6. (visco) plastic deformation models#
The main characteristic of the thermal changes concerned in this type of analysis is that they cover a wide temperature range, which has an important effect on the mechanical behavior of the material that undergoes the thermal evolution. In particular, we are in temperature ranges where viscosity phenomena may no longer be negligible. It may therefore be necessary to use an elasto-viscoplastic behavior model especially when staying in these areas for a long time; for example during stress relieving treatments associated with welding.
We therefore choose a viscoplastic model whose characteristics are such that it makes it possible to describe with the same formalism, therefore without changing the model:
Classic plastic behavior; to model cases at low temperature when viscous effects are still negligible or to model high speed processes (welding);
Collapsible viscoplastic behavior at high temperature, to model the effects of creep and relaxation associated, for example, with stress relieving treatments or multi-pass welding;
A viscous fluid-like behavior for temperatures above the melting point, in order to have a reasonable description of the molten zone.
The viscoplastic model chosen in fact degenerates for certain borderline cases into a plasticity model independent of time, or into a viscous fluid model.
6.1. Deformation partition#
Total deformation \(\varepsilon\) is written as the sum of four components:
Where \({\varepsilon }^{e}\), \({\varepsilon }^{\mathit{th}}\), \({\varepsilon }^{\mathit{vp}}\), and \({\varepsilon }^{\mathit{pt}}\) are, respectively, elastic, thermal, visco‑plastic, and transformation plasticity deformations. The deformations of the plastic type are purely deviatory and the thermal deformation is purely spherical. For the deviatory part, we therefore have:
And for the spherical part:
6.2. Laws of behavior#
6.2.1. The case of isotropic work hardening#
We place ourselves here in the context of von Mises (visco) plasticity with additive isotropic work hardening. The threshold function is written as:
With \({\sigma }_{\text{eq}}\) the equivalent Von Mises stress such as:
\(R\) is the term for isotropic work hardening and \({\sigma }_{c}\) is the initial critical stress. It corresponds to the initial minimum stress to be applied in order to have a visco‑plastic flow. These two quantities may depend on the temperature and on the metallurgical phases. We can rewrite equation () in the form:
That is to say, in this model, the stress can be interpreted as the sum of a flow limit stress (which itself is broken down into an initial limit stress and a work-hardening term) and a « viscous » stress depending on the rate of deformation and zero at zero speed:
The flow rate (visco) plastic is written as:
This tensor is purely deviatoric. Cumulative plastic deformation \(\dot{p}\) is viscous and is written as:
With \(\eta\) and \(n\) the material viscosity coefficients.
6.2.2. Case of kinematic work hardening#
Equivalently to the case with isotropic work hardening, the threshold function is written as:
With \(X\) the work hardening tensor associated with the variable work hardening tensor \(\alpha\) such as:
\({H}_{0}\) is the kinematic work hardening coefficient. In a manner similar to the isotropic case, we can rewrite equation () by introducing the viscous return stress:
: label: eq-50
{f} ^ {text {*}}}mathrm {=}} fmathrm {-} {sigma} _ {v}mathrm {=} (tilde {sigma}mathrm {-} X {)}}mathrm {-} {sigma}} (tilde {sigma}}mathrm {-}} (T, Z)X {)}} _ {mathit {eq}}} _ {c} (T, Z)mathrm {-} {x {)}} {-} {sigma} _ {v}mathrm {=} 0
The flow rate (visco) plastic is written as:
Cumulative plastic deformation \(\dot{p}\) is viscous and is written as:
: label: eq-52
dot {p}mathrm {=} {} {(frac {mathrm {langle} (tilde {sigma}mathrm {-} X {)}} _ {mathit {eq}}}mathrm {eq}}}mathrm {eq}}}mathrm {eq}}}mathrm {eq}}}mathrm {eq}}}mathrm {eq}}}mathrm {-}} {eq}}mathrm {-} {sigma} _ {sigma} _ {c} (T, Z)mathrm {rangle}} {eta})} {eta}})} ^ {n}
6.3. Borderline cases#
6.3.1. Time-independent plastic model#
We want to describe instantaneous elasto-plastic behavior and cancel viscous effects. For this purpose, the viscous parameters \(\eta\) and \(C\) will be taken to be equal to zero. To get rid of the numerical problems that can arise when taking \(\eta\) and \(C\) zero into account, and in a manner similar to the treatment performed for Taheri’s viscoplastic model [R5.03.05], equation () is rewritten in the form:
The strict inequality being obtained in the case \(f<0\) and \(\dot{p}\mathrm{=}0\) which corresponds to the elastic regime. In the purely plastic field of behavior (\(\eta \to 0\)) inequality () is then reduced to:
So \(\dot{p}\) can no longer be determined except by the consistency equation \(\dot{f}\mathrm{=}0\). We therefore find ourselves in the context of instantaneous plasticity independent of time, with digital processing identical to that classically used for the treatment of this plasticity. Note that \({\sigma }_{c}\) then corresponds to the classical definition of the elastic limit \({\sigma }_{y}\). The elastic limit will be noted \({\sigma }_{c}\) in viscoplasticity and \({\sigma }_{y}\) in time-independent plasticity*.*
6.3.2. Viscous fluid behavior model#
At very high temperature we have \(R\to 0\) and \({\sigma }_{c}\to 0\), the threshold function is therefore reduced:
If we take \(n\to 1\), then:
The flow rate (visco) plastic is written as:
Let’s say in one-dimensional:
A Newtonian viscous fluid behavior model, with viscosity \(\eta\), is thus obtained.
6.4. Multiphase plasticity#
Metallurgical transformations lead to changes in the mechanical characteristics of the material. The elastic characteristics (Young’s modulus and Poisson’s ratio) are little affected by changes in metallurgical structures. Therefore, only their dependence on temperature is taken into account. On the other hand, the plastic characteristics (elasticity limit in particular) depend strongly on the metallurgical structure. It is therefore necessary to take into account the differences in plastic characteristics for each of the possible phases. In modeling, deformation and stress are defined at the scale of the material point (macroscopic), which may be multiphase. The aim is to define the equivalent plastic behavior of the material when it has a multiphase structure, in particular with a unique plasticity criterion.
6.4.1. Linear law of mixtures#
The behavior of the equivalent material is defined using a law of mixtures on the characteristics of the phases. More precisely, the definition of this equivalent material would correspond in 1D to a rheological model of \(k\) bars in parallel such as:
6.4.1.1. Viscoplasticity with isotropic work hardening#
In the case of von Mises viscoplasticity with isotropic work hardening:
We write the work hardening of multiphase material \(\overline{R}\) by applying the law of mixtures:
Where \({R}_{k}\) is the work hardening of phase \(k\). The elastic limit of multi-phase material \({\overline{\sigma }}_{c}\) is written in the same way:
: label: eq-62
{overline {sigma}} _ {c} (T, Z)mathrm {=}mathrm {sum} _ {kmathrm {=} 1} ^ {5} {Z} {Z} _ {Z} _ {k} {sigma} _ {c, k} (T)
Where \({\sigma }_{c,k}\) is the elastic limit for phase \(k\). This gives us a new threshold function on multiphase material:
The plastic deformation rate checks the consistency condition () (function \({f}^{\text{*}}\)). With an average taking place over the**five** phases:
The law of mixtures on the viscous return stress is applied:
And so:
That is, when you are in charge, \(\dot{p}\) is such that:
6.4.1.2. Viscoplasticity with kinematic work hardening#
In the case of von Mises viscoplasticity with kinematic work hardening:
We write the work hardening of multiphase material \(\overline{R}\) by applying the law of mixtures:
The elastic limit of the multiphase material \({\overline{\sigma }}_{c}\) is written in the same way as in the isotropic case ():
This gives us a new threshold function on multiphase material:
: label: eq-71
overline {f} (sigma, X; T, Z)mathrm {=} (tilde {sigma}mathrm {-}overline {X} {)}} _ {mathit {eq}}}mathrm {-}} {overline {sigma}} _ {c} (T, Z)
The plastic deformation rate checks the consistency condition () (function \({f}^{\text{*}}\)). In a similar manner, the law of mixtures on viscous stress is applied. That is, when you are in charge, \(\dot{p}\) is such that:
6.4.2. Nonlinear law of mixtures#
Remember that \({Z}_{f}\) is the sum of all ferritic phases, i.e.:
We also give the possibility of using a non-linear law of mixtures [bib9] _ between the hot phase (austenitic) and the cold phases (ferritic), such that in the one-dimensional case we have:
Quantity \({f}_{h}\) is the mixing function. We then have work-hardening which is written:
\({\overline{R}}_{\alpha }\) is the average work hardening of cold phases (ferritic phases):
The initial stress of the multiphase material is written in the same way:
\({\overline{\sigma }}_{c,\alpha }\) is the equivalent initial stress of cold phases (ferritic phases):
\({f}_{h}({Z}_{f})\) is a user defined function. The elastic limit \({\overline{\sigma }}_{y}\) in the elasto-plastic (non-viscous) case uses the same mixing rule as \({\overline{\sigma }}_{c}\). The set is specified in the DEFI_MATERIAU command.
Physical parameter |
Keyword ELAS_META |
|
Elasticity limit of ferritic phase 1 |
\({\sigma }_{y\mathrm{,1}}\) |
F1_SY |
Elasticity limit of ferritic phase 2 |
\({\sigma }_{y\mathrm{,2}}\) |
F2_SY |
Elasticity limit of ferritic phase 3 |
\({\sigma }_{y\mathrm{,3}}\) |
F3_SY |
Elasticity limit of ferritic phase 4 |
\({\sigma }_{y\mathrm{,4}}\) |
F4_SY |
Elasticity limit of the austenitic phase |
\({\sigma }_{y,\gamma }\) |
C_SY |
Initial critical stress of the ferritic phase 1 |
\({\sigma }_{c\mathrm{,1}}\) |
F1_S_VP |
Initial critical stress of ferritic phase 2 |
\({\sigma }_{c\mathrm{,2}}\) |
F2_S_VP |
Initial critical stress of the ferritic phase 3 |
\({\sigma }_{c\mathrm{,3}}\) |
F3_S_VP |
Initial critical stress of ferritic phase 4 |
\({\sigma }_{c\mathrm{,4}}\) |
F4_S_VP |
Initial critical stress of the austenitic phase |
\({\sigma }_{c,\gamma }\) |
C_S_VP |
Mixing function |
\({f}_{h}({Z}_{f})\) |
SY_MELANGE |
Mixing function for the viscous case |
S_ VP_MELANGE |
|
Parameters \({\eta }_{k}\) and \({n}_{k}\) are defined in DEFI_MATERIAU under the keyword factor META_VISC.
Physical parameter |
Keyword META_VISC |
|
Parameter \(\eta\) of ferritic phase 1 |
\({\eta }_{1}\) |
F1_ ETA |
Parameter \(\eta\) of ferritic phase 2 |
\({\eta }_{2}\) |
F2_ ETA |
Parameter \(\eta\) of ferritic phase 3 |
\({\eta }_{3}\) |
F3_ ETA |
Parameter \(\eta\) of ferritic phase 4 |
\({\eta }_{4}\) |
F4_ ETA |
Parameter \(\eta\) of the austenitic phase |
\({\eta }_{\gamma }\) |
C_ ETA |
Parameter \(n\) of ferritic phase 1 |
\({n}_{1}\) |
F1_N |
Parameter \(n\) of ferritic phase 2 |
\({n}_{2}\) |
F2_N |
Parameter \(n\) of ferritic phase 3 |
\({n}_{3}\) |
F3_N |
Parameter \(n\) of ferritic phase 4 |
\({n}_{4}\) |
F4_N |
Parameter \(n\) of the austenitic phase |
\({n}_{\gamma }\) |
C_N |
6.5. Summary of available models#
Viscoplastic model with isotropic work hardening, viscous restoration and transformation plasticity ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~
By bringing together:
The partition of the deformations ();
Thermal deformation ();
Deformation for transformation plasticity ();
Viscoplastic deformation with isotropic work hardening ();
Hooke’s law;
We obtain the following system to be solved:
By adding, the expression of the viscoplastic law with isotropic work hardening, i.e.:
The threshold function () by applying the law of mixtures on work hardening () and critical stress ();
The modified threshold function ();
We have the expression for the criterion of (visco-) plasticity:
Finally, the update of the work hardening in phases with consideration of metallurgical work hardening restoration and viscous restoration allow us to write the update of the work hardening variables:
: label: eq-81
mathrm {{}begin {array} {c} {c} {dot {r}} {dot {r}}} _ {dot {p}} _ {gamma} + {g} _ {gamma} _ {gamma}} ^ {gamma}} ^ {gamma}} ^ {text {re}} ^ {text {re}} _ {gamma} _ {gamma} _ {gamma} _ {gamma} _ {gamma} _ {gamma} _ {gamma} _ {gamma} _ {gamma} _ {gamma} _ {gamma} _ {gamma} _ {gamma} _ {gamma} _ {gamma} _ {gamma} _ {gamma} _ {gamma} _ {gamma} _ {gamma} _} {Z} _ {gamma} >0text {and} {dot {r}}} _ {gamma}mathrm {=} 0text {otherwise}\ {dot {r}}} _ {{k}} _ {k} _ {k} _ {k} _ {k}}} _ {{k}}} _ {{k}} _ {k} _ {f}} _ {k} _ {k} _ {k} _ {k} _ {f}} ^ {text {re}, m}mathrm {-} {overline {g}}} ^ {text {re}, v}text {si} {Z} _ {k} _ {f}}} >0text {f}}} >0text {f}}} >0text {f}}} >0text {f}}} >0text {f}}} >0text {f}}} >0text {otherwise}}} >0text {otherwise}}} >0text {otherwise}}end {array}
Viscoplastic model with kinematic work hardening, viscous restoration and transformation plasticity ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~
By bringing together:
The partition of the deformations ();
Thermal deformation ();
Deformation for transformation plasticity ();
Viscoplastic deformation with kinematic work hardening ();
Hooke’s law;
We obtain the following system to be solved:
: label: eq-82
mathrm {{}begin {array} {c}tilde {varepsilon}tilde {varepsilon}} {tilde {varepsilon}} ^ {e} + {tilde {varepsilon}} + {tilde {varepsilon}} + {tilde {varepsilon}} + {tilde {varepsilon}} + {tilde {varepsilon}} + {tilde {varepsilon}} + {tilde {varepsilon}} + {tilde {varepsilon}} + {tilde {varepsilon}} + {tilde {vare}}\sigmamathrm {=} Amathrm {:} {varepsilon} ^ {e}\ {varepsilon} ^ {mathit {th}}mathrm {=}}mathrm {=} ({=} ({Z}) _ ({Z} _ {Z} _ {Z} _ {left} _ {gamma}left [{alpha} _ {gamma}} (Tmathrm {-} {th}}}}}mathrm {=}} ({=}} ({Z}) ({Z}} _ {Z} _ {Z} _ {gamma}left [{alpha}} _ {gamma}left [{alpha}} _ {gamma}} mathit {ref}})mathrm {-} (1mathrm {-} {-} {Z} _ {gamma} ^ {R})Delta {varepsilon} _ {fgamma} ^ {{gamma}} ^ {{T} ^ {T}} _ {T} _ {T} _ {T} _ {t} _ {f} _ {f} _ {f} _ {f} _ {f} _ {f} _ {f} _ {f} _ {f} _ {f} _ {f} _ {f} _ {f} _ {f} _ {f} _ {f} _ {f} _ {f} _ {f} _ {f} _ {f} _ {f} _ {f} _ {f} _ {f} _ {f} {-} {T} _ {mathit {ref}}) + {mathit {ref}}}) + {Z} _ {mathit {ref}}) + {mathit {ref}}) + {mathit {ref}}}) + {mathit {ref}}} {gamma} ^ {T}})mathit {ref}}}right])mathit {Id}\ {dot {tilde {varepsilon}}}}} ^ {mathit {pt}}mathrm {=}frac {3} {2}tilde {sigma}mathrm {sum}mathrm {sum} _ {sum} _ {sum} _ {sum} _ {sum} _ {sum} _ {sum} _ {{sum} _ {sum} _ {{sum} _ {{sum} _ {{sum} _ {{sum} _ {{sum}} _ {{sum} _ {{sum} _ {{sum}} _ {{sum} _ {{sum}} _ {{sum} _ {{sum}} _ {{sum} _ {{sum}} _ {{sum} _ {{sum}} _ {k} _ {f}} ^ {text {“}}}langle {dot {Z}}} _ {{k} _ {f}}rangle\ {dot {tilde {varepsilon}}}}}} ^ {varepsilon}}}}langle {dot {” psilon}}}}}langle {dot {“psilon}}}}}langle {dot {” varepsilon}}}}langle {dot {“}}}langle {dot {“}}varepsilon}}}}langle {dot {“}}}langle {dot {” psilon}}}}}langle {dot {”} {sigma}mathrm {-} X}} {(tilde {sigma}mathrm {-} X {)}} _ {mathit {eq}}}}}end {array}
By adding, the expression of the viscoplastic law with isotropic work hardening, i.e.:
The threshold function () by applying the law of mixtures on work hardening () and critical stress ();
The modified threshold function e ();
We have the expression for the criterion of (visco-) plasticity:
: label: eq-83
mathrm {{}begin {array} {c}overline {f}overline {f}mathrm {=} (tilde {sigma}mathrm {-}overline {X} {)}}} _ {mathit {eq}}} _ {mathit {eq}}} _ {c}\ {overline {X} {) {)}} _ {mathit {eq}}} _ {mathit {eq}}}\ mathit {eq}}}}mathrm {-}} {overline {f}}} ^ {text {*}}mathrm {=} (tilde {sigma}tilde {sigma}}mathrm {-}overline {X} {)} _ {mathit {eq}}mathrm {-}}mathrm {-}} _ {mathit {eq}}}mathrm {-}} _ {v}\ dot {p}mathrm {=} 0text {si}overline {f}overline {f} <0\dot {p}mathrm {ge} 0text {si}overline {f}mathrm {=}}mathrm {=}}mathrm {=} 0=} 0text {=} 0text {=} 0text {=} 0text {with}} {with} {with} {with}} {overline {f}}} ^ {text {*}}}mathrm {=} 0mathrm {=} 0text {=} 0text {=} 0text {with} {with}}} {overline {f}}}
Finally, the update of the work hardening in phases with consideration of metallurgical work hardening restoration and viscous restoration allow us to write the update of the work hardening variables:
6.5.1. Behaviour selection#
In terms of available behavioral relationships, the modeling implemented offers several possibilities:
Choice of the type of behavior for plastic deformation: plastic independent of time or with consideration of viscous effects;
Choice of isotropic linear, isotropic non-linear or kinematic work hardening;
Whether or not transformation plasticity is taken into account;
Whether or not the restoration of metallurgical work hardening is taken into account.
The choice of the material (steel or zircaloy) and therefore the number of phases is made by entering the keyword KIT of COMPORTEMENT. ACIER for steel with five phases and ZIRC for zircaloy with three phases.
There are 24 combinations in total.
Behavior |
Plastic |
Visco. |
Work hardening |
Plasticity Transform. |
Catering |
||
Isotropic |
Cinematic |
||||||
Linear |
Non Linear |
Linear |
|||||
META_P_CL_PT |
OUI |
NON |
NON |
NON |
OUI |
OUI |
NON |
META_P_CL_PT_RE |
OUI |
NON |
NON |
NON |
OUI |
OUI |
OUI |
META_P_CL |
OUI |
NON |
NON |
NON |
OUI |
NON |
NON |
META_P_CL_RE |
OUI |
NON |
NON |
NON |
OUI |
NON |
OUI |
META_P_IL_PT |
OUI |
NON |
OUI |
NON |
NON |
OUI |
NON |
META_P_IL_PT_RE |
OUI |
NON |
OUI |
NON |
NON |
OUI |
OUI |
META_P_IL |
OUI |
NON |
OUI |
NON |
NON |
NON |
NON |
META_P_IL_RE |
OUI |
NON |
OUI |
NON |
NON |
NON |
OUI |
META_P_INL_PT |
OUI |
NON |
NON |
OUI |
NON |
OUI |
NON |
META_P_INL_PT_RE |
OUI |
NON |
NON |
OUI |
NON |
OUI |
OUI |
META_P_INL |
OUI |
NON |
NON |
OUI |
NON |
NON |
NON |
META_P_INL_RE |
OUI |
NON |
NON |
OUI |
NON |
NON |
OUI |
META_V_CL_PT |
NON |
OUI |
NON |
NON |
OUI |
OUI |
NON |
META_V_CL_PT_RE |
NON |
OUI |
NON |
NON |
OUI |
OUI |
OUI |
META_V_CL |
NON |
OUI |
NON |
NON |
OUI |
NON |
NON |
META_V_CL_RE |
NON |
OUI |
NON |
NON |
OUI |
NON |
OUI |
META_V_IL_PT |
NON |
OUI |
OUI |
NON |
NON |
OUI |
NON |
META_V_IL_PT_RE |
NON |
OUI |
OUI |
NON |
NON |
OUI |
OUI |
META_V_IL |
NON |
OUI |
OUI |
NON |
NON |
NON |
NON |
META_V_IL_RE |
NON |
OUI |
OUI |
NON |
NON |
NON |
OUI |
META_V_INL_PT |
NON |
OUI |
NON |
OUI |
NON |
OUI |
NON |
META_V_INL_PT_RE |
NON |
OUI |
NON |
OUI |
NON |
OUI |
OUI |
META_V_INL |
NON |
OUI |
NON |
OUI |
NON |
NON |
NON |
META_V_INL_RE |
NON |
OUI |
NON |
OUI |
NON |
NON |
OUI |
For all relationships, the internal variables produced in*code_aster* are:
\({r}_{k}\) the effective work hardening variables for the \(k\) phases. They are named by phase. For example, for isotropic work hardening of a steel alloy, we have: FERRITE # EPSPEQ, #, PERLITE # EPSPEQ, BAINITE # EPSPEQ, MARTENSITE # EPSPEQ and AUSTENITE # EPSPEQ. ;
\(d\) the plasticity indicator (0 if the last calculated increment is elastic; 1 if not); It is a global quantity;
\(R\) the work hardening term for the threshold function. It is a global quantity calculated by a law of mixtures (see § 6.4);
In addition, these models can be carried out with the PETIT_REAC geometric update feature. For isotropic work hardening relationships, the SIMO_MIEHE large deformation model is also available (see [R4.04.03]).