Calculus of derivatives =================== Derivative of the criterion with respect to the constraints ---------------------------------------------- We have: :math:`\frac{\partial F}{\partial \sigma }\text{=}\frac{\partial F}{\partial \stackrel{ˉ}{\sigma }}\frac{\partial \stackrel{ˉ}{\sigma }}{\partial \sigma }` where :math:`\stackrel{ˉ}{\sigma }` is the stress tensor expressed in the base of the eigenvectors. If the eigenvalues are ordered in :math:`\stackrel{ˉ}{\sigma }`, the :math:`F` criterion will be written as: :math:`P(\sigma )=(\begin{array}{ccc}{X}_{1}(\sigma )& {X}_{2}(\sigma )& {X}_{3}(\sigma )\end{array})` :math:`F(\stackrel{ˉ}{\sigma },\gamma )\text{=}(\stackrel{ˉ}{{\sigma }_{3}}\text{-}\stackrel{ˉ}{{\sigma }_{1}})\text{-}\sqrt{\text{-}\stackrel{ˉ}{{\sigma }_{3}}\text{.}m{\sigma }_{c}\text{+}S{\sigma }_{c}^{2}}\text{-}b\text{.}(1\text{-}\frac{\stackrel{ˉ}{{\sigma }_{3}}}{\stackrel{ˉ}{{\sigma }_{3}^{b\text{-}d}}})` and :math:`{X}_{2}(\sigma )\ne {X}_{3}(\sigma )` :math:`\frac{\partial F}{\partial \stackrel{ˉ}{{\sigma }_{i}}}\text{=}{\delta }_{\mathrm{i3}}\text{-}{\delta }_{\mathrm{i1}}\text{+}\frac{1}{2}{\delta }_{\mathrm{i3}}{\sigma }_{c}m{[\text{-}\stackrel{ˉ}{{\sigma }_{3}}\text{.}m{\sigma }_{c}\text{+}S{\sigma }_{c}^{2}]}^{\frac{1}{2}}\text{+}b\frac{{\delta }_{\mathrm{i3}}}{{\sigma }_{3}^{b\text{-}d}}`. Derivative of the stress tensor with respect to the principal stresses -------------------------------------------------------------------------- It can be shown (see Appendix 1) that: :math:`\mid \begin{array}{c}\text{Si}\tilde{P}(\sigma )\text{.}\sigma \text{.}P(\sigma )\text{=}\stackrel{ˉ}{\sigma }\text{}\\ \text{où}P(\sigma )=\text{matrice de passage}(\text{matrice des vecteurs propres})\\ \text{et}\stackrel{ˉ}{\sigma }=\text{matrice diagonale des valeurs propres de}\sigma \text{}\\ \text{alors}\frac{\partial {\stackrel{ˉ}{\sigma }}_{k}}{\partial {\sigma }_{\text{ij}}}\text{=}{P}_{\text{ik}}{P}_{\text{jk}}(\text{sans sommation sur les indices})\end{array}` .. _Ref103482072: Special case of multiple eigenvalues ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ In the particular case where several of the main constraints are equal, for example :math:`{\sigma }_{2}\text{=}{\sigma }_{3}`, for example, the previous result will apply to the domains :math:`{\sigma }_{2}<{\sigma }_{3}` and :math:`{\sigma }_{2}>{\sigma }_{3}`. So, in the first domain, we will have :math:`P(\sigma )=(\begin{array}{ccc}{X}_{1}(\sigma )& {X}_{2}(\sigma )& {X}_{3}(\sigma )\end{array})` where :math:`{X}_{2}\ne {X}_{3}` and, in the second domain, :math:`P(\sigma )=(\begin{array}{ccc}{\stackrel{ˉ}{X}}_{1}(\sigma )& {\stackrel{ˉ}{X}}_{2}(\sigma )& {\stackrel{ˉ}{X}}_{3}(\sigma )\end{array})`. Thus, when :math:`{\sigma }_{2}\text{-}{\sigma }_{3}\to {0}^{\text{-}}` (resp. :math:`{\sigma }_{2}\text{-}{\sigma }_{3}\to {0}^{\text{+}}`), the transition matrix will tend to :math:`{P}^{\text{-}}\text{=}(\begin{array}{ccc}{X}_{1}& {X}_{2}& {X}_{3}\end{array})` (resp. to :math:`{P}^{\text{+}}\text{=}(\begin{array}{ccc}{X}_{1}& {X}_{3}& {X}_{2}\end{array})`) with :math:`{X}_{2}\ne {X}_{3}`, the vectors :math:`({X}_{2},{X}_{3})` defining the eigenspace associated with :math:`{\sigma }_{2}\text{=}{\sigma }_{3}`. So we can see that the :math:`\frac{\partial \stackrel{ˉ}{\sigma }}{\partial \sigma }` tensor is not uniquely defined at this point. Moreover, the vector :math:`\frac{\partial {\sigma }_{3}}{\partial {\sigma }_{\text{ij}}}\text{=}\frac{\partial {\sigma }_{2}}{\partial {\sigma }_{\text{ij}}}` is only defined from one of the two vectors :math:`{X}_{2}` or :math:`{X}_{3}` (it is equal to :math:`{P}_{\mathrm{i3}}{P}_{\mathrm{j3}}` or :math:`{P}_{\mathrm{i2}}{P}_{\mathrm{j2}}`), and therefore only corresponds to one of the two directional derivatives. This remark applies in the same way to :math:`\frac{\partial {s}_{3}^{e}}{\partial {\sigma }_{\text{ij}}}` for the calculation of :math:`\frac{\partial \mathrm{\sigma }}{\partial \mathrm{\varepsilon }}` :math:`\frac{\partial \lambda }{\partial {\varepsilon }_{\text{ij}}}` in the coherent tangent matrix (see paragraph :ref:`6 `). Derivative of the criterion with respect to the work-hardening variable ---------------------------------------------------------- :math:`\begin{array}{}\frac{\partial F}{\partial \gamma }\text{=}\text{-}\frac{1}{2}(\text{-}\frac{\partial (m{\sigma }_{c})}{\partial \gamma }{\sigma }_{3}\text{+}\frac{\partial (S{\sigma }_{c}^{2})}{\partial \gamma }){\left[\text{-}{\sigma }_{3}\text{.}m{\sigma }_{c}\text{+}S{\sigma }_{c}^{2}\right]}^{\text{-}\frac{1}{2}}\text{-}\frac{\partial b}{\partial \gamma }\text{.}(1\text{-}\frac{{\sigma }_{3}}{{\sigma }_{3}^{b\text{-}d}})\\ \text{=}\{\begin{array}{cc}\text{-}\frac{1}{2}(\text{-}{p}_{m\sigma }{\sigma }_{3}\text{+}{p}_{S{\sigma }^{2}}){\left[\text{-}{\sigma }_{3}\text{.}m{\sigma }_{c}\text{+}S{\sigma }_{c}^{2}\right]}^{\text{-}\frac{1}{2}}& \text{si}\gamma <{\gamma }^{\text{rup}}\\ \text{-}(\mathrm{2a}\gamma \text{+}d)\text{.}(1\text{-}\frac{{\sigma }_{3}}{{\sigma }_{3}^{b\text{-}d}})& \text{si}{\gamma }^{\text{rup}}<\gamma <\text{=}{\gamma }^{\text{res}}\\ 0& \text{si}\gamma >\text{=}{\gamma }^{\text{res}}\end{array}\end{array}` :math:`\frac{\partial {\sigma }^{\text{'}}}{\partial \varepsilon }` Derivative of the parameters with respect to the work hardening variable -------------------------------------------------------------- 1. :math:`\frac{\partial (m{\sigma }_{c})}{\partial \gamma }(\gamma )=\{\begin{array}{cc}\frac{(m{\sigma }_{c}{)}^{\text{rup}}-(m{\sigma }_{c}{)}^{\text{end}}}{{\gamma }^{\text{rup}}}={p}_{m\sigma }& \begin{array}{cc}\text{si}& \gamma <{\gamma }^{\text{rup}}\end{array}\\ 0& \begin{array}{cc}\text{si}& \gamma >{\gamma }^{\text{rup}}\end{array}\end{array}` 2. :math:`\frac{\partial (S{\sigma }_{c}^{2})}{\partial \gamma }(\gamma )=\{\begin{array}{cc}\frac{(S{\sigma }_{c}^{2}{)}^{\text{rup}}-(S{\sigma }_{c}^{2}{)}^{\text{end}}}{{\gamma }^{\text{rup}}}={p}_{S{\sigma }^{2}}& \begin{array}{cc}\text{si}& \gamma <{\gamma }^{\text{rup}}\end{array}\\ 0& \begin{array}{cc}\text{si}& \gamma >{\gamma }^{\text{rup}}\end{array}\end{array}` 3. :math:`\frac{\partial \varphi }{\partial \gamma }(\gamma )\text{=}\{\begin{array}{cc}\frac{{\phi }^{\text{rup}}\text{-}{\phi }^{\text{end}}}{{\gamma }^{\text{rup}}}& \text{si}\gamma \le {\gamma }^{\text{rup}}\\ \frac{{\phi }^{\text{res}}\text{-}{\phi }^{\text{rup}}}{{\gamma }^{\text{res}}\text{-}{\gamma }^{\text{rup}}}& \text{si}{\gamma }^{\text{rup}}\le \gamma \le {\gamma }^{\text{res}}\\ 0& \text{sinon}\end{array}` :math:`\frac{\partial \sigma }{\partial {p}_{g}}` 4. :math:`\frac{\partial b}{\partial \gamma }(\gamma )=\{\begin{array}{cc}0& \text{si}\gamma <{\gamma }^{\text{rup}}\\ \mathrm{2a}\gamma +d& \text{si}{\gamma }^{\text{rup}}<\gamma \le {\gamma }^{\text{res}}\\ 0& \text{si}\gamma \ge {\gamma }^{\text{res}}\end{array}` :math:`\frac{\partial \sigma }{\partial {p}_{c}}` .. _Ref103481890: