Elastodynamics

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.

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.

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

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, ${\displaystyle (\bullet )^{T}}$ 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).

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