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* BENALLAL A. and COMI C.: The role of deviatoric and volumetric non-associativities in strain localization (1993).


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* RICE JR and RUDNICKI JW: A note on some features of the theory of localization of deformation (1980).


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* DOGHRI I: Study of the location of the damage (1989).


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                                * **r7.01.16** Calculation of partial derivatives of :math:`\Delta p`
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A1.1 Calculation of the partial derivative of the plastic deformation increment in the case of linear work hardening

:math:`R(p)=h\cdot p+{\sigma }^{y}` for :math:`0\le p<{p}_{\text{ultm}}`

:math:`\mathrm{\Delta p}=\frac{{\sigma }_{\text{eq}}^{e}+{\rm A}\cdot \text{Tr}({\sigma }^{e})-h\cdot {p}^{-}-{\sigma }^{Y}}{\mathrm{9K}\cdot {{\rm A}}^{2}+\mathrm{3\mu }+h}`

So:

:math:`\begin{array}{}\frac{\partial \mathrm{\Delta p}}{\partial \Phi }=\frac{1}{\mathrm{9K}\text{.}{{\rm A}}^{2}+\mathrm{3\mu }+h}\cdot (\frac{\partial {\sigma }_{\text{eq}}^{e}}{\partial \Phi }+\frac{\partial {\rm A}}{\partial \Phi }\cdot \text{Tr}({\sigma }^{e})+{\rm A}\cdot \frac{\partial \text{Tr}({\sigma }^{e})}{\partial \Phi }-\frac{\partial h}{\partial \Phi }\cdot {p}^{-}-\frac{\partial {\sigma }^{y}}{\partial \Phi }\\ -\mathrm{\Delta p}\cdot (9\cdot \frac{\partial K}{\partial \Phi }\cdot {{\rm A}}^{2}+\text{18}\cdot K\cdot A\cdot \frac{\partial {\rm A}}{\partial \Phi }+\frac{\partial \mathrm{3\mu }}{\partial \Phi }+\frac{\partial h}{\partial \Phi }))\end{array}`

:math:`\frac{\partial \mathrm{\Delta p}}{\partial \sigma }=\frac{1}{\mathrm{3\mu }+\mathrm{9K}\cdot {{\rm A}}^{2}+h}\cdot (A\cdot \frac{\partial \text{Tr}({\sigma }^{e})}{\partial \sigma }+\frac{\partial {\sigma }_{\text{eq}}^{e}}{\partial \sigma })`

:math:`\frac{\partial \mathrm{\Delta p}}{\partial p}=-h\cdot \frac{1}{\mathrm{3\mu }+\mathrm{9K}\cdot {{\rm A}}^{2}+h}`

:math:`R(p)=h\cdot {p}_{\text{ultm}}+{\sigma }^{y}` for :math:`p>{p}_{\text{ultm}}`

:math:`\mathrm{\Delta p}=\frac{{\sigma }_{\text{eq}}^{e}+{\rm A}\cdot \text{Tr}({\sigma }^{e})-h\cdot {p}_{\text{ultm}}-{\sigma }^{Y}}{\mathrm{9K}\cdot {{\rm A}}^{2}+\mathrm{3\mu }}`

So:

:math:`\begin{array}{}\frac{\partial \mathrm{\Delta p}}{\partial \Phi }=\frac{1}{\mathrm{9K}\cdot {{\rm A}}^{2}+\mathrm{3\mu }}\cdot (\frac{\partial {\sigma }_{\text{eq}}^{e}}{\partial \Phi }+\frac{\partial {\rm A}}{\partial \Phi }\cdot \text{Tr}({\sigma }^{e})+{\rm A}\cdot \frac{\partial \text{Tr}({\sigma }^{e})}{\partial \Phi }-\frac{\partial h}{\partial \Phi }\cdot {p}_{\text{ultm}}-h\cdot \frac{\partial {p}_{\text{ultm}}}{\partial \Phi }-\frac{\partial {\sigma }^{y}}{\partial \Phi }\\ -\mathrm{\Delta p}\cdot (9\cdot \frac{\partial K}{\partial \Phi }\cdot {{\rm A}}^{2}+\text{18}\cdot K\cdot A\cdot \frac{\partial {\rm A}}{\partial \Phi }+\frac{\partial \mathrm{3\mu }}{\partial \Phi }))\end{array}`

:math:`\frac{\partial \mathrm{\Delta p}}{\partial \sigma }=\frac{1}{\mathrm{3\mu }+\mathrm{9K}\cdot {{\rm A}}^{2}}(A\cdot \frac{\partial \text{Tr}({\sigma }^{e})}{\partial \sigma }+\frac{\partial {\sigma }_{\text{eq}}^{e}}{\partial \sigma })`

:math:`\frac{\partial \mathrm{\Delta p}}{\partial p}=0`

**A1.2** **Calculation of the partial derivative of the plastic deformation increment in the case of parabolic work hardening**

:math:`R(p)={\sigma }^{y}\cdot (1-(1-\sqrt{\frac{{\sigma }_{\text{ultm}}^{y}}{{\sigma }^{y}}})\cdot \frac{p}{{p}_{\text{ultm}}}{)}^{2}` for :math:`0\le p<{p}_{\text{ultm}}`

:math:`\begin{array}{}\frac{\partial {\sigma }_{\text{eq}}^{e}}{\partial \Phi }-(\frac{\partial \mathrm{3\mu }}{\partial \Phi }+{\mathrm{9{\rm A}}}^{2}\text{.}\frac{\partial K}{\partial \Phi }+\text{18}K\text{.}{\rm A}\text{.}\frac{\partial {\rm A}}{\partial \Phi })\text{.}\mathrm{\Delta p}-(\mathrm{3\mu }+\mathrm{9K}\text{.}{{\rm A}}^{2})\text{.}\frac{\partial \mathrm{\Delta p}}{\partial \Phi }+\frac{\partial {\rm A}}{\partial \Phi }\text{.}\text{Tr}({\sigma }^{e})+{\rm A}\text{.}\frac{\partial \text{Tr}({\sigma }^{e})}{\partial \Phi }\\ -\frac{\partial {\sigma }^{y}}{\partial \Phi }\text{.}(1-(1-\sqrt{\frac{{\sigma }_{\text{ultm}}^{y}}{{\sigma }^{y}}})\text{.}\frac{{p}^{-}+\mathrm{\Delta p}}{{p}_{\text{ultm}}}{)}^{2}\\ -{\mathrm{2\sigma }}^{y}\text{.}(1-(1-\sqrt{\frac{{\sigma }_{\text{ultm}}^{y}}{{\sigma }^{y}}})\text{.}\frac{{p}^{-}+\mathrm{\Delta p}}{{p}_{\text{ultm}}})\text{.}\\ (\frac{\partial {\sigma }_{\text{ultm}}^{y}}{\partial \Phi }\text{.}\frac{{p}^{-}+\mathrm{\Delta p}}{{\mathrm{2p}}_{\text{ultm}}\text{.}\sqrt{{\sigma }_{\text{ultm}}^{y}\text{.}{\sigma }^{y}}}-\frac{\partial {\sigma }^{y}}{\partial \Phi }\text{.}\frac{{p}^{-}+\mathrm{\Delta p}}{{\mathrm{2p}}_{\text{ultm}}\text{.}{\sigma }^{y}}\text{.}\sqrt{\frac{{\sigma }_{\text{ultm}}^{y}}{{\sigma }^{y}}}+\frac{\partial {p}_{\text{ultm}}}{\partial \Phi }\text{.}(1-\sqrt{\frac{{\sigma }_{\text{ultm}}^{y}}{{\sigma }^{y}}})\text{.}\frac{{p}^{-}+\mathrm{\Delta p}}{{p}_{{\text{ultm}}^{2}}}-\frac{\partial \mathrm{\Delta p}}{\partial \Phi }\frac{1-\sqrt{\frac{{\sigma }_{\text{ultm}}^{y}}{{\sigma }^{y}}}}{{p}_{\text{ultm}}})\\ 0\end{array}`

:math:`\frac{\partial {\sigma }_{\text{eq}}^{e}}{\partial \sigma }-(\mathrm{3\mu }+\mathrm{9K}\cdot {{\rm A}}^{2})\cdot \frac{\partial \mathrm{\Delta p}}{\partial \sigma }+A\cdot \frac{\partial \text{Tr}({\sigma }^{e})}{\partial \sigma }+{\mathrm{2\sigma }}^{y}\cdot (1-(1-\sqrt{\frac{{\sigma }_{\text{ultm}}^{y}}{{\sigma }^{y}}})\cdot \frac{{p}^{-}+\mathrm{\Delta p}}{{p}_{\text{ultm}}})\cdot \frac{1-\sqrt{\frac{{\sigma }_{\text{ultm}}^{y}}{{\sigma }^{y}}}}{{p}_{\text{ultm}}}\cdot \frac{\partial \mathrm{\Delta p}}{\partial \sigma }=0`

:math:`-(\mathrm{3\mu }+\mathrm{9K}\cdot {{\rm A}}^{2})\cdot \frac{\partial \mathrm{\Delta p}}{\partial p}+{\mathrm{2\sigma }}^{y}\cdot (1-(1-\sqrt{\frac{{\sigma }_{\text{ultm}}^{y}}{{\sigma }^{y}}})\frac{{p}^{-}+\mathrm{\Delta p}}{{p}_{\text{ultm}}})\cdot \frac{1-\sqrt{\frac{{\sigma }_{\text{ultm}}^{y}}{{\sigma }^{y}}}}{{p}_{\text{ultm}}}\cdot (1+\frac{\partial \mathrm{\Delta p}}{\partial p})=0`

:math:`R(p)={\sigma }_{\text{ultm}}^{y}` for :math:`p>{p}_{\text{ultm}}`

:math:`\frac{\partial \mathrm{\Delta p}}{\partial \Phi }=\frac{1}{\mathrm{3\mu }+\mathrm{9K}\cdot {{\rm A}}^{2}}(\frac{\partial {\sigma }_{\text{eq}}^{e}}{\partial \Phi }-(\frac{\partial \mathrm{3\mu }}{\partial \Phi }+\frac{\partial \mathrm{9K}}{\partial \Phi }\cdot {{\rm A}}^{2}+\text{18}K\cdot \frac{\partial {\rm A}}{\partial \Phi }\cdot A)\cdot \mathrm{\Delta p}+\frac{\partial {\rm A}}{\partial \Phi }\cdot \text{Tr}({\sigma }^{e})+{\rm A}\cdot \frac{\partial \text{Tr}({\sigma }^{e})}{\partial \Phi }-\frac{\partial {\sigma }_{\text{ultm}}^{y}}{\partial \Phi })`

:math:`\frac{\partial \mathrm{\Delta p}}{\partial \sigma }=\frac{1}{\mathrm{3\mu }+\mathrm{9K}\cdot {{\rm A}}^{2}}(\frac{\partial {\sigma }_{\text{eq}}^{e}}{\partial \sigma }+A\cdot \frac{\partial \text{Tr}({\sigma }^{e})}{\partial \sigma })`

:math:`\frac{\partial \mathrm{\Delta p}}{\partial p}=0`

**A1.3** **Case of the projection at the top of the cone**

The principle of analytical resolution consists in determining the effective constraints such as the projection of the elastic stresses on the criterion.

There may not be a solution.

If condition :math:`\mathrm{\Delta p}\le \frac{{\sigma }_{\text{eq}}^{e}}{\mathrm{3\mu }}` is not met, the effective constraints must be found by projection at the top of cone :math:`\mathrm{\Delta p}=\frac{{\sigma }_{\text{eq}}^{e}}{\mathrm{3\mu }}`.

In this case, we get:

:math:`\frac{\partial \mathrm{\Delta p}}{\partial \Phi }=\frac{1}{\mathrm{3\mu }}\cdot (\frac{\partial {\sigma }_{\text{eq}}^{e}}{\partial \Phi }-\mathrm{\Delta p}\cdot \frac{\partial \mathrm{3\mu }}{\partial \Phi })`

:math:`\frac{\partial \mathrm{\Delta p}}{\partial \sigma }=\frac{1}{\mathrm{3\mu }}\cdot \frac{\partial {\sigma }_{\text{eq}}^{e}}{\partial \sigma }`

:math:`\frac{\partial \mathrm{\Delta p}}{\partial p}=0`