Continuum Mechanics - Formulation of Models

Formulation of Models

Continuum mechanics models begin by assigning a region in three dimensional Euclidean space to the material body being modeled. The points within this region are called particles or material points. Different configurations or states of the body correspond to different regions in Euclidean space. The region corresponding to the body's configuration at time is labeled .

A particular particle within the body in a particular configuration is characterized by a position vector

,

where are the coordinate vectors in some frame of reference chosen for the problem (See figure 1). This vector can be expressed as a function of the particle position in some reference configuration, for example the configuration at the initial time, so that

.

This function needs to have various properties so that the model makes physical sense. needs to be:

• continuous in time, so that the body changes in a way which is realistic,
• globally invertible at all times, so that the body cannot intersect itself,
• orientation-preserving, as transformations which produce mirror reflections are not possible in nature.

For the mathematical formulation of the model, is also assumed to be twice continuously differentiable, so that differential equations describing the motion may be formulated.