Model description
=====================

At any moment, the state of the material is described by the deformation :math:`\varepsilon`, the temperature :math:`T`, the plastic deformation :math:`{\varepsilon }^{p}` and the cumulative plastic deformation :math:`p`. The equations of state then define, according to these state variables, the stress :math:`\sigma \text{=}{\sigma }^{H}\text{Id}+\tilde{\sigma }` (broken down into hydrostatic and deviatoric parts), the isotropic part of the work hardening :math:`R` and the kinematic part :math:`X`, also called the return stress:

:math:`{\sigma }^{H}\text{=}\frac{1}{3}\text{tr}(\sigma )\text{=}K\text{tr}(\varepsilon \text{-}{\varepsilon }^{\text{th}})\text{avec}{\varepsilon }^{\text{th}}\text{=}\alpha ({\text{T-T}}^{\text{réf}})\text{Id}` [:ref:`éq2-1 <éq2-1>`]

:math:`\tilde{\sigma }\text{=}\sigma \text{-}{\sigma }^{H}\text{Id}=2\mu (\tilde{\varepsilon }\text{-}{\varepsilon }^{p})\text{où}\tilde{\varepsilon }\text{=}\varepsilon \text{-}\frac{1}{3}\text{tr}(\varepsilon )\text{Id}` [:ref:`éq2-2 <éq2-2>`]

:math:`R\text{=}R(p)` [:ref:`éq2-3 <éq2-3>`]

:math:`X\text{=}C{\varepsilon }^{p}` [:ref:`éq2-4 <éq2-4>`]

where :math:`K,\mu ,\alpha ,R` and :math:`C` are characteristics of the material that may be temperature dependent.

More precisely, these are respectively the compressibility and shear modules, the mean thermal expansion coefficient (see [:ref:`R4.08.01 <R4.08.01>`]), the isotropic work hardening function and the Prager constant. As for :math:`{T}^{\text{réf}}`, this is the reference temperature, for which thermal deformation is zero.

:math:`K,\mu` are related to the Young's modulus :math:`E` and to the Poisson's ratio by:

:math:`\begin{array}{}\mathrm{3K}\text{=}3\lambda +2\mu \text{=}\frac{E}{1\text{-}2\nu  }\\ 2\mu \text{=}\frac{E}{1\text{+}\nu  }\end{array}`

**Note:**

Concerning the kinematic part of work hardening [é*q], we note that it is linear in this model. In addition, you should take care of the fact that in some references, we call Prager constant* :math:`\mathrm{2C}/3` *and not* :math:`C`\ *. Likewise, for the isotropic work hardening function, the elastic limit is included by* :math:`R(0)\text{=}{\sigma }^{y}` *, some references treating it separately.*

The evolution of internal variables :math:`{\varepsilon }^{p}\text{et}p` is governed by a normal flow law associated with a plasticity criterion :math:`F`:

:math:`F(\sigma ,R,X)={(\tilde{\sigma }-X)}_{\text{eq}}-R` [:ref:`éq2-5 <éq2-5>`]

with :math:`{A}_{\text{eq}}=\sqrt{\frac{3}{2}\tilde{A}\cdot \tilde{A}}`

:math:`{\dot{\varepsilon }}^{p}\text{=}\dot{\lambda }\frac{\partial F}{\partial \sigma }\text{=}\frac{3}{2}\dot{\lambda }\frac{\tilde{\sigma }\text{-}X}{{(\tilde{\sigma }\text{-}X)}_{\text{eq}}}` [:ref:`éq2-6 <éq2-6>`]

:math:`\dot{p}=\dot{\lambda }=\sqrt{\frac{2}{3}{\dot{\varepsilon }}^{p}\cdot {\dot{\varepsilon }}^{p}}` [:ref:`éq2-7 <éq2-7>`]

As for the plastic multiplier :math:`\dot{\lambda }`, it is obtained by the following consistency condition:

:math:`\{\begin{array}{cc}\text{si}F\text{<}0\text{ou}\dot{F}\text{<}0& \dot{\lambda }\text{=}0\\ \text{si}F\text{=}0\text{et}\dot{F}\text{=}0& \dot{\lambda }\text{>=}0\end{array}` [:ref:`éq2-8 <éq2-8>`]