1. Notations#
The notations and sign conventions used are those of the mechanics of continuous media. A bold character represents a tensor of order two and an openwork character represents a tensor of order four.
For a tensor of order two \(\boldsymbol{a}\), its decomposition between deviatory and volume parts, respectively noted \(\boldsymbol{a}_d\) and \(a_v\), is written as follows:
boldsymbol {a} =boldsymbol {a} _d +frac {a_v} {3}boldsymbol {I},quadtext {with}quad a_v =boldsymbol {I} =boldsymbol {I}:boldsymbol {I}:boldsymbol {a} _d=0
where we note \(\boldsymbol{I}=\boldsymbol{\delta}\) the second-order identity tensor. We also write \(a_m=\cfrac{a_v}{3}\).
For a tensor of order two \(\boldsymbol{a}\) of the same nature as a deformation, we denote its equivalent von Mises norm by \(a_{eq}=\sqrt{\cfrac{2}{3}\boldsymbol{a}_d:\boldsymbol{a}_d}\). For a tensor of order two \(\boldsymbol{A}\) of the same nature as a constraint, It says \(A_{eq}=\sqrt{\cfrac{3}{2}\boldsymbol{A}_d:\boldsymbol{A}_d}\).
The notations defined in Tableau 1.1 and Tableau 1.2 will be progressively supplemented by other symbols during the presentation of the CSSM model.
\(\mathbb{C}\) |
Isotropic linear elasticity tensor |
\(f, F_i\) |
Plasticity criteria |
\(\boldsymbol{I}\) |
Second order identity operating on vectors |
\(\mathbb{I}\) |
Fourth-order identity operating on symmetric second-order tensors |
\(\mathbb{J}\) |
Spotlight on the space of hydrostatic tensors (\(\mathbb{J}:\boldsymbol{a}=\cfrac{a_v}{3}\boldsymbol{I}\)) |
\(\mathbb{K}\) |
Projector on the space of symmetric tensors with zero trace (\(\mathbb{K}:\boldsymbol{a}=(\mathbb{I}-\mathbb{J}):\boldsymbol{a}=\boldsymbol{a}_d\)) |
\(\boldsymbol{\alpha}_i,\boldsymbol{\varepsilon}^p,\xi,\gamma\) |
Internal variables |
\(\boldsymbol{\varepsilon}\) |
Total deformation tensor |
\(\varepsilon_v\) |
Total volume deformation |
\(\dot{\lambda},\dot{\lambda}_i\) |
Plastic multipliers respectively associated with the plasticity criteria \(f, F_i\) |
\(\boldsymbol{\sigma}\) |
Stress tensor |
\(\sigma_{eq}\) |
Equivalent von Mises stress |
\(\sigma_m\) |
Average stress |
\(\psi\) |
Energy potential (volume density) |