sectionally holomorphic function. ** The derivations of the results in equation (7) can be found in the appendix.

(9)

55

G.C.Sih For the stresses to be bounded at infinity, the polynomials Pn(zl)=Aoz~ + AIz~ -1 +" • • +An

(10)

Qn(z2)= BoZ~ + BIz~ -1 + . . - + Bn are required to be of degree not greater than n, and from equation (50) in the Appendix, all the coefficients in equation (I0) must be real, i.e.

Ao =Ao,

Al=~xl . . . . . A n = A n

Bo = Bo,

B1 = Bx . . . . .

(11) Bn = Bn

Since for large values of Izj 1,q~j(zj) take the forms

(1)

®j(zj) = rj + 0 7j

j = 1, 2

(12)

in which

I'1= _ _ O Y

P~-

ay

2(s~ - s~)'

1 + s~

~-

(13)

2(s] - s~)

it is evident from equation (7) that A0 and Bo are expressible in terms of the loading at infinity. In fact, by comparing equations (7) and (12) for large Izj I, it is found that Ao = p l ,

Bo =p2

where the shear stress at infinity is assumed to be zero. The remaining coefficients Ax, A2 . . . . . An and B~, B2 . . . . . Bn in equation (I0) may be evaluated from the condition of single-valuedness of the displacement components u and v. This completes the formulation of the moving crack problem. It should be pointed out that the detailed structure of the complex functions Cj(zj) in equation (7) is identical to those for the problem of stationary cracks. For this reason, the problem of a Dugdale(19) crack propagating at a constant speed may be considered as solved. The following examples will illustrate this point. TWO SIMPLE EXAMPLES

Consider a straight crack of length 2a with uniform tractions p applied to its edges. As the crack spreads at a constant rate c along the x-axis, it is assumed that the crack reseals itself spontaneously, i.e., the length Za remains constant at all times. For a single crack, equation (10) simplifies to P l ( z l ) = A o z l +A1,

Q ( z l ) = B o z 2 +B1

If the elastic medium is undisturbed at infinity, then Ao = Bo = O. Further, the single-valuedness condition of the displacements requires A1 = B1 = 0. Hence, the complex functions in equation (7) become 1

f

qb1(zl) - 27ri ~

fl (~) X/~2 - a2 d~ ~-zl

--a a

_

opt(z2)

I

2~i ~

t ~ f 2 ( ~ ) v ~ - a 2 d~ ~ - z2 --llt

(14)

56

G.C.Sih

where 2(1 +s~)p fl(~)=(l+s~) 2-4sis2'

f2(~)-

4sis2 (1 +s~) 2 fl(~)

The Cauchy integrals in equation (14) may be easily evaluated to give (1 +s~)p

[1

z~l

d9, (zl) = (I + s-~) "-2 -- 4s, s2 (15) %(z2) = - ( 1 + s~) [(1 + s~) 2 -4sis2] The solution to the problem of a free-surface crack propagating in a medium subjected to uniform tension p at infinity may also be derived from equation (15). This can be accomplished by adding PI and F2 in equation (13) with ox = Oy = p to the negative ofq~l(zO and ~2(z2) in equation (15), respectively. The resulting expressions agree fully with those found by Yoff4(1) who obtained the solution by superposition of surface waves. The second example deals with the propagation of a semi-infinite crack whose tip coincides with the origin of the moving coordinates. Concentrated forces of equal magnitude P, but opposite in sense, are applied to the crack surfaces at ~ = - a. For this problem, geometric singularity occurs only at ~ = 0. Therefore, the Plemelj functions should read as X(zl)=x/~,

Y(z2)=~¢c~2

It follows that the complex functions qbj(zj) reduce to

_ ~t(zl)

1

0

[ fl (~) x/~" d~

2rri-----~l2~

~-zl (16)

0

~2(z2) - 2~ri------~z2_ ,

~ - z2

in which fl(~)=

2(1 + s~)P 6(~ + a) (1 +s~) 2 - 4 s , s 2 '

f2(~)-

4s,s2 (1 +s~) 2 f'(~)

The symbol 8(~) stands for the Dirac delta generalized function. Upon integration, equation (16) yields ~bl(zx) =Tr [( 1 +s~) 2 - - 4 s l s ~ ] V Z ~ z - - ~ a J 4s~s2P %(z2)='rr(l+s~)[(l+s~)

~a 2-4sas21

~..__2..__1"~

(17)

z2kz2+a/

Equation (I 7) may be further used as the necessary Green's function to obtain the solution to the problem of uniform tractions of intensity p applied to the segment of the crack from ~ = - a to ~ = 0. Letting P = p d~, a = - ~

G.C.Sih

57

in equation (17) and carrying out the integration from - a to 0, the results are

~I(Zl)

--~]"[(1 +2(1sg)+z s~)p -4sxs2]

I ~ z ~ _ tan_l ~ a t zl (18)

8SlS2p

~a

ff2(z2) =-- ~'(1 + s~) [(1 + s~) 2 - 4sls2 ] ~ ~'2 - tan-1

4~..21

This is in agreement with the results of Craggs(2) when it is recognized that in his notation

1 Wl(z,) = - - ~ l ( z ~ ) ,

1 1 W2(z2) = ~ ( s 2 +--) ~2(z2)

and when the identity log :i \ i x/~J x/~j -+ Va~) : 2i tan-I v/'a__-_ zj ' j = l , 2 is employed. Moreover, appropriate superposition of equations (17) and (18)provides the complex functions for the Dugdale model of an infinite plane with a semi-infinite crack whose motion is maintained by concentrated forces.

DYNAMIC STRESS AND DISPLACEMENT FIELDS In problems of brittle fracture, it is informative to find the elastic stresses and displacements in the neighborhood of the ,singular crack point. To this end, a set of polar coordinates r, 0 measured from the front of the moving crack will be defined such that ~ = a +rcosO,

~ =rsin 0

For values of r small in comparison with the half crack length a, the Plemelj functions for a double-ended crack may be approximated by ~/zj2 - a2 ~ V/~-ra(cos 0 + isj sin 0), j = 1,2 and hence equations (15) becomes

(I~l(Zl) ~

(1 +s~)p (i + s ~ ) - 4 s l s 2

~

1

X /2r ~/cos 0 +is1

sin0 (19)

4sls2p

4

~(z2)'~ -(1 + s~) [(1 +s~) 2 -4sis2]

1 x/cos 0 +is2 sin0

Substituting equation (19) into equations (4) and (5), the local stresses around the moving crack are

_ °x

pv~-

1

4sls:i - ( 1 + s~) 2 ~ - R e

[-(1 + s~) (2s] + 1 -s~) .x/cos 0 +is1 sin 0

4SIS2 ]. + O(rO) x/~os 0 + is2 sin

58

G.C.Sih l

Oy-

4sis2 - ( 1 + s]) 2 x/2r

(1 + s~) 2

Re

x/cos 0 + is1 sin 0

4sis2 Ol O(rO) + X/cos 0 + is2 sin 2s1(1 + s~)px/ff 1 Txy =

[

(20) 1

4sl s2 - (1 + s2) 2 X/~ Im L" x/cos 0 + is1 sin 0

x/cos 0 +' is2 sin

o]

+ O(r°)

and the displacements are #u -

px/ 4sis2 - ( 1 + s]) 2

Re [(1 + s]) ~

+ is1 sin 0

+ 2sxs2 X/cos 0 + is2 sin 0] + 0(r)

px/ /av =

4sis2 -(1 + s~)2

Im [-sl(1 +s~) x/cos 0 +is1 sin0

+ 2sl x/cos 0 + is2 sin 0] + 0(r)

(21)

While the dynamic stress singularity of the order 1/x/7 is the same as the static one, the angular variation of the stresses depends on the velocities c, cl and c2 through the parameters sl and s2. It Is this dependency on the velocity that contributes to the tendency of crack branching as explained by Yoff6(1 ). Another point of interest is that in contrast to the static solution of Ox/Oy = 1 for 0 = 0, which corresponds to a state of 'two-dimensional hydrostatic tension', the ratio of the dynamic principal stresses ahead of the crack is xo =(1 + s~) (2s] + 1 - s ~ ) - 4 s l s 2 try

4sis2 - (I + s~) ~

(22)

For a Poisson's ratio of v = 0.25, a graphical representation of the variation of ax/Oy with (c/c2) 2 is shown in Figure 1. As the crack velocity c is increased, the deviation from a state of hydrostatic tension, identified by Ox/Oy = 1, becomes more and more pronounced. Since the analysis is restricted to ideally brittle materials, the result in Figure 1 simply implies that the tendency for incipient yielding becomes greater at higher crack velocities. SCATTERING OF WAVES AROUND CRACKS As mentioned earlier, the problem of stress waves impinging on a crack of finite length is not covered by the classical methods. The technique of Loeber and Sih(11) will be followed since it readily yields the pertinent information in a region close to the crack point. Their method is capable of solving a number of important physical problems such as through cracks in bodies excited by polarized harmonic shear waves or in plates by compressional and out-of-plane flexural waves. The detailed work of these investigations is reported elsewhere. For the mere purpose of illustrating the dynamic character of the finite crack solution, it suffices to discuss the relatively simple case of the diffraction of anti-plane shear waves by a single line of discontinuity.

G.C.Sih

59

4.0

Poisson's

Rot|o

u=0.25

3.01

2.0

I

0

I

l

0.2

I

0.4

I

I

0.6

( C/c a )~ Figure 1. Variation of Ratio of Principal Stresses: with Crack Speed.

A state of anti-plane shear can be realized if the only non-vanishing displacement component is w(x, y, t), i.e., u=v=O, w = w(x, y, t) and hence the corresponding stress field becomes e x = O'y = e z ='rxy = 0 aw

3w

7xz = P ~X '

(23)

Tyz = /S ~Y

Putting equation (23) into the equations of m o t i o n renders the well-known wave equation ~2w - -

~x 2

02w +

- -

ay 2

1 a2w =

C22 ~t 2

(24)

60

G.C.Sih

DUAL INTEGRAL EQUATIONS

A linear elastic medium extending to. infinity in all directions is considered. The medium is uninterrupted except for a crack of length 2a along the x-axis from x = - a to x = a. The polarized harmonic shear waves arriving from infinity impinge normally on the line crack. Thus, in the undisturbed portion of the medium, the displacement component is given by w = Wo exp [ - i (e~y + cot)], as V ~ + y2 __~oo The surfaces of the crack are to be free from tractions such that rxy=0,

for

Ixl
y=0

Because of the steady-state nature of the problem, the time factor exp (- icot) will be introduced. Furthermore, the displacement solution w (x, y, t) may be separated into two parts in the form w (x, y, t) = [w(i) (x, y) + w(r) (x, y)] exp (- icot)

(25)

where exp (iay)

w(i) = Wo

is the displacement of the incident wave and w(r) of the reflected wave. Inserting equation (26) into (24), the Helmholtz equation 02w(r) - -

a2w(r)

+ - Ox2 3y~

+ ol2w(r) = 0

(26)

is obtained. Note that only the reflected portion of the displacement field w(r) appears in equation (26). The incident part of the solution satisfies the wave equation automatically once the wave number a is taken as ¢o/c2. In addition, since w(r) is skew-symmetric with respect to the x-axis, the crack problem reduces to one of specifying mixed boundary conditions on the surface of the half plane y i> 0 as follows: w(r)=0,

Ixl>a;

y=0

r(yr)=-r(~)=iqexp(iay),

(27) Ixl
y=0

in which q =/~o~vo is a measure of the amplitude of the incident waves. By taking a Fourier cosine transform of equation (26) on the variable x, the displacement expression in the transformed domain is obtained. The Fourier inversion theorem then yields w (r)

(x, y) = 2 ~ A(r/) exp (- 7Y) cos xr/d~,

y i> 0

(28)

J

o

From equation (23), the stresses due to the reflected waves may be computed: r(rz) (x, y ) = - ~ ~r/A(r/)exp (-33')sin x~/drl,

y >/0

0

2# ® r(yrz) (x, y) = - ~ ~TA(r/) exp (-TY) cos xr/dr/, 0

(29) y/> 0

G.C.Sih

61

where -t = x / ~ -t~ 2 Substitution of equation (28) and r(y~) in equation (29) into equation (27) gives the system of dual integral equations A@)cosxndn=0,

-

Ixl>a

7r o

(30) 7A(~) cos x~ d~ = - - - , #

Ix I< a

O

which determines the only unknown complex function A00. SINGULAR

SOLUTION

Loeber and Sih(11) have obtained the solution to equation (30) by expressing A(r/) in terms of another complex function, say kt,(p), where p is non-dimensionalized with respect to the half crack length a and the wave number. The real and imaginary parts of xI,(p) can be evaluated from two simultaneous integral equations. In the analysis of brittle fracture, it is only necessary to focus attention on the singular part of A(r~): A0/) -- - rr aq xI,(1) J1 (~a) + . . . 2gn

(31)

In equation (31), Jl0/a) is the Bessel function of the first kind and 4(1) is the value o f ~ ( p ) evaluated at the crack tip. Knowing that the state of stress in an uncragked medium is non-singular, the detailed structure of the stress distribution near the end region of the crack may be found directly from equations (29) and (31) as qx/~ 0 rxz = - " ~ r 4(1) exp ( - icot) cos --2+ O(r°)

(32)

qx/~ Ty z =

4(1) exp ( - i c ~ ) sin 2 + 0(r°)

The dependence of the stresses on r and 0 is the same as the static case. However, the magnitude of the local stress field is modified through ~(1). At this point, it is convenient to define a dynamic stress-intensity factor ka = 4(1) qx/~

(33)

such that when the circular frequency of the incident waves approaches zero, the static value of q x/a-is recovered. Therefore, the function ~(1) may be interpreted simply as the ratio of dynamic to static stress-intensity factors. The modulus of 4(1) as a function of the normalized wave number aa is plotted in Figure 2. A peak in the dynamical stress-intensity factor is observed at aa ~ 0.95, where ka is approximately 2 7 . 5 % higher than that encountered under statical loading. For small wave numbers, the dynamic effect is not significant. In the limit as aa ~ 0% ka tends to zero. This is to be expected

62

G.C.Sih

I.O

0.5

0

I

I

I

0.5

1.0

1.5

o ~/c

2

Figure 2. Dynamical Stress-Intensity Factor versus Normalized Wave Number. since when a becomes very large in comparison with the wave length ~ = 2~r/ct, the crack boundary acts like a plane surface free from tractions. A curve similar in trend to that o f Figure 2 may be obtained for the scattering problem of a finite crack in an infinitely extended plate during the passage of plane compressional waves*. The maximum value of the ratio of dynamic to static stress-intensity factors (opening mode) will occur at a smaller wave number and will be larger in magnitude than the case o f harmonic shear waves impinging on the crack. IMPACT WAVES IN BODIES WITH CRACKS Structural components are often subjected to impulsive loads which generate stress waves. At a certain time, the propagation of these waves can cause high stress intensity in a local region around the geometric singularity such as the crack tip. The magnitude of the dynamical stresses can be considerably larger than the statical ones, and may lead to crack extension. Thus, it is desirable to obtain certain useful relations between the parameters of the transient stress waves and the crack dimension. LAPLACE TRANSFORMATION The problem o f transient shear waves generated in a cracked body by the application of a step-function time dependent load on a finite crack can be solved using the method proposed by Ravera and Sih(17). If the crack is aligned on the x - a x i s from x = - h to x = a, then the con* To keep the crack open, an additional tensile field must be superposed on the solution.

63

G.C.Sih

ditions to be fulfilled at y = 0 are w(x, o, t) = 0,

Ix l > a

(34) Ixl
ryz(X, o, t) = - qoH(t),

Here, qo is the amplitude of the anti-plane shear load and H(t) the Heaviside unit step function. Denoting the operation of Laplace transform by an asterisk on the function under consideration, the wave equation (24), is transformed to

~2W* (~.2) 2

~2W*

~+"~"y2 0x 2

-

W*=0

(35)

and is subjected to the conditions w*(x, o, s) = 0,

Ixl>a

qo r~z(X, o, s ) = - - - ,

(36) Ixl
s

which represent the Laplace transform of equation (34). In equations (35) and (36), s is the transform variable in time. Making use of Fourier cosine transform in x and its inversion theorem as in the steady-state wave problem, the solution to equation (35) becomes w*(X, y, s) -- --

A(~, s) exp (- ey) cos x~ dr?

(37)

lr o

where

e =~/,7 ~ + (sick.?

Differentiating equation (37) with respect to x and y in accordance with equation (23) yields oo

r*z(X, y, s) = -

,

r~A(~, s) exp (- ey) sin x~7 dr~

o~

(38) eA(rh s) exp (- ey) cos xr/d,/

o

Once equation (37) and r~z in equation (38) are put into equation (36), the following system of dual integral equations in the Laplace transformed plane is obtained:

o

(39)

o

In view of the similarity between equations (39) and (30), tlae singular part of A(r/, s) follows immediately from equations (31) as A(n, s)

= -

q°-a ~2"(1, s) Jl(r/a) + . . . 2/l~Ts

(40)

64

G.C.Sih

The function ~2*(p, s), governed by a Fredholm integral equation of t h e s e c o n d kind, is real, and can be calculated numerically. The quantity p takes the value of unity at the crack point, x = a. LOCAL BEHAVIOR OF STRESSES Onceequation (40) is inserted into equation (38), the time inversion on the singular part of the stresses can be carried out by means of the Cagniard technique and the convolution theorem(see Reference 17). The results are t

rxz = j ~2(I, t - r ) 0

I i ( r ) dr

t

ry z = )

(41) ~2(1, t - r) I2(r) dr

0

in which

090

2ty cos co dco I~(t) = ~

(42)

R~x/t~ _ (R/c~) ~

0

and a similar expression follows for I2(t). The upper limit 600 of I, (t) takes the values between 0 and rr, and is determined by 600 = COS " 1 [(X -- 4 ( C 2

02 - y2)/a]

(43)

The parameter R in equation (42) stands for R 2 = (x - a cos 6o)2 + y2 It is clear from equation (41) that the stresses rxz and ry z are generally known for given values of x, y, and t. Furthermore, the structure of the stress expressions in equation (41) reveals that the spatial and time effects are interlaced and there does not appear to be a simple analytical solution unless certain assumptions can be made. With this in mind, a solution near the crack tip will be obtained by letting x -~ a, y ~ 0, and R -~ a (1 - cos co) Now, let c2t be much greater than the maximum value that R can attain, namely 2a. This occurs at 60 = n. Hence, equation (43) fixes the upper limit of I i ( t ) at 60o = 7r. Under such an assumption, 1 1 and I2 in equation (41) become independent of time, and the spatial and time parts of the solution are uncoupled, i.e., rxz(X, y, t) =

q°ayA(t) .~ _ /r

Tyz(X,

where the function

y, t)

qoaA(t) ( ~"

)

COS O9 d60 R2

o (x - a cos 60) cos co d60 R2 ,

(44) c2t N 2a

G.C.Sih

65

t

A(t) = J ~2(1, r) dr o depending on time t and the integrals on space coordinates x, y appear as a product. It is important to recognize that the assumption c2t ~ 2a corresponds to the physical situation where sufficient time has been elapsed so that the disturbance from the crack can be represented by a tingle outgoing cylindrical shear wave emanating from the crack as if the entire crack were a point source. Consequently, the form of equation (44) is not expected to remain valid in the intermediate time range when transient shear stress waves are generated individually from both ends of the crack. The interference pattern of these transient waves is a rather complicated phenomenon, if not an impossible one to trace analytically. In any event, a full account of the time history of the stress solution can be obtained by evaluating the integrals in equation (41) numerically. Returning to equation (44), after the integrals are evaluated analytically, the local stresses become rxz(r, 0 t) = -

A(t) sin- + 0(r °)

'

ryz(r,O,t)=

2

(45)

A(t) cos v- + 0(r°), 2

c~t >> 2a

As usual, r and 0 are the polar eoordinates measured from the crack front. Once again, the functional relationship of rxz and ry z on r and 0 for c2t >> 2a is identical to the static one, but the stress-intensity factor ka = A(t) qoV'a" (46)

z.o

A (t)

1.0

0

I

I

6.0

12.0

.

I

18.0

I

24.0

1.14 C2 ¢/a

Figure 3. OScillationof Stress-lntensity Factor with Time (c2t ~ 2a).

30.0

66

G.C.Sih

has been altered by A(t). When t tends to infinity, the transient character of the solution disappears and equation (46) reduces to the familiar statical result of qoV~ since in that case A(t) -+ 1. In general, A(t) is a function of time and its variation with 1.14 c2t/a is plotted in Figure 3. The dynamical k3-factor reaches a peak very quickly and then oscillates about its statical value. Although the numerical results ofk3 are not accurate for value of c2t smaller than or in the neighborhood of 2a, it nevertheless does illustrate the character of the transient solution. For values of c2t >> 2a, the curve does give a true representation of the values o f k 3 - f a c t o r for the tearing mode of fracture. CONCLUSION The effects of dynamic loading on the distribution of stress around a crack-like imperfection have not received sufficient attention in the past, particularly in seeking effective solution for the geometry of a finite crack. Unlike the static case, solution to the dynamic problem of a doubleended (finite) crack is an order of magnitude more difficult to obtain than the single-end (semiinfinite) crack problem. This paper discusses the dynamic counterpart of the Griffith crack configuration. The crack system may be subjected to fluctuating loads that vary periodically with time or to impulsive loads applied suddenly. The distinct features of the dynamic response to these time-dependent loadings are pointed out. A systematic procedure, based on the solution of the Riemann-Hilbert problem, is also presented for obtaining solutions to problems with moving cracks. Having established the basic mathematical technique for solving elastodynamic problems of cracks, it is now possible to provide useful information to other crack problems that are of fundamental interest in fracture mechanics. For instance, the diffraction of stress waves due to a moving crack may be investigated by incorporating a system of moving coordinates into the method of Loeber and Sih(11). The plasticity model of Dugdale(19) may also be extended to crack systems owing to time-dependent loads since Ravera and Sih(17) have already cleared the way for obtaining stress solutions to crack problems in which the external loading may vary as an arbitrary function of time. The results of these problems will be published separately. ACKNOWLEDGEMENT The financial support of the Office of Naval Research, Washington, D.C., under Contract Nonr-610 (06) is gratefully acknowledged. Received September 8, 1967. REFERENCES 1, Yoff~,E.H.,

Phil. Mag. Vol. 42, p. 739, 1951.

2, Craggs,J.W.,

J. Mech. Phys. Solids, Vol, 8, p. 66, 1960.

3, Bilby, B.A. and Bullough,R.,

Phil. Mag.Vol. 45, p. 631, 1954.

4. McClintock,F.A. and Sukhatme, S.P.,

J. Mech. Phys. Solids, Vol. 8, pag. 187, 1960.

5. Muskhelishvili, N.I.,

Some Basic Problems o f Mathematical Theory o f Elas~7"city,

Ltd., Groningen,~l'he Nethedands, 1953.

Noordhoff

G.C.Sih

67

6. Broberg, B.,

Arkiv, Fysik, Vol. 18, p. 159, 1960.

7. Barenblatt, G.I., Salganik, R.S. and Cherepanov, G.P.,

PMM (j. Appl. Math. Mech.), Vol. 26, p. 469, 1962.

8. Craggs, J.W.,

Fracture o f Solids, John Wiley and Sons, Lew York, New York, p. 51,

1963. 9. Kostrov, B.V., 10. Noble, B.,

PMM (J. Appl. Math. Mech.), Vol. 28, p. 793, 1964. Methods Based on the Wiener-Hopf Technique, Pergamon Press, New

York, New York, p, 48, 1958. 11. Loeber, J.F. and Sih, G.C.,

J. Aeoust. Soe. Am, (ha Press).

12. Papadopoulos, M.,

J. Australian Math. Soc. Vol. 3, p. 325, 1963.

13. Maue, A.W.,

Z. Angew. Math. Mech., Vol. 34, p. 1, 1954.

14. Ang, D.D.,

Some Radiation Problems in Elastoclynamics, Dissertation, Calif.Inst.

Tech., Pasadena, Calif., 1958. 15. Baker, B.R.,

J. Appl. Mech., Vol. 84, p. 449, 1962.

16. Sih, G.C. and Rice, J.R.,

J. Appl. Mech., Vol. 32, p. 464, 1965.

17. Ravera, RJ. and Sih, G.C.,

to be published.

18. Radok, J.R.M.,

Quart. Appl. Math. Vol. 14, p. 289, 1956.

19. Dugdale, D.S.,

J. Mech. Phys. Solids, Vol. 8, p. 100, 1960.

RESUME On diseute les probldmes ~lastodynamiques qui concernent les fissures stationnaires et en propagation. Les travaux precedents sur la dynamique de rupture s'&aient principalement limit,s ~ consid~rer un mod$1e de fissure semi-infinie. Darts le present m~moire, on met sp~cialement l'accent sur une fissure ayant une configuration finie. En hatroduisant une dimension caract6ristique suppl6mentaire darts le probldme de la dynamique - ~ savoir la longueur de la fissure - on peut faire ~tat d'un certain nombre de caract~ristiques typiques d'une solution tenant eompte des contraintes dynamiques. Telles sont notamment les valeurs de crates observ~es pour les codfficients de concentration de coutraintes aux extr~mit~s de la fissure, lots de l'ouverture des fissures sous sollicitations cycliques ~ eertaines fr~quences, ou sous sollicitations brutales de traction ~ certains temps. On fournit ~galement une solution g~nSrale au probldme des fissures se propageant ~ vitesse constante. On montre que l'extension du moddle de Dugdale sur la plasticit6 an cas d'une fissure en propagation est toute hadiqu~e. -

ZUSAMMENFASSUNG - Es wurden elastisch~tynamische Probleme yon station//ren und beweglidaen Rissen diskutiert. Eine vorhergehende Arbeit an dynamischen Frakturen wurde haupts~chlich auf das Model eines halb-infinitiven Risses begrenzt. Es wurde in dieser Abhandlung besonderer Nachdruck auf die Struktur des Risses mit begrenzter L~nge gelegt. In dem man die erg//nzenden characteristischen Dimeusionen in dem dynamischen Problem beibeh~It, besonders die der Rissl~knge,k6nnen verschiedene typische Merkmale der dynamischen Ansparmungslfsung gezeigt werden. Dieses shad die H6chstwerte von den Rissspitzenanspannungsintensit~/tsfaktoren, die bei gewisser Wellenl~nge vorkommen. Sie kommen vor bei Rissen, die sich dutch periodisch, zyklische Ladungen 6ffnen und Risse, die sich zu bestimmter Zeit durch pl6tztich angewandtes Dehnen 6ffnen. Eine allgemeine L6sung zu dem Problem von Rissen, die sich mit konstanter Geschwindigkeit bewegen, wurde auch besprochen. Es wurde gezeigt, dass die Ausdehnung yon Dugdale's Plastizit~tsmodel im Falle eines sich bewegenden Risses, gerade ist.

68

G.C.Sih APPENDIX

The problem is to find the sectionally holomorphic functions qSj(zj) (j = 1,2) for a system of cracks moving collinearly along the x-axis. Let the jth crack Lj = ajbj be delimited by the points (aj,O), (bj ,0), j = 1, 2 . . . . . n, and the union of these cracks be denoted by L = L1 + L~ + . . . + Ln. The xy-plane is divided by the positive direction of the x-axis into left- and right-hand regions S+ and S-, respectively. In the first fundamental problem, tractions are to be specified on the upper and lower edges of each moving crack. Let these tractions be applied normally to L and self-balanced so as to satisfy equation (6). With the help of equation (4), the condition of vanishing shear stress for all values of ~ may be expressed by 4sls2¢l(~) + (1 + s~)2 q~2(~)=O

for

~ =0

(47)

where } represents those values of zj on the x-axis. To be remembered is that Lx, L2, • . . . . Ln are lines of discontinuities and therefore Cj(z]) take different limits depending upon whether zj -> } from S+ or S-. If a + and Oy denote the normal stresses on the upper and lower sides of L, then the application of equation (4) gives I

+

(~:/-- 2(1 + s~) %-

I

[(1 + s~) 2 - 4s,s2 ] [~+(}) + ~bT(})l, on L (48) [(1 + s22)2 - 4sas2] [q57(}) + aS+(})], on L

2(1 + s~)2 in whic.h fi:e function ~52(}) has been eliminated by means of equation (47). Adding and subtracting the two expressions in equation (48) yield

[(I)1 (}) "Jr"~1 (~)1-1- -[- [~1(~) + ~1(~)1- = 2fx (}), on L [ePl(}) - ~ , (})1+" [~I (}) - ~, (})]- = O,

(49)

on L

where Oy + = Oy and f,(}) is given in equation (8). Equation (49) shows that oPt(z,) - dp--1(z,) is holomorphic in the whole plane including L and vanishes at infmity for the present problem. Thus, Liouville'stheor~:~nasserts that

• ,(zl)= ~,(zl)

(50)

Knowing the solution to the non-homogeneous Hilbert problem(5), the remaining boundary condition in equation (49)together with equation (50) give the complex function q>l(zl) as shown in equation (7). A simple check shows that qbl(zl ) indeed satisfies equation (50)by requiring the coefficients in Pn(z) to be real and by recognizing that x+(~) = x'--(~) = x-(~) = - x+(}), on L The function ebb(z2) can be determined in exactly the same way.