Editing Linear elasticity (section)

=== Direct tensor form ===
In direct [[tensor]] form that is independent of the choice of coordinate system, these governing equations are:<ref name="Slau">{{Cite book |last=Slaughter |first=William S. |url=http://link.springer.com/10.1007/978-1-4612-0093-2 |title=The Linearized Theory of Elasticity |date=2002 |publisher=Birkhäuser Boston |isbn=978-1-4612-6608-2 |location=Boston, MA |language=en |doi=10.1007/978-1-4612-0093-2}}</ref>

* [[Cauchy momentum equation]], which is an expression of [[Newton's laws of motion#Newton's second law|Newton's second law]]. In convective form it is written as: <math display="block">\boldsymbol{\nabla} \cdot \boldsymbol{\sigma} + \mathbf{F} = \rho \ddot{\mathbf{u}} </math>
* [[Infinitesimal strain theory|Strain-displacement]] equations: <math display="block">\boldsymbol{\varepsilon} = \tfrac{1}{2} \left[\boldsymbol{\nabla}\mathbf{u} + (\boldsymbol{\nabla}\mathbf{u})^\mathrm{T}\right]</math>
* [[Constitutive equations]]. For elastic materials, [[Hooke's law]] represents the material behavior and relates the unknown stresses and strains. The general equation for Hooke's law is <math display="block"> \boldsymbol{\sigma} = \mathsf{C}:\boldsymbol{\varepsilon},</math>

where <math>\boldsymbol{\sigma}</math> is the [[Cauchy stress tensor]], <math>\boldsymbol{\varepsilon}</math> is the [[infinitesimal strain]] tensor, <math>\mathbf{u}</math> is the [[displacement vector]], <math>\mathsf{C}</math> is the fourth-order [[stiffness tensor]], <math>\mathbf{F}</math> is the body force per unit volume, <math>\rho</math> is the mass density, <math>\boldsymbol{\nabla}</math> represents the [[nabla operator]], <math>(\bullet)^\mathrm{T}</math> represents a [[transpose]], <math>\ddot{(\bullet)}</math> represents the second [[material derivative]] with respect to time, and <math>\mathsf{A}:\mathsf{B} = A_{ij}B_{ij}</math> is the inner product of two second-order tensors (summation over repeated indices is implied).