# Linear elasticity¶

For the linear elasticity model, the strain energy density is given by

The constitutive law (stress-strain relationship) is therefore given by its gradient,

where the colon represents a double contraction (over both indices of \(\bm{\varepsilon}\)), \(\bm{\varepsilon}\) is (small/infintesimal) strain tensor defined by

and the Lamé parameters are given in terms of Young’s modulus \(E\), and Poisson’s ratio \(\nu\) by

The constitutive law (stress-strain relationship) can also be written as

For notational convenience, we express the symmetric second order tensors \(\bm \sigma\) and \(\bm \varepsilon\) as vectors of length 6 using the Voigt notation. Hence, the fourth order elasticity tensor \(\mathsf C\) (also known as elastic moduli tensor or material stiffness tensor) can be represented as

Note that the incompressible limit \(\nu \to \frac 1 2\) causes \(\firstlame \to \infty\), and thus \(\mathsf C\) becomes singular.

## Incompressibility¶

One can see from the above equations that as \(\firstlame \to \infty\), it is necessary that \(\nabla\cdot \bm{u} \longrightarrow 0\) which gives an idea of alternative strategies. One approach is to define an auxiliary variable \(p\), and rewrite constitutive equation (42) as

Alternatively, we can use the definition of hydrostatic pressure i.e., \(p_{\text{hyd}} = - \frac{\trace \bm \sigma}{3}\) and arrive at

where \(\bm \varepsilon_{\text{dev}} = \bm \varepsilon - \frac{1}{3} \trace \bm \varepsilon ~ \bm{I}\) is the deviatoric part of the linear strain tensor and \(\bulk\) is the bulk modulus. We present a general constitutive equation as

where

is the primal portion of the bulk modulus, defined in terms of \(\nu_p\) with \(-1 \leq \nu_p < \nu\), where \(\nu\) is the physical Poisson’s ratio. The standard full-train formulation (46) is obtained using \(\nu_p = 0\), and the deviatoric formulation (47) with \(\nu_p = -1\).