2. Model description

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

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

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

\(R\text{=}R(p)\) [éq2-3]

\(X\text{=}C{\varepsilon }^{p}\) [éq2-4]

where \(K,\mu ,\alpha ,R\) and \(C\) 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 [R4.08.01]), the isotropic work hardening function and the Prager constant. As for \({T}^{\text{réf}}\), this is the reference temperature, for which thermal deformation is zero.

\(K,\mu\) are related to the Young’s modulus \(E\) and to the Poisson’s ratio by:

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

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* \(\mathrm{2C}/3\) and not \(C\). Likewise, for the isotropic work hardening function, the elastic limit is included by \(R(0)\text{=}{\sigma }^{y}\) , some references treating it separately.

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

\(F(\sigma ,R,X)={(\tilde{\sigma }-X)}_{\text{eq}}-R\) [éq2-5]

with \({A}_{\text{eq}}=\sqrt{\frac{3}{2}\tilde{A}\cdot \tilde{A}}\)

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

\(\dot{p}=\dot{\lambda }=\sqrt{\frac{2}{3}{\dot{\varepsilon }}^{p}\cdot {\dot{\varepsilon }}^{p}}\) [éq2-7]

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

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