Material Modelling and Fundamental Solutions - The Boundary Element Method with Programming - page 66

4
Material Modelling and
Fundamental Solutions

If you can measure what
you are speaking about, and
express it in numbers, you
know something about it.

Lord Kelvin

4.1. INTRODUCTION

In addition to specifying the geometry of the problem, it is necessary to describe the
physical response of the material in a mathematical way. This is done by defining the
response characteristics of an infinitesimally small portion of the solid. The constitutive
law establishes a relationship between heat flow and the temperature gradient or
between strain and stress. The constants in such relationships are characteristic values or
properties of the material. We distinguish between material properties which are
direction independent ( isotropic material), and those which are dependent on direction
( anisotropic material). Furthermore, there are problems where the same properties apply
everywhere ( homogeneous problems) and where properties change from location to
location ( non-homogeneous problems).
In the material response we distinguish between linear and non-linear behaviour. For
linear materials we can establish a unique (linear) relationship between stress/strain,
temperature/heat flow or potential/fluid flow. For non-linear material behaviour, this
relationship depends on the current state and can therefore only be written in incremental
form. These problems are therefore dependent on the deformation (thermal) history.
As outlined previously, for the boundary element method a solution of the governing
equation has to be available. In nearly all cases, the solution is obtained for very simple
loading conditions (point load or source) and for infinite or semi-infinite domains. In the