# Linear elasticity

Linear elasticity is the mathematical study of how solid objects deform and become internally stressed due to prescribed loading conditions. Linear elasticity models materials as continua. Linear elasticity is a simplification of the more general nonlinear theory of elasticity and is a branch of continuum mechanics. The fundamental "linearizing" assumptions of linear elasticity are: infinitesimal strains or "small" deformations (or strains) and linear relationships between the components of stress and strain. In addition linear elasticity is valid only for stress states that do not produce yielding. These assumptions are reasonable for many engineering materials and engineering design scenarios. Linear elasticity is therefore used extensively in structural analysis and engineering design, often with the aid of finite element analysis.

## Mathematical formulation

Equations governing a linear elastic boundary value problem are based on three tensor partial differential equations for the balance of linear momentum and six infinitesimal strain-displacement relations. The system of differential equations is completed by a set of linear algebraic constitutive relations.

### Direct tensor form

In direct tensor form that is independent of the choice of coordinate system, these governing equations are:[1]

${\displaystyle {\boldsymbol {\nabla }}\cdot {\boldsymbol {\sigma }}+\mathbf {F} =\rho {\ddot {\mathbf {u} }}}$
${\displaystyle {\boldsymbol {\varepsilon }}={\tfrac {1}{2}}\left[{\boldsymbol {\nabla }}\mathbf {u} +({\boldsymbol {\nabla }}\mathbf {u} )^{T}\right]\,\!}$
• 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
${\displaystyle {\boldsymbol {\sigma }}={\mathsf {C}}:{\boldsymbol {\varepsilon }},}$

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

### Cartesian coordinate form

Template:Einstein summation convention Expressed in terms of components with respect to a rectangular Cartesian coordinate system, the governing equations of linear elasticity are:[1]

${\displaystyle \sigma _{ji,j}+F_{i}=\rho \partial _{tt}u_{i}\,\!}$
where the ${\displaystyle {(\bullet )}_{,j}}$ subscript is a shorthand for ${\displaystyle \partial {(\bullet )}/\partial x_{j}}$ and ${\displaystyle \partial _{tt}}$ indicates ${\displaystyle \partial ^{2}/\partial t^{2}}$, ${\displaystyle \sigma _{ij}=\sigma _{ji}\,\!}$ is the Cauchy stress tensor, ${\displaystyle F_{i}\,\!}$ are the body forces, ${\displaystyle \rho \,\!}$ is the mass density, and ${\displaystyle u_{i}\,\!}$ is the displacement.
These are 3 independent equations with 6 independent unknowns (stresses).
${\displaystyle \varepsilon _{ij}={\frac {1}{2}}(u_{j,i}+u_{i,j})\,\!}$
where ${\displaystyle \varepsilon _{ij}=\varepsilon _{ji}\,\!}$ is the strain. These are 6 independent equations relating strains and displacements with 9 independent unknowns (strains and displacements).
${\displaystyle \sigma _{ij}=C_{ijkl}\,\varepsilon _{kl}\,\!}$
where ${\displaystyle C_{ijkl}}$ is the stiffness tensor. These are 6 independent equations relating stresses and strains. The coefficients of the stiffness tensor can always be specified so that ${\displaystyle C_{ijkl}=C_{klij}=C_{jikl}=C_{ijlk}}$.

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). Specifying the boundary conditions, the boundary value problem is completely 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.

### Cylindrical coordinate form

In cylindrical coordinates (${\displaystyle r,\theta ,z}$) the equations of motion are[1]

{\displaystyle {\begin{aligned}&{\frac {\partial \sigma _{rr}}{\partial r}}+{\cfrac {1}{r}}{\frac {\partial \sigma _{r\theta }}{\partial \theta }}+{\frac {\partial \sigma _{rz}}{\partial z}}+{\cfrac {1}{r}}(\sigma _{rr}-\sigma _{\theta \theta })+F_{r}=\rho ~{\frac {\partial ^{2}u_{r}}{\partial t^{2}}}\\&{\frac {\partial \sigma _{r\theta }}{\partial r}}+{\cfrac {1}{r}}{\frac {\partial \sigma _{\theta \theta }}{\partial \theta }}+{\frac {\partial \sigma _{\theta z}}{\partial z}}+{\cfrac {2}{r}}\sigma _{r\theta }+F_{\theta }=\rho ~{\frac {\partial ^{2}u_{\theta }}{\partial t^{2}}}\\&{\frac {\partial \sigma _{rz}}{\partial r}}+{\cfrac {1}{r}}{\frac {\partial \sigma _{\theta z}}{\partial \theta }}+{\frac {\partial \sigma _{zz}}{\partial z}}+{\cfrac {1}{r}}\sigma _{rz}+F_{z}=\rho ~{\frac {\partial ^{2}u_{z}}{\partial t^{2}}}\end{aligned}}}

The strain-displacement relations are

{\displaystyle {\begin{aligned}\varepsilon _{rr}&={\cfrac {\partial u_{r}}{\partial r}}~;~~\varepsilon _{\theta \theta }={\cfrac {1}{r}}\left({\cfrac {\partial u_{\theta }}{\partial \theta }}+u_{r}\right)~;~~\varepsilon _{zz}={\cfrac {\partial u_{z}}{\partial z}}\\\varepsilon _{r\theta }&={\cfrac {1}{2}}\left({\cfrac {1}{r}}{\cfrac {\partial u_{r}}{\partial \theta }}+{\cfrac {\partial u_{\theta }}{\partial r}}-{\cfrac {u_{\theta }}{r}}\right)~;~~\varepsilon _{\theta z}={\cfrac {1}{2}}\left({\cfrac {\partial u_{\theta }}{\partial z}}+{\cfrac {1}{r}}{\cfrac {\partial u_{z}}{\partial \theta }}\right)~;~~\varepsilon _{zr}={\cfrac {1}{2}}\left({\cfrac {\partial u_{r}}{\partial z}}+{\cfrac {\partial u_{z}}{\partial r}}\right)\end{aligned}}}

and the constitutive relations are the same as in Cartesian coordinates, except that the indices ${\displaystyle 1}$,${\displaystyle 2}$,${\displaystyle 3}$ now stand for ${\displaystyle r}$,${\displaystyle \theta }$,${\displaystyle z}$, respectively.

### Spherical coordinate form

In spherical coordinates (${\displaystyle r,\theta ,\phi }$) the equations of motion are[1]

{\displaystyle {\begin{aligned}\varepsilon _{rr}&={\frac {\partial u_{r}}{\partial r}}\\\varepsilon _{\theta \theta }&={\frac {1}{r}}\left({\frac {\partial u_{\theta }}{\partial \theta }}+u_{r}\right)\\\varepsilon _{\phi \phi }&={\frac {1}{r\sin \theta }}\left({\frac {\partial u_{\phi }}{\partial \phi }}+u_{r}\sin \theta +u_{\theta }\cos \theta \right)\\\varepsilon _{r\theta }&={\frac {1}{2}}\left({\frac {1}{r}}{\frac {\partial u_{r}}{\partial \theta }}+{\frac {\partial u_{\theta }}{\partial r}}-{\frac {u_{\theta }}{r}}\right)\\\varepsilon _{\theta \phi }&={\frac {1}{2r}}\left[{\frac {1}{\sin \theta }}{\frac {\partial u_{\theta }}{\partial \phi }}+\left({\frac {\partial u_{\phi }}{\partial \theta }}-u_{\phi }\cot \theta \right)\right]\\\varepsilon _{r\phi }&={\frac {1}{2}}\left({\frac {1}{r\sin \theta }}{\frac {\partial u_{r}}{\partial \phi }}+{\frac {\partial u_{\phi }}{\partial r}}-{\frac {u_{\phi }}{r}}\right).\end{aligned}}}