Isotropic linear elasticity¶

The linear elastic constitutive formulation in the code assumes an isotropic Hencky material with a linear relationship between elastic logarithmic strains, \(\{\varepsilon^\text{e}\}\), and Kirchhoff stress, \(\{\tau\}\).

The stresses and strains are defined in Vogt notation with, using \(\{\tau\}\) as an example, the following ordering of the components

\[\{\tau\} = \{\begin{array}{cccccc} \tau_{xx} & \tau_{yy} & \tau_{zz} & \tau_{xy} & \tau_{yz} & \tau_{zx} \end{array}\}^T\]

Stress update algorithm¶

Given an initial elastic logarithmic strain, \(\{\varepsilon^\text{e}_{n}\}\), or the converged logarithmic elastic strain from the previous time step, and an increment in the logarithmic strain, \(\{\Delta \varepsilon\}\), the new logarithmic elastic strain is defined as

\[\{\varepsilon^\text{e}_{n+1}\}= \{\varepsilon^\text{e}_{n}\} + \{\Delta \varepsilon\}\]

and the updated Kirchhoff stress calculated using

\[\{\tau_{n+1}\} = [D^{\text{e}}]\{\varepsilon^\text{e}_{n+1}\}\]

where \([D^{\text{e}}]\) is the isotropic elastic stiffness matrix of the material. This stiffness matrix can be expressed as

\[ [D^{\text{e}}] = \frac{E}{(1+\nu)(1-2\nu)}\left[\begin{array}{cccccc} (1-\nu) & \nu & \nu & 0 & 0 & 0\\ \nu & (1-\nu) & \nu & 0 & 0 & 0\\ \nu & \nu & (1-\nu) & 0 & 0 & 0\\ 0& 0& 0& \frac{\displaystyle (1-2\nu)}{\displaystyle 2} & 0 & 0\\ 0& 0& 0& 0& \frac{\displaystyle (1-2\nu)}{\displaystyle 2} & 0\\ 0& 0& 0& 0& 0& \frac{\displaystyle (1-2\nu)}{\displaystyle 2} \end{array}\right]\]

where \(E\) is Young's modulus and \(\nu\) is Poisson's ratio.

Next section: Elasto-plasticity