2. Notations#
We will note by:
\(\mathrm{Id}\) |
second order identity tensor |
\(\mathrm{II}\) |
fourth-order identity tensor |
\(\text{tr}A\) |
trace of the second order tensor \(\mathrm{A}\) |
\(\tilde{A}\) |
deviatory part of the \(\mathrm{A}\) tensor defined by \(\tilde{A}=A-(\frac{1}{3}\text{tr}A)\text{Id}\) |
\({A}_{m}\) |
hydrostatic part of the \(\mathrm{A}\) tensor defined by \({A}_{m}\mathrm{=}\frac{\text{tr}\mathrm{A}}{3}\) |
\({A}_{\text{eq}}\) |
equivalent von Mises value defined by \({A}_{\text{eq}}=\sqrt{\frac{3}{2}\tilde{A}:\tilde{A}}\) |
\(\mathrm{:}\) |
doubly contracted product: \(\mathrm{A}\mathrm{:}\mathrm{B}\mathrm{=}\mathrm{\sum }_{i,j}{A}_{\text{ij}}{B}_{\text{ij}}\mathrm{=}\text{tr}({\text{AB}}^{\mathrm{T}})\) |
\(\mathrm{\otimes }\) |
tensor product: \((\mathrm{A}\mathrm{\otimes }\mathrm{B}{)}_{\text{ijkl}}\mathrm{=}{A}_{\text{ij}}{B}_{\text{kl}}\) |
\(\lambda ,\mu ,E,\nu ,K\) |
isotropic elasticity coefficients |
\(\dot{p}\) |
equivalent plastic deformation rate \(\dot{p}=\sqrt{\frac{2}{3}{\tilde{\dot{\mathrm{\varepsilon }}}}^{p}:{\tilde{\dot{\mathrm{\varepsilon }}}}^{p}}\) |
Moreover, in the context of time discretization, all the quantities \(Q\) evaluated at the previous instant are indexed by \({\text{-}}^{}\), the quantities evaluated at the time \(t={t}^{\text{-}}+\Deltat\) are not indexed and the increments are designated by \(\Delta\). We thus have:
\(Q={Q}^{\text{-}}+\Deltaq\)
The numerical resolution is done by a \(\theta\) -method, with \(0\le \mathrm{\theta }\le 1\). For all quantities, we define:
\({Q}^{\theta }={Q}^{\text{-}}+\theta \Deltaq\)