AD-A131 911 ON DIFFERENTIAL EQUATIONS OF NONLOCAL ELASTICITY AND 1//SOLUTIONS OF SCREW D..(U) PRINCETON UNIV NO DEPT OF

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MICROCOPY RESOLUIION TLS (HARt

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ON DIFFERENTIAL EQUATIONS OF NONLOCAL ELASTICITY

01D AND

SOLUTIONS OF SCREW DISLOCATION AND SURFACE WAVES

A. Cemal Eringen

PRINCETON UNIVERSITY

Technical Report No. 58Civil Engng. Res. Rep. No. 83-SM-10

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underContract N00014-76-C-0240 Mod 4

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'ONLOCAL ELASTICITY &j SOLUJTIONS OF SCREW Technical ReportDISLOCATION & SURFACE WAVES G PERFORMING ORG. REPORT NUMBER

7 AUTHOR(.) . COTA:CT RARAS. MNMER.,

A.C. Eringen N00014-76.-002040 Mod. 4

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IS SUPPLEMENTARY N ,TES

It KEY WORDS (Cwnn ou r0-0., 0,6 I n.,..n eRIA IdeiUII, B. block num,"'I

nonlocal elasticity, waves, dislocation, surface waves

20 ABSTRACT (C.iIIII. An ,9,bl8* sed It ntl***on WrId 1ImfyI &Y bl.05 Omb.,)

Integro-partial differential equations of the linear theory ofnonlocal elasticity are reduced to singular partial differentialequations for a special class of physically admissible kernels.Solutions are obtained for the screw dislocation and surface wave;Experimentai observations and atomic lattice dynamics appear to,support the theoretical results very nicely.

DO 0A",147 tITIN f IJJO 6 1509OLITE SCCURITY CLASIPICATsIA9N THIS PAGE 1,*.AA D~ ~o cE

ON DIFFERENTIAL EQUATIONS OF NONLOCAL ELASTICITY

AND

SOLUTIONS OF SCREW DISLOCATION AND SURFACE WAVES

A. Cemal Eringen

PRINCETON UNIVERSITYPrinceton, NJ 08544

ABSTRACT

Integro-partial differential equations of the linear theory of

nonlocal elasticity are reduced to singular partial differential equations

for a special class of physically admissible kernels. Solutions are obtained

for the screw dislocation and surface waves. Experimental observations and

atomic lattice dynamics appear to support the theoretical results very nicely.

0 i

.3 A

t,, -,

2

1. INTRODUCTION

In the theory of nonlocal elasticity1 '2 the stress at a reference

point x is considered to be a functional of the strain field at every point

X' in the body. For homogeneous, isotropic bodies, the linear theory leads

to a set of integro-partial differential equations for the displacement field,

which are generally difficult to solve. For a spacial class of kernels, these

equations are reducible to a set of singular partial differential equations

forwhich the literature is extensive.

3

The selection of the appropriate class of kernels is not ad hoc

but fairly general, based on mathematical conditions of admissibility and

physical conditions of verifiability. For example, the dispersion curves

available from lattice dynamics and phonon dispersion experiments provide

excellent testing on the success of these kernels. Ultimately, these

kernels should be expressed in terms of interatomic force potentials or

corrolation functions. Presently, several solutions obtained for various

problems support the theory advanced here. For example, the dispersion1,3

curve, obtained for plane waves are in excellent agreement with those of the

Born-Kirmn theory of lattice dynamics. The dislocation core and cohesive

(theoretical) stress predicted by nonlocal theory4'5 are close to those known

in the physics of solids. Moreover, nonlocal theory reduces to classical

(local theory) in the long wave-length limit and to atomic lattice dynamics

in the short wave-length limit. 6 These and several other considerations

lead us to the exciting prospect that by means of nonlocal elasticity, excel-

lent approximation may be provided for a large class of physical phenomena

with characteristic lengths ranging from microscopic to macroscopic

scales. This situation becomes specially promising in dealing with imperfect

solids, dislocations and fracture, since in these cases, the internal (atomic)

state of the body is difficult to characterize.

The solution of nonlocal elasticity problems are however difficult

mathematically, since little is known on the treatment of integro-partial

differential equations, especially for mixed boundary value problems. There-

fore, the treatment of these problems by means of singular differential equa-

tions is promising. This is born out, at least, with the treatment of two

problems here, namely the screw dislocation and Rayleigh surface waves.

4

Results for both problems are gratifying in that they are supported by

atomic lattice dynamics and experiments.

The formulation of nonlocal elasticity in terms of singular

differential equations is reminiscent of other fields of physics and it

brings unification of various methodology. These are indicated briefly

in Section 3.

2. NONLOCAL ELASTICITY

In several previous papers, we developed a theory of nonlocal

elasticity, cf. [1 - 4]. According to this theory, the stress at a

reference point x in the body depends not only on the strains at x

but also on strains at all other points of the body. This observation is

in accordance with atomic theory of lattice dynamics and experimental

observations on phonon dispersion. In the limit when the effects of strains

at points other than x is neglected, one obtains classical (local) theory

of elasticity.

For homogeneous and isotropic elastic solids, the linear theory is

expressed by the set of equations

(2.1) tkz,k + p(ft - ) :

(2.2) tki(x)= a(I!'-xI') a ki(x') dv(x')

(2.3) Gk (') = err( ') 6kz + 2p ekQ(W)

kRr t k

5

(2.4) eka(u') Uk(X') au(x')ek-x + k-I

where t , p , f and u are respectively, the stress tensor,

mass density, body force density and the displacement vector at a reference

point x in the body, at time t. a (') is the macroscopic (classical)

stress tensor at x' which is related to the linear strain tensor ekk(x')

at any point x' in the body at time t , with X and U being Lam'e

constants. The only difference between (2.1) to (2.4) and the corresponding

equations of classical elasticity is in the constitutive equations (2.2)

which replaces Hooke's law (2.3) by (2.2). The volume integral in (2.2) is

over the region V occupied by the body.

Field equations of nonlocal elasticity are obtained by combining

(2.1) to (2.4). lie substitute Eq. (2.2) into (2.1) and use the identity

;x k CkQ( ax~ akkix

a a = + a o( +

k k

to obtain

(2.5) ci(j'-Xj) OkZ( ') nk da(') + J -(I' ) 0 k{,k, dv(x')

av V+ -~ 0~

Here the first integral,over the surface of the body, represents the surface

stresses,(e.g. surface tension). Consequently, nonlocal theory accounts for

6

4uWLace physiac, an important asset not included in classical theories.

Substitution of Eqs. (2.3 and 2.4) into (2.5) gives the field equations

(2.6) " (I'-x) Ix ur 6ki + ' + uk)]nk da'

+ [(xk*) ukk + 'Pu,kk]dv' + p(f, - UZ) 0

V

where a prime (') denotes dependence on x' , e.g. u'= u(x').

Integral equations (2.6) must be solved to determine the displacement

field u(x,t) under appropriate boundary and initial conditions. Boundary

and initial conditions involving the displacement and velocity fields are

identical to those of the classical theory. Boundary conditions involving

tractions is based on the stress tensor tkk , not on k, i.e.

(2.7) tkknk = t(n)P,

where t(n)k are the prescribed boundary tractions.

3. DETERI4INATION OF NONLOCAL MODULUS

From the structure of the constitutive equations (2.2), it is clear

that the nontoca Z modu6u, o (I'-!) has the dimension of (length)3

Therefore, it will depend on a characteristic length ratio a/k , where

a is an internal characteristic length (e.g. lattice parameter, granular

7

distance) and 9 is an external characteristic length (e.g. crack length,

wave length). We may express a in a more appropriate form as

(3.1) a c(IX'-XI,T) ,T eo a/i

where e0 is a constant appropriate to each material.

The nonlocal modulus has the following interesting properties.

(i) It acquires its maximum at x'= x attenuating with Ix'-xl

(ii) When Tr-0, a must revert to the Dirac delta measure so thatclassical elasticity limit is included in the limit of vanishing

internal characteristic length.

(3.2) lim a(Ix'-xl,t) = 6(Ix'-xl)

O

We therefore expect that a is a delta sequence.

(iii) For small internal characteristic lengths, i.e. when T l,nonlocal theory should approximate atomic lattice dynamics.

In fact, by discretizing Eq. (2.2), it can be shown that equations of nonlocal

elasticity reverts to those of atomic lattice dynamics 6

(iv) By matching the dispersion curves of plane waves with those ofatomic lattice dynamics (or experiments), we can determine 0,

for a given material. Several different forms have been found1'3'7.

rA

8

The following are some examples which have found applications:

(a) One-Dimensional ModuZi

(3.3) cdxft) = li

=0 fxf > kT

(3.4) C( xI,-[) -2 e eIXl

(3.5) = exp( _x2/k2)

(b) Two-Dimensional ModuZi

( 3 .6 ) c ( I x l, t ) = (2 7, 2 T ) K 0 ( x,! r - / )

where K0 is the modified Bessel function

= exp(- x 2x/ z2T

(c) Three-Dimensiona! ModuZi

(3.8) C(LxI,t) = 3 exp(-x-x/4t) t 2T/48(rt)

(3.9) ,(4 2 2 l (x x)' exp(- x --x/ )

9

We note that Eq. (3.3) gives a perfect match of the dispersion curve of

one-dimensional plane waves based on the nonlocal elasticity and the Born-

3K~rmIn model of the atomic lattice dynamics . For two-dimensional lattices,

Eq. (3.6) provides an excellent match with atomic dispersion curves, with a

maximum error less than 1.22, Ari and Eringen7 . All other nonlocal moduli

also provide excellent approximation to the atomic dispersion curves, for a

8choice of e0 , Eringen

(v) We observe that all nonlocal moduli given above ar ' lized so

that their integrals over the domain of integration (line, surface, volume)

give unity. Moreover, they are all 6-sequence, i.e. when T - 0 we

obtain the Dirac delta function. Because of this property,

nonlocal elasticity in the limit T- 0 reverts to classical elasticity as

can be seen by letting -- 0 in (2.2), to obtain Hookes law of classical

elasticity.

We now exploit this observation further by assuming that:

(vi) c is Green's function of a linear differential operator:

(3.10) L cdlx'-xl T) = 6(j!'-xI)

If such an operator can be found, then applying L to (2.2), we obtain

(3.11) L t k

10

In particular, if L is a differential operator with constant coefficients,

then

(3.12) (L t ),k L tkk,k

and (2.1) gives

(3.13) 0 k ,k + L(pf - pj) 9 0

In this case, we have partial differential equations to solve, instead

of integro-partial differential equations. This, of course, provides a

great deal of simplicity over the original equations (2.6). In particular,

for static problems with vanishing body forces (or more generally, whet,

L (pf -pU) = 0, we have the classical equations of equilibrium

(3.14) , 0

which upon using (2.3) gives Naviers' equations.

The nonlocal mudulus (3.8) is a Green's function which satisfies the

differential equation

(3.15) ,2 D = ,

The fact that this is the case is well-known since the solution t(x,t) of the

diffusion equation

11

(3.16) 2- - 0

which, fur t=O coincide with a given continuous function a(x), is given by

(3.17) t = a(Ix'-x1,I) o(x') dv(x')

V

10when L' extends to infinity in all directions, cf. Courant •

Eq. (3.17) is valid even when V is finite and the reference point x is V

not too close to the boundaries, since a attenuates rapidly to zero with

iX X' •~

Of course the differential operator L may be different than the

diffusion operator. For example, for Eq. (3.6), one can see that

(3-18) L = 1 -

i.e., we have

(3.19) (1 )t

In fact, this result can be justified by an approximation of the atomic

dispersion relations. To see this, consider the Born-Karman model of

lattice dynamics and equate the expression of the frequency given there

to that of nonlocal theory for plane waves.

(3,20) (k) (W/O)2 (2/k2a2)(l - cos ka)

12

We rewrite this expression in the form

2a2 k2a2 k 4a4(3.21) 2(I - cos ka)/k a 1 - T + 3-60

= 2 2 - I1

( + e0a 2 kk

for k2 a2 << 1. This is permissible since according to

Tauberian theorems, the behavior of a function near infinity is reflected

in the behavior of its Fourier transform near the origin. Thus, this

approximation is tentamount to approximating c(IxI) for large x "

In fact, by this process, the boundary of the Brillouin zone is thrown to

infinity so that a does not have a finite support. By matching =

2 2 2 -1(l + eoa k ) obtained this way with Eq. (3,20) at the end of the

Brillouin zone, we obtain a curve which approximates the atomic dispersion

curve quite well-

In Figure 1, the comparison is made for this matching, i.e.

2 2 2,-'-(3.22) wa/c 0 = k a (1 + e0k a ) (Nonlocal)

wa/c 0 = 2 sin(ka/2) (Lattice Dynamics)

The matching is perfect at ka=1T, which leads to

(3.23) e0 = 0.39

The maximum error is of the order of 6% in Ikal < 7 . Note, however,

that the group velocity at ka=,r is badly off and this is the price we

have to pay.

13

If we accept this approximation, Eq. (3.21) upon inversion, gives

(3.24) (1 - 22 )0 = 6(1x1) ; % e 0 a

The application of this operator to Eq. (3.17) then, leads to (3.19). Of

course, other types of approximations are possible.

It is not difficult to determine L for other moduli. In fact, for

Gaussian kernels, (3.5) and (3.7) are similar to (3.15).

(vii) The above considerations further suggest that Eq. (3.17) may be

considered as the probablistic average of a , if a is considered to

be a probability density function. In fact, (3.8) is non-other than the

Gaussian density function. Such a consideration resembles the method of

analysis of quantum mechanics with the probability density function satis-

fying a diffusion equation.

(viii) A somewhat different interpretation of (3.16) is made by

considering a. as a correlation function,in which case, it should be

possible to determine a from statistical mechanical considerations. Such

a study is now underway.

14

4. SCREW DISLOCATION REVISITED

Consider a screw dislocation, in the sense of Volterra, located

at the plane x3 =O of rectangular coordinates xk , Fig. 2. The displace-

ment field has only single component u3 (xl,x 2 ,t). The stress field is

determined by solving Eq. (3.16) whose Laplace transform with respect to t is

(4.1) V2 " -tkk - s tkk Okk

where 0k. has only two now-vanishing components

(4.2) 031 032 3

In (4.1), a superposed bar indicates the Laplace transform and s is the

transform variable,

The divergence of Eq. (4,l),upon using (2.1) and (4.2), gives

(4.3) Okk,k = (V2 - s) Puk

For the static case, Eq. (4.3) gives

(4.4) V2 u3 = 0

whose solution is

(4.5) u b tan- 1 (x2/xl)

where b is the Burger's vector. In polar coordinates (r,e,z) this is

equivalent to

15

(4.6) b b

(4.7) s31 E " , 32 - Cos

Carrying (4.7) into (4.1), we obtain differential equations for tj31

and t32 , whose general solutions, having proper symmetry with respect to +e,are given by

(4.8) t 31 : 1 (p) sine, = T2 (c ) cos e

where

(4.9) = A0 KI + B"' 11(P) + (.) C , Cl,2 j(4.10) p= r, C = Pb/27T's

Here, If(p) and Kl(p) are modified Bessels' functions and A and B are

arbitrary constants. The stress field must vanish as p,= . This implies

that B :0. In cyclindrical coordinates (r,e,z), the stress field is

given by

1zr = (A1 + A2) K,(p) sine cosetz = (-A1 t

in2e + A2 Cos 6) KI(M) + C p

If we imagine the edge line p-0 of the dislocation as a limit c40 of a

small cylindrical surface with radius r=c , then tzr must vanish as c-0.

This, through (4.11), gives A2 a-A 1 so that

I ..., .........l r. ........ .." ...I .. .......I' .......I[ ... ..." -.....I ..

16

(4.12) tzr 0 z "A1 KI(P) + C p-1

The hoop stress tze will be regular at p-O, if and only if, AlC.

Consequently,

(4.13) = r 1 r Kl(rv))(.3 tze 2rr s rS

(4.14) te b (1 - er2/4t)

2r

Remembering t =I 2 /4 and in terms of the definition of ci given in Ref. [4J,

t= a2 /4k2 , this result is in agreement with that found earlier in a

4different way. We observe that the displacement field (4.5) of a screw

dislocation in a nonlocal elastic medium is the same as in a classical

elasticity, even though stress field is different. This is the result

of the particular choice of the kernel. The effects of anholonomicity of

the dispersion curve near the boundaries of the Brillouin zone and the

nonlinear force law are ignored.

Had we employed the operator (3.18) instead of (3.15), we would

have obtained

(4.15) tr b[ - - Kl(r/Tk) ] all other tk= 0(415 ze r Tk [1- k

This is also regular for all r in the interval 0 < r •<.

In non-dimensional form, (4.14) may be written as

(4 c) - p 1(1 - P )

wh!

(4.17) Te = 2ra tze/ubk - pl e( 2), p kr/a

" . * t4.i.q *

17

To(p) possesses a maximum at PC which is the root of

2(4.18) 1 + 2 Pc ' PC 1.1209

and temax is given by

2 Pc

(4.19) Te '- 2 0.6332(4]96Tmax ]+ 2 Pc2

PC

The plot of Te(P) versus p is shown in Fig. 3. For a perfectly brittle

crystal, the theoretical shear strength t is reached when tze =ty. Con-y z

sequently,

(4.20) ty = 0.6382 2ibk2Tna

In a previous paper, we have shown that k= 1.65 gives a perfect match in

the entire Brillouin zone of Born-Kgrman lattice with an error less than

0.2%. Using this value and b/a=I/v'7 for fcc materials in (4.20) we find

(4.21) t y/V = 0.12

This result compares well with the value 0.11 for Ak (f.c.c.); W, a-Fe

(b.c.c.) and 0.12 for Na CZ, MgO. (cf. Lawn and Wilshaw, p. 160).

In examining Fig. 2 closely, we note that when ty tzemax a brittle

perfect crystal will rupture. However, if the crystal is ductile at this

point, dislocations will be produced. Thus the region around the crack tip

0 < P<P c is a dislocation-free zone, i.e. dislocations will emerge at p =c c

and will pile up in a region p> Pc" This prediction of the present theoryiaIs supported by the recent observations made in electron microscopy.

18

According to the present theory, the rupture or dislocation initia-

tion then begins not at the center of the dislocation p -0 but at a critical

distance p 0-c> . This is against our previous understanding of the dis-

location and fracture mechanism but supported by experiments.

Finally, we may wish to enquire whether these predictions are

greatly affected by the kernel chosen. In the case of the kernel (3.6),

leading to Eq. (4.15), the T(P)-curve is also shown in Fig. 2 with Pc

T and ty given by6ma x x

(4.22) c .1, Temax z 0.3995

(4.23) t = 0.3996 - jib2r eoa

If we write h=e0 a/0.3995, Eq. (4.23) agrees with the estimate of Frenkel12

based on an atomic model (cf. Kelly , p. 12). In the case of the kernel

(3.6), the match of the dispersion curves were provided for the value of

e0 = 0.39 with an error less than 6% at a point in the Brillouin zone (Eq.

3.23). Using this value, we again obtain t y/p= 0.12. Thus, in spite of

the difference in the two curves in Fig. 2, the resulting theoretical

strengths are not too far off from each other. Moreover, the location of

the maxima are nearly coincident so that the origin of the dislocation

generation or rupture are predicted to be the same.

19

5. RAYLEIGH SURFACE WAVES

13In a previous paper , we have determined the phase velocity of

Rayleigh surface waves and found that they are dispersive. In these calcu-

lations, one-dimensional kernel (3.3) was used with nonlocal effects taken

only in the direction of boundary line, x2= 0 in two-dimensional medium

0 X2 c -O <x 1 < . Here, we take advantage of the two-dimensional

kernel (3.6) which reduces constitutive equations to the form (3.19). Upon

taking the divergence of (3.19) and using (2.1), (2.3) and (2.4) with f=0,

we obtain

(5.1) (X+) u + IjUkk - (1 - . 22 2 )u£ = 0

For the plane-strain, we introduce Lam potentials O(x,t) and q,(x,t),

(_x = {xl,x 2 }) by

(5.2) ul 21- + 2 u =I ax1I ax2 2 ax2 ax1

leading to

2 2 (Cl V C ) ( T 0,

(5.3)

c2 V2 2 2V2) V 2

where cI and c2 are respectively, classical phase velocities defined by

(5,4) C = '2 =

20

We now try solutions of (5.3) in the forms of surface waves characterized by

= A exp[- kvix 2 + i k(x I - ct)]

(5.5)

= B exp[- kv2x2 + i k (x2 - ct)]

Equations (5.3) are satisfied if v are given by

(5.6) 2 = 1 - (C/C) 2 [l - k2 r2k2(c/c) 2]- , a=1,2

Using (5.5) in (5.2), we obtain the displacement field and carrying (5.2) into

the constitutive equations (2.2) to (2.5), we arrive at the stress field.

Thus, for example, at x2 =0 we have

t22 = [(Cl/c2)2(V1_l)+2]MlA + 2iv 2M2B

(5.7) t21 = -2iMA + (1 +,)2 )M2B

where

21j

M, Jdx Ldxi K0 {[(xj - 2l + .21/2

*exp [-k vixi + 2k(xi - ct)]

(5.8)

M - dx2 dxl Ko{[(x xl)2 + x21/211

2 1 x

* ex, [k,,2x2,, + ik(xi - ct)]

But t22 and t21 must vanish at x2 0 Hence, we must have

(5.9) [(c 1 /c 2 )2 ( 21)+21(l + 2 - 4 v'2

provided MIM 2 # 0. This is the Rayleigh determinant for nonlocal elastic

surface waves. Upon carrying (5,6) into (5.9), eventually we rearrange (5.9)

into the form

(5.10) Y[al )'3 + a2 2 + a3, + a4 ) : 0

wherea+l 2 + +4-3M24 + m(l-m)c6

I T- 4 4

22

(5.11) 82 = + 2m2 "m-3 2 2 42 " (l+m-2m )c

m + (2-m-m2 2)a 3 2

a4 = 1-m

2 =ki, m=1-2vy= (c/c2)2 2(1-v)

and v is Poisson's ratio.

It is not difficult to verify that for E=0, (5.10) reduces to the

the classical Rayleigh function whise roots are recorded in Ref. 114] for

various v . It is also clear that the roots of (5.10) is a function of E ,

consequently, the Rayleigh wave velocity is dispersive. In Table 1 below,

we give values of /7 = cR/C2 as a function of c= e0ka, for various

Poisson's ratios v . Phase velocities cR/c2 versus c=e 0 ka is plotted

in Fig. 4 for various values of v . The non-dimensional frequency _z/c 2 .Table 2 and

versus e are displayed in/Fig. 5. To provide a comparison of these re-

sults with the lattice dynamic calculations, we divide the abscissa and

ordinate of Fig. 4 by e0 = 0.31. Fig. 6 shows ka/c 2 versus ka of nonlocal

results. Comparison with atomic lattice dynanic calculations carried out by15

Wallis and Gazis for KC is excellent. lie have used v = 0.3 to approxi-

mate the KC molecules with an isotropic solid. The lattice dynamic cal-

culations were made for waves propagating in the (100) direction on a (001)

surface.

23

REFERENCES

[1] A.C. Eringen. Int. J. Engnq. Sci., 10, 425, 1972.

[2) A.C. Eringen, Nonlocal Polar Field Theories, Vol. 4, 205. Academic Press,1976.

[3] A.C. Eringen, Continuum Mechanics Aspects of Geodynamics and Rock FractureMechanics (Edit. P. Thoft Christensen), 81, D. Reidel, 1974.

[4] A.C. Eringen, J. Phys. D: Appl. Phys., 10, 671, 1977.

[5] A.C. Eringen, Int. J. of Fracture, 14, 367, 1978.

[6] A.C. Eringen and B.S. Kim, Crystal Lattice Defects, 7, 51, 1977.

[7] N. Ari and A.C. Eringen, Crystal Lattice Defects.(to be published1982).

[8] A.C. Eringen, Nonlinear Equations of Physics and Mathematics, 271 (Ed.,A.O. Barut), D. Reidel, 1978.

[9] 1 am indebted to a referee who pointed out that the device of Green-

function method for reduction of nonlocal operators has beenused in other parts of physics (Ph. Rev. B 5, 4637, (1972),Ph . Rev., B 7, 2787 (1973)).

L10] R. Courant, Methods of Mathematical Physics, Vol. II, Interscience, p. 199,1965.

[ll] S.M. Ohr and S. Chang, J. Appl. Phys., 53, 5645, 1982.

[12] A. Kelly, Strong Solids, 12, Oxford, 1966.

[13] A.C. Eringen, Letters in Appl. Engng. Sci., 1, 1, 1973.

[14: A.C. Eringen and E.S. Suhubi, Elastodynamics, Vol. 2, Academic Press,p. 521, 1975.

[15] R.F. Wallis and D.C. Gazis, Lattice Dynamics, 537 (Edit. by R.F. Wallis),Pergamon Press, 1965.

TABLE I

Phase Velocity of Rayleigh Surface Waves

CR/C2

c eoka v 0.0 v =0. 1 D.2 v 0.3 v = 0.4 v 0.5

0 0.87402 0.89311 0.91099 0.9274 0.9422 0.95532

0.1 0.86382 0.88418 0.90337 0.92096 0.9366 0.94998

0.2 0.8356 0.85902 0.88162 0.90242 0.92042 0.93446

0.3 0.79476 0.82176 0.84857 0.87377 0.89519 0.91025

0.4 0.74719 0.77715 0.80798 0.83775 0.86318 0.8794

0.5 0.69757 0.72944 0.76336 0.79724 0.82664 0.84415

0.6 0.64889 0.68173 0.71768 0.75479 0.78772 0.80646

0.7 0.60288 0.63598 0.67295 0.71226 0.74819 0.76809

0.8 0.56044 0.59312 0.63039 0.67105 0.7093 0.73021

0.9 0.52176 0.55374 0.5907 0.63193 0.67187 0.69363

1.0 0.4869 0.51786 0.55407 0.59537 0.63637 0.65882

TABLE II

Dispersion of Rayleigh Surface Waves

wa/c 2

c eoka v0.O v=0. 1 v = 0.2 v = 0.3 v = 0.4 v = 0.5

0 0 0 0 0 0 0

0.1 0.086382 0.08842 0.09034 0.09210 0.09366 0.09500

0.2 0.16712 0.1718 0.17632 0.18048 0.18408 0.18689

0.3 0.23843 0.24653 0.25457 0.26213 0.26856 0.27307

0.4 0.29888 0.31086 0.32319 0.3351 0.34527 0.35176

0.5 0.34878 0.36472 0.38168 0.39862 0.41332 0.42207

0.6 0.38933 0.40904 0.43061 0.45288 0.47263 0.48388

0.7 0.42202 0.44518 0.47106 0.49858 0.52373 0.53766

0.8 0.44835 0.4745 0.50431 0.53684 0.56744 0.58417

0.9 0.4698 0.49837 0.53163 0.56B74 0.60469 0.62427

1.0 0.4869 0.51786 0.55407 0.59537 0.63637 0.65882

FIGURE CAPTIONS

Figure

1 Dispersion Curves

2 Screw Dislocation

3 Non-Dimensional Hoop Stress

4 Phase Velocity of Rayleigh Surface Waves

5 Dispersion of Rayleigh Surface Waves

6 Dispersion Relations for Rayleigh Surface Waves.4

ar-z C,,

0

0nNQ S Q

x 23

FIGURE 2 :SCREW DISLOCATION

09

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0.7- PC8 1* 112

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0.4-

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IdON-DWMESIONAL 1400P STRESSFIGUE 3

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d C5 d 0 0 d

IAJ.

0 F@6 V)

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i N

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474:NP:716:lab78u474-619

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