Editing Linear elasticity (section)

=== Cartesian coordinate form ===
{{Einstein_summation_convention}}
Expressed in terms of components with respect to a rectangular [[Cartesian coordinate]] system, the governing equations of linear elasticity are:<ref name="Slau" />

* [[Cauchy momentum equation|Equation of motion]]: <math display="block"> \sigma_{ji,j} + F_i = \rho \partial_{tt} u_i</math> where the <math>{(\bullet)}_{,j}</math> subscript is a shorthand for <math>\partial{(\bullet)} / \partial x_j</math> and <math>\partial_{tt}</math> indicates <math>\partial^2 / \partial t^2</math>, <math> \sigma_{ij} = \sigma_{ji}</math> is the Cauchy [[Stress (physics)|stress]] tensor, <math> F_i</math> is the body force density, <math> \rho</math> is the mass density, and <math> u_i</math> is the displacement.{{pb}}These are 3 [[System of linear equations#Independence|independent]] equations with 6 independent unknowns (stresses).{{pb}} In engineering notation, they are: <math display="block">\begin{align}
\frac{\partial \sigma_x}{\partial x} + \frac{\partial \tau_{yx}}{\partial y} + \frac{\partial \tau_{zx}}{\partial z} + F_x = \rho \frac{\partial^2 u_x}{\partial t^2} \\
\frac{\partial \tau_{xy}}{\partial x} + \frac{\partial \sigma_y}{\partial y} + \frac{\partial \tau_{zy}}{\partial z} + F_y = \rho \frac{\partial^2 u_y}{\partial t^2} \\
\frac{\partial \tau_{xz}}{\partial x} + \frac{\partial \tau_{yz}}{\partial y} + \frac{\partial \sigma_z}{\partial z} + F_z = \rho \frac{\partial^2 u_z}{\partial t^2}
\end{align}</math>
* [[Deformation (mechanics)#Strain|Strain-displacement]] equations: <math display="block">\varepsilon_{ij} =\frac{1}{2} (u_{j,i} + u_{i,j})</math> where <math> \varepsilon_{ij}=\varepsilon_{ji}\,\!</math> is the strain.  These are 6 independent equations relating strains and displacements with 9 independent unknowns (strains and displacements).{{pb}} In engineering notation, they are: <math display="block">\begin{align}
\epsilon_x=\frac{\partial u_x}{\partial x} \\
\epsilon_y=\frac{\partial u_y}{\partial y} \\
\epsilon_z=\frac{\partial u_z}{\partial z}
\end{align}
\qquad
\begin{align}
\gamma_{xy}=\frac{\partial u_x}{\partial y}+\frac{\partial u_y}{\partial x} \\
\gamma_{yz}=\frac{\partial u_y}{\partial z}+\frac{\partial u_z}{\partial y} \\
\gamma_{zx}=\frac{\partial u_z}{\partial x}+\frac{\partial u_x}{\partial z}
\end{align}</math>
* [[Constitutive equations]].  The equation for Hooke's law is: <math display="block"> \sigma_{ij} = C_{ijkl} \, \varepsilon_{kl} </math> where <math>C_{ijkl}</math>  is the stiffness tensor.  These are 6 independent equations relating stresses and strains.  The requirement of the symmetry of the stress and strain tensors lead to equality of many of the elastic constants, reducing the number of different elements to 21<ref>{{cite journal |last1=Belen'kii |last2= Salaev|date= 1988|title= Deformation effects in layer crystals|journal= Uspekhi Fizicheskikh Nauk|volume= 155|issue= 5|pages= 89–127|doi= 10.3367/UFNr.0155.198805c.0089|doi-access= free}}</ref> <math> C_{ijkl} = C_{klij} = C_{jikl} = C_{ijlk}</math>.

An elastostatic boundary value problem for an isotropic-homogeneous media is a system of 15 independent equations and equal number of unknowns (3 equilibrium equations, 6 strain-displacement equations, and 6 constitutive equations). By specifying the boundary conditions, the boundary value problem is fully defined. To solve the system two approaches can be taken according to boundary conditions of the boundary value problem: a '''displacement formulation''', and a '''stress formulation'''.