QUARTERLY OF APPLIED MATHEMATICS
JANUARY, 1972
ON AN INTEGRAL EQUATION APPROACH TO
DISPLACEMENT PROBLEMS OF CLASSICAL ELASTICITY*
By M. MAITI and G. R. MAKAN (Indian Institute of Technology, Kharagpur)
1. Introduction. In a recent paper Kanwal [1] has established an integral equation
method for solving displacement problems of elasticity. The method is based on the
generalization of Green's procedure in potential theory. But it appears to have gone
unnoticed by Kanwal that his formulation has a direct bearing on Betti's method of
elasticity [2], which has long been known as a tensor counterpart of Green's procedure.
The purpose of the present note is to point this fact out and to show further that
the result obtained by Kanwal is identical with that of Betti. The note deals only with
the static case, but the analysis may be extended easily to the dynamic case.
2. Integral equation method. In the absence of body forces the Navier-Cauchy
equations in elastostatics are
(X + /i) grad + mV2u = 0, # = div u, (1)
where u (u{ ; i = 1, 2, 3) is the displacement vector and X, fx are Lame constants of the
material medium. The components u, are functions of Cartesian coordinates x, . Let
the region under consideration be denoted by V and its bounding surface by S. The
unit normal n (rc; ; i = 1, 2, 3) will be directed outward to S. A brief description of
Kanwal's procedure in deriving the solutions of Eqs. (1) follows.
Choose a tensor function U (Uu ; i, j = 1, 2, 3) such that
u« - 2.) + 3*> iP^ + <> + »> ~|P-¥ Xl>} • m
which corresponds to the ith component of the displacement at a point P(x{) by a unit
force applied in the ith direction at a point q(x°); <5,v is the Kronecker delta. From U
we can compute dilatations 6, which are given by
dUu = 1 Xi ~ n\
dxk 4tt(X + 2/0 |P - q|3' W
To find the displacement u at an interior point P we surround the point P with an
infinitesimal sphere S„ (since d{ and Ua are singular at P = q) of radius a so that the
sphere lies entirely in V. Let the volume enclosed by the sphere be denoted by V, .
Applying Green's second identity to the functions Ui , Uu , we obtain
L. ,u-v'u" - u"v'u'>dV - L. r aJt -u-1?}<«
Received August 10, 1970; revised version received December 25, 1970.
558 M. MAITI AND G. R. MAKAN
whence it can be derived that
l+s. [U'{M ir+ (x + M)to} ~ fn+ (x + ^n}]& = °- (5)
Evaluating the integral over S„ and letting a —» 0, we may derive the displacement
at any point P (00 as
U(P) = ~ I [U'{M S? + (X + ~ U'{M fn + (X +
(6)
Now we show that Kanwal's formulation (6) is identical with the classic result of
Betti, which expresses the displacement at the point P as
w,(P) = f {tjUii — UjTij) ds, (7)
J S
better known as Betti's second identity [3]. The boundary tractions t (f,- ; i = 1, 2, 3)
and T (Tu ; i, j, = 1, 2, 3) are computed from the displacements u and U respectively.
Eq. (5) may be rewritten as
f {u-T — t-Uj ds + fi f {u-(a A U) — U-(a A u)j ds = 0, (8)
•'S+S. Js + s,
where a = V A n. The second integral in Eq. (8) vanishes identically by Gauss' di-
vergence theorem, and thus Eq. (8) is identical with Eq. (2.3) of Rizzo [4] from which
Betti's second identity (7) follows immediately.
References
[1] R. P. Kanwal, Integral equation formulation of classical elasticity, Quart. Appl. Math. 27, 57-65 (1969)
[2] A. E. H. Love, Mathematical theory of elasticity, 4th ed. Dover, New York, 1944
[3] S. Bergmann and M. Schififer, Kernel functions and elliptic differential equations in mathematical
physics, Academic Press, New York, 1953
[4] F. J. Rizzo, An integral equation approach to boundary value problems of classical elastostatics, Quart.
Appl. Math. 25, 83-95 (1967)