Model formulation ===================== Model CSSM is based on the association of behavioral equations similar to the modified Cam-Clay and Iwan models. These are detailed respectively in the [:ref:`r7.01.38 `] and [:ref:`r7.01.48 `] documentation. A rheological description of model CSSM consists in superimposing these two previous models as two components, the choice being: * A parallel association of deviatoric behavior. * A serial association of volume behavior. These associations are very schematically represented on :numref:`r7.01.44-schema_rheo`. .. _r7.01.44-schema_rheo: .. figure:: images/schema_rheo.png :align: center :width: 704 :height: 163 Rheological representation of components put in parallel (deviatoric behavior) and in series (volume behavior). Components 1 and 2 see the same mean constraint :math:`\sigma_m` but not the same deviator from constraints :math:`\boldsymbol{\sigma}_d`. Energy and state laws ---------------------- The state variables in model CSSM are the tensor of total deformations :math:`\boldsymbol{\varepsilon}` and a collection of own internal variables to its two components: * For the first component: three internal variables, tensor and scalar respectively, noted :math:`\boldsymbol{\varepsilon}^p,\xi` and :math:`\gamma`. * For the second component: :math:`N` tensor internal variables, noted :math:`\left(\boldsymbol{\alpha}^i\right)_{1\leq i\leq N}`. The free energy potential :math:`\psi` is defined as follows: .. math:: \ psi (\ boldsymbol {\ varepsilon},\ boldsymbol {\ varepsilon} ^p,\ boldsymbol {\ alpha} ^1,\ dots,\ boldsymbol {\ alpha} ^N,\ alpha} ^N,\ alpha} ^N,\ xi,\ ^N,\ N,\ xi,\ gamma) =\ psi_ {\ varepsilon} ^p) =\ psi_ {e, v}\ left (\ varepsilon_v-\ varepsilon_v^p-\ sum_ {i=1} ^N\ alpha^i_v\ right) +\ left [\ rho\ psi_ {e, d} (\ boldsymbol {\ varepsilon} _d-\ d-\ boldsymbol {\ varepsilon} _d^p) + (1-\ rho)\ psi_ {e, d}\ left (\ boldsymbol {\ varepsilon} _d-\ sum_ {i=1} ^N\ boldsymbol {\ alpha} ^i_d\ right)\ right)\ right)\ right)\ right)\ right]\ right] +\ psi_ {h,1} (\ boldsymbol {\ gamma) +\ psi_ {h,2} (\ boldsymbol {\ right)\ right)\ right]\ right] +\ psi_ {h,1} (\ boldsymbol {\ gamma) alpha} ^1,\ dots,\ boldsymbol {\ alpha} ^N) :label: free_energy In this potential, we note that: * The first term :math:`\psi_{e,v}` takes the place of elastic energy by volume. *The second term, in square brackets, represents the elastic shear energy, which is weighted between components 1 and 2, :math:`\rho\psi_{e,d}` and :math:`(1-\rho)\psi_{e,d}`, in*proporata* of a coefficient :math:`\rho`. * The last two terms :math:`\psi_{h,1}` and :math:`\psi_{h,2}` are the energies stored by work hardening. The following sections specify the expressions for functions :math:`\psi_{e,v},\psi_{e,d},\psi_{h,1}` and :math:`\psi_{h,2}`. Elastic energy ^^^^^^^^^^^^^^^^^^ The elastic volume and shear energies are quadratic: .. math:: \ psi_ {e, v} (x) =\ frac {K} {2} x^2,\ quad\ psi_ {e, d} (\ boldsymbol {x}) =\ mu\ boldsymbol {x}) =\ mu\ boldsymbol {x}) =\ mu\ boldsymbol {x}) :label: elastic_energy :math:`K` is the compression modulus and :math:`\mu` is the shear modulus. Energy stored by working the first component ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ The energy stored by work hardening :math:`\psi_{h,1}`, controlled by the work hardening variables :math:`\xi` and :math:`\gamma`, is defined as: .. math:: \ psi_ {h,1} (\ xi,\ gamma) = p_ {c0}\ left (\ frac {\ exp\ left (-\ beta\ xi\ right) -1} {\ beta} +\ eta\ frac {\ frac {\ frac {\ frac {\ exp\ left (-\ omega\ gamma\ right) -1} {\ omega}\ right) :label: energie_ecrouissage_1 The parameters :math:`\beta` (plastic incompressibility index) and :math:`p_{c0}` (initial critical pressure) are those invoked in the expression of the critical pressure of the modified Cam-Clay model [:ref:`r7.01.48 `]. The indices :math:`\eta` and :math:`\omega`, which relate to work hardening controlled by the variable :math:`\gamma`, in particular have the effect of modulating the responses of the model during isotropic compression loads. This is illustrated in :numref:`r7.01.44-cas_tests`. Energy stored by working the second component ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ The energy stored by work hardening :math:`\psi_{h,2}`, driven by the work hardening variables :math:`\left(\boldsymbol{\alpha}^i\right)_{1\leq i\leq N}`, is defined as: .. math:: \ psi_ {h,2} (\ boldsymbol {\ alpha} ^1,\ dots,\ boldsymbol {\ alpha} ^N) =\ sum_ {i=1} ^N\ left (\ frac {h^i_D} {2}\ frac {H^i_D} {2} {2}\ boldsymbol {\ alpha} ^i_v} ^i_ d:\boldsymbol {\ alpha} ^i_d+\ frac {H^i_D} {2}\ frac {H^i_D} {2}} (\ alpha^i_v) ^2\ right) :label: energie_ecrouissage_2 where :math:`H_d^i` and :math:`H_v^i` are respectively the deviatoric and volume kinematic work hardening modules associated with :math:`\boldsymbol{\alpha}^i`. State laws ^^^^^^^^^^^ Having established the expression of energy potential via its components, we can write the intrinsic dissipation volume density :math:`D` as: .. math:: D=\ boldsymbol {\ sigma}:\ dot {\ boldsymbol {\ boldsymbol {\ varepsilon}} -\ dot {\ psi} =\ boldsymbol {X}:\ dot {\ boldsymbol {\ boldsymbol {\ varepsilon}} ^p+ \ boldsymbol {A} ^ i:\dot {\ boldsymbol {\ alpha}} ^i+p_c\ dot {\ xi} +S\ dot {\ gamma} :label: intrinsic_dissipation by defining the stress tensor :math:`\boldsymbol{\sigma}` and the irreversible forces :math:`\boldsymbol{X},\boldsymbol{A}^i,p_c` and :math:`S` combined with the internal variables by state laws: .. math:: \ boldsymbol {\ sigma} =\ frac {\ partial\ psi} {\ partial\ partial\ boldsymbol {\ varepsilon}},\ quad\ boldsymbol {X} =-\ frac {\ partial\ psi} {\ partial\ psi} {\ partial\ partial\ psi} {\ partial\ boldsymbol {\ varepsilon} ^p} =-\ frac {\ partial\ psi} {\ partial\ psi} {\ partial\ psi} {\ partial\ psi} {\ partial\ psi} {\ partial\ psi} {\ partial\ psi} {\ partial\ psi} {\ partial\ psi} {\ partial\ psi} \ boldsymbol {A} ^i =-\ frac {\ frac {\ partial\ psi} {\ partial\ boldsymbol {\ alpha} ^i},\ quad p_c = -\ frac {\ partial\ psi} {\ partial\ xi},\ quad S =-\ frac {\ partial\ psi} {\ partial\ gamma} :label: law_state_1 These laws can be specified from :eq:`energie_libre`, :eq:`energie_elastique`,, :eq:`energie_ecrouissage_1`, and :eq:`energie_ecrouissage_2`: .. math:: \ begin {align} \ boldsymbol {\ sigma} &= K\ left (\ varepsilon_v-\ varepsilon_v-\ v^p-\ sum_ {j=1} ^N\ alpha^j_v\ right)\ boldsymbol {I} + 2\ mu\ left (\ mu\ left (\ boldsymbol {\ varepsilon} _d-\ boldsymbol {I}} + 2\ mu\ left (\ rho\ left (\ boldsymbol {\ varepsilon} _d-\ boldsymbol {\ varepsilon} _d^p\ right) + (1-\ rho) + (1-\ rho)\ left (\ boldsymbol {\ varepsilon} _d-\ sum_ {j=1} ^N\ boldsymbol {\ alpha} {\ alpha}} ^j_d\ right)\ right)\ vphantom {\ Big {(}}\\ \ boldsymbol {X} &= K\ left (\ varepsilon_v-\ varepsilon_v^p-\ sum_ {j=1} ^N\ alpha^j_v\ right)\ boldsymbol {I} + 2\ mu\ rho\ left (\ boldsymbol {\ varepsilon} _d-\ boldsymbol {\ varepsilon} _d-\ boldsymbol {I} + 2\ mu\ rho\ left (\ boldsymbol {\ varepsilon} _d-\ boldsymbol {\ varepsilon} on} _d^p\ right)\ vphantom {\ Big {(}}\\ \ boldsymbol {A} ^i &= K\ left (\ varepsilon_v-\ varepsilon_v-\ v^p-\ sum_ {j=1} ^N\ alpha^j_v\ right)\ boldsymbol {I} + 2\ boldsymbol {I} + 2\ mu (1-\ rho)\ left (\ boldsymbol {\ varepsilon} _d-\ sum_ {j=1} ^N\ boldsymbol {\ alpha} ^j_d\ right) -H_d^i\ boldsymbol {\ alpha} ^i_d-h_v^i\ alpha^i_v\ boldsymbol {I}\ boldsymbol {I}\ vphantom {\ big {(}}\\ p_c &= p_ {c0}\ exp\ left (-\ beta\ xi\ right)\ vphantom {\ big {(}}}\\ S &=\ eta p_ {c0}\ exp\ left (-\ omega\ gamma\ right)\ vphantom {\ Big {(}}} \ end {align} :label: law_state_2 .. How to get the gray box here-under .. **Note:** For convenience in solving behavior equations with MFront in :numref:`r7.01.44-integration_numerique`, we will advantageously write the first three state laws :eq:`lois_etat_2` -1, :eq:`lois_etat_2` -2 and :eq:`lois_etat_2` -3 in the form: .. math: \ boldsymbol {\ sigma} =\ boldsymbol {\ sigma} (\ boldsymbol {\ varepsilon} ^e) =K\ varepsilon_v^e\ boldsymbol {I} + 2\ mu\ boldsymbol {\ sigma} (\ boldsymbol {\ varepsilon} _d^e,\ quad \ boldsymbol {X} =\ boldsymbol {X} (\ boldsymbol {\ varepsilon} ^x) = K\ varepsilon_v^x\ boldsymbol {I} + 2\ mu\ rho\ boldsymbol {\ varepsilon} _d^x,\ quad \ boldsymbol {A} ^i =\ boldsymbol {\ sigma} -\ boldsymbol {X} _d-h_d^i\ boldsymbol {\ alpha} ^i_d-h_v^i\ alpha^i_v\ boldsymbol {I} :label: rewrite_state_law where two elastic deformations, :math:`\boldsymbol{\varepsilon}^e` and :math:`\boldsymbol{\varepsilon}^x`, are introduced, respectively in bijection to :math:`\boldsymbol{\sigma}` and :math:`\boldsymbol{X}`. These are easily expressed as a function of the tensor of the total deformations and the internal variables by the relationships: .. math: \ boldsymbol {\ varepsilon} ^e =\ boldsymbol {\ varepsilon} -\ left (\ mathbb {J} +\ rho\ mathbb {K}\ right):\ boldsymbol {\ varepsilon} ^p -\ left (\ mathbb {J} + (1-\ rho)\ mathbb {K}\ right):\ sum_ {i=1} ^N\ boldsymbol {\ alpha} ^i,\ quad \ boldsymbol {\ varepsilon} ^x =\ boldsymbol {\ varepsilon} ^e -\ left (1-\ rho\ right)\ mathbb {K}:\ left (\ boldsymbol {\ varepsilon} ^p-\ varepsilon} ^p-\ sum_ {i=1} ^N\ boldsymbol {\ alpha} ^i\ right) :label: elastic_deformations Laws of evolution ---------------- The laws of evolution are deduced from the plasticity criteria specific to the two components of model CSSM defined below. Plasticity criterion and flow rule for the first component ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ The expression for the plasticity criterion is the equation of an ellipse in plane :math:`(X_m,X_{eq})`: .. math:: f (\ boldsymbol {X}, p_c, S;\ xi) =\ sqrt {\ left (\ frac {X_ {eq}} {M}\ right) ^2 +\ left (X_M+P_C-S\ right) ^2} +S-R (\ xi),\ quad\ text {with}\ quad R (\ xi) = p_ {c0}\ exp (-\ beta\ xi) :label: crite_plasticity_1 The parameter :math:`M`, called the critical state slope, has the same interpretation as in the modified Cam-Clay model [:ref:`r7.01.48 `]. Geometrically, :math:`p_c-S` positions the center of the ellipse along the hydrostatic axis, and :math:`S-R(\xi)` specifies its size. **Note:** Following the expression of plasticity criterion :eq:`critere_plasticite_1`, it should be noted that :math:`p_c` plays the role of hydrostatic restoring force. Its variation therefore predicts kinematic work hardening along the hydrostatic axis. In addition, the plasticity criterion is parameterized, as specified in the notation :math:`f( ; \xi)`, via the :math:`R(\xi)` function. Its evolution therefore leads to the prediction of work hardening. isotropic type. Thus, as for the modified Cam-Clay model [:ref:`r7.01.48 `], the first component of model CSSM has a combined kinematic-isotropic work hardening controlled by :math:`\xi`. To this work hardening is added that controlled by the variable of work hardening :math:`\gamma` due to the intervention of its combined force :math:`S` in the plasticity criterion :eq:`critere_plasticite_1`. The initial reversibility domain associated with plasticity criterion :eq:`critere_plasticite_1` is explained using state laws :eq:`lois_etat_2` by: .. math:: f_ {0} =\ sqrt {\ left (\ frac {\ rho\ sigma_ {eq}} {M}\ right) ^2 +\ left (\ sigma_m+p_ {c0} (1-\ c0} (1-\ eta}) (1-\ eta)\ right) ^2} -p_ {c0} (1-\ eta) :label: crite_plasticite_1_initial When :math:`\rho=0`, the initial plasticity criterion therefore no longer depends on the equivalent von Mises stress :math:`\sigma_{eq}`. However, regardless of the value of :math:`\rho`, The isotropic tension limit is zero, and is :math:`2p_{c0}(1-\eta)` in isotropic compression. The :numref:`r7.01.44-camclay_surface` represents the effect of :math:`\eta` on the initial reversibility domain. The closer this one approaches one, the more the size of the domain tends to zero by homothetic reduction around from the origin. .. _r7.01.44-camclay_surface: .. figure:: images/camclay_surface.svg :align: center :width: 400 Reversibility domains defined by plasticity criterion :eq:`critere_plasticite_1_initial` for three values of :math:`\eta`. The evolution of the three internal variables :math:`\boldsymbol{\varepsilon}^p,\xi` and :math:`\gamma` respects the rule of normality to the plasticity criterion :eq:`critere_plasticite_1`: .. math:: \ begin {align} \ dot {\ boldsymbol {\ varepsilon}}} ^p &=\ dot {\ lambda}\ frac {\ partial f} {\ partial\ boldsymbol {X}} =\ dot {\ lambda}\ frac}\ frac {\ cfrac {3} {3} {2M^2}\ frac {3} {2M^2}\ boldsymbol {X} _d +\ left (x_m+p_c-s\ right)\ cfrac {\ boldsymbol {I}} {3}} {3}} {\ sqrt {\ left (\ cfrac {X_ {eq}} {M}\ right) ^2 +\ left (X_m+p_c-s\ right) ^2}}\\ \ dot {\ xi} &=\ dot {\ lambda}\ frac {\ partial f} {\ partial p_c} =\ dot {\ lambda}\ frac {x_m+p_c-s} {\ sqrt {\ lambda}}\ frac {\ lambda}\ left {\ left (\ cfrac {\ partial f}} {\ partial p_c}} =\ dot {\ lambda}}\ frac {x_m+p_c-s} {\ left (\ cfrac {X_ {eq}}} {M}\ right) ^2 +\ left (x_m+p_c-s\ right) ^2 +\ left (x_m+p_c-s\ right) ^2 +\ left (x_m+p_c-s\ right) ^2 2}}\\ \ dot {\ gamma} &=\ dot {\ lambda}\ frac {\ partial f} {\ partial S} =\ dot {\ lambda} -\ dot {\ lambda}\ frac {x_M+p_c-s} {\ sqrt {\ p_c-s} {\ frac {\ partial f}} {\ partial S} =\ dot {\ lambda} -\ dot {\ lambda}}\ frac {x_M+P_c-s} {\ frac {X_M+P_c-s} {\ left} {\ sqrt {\ partial f} {\ left (\ cfrac {X_ {eq}}} {M}\ right) ^2 +\ left (X_M+P_C-S} {\ left C-s\ right) ^2}} \ end {align} :label: flow_normal_1 where :math:`\dot{\lambda}` is the plastic multiplier satisfying the following consistency condition: .. math:: \ dot {\ lambda}\ geq 0,\ quad f\ leq 0,\ quad\ dot {\ lambda} f=0 :label: condition_coherence_1 So as a result of :eq:`ecoulement_normal_1`, we get the relationships: .. math:: \ dot {\ xi} =\ dot {\ varepsilon} _v^p,\ quad\ dot {\ gamma} =\ dot {\ lambda} -\ dot {\ varepsilon} _v^p,\ quad\ dot {\ varepsilon} _v^p,\ quad\ dot {\ varepsilon} _v^p,\ quad\ dot {\ lambda} =\ sqrt {\ lambda} =\ sqrt {\ gamma} =\ sqrt {\ varepsilon} _v^p) ^2+ (M\ dot {\ varepsilon} _v^p) ^2+ (M\ dot {\ varepsilon} _v^p) ^2+ (M\ dot {\ varepsilon} _v^p) ^2+ psilon} _ {eq} ^p) ^2} :label: flowment_normal_1_2 The first work-hardening variable :math:`\xi` is volume plastic deformation. The second :math:`\gamma` is the difference between the cumulative plastic deformation :math:`\lambda` and the volume one. Since :eq:`ecoulement_normal_1_2` shows that :math:`\dot{\gamma}\geq 0`, the work hardening carried out by its combined force :math:`S` is monotonic decreasing and evanescent according to :eq:`lois_etat_2` -5 (:math:`\dot{S}\leq 0` and :math:`\lim_{\gamma\rightarrow\infty} S = 0`). Plasticity criterion and flow rule for the second component ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ The law of evolution of internal variables :math:`\left(\boldsymbol{\alpha}^i\right)_{1\leq i\leq N}` comes from plasticity criteria that differ only in their size. These criteria are hemi-elliptical in planes :math:`(A^i_m,A^i_{eq})`: .. math:: F_i (\ boldsymbol {A} ^i) =\ sqrt {\ left (A^i_ {eq}\ right) ^2 +\ left (\ frac {R_i} {C}\ left\ left\ langle A^i_m+c\ right\ r_M+c\ right\ rangle\ m+c\ right\ right\ right) ^2} -R_i,\ quad\ text {with}\ quad\ langle x\ rangle =\ frac {x+|x|} {2} :label: crite_plasticity_2 The constant :math:`C` positions the transition between the ellipses and the lines of the plasticity criteria, this value being assumed to be common to the :math:`N` criteria. The constants :math:`R_i` are the maximum thresholds achievable by the equivalent von Mises :math:`A^i_{eq}` standards. The :numref:`r7.01.44-iwan_surfaces` illustrates three of the :eq:`critere_plasticite_2` criteria for increasing values :math:`R_1,R_2` and :math:`R_3`. .. _r7.01.44-iwan_surfaces: .. figure:: images/iwan_surfaces.svg :align: center :width: 400 Reversibility domains defined by plasticity criteria :eq:`critere_plasticite_2` for three increasing values of :math:`R_1,R_2` and :math:`R_3`. .. How to get the gray box here-under .. **Note:** When :math:`C\rightarrow 0`, the plasticity criteria :eq:`critere_plasticite_2` approach those used in the Iwan model [:ref:`r7.01.38 `] (von Mises criteria), with in this case a cut prohibiting the traction half-space (:math:`A^i_m>0`). The initial reversibility domain associated with plasticity criteria :eq:`critere_plasticite_2` is explained using state laws :eq:`lois_etat_2` by: .. math:: F_ {1.0} =\ sqrt {\ left ((1-\ rho)\ sigma_ {eq}\ right) ^2 +\ left (\ frac {R_1} {C}\ left\ langle\ langle\ sigma_M+C\ right\ rangle\ right) ^2} -R_1 :label: crite_plasticite_2_initial having assumed :math:`R_1` to be the lowest value of :math:`R_i`. When :math:`\rho=1`, the initial plasticity criterion therefore no longer depends on the equivalent stress by von Mises :math:`\sigma_{eq}`. By combining :eq:`critere_plasticite_1_initial` and :eq:`critere_plasticite_2_initial`, the initial elasticity domain of the CSSM model is therefore defined by intersection :math:`\left\{f_{0}\leq 0 \cap F_{1,0}\leq 0\right\}`. The evolution of the internal variables :math:`\left(\boldsymbol{\alpha}^i\right)_{1\leq i\leq N}` respects the rule of normality with the :eq:`critere_plasticite_2` plasticity criteria: .. math:: \ dot {\ boldsymbol {\ alpha}} ^i} ^i &=\ dot {\ lambda} _i\ frac {\ partial F_i} {\ partial\ boldsymbol {A} ^i} =\ dot {\ lambda} _i _i\ frac {\ frac {3} {2}\ boldsymbol {A} ^i_d +\ left (\ cfrac {R__I} {C}\ right) ^2\ left\ langle A^i_M+C\ right\ right\ rangle\ cfrac {\ boldsymbol {I}} {3}} {\ sqrt {\ left (A^i_ {eq}\ right) ^2 +\ left (\ cfrac {R_i} {C}}\ left\ langle A^i_m+c\ right) ^2 +\ left (\ cfrac {R_i}} {C}\ left\ langle A^i_m+c\ right\ rangle\ right) ^2}}\\ :label: flow_normal_2 where :math:`\dot{\lambda}_i` are the plastic multipliers satisfying the following consistency conditions: .. math:: \ dot {\ lambda} _i\ geq 0,\ quad F_i\ leq 0,\ quad\ dot {\ lambda} _if_i=0 :label: condition_coherence_2 Determination of kinematic work hardening modules and associated thresholds -------------------------------------------------------------------------- In this part, we explain how to determine the modules (kinematic hardening modules :math:`H_d^i,H_v^i`) and thresholds :math:`R_i` can be simplified by relying on two alternative parameters: :math:`\gamma_{\mathrm{hyp}}`, :math:`n_\mathrm{hyp}`. Calculation of deviatory work hardening modules and thresholds ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ The calculation of deviatoric work hardening modules and thresholds follows the same methodology as that presented in the documentation [:ref:`r7.01.38 `] assuming that the following conditions are met: * The plasticity surface of the first component is not reached (:math:`f<0`). * The volume deformations of each :math:`i`]. Expression of volume work hardening modules ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Obtaining the :math:`H_v^i` volume work hardening modules is justified by the desire to effectively integrate the evolution equations present in :eq:`critere_plasticite_2` and :eq:`condition_coherence_2`, without requiring the resolution of a non-linear equation for plastic multipliers. In fact, it is proved that this is possible when the following condition is met: .. math:: H_v^i =\ frac {3} {2} {2}\ left (\ frac {C} {R_i}\ right) ^2H_d^i :label: expression_module_volume_hardening_volume Under this condition, the numerical integration of model CSSM then makes it possible to reduce the number of equations to be solved. This procedure is detailed in :numref:`r7.01.44-integration_numerique`. Critical state equations ---------------------------- In this paragraph, we seek to establish the equations of model CSSM corresponding to the critical state. This state is defined by constant stresses and volume deformation, asympotically achieved in a monotonic loading. We assume that this state is reached for each of components 1 and 2 of the model. For the first component, if the stresses remain constant, there is no more work hardening due to :math:`\xi` or :math:`\gamma`. In this situation, it is proved that the following conditions are met: .. math:: X_ {eq} +M x_M=0,\ when X_M+P_C=0 :label: critical_status_1 Now, state laws :eq:`lois_etat_2` -1 and :eq:`lois_etat_2` -2 show that :math:`X_m=\sigma_m`. So :eq:`etat_critique_1` is equivalent to: .. math:: X_ {eq} +M\ sigma_m=0,\ quad\ sigma_m+p_c=0 :label: critical_status_2 As for the second component, following :eq:`calibration_cisaillement_2` -2 and :eq:`expression_module_ecrouissage_volumique`, :math:`H_d^{m}=H_v^{m}=0=0` for a certain :math:`m\leq N`. So state laws :eq:`lois_etat_2` -1 and :eq:`lois_etat_2` -3 show that :math:`A^m_m=\sigma_m`. The verification of the criterion :eq:`critere_plasticite_2` then leads to: .. math:: A^m_ {eq} = R_m\ sqrt {1-\ left\ langle\ sigma_m/C+1\ right\ rangle^2} :label: critical_status_3 In addition, we recall :eq:`lois_etat_2` -1, :eq:`lois_etat_2` -2 and :eq:`lois_etat_2` -3 which establish that :math:`\boldsymbol{\sigma}_d=\boldsymbol{X}_d+\boldsymbol{A}^m_d`. By admitting that these last two tensors are collinear and in the same sense, equations :eq:`etat_critique_2` -1 and :eq:`etat_critique_3` imply that: .. math:: \ sigma_ {eq} +M\ sigma_m=r_m\ sqrt {1-\ left\ langle\ sigma_m/C+1\ right\ rangle^2} :label: critical_status_4 Finally, it is shown that the total volume deformation remains constant only if :math:`\sigma_m+C<0`. Otherwise, it only evolves like :math:`\dot{\varepsilon}_v=\dot{\alpha}_v^m>0`. In short, it should be noted that the critical state predicted by model CSSM can be defined by the following set of equations: .. math:: \ begin {align} \ dot {\ boldsymbol {\ sigma}} =\ boldsymbol {0}\\ \ dot {\ varepsilon} _v=0\\ \ sigma_m+p_c = 0\\ \ sigma_M+C<0\\ \ sigma_ {eq} +M\ sigma_m=r_m\ sqrt {1-\ left\ langle\ sigma_m/C+1\ right\ rangle^2} \ end {align} :label: critical_status_system :numref:`r7.01.44-critical_state` represents three :eq:`systeme_etat_critique` -5 critical state lines based on values of :math:`R_m` in the stress meridian plane. For :math:`\sigma_m+C<0`, it indicates that the model CSSM predicts a dilated state (:math:`\dot{\varepsilon}_v=\dot{\alpha}_v^m>0`). .. _r7.01.44-critical_state: .. figure:: images/critical_state.svg :align: center :width: 400 Examples of critical state lines :eq:`systeme_etat_critique` -5 for three values of :math:`R_m` in the stress meridian plane. Eliminating negative work hardening from the model ---------------------------------------------- Model CSSM inherits the same work hardening as that of the modified Cam-Clay model, driven by the :math:`\xi` variable. Now we have shown after the criterion of plasticity :eq:`critere_plasticite_1` that this variable produces two workings: * A kinematic one: this mechanism is associated with its conjugate force :math:`p_c` conjugate by the state law :eq:`lois_etat_2` -4, playing the role of a hydrostatic restoring force. This work hardening moves the plasticity surface into the stress meridian plane without changing its size or shape. * The other isotropic: this mechanism is linked to the parametrization of the criterion through the function :math:`R(\xi)`. This work hardening changes the size of the plasticity surface. Kinematic work hardening alone cannot produce negative work hardening, due to the convexity of the potential from which :math:`p_c` drifts. On the other hand, negative isotropic work hardening can occur if :math:`R(\xi)` decreases, which potentially poses numerical stability problems in a finite element calculation. To avoid this situation, it is proposed to replace function :math:`R(\xi)`, after discretization incremental loading times, by: .. math:: R_ {n+1} = R_n\ max\ max\ left\ {1,\ exp\ left (-\ beta\ Delta\ xi\ right)\ right\},\ quad\ text {with}\ quad R_0=p_ {c0},\ quad\ Delta\ xi =\ xi_ {n+1} -\ xi_n :label: suppression_softening :math:`\left(R_n\right)_{n\geq 0}` is a growing number. We then only take into account cumulative isotropic positive work hardening that would have been predicted by the initial function :math:`R(\xi)` :eq:`critere_plasticite_1` -2. **Note:** If applications of the CSSM model require an interest in predicting negative work hardening, the restriction imposed by :eq:`suppression_adoucissement` can, if necessary, be removed. in later developments.