2. Description of the models#
2.1. Description of the models#
At any moment, the state of the material is described by the deformation \(\varepsilon\), the temperature \(T\), the plastic deformation \({\varepsilon }^{p}\), the cumulative plastic deformation \(p\) and the reminder tensor \(X\). The equations of state then define, according to these state variables, the stress \(\sigma \mathrm{=}{\sigma }^{H}\text{Id}+\tilde{\sigma }\) (broken down into hydrostatic and deviatoric parts), the isotropic part of the work hardening \(R\) and the kinematic part \(X\):
\({\sigma }^{H}\mathrm{=}\frac{1}{3}\text{tr}(\sigma )\mathrm{=}K\text{tr}(\varepsilon \mathrm{-}{\varepsilon }^{\text{th}})\) with \({\varepsilon }^{\text{th}}\mathrm{=}\alpha ({\text{T-T}}^{\text{ref}})\text{Id}\) eq 2.1-1
\(\tilde{\sigma }\mathrm{=}\sigma \mathrm{-}{\sigma }^{H}\text{Id}\mathrm{=}2\mu (\tilde{\varepsilon }\mathrm{-}{\varepsilon }^{p})\) eq 2.1-2
\(R\mathrm{=}R(p)\) eq 2.1-3
\(X\mathrm{=}X(p,{\varepsilon }^{p})\mathrm{=}{X}_{1}(p,{\varepsilon }^{p})+{X}_{2}(p,{\varepsilon }^{p})\) eq 2.1-4
where \(K,\mu ,\alpha\) and the coefficients of \(X(p)\) and \(R(p)\) are characteristics of the material that may depend on temperature. More precisely, they are respectively the compressibility and shear modules, the thermal expansion coefficient, and the isotropic and kinematic work hardening functions. As for \({T}^{\text{réf}}\), this is the reference temperature, for which thermal deformation is considered to be zero.
Note:
For the model VISC_CIN1_CHAB we only consider the only tensor variable \({X}_{1}(p)\) so \({X}_{2}(p)\mathrm{=}0\). This remains valid for the rest of the following: we will formally describe the two models in the same way, the model VISC_CIN1_CHAB deriving from VISC_CIN2_CHAB assuming \({X}_{2}(p)\mathrm{=}0\) .
The evolution of plastic deformation is governed by a flow law normal to a von Mises plasticity criterion:
\(F(\sigma ,R,X)\mathrm{=}{(\tilde{\sigma }\mathrm{-}{X}_{1}\mathrm{-}{X}_{2})}_{\text{eq}}\mathrm{-}R(p)\) with \({A}_{\text{eq}}\mathrm{=}\sqrt{\frac{3}{2}\tilde{A}\mathrm{:}\tilde{A}}\) eq 2.1-5
\({\dot{\varepsilon }}^{p}\mathrm{=}\dot{\lambda }\frac{\mathrm{\partial }F}{\mathrm{\partial }\sigma }\mathrm{=}\frac{3}{2}\dot{\lambda }\frac{\tilde{\sigma }\mathrm{-}{X}_{1}\mathrm{-}{X}_{2}}{{(\tilde{\sigma }\mathrm{-}{X}_{1}\mathrm{-}{X}_{2})}_{\text{eq}}}\) eq 2.1-6
\(\dot{p}\mathrm{=}\dot{\lambda }\mathrm{=}\sqrt{\frac{2}{3}{\dot{\varepsilon }}^{p}\mathrm{:}{\dot{\varepsilon }}^{p}}\) eq 2.1-7
As for the plastic multiplier \(\dot{\lambda }\), it is obtained by the consistency condition:
\(\mathrm{\{}\begin{array}{cc}\text{si}F<0\text{ou}\stackrel{}{\mathrm{\dot{}}}F<0& \dot{\lambda }\mathrm{=}0\\ \text{si}F\mathrm{=}0\text{et}\stackrel{}{\mathrm{\dot{}}}F\mathrm{=}0& \dot{\lambda }\mathrm{\ge }0\end{array}\) eq 2.1-8
Note:
The evolution of the variables \({\mathrm{X}}_{1}\) and \({\mathrm{X}}_{2}\) is given by:
\(\begin{array}{c}{X}_{1}\mathrm{=}\frac{2}{3}{C}_{1}(p){\alpha }_{1}\\ {X}_{2}\mathrm{=}\frac{2}{3}{C}_{2}(p){\alpha }_{2}\\ {\dot{\alpha }}_{1}\mathrm{=}{\dot{\varepsilon }}^{p}\mathrm{-}{\gamma }_{1}(p){\alpha }_{1}\dot{p}\\ {\dot{\alpha }}_{2}\mathrm{=}{\dot{\varepsilon }}^{p}\mathrm{-}{\gamma }_{2}(p){\alpha }_{2}\dot{p}\end{array}\) eq 2.1-9
The functions \(C(p)\), \(\gamma (p)\), and \(R(p)\) are defined, in accordance with [bib2], by:
\(\begin{array}{c}R(p)\mathrm{=}{R}_{\mathrm{\infty }}+({R}_{0}\mathrm{-}{R}_{\mathrm{\infty }}){e}^{\mathrm{-}\text{bp}}\\ {C}_{1}(p)\mathrm{=}{C}_{1}^{\mathrm{\infty }}(1+(k\mathrm{-}1){e}^{\mathrm{-}\text{wp}})\\ {C}_{2}(p)\mathrm{=}{C}_{2}^{\mathrm{\infty }}(1+(k\mathrm{-}1){e}^{\mathrm{-}\text{wp}})\end{array}\)
\(\begin{array}{c}{\gamma }_{1}(p)\mathrm{=}{\gamma }_{1}^{0}({a}_{\mathrm{\infty }}+(1\mathrm{-}{a}_{\mathrm{\infty }}){e}^{\mathrm{-}\text{bp}})\\ {\gamma }_{2}(p)\mathrm{=}{\gamma }_{2}^{0}({a}_{\mathrm{\infty }}+(1\mathrm{-}{a}_{\mathrm{\infty }}){e}^{\mathrm{-}\text{bp}})\end{array}\)
The evolution of these coefficients makes it possible to represent work hardening in several ways: classical isotropic work hardening (monotonic or cyclic) by \(R(p)\), « hardening » of the coefficients relating to kinematic terms by \(C(p)\) and \(\gamma (p)\). (cf. [bib 11]). Exponential expressions are similar to the definition of nonlinear kinematic work hardening (eq.2,1,9), and (in principle) represent a variation in coefficients from the value indexed by \(0\) (for \(p=0\)) to the value indexed by \(\infty\) to the value indexed by when \(p\) becomes large.
This implies that the coefficients \(b\) and \(w\) are assumed to be positive. Otherwise, an alarm message is sent, as the calculated solution may be non-physical.
The presence of viscosity can be modelled in a simple way [bib2] by replacing the consistency condition [éq 2.1-8] by:
\(\dot{p}\mathrm{=}{(\frac{\mathrm{\langle }F\mathrm{\rangle }}{K})}^{N}\) eq 2.1-10
\(\mathrm{\langle }F\mathrm{\rangle }\) positive part of \(F\) (Macauley hooks), \(K,N\) viscosity characteristics (Norton) of the material. All the other equations in the model are left unchanged. It will be seen that such introduction of viscosity only leads to minor modifications of the algorithm for the implicit integration of the law of behavior.
The memory effect consists in replacing the evolution of isotropic work hardening by:
\(F(\sigma ,R,X)\mathrm{=}{(\tilde{\sigma }\mathrm{-}{X}_{1}\mathrm{-}{X}_{2})}_{\text{eq}}\mathrm{-}{R}_{0}\mathrm{-}R(p)\)
\(\dot{R}\mathrm{=}b(Q\mathrm{-}R)\dot{p}\)
\(Q\mathrm{=}Q{}_{0}\text{}+({Q}_{m}\mathrm{-}{Q}_{0})(1\mathrm{-}{e}^{\mathrm{-}2\mu q})\)
\(f({\varepsilon }^{p},\xi ,q)\mathrm{=}\frac{2}{3}{J}_{2}({\varepsilon }^{p}\mathrm{-}\xi )\mathrm{-}q\mathrm{\le }0\) defining a domain characterizing the maximum plastic deformations, of which \(q\) measures the radius and \(\xi\) the center, calculated according to a law of normality, i.e. with the law of evolution: \(\dot{\xi }\mathrm{=}\frac{1\mathrm{-}\eta }{\eta }\dot{q}{n}^{\text{*}}\). The parameter \(\eta\) (which does not exist in the initial formulation [bib.2]), i allows the memory effect to be partially taken into account. If it is equal to 0.5, the initial formulation is returned. If it is 1, \(q\) is equal to the norm of the greatest plastic deformation reached. If it is much less than 0.5, the memory effect is only partially taken into account.
Notes:
The definition of \({\mathrm{X}}_{1}\) and \({\mathrm{X}}_{2}\) in the form [éq 2.1-9]:
allows you to keep a formulation that takes into account the variations of parameters with temperature without introducing a term in \(\dot{T}\) as in [bib.4], in the same way as the viscoplastic Chaboche model. These terms are necessary because not taking them into account would lead to inaccurate results [bib4].
allows to have a form consistent with the thermodynamic expression of plastic potential [bib2] (p.221).
It can be seen that the functions \({C}_{1}(p),{\gamma }_{1}(p),{C}_{2}(p),{\gamma }_{2}(p),R(p)\) used in the previous equations all three make it possible to model different non**linear work hardening effects. The introduction of work hardening, either at the level of the kinematic part, by\(C(p)`*, or at the level of the recovery term, by the function*:math:\)gamma (p)`, does not have the same effect on identification tests [bib2]. In particular, the use of a model with \(\gamma (p)`* makes it easier to identify strong cyclical workings. Several studies to identify the coefficients of the Chaboche models have in fact been carried out on the basis of the model with work hardening represented by* :math:\)gamma (p)` ([bib5], [bib6]), in particular for stainless steels.
2.2. Add the memory effect#
The implicit discretization of the problem with memory effect leads to a system of 20 equations with 20 unknowns [7]:
6 eq: \(\tilde{\sigma }\mathrm{=}\frac{\mu }{{\mu }^{\mathrm{-}}}{\tilde{\sigma }}^{\mathrm{-}}+2\mu (\Delta \tilde{\varepsilon }\mathrm{-}\Delta {\varepsilon }^{p})\)
1 eq: \({(\tilde{\sigma }\mathrm{-}\frac{2}{3}{C}_{1}{\alpha }_{1}^{\mathrm{-}}\mathrm{-}\frac{2}{3}{C}_{2}{\alpha }_{2}^{\mathrm{-}}\mathrm{-}\frac{2}{3}{C}_{1}\Delta {\alpha }_{1}(\Delta {\varepsilon }^{p})\mathrm{-}\frac{2}{3}{C}_{2}\Delta {\alpha }_{2}(\Delta {\varepsilon }^{p}))}_{\text{eq}}\mathrm{=}{R}_{0}+{R}^{\mathrm{-}}+\Delta R+K{(\frac{\Delta p}{\Delta t})}^{1\mathrm{/}N}\)
6 eq: \(\Delta {\varepsilon }^{p}\mathrm{=}\frac{3}{2}\Delta p\frac{\tilde{\sigma }\mathrm{-}\frac{2}{3}{C}_{1}{\alpha }_{1}^{\mathrm{-}}\mathrm{-}\frac{2}{3}{C}_{2}{\alpha }_{2}^{\mathrm{-}}\mathrm{-}\frac{2}{3}{C}_{1}\Delta {\alpha }_{1}\mathrm{-}\frac{2}{3}{C}_{2}\Delta {\alpha }_{2}}{{R}_{0}+R(p)+K{(\frac{\Delta p}{\Delta t})}^{1\mathrm{/}N}}\mathrm{=}\Delta pn\)
1 eq: \(f({\varepsilon }^{p},\xi ,q)\mathrm{=}\frac{2}{3}{J}_{2}({\epsilon }^{p}\mathrm{-}\xi )\mathrm{-}q\mathrm{=}\frac{2}{3}\sqrt{\frac{3}{2}({\varepsilon }^{p}\mathrm{-}\xi )\mathrm{:}({\varepsilon }^{p}\mathrm{-}\xi )}\mathrm{-}q\mathrm{\le }0\)
6 eq: \(\Delta \xi \mathrm{=}(1\mathrm{-}\eta )\frac{\Delta q}{\eta }{n}^{\text{*}}\)
with \(\Delta R\mathrm{=}b(Q\mathrm{-}R)\Delta p\) \(Q\mathrm{=}Q{}_{0}\text{}+({Q}_{m}\mathrm{-}{Q}_{0})(1\mathrm{-}{e}^{\mathrm{-}\mathrm{2\mu }({q}^{\mathrm{-}}+\mathit{\Delta q})})\) \(\Delta q\mathrm{=}\eta H(F)\mathrm{\langle }{\text{n:n}}^{\text{*}}\mathrm{\rangle }\Delta p\)
\(\Delta {\alpha }_{i}\mathrm{=}\frac{\Delta {\varepsilon }^{p}\mathrm{-}{\gamma }_{i}{\alpha }_{i}^{\mathrm{-}}\Delta p}{1+{\gamma }_{i}\Delta p}\) \({n}^{\text{*}}\mathrm{=}\frac{3}{2}\frac{{\varepsilon }^{p}\mathrm{-}\xi }{{J}_{2}({\varepsilon }^{p}\mathrm{-}\xi )}\) \(n\mathrm{=}\frac{3}{2}\frac{\tilde{\sigma }\mathrm{-}\frac{2}{3}{C}_{1}{\alpha }_{1}\mathrm{-}\frac{2}{3}{C}_{2}{\alpha }_{2}}{(\tilde{\sigma }\mathrm{-}\frac{2}{3}{C}_{1}{\alpha }_{1}\mathrm{-}\frac{2}{3}{C}_{2}{\alpha }_{2})\underset{}{\text{eq}}}\)
The 20 unknowns are: \(\tilde{\sigma },\Delta {\varepsilon }^{p},\Delta \xi ,\Delta p,\Delta q\)
2.3. Inserting the effect of non-proportionality of loading#
In a similar way to model VISCOCHAB, equations translating the non-proportional effect can be inserted into VISC_CIN2_CHAB/MEMO. The model obtained is referred to here as VISC/VMIS_CIN2_NRAD, or VISC/, or/VMIS_MEMO_NRAD (depending on whether or not the memory effect is taken into account).
\(\begin{array}{c}\dot{{\alpha }_{1}}\mathrm{=}{\dot{\varepsilon }}^{p}\mathrm{-}{\gamma }_{1}(p){\alpha }_{1}\dot{p}\\ {\dot{\alpha }}_{2}\mathrm{=}{\dot{\varepsilon }}^{p}\mathrm{-}{\gamma }_{2}(p){\alpha }_{2}\dot{p}\end{array}\) becomes: \(\begin{array}{c}\dot{{\alpha }_{1}}\mathrm{=}{\dot{\varepsilon }}^{p}\mathrm{-}{\gamma }_{1}(p)({\delta }_{1}{\alpha }_{1}+(1\mathrm{-}{\delta }_{1})({\alpha }_{1}\mathrm{:}n)n)\dot{p}\\ {\dot{\alpha }}_{2}\mathrm{=}{\dot{\varepsilon }}^{p}\mathrm{-}{\gamma }_{2}(p)({\delta }_{2}{\alpha }_{2}+(1\mathrm{-}{\delta }_{2})({\alpha }_{2}\mathrm{:}n)n)\dot{p}\end{array}\)
with \(n\mathrm{=}\sqrt{\frac{3}{2}}\frac{\tilde{\sigma }\mathrm{-}{X}_{1}\mathrm{-}{X}_{2}}{{(\tilde{\sigma }\mathrm{-}{X}_{1}\mathrm{-}{X}_{2})}_{\mathit{eq}}}\) so \(n\mathrm{:}n\mathrm{=}1\) and in particular \(\dot{{\varepsilon }^{p}}\mathrm{=}\sqrt{\frac{3}{2}}\Delta pn\)
It is easy to verify that this new expression of the evolution of the internal variables \({\alpha }_{i}\) returns to the previous expression in the case where \({\delta }_{i}\mathrm{=}1\), or in the case of radial situation, where one can set \({\alpha }_{i}\mathrm{=}\xi n\).
Then it comes: \(\dot{{\alpha }_{i}}\mathrm{=}\dot{{\varepsilon }^{p}}\mathrm{-}{\gamma }_{i}\dot{p}({\delta }_{i}\xi n+(1\mathrm{-}{\delta }_{i})\xi n)\mathrm{=}\dot{{\varepsilon }^{p}}\mathrm{-}{\gamma }_{i}\dot{p}{\alpha }_{i}\).