2. Notations

We will note by:

\(\mathrm{Id}\)	identity matrix
\(\text{tr}A\)	trace of tensor \(A\)
\({A}^{T}\)	transposed from the \(A\) tensor
\(\text{det}A\)	determinant of \(A\)
\(\langle X\rangle\)	positive part of*X*
\(\tilde{A}\)	deviatory part of the \(A\) tensor defined by \(\tilde{A}=A-(\frac{1}{3}\text{tr}A)\text{Id}\)
:	doubly contracted product: \(A:B=\sum _{i,j}{A}_{\text{ij}}{B}_{\text{ij}}=\text{tr}({\text{AB}}^{T})\)
\(\otimes\)	tensor product: \((A\otimes B{)}_{\text{ijkl}}={A}_{\text{ij}}{B}_{\text{kl}}\)
\({A}_{\text{eq}}\)	equivalent von Mises value defined by \({A}_{\text{eq}}=\sqrt{\frac{3}{2}\tilde{A}:\tilde{A}}\)
\({\nabla }_{X}A\)	gradient: \({\nabla }_{X}A=\frac{\partial A}{\partial X}\)
\({\text{div}}_{x}A\)	discrepancy: \(({\text{div}}_{x}A{)}_{i}=\sum _{j}\frac{\partial {A}_{\text{ij}}}{\partial {x}_{j}}\)
\(\lambda ,\mu ,E,\nu ,K\)	isotropic elasticity coefficients
\({\sigma }_{y}\)	elastic limit
\(\alpha\)	thermal expansion coefficient
\(T\)	temperature
\({T}_{\text{ref}}\)	reference temperature

Moreover, in the context of time discretization, all the quantities evaluated at the previous instant are indexed by \({}^{-}\), the quantities evaluated at the instant \(t+\Delta t\) are not indexed and the increments are designated by \(\Delta\). We thus have:

\(\Delta Q=Q-{Q}^{-}\)