4. Rousselier model#
We now describe the application of large deformations to the Rousselier model presented in the introduction.
4.1. Model equations#
To describe a thermoelastoplastic model with isotropic work hardening (the equivalent in small deformations to the isotropic work hardening model and Von Mises criterion), Simo and Miehe propose a polyconvex elastic potential. For the sake of simplicity, we choose here the potential of Saint Venant, which is written as:
\(\Phi (e,p)=\frac{1}{2}\left[K{(\text{tr}e)}^{2}+2\mu \tilde{e}:\tilde{e}+\mathrm{6K}\alpha \Delta T\text{tr}e\right]+\underset{0}{\overset{p}{\int }}R(u)\text{du}\) eq 4.1-1
In accordance with equations [éq 3.3-8] and [éq 3.3-9], the state laws that derive from the elastic potential above are then written:
\(s=-\left[K\text{tr}e\text{Id}+2\mu \tilde{e}+\mathrm{3K}\alpha \Delta T\text{Id}\right]\) eq 4.1-2
\(A=-R(p)\) eq 4.1-3
The elasticity threshold is given by:
\(F(s,R)={s}_{\text{eq}}+{\sigma }_{1}\text{Df}\text{exp}(\frac{{s}_{H}}{{\sigma }_{1}})-R-{\sigma }_{y}\) eq 4.1-4
According to equations [éq 3.3.2-1] and [éq 3.3.2-2], the flow laws are defined by:
\(-\frac{1}{2}F{\dot{G}}^{p}{F}^{T}=\dot{\lambda }\left[\frac{3\tilde{s}}{2{s}_{\text{eq}}}+\frac{\text{Df}}{3}\text{exp}(\frac{{s}_{H}}{{\sigma }_{1}})\text{Id}\right]\) eq 4.1-5
\(\dot{p}=\dot{\lambda }\) eq 4.1-6
\(\dot{\lambda }\ge 0\text{}F\le 0\text{}F\dot{\lambda }=0\) eq 4.1-7
4.2. Treatment of singular points#
In fact, the flow equation [éq 4.1-5] reflects the belonging of the flow direction to the normal cone at the surface of the elasticity domain. It is only valid at regular points, characterized by:
\({s}_{\text{eq}}\ne 0\) eq 4.2-1
It therefore remains to characterize the cone normal to the singular points, that is to say verifying:
\(\tilde{s}=0\text{et}{\sigma }_{1}Df\text{exp}(\frac{{s}_{H}}{{\sigma }_{1}})-R={\sigma }_{y}\) eq 4.2-2
The normal to convex elasticity cone at such a point is the set of flow directions that achieve the following maximization problem:
\({\Delta }^{\text{*}}(s\text{,R})=\underset{{D}^{p},\dot{p}}{\text{sup}}\left[s:{D}^{p}-R\dot{p}-\Delta ({D}^{p},\dot{p})\right]\) eq 4.2-3
where \({\Delta }^{\text{*}}\) is the indicative function of the convex \(F\) and \(\Delta ({D}^{p},\dot{p})\) is the dissipation potential obtained by the Legendre-Fenchel transform of the indicator function of \(F\):
\(\Delta ({D}^{p},\dot{p})=\underset{\begin{array}{}s,R\\ F(s,R)\le 0\end{array}}{\text{Sup}}\left[s:{D}^{p}-R\dot{p}\right]\) eq 4.2-4
After some calculations, we get:
\(\Delta ({D}^{p},\dot{p})={\sigma }_{y}\dot{p}+{\sigma }_{1}\text{tr}{D}^{p}(\text{ln}\frac{\text{tr}{D}^{p}}{Df\dot{p}}-1)+{I}_{{ℝ}^{+}}(\text{tr}{D}^{p})+{I}_{{ℝ}^{+}}(\dot{p}-\frac{2}{3}{D}_{\text{eq}}^{p})\) eq 4.2-5
with
\({I}_{{ℝ}^{+}}(x)=\{\begin{array}{}0\text{}\text{si}\text{}x\ge 0\\ \text{+}\infty \text{sinon}\end{array}\) eq 4.2-6
For \(\tilde{s}=0\), \({\Delta }^{\text{*}}\) is equivalent to:
\({\Delta }^{\text{*}}(s,R)=\underset{\begin{array}{}{D}^{p},\dot{p}\\ \text{tr}{D}^{p}\ge 0\\ \dot{p}-\frac{2}{3}{D}_{\mathrm{eq}}^{p}\ge 0\end{array}}{\text{Sup}}\left[\underset{G(\text{tr}{D}^{p})}{\underset{\underbrace{}}{{s}_{H}\text{tr}{D}^{p}-{\sigma }_{1}\text{tr}{D}^{p}(\text{ln}\frac{\text{tr}{D}^{p}}{Df\dot{p}}-1)}}-R\dot{p}-{\sigma }_{y}\dot{p}\right]\) eq 4.2-7
Noting that for \(\text{tr}{D}^{p}\ge 0\), the function \(G(\text{tr}{D}^{p})\) is concave, the supremacy with respect to the trace of the plastic deformation rate \({D}^{p}\) is obtained for:
\({G}^{\text{'}}(\text{tr}{D}^{p})=0\text{}\text{d'où}\text{}\text{tr}{D}^{p}=\mathrm{Df}\dot{p}\text{exp}(\frac{{s}_{H}}{{\sigma }_{1}})\) eq 4.2-8
Note:
We then find again for the function indicative of the elasticity threshold \(F\) .
\({D}^{\text{*}}(s\text{,R})=\underset{\begin{array}{}\dot{p}\\ \dot{p}\ge \frac{2}{3}{D}_{\text{eq}}^{p}\ge 0\end{array}}{\text{Sup}}\left[F\dot{p}\right]=\{\begin{array}{cc}0& \text{si}F\le 0\\ \text{+}\infty & \text{sinon}\end{array}\) eq 4.2-9
At a singular point, the normal cone, which is the set of admissible directions of flow, is therefore characterized by:
\(\text{tr}{D}^{p}=Df\dot{p}\text{exp}(\frac{{s}_{H}}{{\sigma }_{1}})\) eq 4.2-10
\(\dot{p}\ge \frac{2}{3}{D}_{\text{eq}}^{p}\ge 0\) eq 4.2-11
4.3. Expression of porosity#
We saw in the introduction that the microscopic inspiration of the Rousselier model is based on a microstructure consisting of a cavity and a rigid plastic matrix, and therefore isochoric. In this case, the porosity is directly linked to the macroscopic deformation by the relationship eq. 1-1.
In this expression, the change in volume of elastic origin is overlooked. Without particular precautions, this approximation may prove to be penalizing in the presence of elastic compression, even reasonable, because it leads to possibly negative porosity.
The following equivalent expression is therefore preferred, accompanied by an explicit reduction in the initial porosity:
\(f=\text{max}({f}_{\mathrm{0,}}1-\frac{1-{f}_{0}}{J})\) eq 4.3-1
For his part, Rousselier proposes to express porosity based on the plastic deformation rate \({D}^{p}\). The relationship is written in incremental form:
\(\dot{f}=(1-f)\text{tr}{D}^{p}\) eq 4.3-2
This means that the porosity variable used to set the plasticity criterion \(F\) only depends on the plastic deformation. In fact, the plastic deformation rate is an evaluated quantity in the relaxed configuration. Its transport in the current configuration (like \(D\)) is still expressed:
\({F}^{e}{D}^{p}{F}^{{e}^{T}}=-\frac{1}{2}F{\dot{G}}^{p}{F}^{T}\) eq 4.3-3
In this case, the law of evolution of porosity is expressed:
\(\dot{f}=(1-f)\text{tr}(-\frac{1}{2}F{\dot{G}}^{p}{F}^{T})\) eq 4.3-4
To prevent the integration of porosity from interfering with that of plasticity (since the two variables are coupled), it is necessary to separate the integration of the law of behavior into two stages: integration of plasticity with porosity fixed to its value at the beginning of the time step, then integration of porosity by means of equation 4.3-4 where the plastic evolution is that calculated in the previous phase.
Important note:
There are thus two possible versions of the model (PORO_TYPE = 1 or 2, cf. U4.43.01), depending on whether total deformation or plastic deformation is chosen respectively in the evolution of porosity. It has been noticed that for a low initial porosity f0, the behavior at the beginning of evolution changes strongly as a function of this parameter. Indeed, the choice seems to have a decisive impact on the response at the scale of the structure, since it favors or not the bifurcations of the zone where the deformations are located. Thus, this high sensitivity should lead the user to be very careful in using this model. Research is ongoing to understand this sensitivity and to discriminate between the two variants for the evolution of porosity.
4.4. Relationship “ROUSSELIER”#
This behavior relationship is available via the “ROUSSELIER” argument of the COMPORTEMENT keyword under the STAT_NON_LINE operator, with the “SIMO_MIEHE” argument of the keyword factor DEFORMATION.
All the parameters of the model are provided under the keywords factors” ROUSSELIER “or” ROUSSELIER_FO “or” “and” TRACTION “(to define the traction curve) of the control DEFI_MATERIAU ([U4.43.01]).
Note:
The user must ensure that the « experimental » tensile curve used, either directly or to derive/to establish the work-hardening slope, is given in the rational stress plane \(\sigma =F/S\) - logarithmic deformation \(\text{ln}(1+\Delta l/{l}_{0})\) where where where \({l}_{0}\) is the initial length of the useful part of the specimen, \(\Delta l\) the initial length of the useful part of the specimen, the variation in length after deformation, Fla force applied and Sthe current surface.
4.5. Constraints and internal variables#
The constraints are the Cauchy constraints, \(\sigma\) calculated therefore on the current configuration (six components in 3D, four in 2D).
The internal variables produced in the*Code_Aster* are:
V1, the cumulative plastic deformation \(p\),
V2, porosity \(f\),
V3, the plasticity indicator (0 if the last calculated increment is elastic, 1 if a regular plastic solution, 2 if a singular plastic solution),
V4 to V9, the :math:`e` elastic deformation tensor* . **
Note:
If the user wants to possibly recover deformations by post-processing his calculation, the Green-Lagrange \(E\) deformations must be plotted, which represents a measure of deformations into large deformations (option EPSG_ELGA or or EPSG_ELNO of CALC_CHAMP). Classical linearized \(\varepsilon\) deformations measure deformations under the assumption of small deformations and do not make sense in large deformations.