8. Appendices: Jacobian matrix terms#
Evaluating terms for \(\frac{d{({R}_{1})}_{\mathit{ij}}}{d({(\Delta {Y}_{1})}_{\mathit{mn}})}\):
\(\begin{array}{c}\frac{d{({R}_{1})}_{\mathit{ij}}}{d({(\Delta {Y}_{1})}_{\mathit{mn}})}\mathrm{=}{I}_{\mathit{ilkl}}\mathrm{-}\frac{\mathrm{\partial }{C}_{\mathit{ijkl}}^{e}}{\mathrm{\partial }{\sigma }_{\mathit{mn}}}\mathrm{:}\left[\Delta {\epsilon }_{\mathit{kl}}\mathrm{-}\Delta \lambda \mathrm{\cdot }{G}_{\mathit{kl}}^{p}\mathrm{-}\Delta {\epsilon }_{\mathit{kl}}^{\mathit{vp}}\right]+\Delta \lambda {C}_{\mathit{ijkl}}^{e}\mathrm{:}\frac{\mathrm{\partial }{G}_{\mathit{kl}}^{p}}{\mathrm{\partial }{\sigma }_{\mathit{mn}}}+\\ {C}_{\mathit{ijkl}}^{e}\mathrm{:}{G}_{\mathit{kl}}^{\mathit{vp}}\mathrm{\otimes }\frac{\mathrm{\partial }({\mathrm{\langle }\phi ({f}^{\mathit{vp}})\mathrm{\rangle }}^{\text{+}})}{\mathrm{\partial }{\sigma }_{\mathit{mn}}}\Delta t+{\mathrm{\langle }\phi ({f}^{\mathit{vp}})\mathrm{\rangle }}^{\text{+}}\mathrm{\cdot }\Delta t\mathrm{\cdot }{C}_{\mathit{ijkl}}^{e}\mathrm{:}\frac{\mathrm{\partial }{G}_{\mathit{kl}}^{\mathit{vp}}}{\mathrm{\partial }{\sigma }_{\mathit{mn}}}\end{array}\)
\(\frac{\mathrm{\partial }{C}_{\mathit{ijkl}}^{e}}{\mathrm{\partial }{\sigma }_{\mathit{mn}}}\mathrm{=}\frac{{n}_{\mathit{elas}}}{{I}_{1}}\mathrm{\cdot }{C}_{\mathit{ijkl}}^{e}\mathrm{\otimes }{\delta }_{\mathit{mn}}\)
\(\frac{\mathrm{\partial }{G}_{\mathit{ij}}^{p}}{\mathrm{\partial }{\sigma }_{\mathit{kl}}}\mathrm{=}\frac{\mathrm{\partial }}{{\sigma }_{\mathit{kl}}}(\frac{\mathrm{\partial }{f}^{p}}{\mathrm{\partial }{\sigma }_{\mathit{ij}}})\mathrm{-}(\frac{\mathrm{\partial }}{{\sigma }_{\mathit{kl}}}(\frac{\mathrm{\partial }{f}^{p}}{\mathrm{\partial }{\sigma }_{\mathit{ij}}})\mathrm{:}{n}_{\mathit{mn}})\mathrm{\otimes }{n}_{\mathit{ij}}\mathrm{-}(\frac{\mathrm{\partial }{f}^{p}}{\mathrm{\partial }{\sigma }_{\mathit{mn}}}\mathrm{:}\frac{\mathrm{\partial }{n}_{\mathit{mn}}}{\mathrm{\partial }{\sigma }_{\mathit{kl}}})\mathrm{\otimes }{n}_{\mathit{ij}}\mathrm{-}(\frac{\mathrm{\partial }{f}^{p}}{\mathrm{\partial }{\sigma }_{\mathit{mn}}}\mathrm{:}{n}_{\mathit{mn}})\mathrm{\cdot }\frac{\mathrm{\partial }{n}_{\mathit{ij}}}{\mathrm{\partial }{\sigma }_{\mathit{kl}}}\)
with \(\begin{array}{}\frac{\partial {f}^{d}}{\partial {\sigma }_{\text{ij}}}=\\ \frac{\partial ({s}_{\text{II}}H(\theta ))}{\partial {\sigma }_{\text{ij}}}-{a}^{d}({\xi }_{p}){\sigma }_{c}{H}_{0}^{c}{\left[{A}^{d}({\xi }_{p}){s}_{\text{II}}H(\theta )+{B}^{d}({\xi }_{p}){I}_{1}+{D}^{d}({\xi }_{p})\right]}^{{a}^{d}({\xi }_{p})-1}\\ ({A}^{d}({\xi }_{p})\frac{\partial ({s}_{\text{II}}H(\theta ))}{\partial {\sigma }_{\text{ij}}}+{B}^{d}({\xi }_{p}){I}_{d})\end{array}\) either
\(\begin{array}{c}\frac{\mathrm{\partial }}{{\sigma }_{\mathit{kl}}}(\frac{\mathrm{\partial }{f}^{p}}{\mathrm{\partial }{\sigma }_{\mathit{ij}}})\mathrm{=}\frac{\mathrm{\partial }}{\mathrm{\partial }{\sigma }_{\mathit{kl}}}(\frac{\mathrm{\partial }{s}_{\mathit{II}}H(\theta )}{\mathrm{\partial }{\sigma }_{\mathit{ij}}})\mathrm{-}{a}^{d}({\xi }^{p}){\sigma }_{c}{H}_{0}^{c}{\left[{A}^{d}({\xi }^{p}){s}_{\mathit{II}}H(\theta )+{B}^{d}({\xi }^{p}){I}_{1}+{D}^{d}({\xi }^{p})\right]}^{{a}^{d}({\xi }^{p})\mathrm{-}1}\\ \mathrm{\cdot }{A}^{d}({\xi }^{p})\frac{\mathrm{\partial }}{\mathrm{\partial }{\sigma }_{\mathit{kl}}}(\frac{\mathrm{\partial }{s}_{\mathit{II}}H(\theta )}{\mathrm{\partial }{\sigma }_{\mathit{ij}}})\mathrm{-}{a}^{d}({\xi }^{p})({a}^{d}({\xi }^{p})\mathrm{-}1){\sigma }_{c}{H}_{0}^{c}{({A}^{d}({\xi }^{p}){s}_{\mathit{II}}H(\theta )+{B}^{d}({\xi }^{p}){I}_{1}+{D}^{d}({\xi }^{p}))}^{{a}^{d}({\xi }^{p})\mathrm{-}2}\\ ({A}^{d}({\xi }^{p})\frac{\mathrm{\partial }{s}_{\mathit{II}}H(\theta )}{\mathrm{\partial }{\sigma }_{\mathit{ij}}}+{B}^{d}({\xi }^{p}){\delta }_{\mathit{ij}})\mathrm{\otimes }({A}^{d}({\xi }^{p})\frac{\mathrm{\partial }{s}_{\mathit{II}}H(\theta )}{\mathrm{\partial }{\sigma }_{\mathit{kl}}}+{B}^{d}({\xi }^{p}){\delta }_{\mathit{kl}})\end{array}\)
Gold \(\frac{\mathrm{\partial }({s}_{\text{II}}H(\theta ))}{\mathrm{\partial }{\sigma }_{\text{ij}}}\mathrm{=}((\frac{{H}_{0}^{c}\mathrm{-}{H}_{0}^{e}}{{h}_{0}^{c}\mathrm{-}{h}_{0}^{e}})\frac{\mathrm{\partial }h(\theta )}{\mathrm{\partial }{s}_{\text{kl}}}{s}_{\text{II}}+H(\theta )\frac{{s}_{\text{kl}}}{{s}_{\text{II}}})\text{.}({\delta }_{\text{ik}}\text{.}{\delta }_{\text{jl}}\mathrm{-}\frac{1}{3}{\delta }_{\text{ij}}\text{.}{\delta }_{\text{kl}})\) from where
\(\begin{array}{c}\frac{\mathrm{\partial }}{\mathrm{\partial }{\sigma }_{\mathit{kl}}}(\frac{\mathrm{\partial }{s}_{\mathit{II}}H(\theta )}{\mathrm{\partial }{\sigma }_{\mathit{ij}}})\mathrm{=}(\frac{{H}_{0}^{c}\mathrm{-}{H}_{0}^{e}}{{h}_{0}^{c}\mathrm{-}{h}_{0}^{e}})\frac{\mathrm{\partial }}{\mathrm{\partial }{\sigma }_{\mathit{kl}}}(\frac{\mathrm{\partial }h(\theta )}{\mathrm{\partial }{s}_{\mathit{mn}}}){s}_{\mathit{II}}\frac{\mathrm{\partial }{s}_{\mathit{mn}}}{\mathrm{\partial }{\sigma }_{\mathit{ij}}}+(\frac{{H}_{0}^{c}\mathrm{-}{H}_{0}^{e}}{{h}_{0}^{c}\mathrm{-}{h}_{0}^{e}})\frac{\mathrm{\partial }h(\theta )}{\mathrm{\partial }{s}_{\mathit{mn}}}\frac{\mathrm{\partial }{s}_{\mathit{mn}}}{\mathrm{\partial }{\sigma }_{\mathit{ij}}}\frac{\mathrm{\partial }{s}_{\mathit{II}}}{\mathrm{\partial }{s}_{\mathit{pq}}}\frac{\mathrm{\partial }{s}_{\mathit{pq}}}{\mathrm{\partial }{\sigma }_{\mathit{kl}}}+\\ \frac{\mathrm{\partial }H(\theta )}{\mathrm{\partial }{\sigma }_{\mathit{kl}}}\frac{{s}_{\mathit{mn}}}{{s}_{\mathit{II}}}\frac{\mathrm{\partial }{s}_{\mathit{mn}}}{\mathrm{\partial }{\sigma }_{\mathit{ij}}}+\frac{H(\theta )}{{s}_{\mathit{II}}}\frac{\mathrm{\partial }{s}_{\mathit{mn}}}{\mathrm{\partial }{\sigma }_{\mathit{kl}}}\frac{\mathrm{\partial }{s}_{\mathit{mn}}}{\mathrm{\partial }{\sigma }_{\mathit{ij}}}\mathrm{-}\frac{H(\theta ){s}_{\mathit{mn}}}{{s}_{\mathit{II}}^{2}}\frac{\mathrm{\partial }{s}_{\mathit{II}}}{\mathrm{\partial }{s}_{\mathit{pq}}}\frac{\mathrm{\partial }{s}_{\mathit{pq}}}{\mathrm{\partial }{\sigma }_{\mathit{kl}}}\frac{\mathrm{\partial }{s}_{\mathit{mn}}}{\mathrm{\partial }{\sigma }_{\mathit{ij}}}\end{array}\)
We have more \(\frac{\mathrm{\partial }h(\theta )}{\mathrm{\partial }{s}_{\text{kl}}}\mathrm{=}\frac{\gamma \text{cos}(\mathrm{3\theta })}{\mathrm{6h}{(\theta )}^{5}}\frac{{\mathrm{3s}}_{\text{kl}}}{{s}_{\text{II}}^{2}}\mathrm{-}\frac{\gamma \sqrt{\text{54}}}{\mathrm{6h}{(\theta )}^{5}{s}_{\text{II}}^{3}}(\frac{\mathrm{\partial }\text{det}(\underline{\underline{s}})}{\mathrm{\partial }{s}_{\text{kl}}})\)
\(\frac{\mathrm{\partial }}{\mathrm{\partial }{\sigma }_{\mathit{kl}}}(\frac{\mathrm{\partial }h(\theta )}{\mathrm{\partial }{s}_{\mathit{mn}}})\mathrm{=}\frac{\mathrm{\partial }}{\mathrm{\partial }{s}_{\mathit{pq}}}(\frac{\mathrm{\partial }h(\theta )}{\mathrm{\partial }{s}_{\mathit{mn}}})\frac{\mathrm{\partial }{s}_{\mathit{pq}}}{\mathrm{\partial }{\sigma }_{\mathit{kl}}}\)
\(\begin{array}{c}\frac{\mathrm{\partial }}{\mathrm{\partial }{s}_{\mathit{pq}}}(\frac{\mathrm{\partial }h(\theta )}{\mathrm{\partial }{s}_{\mathit{mn}}})\mathrm{=}\frac{\sqrt{54}\gamma }{2{h}^{5}(\theta ){s}_{\mathit{II}}^{5}}{s}_{\mathit{mn}}\mathrm{\otimes }\frac{\mathrm{\partial }\mathit{det}({s}_{\mathit{ij}})}{\mathrm{\partial }{s}_{\mathit{pq}}}\mathrm{-}\frac{3\gamma \sqrt{54}\mathit{det}({s}_{\mathit{ij}})}{2{h}^{5}(\theta ){s}_{\mathit{II}}^{7}}{s}_{\mathit{mn}}\mathrm{\otimes }{s}_{\mathit{pq}}+\frac{\gamma \mathrm{cos}3\theta }{2{h}^{5}(\theta ){s}_{\mathit{II}}^{2}}{I}_{\mathit{mnpq}}\mathrm{-}\\ \frac{\gamma \mathrm{cos}3\theta }{{h}^{5}(\theta ){s}_{\mathit{II}}^{4}}{s}_{\mathit{mn}}\mathrm{\otimes }{s}_{\mathit{pq}}\mathrm{-}\frac{5\gamma \mathrm{cos}3\theta }{2{h}^{6}(\theta ){s}_{\mathit{II}}^{2}}{s}_{\mathit{mn}}\mathrm{\otimes }\frac{\mathrm{\partial }h(\theta )}{\mathrm{\partial }{s}_{\mathit{pq}}}+\frac{5\gamma \sqrt{54}}{6{s}_{\mathit{II}}^{3}{h}^{6}(\theta )}\frac{\mathrm{\partial }\mathit{det}({s}_{\mathit{ij}})}{\mathrm{\partial }{s}_{\mathit{mn}}}\mathrm{\otimes }\frac{\mathrm{\partial }h(\theta )}{\mathrm{\partial }{s}_{\mathit{pq}}}+\\ \frac{\gamma \sqrt{54}}{2{h}^{5}(\theta ){s}_{\mathit{II}}^{5}}\frac{\mathrm{\partial }\mathit{det}({s}_{\mathit{ij}})}{\mathrm{\partial }{s}_{\mathit{mn}}}\mathrm{\otimes }{s}_{\mathit{pq}}\mathrm{-}\frac{\gamma \sqrt{54}}{6{h}^{5}(\theta ){s}_{\mathit{II}}^{3}}\frac{{\mathrm{\partial }}^{2}\mathit{det}({s}_{\mathit{ij}})}{\mathrm{\partial }{s}_{\mathit{mn}}\mathrm{\partial }{s}_{\mathit{pq}}}\end{array}\)
with
\(\frac{{\mathrm{\partial }}^{2}\mathit{det}({s}_{\mathit{ij}})}{\mathrm{\partial }{s}_{\mathit{mn}}\mathrm{\partial }{s}_{\mathit{pq}}}\mathrm{=}\left[\begin{array}{cccccc}0& {s}_{33}& {s}_{22}& 0& 0& \mathrm{-}\sqrt{2}{s}_{23}\\ {s}_{33}& 0& {s}_{11}& 0& \mathrm{-}\sqrt{2}{s}_{13}& 0\\ {s}_{22}& {s}_{11}& 0& \mathrm{-}\sqrt{2}{s}_{12}& 0& 0\\ 0& 0& \mathrm{-}\sqrt{2}{s}_{12}& \mathrm{-}{s}_{33}& {s}_{13}& {s}_{23}\\ 0& \mathrm{-}\sqrt{2}{s}_{13}& 0& {s}_{13}& \mathrm{-}{s}_{11}& {s}_{12}\\ \mathrm{-}\sqrt{2}{s}_{23}& 0& 0& {s}_{23}& {s}_{12}& \mathrm{-}{s}_{22}\end{array}\right]\)
Remember that \({n}_{\text{ij}}\mathrm{=}\frac{{\beta }^{\text{'}}\frac{{s}_{\text{ij}}}{{s}_{\text{II}}}\mathrm{-}{\delta }_{\text{ij}}}{\sqrt{{\beta }^{\text{'}2}+3}}\) and \({\beta }^{\text{'}}\mathrm{=}\mathrm{-}\frac{2\sqrt{6}\text{sin}(\Psi )}{3\mathrm{-}\text{sin}(\Psi )}\)
\(\frac{\mathrm{\partial }{n}_{\mathit{ij}}}{\mathrm{\partial }{\sigma }_{\mathit{kl}}}\mathrm{=}\frac{\left[\frac{\mathrm{\partial }{\beta }^{\text{'}}}{\mathrm{\partial }{\sigma }_{\mathit{kl}}}\frac{{s}_{\mathit{ij}}}{{s}_{\mathit{II}}}+\frac{{\beta }^{\text{'}}}{{s}_{\mathit{II}}}\frac{\mathrm{\partial }{s}_{\mathit{ij}}}{\mathrm{\partial }{\sigma }_{\mathit{kl}}}\mathrm{-}\frac{{\beta }^{\text{'}}{s}_{\mathit{ij}}}{{s}_{\mathit{II}}^{2}}\frac{\mathrm{\partial }{s}_{\mathit{II}}}{\mathrm{\partial }{\sigma }_{\mathit{kl}}}\right]({\beta }^{\text{'2}}+3)\mathrm{-}{\beta }^{\text{'}}({\beta }^{\text{'}}\frac{{s}_{\mathit{ij}}}{{s}_{\mathit{II}}}\mathrm{-}{\delta }_{\mathit{ij}})\mathrm{\otimes }\frac{\mathrm{\partial }{\beta }^{\text{'}}}{\mathrm{\partial }{\sigma }_{\mathit{kl}}}}{({\beta }^{\text{'2}}+3)\sqrt{{\beta }^{\text{'2}}+3}}\)
\(\frac{\mathrm{\partial }{\beta }^{\text{'}}}{\mathrm{\partial }{\sigma }_{\mathit{kl}}}\mathrm{=}\frac{\mathrm{\partial }{\beta }^{\text{'}}}{\mathrm{\partial }{s}_{\mathit{mn}}}\frac{\mathrm{\partial }{s}_{\mathit{mn}}}{\mathrm{\partial }{\sigma }_{\mathit{kl}}}+\frac{\mathrm{\partial }{\beta }^{\text{'}}}{\mathrm{\partial }{I}_{1}}\frac{\mathrm{\partial }{I}_{1}}{\mathrm{\partial }{\sigma }_{\mathit{kl}}}\)
\(\frac{\mathrm{\partial }{\beta }^{\text{'}}}{\mathrm{\partial }{s}_{\mathit{mn}}}\mathrm{=}\frac{\mathrm{-}6\sqrt{6}}{{(3\mathrm{-}\mathrm{sin}\psi )}^{2}}\frac{\mathrm{\partial }\mathrm{sin}\psi }{\mathrm{\partial }{s}_{\mathit{mn}}}\) and \(\frac{\mathrm{\partial }{\beta }^{\text{'}}}{\mathrm{\partial }{I}_{1}}\mathrm{=}\frac{\mathrm{-}6\sqrt{6}}{{(3\mathrm{-}\mathrm{sin}\psi )}^{2}}\frac{\mathrm{\partial }\mathrm{sin}\psi }{\mathrm{\partial }{I}_{1}}\)
Distinction of expressions between pre-peak or visco-plastic and post-peak behavior
Expression of derivatives in pre-peak or visco-plastic
\(\begin{array}{ccc}\frac{\mathrm{\partial }\mathrm{sin}\psi }{\mathrm{\partial }{s}_{\mathit{mn}}}& \mathrm{=}& \frac{\mathrm{\partial }\mathrm{sin}\psi }{\mathrm{\partial }{\sigma }_{\mathit{max}}}\frac{\mathrm{\partial }{\sigma }_{\mathit{max}}}{\mathrm{\partial }{s}_{\mathit{mn}}}+\frac{\mathrm{\partial }\mathrm{sin}\psi }{\mathrm{\partial }{\sigma }_{\text{lim}}}\frac{\mathrm{\partial }{\sigma }_{\text{lim}}}{\mathrm{\partial }{s}_{\mathit{mn}}}\\ & \mathrm{=}& {\mu }_{\mathrm{0,}v}(\frac{(1+{\xi }_{\text{0,v}}){\sigma }_{\text{lim}}}{{({\xi }_{\text{0,v}}{\sigma }_{\mathit{max}}+{\sigma }_{\text{lim}})}^{2}}\frac{\mathrm{\partial }{\sigma }_{\mathit{max}}}{\mathrm{\partial }{s}_{\mathit{mn}}}\mathrm{-}\frac{(1+{\xi }_{\text{0,v}}){\sigma }_{\text{max}}}{{({\xi }_{\text{0,v}}{\sigma }_{\mathit{max}}+{\sigma }_{\text{lim}})}^{2}}\frac{\mathrm{\partial }{\sigma }_{\text{lim}}}{\mathrm{\partial }{s}_{\mathit{mn}}})\end{array}\)
and
\(\begin{array}{ccc}\frac{\mathrm{\partial }\mathrm{sin}\psi }{\mathrm{\partial }{I}_{1}}& \mathrm{=}& \frac{\mathrm{\partial }\mathrm{sin}\psi }{\mathrm{\partial }{\sigma }_{\mathit{max}}}\frac{\mathrm{\partial }{\sigma }_{\mathit{max}}}{\mathrm{\partial }{I}_{1}}+\frac{\mathrm{\partial }\mathrm{sin}\psi }{\mathrm{\partial }{\sigma }_{\text{lim}}}\frac{\mathrm{\partial }{\sigma }_{\text{lim}}}{\mathrm{\partial }{I}_{1}}\\ & \mathrm{=}& \frac{{\mu }_{\mathrm{0,}v}}{3}(\frac{(1+{\xi }_{\text{0,v}}){\sigma }_{\text{lim}}}{{({\xi }_{\text{0,v}}{\sigma }_{\mathit{max}}+{\sigma }_{\text{lim}})}^{2}}\mathrm{-}\frac{(1+{m}_{\text{v,max}})(1+{\xi }_{\text{0,v}}){\sigma }_{\text{max}}}{{({\xi }_{\text{0,v}}{\sigma }_{\mathit{max}}+{\sigma }_{\text{lim}})}^{2}})\end{array}\)
\(\frac{\mathrm{\partial }{\sigma }_{\mathit{max}}}{\mathrm{\partial }{s}_{\mathit{mn}}}\mathrm{=}\frac{1}{\sqrt{6}}\left[\frac{{s}_{\mathit{II}}}{{H}_{0}^{c}\mathrm{-}{H}_{0}^{e}}\frac{\mathrm{\partial }H(\theta )}{\mathrm{\partial }{s}_{\mathit{mn}}}+(\frac{3}{2}+\frac{2H(\theta )\mathrm{-}({H}_{0}^{c}+{H}_{0}^{e})}{2({H}_{0}^{c}\mathrm{-}{H}_{0}^{e})})\frac{{s}_{\mathit{mn}}}{{s}_{\mathit{II}}}\right]\) and \(\frac{\mathrm{\partial }{\sigma }_{\mathit{max}}}{\mathrm{\partial }{I}_{1}}\mathrm{=}\frac{1}{3}\)
\(\frac{\mathrm{\partial }{\sigma }_{\text{lim}}}{\mathrm{\partial }{s}_{\mathit{mn}}}\mathrm{=}\frac{(1+{m}_{\text{v,max}})}{\sqrt{6}}\left[\frac{{s}_{\mathit{II}}}{{H}_{0}^{c}\mathrm{-}{H}_{0}^{e}}\frac{\mathrm{\partial }H(\theta )}{\mathrm{\partial }{s}_{\mathit{mn}}}\mathrm{-}(\frac{3}{2}\mathrm{-}\frac{2H(\theta )\mathrm{-}({H}_{0}^{c}+{H}_{0}^{e})}{2({H}_{0}^{c}\mathrm{-}{H}_{0}^{e})})\frac{{s}_{\mathit{mn}}}{{s}_{\mathit{II}}}\right]\) and \(\frac{\mathrm{\partial }{\sigma }_{\text{lim}}}{\mathrm{\partial }{I}_{1}}\mathrm{=}\frac{(1+{m}_{\text{v,max}})}{3}\)
Expression of post-peak derivatives for the law of plastic dilatance
\(\frac{\mathrm{\partial }\mathrm{sin}\psi }{\mathrm{\partial }{s}_{\mathit{mn}}}\mathrm{=}\frac{\mathrm{\partial }\mathrm{sin}\psi }{\mathrm{\partial }\alpha }(\frac{\mathrm{\partial }\alpha }{\mathrm{\partial }{\sigma }_{\mathit{min}}}\frac{\mathrm{\partial }{\sigma }_{\mathit{min}}}{\mathrm{\partial }{s}_{\mathit{mn}}}+\frac{\mathrm{\partial }\alpha }{\mathrm{\partial }{\sigma }_{\mathit{max}}}\frac{\mathrm{\partial }{\sigma }_{\mathit{max}}}{\mathrm{\partial }{s}_{\mathit{mn}}})\)
and
\(\frac{\mathrm{\partial }\mathrm{sin}\psi }{\mathrm{\partial }{I}_{1}}\mathrm{=}\frac{\mathrm{\partial }\mathrm{sin}\psi }{\mathrm{\partial }\alpha }(\frac{\mathrm{\partial }\alpha }{\mathrm{\partial }{\sigma }_{\mathit{min}}}\frac{\mathrm{\partial }{\sigma }_{\mathit{min}}}{\mathrm{\partial }{I}_{1}}+\frac{\mathrm{\partial }\alpha }{\mathrm{\partial }{\sigma }_{\mathit{max}}}\frac{\mathrm{\partial }{\sigma }_{\mathit{max}}}{\mathrm{\partial }{I}_{1}})\)
\(\frac{\mathrm{\partial }\mathrm{sin}\psi }{\mathrm{\partial }{s}_{\mathit{mn}}}\mathrm{=}{\mu }_{1}\frac{(1+{\xi }_{1}){\alpha }_{\mathit{res}}}{{({\xi }_{1}\alpha +{\alpha }_{\mathit{res}})}^{2}}(\mathrm{-}\frac{{\sigma }_{\mathit{max}}+\tilde{\sigma }}{{({\sigma }_{\mathit{min}}+\tilde{\sigma })}^{2}}\frac{\mathrm{\partial }{\sigma }_{\mathit{min}}}{\mathrm{\partial }{s}_{\mathit{mn}}}+\frac{1}{{\sigma }_{\mathit{min}}+\tilde{\sigma }}\frac{\mathrm{\partial }{\sigma }_{\mathit{max}}}{\mathrm{\partial }{s}_{\mathit{mn}}})\)
and
\(\frac{\mathrm{\partial }\mathrm{sin}\psi }{\mathrm{\partial }{I}_{1}}\mathrm{=}\frac{{\mu }_{1}{\alpha }_{\mathit{res}}(1+{\xi }_{1})({\sigma }_{\mathit{min}}\mathrm{-}{\sigma }_{\mathit{max}})}{3{({\xi }_{1}\alpha +{\alpha }_{\mathit{res}})}^{2}{({\sigma }_{\mathit{min}}+\tilde{\sigma })}^{2}}\)
\(\frac{\mathrm{\partial }{\sigma }_{\mathit{min}}}{\mathrm{\partial }{s}_{\mathit{mn}}}\mathrm{=}\frac{1}{\sqrt{6}}\left[\frac{{s}_{\mathit{II}}}{{H}_{0}^{c}\mathrm{-}{H}_{0}^{e}}\frac{\mathrm{\partial }H(\theta )}{\mathrm{\partial }{s}_{\mathit{mn}}}\mathrm{-}(\frac{3}{2}\mathrm{-}\frac{2H(\theta )\mathrm{-}({H}_{0}^{c}+{H}_{0}^{e})}{2({H}_{0}^{c}\mathrm{-}{H}_{0}^{e})})\frac{{s}_{\mathit{mn}}}{{s}_{\mathit{II}}}\right]\) and \(\frac{\mathrm{\partial }{\sigma }_{\mathit{min}}}{\mathrm{\partial }{I}_{1}}\mathrm{=}\frac{1}{3}\)
The evaluation of the term \(\frac{\mathrm{\partial }{G}_{\mathit{ij}}^{\mathit{vp}}}{\mathrm{\partial }{\sigma }_{\mathit{kl}}}\) is the same as that of \(\frac{\mathrm{\partial }{G}_{\mathit{ij}}^{p}}{\mathrm{\partial }{\sigma }_{\mathit{kl}}}\) except for the distinctions already specified above.
We are now approaching the evaluation of the term \(\frac{\mathrm{\partial }({\mathrm{\langle }\phi ({f}^{\mathit{vp}})\mathrm{\rangle }}^{\text{+}})}{\mathrm{\partial }{\sigma }_{\mathit{mn}}}\) with \(\Phi ({f}^{\text{vp}})={A}_{v}{(\frac{{f}^{\text{vp}}}{\text{Pa}})}^{{n}_{v}}\).
We then obtain:
\(\frac{\mathrm{\partial }({\mathrm{\langle }\phi ({f}^{\mathit{vp}})\mathrm{\rangle }}^{\text{+}})}{\mathrm{\partial }{\sigma }_{\mathit{mn}}}\mathrm{=}\frac{\mathrm{\partial }{f}^{\mathit{vp}}}{\mathrm{\partial }{\sigma }_{\mathit{mn}}}\mathrm{\cdot }\frac{{A}_{v}{n}_{v}}{{P}_{\text{atm}}}{(\frac{{f}^{\mathit{vp}}}{{P}_{\text{atm}}})}^{{n}_{v}\mathrm{-}1}\)
Evaluating terms for \(\frac{d{({R}_{1})}_{\mathit{ij}}}{d(\Delta {Y}_{3})}\mathrm{=}\Delta \lambda \mathrm{\cdot }{C}_{\mathit{ijkl}}^{e}\mathrm{:}\frac{\mathrm{\partial }{G}_{\mathit{kl}}^{p}}{\mathrm{\partial }{\xi }^{p}}\):
\(\frac{\mathrm{\partial }{G}_{\mathit{ij}}^{p}}{\mathrm{\partial }{\xi }^{p}}\mathrm{=}\frac{\mathrm{\partial }}{{\xi }^{p}}(\frac{\mathrm{\partial }{f}^{p}}{\mathrm{\partial }{\sigma }_{\mathit{ij}}})\mathrm{-}(\frac{\mathrm{\partial }}{{\xi }^{p}}(\frac{\mathrm{\partial }{f}^{p}}{\mathrm{\partial }{\sigma }_{\mathit{ij}}})\mathrm{:}{n}_{\mathit{mn}})\mathrm{\otimes }{n}_{\mathit{ij}}\mathrm{-}(\frac{\mathrm{\partial }{f}^{p}}{\mathrm{\partial }{\sigma }_{\mathit{mn}}}\mathrm{:}\frac{\mathrm{\partial }{n}_{\mathit{mn}}}{\mathrm{\partial }{\xi }^{p}})\mathrm{\otimes }{n}_{\mathit{ij}}\mathrm{-}(\frac{\mathrm{\partial }{f}^{p}}{\mathrm{\partial }{\sigma }_{\mathit{mn}}}\mathrm{:}{n}_{\mathit{mn}})\mathrm{\cdot }\frac{\mathrm{\partial }{n}_{\mathit{ij}}}{\mathrm{\partial }{\xi }^{p}}\)
with
\(\begin{array}{c}\frac{\mathrm{\partial }}{{\xi }^{p}}(\frac{\mathrm{\partial }{f}^{p}}{\mathrm{\partial }{\sigma }_{\mathit{ij}}})\mathrm{=}\frac{\mathrm{\partial }{a}^{d}}{\mathrm{\partial }{\xi }^{p}}{\sigma }_{c}{H}_{0}^{c}{({A}^{d}{s}_{\mathit{II}}H(\theta )+{B}^{d}{I}_{1}+{D}^{d})}^{{a}^{d}\mathrm{-}1}\mathrm{\cdot }({A}^{d}\frac{\mathrm{\partial }({s}_{\mathit{II}}H(\theta ))}{\mathrm{\partial }{\sigma }_{\mathit{ij}}}+{B}^{d}{\delta }_{\mathit{ij}})\\ \mathrm{-}{a}^{d}{\sigma }_{c}{H}_{0}^{c}\mathrm{\{}\frac{\mathrm{\partial }{a}^{d}}{{\xi }^{p}}\mathrm{ln}({A}^{d}{s}_{\mathit{II}}H(\theta )+{B}^{d}{I}_{1}+{D}^{d})\mathrm{...}\\ \mathrm{...}+\frac{({a}^{d}\mathrm{-}1)}{{A}^{d}{s}_{\mathit{II}}H(\theta )+{B}^{d}{I}_{1}+{D}^{d}}(\frac{\mathrm{\partial }{A}^{d}}{\mathrm{\partial }{\xi }^{p}}{s}_{\mathit{II}}H(\theta )+\frac{\mathrm{\partial }{B}^{d}}{\mathrm{\partial }{\xi }^{p}}{I}_{1}+\frac{\mathrm{\partial }{D}^{d}}{\mathrm{\partial }{\xi }^{p}})\mathrm{\}}\mathrm{\cdot }\\ {({A}^{d}{s}_{\mathit{II}}H(\theta )+{B}^{d}{I}_{1}+{D}^{d})}^{{a}^{d}\mathrm{-}1}\left[{A}^{d}\frac{\mathrm{\partial }({s}_{\mathit{II}}H(\theta ))}{\mathrm{\partial }{\sigma }_{\mathit{ij}}}+{B}^{d}{\delta }_{\mathit{ij}}\right]\\ \mathrm{-}{a}^{d}{\sigma }_{c}{H}_{0}^{c}{\left[{A}^{d}{s}_{\mathit{II}}H(\theta )+{B}^{d}{I}_{1}+{D}^{d}\right]}^{{a}^{d}\mathrm{-}1}\mathrm{\cdot }(\frac{\mathrm{\partial }{A}^{d}}{\mathrm{\partial }{\xi }^{p}}\frac{\mathrm{\partial }({s}_{\mathit{II}}H(\theta ))}{{\sigma }_{\mathit{ij}}}+\frac{\mathrm{\partial }{B}^{d}}{\mathrm{\partial }{\xi }^{p}})\end{array}\)
and
\(\begin{array}{ccc}\frac{\mathrm{\partial }{n}_{\mathit{kl}}}{{\xi }^{p}}& \mathrm{=}& \frac{\mathrm{\partial }{n}_{\mathit{kl}}}{\mathrm{\partial }\beta \text{'}}\frac{\mathrm{\partial }\beta \text{'}}{\mathrm{\partial }{\xi }^{p}}\\ & \mathrm{=}& \frac{(\frac{{s}_{\mathit{kl}}}{{s}_{\mathit{II}}}(\beta {\text{'}}^{2}+3)\mathrm{-}2\beta {\text{'}}^{2}\frac{{s}_{\mathit{kl}}}{{s}_{\mathit{II}}}+2\beta \text{'}{\delta }_{\mathit{kl}})}{{(\beta {\text{'}}^{2}+3)}^{3\mathrm{/}2}}\frac{\mathrm{\partial }\beta \text{'}}{\mathrm{\partial }{\xi }^{p}}\\ & \mathrm{=}& \frac{(\frac{{s}_{\mathit{kl}}}{{s}_{\mathit{II}}}(\beta {\text{'}}^{2}+3)\mathrm{-}2\beta {\text{'}}^{2}\frac{{s}_{\mathit{kl}}}{{s}_{\mathit{II}}}+2\beta \text{'}{\delta }_{\mathit{kl}})}{{(\beta {\text{'}}^{2}+3)}^{3\mathrm{/}2}}\frac{\mathrm{-}6\sqrt{6}\mathrm{sin}\psi }{{(3\mathrm{-}\mathrm{sin}\psi )}^{2}}\frac{\mathrm{\partial }\mathrm{sin}\psi }{\mathrm{\partial }{\xi }^{p}}\end{array}\)
if the dilatance law followed corresponds to the pre-peak domain, \(\frac{\mathrm{\partial }\mathrm{sin}\psi }{\mathrm{\partial }{\xi }^{p}}\mathrm{=}0\)
if the dilatance law followed corresponds to the post-peak domain, the following operations are necessary:
\(\begin{array}{ccc}\frac{\mathrm{\partial }\mathrm{sin}\psi }{\mathrm{\partial }{\xi }^{p}}& \mathrm{=}& \frac{\mathrm{\partial }\mathrm{sin}\psi }{\mathrm{\partial }\alpha }\frac{\mathrm{\partial }\alpha }{\mathrm{\partial }{\xi }^{p}}\\ & \mathrm{=}& {\mu }_{1}\frac{(1+{\xi }_{1}){\alpha }_{\text{res}}}{{({\xi }_{1}\alpha +{\alpha }_{\text{res}})}^{2}}\frac{\mathrm{\partial }\alpha }{\mathrm{\partial }{\xi }^{p}}\\ & \mathrm{=}& {\mu }_{1}\frac{(1+{\xi }_{1}){\alpha }_{\text{res}}}{{({\xi }_{1}\alpha +{\alpha }_{\text{res}})}^{2}}\frac{({\sigma }_{\text{min}}\mathrm{-}{\sigma }_{\text{max}})}{{({\sigma }_{\text{min}}+\tilde{\sigma })}^{2}}\frac{\mathrm{\partial }\tilde{\sigma }}{\mathrm{\partial }{\xi }^{p}}\end{array}\)
\(\begin{array}{ccc}\frac{\mathrm{\partial }\tilde{\sigma }}{\mathrm{\partial }{\xi }^{p}}& \mathrm{=}& \frac{\mathrm{\partial }\tilde{\sigma }}{\mathrm{\partial }\tilde{c}}\frac{\mathrm{\partial }\tilde{c}}{\mathrm{\partial }{\xi }^{p}}+\frac{\mathrm{\partial }\tilde{\sigma }}{\mathrm{\partial }\mathrm{tan}(\phi )}\frac{\mathrm{\partial }\mathrm{tan}(\phi )}{\mathrm{\partial }{\xi }^{p}}\\ & \mathrm{=}& \frac{1}{\mathrm{tan}(\phi )}\frac{\mathrm{\partial }\tilde{c}}{\mathrm{\partial }{\xi }^{p}}\mathrm{-}\frac{\tilde{c}}{\mathrm{tan}{(\phi )}^{2}}\frac{\mathrm{\partial }\mathrm{tan}(\phi )}{\mathrm{\partial }{\xi }^{p}}\end{array}\)
with
\(\begin{array}{c}\frac{\mathrm{\partial }\tilde{c}}{\mathrm{\partial }{\xi }^{p}}\mathrm{=}\frac{({\sigma }_{c}{({s}^{d})}^{{a}^{d}}\left[\frac{\mathrm{\partial }{a}^{d}}{\mathrm{\partial }{\xi }^{p}}\mathrm{ln}({s}^{d})+\frac{{a}^{d}}{{s}^{d}}\frac{\mathrm{\partial }{s}^{d}}{\mathrm{\partial }{\xi }^{p}}\right])}{2\sqrt{1+{a}^{d}{m}^{d}{({s}^{d})}^{{a}^{d}\mathrm{-}1}}}+\mathrm{...}\\ \mathrm{...}\frac{{\sigma }_{c}{({s}^{d})}^{{a}^{d}}(\frac{\mathrm{\partial }{a}^{d}}{\mathrm{\partial }{\xi }^{p}}{m}^{d}{({s}^{d})}^{{a}^{d}\mathrm{-}1}+{a}^{d}\frac{\mathrm{\partial }{m}^{d}}{\mathrm{\partial }{\xi }^{p}}{({s}^{d})}^{{a}^{d}\mathrm{-}1}+{a}^{d}{m}^{d}(\frac{\mathrm{\partial }{a}^{d}}{\mathrm{\partial }{\xi }^{p}}\mathrm{ln}{s}^{d}+\frac{{a}^{d}\mathrm{-}1}{{s}^{d}}\frac{\mathrm{\partial }{s}^{d}}{\mathrm{\partial }{\xi }^{p}}){({s}^{d})}^{{a}^{d}\mathrm{-}1})}{4{(1+{a}^{d}{m}^{d}{({s}^{d})}^{{a}^{d}\mathrm{-}1})}^{3\mathrm{/}2}}\end{array}\)
\(\begin{array}{ccc}\frac{\mathrm{\partial }\mathrm{tan}\phi }{\mathrm{\partial }{\xi }^{p}}& \mathrm{=}& (1+{\mathrm{tan}}^{2}\phi )\frac{\mathrm{\partial }\phi }{\mathrm{\partial }{\xi }^{p}}\\ & \mathrm{=}& \frac{(1+{\mathrm{tan}}^{2}\phi )\left[\frac{\mathrm{\partial }{a}^{d}}{\mathrm{\partial }{\xi }^{p}}{m}^{d}{({s}^{d})}^{{a}^{d}\mathrm{-}1}+{a}^{d}\frac{\mathrm{\partial }{m}^{d}}{\mathrm{\partial }{\xi }^{p}}{({s}^{d})}^{{a}^{d}\mathrm{-}1}+{a}^{d}{m}^{d}{({s}^{d})}^{{a}^{d}\mathrm{-}1}(\frac{\mathrm{\partial }{a}^{d}}{\mathrm{\partial }{\xi }^{p}}\mathrm{ln}{s}^{d}+\frac{{a}^{d}\mathrm{-}1}{{s}^{d}}\frac{\mathrm{\partial }{s}^{d}}{\mathrm{\partial }{\xi }^{p}})\right]}{(2+{a}^{d}{m}^{d}{({s}^{d})}^{{a}^{d}\mathrm{-}1})\sqrt{(1+{a}^{d}{m}^{d}{({s}^{d})}^{{a}^{d}\mathrm{-}1})}}\end{array}\)
Calculation of terms relating to \(\frac{d{({R}_{1})}_{\mathit{ij}}}{d(\Delta {Y}_{4})}\mathrm{=}{C}_{\mathit{ijkl}}^{e}\mathrm{:}\left[\frac{{\mathrm{\partial }\mathrm{\langle }\phi ({f}^{\mathit{vp}})\mathrm{\rangle }}^{\text{+}}}{\mathrm{\partial }{\xi }^{\mathit{vp}}}{G}_{\mathit{kl}}^{\mathit{vp}}+{\mathrm{\langle }\phi ({f}^{\mathit{vp}})\mathrm{\rangle }}^{\text{+}}\mathrm{\cdot }\frac{\mathrm{\partial }{G}_{\mathit{kl}}^{\mathit{vp}}}{\mathrm{\partial }{\xi }^{\mathit{vp}}}\right]\mathrm{\cdot }\Delta t\):
The evaluation of the term \(\frac{\mathrm{\partial }{G}_{\mathit{kl}}^{\mathit{vp}}}{\mathrm{\partial }{\xi }^{\mathit{vp}}}\) is identical in form to the previous calculation for \(\frac{\mathrm{\partial }{G}_{\mathit{kl}}^{p}}{\mathrm{\partial }{\xi }^{p}}\).
\(\frac{\mathrm{\partial }{\mathrm{\langle }\phi ({f}^{\mathit{vp}})\mathrm{\rangle }}^{\text{+}}}{\mathrm{\partial }{\xi }^{\mathit{vp}}}\mathrm{=}\frac{{A}_{v}{n}_{v}}{{P}_{\text{atm}}}{(\frac{{f}^{\mathit{vp}}}{{P}_{\text{atm}}})}^{{n}_{v}\mathrm{-}1}\frac{\mathrm{\partial }{f}^{\mathit{vp}}}{\mathrm{\partial }{\xi }^{\mathit{vp}}}\)
The evaluation of the term \(\frac{\mathrm{\partial }{f}^{\mathit{vp}}}{\mathrm{\partial }{\xi }^{\mathit{vp}}}\) is identical in its form to the previously calculated term \(\frac{\mathrm{\partial }{f}^{p}}{\mathrm{\partial }{\xi }^{p}}\).
Evaluating terms relating to \(\frac{d({R}_{3})}{d{(\Delta {Y}_{1})}_{\mathit{ij}}}\) and \(\frac{d({R}_{4})}{d{(\Delta {Y}_{1})}_{\mathit{ij}}}\)
\(\begin{array}{ccc}\frac{\mathrm{\partial }{\tilde{G}}_{\mathit{II}}^{p}}{\mathrm{\partial }{\sigma }_{\mathit{ij}}}& \mathrm{=}& \frac{\mathrm{\partial }{\tilde{G}}_{\mathit{II}}^{p}}{\mathrm{\partial }{\tilde{G}}_{\mathit{kl}}^{p}}\frac{\mathrm{\partial }{\tilde{G}}_{\mathit{kl}}^{p}}{\mathrm{\partial }{G}_{\mathit{mn}}^{p}}\frac{\mathrm{\partial }{G}_{\mathit{mn}}^{p}}{\mathrm{\partial }{\sigma }_{\mathit{ij}}}\\ & \mathrm{=}& \frac{{\tilde{G}}_{\mathit{kl}}^{p}}{{\tilde{G}}_{\mathit{II}}^{p}}({\delta }_{\mathit{mk}}{\delta }_{\mathit{nl}}\mathrm{-}\frac{1}{3}{\delta }_{\mathit{mn}}{\delta }_{\mathit{kl}})\frac{\mathrm{\partial }{G}_{\mathit{mn}}^{p}}{\mathrm{\partial }{\sigma }_{\mathit{ij}}}\end{array}\) and \(\begin{array}{ccc}\frac{\mathrm{\partial }{\tilde{G}}_{\mathit{II}}^{\mathit{vp}}}{\mathrm{\partial }{\sigma }_{\mathit{ij}}}& \mathrm{=}& \frac{\mathrm{\partial }{\tilde{G}}_{\mathit{II}}^{\mathit{vp}}}{\mathrm{\partial }{\tilde{G}}_{\mathit{kl}}^{\mathit{vp}}}\frac{\mathrm{\partial }{\tilde{G}}_{\mathit{kl}}^{\mathit{vp}}}{\mathrm{\partial }{G}_{\mathit{mn}}^{\mathit{vp}}}\frac{\mathrm{\partial }{G}_{\mathit{mn}}^{\mathit{vp}}}{\mathrm{\partial }{\sigma }_{\mathit{ij}}}\\ & \mathrm{=}& \frac{{\tilde{G}}_{\mathit{kl}}^{\mathit{vp}}}{{\tilde{G}}_{\mathit{II}}^{\mathit{vp}}}({\delta }_{\mathit{mk}}{\delta }_{\mathit{nl}}\mathrm{-}\frac{1}{3}{\delta }_{\mathit{mn}}{\delta }_{\mathit{kl}})\frac{\mathrm{\partial }{G}_{\mathit{mn}}^{\mathit{vp}}}{\mathrm{\partial }{\sigma }_{\mathit{ij}}}\end{array}\)
Evaluating terms for \(\frac{d({R}_{3})}{d(\Delta {Y}_{3})}\)
\(\frac{\mathrm{\partial }{\tilde{G}}_{\mathit{II}}^{p}}{\mathrm{\partial }{\xi }^{p}}\mathrm{=}\frac{{\tilde{G}}_{\mathit{kl}}^{p}}{{\tilde{G}}_{\mathit{II}}^{p}}({\delta }_{\mathit{mk}}{\delta }_{\mathit{nl}}\mathrm{-}\frac{1}{3}{\delta }_{\mathit{mn}}{\delta }_{\mathit{kl}})\frac{\mathrm{\partial }{G}_{\mathit{mn}}^{p}}{\mathrm{\partial }{\xi }^{p}}\)
Evaluating terms relating to \(\frac{d({R}_{3})}{d(\Delta {Y}_{4})}\) and \(\frac{d({R}_{4})}{d(\Delta {Y}_{4})}\)
\(\frac{\mathrm{\partial }{\tilde{G}}_{\mathit{II}}^{\mathit{vp}}}{\mathrm{\partial }{\xi }^{\mathit{vp}}}\mathrm{=}\frac{{\tilde{G}}_{\mathit{kl}}^{\mathit{vp}}}{{\tilde{G}}_{\mathit{II}}^{\mathit{vp}}}({\delta }_{\mathit{mk}}{\delta }_{\mathit{nl}}\mathrm{-}\frac{1}{3}{\delta }_{\mathit{mn}}{\delta }_{\mathit{kl}})\frac{\mathrm{\partial }{G}_{\mathit{mn}}^{\mathit{vp}}}{\mathrm{\partial }{\xi }^{\mathit{vp}}}\)