Material Modelling and Fundamental Solutions - The Boundary Element Method with Programming - page 73

For anisotropic problems the fundamental solutions have been presented by Bonnet 1 .
For example, the solution for temperature/potential is given by

1
S

1

U P , ln

(4.21)

r

2 k

for two-dimensional problems and

1

U P , Q

(4.22)

4 r k

for three-dimensional problems, where

T 1

k det D and r r D r (4.23)

For general anisotropy in three dimensions, D has 9 material parameters but, because
of the property of symmetry, only 6 components need to be input. A special case of
anisotropy exists where the material parameters are different in three orthogonal
coordinate directions. This is known as orthotropic material. If these conductivities are
defined in the direction of global coordinates, then all off-diagonal elements of D are
zero. If we denote the conductivities in x,y and z-directions as k 1 , k 2 , k 3 then

ª k 0 0 º

1

D

«

0 k 0

»

(4.24)

2

0 0 k

¬ ¼

3

For this case the values in equation (4.23) are given by:

1 1 1 2 2 2

k k k k and r r r r

(4.25)

1 2 x y z

3

k k k

1 2 3

4.3. STATIC ELASTICITY PROBLEMS

In solid mechanics applications, a relationship between stress and strain must be
established. Stresses are forces per unit area inside a solid. They can be visualised by
cutting the solid on planes parallel to the axes and by showing the traction vectors acting
on these planes (in Figure 4.5).