Editing Continuum mechanics (section)

==Formulation of models==
[[Image:Continuum body.svg|class=skin-invert-image|200px|right|thumb|Figure 1. Configuration of a continuum body.]]
Continuum mechanics models begin by assigning a region in three-dimensional [[Euclidean space]] to the material body <math>\mathcal B</math> 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 <math>t</math> is labeled <math>\kappa_t(\mathcal B)</math>.

A particular particle within the body in a particular configuration is characterized by a position [[Vector space|vector]] <br />
:<math>\mathbf x = \sum_{i=1}^3 x_i \mathbf e_i,</math>
where <math>\mathbf e_i</math> are the [[coordinate vector]]s in some [[frame of reference]] chosen for the problem (See figure 1). This vector can be expressed as a [[function (mathematics)|function]] of the particle position <math>\mathbf X</math> in some ''reference configuration'', for example the configuration at the initial time, so that

:<math>\mathbf{x}=\kappa_t(\mathbf X).</math>

This function needs to have various properties so that the model makes physical sense. <math>\kappa_t(\cdot)</math> needs to be:
* [[continuity (mathematics)|continuous]] in time, so that the body changes in a way which is realistic,
* globally [[inverse function|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, <math>\kappa_t(\cdot)</math> is also assumed to be [[continuously differentiable|twice continuously differentiable]], so that differential equations describing the motion may be formulated.