Kroner

For a constant point force F acting at x ′ , the displacement at x is given by:
u m (x) = G mr (x − x ′ )F r (x ′ ) (31)
So that the displacement gradient and the strain take the form:
u m,n = G mr,n (x − x ′ )F r (32)
ɛ mn = 1 2 (u m,n + u n,m ) (33)
and from the constitutive relation, the stress is given by:
σ kl = C 0 klmnɛ mn (34)
because of the symmetry of the linear elastic coefficients:
σ kl = C 0 klmnu m,n (35)
σ kl = C 0 klmnG mr,n (x − x ′ )F r (36)
Now, following Weinberger et al [4], for a given volume V , from the definition
of the Dirac delta function, the k component of the force F acting at x ′ can be
described by:
∫
F k δ(x − x ′ )dV (37)
V
Equilibrium requires that the force F be balanced by tractions acting on the
surface S enclosing this volume V , so that:
∫
∫
F k δ(x − x ′ )dV + σ kl n l dS = 0 (38)
V
S
∫
∫
F k δ(x − x ′ )dV + CklmnG 0 mr,n (x − x ′ )n l F r dS = 0 (39)
V
S
Applying Gauss’s theorem:
∫
∫
F k δ(x − x ′ )dV + CklmnG 0 mr,nl (x − x ′ )F r dV = 0 (40)
V
V
∫
[F r δ kr δ(x − x ′ ) + CklmnG 0 mr,nl (x − x ′ )F r ]dV = 0 (41)
from which:
V
−C 0 klmnG mr,nl (x − x ′ ) = δ kr δ(x − x ′ ) (42)
For an unbounded isotropic matrix, Weinberger et al [4] derive that Green
tensor function for x ′ = 0 in their Eq. (1.79) as:
G mr (x) =
1
(
(3 − 4ν)δ mr + x )
mx r
16πµ(1 − ν)x
x 2
(43)