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:

\({Q}^{\theta }={Q}^{\text{-}}+\theta \Deltaq\)