Infinitesimal strain tensor ($\epsilon =\mathrm{sym}\nabla u$)

The infinitesimal strain tensor is the symmetric part of a displacement gradient tensor. It is the linear approximation to a finite strain measure for small displacement gradients. An infinitesimal rigid rotation contributes no strain, so the elastic energy depends on the symmetric part rather than the entire gradient.

The deviatoric strain is the trace-free part of the infinitesimal strain tensor. It removes the isotropic volume change. The decomposition is orthogonal for the Frobenius inner product, because a trace-free matrix has zero Frobenius inner product with $I$. ConsequentlyFor $\epsilon =\frac{1}{2}(\nabla v+(\nabla v{)}^{\mathsf{T}})$, its trace is $\mathrm{div}v$.