Material Modelling and Fundamental Solutions - The Boundary Element Method with Programming - page 87

The Kelvin solutions for displacements are plotted in Figures 4.13 and 4.14. A small
circle of exclusion is used to avoid plotting very high values near the singularity. The
variation of the displacement in x-direction shows symmetry about the x- and y-axes.
The variation of the displacements in y-direction shows anti-symmetry about both axes.
The influence of the Poisson's ratio on the displacements perpendicular to the load axis
can be clearly seen in Figure 4.14. Note that the finite element method has difficulty
dealing with a Poisson's ratio of 0.5 (incompressible material) because of the definition
of C 1 in equation (4.47) which would give an infinite value for Q= 0.5.

U xx (P,Q)

P x = 1.0

y

x

Figure 4.13 3-D Kelvin solution: variation of displacements in x-direction due to P x = 1.0

Figure 4.15 shows the variation of the fundamental solution for the boundary traction
in x-direction assuming that the vector normal to the boundary, n , points in the x-
direction (this means that the computed traction is equivalent to the stress in the x-
direction). We can see clearly that the fundamental solution is anti-symmetric about the
y-axis and decays very rapidly from the singularity.
To implement the above equations in F90 we define functions UK and TK which
return rank two arrays of dimension 2 or 3. The function only provides solutions for
plane strain and 3-D problems. To obtain the solutions for plane stress problems simply
substitute an effective Poisson´s ration of

Q Q ( Q )

.