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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
Research Sponsored by theOFFICE OF NAVAL RESEARCH
underContract N00014-76-C-0240 Mod 4
Task No. NR 064-410 j T IC
CL)C
August 1983 AUG 3 01983LU
L.-Approved for public release:Adistribution unlimited
Reproduction in whole or in part is permittedfor any purpose of the United States Government.
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TITE *hIION DIFFERENTIAL EQUATIONS OF & TYP OF REPORT & PERIOD COVERED
'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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unclassifiedIIL* OECLASSIFICATION OWNGRADING
SIhEDULE
14 DISTRIBUTION STATEMENT f.0 Ih,. P.p.,I)
17 DIST RIOUTIOI ST ATtMENT eof rho ba.i,oc f*do Bloch 20. I11 -lIo, fR opW,')
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
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Part I - Government Navy (Con't.)Administrative and Liaison Activities
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A.- -_ . ... .. .. . . . . . .. . . . . ... . .. ... . ...!. .. . ..,, ,
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Navy (Con't.) navy (Con't.)
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~ ~ .*___________
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