On the three-dimensional Boussinesq problem for an elastic nonlocal medium

0020-7225/W$3.00+ 0.00

hr. J. EngngSci.Vol. 28, No. 12,pp. 1245-1251, 1990

Copyright @ 1990Pergamon Press plc

Printed in Great Britain. All rights reserved

ON THE THREE-DIMENSIONAL BOUSSINESQ PROBLEM FOR AN ELASTIC NONLOCAL MEDIUM? J. L. NOWINSKI Department

of Mechanical Engineering,

University of Delaware, Newark, DE 19716,U.S.A.

Ahatract-After determining the nonlocal elastic moduli and the constitutive equations used, a brief review of the Kelvin problem in nonlocal setting is given. The Westergaard procedure of transition from the classical Kelvin problem to the classical Boussinesq problem is discussed, and applied to the nonlocal case using Fourier’s exponential transformation. An example illustrating the application of the method to calculate the stress system in a nonlocal half-space is given.

1. INTRODUCTION

It is well known that one of the fundamental assumptions of the nonlocal theories of matter is that interactions of the elementary particles are not limited to their infinitestimal neighborhoods, but extend over distances many times exceeding the spaces between the particles. The diversity of nonlocal problems encountered both in physics and in technology is considerable, and early observations on nonlocal phenomena go far back to Duhem in 1893 and Rayleigh in 1918 (cf. [l]). Nonlocal situations appear in fact in branches of science so widely apart as quantum mechanics, relativistic particle dynamics, atomic lattice dynamics, and continuum mechanics. A nonlocal aspect of the last named discipline, in particular of mechanics of elastic media-the subject of the present study-is of more recent vintage, and owes its development primarily to Edelen, Eringen, Kroener, and Kunin (see, e.g. Refs [2-51, respectively). The nonlocality enters the nonlocal theory of elasticity through the assumed integral form of the constitutive equations. One version of these equations proposed by Eringen and Kroener, and adopted in the present study, is rij(Z) = “[2~‘(12’ - Z’l)eb(Z’) + n’(lz’ - ?l)e;,(Y) I

S,] du’(T’),

(1.1)

where rij and eij are the stress and the strain tensor, respectively, A(]Z - 2’1) and ~(lz’ - .?‘I) are the nonlocal elastic moduli, V is the volume of the body, a repeated index denotes the Einstein summation, the prime marks a generic point of the body, and the symbol Z refers to the point of observation. The calculation of the nonlocal moduli is most often done by comparison with the corresponding coefficients appearing in the atomic lattice dynamics. This idea and other possibilities were suggested by Eringen (cf., e.g. Ref. [6]). Different lines of approach were proposed by physicists; they are summed up in Ref. [7]. As shown in Ref. [8], a frequently selected expression for the Fourier exponential transforms of the nonlocal moduli has the form

.*ky

w +2jiW) = I+2p

1

i

(A + 2P)U n=l

( >, c”n2s1nko22 i

-

(l-2)

PI )

where il and ~1 are Lame’s constants, a the interparticle distance, k the Fourier transform coefficient, C,, the so-called force constant (the value of which depends on the modulus involved), Nu the range of cohesion, and the bar denotes the transform: f(k)

= lrn f(z)eikz dz,

-m

f(z) = & 1: f(k)epik’ m t Prepared with partial support of the University of Delaware. 1245

dk.

(I-3)

1246

J. L. NOWINSKI

Formulas similar to (1.3) hold true for the ratios &k)lL and F(k)/p with an appropriate change of the value of Cn’s, It is not difficult to invert the equation (1.2), and obtain the actual values of the nonlocal moduli, A’(12 - 2’1) + 2p’(lz - z’l) =&$,G-$(l-F), 1+2p n

(1.4)

where ]z - z’] I na. The present study is concerned with the Boussinesq problem in which a half-infinite nonlocal space is acted upon by a concentrated force applied normally at the plane boundary of the medium. We first give a brief review of the known solution to the nonlocal Kelvin problem [9] for the reason that it serves as a basis of the Westergaard procedure. A version of the latter is then given in terms of the Fourier exponential transforms, and applied to the problem in hand. The study ends with an example illustrating computation of the state of stress.

2. BRIEF

REVIEW

OF KELVIN’S

NONLOCAL

PROBLEM

Let a nonlocal medium fill a half-infinite space z 2 0 and be acted upon by a force P. The force is applied at the origin 0 of a cylindrical coordinate system r, (Y, z, and acts in the positive direction of the z-axis (Fig. 1). Clearly the problem exhibits a cylindrical symmetry. There are several ways of arriving at the solution to the so posed Boussinesq problem within the framework of the local theory of elasticity. One of those, modified in the present study, so as to work in the nonlocal case, makes use of the solution of the Kelvin problem. It is known that the latter involves the action of a force (2P, say) applied at a point (r = z = 0, say) of an infinite medium, and generating in the classical case the stress components such as r,,(r, z) = -B

(l-2v)z+3t3 R3

1

R5 ’

t,(r,

z) = -B

1

(1 - 2v)r I 3rz2 R3 R5 ’

(2.1)

where B = P/4~r(lY) and R = (r* + z~)~‘~. Now, it so happens that if Poisson’s ratio of the medium, Y, equals l/2, then the Kelvin problem automatically splits into a pair of Boussinesq’s problems, with the half-space z z 0 acted upon by the force +P, and the half-space z I 0 by the force -P. In the former case we have, of course, from (2.1), tlL(r, z) = --

3Pz3 2zrR5’

z,,(r, z) = --

3Prz2 27rR5 ’

(2.2)

so that the surface tractions vanish at the plane z = 0 (except at the point of application of the force) as required in the Boussinesq case. This being so in the local case, we decide to suppose temporarily that the same holds also true in the nonlocal case. As will be shown shortly, our conjecture proves in fact to be correct with a high degree of accuracy. To this end we first take advantage of the solution of the nonlocal Kelvin problem given in [9], and for future reference, and to make the present study more self-contained, we cite the following equations:

Fig. 1. Geometry of the problem.

On the three-dimensional

1247

Boussinesq problem

(1) Stress components frr(r, k) = &ii,, + &ii/r - ik&W, &(r,

k) = &ii/r + d29,, - ik&i+‘,

TZr(r, k) = &(ii,, + ii/r) - ikaI@, TJr,

k) = fi,(Gir - ikti),

(2.3)

where fil = i(k) + 2p(k), a2 = i(k), a3 = P(k), ii and %Jare displacement components in the radial and axial direction, respectively, and the index after the comma denotes differentiation (e.g. ii,, = &I&). (2) Dilatation ti(r, k) = ii,, + ii/r - ik@.

(2.4)

w,=k2z+-

(2.5)

(3) Displacement components ii = ikX,r,

where j(r, k) is the transform of a displacement

a’ 022, a2+a3

potential X(r, k) such that

0202x = 0

(2.6)

0’ = d2/dr2 + i dldr - k2.

(2.7)

with

A particular solution of (2.6), of interest in the problem in question, is g(r, k) = K,(kr), K0 is the modified Bessel function of the second kind and of zero order; its inverse is 1

1

X(r, z) =

where

(2.8)

2(r2 + z2)ln = 2R’

With the foregoing in mind, it is readily found that the transforms of the stress components interest become,

where K~ = (3a2 + 4a,)/(a,

&Jr,

k) = -4ik&[K1K,,(kr)

?Jr,

k) = 4ti,[rk2KO(kr) - K2kK,(kr)]B,

+ a3) and rc2= a,/(a,

r&r,

z) = -

r,,(r,

z) = -

P 4J6(1-

- rkKl(kr)]B, (2.9)

+ a3). Inversion of the equations

(1- 2v)(z + 28a) + 3(2 + 28a)3 Y) C

of

R;

R: 3r(z + 2Ba)’ R;

1’

above gives?

1’ (2.10)

where 0 5 8 I 1 and R, = [r2 + (z + 2Ba)2]1n. It is immediately seen that if the nonlocal medium converts into the classical continuum, that is, if the interparticle distance, a, is set equal to zero, then the equations (2.10) reduce into their well-known classical counterparts (cf. Ref. [lo], p. 199). Returning now to our earlier conjecture, a glance at the equations (2.10) shows that for v = l/2: (a) The surface tractions actually vanish at the plane z = -2&r, that is at the distance of the order of 10e8 cm from the plane t = 0$-a distance of no significance if compared with an infinite depth of the half-space. (b) At the plane z = 0 even at the distance 10-l cm from the load, the tractions rZZand rZ, become so small as to be of the order of lo-l4 kgf/cm2 and lOmEkgf/cm’ per unit load, respectively. t Note that in Ref. [9] the assumed load is P. $ At which they should vanish according to the conjecture.

1248

J. L. NOWINSKI

Taking these two facts into account it is possible to assume that for Poisson’s ratio Y = 112 the solution of the Kelvin problem furnishes the solution of the Bousinesq problem not only in the local but with a high degree of accuracy in the nonlocal case as well, the respective stress components in both cases being z;&, 2) = -jd

P 3(2 + 26a)3 R; ’ (2.11)

There is hardly need to say that for Poisson’s ratio different from l/2 the equations above are at fault, and have either to be corrected or completed. The last named operation is actually carried out in the subsequent section using the idea of Westergaard.

3. FOURlER

TRANSFORM

APPROACH

TO W~ST~R~AARD’S

PROCEDURE

The procedure proposed by Westergaard [ 111, that may prove to be useful in solution of certain class of more complicated problems of elastomechanics, enables one to determine how the change of Poisson’s ratio from its value Y = l/2 to some other appropriate selected value Y # l/2 effects the state of stress in the body. In this section we give a translation of the Westergaard procedure to the space of Fourier exponential transforms. To do this it is required to consider three different cylindrically symmetrical problems. (I) First of those marked by the minuscule 0 represents the problem at hand the solution of which is sought. Referring to equations (2.3) we write down the associated constitutive equations, %$= -0 t (YP= TZ:‘, = pz =

6&

+ &8/r

- &G*6J0,

di ii”/r + &iipr - ik& $‘, &(iq + G”/r) - ikiil I@, a,(t?f, - ikii’),

(3*1>

as well as the equations of equilibrium including Poisson’s coefftcient Y,

2(1- ” l-2Y

eO _ ik~,O

I= 0

*?

2(1-

,

1-

Y)

ik6jO-bf (r15”),~= 0,

(3.2)

where 8” = Ly, + ii’/r - ik#’ and 3’ = -(iki? + Spr) is the dilatation and the rotation, respectively. To these we add the actually prescribed boundary conditions denoted BC?, say. (II) The second problem includes the same? boundary conditions BC’, and the same value of the Lame constant p as the problem (I). The value m of the Poisson ratio, however, differs from the value Y adopted in the problem (I), so that the Lame coefficients h appearing in problems (I) and (II) differ. All quantities associated with problem (II} are singled out by an asterisk, and so, for example, we have,

2’1-m’e*_ik~“=0 , l-2m

2(1-m).

-

1

1 _ 2m Ike* + ; (ri;l*),F= 0.

”

(3.3)

(III) We construct the third problem as a difference between problems (I) and (II), and distinguish it by two asterisks. This gives, say, u** = u” - u*, rv** = w” - w*, G** = 6” - 8*, -**=&o-&j*, and CO 211-v) ____l-2Y

--

2(1 - v) l-2Y

2(1-m) l_2m 2(1 -m) l-2m

-* 2(1-v) 8,,+~

s** ,r -ik&**=O,

-**’ 1&&*+2(1-y). I

1

Ek@ + ; (a**),,

t This restriction suitable for the purposes of this text may be waived.

= 0.

(3.4)

On the three-dimensional

1249

Boussinesq problem

Clearly, the solution of the problem (III) satisfies the zero boundary conditions, and added to the solution of the problem (II) yields the solution of the problem (I). The trick now consists in the assumption that the displacements in the problem (III) are derivable from a potential &r, k) such that ii**@, k) = & We next refer to equation stress components,

@**(T, k) = ik$

and

(2.3) and after some manipulation 1 rt:**

=~~a* +

(3.5)

arrive at the equations

of the

K28**-(li**/r-i/Cm**),

a3

1

1 -** _ -** _ ikQ**, r r,z - w,, a3

(3.6)

where 1-v

v-m K1=(l - 2v)(l-

e** =

Kz=~,

2m)’

q*+ + 2k*cj.

(3.7)

Having the freedom to select the displacement potential $(r, k) in the form appropriate the intended purposes, we set the stress component t:$ = 0. This gives

s

a*= -(vO*cj

+ k*t$)

for

(3.8)

as the first condition imposed on the function 6. The second condition is gained by substituting the expressions (3.6) into the equations equilibrium. We find easily that I%$ = 0,

(3.9)

which makes (3.8) read k*$,=---

m--Y l-2m

e*.

(3.10)

At this point we apply the procedure just explained to the problem of Boussinesq, and make the following crucial decisions: (4 We identify problem (I) with the Boussinesq problem involving an arbitrary value, v, of Poisson’s ratio. (b) We identify problem (II) with the Kelvin problem involving a particular value, m, (actually, one half) of Poisson’s ratio. This simultaneously furnishes the solution to a specific Boussinesq problem in which Poisson’s ratio is v = m = l/2. (cl We apply Westergaard’s procedure pertaining to the problem (III), and find that portion of the desired solution that added to the solution of the problem (II) yields a complete solution of the problem (I). To carry out the just formulated plan we refer to the equation (2.4) and with the function z(r, k) equal to &(kr) find easily that 8*(r, k) = -4K,ikK,(kr)

(3.11)

where ~~ = a,/(a, + a3). Equation (3.10) then gives for m = l/2 $ = B;

(2v - l)K,(kr),

where

B=

P

4;7G(l- m)p

=-

P

2np’

(3.12)

1250

J. L. NOWINSKI

We are now in a position to find the stress system associated with the title problem and designated (I). We first observe that from the equations (3.5) and (3.6) there follows that r-Fz*= 0. Again by assumption there is t,*,*= 0. In view of these two facts the equations (2.11) remain true for any value of Poisson’s constant, and thus provide the final solution to the title problem. This is not the case, however, as regards the remaining stress components, ?rr and LX* This will be illustrated in the succeeding section.

ILLUSTRATIVE

4.

EXAMPLE

In order to find the final value of the stress component equation (3.6) and after some manipulations find that & T;*(*(T,k) = B

2(2v - 1)

T,:*, for example,

we refer to

(4.1)

iK,(kr),

r

3

where again B =-

P 2nry’

We invert the equation above to get

z;*(r,

z) = 4j~B -

and take advantage of the convolution

-

-zKl(kt)e-ik’

dk,

(4.2)

theorem

&1:m@%XkFik’dk = /--00f(z -

q)g(q) dn,

(4.3)

where in the present case

and

Some manip~ations

g(k) = iKl(kr).

(4.4)

using the data in the tables of transforms ([12], p. 163) yield ,

(4.5)

with R = (q2 + r2)ln. A rather long calculation, with the restriction Iz - n/ : a in mind, provides the desired result, P(1- 2Y) (4.6) rz*(*(T,z) = 23GrZa2 A$” - zr - r2 log@ + z)], where A; denotes the second difference with respect to the variable z, and AZ = a. In order to arrive at a more tangible resuit it is helpful to take recourse to a theorem by de la Valle Poussin ([13], p. 59) stating that A’f(x) = Ax”f(“)(x + n8 d..~),

(4.7)

where the coefficient 8 remains within the bounds 0 and 1. Applying (4.7)-(4.6) that l-2Y P rF(r, 2) 21dR, (R, + z + 2t3a) with R, = [rz -t (Z + 28a)2]1’2. In the classical case, of course,

we readily find

a = 0, and we recover

(4.8) the

On the three-dimensional

Boussinesq problem

1251

well-known result ([lo], p. 205) tr*l*(r, 2) =

P(l - 2v) 2JrR(R + 2)’

(4.9)

The stress component above is, of course, one portion of the actual stress, ty,(r, z), generated in the half-space. The remaining portion, tF,(r, z), coincides, as already explained, with the respective component associated with the Kelvin problem, provided the Poisson ratio Y = m is set equal to l/2. It is not difficult to show, by carrying the calculations along the line of argument given in [9], that rMr, 2) = -

5.

3W(z + 26%) 21rR: ’

(4.10)

CONCLUSIONS

It seems of interest to sum up the following results of this study already pointed out in [9]: (1) First, that the stress concentration at the point of application of the load (r = z = 0) predicted by the nonlocal theory is finite, and not infinite as claimed by the local approach. (2) Second, that although this prediction is more realistic than the one of the classical theory, still the concentration foreseen by the nonlocal theory is extremely high, being of the order of 1016kgf/cm2 (per unit load). (3) In defense of the classical theory, therefore, it is only too fair to state that from a strictly practical point of view the classical solution may be considered fully acceptable. On the other hand, from a purely methodological standpoint, of major consequence for any rationally constructed scientific theory, the nonlocal approach has probably to be rated higher.

REFERENCES [l] D. G. B. EDELEN, Nonlocal field theories. In Continuum Physics (Edited by A. C. ERINGEN), Vol. 4, pp. 75-204. Academic Press, New York (1976). [2] A. C. ERINGEN, and D. G. B. EDELEN, Int. 1. Engng Sci. 10,233(1972). [3] A. C. ERINGEN, Nonlocal Polar Field Theories. In Continum Physics (Edited by A. C. ERINGEN), Vol. 4, pp. 205-267 (1976). [4] E. KROENER, Continuum mechanics and range of atomic cohesion forces. Proc. Inf. Conf Fruct., Vol. 1, Jap. Sot. Strength Fract. Mat., Sandai (1966). [S] I. A. KUNIN, P&l. Mat. Mech. 30, 542 (in Russian) (1966). [6] A. C. ERINGEN, Continuum mechanics at the atomic scale. In Crystal Lattice Defects, Vol. 7, pp. 109-130. Gordon Jr Breach, New York (1977). [7] G. LEIBFRIED, Mechanical and thermal properties of crystals. In Encyclopedia of Physics (Edited by S. FLUEGGE), VII.1. Springer, Berlin (in German) (1955). [8] J. L. NOWINSKI, Acta. Mech. 78, 209 (1989). (91 J. L. NOWINSKI, On a three-dimensional Kelvin problem for an elastic nonlocal medium. Acta Mech. To appear. [lo] Y. C. FUNG, Foundations of Solid Mechanics. Prentice-Hall, Englewood Cliffs, NJ (1965). [ll] H. M. WESTERGAARD, Theory of Elasticity and Plasticity. Harvard University Press, Cambridge, MA (1952). [12] W. MAGNUS and F. OBERHETTINGER, Formulae and Theorems for Special Functions of Mathematical Physics. Springer, Berlin (in German) (1948). [13] J. B. SCARBOROUGH, Numerical Mathematical Analysis, Johns Hopkins Press, Baltimore, MD (1958). (Received 20 February

1990; accepted 9 May 1990)