Hooke's Law - Tensor Expression

Tensor Expression

Note: the Einstein summation convention of summing on repeated indices is used below.

When working with a three-dimensional stress state, a 4th order tensor containing 81 elastic coefficients must be defined to link the stress tensor (σ_ij) and the strain tensor .

Expressed in terms of components with respect to an orthonormal basis, the generalized form of Hooke's law is written as (using the summation convention)

The tensor is called the stiffness tensor or the elasticity tensor. Due to the symmetry of the stress tensor, strain tensor, and stiffness tensor, only 21 elastic coefficients are independent. As stress is measured in units of pressure and strain is dimensionless, the entries of are also in units of pressure.

The expression for generalized Hooke's law can be inverted to get a relation for the strain in terms of stress:

$\boldsymbol{\epsilon} = \mathsf{s}:\boldsymbol{\sigma} \qquad {\rm or} \qquad \epsilon_{ij} = s_{ijk\ell}~\sigma_{k\ell} ~.$

The tensor is called the compliance tensor.

Generalization for the case of large deformations is provided by models of neo-Hookean solids and Mooney-Rivlin solids.

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