Work hardening restoration ========================== Usually, the work-hardened state of a material is characterized by its plastic history. Thus, for example, in the case of plasticity with isotropic*linear* work hardening, the cumulative plastic deformation noted :math:`p` is generally taken as a work-hardening variable. The term work hardening is then written: .. math:: :label: eq-13 R\ mathrm {=} {R} _ {o} p Where :math:`{R}_{o}` is the linear work hardening coefficient. This model is inadequate in two cases: 1. For viscoplastic materials, under the action of thermal agitation, there is a slow restoration of the crystalline structure of the metal by annihilation of dislocations and relaxation of internal stresses; 2. During metallurgical transformations, there are more or less important atom displacements within the material. These movements of atoms can annihilate dislocations that are at the origin of work hardening; Viscous work-hardening restoration ------------------------------------ Monophasic model ~~~~~~~~~~~~~~~~~~~~~ The phenomenon of viscous restoration of work hardening is introduced into the modeling, which leads to a partial evanescence of the work hardening. The model used to describe this phenomenon is the following in the case of isotropic work hardening: .. math:: :label: eq-14 \ dot {r} =\ dot {p} - {g} ^ {\ text {re}, v} The term for the evolution of the work-hardening variable :math:`r` therefore includes a term for work hardening due to plastic deformation :math:`\dot{p}` and a term for viscous restoration that will be noted :math:`{g}^{\text{re},v}`: .. math:: :label: eq-15 {g} ^ {\ text {re}, v}\ mathrm {=} (\ mathit {Cr} {)} ^ {m} The model thus makes it possible to describe the phenomenon of primary creep (work hardening) and secondary creep (stabilization of work hardening) with the parameters :math:`C` and :math:`m`. In the case of kinematic work hardening, the model will be written as: .. math:: :label: eq-16 \ dot {\ alpha}\ mathrm {=} {\ dot {\ varepsilon}} ^ {\ mathit {VP}}} + {h} ^ {\ text {re}, v} With the term restoration that will be noted :math:`{h}^{\text{re},v}`: .. math:: :label: eq-17 {h} ^ {\ text {re}, v} =\ frac {3} {2} (C {\ alpha} _ {\ mathit {eq}} {)} ^ {m}\ frac {\ alpha}\ frac {\ alpha} {\ alpha} {{\ alpha}} _ {\ mathit {eq}}} .. _RefNumPara__10611_18946486181: Multiphasic model ~~~~~~~~~~~~~~~~~~~~~~ In the multiphase case, the viscous part of the restoration is expressed by replacing the work-hardening variable by an average value over the phases: .. math:: :label: eq-18 r\ to\ overline {r} With an average taking place over the **five** phases: .. math:: :label: eq-19 \ overline {r}\ mathrm {=}\ mathrm {\ sum} _ {k\ mathrm {=} 1} ^ {5} {Z} _ {k} {Z} _ {r} _ {k} And so: .. math:: :label: eq-20 {\ overline {g}} ^ {\ text {re}, v}\ mathrm {=} (\ overline {C}\ overline {r} {)}} ^ {\ text {re}}} ^ {\ overline {m}} In the case of kinematic work hardening, according to (), we have: .. math:: : label: eq-21 {h} ^ {\ text {re}, v}\ mathrm {=}\ mathrm {=}\ frac {3} {2} (C {\ alpha} _ {\ mathit {eq}} {)}} {)} ^ {m}}\ frac {\ alpha}}\ frac {\ alpha} {\ alpha} _ {\ mathit {eq}}} {)} ^ {m}}\ frac {\ alpha}} The procedure is the same as for isotropic work hardening: .. math:: :label: eq-22 \ alpha\ to\ overline {\ alpha} With: .. math:: :label: eq-23 \ overline {\ alpha}\ mathrm {=}\ mathrm {\ sum} _ {k\ mathrm {=} 1} ^ {5} {Z} _ {k} _ {k} {k} {\ overline {\ alpha}} And so: .. math:: :label: eq-24 {\ overline {h}} ^ {\ text {re}, v}\ mathrm {=}\ frac {3} {2} (\ overline {C} {\ overline {\ alpha}}} _ {\ mathit {eq}}} {)} {)} {)}} {\ mathit {eq}} {)} {\ mathit {eq}} {)} {)} {)} ^ {\ overline {eq}} {)} {)} ^ {\ overline {eq}} {)} {)} ^ {\ overline {\ alpha}} {)} {)} ^ {\ overline {\ alpha}} {)} {)} ^ {\ overline {\ alpha}} {)} {)} {\ mathit {eq}} {\ mathit {eq}}} Metallurgical isotropic work hardening restoration ------------------------------------------------- In the multi-phase case, the term work hardening is written for each phase: .. math:: :label: eq-25 {R} _ {k}\ mathrm {=} {R} _ {\ mathrm {0,} k} p Where :math:`{R}_{\mathrm{0,}k}` is the linear work hardening coefficient for phase :math:`k`. During metallurgical transformations, there are more or less important atom displacements within the material. These movements of atoms can annihilate dislocations that are at the origin of work hardening. In these cases, the work hardening of the mother phase is not transmitted to the phase produced, it is the restoration of work hardening. The new phase can then be born with a virgin plastic state or inherit only a portion, possibly all, of the work-hardening of the mother phase. Cumulative plastic deformation :math:`p` is no longer characteristic of the work-hardening state and it is necessary to define other work hardening variables for each phase, noted :math:`{r}_{k}`, which take into account the work-hardening restoration. The work-hardening term for phase :math:`k` is then written: .. math:: :label: eq-26 {R} _ {k}\ mathrm {=} {R} _ {\ mathrm {0,} k} {r} _ {k} Two-phase model with a sense of transformation ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ To define variables :math:`{r}_{k}`, we choose the model proposed by Leblond [bib11] _. We consider a two-phase volume element :math:`V` that undergoes metallurgical transformation and plastic deformation. Phase 1 is the mother phase, characterized by its volume fraction :math:`{V}_{1}`, its phase proportion :math:`(1\mathrm{-}z)` and a work-hardening variable :math:`{r}_{1}`. Phase 2 is the phase produced, with its volume fraction :math:`{V}_{2}`, its phase proportion :math:`z` and its work-hardening variable :math:`{r}_{2}`. The equations for the evolution of :math:`{r}_{i}` obtained by derivation with respect to time are written: .. math:: : label: eq-27 \ mathrm {\ {}\ begin {array} {c} {c} {\ dot {r}}} _ {1}\ mathrm {=}\ dot {p}\\ {\ dot {r}} _ {2}\ mathrm {=}\ mathrm {=}}\ dot {p}}\ mathrm {-}\ frac {\ dot {z}} {z} {r} {r}} {2}} _ {2} +\ frac {\ dot {z}} {z} {z}\ theta {r} _ {1}\ end {array} The parameter :math:`\theta` characterizes the proportion of work hardening transmitted from the mother phase to the produced phase and :math:`\dot{p}` is the equivalent plastic deformation rate. :math:`p` here is no longer an internal variable of the problem as such. The only meaning of :math:`\dot{p}` here is to be the plastic multiplier and it is equal to the equivalent plastic deformation rate. Sjöström obtains the same equations using phenomenological reasoning that is reported here to explain the [bi612] _ model. Let it be a :math:`\Delta t` time increment, such as between :math:`t` and :math:`t+\Delta t`: * A fraction :math:`\Delta {V}_{2}` of the mother phase is transformed into phase 2 and is therefore added to volume :math:`{V}_{2}` of this product phase; * Volume element :math:`V` undergoes plastic deformation :math:`\Delta p`. +-------------------------------------------------------------------------------------+----------------------------------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------------------+ | | +-------------------------------------------------------------------------------------+----------------------------------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------------------+ |:math:`\mathrm{\{}\begin{array}{c}{V}_{2},{r}_{2}(t)\\ {V}_{1},{r}_{1}(t)\end{array}`|:math:`\mathrm{\{}\begin{array}{c}\Delta t\\ \Delta p\\ \Delta {V}_{2}\end{array}`|:math:`\{\begin{array}{}{V}_{2},{r}_{2}(t)+\Delta p\\ \Delta {V}_{2},\theta {r}_{1}(t)+\Delta p\\ {V}_{1},{r}_{1}(t)+\Delta p\end{array}`| +-------------------------------------------------------------------------------------+----------------------------------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------------------+ It is assumed that during metallurgical transformation, the transformed fraction :math:`\Delta {V}_{2}` inherits only part :math:`\theta {r}_{1}` of the work-hardening of the mother phase :math:`0\le \theta \le 1`. So the work hardening variables :math:`{r}_{i}` at the moment :math:`t+\Delta t` are such that: .. math:: : label: eq-28 \ mathrm {\ {}\ begin {array} {c} {c} {r} {r} _ {1} (t+\ Delta t)\ mathrm {=} {r} _ {1} (t) +\ Delta p\\\\ {r} {r} {r} {r} {r} {r} {r} {r} {r} {r} {r} {r} {r} {r} {r} {r} {r} {r} {r} {r} {r} {r} {r} {r} {r} {r} {r} {r} {r} {r} {r} {r} {r}} {r} {r} {r} {r} {r} {r} {r} {r} {r} {r} (t) +\ Delta p) +\ Delta {V} _ {2} _ {2} (\ theta {r} _ {1} (t) +\ Delta p)} {{V} _ {2}} +\ Delta {V} _ {2}}\ end {array} +\ Delta {V} _ {2}} That is, considering a standard incremental writing of the work hardening variables such as: .. math:: :label: eq-29 \ Delta {r} _ {i}\ mathrm {=} {r} _ {r} _ {i} (t+ {\ Delta} _ {t})\ mathrm {-} {r} {r} _ {i} (t) We get: .. math:: : label: eq-30 \ mathrm {\ {}\ begin {array} {c} {c}\ Delta {r} _ {1}\ mathrm {=}\ Delta p\\\ Delta {r} _ {2}\ mathrm {=}\ mathrm {=}}\\ Delta p+\ frac {\ frac {\ frac {\\ Delta {V}} _ {2}}\ theta {r} _ {1} ^ {\ mathrm {-}}\ mathrm {-}}\ mathrm {-}}\ frac {\ Delta {V} _ {2}} +\ Delta {V} _ {2} _ {2}}} {2}}\ delta {V} _ {2}} {\ mathrm {-}}\ end {array} _ {2}} +\ Delta {V} _ {2}} +\ Delta {V} _ {2}} +\ Delta {V} _ {2}} +\ Delta {V} _ {2}} +\ Delta {V} _ {2}} +\ Delta {V} _ { We get the equations () by going to the limit. For the discretization of the laws of evolution of :math:`{r}_{i}`, we choose an explicit integration scheme using the equations () directly. Generalization of the n-phase model with two-way transformations ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ In the case of steel, the existing phases are ferrite, pearlite, pearlite, bainite, martensite and austenite :math:`\gamma` with respective proportions :math:`{Z}_{{k}_{f}\mathrm{=}\mathrm{1,4}}` and :math:`{Z}_{\gamma }`. When heating, the metallurgical transformations to consider are transformations from ferritic phases (ferrite, pearlite, bainite and martensite) to the austenitic phase. The proportion of austenite increases and therefore: .. math:: : label: eq-31 \ mathrm {\ {}\ begin {array} {cc} {cc}\ text {Si} {Z} _ {\ gamma} >0& {\ dot {r}} _ {\ gamma}\ mathrm {=} {\ dot {p}} {\ dot {p}}}} _ {\ gamma}} _ {\ gamma} >0& {\ dot {r} ^ {\ text {re}, m}\\\ text {Otherwise} & {\ dot {r}}} _ {\ gamma}\ mathrm {=} 0\ end {array} With :math:`{g}_{\gamma }^{\text{re},m}` the metallurgical isotropic work hardening restoration function for austenite: .. math:: : label: eq-32 {g} _ {\ gamma} ^ {\ text {re}, m}\ mathrm {=}\ frac {\ mathrm {\ sum} _ {f}\ mathrm {=} 1} 1} ^ {4}\ mathrm {4}\ mathrm {\ 4}}\ mathrm {\ langle} {\ langle}}\ mathrm {-}\ dot {Z}}} _ {{k} _ {=} 1} 1} ^ {4}}\ mathrm {4}\ mathrm {4}}\ mathrm {\ langle} {\ langle}}\ mathrm {-}\ dot {Z}} _ {{k} _ {=}} 1} ^ {4}}\ mathrm {4}}\ mathrm m {\ rangle} {\ theta} _ {{k} _ {f} _ {f},\ gamma} {r} _ {f}}\ mathrm {-}\ mathrm {\ sum} _ {\ sum} _ {k} _ {f} _ {f}\ mathrm {=} {f}\ mathrm {=} 1}} ^ {4}\ mathrm {\ langle} {\ mathrm {-}\ sum} _ {sum} _ {sum} _ {k} _ {f} _ {f}\ dot {Z}}} _ {{k} _ {f}}}\ mathrm {\ rangle} {r} _ {\ gamma}} {{Z} _ {\ gamma}} Where :math:`{\theta }_{{k}_{f},\gamma }` is the proportion of work-hardening restoration during the transformation from a ferritic phase to the austenitic phase. In the case of cooling, the metallurgical transformations to be considered are the transformations from austenite to ferritic phases (ferrite, pearlite, bainite and martensite). It is the proportions of ferritic phases that increase and we have: .. math:: :label: eq-33 \ mathrm {\ {}\ begin {array} {cc} {cc}\ text {Si} {Z}} _ {{k}}} >0& {\ dot {r}}} _ {{k} _ {f}}}\ mathrm {=}}\ mathrm {=}}\ mathrm {=}} {=}} {=} {\ dot {p}}} _ {{k} _ {f}}}}\ mathrm {=}} {=} {=} {\ dot {p}}} _ {{k} _ {f}}}}\ mathrm {=}} {=} {=} {\ dot {p}}} _ {{k} _ {f}}}}\ mathrm {=}} {=} {=} {text {re}, m}\\\ text {Otherwise} & {\ dot {r}} _ {{k} _ {f}}\ mathrm {=} 0\ end {array}} 0\ end {array} With :math:`{g}_{{k}_{f}}^{\text{re},m}` the isotropic metallurgical work hardening restoration function for ferritic phases: .. math:: :label: eq-34 {g} _ {{k} _ {f}}} ^ {\ text {re}} ^ {\ text {re}, m}}\ mathrm {=}\ frac {\ mathrm {\ langle} {\ dot {Z}}} _ {{k}} _ {{k} _ {k}} _ {f}} _ {f}} _ {f}} _ {f}} _ {f}} _ {f}} _ {f}} _ {f}} {f}} {r} _ {f}}} {f}} _ {f}} {f}} {r} _ {f}}} {f}} _ {f}} {f}} {r} _ {f}}} {f}} _ {f}} {f}} {f}}\ mathrm {-}\ mathrm {\ langle} {\ dot {Z}} {\ dot {Z}}} _ {{k} _ {f}}\ mathrm {\ rangle} {r} _ {k} _ {f}} _ {f}}} {{f}}} :math:`{\theta }_{\gamma ,k}` is the proportion of work-hardening restoration during the transformation of the austenitic phase into a ferritic phase. For transformations with diffusion (for example: austenite to ferrite, pearlite and bainite), involving significant movements of atoms, we can take :math:`\theta \mathrm{=}0`; the dislocations at the origin of plastic work hardening are completely annihilated by the transformation. For transformations without diffusion (for example a martensitic transformation), we can take :math:`\theta \mathrm{=}1`, with the work hardening being completely transmitted. Metallurgical kinematic work hardening restoration ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ By analogy with the case of isotropic work hardening, the restoration of kinematic work hardening in the multiphase case will be written, for the case of austenite: .. math:: :label: eq-35 \ mathrm {\ {}\ begin {array} {cc}\ text {Si} {C}\ text {Si} {Z} _ {\ gamma}} _ {\ gamma}\ mathrm {=} {=} {\ dot {\ varepsilon}}\ {\ dot {\ varepsilon}}}} ^ {\ mathit {VP}}} + {\ dot {\ alpha}}} _ {\ gamma}} ^ {\ text {re}, m}\\ text {Otherwise} & {\ dot {\ alpha}}} _ {\ gamma}\ mathrm {=} 0\ end {array}} With :math:`{h}_{\gamma }^{\text{re},m}` the metallurgical isotropic work hardening restoration function for austenite: .. math:: :label: eq-36 {h} _ {\ gamma} ^ {\ text {re}, m}\ mathrm {=}\ frac {\ mathrm {\ sum} _ {f}\ mathrm {=} 1} 1} ^ {4}\ mathrm {4}\ mathrm {\ 4}}\ mathrm {\ langle} {\ langle}}\ mathrm {-}\ dot {Z}}} _ {{k} _ {=} 1}} ^ {4}}\ mathrm {4}}\ mathrm {\ langle} {\ langle} {\ langle} {\ m {\ rangle} {\ theta} _ {{k} _ {f} _ {f},\ gamma} {\ alpha} _ {f}}\ mathrm {-}\ mathrm {\ sum} _ {\ sum} _ {m} _ {f} _ {f}\ mathrm {=} {f}\ mathrm {-}\ dot Z}} _ {{k} _ {f}}}\ mathrm {\ rangle} {\ rangle} {\ alpha} _ {\ gamma}} {{Z} _ {\ gamma}} And for the ferritic phases: .. math:: :label: eq-37 \ mathrm {\ {}\ begin {array} {cc} {cc}\ text {Si} {Z}} _ {{k} _ {f}} >0& {\ dot {\ alpha}} _ {f}} _ {f}}\ mathrm {=}}\ mathrm {=}} {=} {\ dot {\ varepsilon}}} ^ {\ mathit {VP}}} + {h} _ {f} _ {f}}}\ mathrm {=}} {=} {=} {\ dot {varepsilon}}} ^ {\ mathit {VP}}} + {h} _ {k} _ {f}}}}} ^ {\ text {re}, m}\\\ text {Otherwise} & {\ dot {\ alpha}} _ {{k} _ {f}}}\ mathrm {=}} 0\ end {array}} With :math:`{h}_{{k}_{f}}^{\text{re},m}` the kinematic metallurgical work hardening restoration function for ferritic phases: .. math:: :label: eq-38 {h} _ {{k} _ {f}}} ^ {\ text {re}} ^ {\ text {re}, m}\ mathrm {=}\ frac {\ mathrm {\ langle} {\ dot {Z}}} _ {{k}} _ {{k} _ {k}} _ {k} _ {f}} _ {f}} _ {f}} _ {f} _ {f}} _ {f} _ {f}} _ {f} _ {f}} _ {f} _ {f}} _ {f}} _ {f} _ {f}} _ {f}} _ {f} _ {f}} _ {f}} _ {f} _ {f}} _ {f}} _ {f} _ {f}} _ {f}}\ mathrm {-}\ mathrm {\ langle} {\ langle} {\ dot {Z}} {\ dot {Z}} _ {{k}} {\ alpha} _ {{k} _ {f}}} {\ dot {Z}}} {{z} _ {f}}} Material parameters -------------------- Parameters :math:`\theta` for metallurgical work hardening restoration are provided by the user in operator DEFI_MATERIAU under the keyword META_RE. +-------------------------------------------------------------------------+--------------------------------------+------------------------+ |**Physical parameter for metallurgical work hardening restoration** |**Keyword** **MET A_RE**| +-------------------------------------------------------------------------+--------------------------------------+------------------------+ |Rate of work hardening transmitted from austenite to the ferritic phase 1|:math:`{\theta }_{\gamma ,1}` |C_F1_ THETA | +-------------------------------------------------------------------------+--------------------------------------+------------------------+ |Rate of work hardening transmitted from austenite to the ferritic phase 2|:math:`{\theta }_{\gamma ,2}` |C_F2_ THETA | +-------------------------------------------------------------------------+--------------------------------------+------------------------+ |Rate of work hardening transmitted from austenite to the ferritic phase 3|:math:`{\theta }_{\gamma ,3}` |C_F3_ THETA | +-------------------------------------------------------------------------+--------------------------------------+------------------------+ |Rate of work hardening transmitted from austenite to the ferritic phase 4|:math:`{\theta }_{\gamma ,4}` |C_F4_ THETA | +-------------------------------------------------------------------------+--------------------------------------+------------------------+ |Hardening rate transmitted from ferritic phase 1 to austenite |:math:`{\theta }_{\mathrm{1,}\gamma }`|F1_C_ THETA | +-------------------------------------------------------------------------+--------------------------------------+------------------------+ |Hardening rate transmitted from ferritic phase 2 to austenite |:math:`{\theta }_{\mathrm{2,}\gamma }`|F2_C_ THETA | +-------------------------------------------------------------------------+--------------------------------------+------------------------+ |Hardening rate transmitted from ferritic phase 3 to austenite |:math:`{\theta }_{\mathrm{3,}\gamma }`|F3_C_ THETA | +-------------------------------------------------------------------------+--------------------------------------+------------------------+ |Hardening rate transmitted from ferritic phase 4 to austenite |:math:`{\theta }_{\mathrm{4,}\gamma }`|F4_C_ THETA | +-------------------------------------------------------------------------+--------------------------------------+------------------------+ The parameters :math:`C` and :math:`m` for the viscous work hardening restoration are provided by the user in the DEFI_MATERIAU operator under the keyword META_VISC. +-------------------------------------------------------------+---------------------+-------------------------+ |**Physical parameter for viscous work hardening restoration** |**Keyword** **META_VISC**| +-------------------------------------------------------------+---------------------+-------------------------+ |Parameter :math:`m` of ferritic phase 1 |:math:`{m}_{1}` |F1_M | +-------------------------------------------------------------+---------------------+-------------------------+ |Parameter :math:`m` of ferritic phase 2 |:math:`{m}_{2}` |F2_M | +-------------------------------------------------------------+---------------------+-------------------------+ |Parameter :math:`m` of ferritic phase 3 |:math:`{m}_{3}` |F3_M | +-------------------------------------------------------------+---------------------+-------------------------+ |Parameter :math:`m` of ferritic phase 4 |:math:`{m}_{4}` |F4_M | +-------------------------------------------------------------+---------------------+-------------------------+ |Parameter :math:`m` of the austenitic phase |:math:`{m}_{\gamma }`|C_M | +-------------------------------------------------------------+---------------------+-------------------------+ |Parameter :math:`C` of ferritic phase 1 |:math:`{C}_{1}` |F1_C | +-------------------------------------------------------------+---------------------+-------------------------+ |Parameter :math:`C` of ferritic phase 2 |:math:`{C}_{2}` |F2_C | +-------------------------------------------------------------+---------------------+-------------------------+ |Parameter :math:`C` of ferritic phase 3 |:math:`{C}_{3}` |F3_C | +-------------------------------------------------------------+---------------------+-------------------------+ |Parameter :math:`C` of ferritic phase 4 |:math:`{C}_{4}` |F4_C | +-------------------------------------------------------------+---------------------+-------------------------+ |Parameter :math:`C` of the austenitic phase |:math:`{C}_{\gamma }`|C_C | +-------------------------------------------------------------+---------------------+-------------------------+