126 ELASTOSTATICS
The fundamental solution used for a three-dimensional isotropic
body is the Kelvin solution and corresponds to a concentrated force
acting at a point in the infinite elastic space. For the displacements it
can be written,
1 f(3-4v)ö
lk t
dr dr
16πΟ(1
— v)
r cx
l
ox
k
where we interpret the derivatives as
dr
=
r
k
dx
k
r
(5.25)
where r
k
are the projections defined in Figure 5.2.
The traction components corresponding to the Kelvin solution are
Ρ
*
β
-8π(1-ννΚ(
(1
"
2ν)
**
+ 3
^έ)
+
. dr dr
+ (1
-
2V)
^-^
(5.26)
n is the normal to the surface of the body.
Equation (5.23) can be specialised for a boundary point in the same
way as the potential problem. Due to the singularities existing in the
left-hand side integrals, we obtain a coefficient such that,
r
hP*k
άΓ +
r
2
CiU+ U
= I MÄdß+ I
p
k
ur
k
dr
+
Ja ύΓ
χ
p
k
uf
k
af
(5.27)
r,
The c
{
coefficient is equal to for a smooth boundary but generally
will be different from this value. Fortunately explicit calculation of
this value is not necessary as it can be obtained using the rigid body
motions in a way similar to that explained in Chapter 2 for potential
problems. We will come back to this in Section 5.4.
5.4 SOURCE APPROACH
Boundary solutions can also be expressed using a source of dipole
type formulation. Here we will discuss only the source one, the other