4. Extension of the large deformations model#
The objective of this paragraph is to justify the extension of the model written in large deformations to take account of metallurgical transformations. In particular, to take into account transformation plasticity, we cannot add up as in small deformations an additional term of deformation linked to transformation plasticity. In fact, on the kinematic decomposition aspect, taking into account transformation plasticity does not change anything. We still have decomposition \(F={F}^{e}{F}^{p}\) where \({F}^{p}\) contains all the information on « anelastic » deformation (therefore including that related to transformation plasticity). In particular, it is only at the behavioral level that transformation plasticity is treated.
First, we recall some theoretical elements that make it possible to write the model without metallurgical effects and then we show the modifications to be made to take into account metallurgical effects and transformation plasticity in particular.
4.1. Thermodynamic aspect#
The writing of the law of large deformations behavior comes from the thermodynamic framework with internal variables. The thermodynamic formalism is based on two hypotheses. The first is that free energy depends only on elastic deformations \({b}^{e}\) and on internal variables related to the hardening of the material (here the cumulative plastic deformation associated with the isotropic work hardening variable \(R\)). This makes it possible, thanks to the Clausius-Duhem inequality, to obtain state laws. The second hypothesis is the principle of maximum dissipation, which corresponds to the data of a dissipation potential, which then makes it possible to determine the laws of evolution of the internal variables.
Free energy is given by:
\(\psi =\psi ({b}^{e},p)={\psi }^{e}({b}^{e})+{\psi }^{p}(p)\)
By the first hypothesis, we obtain state laws, i.e.:
\(\tau =2{\rho }_{0}\frac{\partial {\psi }^{e}}{\partial {b}^{e}}{b}^{e}\) and \(R={\rho }_{0}\frac{\partial {\psi }^{p}}{\partial p}\)
There is left for the dissipation:
\(\tau :(-\frac{1}{2}F{\dot{G}}^{p}{F}^{T}{b}^{e-1})-R\dot{p}\ge 0\)
By introducing a threshold function such as \(f(\tau ,R)\le 0\), the principle of maximum dissipation (or equivalently the data of a pseudo-dissipation potential [bib3]) makes it possible to deduce, through the property of normality, the laws of evolution, i.e.:
\(-\frac{1}{2}F{\dot{G}}^{p}{F}^{T}{b}^{e-1}=\dot{\lambda }\frac{\partial f}{\partial \tau }\) and \(\dot{p}=-\dot{\lambda }\frac{\partial f}{\partial R}\)
This is a model of associated plasticity.
4.2. Extension#
For the restoration of work hardening, there are no particular difficulties associated with large deformations. It is sufficient for the free energy to depend, no longer on the cumulative plastic deformation, but on the internal work-hardening variables \({r}_{k}\) associated with the work hardening variables \({Z}_{k}\text{.}{R}_{k}\) of each of the metallurgical phases.
To now take into account deformations due to transformation plasticity, it is proposed to add an additional term in the plastic deformation flow law \({G}^{p}\) which derives from a dissipation potential \(\Omega\).
For state laws, we thus obtain:
\(\tau =2{\rho }_{0}\frac{\partial {\psi }^{e}}{\partial {b}^{e}}{b}^{e}\) and \({Z}_{k}\text{.}{R}_{k}={\rho }_{0}\frac{\partial {\psi }^{p}}{\partial {r}_{k}}\)
and for the laws of evolution:
\(-\frac{1}{2}F{\dot{G}}^{p}{F}^{T}{b}^{e-1}=\dot{\lambda }\frac{\partial f}{\partial \tau }+\underset{\text{plasticité de transformation}}{\underset{\underbrace{}}{\frac{\partial {\Omega }^{\text{pt}}}{\partial \tau }}}\)
\({\dot{r}}_{k}=-\dot{\lambda }\frac{\partial f}{\partial ({Z}_{k}\text{.}{R}_{k})}-\underset{\begin{array}{}\text{restauration d'écrouissage}\\ \text{métallurgique et visqueux}\end{array}}{\underset{\underbrace{}}{\frac{\partial {\Omega }^{r}}{\partial ({Z}_{k}\text{.}{R}_{k})}}}\)
\(\Omega ={\Omega }^{\text{pt}}(t)+{\Omega }^{r}\)
The potentials \({\Omega }^{\text{pt}}\) and \({O}^{r}\), linked respectively to the plasticity of transformation and to the restoration of work-hardening, are chosen in such a way as to find, under the hypothesis of small deformations, the same laws of evolution as those of the model with metallurgical effects written in small deformations.
4.3. Behavioral relationships#
This is the case of linear isotropic work hardening.
The partitioning of deformations involves:
\({\stackrel{ˉ}{b}}^{e}=\stackrel{ˉ}{F}{G}^{p}{\stackrel{ˉ}{F}}^{T}\) with \(\stackrel{ˉ}{F}={J}^{-1/3}F\), \(J=\text{det}F\), and \({\stackrel{ˉ}{b}}^{e}={J}^{-2/3}{b}^{e}\)
Behavioral relationships are given by:
Stress-thermoelastic deformation relationship:
\(\tilde{\tau }=\mu {\tilde{\stackrel{ˉ}{b}}}^{e}\)
\(\text{tr}\tau =\frac{\mathrm{3K}}{2}({J}^{2}-1)-\frac{\mathrm{9K}}{2}{\varepsilon }^{\text{th}}(J+\frac{1}{J})\)
\({\epsilon }^{\text{th}}={Z}_{\gamma }\left[{\alpha }_{\gamma }(T-{T}_{\text{ref}})-(1-{Z}_{\gamma }^{r})\Delta {\varepsilon }_{f\gamma }^{T\text{ref}}\right]+(\sum _{i=1}^{4}{Z}_{i})\left[{\alpha }_{f}(T-{T}_{\text{ref}})+{Z}_{\gamma }^{r}\Delta {\varepsilon }_{f\gamma }^{T\text{ref}}\right]\)
where: \({Z}_{\gamma }^{r}\) characterizes the reference metallurgical phase
\({Z}_{\gamma }^{r}=1\) when the reference phase is the austenitic phase,
\({Z}_{\gamma }^{r}=0\) when the reference phase is the ferritic phase.
\(\Delta {\varepsilon }_{f\gamma }^{{T}_{\text{ref}}}={\varepsilon }_{f}^{\text{th}}({T}_{\text{ref}})-{\varepsilon }_{\gamma }^{\text{th}}({T}_{\text{ref}})\) reflects the difference in compactness between the ferritic and austenitic phases at the reference temperature \({T}_{\text{ref}}\),
\({\alpha }_{f}\) is the expansion coefficient of the four ferritic phases and \({\alpha }_{\gamma }\) that of the austenitic phase.
Plasticity threshold:
\(f={\tau }_{\text{eq}}-R-{\sigma }_{y}\)
R is the work-hardening variable of the multiphase material, which is written as:
\(R=(1-\stackrel{ˉ}{f}(Z)){R}_{\gamma }+\frac{\stackrel{ˉ}{f}(Z)}{Z}\sum _{i=1}^{4}{Z}_{i}\text{.}{R}_{i}\), \(Z=\sum _{i=1}^{4}{Z}_{i}\)
where \({R}_{k}\) is the hardening variable for phase \(k\) which can be linear or non-linear with respect to \({r}_{k}\) and \(\stackrel{ˉ}{f}(Z)\) a function dependent on \(Z\) such as \(\stackrel{ˉ}{f}(Z)\in \left[\mathrm{0,1}\right]\).
In the linear case, we have \({R}_{k}={R}_{\mathrm{0k}}{r}_{k}\) where \({R}_{\mathrm{0k}}\) is the work-hardening slope of phase \(k\).
In the non-linear case, we write: \({R}_{k}={R}_{k}^{(i)}+{R}_{\mathrm{0k}}^{(i)}(r-{r}_{k}^{(i)})\) where the meanings of \({R}_{k}^{(i)}\), \({R}_{\mathrm{0k}}^{(i)}\) and \({r}_{k}^{(i)}\) are shown in the figure below.
Nonlinear work hardening curve
The elastic limit \({\sigma }_{y}\) is equal to:
If \(Z\ne 0\), \({\sigma }_{y}=(1-\stackrel{ˉ}{f}(Z)){\sigma }_{y\gamma }+\stackrel{ˉ}{f}(Z){\sigma }_{\mathrm{y\alpha }}\), \({\sigma }_{\mathrm{y\alpha }}=\frac{\sum _{i=1}^{4}{Z}_{i}{s}_{\mathrm{y\alpha i}}}{Z}\)
If \(Z=0\), \({\sigma }_{y}={\sigma }_{y\gamma }\)
where \({\sigma }_{\mathrm{y\alpha i}}\) are the four elastic limits of ferritic phases, \({\sigma }_{y\gamma }\) that of the austenitic phase.
Laws of evolution:
\(\stackrel{ˉ}{F}{\dot{G}}^{p}{\stackrel{ˉ}{F}}^{T}=-\dot{p}\frac{3}{{\tau }_{\text{eq}}}\tilde{\tau }{\stackrel{ˉ}{b}}^{e}-3\tilde{\tau }{\stackrel{ˉ}{b}}^{e}\sum _{i=1}^{4}{K}_{i}{F}_{i}^{\text{'}}(1-{Z}_{\gamma })\langle {\dot{Z}}_{i}\rangle\)
\({\dot{r}}_{\gamma }=\dot{p}+\frac{\sum _{i=1}^{4}\langle -{\dot{Z}}_{i}\rangle ({\theta }_{i\gamma }{r}_{i}-{r}_{\gamma })}{{Z}_{\gamma }}\underset{\text{uniquement en viscosité}}{\underset{\underbrace{}}{-({\text{Cr}}_{\text{moy}}{)}^{m}}}\text{si}{Z}_{\gamma }>0\)
\({\dot{r}}_{i}=\dot{p}+\frac{\langle {\dot{Z}}_{i}\rangle ({\theta }_{\gamma i}{r}_{\gamma }-{r}_{i})}{{Z}_{i}}\underset{\text{uniquement en viscosité}}{\underset{\underbrace{}}{-({\text{Cr}}_{\text{moy}}{)}^{m}}}\text{si}{Z}_{i}>0\)
\({r}_{\text{moy}}=\sum _{k=1}^{5}{Z}_{k}{r}_{k}\), \(C=\sum _{k=1}^{5}{Z}_{k}{C}_{k}\), \(m=\sum _{k=1}^{5}{Z}_{k}{m}_{k}\)
where \({K}_{i}\), \({F}_{i}^{\text{'}}\), \({C}_{i}\), and \({m}_{i}\) are material data associated with phase \(i\), \({\theta }_{\gamma i}\) the work-hardening restoration coefficient during transformation \(\gamma\) in \(i\) (\({\theta }_{\gamma i}\in \left[\mathrm{0,1}\right]\)) and \({\theta }_{i\gamma }\) the work-hardening restoration coefficient when transforming \(i\) into \(\gamma\) ( \({\theta }_{i\gamma }\in \left[\mathrm{0,1}\right]\)) .
All material data is entered in the operator DEFI_MATERIAU ([U4.43.01]) under the various keyword factors ELAS_META (_F0) and META_ **.
For a plasticity model, the plastic multiplier is obtained by writing the coherence condition \(\dot{f}=0\) and we have:
\(\dot{p}\ge \mathrm{0,}f\le 0\text{et}\dot{p}f=0\)
In the viscous case, \(\dot{p}\) is written as:
\(\dot{p}={(\frac{\langle f\rangle }{h})}^{n}\)
or equivalently:
\(\langle f\rangle =(1-\stackrel{ˉ}{f}(Z)){\eta }_{\gamma }{\dot{p}}^{1/{n}_{\gamma }}+\frac{\stackrel{ˉ}{f}}{Z}\sum _{i=1}^{4}{Z}_{i}{\eta }_{i}{\dot{}p}^{1/{n}_{i}}\)
where \({n}_{i}\) and \({\eta }_{i}\) are the viscosity coefficients of the material associated with phase \(i\) which possibly depend on the temperature.
Calculating \(\stackrel{ˉ}{F}{\dot{G}}^{p}{\stackrel{ˉ}{F}}^{T}\) gives:
\(\stackrel{ˉ}{F}{\dot{G}}^{p}{\stackrel{ˉ}{F}}^{T}=-3(A{\tau }_{\text{eq}}+\dot{p})(\frac{1}{3}\text{tr}\stackrel{ˉ}{{b}^{e}}\frac{\tilde{\tau }}{{\tau }_{\text{eq}}}+\frac{{\tau }_{\text{eq}}}{\mu }\frac{\tilde{\tau }\tilde{\tau }}{{t}_{\text{eq}}^{2}})\)
where we posed \(A=\sum _{i=1}^{4}{K}_{i}{F}_{i}^{\text{'}}\langle {\dot{Z}}_{i}\rangle\).
Since \(\parallel \tilde{\tau }/{\tau }_{\text{eq}}\parallel \le 1\) and \(\parallel \tilde{\tau }\tilde{\tau }/{\tau }_{\text{eq}}^{2}\parallel \le 1\), the second term in the above expression can be overlooked (in front of 1) for metallic materials insofar as:
\(\frac{{\tau }_{\text{eq}}}{\mu }=\frac{R+{\sigma }_{y}}{\mu }\approx {\text{10}}^{-3}\ll 1\le \text{tr}\stackrel{ˉ}{{b}^{e}}\)
\(\text{tr}\stackrel{ˉ}{{b}^{e}}\ge 1\) because the tensor \({\stackrel{ˉ}{b}}^{e}\) is symmetric, positive definite, and \(\text{det}\stackrel{ˉ}{{b}^{e}}=1\).
It is this simplification of the law of evolution of \({G}^{p}\) that makes it possible to easily integrate the law of behavior, i.e. to bring it back to the resolution of a non-linear scalar equation. We will therefore then take:
\(\stackrel{ˉ}{F}{\dot{G}}^{p}{\stackrel{ˉ}{F}}^{T}\approx -(\dot{p}+{\tau }_{\text{eq}}A)\frac{\text{tr}\stackrel{ˉ}{{b}^{e}}}{{\tau }_{\text{eq}}}\tilde{\tau }\) eq 4.3-1
4.4. The different relationships#
In operator STAT_NON_LINE, these different models are accessed using the following keyword factors:
| BEHAVIOR: (
RELATION:/”META_P_IL” /”META_P_INL” /”META_P_IL_PT” /”META_P_INL_PT” /”META_P_IL_RE” /”META_P_INL_RE” /”META_P_IL_PT_RE” /”META_P_INL_PT_RE” /”META_V_IL” /”META_V_INL” /”META_V_IL_PT” /”META_V_INL_PT” /”META_V_IL_RE” /”META_V_INL_RE” /”META_V_IL_PT_RE” /”META_V_INL_PT_RE”
DEFORMATION: /” SIMO_MIEHE “
)
Here we only recall the meaning of the letters for behaviors META:
P_IL: linear isotropic work hardening plasticity,
P_ INL: non-linear isotropic work hardening plasticity,
V_IL: viscoplasticity with linear isotropic work hardening,
V_ INL: viscoplasticity with nonlinear isotropic work hardening,
PT: transformation plasticity,
RE: restoration of metallurgical work hardening.
Example: “META_V_INL_RE” = elastoviscoplastic law with nonlinear isotropic work hardening with hardening restoration but without taking into account transformation plasticity
The various characteristics of the material are given in operator DEFI_MATERIAU. The reader is referred to the note [R4.04.02] for the meaning of the keyword factors of this operator.
Attention:
If isotropic work hardening is linear, we enter under the keyword META_ECRO_LINE of DEFI_MATERIAU, the work-hardening module, i.e. the slope in the stress-deformation plane.
On the other hand, if isotropic work hardening is non-linear, we give directly under the keyword META_TRACTION of DEFI_MATERIAU, the isotropic work hardening curve \(R\) ( \(R=\tau -{\sigma }_{y}\) ) as a function of the cumulative plastic deformation \(p\) ( \(p=\epsilon -\frac{\tau }{E}\) ) .
Note:
The user must ensure that the « experimental » tensile curve used 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 \({l}_{0}\) 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, \(F\) the force applied and \(S\) the current area. Note that \(\sigma =F/S=\frac{F}{{S}_{0}}\frac{l}{{l}_{0}}\frac{1}{J}\) whereuse \(\tau =J\sigma =\frac{F}{{S}_{0}}\frac{l}{{l}_{0}}\). In general, it is the quantity \(\frac{F}{{S}_{0}}\frac{l}{{l}_{0}}\) that is measured by the experimenters and this directly gives the Kirchhoff constraint used in the Simo and Miehe model.
4.5. Constraints and internal variables#
The output constraints are Cauchy \(\sigma\) constraints, therefore measured on the current configuration.
For all META_ **relationships, the internal variables produced in*Code_Aster* are:
V1: \({r}_{1}\) work-hardening variable for ferrite,
V2: \({r}_{2}\) work hardening variable for pearlite,
V3: \({r}_{3}\) work-hardening variable for bainite,
V4: \({r}_{4}\) work-hardening variable for martensite,
V5: \({r}_{5}\) variable work hardening for austenite,
V6: plasticity indicator (0 if the last calculated increment is elastic; 1 otherwise),
V7: \(R\) the isotropic work hardening term for the threshold function,
V8: the trace divided by three of the elastic deformation tensor \({\stackrel{ˉ}{b}}^{e}\), which is \(\frac{1}{3}\text{tr}\stackrel{ˉ}{{b}^{e}}\).