Editing Continuum mechanics (section)

===Displacement field===
{{main|Displacement field (mechanics)}}

The vector joining the positions of a particle <math>P</math> in the undeformed configuration and deformed configuration is called the [[displacement (vector)|displacement vector]] <math>\mathbf u(\mathbf X,t)=u_i\mathbf e_i</math>, in the Lagrangian description, or <math>\mathbf U(\mathbf x,t)=U_J\mathbf E_J</math>, in the Eulerian description.

A ''displacement field'' is a vector field of all displacement vectors for all particles in the body, which relates the deformed configuration with the undeformed configuration. It is convenient to do the analysis of deformation or motion of a continuum body in terms of the displacement field,  In general, the displacement field is expressed in terms of the material coordinates as

:<math>\mathbf u(\mathbf X,t) = \mathbf b+\mathbf x(\mathbf X,t) - \mathbf X \qquad \text{or}\qquad u_i = \alpha_{iJ}b_J + x_i - \alpha_{iJ}X_J</math>

or in terms of the spatial coordinates as

:<math>\mathbf U(\mathbf x,t) = \mathbf b+\mathbf x - \mathbf X(\mathbf x,t) \qquad \text{or}\qquad U_J = b_J + \alpha_{Ji}x_i - X_J \,</math>

where <math>\alpha_{Ji}</math> are the direction cosines between the material and spatial coordinate systems with unit vectors <math>\mathbf E_J</math> and <math>\mathbf e_i</math>, respectively. Thus

:<math>\mathbf E_J \cdot \mathbf e_i = \alpha_{Ji}=\alpha_{iJ}</math>

and the relationship between <math>u_i</math> and <math>U_J</math> is then given by

:<math>u_i=\alpha_{iJ}U_J \qquad \text{or} \qquad U_J=\alpha_{Ji}u_i</math>

Knowing that
:<math>\mathbf e_i = \alpha_{iJ}\mathbf E_J</math>
then
:<math>\mathbf u(\mathbf X,t)=u_i\mathbf e_i=u_i(\alpha_{iJ}\mathbf E_J)=U_J\mathbf E_J=\mathbf U(\mathbf x,t)</math>

It is common to superimpose the coordinate systems for the undeformed and deformed configurations, which results in <math>\mathbf b=0</math>,  and the direction cosines become [[Kronecker delta]]s, i.e.

:<math>\mathbf E_J \cdot \mathbf e_i = \delta_{Ji}=\delta_{iJ}</math>

Thus, we have

:<math>\mathbf u(\mathbf X,t) = \mathbf x(\mathbf X,t) - \mathbf X \qquad \text{or}\qquad u_i = x_i - \delta_{iJ}X_J</math>

or in terms of the spatial coordinates as

:<math>\mathbf U(\mathbf x,t) = \mathbf x - \mathbf X(\mathbf x,t) \qquad \text{or}\qquad U_J = \delta_{Ji}x_i - X_J </math>
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==Fundamental laws==
===Conservation of mass===
===Conservation of momentum===
P<sub>i</sub>=P<sub>f</sub>

===Conservation of energy===
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