124 ELASTOSTATICS
consideration the constitutive relations (5.10) we have,
Ό
dx
J
=

Pic"*
J
r
2
h
ef
k
dQ + b
k
uidQ
P
k
utdr +
Jr. Jr.
Ptdr
(5.18)
For temperature effects for instance this equation becomes (see
equations (5.11) and (5.12)),
^u
fc
d0
4
ü
8x
J
XoLTÖj
k
sf
k
dQ
+
b
k
ui άΩ +
Pk"k
df +
P
k
u*dr =
i
u
k
p*
k
dr+\ u
k
ptdr (5.19)
This shows that the initial stress field can be treated in a similar way to
the body force field b
k
. In addition to being used for temperature and
other problems the initial stress fields are important because they can
be applied to introduce the effect of nonlinearities into the
formulation. These two effects do not produce any internal unknowns
and they relate to the boundary values in a similar way to the p term in
the Poisson equation.
The problem is to find a solution such that
dafjdxj = 0
(5.20)
In this way the first integral in equation (5.19) disappears which
reduces the problem to a boundary problem. We need to find the
solution to this homogeneous equation in order to apply (5.19)
without having to integrate the first term in equation (5.19) over the Ω
domain, which would produce internal unknowns.
5.3 FUNDAMENTAL SOLUTION
A way of applying (5.19) is to use the fundamental solution for the
elasticity problem, i.e. the solution corresponding to the equation
daf
k
dx
J
(5.21)
where δ\ is the Dirac delta function and represents a unit load at i
acting in one of the x
t
directions. This type of solution will produce for
each direction / the following equation (the initial stress term has not