5. Tangent matrix

By introducing the elastic shear module \(\mu\), the stress deflector at time \(n+1\) is written as a function of the elastic deformation deflector:

\({\underline{\underline{\sigma }}}_{n+1}^{d}=\mathrm{2\mu }{\underline{\underline{\epsilon }}}_{n+1}^{\text{ed}}={\underline{\underline{\sigma }}}_{n}^{d}+\mathrm{2\mu \Delta }{\underline{\underline{\epsilon }}}_{n}^{d}-\mathrm{2\mu \Delta }{\underline{\underline{\epsilon }}}_{n}^{f,d}\) eq 5-1

By substituting the deviatoric part of the natural creep deformation with the expression [éq 4.2-1], the following relationship arises:

\({\underline{\underline{\sigma }}}_{n+1}^{d}(1+{\mathrm{2\mu c}}^{d})={\underline{\underline{\sigma }}}_{n}^{d}(1-{\mathrm{2\mu b}}^{d})+\mathrm{2\mu \Delta }{\underline{\underline{\epsilon }}}_{n}^{d}-{\mathrm{2\mu a}}^{d}\underline{\underline{1}}\) eq 5-2

Expression that induces by derivation with respect to \({\underline{\underline{\epsilon }}}_{n+1}^{d}\):

\(\frac{\partial {\underline{\underline{\sigma }}}_{n+1}^{d}}{\partial {\underline{\underline{\epsilon }}}_{n+1}^{d}}(1+{\mathrm{2\mu c}}^{d})=\mathrm{2\mu }\underline{\underline{\underline{\underline{1}}}}\) eq 5-3

By carrying out a similar approach for the spherical part and by introducing the expansion stiffness module \(K\), it follows the following three relationships:

\(\text{tr}{\underline{\underline{\sigma }}}_{n+1}=\mathrm{3K}\text{tr}{\underline{\underline{\epsilon }}}_{n+1}^{e}=\text{tr}{\underline{\underline{\sigma }}}_{n}+\mathrm{3K}\text{tr}(\Delta {\underline{\underline{\epsilon }}}_{n})-\mathrm{3K}\text{tr}(\Delta {\underline{\underline{\epsilon }}}_{n}^{f})\) eq 5-4

\(\text{tr}{\underline{\underline{\sigma }}}_{n+1}(1+3{\text{Kc}}^{s})=\text{tr}{\underline{\underline{\sigma }}}_{n}(1-3{\text{Kb}}^{s})+\mathrm{3K}\text{tr}(\Delta {\underline{\underline{\epsilon }}}_{n})-{\text{Ka}}^{s}\) eq 5-5

\(\frac{\partial (\text{tr}{\underline{\underline{\sigma }}}_{n+1})}{\partial (\text{tr}{\underline{\underline{\epsilon }}}_{n+1})}(1+3{\text{Kc}}^{s})=\mathrm{3K}\) eq 5-6

The tangent matrix is finally written as:

\(\frac{\partial \underline{\underline{\sigma }}}{\partial \underline{\underline{\epsilon }}}=\frac{\partial {\underline{\underline{\sigma }}}^{d}}{\partial \underline{\underline{\epsilon }}}+\frac{1}{3}\frac{\partial (\text{tr}\underline{\underline{\sigma }})}{\partial \underline{\underline{\epsilon }}}\underline{\underline{1}}=\frac{\partial {\underline{\underline{\sigma }}}^{d}}{\partial {\underline{\underline{\epsilon }}}^{d}}\frac{\partial {\underline{\underline{\epsilon }}}^{d}}{\partial \underline{\underline{\epsilon }}}+\frac{1}{3}\frac{\partial (\text{tr}\underline{\underline{\sigma }})}{\partial (\text{tr}\underline{\underline{\epsilon }})}\frac{\partial (\text{tr}\underline{\underline{\epsilon }})}{\partial \underline{\underline{\epsilon }}}\underline{\underline{1}}\) eq 5-7

That is to say:

\(\frac{\partial \underline{\underline{\sigma }}}{\partial \underline{\underline{\epsilon }}}=\underset{\chi }{\underset{\underbrace{}}{\frac{\mathrm{2\mu }}{1+{\mathrm{2\mu c}}^{d}}}}(\underline{\underline{\underline{\underline{1}}}}-\frac{1}{3}\underline{\underline{1}}\otimes \underline{\underline{1}})+\underset{\xi }{\underset{\underbrace{}}{\frac{K}{1+3{\text{Kc}}^{s}}}}\underline{\underline{1}}\otimes \underline{\underline{1}}\) eq 5-8

After linearization, the tangent matrix develops as follows:

\((\begin{array}{c}{\mathrm{\Delta \sigma }}_{\text{11}}\\ {\mathrm{\Delta \sigma }}_{\text{22}}\\ {\mathrm{\Delta \sigma }}_{\text{33}}\\ \sqrt{2}{\mathrm{\Delta \sigma }}_{\text{12}}\\ \sqrt{2}{\mathrm{\Delta \sigma }}_{\text{13}}\\ \sqrt{2}{\mathrm{\Delta \sigma }}_{\text{23}}\end{array})=\left[\begin{array}{cccccc}\xi +\frac{2}{3}\chi & \xi -\frac{1}{3}\chi & \xi -\frac{1}{3}\chi & 0& 0& 0\\ \xi -\frac{1}{3}\chi & \xi +\frac{2}{3}\chi & \xi -\frac{1}{3}\chi & 0& 0& 0\\ \xi -\frac{1}{3}\chi & \xi -\frac{1}{3}\chi & \xi +\frac{2}{3}\chi & 0& 0& 0\\ 0& 0& 0& \chi & 0& 0\\ 0& 0& 0& 0& \chi & 0\\ 0& 0& 0& 0& 0& \chi \end{array}\right]\cdot (\begin{array}{c}{\mathrm{\Delta \epsilon }}_{\text{11}}\\ {\mathrm{\Delta \epsilon }}_{\text{22}}\\ {\mathrm{\Delta \epsilon }}_{\text{33}}\\ \sqrt{2}{\mathrm{\Delta \epsilon }}_{\text{12}}\\ \sqrt{2}{\mathrm{\Delta \epsilon }}_{\text{13}}\\ \sqrt{2}{\mathrm{\Delta \epsilon }}_{\text{23}}\end{array})\) eq 5-9