3. Numerical integration of second gradient models#

The fields of deformations, associated stresses and the tangent matrix in mechanics are detailed below with the historical formulation of the second expansion gradient model. There are no internal variables specific to the second gradient. As a reminder, in mechanics, the nodal variables defining the degrees of freedom are as follows: \(\left({u}_{i},\chi ,\lambda \right)\).

For THM models with a second gradient, there is no coupling between the generalized deformations and stresses of the second gradient and those of thermo-hydric. Thus, the fields of generalized deformations and stresses, as well as the elementary stiffness matrices associated with the second gradient are identical to those of the demechanical models.

3.1. The field of generalized deformations#

\(E=\left(\begin{array}{c}\epsilon \\ {\epsilon }_{V}\\ \chi \\ \frac{\partial \chi }{\partial {x}_{j}}\\ \lambda \end{array}\right)\)

3.2. The field of generalized constraints#

\(\Sigma =\left(\begin{array}{c}\sigma \\ r\left({\epsilon }_{V}-\chi \right)+\lambda \\ -r\left({\epsilon }_{V}-\chi \right)-\lambda \\ {S}_{j}\\ \left({\epsilon }_{V}-\chi \right)\end{array}\right)\)

3.3. The tangent matrix#

The elementary tangent matrix of second gradient modeling is composed, among others:

  1. of the elementary stiffness tangent matrix \({D}^{1d}\) associated with the law of first gradient behavior

  2. of the tangent elementary stiffness matrix \({D}^{2d}\) associated with the law of behavior second expansion gradient which links the double stresses \({S}_{j}\) to the gradients of microscopic volume deformations \(\frac{\partial \chi }{\partial {x}_{j}}\).

\({K}^{\text{el}}=\left(\begin{array}{ccccc}{\left({D}^{1d}\right)}_{\text{dim}\left(j\right)}& 0& 0& 0& 0\\ 0& r& -r& 0& 1\\ 0& -r& r& 0& -1\\ 0& 0& 0& {\left({D}^{2d}\right)}_{\text{dim}\left(j\right)}& 0\\ 0& 1& -1& 0& 0\end{array}\right)\)