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. 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 σ = C : ε , {\displaystyle {\boldsymbol {\sigma }}={\mathsf {C}}:{\boldsymbol {\varepsilon }},}
where σ {\displaystyle {\boldsymbol {\sigma }}} is the Cauchy stress tensor, ε {\displaystyle {\boldsymbol {\varepsilon }}} is the infinitesimal strain tensor, u {\displaystyle \mathbf {u} } is the displacement vector, C {\displaystyle {\mathsf {C}}} is the fourth-order stiffness tensor, F {\displaystyle \mathbf {F} } is the body force per unit volume, ρ {\displaystyle \rho } is the mass density, ∇ {\displaystyle {\boldsymbol {\nabla }}} represents the nabla operator, ( ∙ ) T {\displaystyle (\bullet )^{\mathrm {T} }} represents a transpose, ( ∙ ) ¨ {\displaystyle {\ddot {(\bullet )}}} represents the second material derivative with respect to time, and A : B = A i j B i j {\displaystyle {\mathsf {A}}:{\mathsf {B}}=A_{ij}B_{ij}} is the inner product of two second-order tensors (summation over repeated indices is implied).
Cartesian coordinate form
Expressed in terms of components with respect to a rectangular Cartesian coordinate system, the governing equations of linear elasticity are:
… excerpt ends here. Continue reading the full article.
