Linear elasticity

Linear elasticity is a mathematical model of how solid objects deform and become internally stressed by prescribed loading conditions. It is a simplification of the more general nonlinear theory of elasticity and a branch of continuum mechanics.

The fundamental assumptions of linear elasticity are infinitesimal strains — meaning, "small" deformations — and linear relationships between the components of stress and strain — hence the "linear" in its name. Linear elasticity is valid only for stress states that do not produce yielding. Its 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:

Cauchy momentum equation, which is an expression of Newton's second law. In convective form it is written as:

∇

⋅

σ

+

F

=

ρ

u

¨

{\displaystyle {\boldsymbol {\nabla }}\cdot {\boldsymbol {\sigma }}+\mathbf {F} =\rho {\ddot {\mathbf {u} }}}

Strain-displacement equations:

ε

=

1

2

[

∇

u

+

(

∇

u

)

T

]

{\displaystyle {\boldsymbol {\varepsilon }}={\tfrac {1}{2}}\left[{\boldsymbol {\nabla }}\mathbf {u} +({\boldsymbol {\nabla }}\mathbf {u} )^{\mathrm {T} }\right]}

Constitutive equations.