Korn’s Inequality

Synonyms

L² estimate for components of linearly elastic displacement

Definitions

Korn’s inequality estimates the integral value of the squared displacement through the strain energy of an elastic body occupying volume V . This result was central to the development of linear elasticity theory.

A Version of Korn’s Inequality

The present article highlights the inequality

$$\displaystyle \begin{aligned} \int_V ( |\boldsymbol{u}|{}^2 + \| \nabla \boldsymbol{u} \|{}^2 ) \,dV \le c \int_V W({\boldsymbol{\varepsilon}}) \,dV \, , \end{aligned} $$

(1)

where $\boldsymbol{u}= \boldsymbol{u}( \boldsymbol{r})$ is the displacement field of a three-dimensional elastic body, V is the volume region occupied by the body, ${\boldsymbol {\varepsilon }} = {\boldsymbol {\varepsilon }}( \boldsymbol{u}) = \tfrac {1}{2} [\nabla \boldsymbol{u} + (\nabla \boldsymbol{u})^T]$ is the linear strain tensor, W(ε) is the strain energy function—which is a positive definite quadratic form in terms of the Cartesian...