5. Calculation of the tangent operator

In the case where \({\sigma }_{\mathrm{eq}}^{e}=0\) and \(x=0\), the elasticity tensor is taken as a tangent operator.

Otherwise, we get this operator by deriving the equation [éq 4.3-1] with respect to \(\mathrm{\Delta }\mathrm{\epsilon }\):

\(\frac{\partial \stackrel{~}{\mathrm{\sigma }}}{\partial \mathrm{\Delta }\mathrm{\epsilon }}=\frac{\partial \mathrm{\Delta }\stackrel{~}{\mathrm{\sigma }}}{\partial \mathrm{\Delta }\mathrm{\epsilon }}=\frac{\partial b\left(x,{\sigma }_{\text{eq}}^{e}\right)}{\partial \mathrm{\Delta }\mathrm{\epsilon }}{\stackrel{~}{\mathrm{\sigma }}}^{e}+b\left(x,{\sigma }_{\text{eq}}^{e}\right)\frac{\partial {\stackrel{~}{\mathrm{\sigma }}}^{e}}{\partial \mathrm{\Delta }\mathrm{\epsilon }}\)

then also deriving [éq 4.3-2] with respect to \(\mathrm{\Delta }\mathrm{\epsilon }\):

\(\frac{\partial \mathrm{\Delta }\mathrm{\sigma }}{\partial \mathrm{\Delta }\mathrm{\epsilon }}=\frac{\partial \mathrm{\Delta }\stackrel{~}{\mathrm{\sigma }}}{\partial \mathrm{\Delta }\mathrm{\epsilon }}+K{\mathrm{I}}_{3}\frac{\partial \text{Tr}\left(\mathrm{\Delta }\mathrm{\epsilon }\right)}{\partial \mathrm{\Delta }\mathrm{\epsilon }}=\frac{\partial \mathrm{\Delta }\stackrel{~}{\mathrm{\sigma }}}{\partial \mathrm{\Delta }\mathrm{\epsilon }}+K\mathrm{.}{\mathrm{I}}_{3}^{\mathrm{t}}{\mathrm{I}}_{3}\)

It will be noted that, in these equations, the tensors of order 2 and of order 4 are respectively assimilated to vectors and to matrices. \({\mathrm{I}}_{3}\) is here a tensor of order 2, similar to a vector:

\({}^{t}\text{}{\mathrm{I}}_{3}=\left(\mathrm{1,1,1,0,0,0}\right)\)

In addition, we have:

\(\frac{\partial b(x,{\sigma }_{\text{eq}}^{e})}{\partial \Delta \varepsilon }=\frac{\partial b}{\partial x}(x,{\sigma }_{\text{eq}}^{e})\frac{\partial x}{\partial \Delta \varepsilon }+\frac{\partial b}{\partial {\sigma }_{\text{eq}}^{e}}(x,{\sigma }_{\text{eq}}^{e})\frac{\partial {\sigma }_{\text{eq}}^{e}}{\partial \Delta \varepsilon }\)

So you have to calculate \(\frac{\partial x}{\partial \Delta \varepsilon }\). To do this, we implicitly derive the scalar equation with respect to \(\Delta \varepsilon\).

To simplify, the parameter \(T\) will be omitted later in the writing of \(g\) and its derivatives.

We then have:

\(\left[3\mu \Delta t{G}^{\text{'}}(x)+1\right]\frac{\partial x}{\partial \Delta \varepsilon }+\Delta t\frac{\partial g}{\partial y}(x,y)\frac{\partial {\sigma }_{\text{eq}}^{e}}{\partial \Delta \varepsilon }=\frac{\partial {\sigma }_{\text{eq}}^{e}}{\partial \Delta \varepsilon }\)

From where:

\(\frac{\partial x}{\partial \Delta \varepsilon }=\frac{1-\Delta t\frac{\partial g}{\partial y}(x,y)}{1+3\mu \Delta t{G}^{\text{'}}(x)}\frac{\partial {\sigma }_{\text{eq}}^{e}}{\partial \Delta \varepsilon }\)

\(\frac{\partial x}{\partial \Delta \varepsilon }=\frac{1-\Delta t\frac{\partial g}{\partial y}(x,y)}{1+3\mu \Delta t{G}^{\text{'}}(x)}\frac{3\mu }{{\sigma }_{\mathrm{eq}}^{e}}{\tilde{\sigma }}^{e}\)

with the expression for \(\text{G'}(x)\) obtained in [§4.3].

The following expression for the tangent operator is finally obtained:

\(\frac{\partial \Delta \sigma }{\partial \Delta \varepsilon }=K{I}_{3}^{t}{I}_{3}+2\mu \left[\gamma {\tilde{\sigma }}^{e}{}^{t}\text{}{\tilde{\sigma }}^{e}+b(x,{\sigma }_{\text{eq}}^{e})A\right]\)

with

\(A=\frac{\partial \Delta \tilde{\varepsilon }}{\partial \Delta \varepsilon }={J}_{6}-\frac{1}{3}{I}_{3}{}^{t}\text{}{I}_{3}\) where \({J}_{6}\) is the rank 6 identity matrix.

\(\gamma =\frac{3}{2{({\sigma }_{\mathrm{eq}}^{e})}^{3}}\left[{\sigma }_{\mathrm{eq}}^{e}\frac{1-\Delta t\frac{\partial g}{\partial y}(x,y)}{1+3\mu \Delta t{G}^{\text{'}}(x)}-x\right]\)

Note:

In the case of the law VISC_IRRA_LOG, it is easy to verify that:

\(\begin{array}{}{G}^{\text{'}}(x)=\frac{1}{{f}_{1}^{\text{'}}{g}_{1}+{f}_{2}^{\text{'}}{g}_{2}}[{g}_{1}{g}_{1}^{\text{'}}({f}_{1}^{{\text{'}}^{2}}-{f}_{1}{f}_{1}^{\text{''}})+{g}_{2}{g}_{2}^{\text{'}}({f}_{2}^{{\text{'}}^{2}}-{f}_{2}{f}_{2}^{\text{''}})+{g}_{1}{g}_{2}^{\text{'}}({f}_{1}^{\text{'}}{f}_{2}^{\text{'}}-{f}_{1}^{\text{''}}{f}_{2})\\ +{g}_{2}{g}_{1}^{\text{'}}({f}_{1}^{\text{'}}{f}_{2}^{\text{'}}-{f}_{1}{f}_{2}^{\text{''}})-\frac{1}{\mathrm{3m}}({f}_{1}^{\text{''}}{g}_{1}+{f}_{2}^{\text{''}}{g}_{2})]\\ \frac{\partial g}{\partial \lambda }(x,y,T)=\frac{{f}_{1}^{\text{''}}{g}_{1}+{f}_{2}^{\text{''}}{g}_{2}}{{f}_{1}^{\text{'}}{g}_{1}+{f}_{2}^{\text{'}}{g}_{2}}\end{array}\)

where \({f}_{\mathrm{1,}}f{\text{'}}_{\mathrm{1,}}f\text{'}{\text{'}}_{\mathrm{1,}}{f}_{\mathrm{2,}}f{\text{'}}_{\mathrm{2,}}f\text{'}{\text{'}}_{2}\) denote the values of \({f}_{1}\) and \({f}_{2}\) and their derivatives at the point \(t(x,y,T)\) and where \({g}_{\mathrm{1,}}g{\text{'}}_{\mathrm{1,}}{g}_{\mathrm{2,}}g{\text{'}}_{2}\) refer to the values of \({g}_{1}\) and \({g}_{2}\) and their derivative with respect to \(s\) * at the point \((x,T)\) (see [bib1]).