Numerical formulation ===================== Keywords, material data and internal variables --------------------------------- For foreseeable applications, the model was implemented under two distinct keywords: 'ROUSS_PR' for the plastic model with cavity nucleation or 'ROUSS_VISC' for the viscoplastic model without nucleation. This makes it possible to avoid unnecessary numerical calculations. The corresponding simplified equations are obtained from the general equations by setting :math:`{\mathrm{\sigma }}_{0}=0` or :math:`{A}_{n}=0` respectively. 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 ([:external:ref:`U4.43.01 `]). The parameters of the viscoplastic model (:math:`{\mathrm{\sigma }}_{0}`, :math:`{\dot{\mathrm{\varepsilon }}}_{0}`, and :math:`m`) are provided by the keyword 'VISC_SINH'. The internal variables produced in*Code_Aster* are: * V1, the cumulative plastic deformation :math:`p`, * V2, porosity :math:`f`, * V3 to V8, the :math:`{\mathrm{\varepsilon }}^{e}` elastic deformation tensor, * V9, the plasticity indicator (0 if the last calculated increment is elastic, 1 if a regular plastic solution, 2 if a singular plastic solution). We now present the numerical integration of the law of behavior and give the expression for the tangent matrix (options FULL_MECA and RIGI_MECA_TANG). Expression of the discretized model ------------------------------- The numerical resolution is done by a :math:`\mathrm{\theta }` -method, with :math:`0\mathrm{\le }\theta \mathrm{\le }1`, and incrementally. For all quantities :math:`Q`, we define: :math:`Q={Q}^{\text{-}}+\Deltaq` :math:`{Q}^{\theta }={Q}^{\text{-}}+\theta \Deltaq` Incremental writing requires taking into account the possible variation in material properties (due, for example, to a change in temperature during the time step). The discretized system of equations is: .. _RefEquation 4.2-1: :math:`{\tilde{\tau }}^{\theta }=2\mu \theta \Delta \tilde{{\varepsilon }^{e}}+\frac{2\mu \theta +(1-\theta )2{\mu }^{\text{-}}}{2{\mu }^{\text{-}}}\tilde{{\tau }^{\text{-}}}\text{=}2\mu \theta (\Delta \tilde{\varepsilon }-\Delta {\tilde{\varepsilon }}^{p})+\frac{2\mu \theta +(1-\theta )2{\mu }^{\text{-}}}{2{\mu }^{\text{-}}}\tilde{{\tau }^{\text{-}}}` eq 4.2-1 .. _RefEquation 4.2-2: :math:`{\tau }_{m}^{\theta }=K\theta \text{tr}\Delta {\varepsilon }^{e}+\frac{\mathrm{3K}\ast \theta +(1-\theta ){\mathrm{3K}}^{\text{-}}}{{\mathrm{3K}}^{\text{-}}}{\tau }_{m}^{\text{-}}=K\theta (\text{tr}\Delta \varepsilon -\text{tr}\Delta {\varepsilon }^{p})\frac{\mathrm{3K}\ast \theta +(1-\theta ){\mathrm{3K}}^{\text{-}}}{{\mathrm{3K}}^{\text{-}}}{\tau }_{m}^{\text{-}}` eq 4.2-2 .. _RefEquation 4.2-3: :math:`\Delta {\tilde{\varepsilon }}^{p}=\Delta p\frac{3{\tilde{\tau }}^{\theta }}{2{\tau }_{\text{eq}}^{\theta }}` eq 4.2-3 .. _RefEquation 4.2-4: :math:`\text{tr}\Delta {\varepsilon }^{p}=\Delta {\text{pD}}_{1}({f}^{\theta }+{A}_{n}{p}^{\theta })\text{exp}(\frac{{\tau }_{m}^{\theta }}{{\sigma }_{1}})` eq 4.2-4 .. _RefEquation 4.2-5: :math:`\Deltaf ={A}_{1}(1-{f}^{\theta })\text{tr}\Delta {\varepsilon }^{p}` eq 4.2-5 :math:`{\Phi }_{\text{vp}}^{\theta }={\tau }_{\text{eq}}^{\theta }+{\sigma }_{1}{D}_{1}({f}^{\theta }+{A}_{n}{p}^{\theta })\text{exp}(\frac{{\tau }_{m}^{\theta }}{{\sigma }_{1}})-R({p}^{\theta })-{\sigma }_{0}{\text{sh}}^{\text{-}1}\left[{(\frac{\Delta p}{{\dot{\varepsilon }}_{0}\Delta t})}^{\frac{1}{m}}\right]=0` **eq 4.2-6** This system comes down to solving a single scalar equation for the unknown :math:`\Deltaf`, knowing :math:`\Delta \varepsilon`, :math:`\Deltat` and the quantities :math:`{Q}^{\text{-}}`. Note that :math:`\rho` does not intervene in the algorithm, on the other hand it will intervene in the calculation of the coherent tangent matrix. The following are calculated successively: .. _RefEquation 4.2-7: :math:`{\tau }_{m}^{\theta }=\frac{\mathrm{3K}\ast \theta +(1-\theta ){\mathrm{3K}}^{\text{-}}}{{\mathrm{3K}}^{\text{-}}}{\tau }_{{m}^{\text{-}}}+K\theta (\text{tr}\Delta \varepsilon -\frac{\Delta f}{{A}_{1}(1-{f}^{\theta })})` eq 4.2-7 :math:`\Deltap` is the positive root of the quadratic equation: .. _RefEquation 4.2-8: :math:`{A}_{n}\theta {(\Delta p)}^{2}+({f}^{\theta }+{A}_{n}{p}^{\text{\_}})\Delta p-\frac{\Delta f}{{A}_{1}(1-{f}^{\theta })}\frac{1}{{D}_{1}\text{exp}({\tau }_{m}^{\theta }/{\sigma }_{1})}=0` eq 4.2-8 :math:`{\tilde{\tau }}^{\theta }=(1-\frac{3\mu \theta \Delta p}{{\left[\frac{2\mu \theta +(1-\theta )2{\mu }^{\text{-}}}{2{\mu }^{\text{-}}}{\tilde{\tau }}^{\text{-}}+2\mu \theta \Delta \tilde{\varepsilon }\right]}_{\text{eq}}})(\frac{2\mu \theta +(1-\theta )2{\mu }^{\text{-}}}{2{\mu }^{\text{-}}}{\tilde{\tau }}^{\text{-}}+2\mu \theta \Delta \tilde{\varepsilon })` eq 4.2-9 :math:`{\tau }_{\text{eq}}^{\theta }={\left[\frac{2\mu \theta +(1-\theta )2{\mu }^{\text{-}}}{2{\mu }^{\text{-}}}{\tilde{\tau }}^{\text{-}}+2\mu \theta \Delta \tilde{\varepsilon }\right]}_{\text{eq}}-3\mu \theta \Delta p` eq 4.2-10 The scalar equation for :math:`\Deltaf` is the equation [:ref:`éq 4.2-6 <éq 4.2-6>`] :math:`{\Phi }_{\text{vp}}^{\theta }=0`. **Note 1**: *As* :math:`\Deltaf` *is very weak in most of the structure, it would be best to use* :math:`\Deltap` *as the main unknown. But in this case it is not possible to reduce yourself to a scalar equation, which makes it more difficult to use a Newtons-type method. This is also one reason why the equations* [:ref:`éq 1-1 <éq 1-1>`] *,* [:ref:`éq 3.2-6 <éq 3.2-6>`], and [:ref:`éq4.2-5 <éq4.2-5>`] *have not been modified by the introduction of cavity nucleation.* **Note 2**: *The equation* [:ref:`éq 3.2-6 <éq 3.2-6>`] *can be integrated exactly:* :math:`\text{tr}{\varepsilon }^{p}=\frac{1}{{A}_{1}}\text{ln}(\frac{1-{f}_{0}}{1-f})` *from where:* :math:`\text{tr}\Delta {\varepsilon }^{p}=\frac{1}{{A}_{1}\theta }\text{ln}(\frac{1-{f}^{\text{-}}}{1-{f}^{\theta }})` *As the numerical parameter* :math:`{A}_{1}` *can be changed discontinuously, the derived form* [:ref:`éq 4.2-5 <éq 4.2-5>`] *has been retained, including in the calculation of the coherent tangent matrix. If the use of the parameter* :math:`{A}_{1}` *were to be abandoned in a later release, consideration should be given to using the built-in form.* **Note 3**: *The integrated form* :math:`{\Phi }_{\text{vp}}^{\theta }=0` *is used, including in plasticity instead of the consistency relationship* :math:`\dot{F}=0` *which gives* :math:`\dot{p}` *. The coherent tangent matrix is calculated with this integrated form.* Solving the nonlinear scalar equation ---------------------------------------------- Equation :math:`{\Phi }_{\text{vp}}^{\theta }(\Deltaf )=0` is solved by a Newton algorithm with controlled terminals in routine LCROUS. :math:`{\Phi }_{\text{vp}}^{\theta }(\Deltaf )` and its derivative with respect to :math:`\Deltaf` are calculated in the routine RSLPHI called by LCROUS. The initial values of the terminals are: * lower bound: :math:`\Delta {f}_{1}=0` since :math:`{\Phi }_{\text{vp}}^{\theta }(0)<0` (we checked beforehand that the elastic branch (negative threshold) is not a solution), * upper bound: :math:`\Delta {f}_{2}` such as :math:`{\Phi }_{\text{vp}}^{\theta }(0)>0` searched by dichotomy between 0 and :math:`1-{f}^{\text{-}}` (first value for this search: :math:`\frac{1-{f}^{\text{-}}}{2}`). Newton's algorithm starts with the value :math:`\Deltaf =0`. Whatever the value found for :math:`\Deltaf`, we therefore note that the function :math:`{\Phi }_{\text{vp}}^{\theta }(\Deltaf )` and its derivative with respect to :math:`\Deltaf` are at least calculated for :math:`\Deltaf =0` and :math:`\frac{1-{f}^{\text{-}}}{2}`. Developments made to improve the convergence and robustness of the algorithm are described in [:ref:`bib5 `]. .. _Ref521234688: .. _Ref521234659: .. _Ref521234468: .. _Ref521234429: .. _Ref521234365: .. _Ref521234337: .. _Ref521234296: .. _Ref521234197: .. _Ref521234080: Behavior tangent matrix expression --------------------------------- One gives here the expression for the tangent matrix (option FULL_MECA during Newton iterations, option RIGI_MECA_TANG for the first iteration). For option RIGI_MECA_TANG, the tangent operator is the same operator that connects :math:`{\varepsilon }^{e}` to :math:`\sigma` in [:ref:`éq3.2-2 <éq3.2-2>`]. For option FULL_MECA, the tangent matrix is obtained by linearizing the system of equations that govern the law of behavior: [:ref:`éq 4.2-1 <éq 4.2-1>`] to [:ref:`éq 4.2-6 <éq 4.2-6>`]. So it is a *coherent* tangent matrix. To simplify the expressions, in this paragraph we note [:ref:`§4.5 <§4.5>`]: :math:`Q` for :math:`{Q}^{\mathrm{\theta }}`, the quantities all being expressed at time :math:`{t}^{\theta }={t}^{\text{-}}+\theta \Deltat`. The coherent tangent matrix is: :math:`\frac{\delta \sigma }{\delta \varepsilon }=\rho \left[{a}_{3}\mathrm{II}+\mathrm{Id}\otimes (\frac{{a}_{1}-{a}_{3}}{3}\mathrm{Id}+{a}_{2}\tilde{\tau })+\tilde{\tau }\otimes ({a}_{4}\tilde{\tau }+\frac{{a}_{5}}{3}\mathrm{Id})+\tau \otimes ({y}_{4}(\frac{{a}_{1}}{\mathrm{3K}}-1)\mathrm{Id}+\frac{{y}_{5}}{\text{K}}\tilde{\tau })\right]` eq 4.4-1 This operator is calculated in routine RSLJPL. The coefficients are calculated as follows: .. _RefEquation 4.4-2: :math:`{a}_{1}=\mathrm{3K}+{y}_{1}K{\tau }_{\text{eq}}({z}_{7}+{z}_{2}\theta \Deltap )` eq 4.4-2 .. _RefEquation 4.4-3: :math:`{a}_{2}=\mathrm{\mu }({y}_{1}+{y}_{3}){\mathrm{\sigma }}_{1}` eq 4.4-3 .. _RefEquation 4.4-4: :math:`{a}_{3}=\frac{2\mu {\tau }_{\text{eq}}}{{z}_{5}}` eq 4.4-4 .. _RefEquation 4.4-5: :math:`{a}_{4}=3\mu {y}_{2}{x}_{2}` eq 4.4-5 .. _RefEquation 4.4-6: :math:`{a}_{5}=3\mu {y}_{1}{\sigma }_{1}` eq 4.4-6 .. _RefEquation 4.4-7: :math:`{a}_{6}=3\muk \theta \Deltap -{a}_{2}{\tau }_{\text{eq}}{\sigma }_{1}` eq 4.4-7 .. _RefEquation 4.4-8: :math:`{y}_{1}=-\frac{3{\text{Kz}}_{6}{z}_{1}(f+{A}_{n}p)}{{x}_{1}{\tau }_{\text{eq}}}` eq 4.4-8 .. _RefEquation 4.4-9: :math:`{y}_{2}=-\frac{\mathrm{3\mu }}{{x}_{1}{z}_{5}{\mathrm{\tau }}_{\text{eq}}^{2}}` eq 4.4-9 .. _RefEquation 4.4-10: :math:`{y}_{3}=-\frac{3{\text{Kz}}_{6}{z}_{1}{A}_{n}\theta \Deltap }{{x}_{1}{\tau }_{\text{eq}}}` eq 4.4-10 .. _RefEquation 4.4-11: :math:`{y}_{4}=\frac{{A}_{1}{z}_{8}}{{z}_{1}}+\frac{{z}_{9}{\sigma }_{1}}{{z}_{7}+{z}_{2}\theta \Deltap }` eq 4.4-11 .. _RefEquation 4.4-12: :math:`{y}_{5}=\frac{{A}_{1}{a}_{2}{z}_{8}}{{z}_{1}}-\frac{{z}_{9}{a}_{6}}{{\tau }_{\text{eq}}({z}_{7}+{z}_{2}\theta \Deltap )}` eq 4.4-12 .. _RefEquation 4.4-13: :math:`{z}_{1}=1+{A}_{1}\theta \Delta {\text{pD}}_{1}(f+{A}_{n}p)\text{exp}(\frac{{\tau }_{m}}{{\sigma }_{1}})` eq 4.4-13 .. _RefEquation 4.4-14: :math:`{z}_{2}=\mathrm{3\mu }+{R}_{\text{vp}}` eq 4.4-14 .. _RefEquation 4.4-15: :math:`{z}_{3}=K(f+{A}_{n}p){z}_{1}-{A}_{1}{\sigma }_{1}(1-f)` eq 4.4-15 .. _RefEquation 4.4-16: :math:`{z}_{4}={R}_{\text{vp}}\theta \Deltap -{\tau }_{\text{eq}}` eq 4.4-16 .. _RefEquation 4.4-17: :math:`{z}_{5}={\tau }_{\text{eq}}+3\mu \theta \Deltap` eq 4.4-17 .. _RefEquation 4.4-18: :math:`{z}_{6}={D}_{1}\text{exp}(\frac{{\tau }_{m}}{{\sigma }_{1}})` eq 4.4-18 .. _RefEquation 4.4-19: :math:`{z}_{7}={z}_{6}{\sigma }_{1}(f+{A}_{n}p)` eq 4.4-19 .. _RefEquation 4.4-20: :math:`{z}_{8}=\frac{1-f}{1-f-{A}_{n}p}` eq 4.4-20 .. _RefEquation 4.4-21: :math:`{z}_{9}=\frac{{A}_{n}}{1-f-{A}_{n}p}` eq 4.4-21 .. _RefEquation 4.4-22: :math:`{x}_{1}={z}_{3}{z}_{6}({z}_{7}+{z}_{2}\theta \Deltap )+{z}_{1}{z}_{2}{\sigma }_{1}-{x}_{3}` eq 4.4-22 .. _RefEquation 4.4-23: :math:`{x}_{2}=-{z}_{3}{z}_{6}\theta \Deltap ({z}_{4}+{z}_{7})-{z}_{1}{z}_{4}{\sigma }_{1}+{x}_{3}\theta \Deltap` eq 4.4-23 .. _RefEquation 4.4-24: :math:`{x}_{3}={A}_{n}{z}_{1}{z}_{6}{\mathrm{\sigma }}_{1}^{2}` eq 4.4-24 .. _RefEquation 4.4-25: :math:`{R}_{\text{vp}}=\frac{\text{dR}(p)}{\text{dp}}+\frac{1}{\theta \Deltat }\frac{\text{dS}(\Deltap /\Deltat )}{d\dot{p}}` eq 4.4-25 For the plastic model with nucleation of cavities' ROUSSELIER_PR 'and for the viscoplastic model without nucleation' ROUSSELIER_VISC ', the corresponding simplified equations are obtained from the equations above by setting :math:`{R}_{\text{vp}}=\text{dR}(p)/\text{dp}` and :math:`{A}_{n}=0` respectively. .. _Ref375039773: .. _Ref374957480: