5. Calculus of derivatives#

5.1. Derivative of the criterion with respect to the constraints#

We have: \(\frac{\partial F}{\partial \sigma }\text{=}\frac{\partial F}{\partial \stackrel{ˉ}{\sigma }}\frac{\partial \stackrel{ˉ}{\sigma }}{\partial \sigma }\) where \(\stackrel{ˉ}{\sigma }\) is the stress tensor expressed in the base of the eigenvectors. If the eigenvalues are ordered in \(\stackrel{ˉ}{\sigma }\), the \(F\) criterion will be written as:

\(P(\sigma )=(\begin{array}{ccc}{X}_{1}(\sigma )& {X}_{2}(\sigma )& {X}_{3}(\sigma )\end{array})\) \(F(\stackrel{ˉ}{\sigma },\gamma )\text{=}(\stackrel{ˉ}{{\sigma }_{3}}\text{-}\stackrel{ˉ}{{\sigma }_{1}})\text{-}\sqrt{\text{-}\stackrel{ˉ}{{\sigma }_{3}}\text{.}m{\sigma }_{c}\text{+}S{\sigma }_{c}^{2}}\text{-}b\text{.}(1\text{-}\frac{\stackrel{ˉ}{{\sigma }_{3}}}{\stackrel{ˉ}{{\sigma }_{3}^{b\text{-}d}}})\)

and \({X}_{2}(\sigma )\ne {X}_{3}(\sigma )\) \(\frac{\partial F}{\partial \stackrel{ˉ}{{\sigma }_{i}}}\text{=}{\delta }_{\mathrm{i3}}\text{-}{\delta }_{\mathrm{i1}}\text{+}\frac{1}{2}{\delta }_{\mathrm{i3}}{\sigma }_{c}m{[\text{-}\stackrel{ˉ}{{\sigma }_{3}}\text{.}m{\sigma }_{c}\text{+}S{\sigma }_{c}^{2}]}^{\frac{1}{2}}\text{+}b\frac{{\delta }_{\mathrm{i3}}}{{\sigma }_{3}^{b\text{-}d}}\).

5.2. Derivative of the stress tensor with respect to the principal stresses#

It can be shown (see Appendix 1) that:

\(\mid \begin{array}{c}\text{Si}\tilde{P}(\sigma )\text{.}\sigma \text{.}P(\sigma )\text{=}\stackrel{ˉ}{\sigma }\text{}\\ \text{où}P(\sigma )=\text{matrice de passage}(\text{matrice des vecteurs propres})\\ \text{et}\stackrel{ˉ}{\sigma }=\text{matrice diagonale des valeurs propres de}\sigma \text{}\\ \text{alors}\frac{\partial {\stackrel{ˉ}{\sigma }}_{k}}{\partial {\sigma }_{\text{ij}}}\text{=}{P}_{\text{ik}}{P}_{\text{jk}}(\text{sans sommation sur les indices})\end{array}\)

5.2.1. Special case of multiple eigenvalues#

In the particular case where several of the main constraints are equal, for example \({\sigma }_{2}\text{=}{\sigma }_{3}\), for example, the previous result will apply to the domains \({\sigma }_{2}<{\sigma }_{3}\) and \({\sigma }_{2}>{\sigma }_{3}\). So, in the first domain, we will have \(P(\sigma )=(\begin{array}{ccc}{X}_{1}(\sigma )& {X}_{2}(\sigma )& {X}_{3}(\sigma )\end{array})\) where \({X}_{2}\ne {X}_{3}\) and, in the second domain, \(P(\sigma )=(\begin{array}{ccc}{\stackrel{ˉ}{X}}_{1}(\sigma )& {\stackrel{ˉ}{X}}_{2}(\sigma )& {\stackrel{ˉ}{X}}_{3}(\sigma )\end{array})\). Thus, when \({\sigma }_{2}\text{-}{\sigma }_{3}\to {0}^{\text{-}}\) (resp. \({\sigma }_{2}\text{-}{\sigma }_{3}\to {0}^{\text{+}}\)), the transition matrix will tend to \({P}^{\text{-}}\text{=}(\begin{array}{ccc}{X}_{1}& {X}_{2}& {X}_{3}\end{array})\) (resp. to \({P}^{\text{+}}\text{=}(\begin{array}{ccc}{X}_{1}& {X}_{3}& {X}_{2}\end{array})\)) with \({X}_{2}\ne {X}_{3}\), the vectors \(({X}_{2},{X}_{3})\) defining the eigenspace associated with \({\sigma }_{2}\text{=}{\sigma }_{3}\). So we can see that the \(\frac{\partial \stackrel{ˉ}{\sigma }}{\partial \sigma }\) tensor is not uniquely defined at this point.

Moreover, the vector \(\frac{\partial {\sigma }_{3}}{\partial {\sigma }_{\text{ij}}}\text{=}\frac{\partial {\sigma }_{2}}{\partial {\sigma }_{\text{ij}}}\) is only defined from one of the two vectors \({X}_{2}\) or \({X}_{3}\) (it is equal to \({P}_{\mathrm{i3}}{P}_{\mathrm{j3}}\) or \({P}_{\mathrm{i2}}{P}_{\mathrm{j2}}\)), and therefore only corresponds to one of the two directional derivatives. This remark applies in the same way to \(\frac{\partial {s}_{3}^{e}}{\partial {\sigma }_{\text{ij}}}\) for the calculation of \(\frac{\partial \mathrm{\sigma }}{\partial \mathrm{\varepsilon }}\) \(\frac{\partial \lambda }{\partial {\varepsilon }_{\text{ij}}}\) in the coherent tangent matrix (see paragraph 6).

5.3. Derivative of the criterion with respect to the work-hardening variable#

\(\begin{array}{}\frac{\partial F}{\partial \gamma }\text{=}\text{-}\frac{1}{2}(\text{-}\frac{\partial (m{\sigma }_{c})}{\partial \gamma }{\sigma }_{3}\text{+}\frac{\partial (S{\sigma }_{c}^{2})}{\partial \gamma }){\left[\text{-}{\sigma }_{3}\text{.}m{\sigma }_{c}\text{+}S{\sigma }_{c}^{2}\right]}^{\text{-}\frac{1}{2}}\text{-}\frac{\partial b}{\partial \gamma }\text{.}(1\text{-}\frac{{\sigma }_{3}}{{\sigma }_{3}^{b\text{-}d}})\\ \text{=}\{\begin{array}{cc}\text{-}\frac{1}{2}(\text{-}{p}_{m\sigma }{\sigma }_{3}\text{+}{p}_{S{\sigma }^{2}}){\left[\text{-}{\sigma }_{3}\text{.}m{\sigma }_{c}\text{+}S{\sigma }_{c}^{2}\right]}^{\text{-}\frac{1}{2}}& \text{si}\gamma <{\gamma }^{\text{rup}}\\ \text{-}(\mathrm{2a}\gamma \text{+}d)\text{.}(1\text{-}\frac{{\sigma }_{3}}{{\sigma }_{3}^{b\text{-}d}})& \text{si}{\gamma }^{\text{rup}}<\gamma <\text{=}{\gamma }^{\text{res}}\\ 0& \text{si}\gamma >\text{=}{\gamma }^{\text{res}}\end{array}\end{array}\) \(\frac{\partial {\sigma }^{\text{'}}}{\partial \varepsilon }\)

5.4. Derivative of the parameters with respect to the work hardening variable#

  1. \(\frac{\partial (m{\sigma }_{c})}{\partial \gamma }(\gamma )=\{\begin{array}{cc}\frac{(m{\sigma }_{c}{)}^{\text{rup}}-(m{\sigma }_{c}{)}^{\text{end}}}{{\gamma }^{\text{rup}}}={p}_{m\sigma }& \begin{array}{cc}\text{si}& \gamma <{\gamma }^{\text{rup}}\end{array}\\ 0& \begin{array}{cc}\text{si}& \gamma >{\gamma }^{\text{rup}}\end{array}\end{array}\)

  2. \(\frac{\partial (S{\sigma }_{c}^{2})}{\partial \gamma }(\gamma )=\{\begin{array}{cc}\frac{(S{\sigma }_{c}^{2}{)}^{\text{rup}}-(S{\sigma }_{c}^{2}{)}^{\text{end}}}{{\gamma }^{\text{rup}}}={p}_{S{\sigma }^{2}}& \begin{array}{cc}\text{si}& \gamma <{\gamma }^{\text{rup}}\end{array}\\ 0& \begin{array}{cc}\text{si}& \gamma >{\gamma }^{\text{rup}}\end{array}\end{array}\)

  3. \(\frac{\partial \varphi }{\partial \gamma }(\gamma )\text{=}\{\begin{array}{cc}\frac{{\phi }^{\text{rup}}\text{-}{\phi }^{\text{end}}}{{\gamma }^{\text{rup}}}& \text{si}\gamma \le {\gamma }^{\text{rup}}\\ \frac{{\phi }^{\text{res}}\text{-}{\phi }^{\text{rup}}}{{\gamma }^{\text{res}}\text{-}{\gamma }^{\text{rup}}}& \text{si}{\gamma }^{\text{rup}}\le \gamma \le {\gamma }^{\text{res}}\\ 0& \text{sinon}\end{array}\) \(\frac{\partial \sigma }{\partial {p}_{g}}\)

  4. \(\frac{\partial b}{\partial \gamma }(\gamma )=\{\begin{array}{cc}0& \text{si}\gamma <{\gamma }^{\text{rup}}\\ \mathrm{2a}\gamma +d& \text{si}{\gamma }^{\text{rup}}<\gamma \le {\gamma }^{\text{res}}\\ 0& \text{si}\gamma \ge {\gamma }^{\text{res}}\end{array}\) \(\frac{\partial \sigma }{\partial {p}_{c}}\)