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 is written \(A_{eq}=\sqrt{\cfrac{3}{2}\boldsymbol{A}_d:\boldsymbol{A}_d}\).
The notations defined in Tableau 1.1 and Tableau 1.2 will be progressively supplemented by other symbols during the presentation of the Barcelona model.
\(e\) |
Void index |
\(e^p\) |
Plastic void index |
\(f_1, f_2\) |
Plasticity criteria |
\(f_1, f_2\) |
Flow potentials respectively associated with plasticity criteria \(f_1, f_2\) |
\(\boldsymbol{I}\) |
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 of second order (\(\mathbb{J}:\boldsymbol{a}=\cfrac{a_v}{3}\boldsymbol{I}\)) |
\(\mathbb{K}\) |
Projector on the space of symmetric tensors with zero trace (\(\mathbb{K}:\boldsymbol{a}=(\mathbb{I}-\mathbb{J}):\boldsymbol{a}=\boldsymbol{a}_d\)) |
\(p_c\) |
Capillary pressure, otherwise called suction (\(p_c=p_g-p_l\)) |
\(p_g\) |
Pressure of the gas phase |
\(p_l\) |
Liquid phase pressure |
\(\boldsymbol{\varepsilon}\) |
Total deformation tensor |
\(\varepsilon_v\) |
Total volume deformation |
\(\dot{\lambda}_1,\dot{\lambda}_2\) |
Plastic multipliers respectively associated with the plasticity criteria \(f_1, f_2\) |
\(\boldsymbol{\sigma}\) |
Stress tensor |
\(\boldsymbol{\sigma}''\) |
Net stress tensor (\(\boldsymbol{\sigma}''=\boldsymbol{\sigma}+p_g\boldsymbol{I}\)) |
\(\sigma''_{eq}\) |
Net von Mises equivalent stress (\(\sigma''_{eq}=\sigma_{eq}\)) |
\(\sigma''_m\) |
Net mean stress |