6. Bibliography

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

• CANO V: Instabilities and breakages in elasto-visco-plastic solids (1996).

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

• RICE JR: The localization of plastic deformations, in Theoretical and Applied Mechanics (1976).

• HILL R: A general theory of uniqueness and stability in elastic-plastic solids (1958).

• ORTIZ M: An analytical study of the localized failure modes of concrete (1987).

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

1. • • • • • • • • r7.01.16 Calculation of partial derivatives of \(\Delta p\)

A1.1 Calculation of the partial derivative of the plastic deformation increment in the case of linear work hardening

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

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

So:

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

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

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

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

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

So:

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

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

\(\frac{\partial \mathrm{\Delta p}}{\partial p}=0\)

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

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

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

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

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

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

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

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

\(\frac{\partial \mathrm{\Delta p}}{\partial p}=0\)

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

In this case, we get:

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

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

\(\frac{\partial \mathrm{\Delta p}}{\partial p}=0\)