2. Formulation of the model

The modified Cam-Clay model is formulated according to the construction usual for Generalized Standard Materials, based on an energy potential and a plasticity criterion (or a dissipation potential). The case of linearized deformations and isothermal conditions is assumed.

2.1. State laws

The state variables are the total strain tensor \(\boldsymbol{\epsilon}\), the plastic strain tensor \(\boldsymbol{\epsilon}^p\), and a scalar work hardening variable \(\xi\). Its interpretation will be specified when the laws of evolution are established.

The free energy potential \(\psi\) (volume density) is written in the form:

\[\]

psi (boldsymbol {epsilon}},boldsymbol {epsilon} ^p,xi) =psi_e (boldsymbol {epsilon} -boldsymbol {epsilon} ^p) +psi_h (xi)

label:

free_energy

where \(\psi_e\) refers to the energy returned by elastic discharge and \(\psi_h\) represents the energy stored by work hardening. Their expressions are indicated later.

The expression for intrinsic dissipation volume density \(D\) is obtained as:

\[D=\ mathbf {\ boldsymbol {\ sigma}}:\ dot {\ boldsymbol {\ epsilon}} -\ dot {\ psi} = \ mathbf {\ boldsymbol {\ sigma}}:\ dot {\ boldsymbol {\ epsilon}} -\ left (\ frac {\ partial\ psi} {\ partial\ sigma}} {\ partial\ boldsymbol {\ epsilon}}}: \ dot {\ boldsymbol {\ epsilon}}} +\ frac {\ partial\ psi} {\ partial\ boldsymbol {\ epsilon} ^p}:\ dot {\ boldsymbol {\ epsilon}} ^p +\ frac {\ partial\ psi} {\ partial\ xi} {\ partial\ xi}}\ right) =\ boldsymbol {\ sigma}:\ dot {\ boldsymbol {\ epsilon}}} ^p + p_c\ dot {\ xi} :label: intrinsic_dissipation\]

the latter equality being due to the non-dissipative nature of the total deformation \(\boldsymbol{\epsilon}\) (the plasticity has no viscous effect).

Above, we will also have defined \(\boldsymbol{\sigma}\) the stress tensor and \(p_c\), which is called the critical pressure. These two thermodynamic forces are naturally derived by the following state laws:

\[\]

begin {align}

boldsymbol {sigma} &=frac {partialpsi} {partialpsi} {partialboldsymbol {epsilon}} =frac {partialpsi_e} {partialboldsymbol {boldsymbol {epsilon}}} =-frac {partialpsi_e} {partial boldsymbol {epsilon} ^p}\ p_c &=-frac {partialpsi} {partialpsi} {partialpsi} {partialpsi_h} {partialpsi} {partialxi} end {align} :label: state_law

2.1.1. Elastic energy

The elastic energy potential \(\psi_e\), established in [Bour97] _, and taken advantage of here, takes the following expression:

\[\]

psi_e (boldsymbol {epsilon} ^e) =frac {epsilon} ^e) =frac {K} {K} {kappa^2}left (boldsymbol {epsilon} ^e) -1upsilon (boldsymbol {epsilon} ^e) -1right)qquadtext {with}qe) +kappamathrm {tr} (r) (boldsymbol {epsilon} ^e) -1right)qquadtext {with}qquadUpsilon (boldsymbol {epsilon} ^e) =expleft (-kappamathrm {tr} (boldsymbol {epsilon} ^e) +frac {mu} {K} (kappaboldsymbol {epsilon} ^e_d) :(kappaboldsymbol {epsilon} ^e_d) :(kappaboldsymbol {boldsymbol {epsilon} ^e) :(kappaboldsymbol {epsilon} ^e) :(kappaboldsymbol {epsilon} ^e) :(kappaboldsymbol {epsilon} ^e) :(kappaboldsymbol {epsilon} ^e_d)right)

label:

elastic_energy

where \(\boldsymbol{\epsilon}^e=\boldsymbol{\epsilon}-\boldsymbol{\epsilon}^p\) designates the elastic deformation tensor, having noted \(\boldsymbol{\epsilon}^e_d=\boldsymbol{\epsilon}^e-\cfrac{\mathrm{tr}(\boldsymbol{\epsilon}^e)}{3}\boldsymbol{I}\) as its deviator (\(\boldsymbol{I}\) tensor second order identity).

The expression for the stress tensor is derived from the state law lois_etat -1:

\[\]

boldsymbol {sigma} =Upsilon (boldsymbol {epsilon} ^e)left (-frac {K} {kappa}boldsymbol {I} +2muboldsymbol {epsilon} {epsilon} ^e_dright) +frac {K} {kappa}boldsymbol {I} +2muboldsymbol {I} +2muboldsymbol {I} +2muboldsymbol {I}

label:

constraint-expression

In lois_etat and expression_contrainte, \(K\) and \(\mu\) are the initial compressibility and shear modules respectively. The coefficient \(\kappa\geq 0\) introduces non-linearity through the effect of the mean stress \(\sigma_m\) on the tangent compressibility and shear moduli. In particular, this non-linear elasticity generalizes the more classical elasticity of the modified Cam-Clay model, only on the volume behavior (see [r7.01.14] _), to the deviatoric behavior. This effect is detailed [Bacq23] _ on isotropic and shear loads.

Note:

The elastic energy potential energie_elastique is well defined for \(\kappa=0\) since using an expansion limited to order two in \(\kappa\boldsymbol{\epsilon}^e\), we have:

So for \(\kappa=0\), the elastic energy potential energie_elastique predicts isotropic elastic linear behavior.

2.1.2. Energy stored by work hardening

The energy potential stored by work hardening \(\psi_h\) is written as:

\[\]

psi_h (xi) =frac {p_ {c0}} {beta}left (exp (-betaxi) -1right)

label:

energie_stored

From this we deduce the expression of critical pressure according to state law lois_etat -2:

\[p_c = p_ {c0}\ exp (-\ beta\ xi) :label: critical_pressure_expression\]

\(\beta\geq 0\) is called the plastic incompressibility index and \(p_{c0}>0\) is the initial critical pressure.

2.2. Laws of evolution

The expression for the plasticity criterion is the equation of an ellipse in the stress meridian plane:

\[f (\ boldsymbol {\ sigma}, p_c;\ xi) =\ sqrt {\ left (\ frac {\ sigma_ {eq}} {M}\ right) ^2 +\ left (\ sigma_m+p_c\ right) ^2} - R (\ xi) :label: plasticity_criteria\]

with \(\sigma_m=\cfrac{\mathrm{tr}(\boldsymbol{\sigma})}{3}\) the hydrostatic stress and \(\sigma_{eq}=\sqrt{\cfrac{3}{2}\boldsymbol{\sigma}_d:\boldsymbol{\sigma}_d}\) the equivalent von Mises stress, having noted \(\boldsymbol{\sigma}_d=\boldsymbol{\sigma}-\sigma_m \boldsymbol{I}\) the stress tensor deviator.

In critere_plasticite, the parameter \(M\), called the critical state slope, modulates the ratio of the axes of the reversibility domain in the stress meridian plane. Geometrically, the critical pressure \(p_c\) positions the center of the ellipse along the hydrostatic axis, and finally the function \(R(\xi)\) specifies its size.

Note:

Following the expression of criterion critere_plasticite, it should be noted that \(p_c\) acts as a hydrostatic restoring force. Its variation therefore predicts kinematic work hardening along the hydrostatic axis. In addition, the criterion is set, as specified in the rating \(f( ; \xi)\), via the \(R(\xi)\) function. Its evolution therefore leads to the prediction of work hardening. of the isotropic type. Thus, the modified Cam-Clay model has combined kinematic-isotropic work hardening. These two mechanisms are both driven by the scalar work hardening variable \(\xi\).

The flow rule of the modified Cam-Clay model respects the law of normality, so the plastic deformation tensor \(\boldsymbol{\epsilon}^p\) and the variable \(\xi\) evolve as:

\[\]

begin {align}

dot {boldsymbol {epsilon}}} ^p &=dot {lambda}frac {partial f} {partialboldsymbol {sigma}} =frac {cfrac {3} {3} {2M^2} {2M^2}boldsymbol {sigma} _d +left (sigma_m+p_cright)cfrac {cright)cfrac {cright)cfrac {cright boldsymbol {I}} {3}} {sqrt {left (cfrac {sigma_ {eq}} {M}right) ^2 +left (sigma_m+p_cright) ^2}}\ dot {xi} &=dot {lambda}frac {partial f} {partial p_c} =frac {sigma_m+p_c} {sqrt {left (cfrac {sigma_ {eq}}frac {sigma_ {eq}}} {M}right) ^2}} end {align} :label: normal_flow

where \(\dot{\lambda}\) is the plastic multiplier given by the following consistency condition:

\[\]

dot {lambda}geq 0,quad fleq 0,quaddot {lambda} f=0

label:

condition_coherence

From flow law ecoulement_normal, we deduce that \(\dot{\xi}=\mathrm{tr}(\dot{\boldsymbol{\epsilon}}^p)\). The scalar work hardening variable of the modified Cam-Clay model is therefore the volume plastic deformation.

Note:

It is proposed to establish the expression of the dissipation potential \(\phi\) of the model, using the Legendre-Fenchel transformation of the indicator function of the reversibility domain defined by the criterion of plasticity in critere_plasticite. We get:

with the indicator function \(I_{\{0\}}(x)=0\) if \(x=0\), \(I_{\{0\}}(x)=+\infty\) otherwise, and \(\dot{\epsilon}_{eq}^p=\sqrt{\cfrac{2}{3}\dot{\boldsymbol{\epsilon}}^p_d: \boldsymbol{\epsilon^p_d}}\) the equivalent von Mises plastic deformation rate, having noted \(\dot{\boldsymbol{\epsilon}}^p_d=\dot{\boldsymbol{\epsilon}}^p-\cfrac{\mathrm{tr}(\dot{\boldsymbol{\epsilon}}^p)}{3}\boldsymbol{I}\) its deviator.

2.2.1. Expression of parametrization to the state

The formulation of the modified Cam-Clay model ends by specifying the definition of function \(R(\xi)\), which is responsible for isotropic work hardening with the volume plastic deformation \(\xi=\mathrm{tr}(\boldsymbol{\epsilon}^p)\). Recall that this work hardening comes from a parametrization to the state in the plasticity criterion critere_plasticite (and in the dissipation potential potentiel_dissipation). Its expression is deduced from the constant maintenance of a traction limit noted \(\sigma_0\geq 0\) regardless of the value of \(\xi\). This condition is therefore expressed as follows:

\[\]

forallxi,quad f (boldsymbol {sigma} =sigma_0boldsymbol {I}, p_c;xi) = 0

label:

condition_sig0

Based on the expression of the plasticity criterion critere_plasticite as well as the expression of the critical pressure expression_pression_critique, the equality condition_sig0 is verified if:

\[R (\ xi) =\ sigma_0+p_c=\ sigma_0+p_ {c0}\ exp (-\ beta\ xi) :label: Expression_r\]

Geometrically, the reversibility domain of the modified Cam-Clay model is therefore delimited along the hydrostatic axis by segment \([-p_c-R(\xi),-p_c+R(\xi)]=[-\sigma_0-2p_c,\sigma_0]\) in the meridian plane of the stresses. The following figure shows it for various values of volume plastic deformation \(\xi\).

Fig. 2.1 Reversibility domain of the modified Cam-Clay model in the stress meridian plane for several values of volume plastic deformation \(\xi\). A decrease in \(\xi\) (contracency) leads to an increase in the range of reversibility. An increase in \(\xi\) (dilatance) reduces it. All along, the isotropic tensile limit remains constant equal to \(\sigma_0\) because condition_sig0 is verified.

2.3. Critical state equations

In this section, the critical state equations for the modified Cam-Clay model are established. This state corresponds to a monotonous loading in deformation for which the state of the stresses and the volume deformation remain constant. For this, the volume of plastic deformation rate \(\dot{\xi}=\mathrm{tr}(\dot{\boldsymbol{\epsilon}}^p)\) is cancelled out. Starting with flow law ecoulement_normal -2, this condition occurs when:

\[\]

sigma_m+p_c=0

label:

deformation_volumic_critical_state

The average pressure \(-\sigma_m\) is thus equal to the critical pressure \(p_c\). In addition, using the expression for plasticity criterion critere_plasticite, we deduce that the equivalent stress \(\sigma_{eq}\) is in this situation:

\[\]

sigma_ {eq} = MR (xi)

label:

critical_status_equivalent_constraint

The function expression \(R(\xi)\) in expression_R allows us to conclude using deformation_volumique_etat_critique:

\[\]

sigma_ {eq} = M (sigma_0+p_ {c})Longrightarrowsigma_ {eq} + M (sigma_m-sigma_0) =0

label:

right_critical_state

In summary, the critical state predicted by the modified Cam-Clay model is defined by the following set of equations:

\[\]

begin {align}

dot {boldsymbol {sigma}} =boldsymbol {0}\ mathrm {tr} (dot {boldsymbol {epsilon}}) =0\ sigma_m+p_c = 0\ sigma_ {eq} +M (sigma_m-sigma_0) =0 end {align} :label: critical_status_system

It is an isochore state of plasticity (without volume variation) and perfect (without stress variation). Geometrically, the state of the stresses in the critical state is positioned on the half-line with slope \(M\) intersecting the surface of the elliptical reversibility domain at its vertex of abscissa \(-\sigma_m=p_c\).