Description of document versions ==================================== .. csv-table:: "**Version** **Aster**", "**Author (s)** **Organization (s)**", "**Description of changes**" "8.5", "C.Chavant, J.El-Gharib EDF -R&D/ AMA V.Gervais, CS", "Initial text" .. _Ref100118895: **Derivation of main stress** Let :math:`\sigma` be a symmetric tensor and :math:`{\sigma }_{d}` this tensor in the base that diagonalizes it. Let's denote by :math:`P(\sigma )` the passage matrix that diagonalizes the :math:`\mathrm{\sigma }`: :math:`\sigma \text{:}\sigma \text{=}P(\sigma )\text{.}{\sigma }_{d}\text{.}\tilde{P}(\sigma )` tensor. In index writing, we adopt the following convention for raster entries: .. image:: images/10000234000069D500002972DB170B0A78C3E88D.svg :width: 67 :height: 26 .. _RefImage_10000234000069D500002972DB170B0A78C3E88D.svg: so that the matrix product is written: :math:`{(A\text{.}B)}_{j}^{i}\text{=}{A}_{m}^{i}{B}_{j}^{m}` with the rule of summation of repeated indices. So we have the relationship: :math:`\frac{\partial {\sigma }_{d}}{\partial {\sigma }_{j}^{i}}\text{=}\tilde{P}(\sigma )\text{.}\frac{\partial \sigma }{\partial {\sigma }_{j}^{i}}\text{.}P(\sigma )` :math:`\frac{\partial {\sigma }_{{d}_{k}}}{\partial {\sigma }_{j}^{i}}={P}_{k}^{i}{P}_{k}^{j}` or in index form*without index indication* :math:`k` Demonstration: In what follows, we'll note :math:`\mathrm{\sigma }` any component of the :math:`\sigma` tensor without specifying the indices when they don't play a role. We have :math:`\sigma \text{=}P(\sigma )\text{.}{\sigma }_{d}\text{.}\tilde{P}(\sigma )`, and as a result: 1. :math:`\frac{\partial {\sigma }_{d}}{\partial \sigma }\text{=}\tilde{P}(\sigma )\text{.}\frac{\partial \sigma }{\partial \sigma }\text{.}P(\sigma )+\frac{\partial \tilde{P}(\sigma )}{\partial \sigma }\text{.}\sigma \text{.}P(\sigma )+\tilde{P}(\sigma )\text{.}\sigma \text{.}\frac{P(\sigma )}{\partial \sigma }` 2.. By reporting the :math:`\sigma \text{=}P(\sigma )\text{.}{\sigma }_{d}\text{.}\tilde{P}(\sigma )` equality in the last two terms, we get: 1. :math:`\frac{\partial {\sigma }_{d}}{\partial \sigma }\text{=}\tilde{P}(\sigma )\text{.}\frac{\partial \sigma }{\partial \sigma }\text{.}P(\sigma )+\frac{\partial P(\sigma )}{\partial \sigma }\text{.}\tilde{P}(\sigma )\text{.}{\sigma }_{d}\text{.}P(\sigma )\text{.}\tilde{P}(\sigma )+\tilde{P}(\sigma )\text{.}P(\sigma )\text{.}{\sigma }_{d}\text{.}\tilde{P}(\sigma )\text{.}\frac{\partial P(\sigma )}{\partial \sigma }` that is to say: 1. :math:`\frac{\partial {\sigma }_{d}}{\partial \sigma }=\tilde{P}(\sigma )\text{.}\frac{\partial \sigma }{\partial \sigma }\text{.}P(\sigma )+\frac{\partial \tilde{P}(\sigma )}{\partial \sigma }\text{.}P(\sigma )\text{.}{\sigma }_{d}+{\sigma }_{d}\text{.}\tilde{P}(\sigma )\text{.}\frac{\partial P(\sigma )}{\partial \sigma }` In matrix writing, this is written: :math:`\frac{\partial {\sigma }_{{}_{d}}^{i}}{\partial \sigma }\text{=}{(\tilde{P}(\sigma )\text{.}\frac{\partial \sigma }{\partial \sigma }\text{.}P(\sigma )+\frac{\partial \tilde{P}(\sigma )}{\partial \sigma }\text{.}P(\sigma )\text{.}{\sigma }_{d}+{\sigma }_{d}\text{.}\tilde{P}(\sigma )\text{.}\frac{\partial P(\sigma )}{\partial \sigma })}_{i}^{i}` Let's show that the sum of the last two terms in this expression is zero: .. _Ref96362860: * :math:`{(\frac{\partial \tilde{P}(\sigma )}{\partial \sigma }\text{.}P(\sigma )\text{.}{\sigma }_{d}+{\sigma }_{d}\text{.}\tilde{P}(\sigma )\text{.}\frac{\partial P(\sigma )}{\partial \sigma })}_{j}^{i}=\partial {\tilde{P}}_{m}^{i}\text{.}{P}_{l}^{m}\text{.}{\sigma }_{{d}_{j}^{l}}+{\sigma }_{{d}_{m}^{i}}\text{.}{\tilde{P}}_{l}^{m}\text{.}\partial {P}_{j}^{l}` where :math:`\partial a` refers to :math:`\frac{\partial a}{\partial \mathrm{\sigma }}` in order to simplify the writing. We then write that :math:`i=j` and that only the terms :math:`{\sigma }_{{d}_{p}}^{p}` are non-zero. We get: :math:`\begin{array}{}{(\frac{\partial \tilde{P}(\sigma )}{\partial \sigma }\text{.}P(\sigma )\text{.}{\sigma }_{d}+{\sigma }_{d}\text{.}\tilde{P}(\sigma )\text{.}\frac{\partial P(\sigma )}{\partial \sigma })}_{i}^{i}\text{=}\partial {\tilde{P}}_{m}^{i}\text{.}{P}_{i}^{m}\text{.}{\sigma }_{{d}_{i}^{i}}+{\sigma }_{{d}_{i}^{i}}\text{.}{\tilde{P}}_{l}^{i}\text{.}\partial {P}_{i}^{l}\\ \text{=}(\partial {\tilde{P}}_{m}^{i}\text{.}{P}_{i}^{m}+{\tilde{P}}_{m}^{i}\text{.}\partial {P}_{i}^{m}){\sigma }_{{d}_{i}^{i}}\underline{\text{sans}\text{sommation}\text{sur}\text{l'indice}i}\end{array}` which is clearly rubbish since :math:`\tilde{P}\text{.}P\text{=}I`. Hence finally :math:`\frac{\partial {\mathrm{\sigma }}_{{d}_{k}}}{\partial {\mathrm{\sigma }}_{j}^{i}}=\frac{\partial {\mathrm{\sigma }}_{{d}_{k}^{k}}}{\partial {\mathrm{\sigma }}_{j}^{i}}={P}_{k}^{i}{P}_{k}^{j}`. **Derived from the function** :math:`\stackrel{ˉ}{F}(\mathrm{\Delta \lambda })` :math:`\begin{array}{}\frac{\partial h}{\partial \Delta \gamma }\text{=}\frac{\partial }{\partial \Delta \gamma }(\frac{3\text{+}\text{sin}\phi ({\gamma }^{\text{-}}\text{+}\Delta \gamma )}{3(1\text{+}\text{sin}\phi ({\gamma }^{\text{-}}\text{+}\Delta \gamma ))})\text{=}\{\begin{array}{cc}\text{-}\frac{2\frac{\partial \phi }{\partial \gamma }\text{cos}\phi (\gamma )}{3(1\text{+}\text{sin}\phi (\gamma ){)}^{2}}& \text{si}\gamma <{\gamma }^{\text{res}}\\ 0& \text{sinon}\end{array}\text{}\\ \frac{\partial g}{\partial \Delta \gamma }\text{=}\frac{\partial }{\partial \Delta \gamma }\left[\frac{3\text{+}\text{sin}\phi ({\gamma }^{\text{-}}\text{+}\Delta \gamma )}{3(1\text{+}\text{sin}\phi ({\gamma }^{\text{-}}\text{+}\Delta \gamma ))}(\frac{\mathrm{6K}\text{sin}\phi ({\gamma }^{\text{-}}\text{+}\Delta \gamma )}{3\text{+}\text{sin}\phi ({\gamma }^{\text{-}}\text{+}\Delta \gamma )}\text{+}\frac{3{m}_{3}^{e}}{{\sigma }_{\text{eq}}^{e}})\right]\\ \text{}\text{=}\frac{\mathrm{6K}\frac{\partial \phi }{\partial \gamma }\text{cos}\phi (\gamma )}{(3\text{+}\text{sin}\phi (\gamma ))(1\text{+}\text{sin}\phi (\gamma ))}\text{-}\frac{2\frac{\partial \phi }{\partial \gamma }\text{cos}\phi (\gamma )}{3(1\text{+}\text{sin}\phi (\gamma ){)}^{2}}(\frac{\mathrm{6K}\text{sin}\phi (\gamma )}{3\text{+}\text{sin}\phi (\gamma )}\text{+}\frac{3{m}_{3}^{e}}{{\sigma }_{\text{eq}}^{e}})\end{array}` :math:`\begin{array}{}\frac{\partial \stackrel{ˉ}{F}}{\partial \Delta \gamma }\text{=}2\left[\text{-}({s}_{3}^{e}\text{-}{s}_{1}^{e})\frac{3\mu }{{\sigma }_{\text{eq}}^{e}}(\frac{\partial h}{\partial \Delta \gamma }\Delta \gamma \text{+}h)\text{-}\frac{\partial b}{\partial \Delta \gamma }(1\text{-}\frac{1}{{\sigma }_{3}^{b\text{-}d}}\left[{s}_{3}^{e}\text{+}\frac{{I}_{1}^{e}}{3}\text{-}g\Delta \gamma \right])\text{-}\frac{b}{{\sigma }_{3}^{b\text{-}d}}(\frac{\partial g}{\partial \Delta \gamma }\Delta \gamma \text{+}g)\right]\\ \times \left[({s}_{3}^{e}\text{-}{s}_{1}^{e})\left[1\text{-}\frac{3\mu }{{\sigma }_{\text{eq}}^{e}}h\Delta \gamma \right]\text{-}b\left[1\text{-}\frac{1}{{\sigma }_{3}^{b\text{-}d}}({s}_{3}^{e}\text{+}\frac{{I}_{1}^{e}}{3}\text{-}g\Delta \gamma )\right]\right]\\ \text{-}(\frac{\partial (S{\sigma }_{c}^{2})}{\partial \Delta \gamma }\text{-}\frac{\partial (m{\sigma }_{c})}{\partial \Delta \gamma }({s}_{3}^{e}\text{+}\frac{{I}_{1}^{e}}{3}\text{-}g\Delta \gamma )\text{+}{\sigma }_{c}m(\frac{\partial g}{\partial \Delta \gamma }\Delta \gamma \text{+}g))\end{array}` .. _Ref121545567: **Calculating derivatives** :math:`\frac{\partial {\sigma }_{p}}{\partial {p}_{g}}` **and** :math:`\frac{\partial {\sigma }_{p}}{\partial {p}_{c}}` * LIQU_SATU (PRE1 = :math:`{p}_{\mathrm{lq}}`): :math:`\frac{\partial {\sigma }_{p}}{\partial {p}_{c}}\text{=}\text{-}\frac{\partial {\sigma }_{p}}{\partial {p}_{\text{lq}}}\text{=}\mathrm{bS}` * LIQU_GAZ_ATM (PRE1 =- :math:`{p}_{\mathrm{lq}}`): :math:`\frac{\partial {\sigma }_{p}}{\partial {p}_{c}}\text{=}\text{-}\frac{\partial {\sigma }_{p}}{\partial {p}_{\text{lq}}}\text{=}\mathrm{bS}` * GAZ (PRE1 = :math:`{p}_{g}`): :math:`\frac{\partial {\sigma }_{p}}{\partial {p}_{g}}\text{=}\text{-}b(1\text{-}S)` * LIQU_VAPE_GAZ (PRE1 = :math:`{p}_{c}`, PRE2 = :math:`{p}_{g}`): :math:`\frac{\partial {\sigma }_{p}}{\partial {p}_{g}}\text{=}\text{-}b`, :math:`\frac{\partial {\sigma }_{p}}{\partial {p}_{c}}\text{=}\mathrm{bS}` * LIQU_GAZ (PRE1 = :math:`{p}_{c}`, PRE2 = :math:`{p}_{g}`): :math:`\frac{\partial {\sigma }_{p}}{\partial {p}_{g}}\text{=}\text{-}b`, :math:`\frac{\partial {\sigma }_{p}}{\partial {p}_{c}}\text{=}\mathrm{bS}` * LIQU_VAPE (PRE1 = :math:`{p}_{\mathrm{lq}}`): :math:`\frac{\partial {\sigma }_{p}}{\partial {p}_{c}}\text{=}\text{-}\frac{\partial {\sigma }_{p}}{\partial {p}_{\text{lq}}}\text{=}\mathrm{bS}` * LIQU_AD_GAZ_VAPE (PRE1 = .. image:: images/Object_314.svg :width: 67 :height: 26 .. _RefImage_Object_314.svg: , PRE2 = .. image:: images/Object_315.svg :width: 67 :height: 26 .. _RefImage_Object_315.svg: ): :math:`\frac{\partial {\sigma }_{p}}{\partial {p}_{g}}\text{=}\text{-}b`, :math:`\frac{\partial {\sigma }_{p}}{\partial {p}_{c}}\text{=}\mathrm{bS}`