3. Rousselier model

We now describe the derivation of the equations of the Rousselier model presented in the introduction.

3.1. Derivation of model equations

It is assumed that specific free energy is divided into three parts: a hyperelastic part that only depends on elastic deformation, a part related to the work hardening mechanism and a part related to damage:

\(\Phi ({\varepsilon }^{e},p,f)={\Phi }^{e}({\varepsilon }^{e})+{\Phi }^{p}(p)+{\Phi }^{f}(f)\) eq 3.1-1

The Clausius-Duhem inequality is written (we do not consider the thermal part):

\(\tau :\dot{\varepsilon }-\dot{\Phi }\ge 0\) eq 3.1-2

An expression in which \(\dot{\varepsilon }={\dot{\varepsilon }}^{e}+{\dot{\varepsilon }}^{p}\) represents the deformation rate.

Dissipation is still written as:

\((\tau -\frac{\partial \Phi }{\partial {\varepsilon }^{e}}):{\dot{\varepsilon }}^{e}+\tau :{\dot{\varepsilon }}^{p}-\frac{\partial \Phi }{\partial p}\dot{p}-\frac{\partial \Phi }{\partial f}\dot{f}\ge 0\) eq 3.1-3

The second principle of thermodynamics then requires the following expression for the elastic stress-strain relationship:

\(\tau =\frac{\partial \Phi }{\partial {\varepsilon }^{e}}\) eq 3.1-4

The thermodynamic forces associated with elastic deformation, cumulative plastic deformation, and porosity are defined in accordance with the framework of generalized standard materials:

\(\tau ({\varepsilon }^{e})=\frac{\partial \Phi }{\partial {\varepsilon }^{e}}\) eq 3.1-5

\(A(p)=\frac{\partial \mathrm{\Phi }}{\partial p}\) eq 3.1-6

\(B(f)=\frac{\partial \mathrm{\Phi }}{\partial f}\) eq 3.1-7

There is then left for the dissipation:

\(\tau :{\dot{\varepsilon }}^{p}-A\dot{p}-B\dot{f}\ge 0\) eq 3.1-8

The principle of maximum dissipation applied from the viscoplastic potential \({F}^{\text{vp}}(\tau \text{,A,B})\) makes it possible to deduce the laws of evolution of plastic deformation, cumulative plastic deformation and porosity, i.e.:

\({\dot{\varepsilon }}^{p}=\frac{\partial {F}^{\text{vp}}}{\partial \tau }\) eq 3.1-9

\(\dot{p}=-\frac{\partial {F}^{\text{vp}}}{\partial A}\) eq 3.1-10

\(\dot{f}\mathrm{=}\mathrm{-}\frac{\mathrm{\partial }{F}^{\text{vp}}}{\mathrm{\partial }B}\) eq 3.1-11

It is assumed that \({F}^{\text{vp}}(\tau ,A,B)\) is a function of the plastic potential \(F(\tau ,A,B)\) and that the latter is broken down into two terms depending respectively on the second invariant of \(\tau\) coupled to \(A\) and on the first invariant of \(\tau\) coupled to \(B\):

\({F}^{\text{vp}}=\Lambda (F)=\Lambda ({F}_{\text{vM}}({\tau }_{\text{eq}},A)+{F}_{m}({\tau }_{m},B))\) eq 3.1-12

Hypothetically, the first term is broken down additively like the von Mises potential:

\({F}_{\text{vM}}({\tau }_{\text{eq}},A)={\tau }_{\text{eq}}-A(p)-{R}_{0}={\tau }_{\text{eq}}-R(p)\) eq 3.1-13

In order not to obtain a trivial result, the decomposition of the second term must be multiplicative:

\({F}_{m}({\tau }_{m},B)=g({\tau }_{m})h(B)\) eq 3.1-14

Given the equation [éq 1-1], the laws of evolution for \(\text{tr}{\dot{\varepsilon }}^{p}\) and \(\dot{f}\) lead to equality:

\(\frac{g\text{'}({\tau }_{m})}{g({\tau }_{m})}=(\frac{-1}{1-f})\frac{{h}^{\text{'}}(B(f))}{h(B(f))}\) eq 3.1-15

Both members of this equation are functions of the two independent variables \({\tau }_{m}\) and \(f\), so they are equal to a dimensional constant the inverse of a constraint, this is the material parameter \(1/{\sigma }_{1}\). The dimensionless parameter \({D}_{1}\) appears in the \(g\text{'}\mathrm{/}g\) integration:

\(g({\mathrm{\tau }}_{m})={D}_{1}{\mathrm{\sigma }}_{1}\text{exp}(\frac{{\mathrm{\tau }}_{m}}{{\mathrm{\sigma }}_{1}})\) eq 3.1-16

The function \(B(f)\) and the inverse function \(f={h}_{1}(B)\) are unknown. The easiest and most natural choice is to take \({h}_{1}\equiv h\), which results in:

\(h(B)\equiv {h}_{1}(B)=f\) eq 3.1-17

\({h}^{\text{'}}(B)=\frac{\mathrm{df}}{\text{dB}}=-\frac{1}{{\mathrm{\sigma }}_{1}}f(1-f)\) eq 3.1-18

The plastic potential is finally written:

\(F={\tau }_{\text{eq}}+{\sigma }_{1}{D}_{1}f\text{exp}(\frac{{\tau }_{m}}{{\sigma }_{1}})-R(p)\) eq 3.1-19

The law of evolution for \(\dot{p}\) gives:

\(\dot{p}=\frac{d\Lambda (F)}{\text{dF}}=V(F)\) eq 3.1-20

Function \(V(F)\) defines the viscosity of the material. We will only consider the particular case such as:

\(V(F)={\dot{\varepsilon }}_{0}{\left[\text{sh}(\frac{F}{{\sigma }_{0}})\right]}^{m}\) eq 3.1-21

which is reduced to a power function (Norton law) when the two parameters of the material \({\dot{\varepsilon }}_{0}\) and \({\mathrm{\sigma }}_{0}\) are very large. Conversely we have:

\(F-S(\dot{p})=0\) eq 3.1-22

\(S(\dot{p})={\sigma }_{0}{\text{sh}}^{\text{-}1}\left[{(\frac{\dot{p}}{{\dot{\varepsilon }}_{0}})}^{\frac{1}{m}}\right]\) eq 3.1-23

In the case of plasticity independent of time, the previous equation becomes \(F=0\) (plasticity criterion or threshold) and \(\dot{p}\) is given by the consistency equation \(\dot{F}=0\) if \(F=0\) and \(\dot{p}=0\) if \(F<0\).

The equations of the model are now completely defined, in the case without nucleation of new cavities. In the case of nucleation of new cavities, it is considered that the volume fraction created is proportional to the cumulative plastic deformation. So all you have to do is replace \(f\) by \(f+{A}_{n}p\) in the model equations. \({A}_{n}\) is a material parameter. Equation [éq 1-1] is not changed.

3.2. Model equations

The model equations deduced from the above thermodynamic and phenomenological analysis are summarized:

\({\Phi }_{\text{vp}}={\tau }_{\text{eq}}+{\sigma }_{1}{D}_{1}(f+{A}_{n}p)\text{exp}(\frac{{\tau }_{m}}{{\sigma }_{1}})-R(p)-{\sigma }_{0}{\text{sh}}^{\text{-}1}\left[{(\frac{\dot{p}}{{\dot{\varepsilon }}_{0}})}^{\frac{1}{m}}\right]=0\) eq 3.2-1

\(\tau =\frac{\sigma }{\rho }=\left[\lambda (\mathrm{Id}\otimes \mathrm{Id})+2\mu \mathrm{II}\right]:{\varepsilon }^{e}\) eq 3.2-2

\(\mathrm{\rho }=\frac{1-f-{A}_{n}p}{1-{f}_{0}}\) eq 3.2-3

\({\tilde{\dot{\varepsilon }}}^{p}=\dot{p}\frac{3\tilde{\sigma }}{2{\sigma }_{\text{eq}}}=\dot{p}\frac{3\tilde{\tau }}{2{\tau }_{\text{eq}}}\) eq 3.2-4

\(\mathrm{tr}{\dot{\varepsilon }}^{p}=\dot{p}{D}_{1}(f+{A}_{n}p)\text{exp}(\frac{{\tau }_{m}}{{\sigma }_{1}})\) eq 3.2-5

\(\dot{f}={A}_{1}(1-f)\mathrm{tr}{\dot{\varepsilon }}^{p}\) eq 3.2-6

with \({A}_{1}=1\), this parameter being introduced for numerical reasons only.