.. _r7.01.17-formulation_tridimensionnelle: Three-dimensional formulation of the model ======================================= The presentation of the Barcelona model in a three-dimensional formulation extends that established previously in the case of isotropic loads. To do this, we restrict ourselves here to the equations allowing the numerical integration of the model, without justifying the Barcelona model based on the thermodynamic principles applied to the modeling of unsaturated soils. The reader interested in this topic will find some elements related to these difficulties in reference ([bib3] __, [bib4] __). Elastic behavior ---------------------- Note the total strain tensor :math:`\boldsymbol{\varepsilon}`, the plastic strain tensor :math:`\boldsymbol{\varepsilon}^p`, and :math:`\boldsymbol{\varepsilon}^e=\boldsymbol{\varepsilon}-\boldsymbol{\varepsilon}^p` the elastic strain tensor. The net stress tensor expression :math:`\boldsymbol{\sigma}''=\boldsymbol{\sigma}+p_g\boldsymbol{I}` is written as: .. math:: \ boldsymbol {\ sigma}" = K\ tilde\ kappa\ left ( 1-\ exp\ left (-\ frac {\ varepsilon^e_v} {\ tilde\ kappa} -\ frac {\ tilde\ kappa_s} {\ tilde\ kappa}\ ln\ left}\ ln\ left (1+\ frac {p_c} {\ frac {p_c} {K_s\ tilde\ kappa_s} {\ tilde\ kappa}}\ right)\ left (1+\ frac {p_c} {\ tilde\ kappa_s}\ right) \ right)\ boldsymbol {I} +2\ mu\ boldsymbol {\ varepsilon} ^e_d :label: constraint-expression Note that shear elasticity is linear, where :math:`\mu` is the shear modulus. The visualization of volume elasticity, which is clearly non-linear, is analyzed by differentiating and inverting :eq:`expression_contrainte`, leading to: .. math:: d\ varepsilon_v^e =-\ tilde\ kappa\ frac {dp "} {p"+k\ tilde\ kappa} -\ tilde\ kappa_s\ frac {dp_c} {p_c+k_s\ tilde\ kappa_s} :label: elasticite_incrementale_2 having again noted :math:`p''=-\sigma_m''` the net pressure. Comparing :eq:`elasticite_incrementale_1` and :eq:`elasticite_incrementale_2`, using :eq:`equivalence_deformation_indice_des_vides_2` -a, the equality of the two expressions occurs when: .. math:: \ tilde\ kappa =\ frac {\ kappa} {1+e_0},\ quad\ tilde\ kappa_s =\ frac {\ kappa_s} {1+e_0},\ quad K_s\ tilde\ kappa} {1+e_0},\ quad K_S\ tilde\ kappa\ rightarrow 0 :label: equivalence_parameters Satisfying the first three expressions poses *a priori* no particular difficulty. On the other hand, you cannot strictly check the last one with :math:`K=0`. This impossibility comes from the fact that the incremental relationship :eq:`elasticite_incrementale_1` assumes an initial state with zero volume deformation but non-zero net pressure, which is not the case in the :eq:`elasticite_incrementale_2` relationship. In the present case, :math:`K` acts as an initial compressibility module under net stress, since :eq:`elasticite_incrementale_2` results in: .. math:: d\ varepsilon_v^e\ big {|} {|} _ {p"=0, dp_c=0} =-\ frac {dp "} {dp"} {K} :label: initial_compressibility_module Range of elasticity -------------------- The elasticity domain of the Barcelona model is deduced from the two plasticity criteria introduced :eq:`domaine_elasticite_isotrope`, now taking into account the deviatoric component of the net stress tensor. These two plasticity criteria can be written as: .. math:: f_1 (\ boldsymbol {\ sigma}}", p_c, e_p) =\ sqrt {\ left (\ frac {\ sigma_ {eq} "} {M}\ right) ^2+\ left (\ sigma_m"+\ frac {p_ {con} (p_c, e^p) -kp_c} {2}\ right) ^2}} - \ frac {p_ {con} (p_c, e^p) +kp_c} {2},\ quad f_2 (p_c, e_p) = p_c - s_0 (e^p) :label: elasticity_domain The parameter :math:`M`, called the critical state slope, has the same interpretation as in the modified Cam-Clay model ([:ref:`r7.01.48 `]). A three-dimensional view of the elasticity domain is shown on :numref:`r7.01.17-domaine_elasticite` for a fixed :math:`e^p` plastic void index. .. _r7.01.17-domaine_elasticite: .. figure:: images/domaine_elasticite.png :align: center :width: 500 Range of elasticity with a constant plastic void index. .. How to get the gray box here-under .. note In :eq:`domaine_elasticite`, the same punctuation (",") is used to refer to the intervention of :math:`\boldsymbol{\sigma}'',p_c` and :math:`e_p` in both plasticity criteria, without distinguishing the role of each as an argument or parameter. This simplified notation is accepted insofar as the Barcelona model presented here does not claim to demonstrate its thermodynamic accuracy. Work hardening functions ----------------------- The two work hardening functions, which are involved in the two plasticity criteria :eq:`domaine_elasticite`, take the same forms as those obtained in :numref:`r7.01.17-introduction` taking into account the :eq:`equivalence_deformation_indice_des_vides_2` -b relationship. So we write: .. math:: p_ {con} (p_c, e^p) =\ tilde {p} _ {con} (p_c,\ varepsilon_v^p) = p_r\ left (\ cfrac {p^0_ {con}\ exp\ left}\ exp\ left (-\ cfrac {\ varepsilon_v^p} {\ tilde {\ lambda} _0-\ tilde {\ kappa}}\ right)} {p_r}\ right) ^\ cfrac {\ tilde {\ lambda} _0-\ tilde {\ kappa}}} {\ tilde {\ lambda}} (p_c) -\ tilde {\ kappa}},\ quad s_0 (e^p) =\ tilde {s} _0 (\ varepsilon_v^p) =\ left (s_0^0+k_s\ tilde\ kappa_s\ right)\ exp\ left (-\ frac {\ varepsilon_v^p} {\ tilde\ lambda_s-\ tilde\ kappa_s\ right)\ exp\ left (-\ frac {\ varepsilon_v^p} {\ frac {\ varepsilon_v^p} {\ tilde\ lambda_s-\ tilde\ kappa_s\ right) - K_s\ tilde\ kappa_s :label: ecrouissage_expression where: .. math:: p^0_ {con} = p_ {con} (0,0),\ quad\ tilde {\ lambda} _0=\ frac {\ lambda_0} {1+e_0},\ quad\ tilde {\ tilde {\ lambda} (p_c) =\ frac {\ lambda (p_c)} {1+e_0},\ quad s^0_0 = s_0 (0),\ quad\ tilde\ tilde\ lambda_s=\ frac {\ lambda_s} {1+e_0} :label: working_equivalence Flow rules ------------------- The flow rule relating to the plastic deformation tensor rate is written using two flow potentials :math:`g_1` and :math:`g_2`: .. math:: \ dot {\ boldsymbol {\ varepsilon}}} ^p =\ dot {\ lambda} _1\ frac {\ partial g_1} {\ partial\ boldsymbol {\ sigma} "}"} +\ dot {\ lambda}} _2\ dot {\ lambda} _2\ frac {\ partial g_2} {\ partial\ boldsymbol {\ sigma} "} +\ dot {\ lambda} _2\ frac {\ partial g_2} {\ partial\ boldsymbol {\ sigma}"} :label: plastic_flow where the plastic multipliers :math:`\dot{\lambda}_1` and :math:`\dot{\lambda}_2` verify the consistency conditions: .. math:: \ begin {align} &\ dot {\ lambda} _1\ geq 0,\ quad f_1\ leq 0,\ quad\ dot {\ lambda} _1f_1 = 0\\ &\ dot {\ lambda} _2\ geq 0,\ quad f_2\ leq 0,\ quad\ dot {\ lambda} _2f_2 = 0 \ end {align} :label: conditions_consistency The two flow potentials in :eq:`ecoulement_plastique` are written as: .. math:: g_1 (\ boldsymbol {\ sigma}}", p_c, e_p) =\ sqrt {\ alpha\ left (\ frac {\ sigma_ {eq} "} {M}\ right) ^2+\ left (\ sigma_m"+\ frac {p_ {con} (p_c, e^p) -kp_c} {2}\ right) ^2} -\ frac {p_ {con} (p_c, e^p) +kp_c} {2},\ quad g_2 (\ boldsymbol {\ sigma}") = -\ sigma_m" :label: flow_potentials The coefficient :math:`\alpha`, when taken to be different from one, is suggested to take a value that makes it possible to predict a state of oedometric deformation for a state of stress defined by the earth pressure coefficient at rest given by Jacky's formula [bib2] __, which leads to: .. math:: \ alpha =\ frac {M (9-M) (3-M)} {9 (6-M)}\ frac {\ lambda_0} {\ lambda_0-\ kappa} :label: alpha_expression Critical state equations ---------------------------- In this paragraph, the critical state equations of the Barcelona model are established. This state corresponds to a monotonous deformation loading for which the state of the net stresses and the volume deformation remain constant. For this, it is assumed that the capillary pressure is constant, and that only the surface of the first plasticity criterion :eq:`domaine_elasticite` -a is reached. Since the volume of plastic deformation should cancel out, the direction of flow given by :eq:`ecoulement_plastique` and :eq:`potentiels_ecoulement` -a implies: .. math:: \ sigma_m+\ frac {p_ {con} (p_c, e^p) -kp_c} {2} =0 :label: critical_state_pressure In this situation, the expression of the first plasticity criterion :eq:`domaine_elasticite` -led to: .. math:: \ sigma"_ {eq} = M\ frac {p_ {con} (p_c, e^p) +kp_c} {2} :label: critical_status_equivalent_constraint The two previous relationships allow us to conclude: .. math:: \ sigma"_ {eq} + M (\ sigma_m" -kp_c) =0 :label: right_critical_state Thus, in summary, the critical state predicted by the Barcelona model, with capillary pressure assumed to be constant, is defined by the following set of equations: .. math:: \ begin {align} \ dot {\ boldsymbol {\ sigma}} "=\ boldsymbol {0}\\ \ dot\ varepsilon_v=0\\ \ sigma_m+\ frac {p_ {con} (p_c, e^p) -kp_c} {2} =0\\ \ sigma"_ {eq} + M (\ sigma_m" -kp_c) =0 \ end {align} :label: critical_status_system In particular, :eq:`systeme_etat_critique` -d highlights the gain in cohesion that results from a higher value of the imposed capillary pressure.