3. Cam Clay’s law as amended

3.1. Modeling hypotheses

The model is written in small disturbances.

The coefficients of the model do not depend on temperature.

3.2. Charging surface

The expression for the load area is written as follows:

\(f(P,Q,{P}_{\text{cr}})={Q}^{2}+{M}^{2}{(P-{P}_{\text{trac}})}^{2}-{\mathrm{2M}}^{2}(P-{P}_{\text{trac}}){P}_{\text{cr}}\) \(\le 0\) eq 3.2-1

In plane \((P,Q)\), the expression represents a family of ellipses, centered on \({P}_{\text{cr}}\), which is linked to consolidation pressure: \({P}_{\text{cons}}=2{P}_{\text{cr}}-{P}_{\text{trac}}\) (cf. [Figure 3.2-a). \({P}_{\text{cr}}\) will be the model’s work hardening parameter.

Figure 3.2-a: Family of collapsable load surfaces

When \(f=0\) and \(P-{P}_{\text{trac}}<{P}_{\text{cr}}\) the material is dilating (\({\dot{\epsilon }}_{v}^{p}<0\)) and \({P}_{\text{cr}}\) is decreasing (softening).

When \(f=0\) and \(P-{P}_{\text{trac}}>{P}_{\text{cr}}\) the material is contacting (\({\dot{\epsilon }}_{v}^{p}>0\)) and \({P}_{\text{cr}}\) is increasing (hardening).

3.3. Elastic law and work hardening law

The hypothesis is made of the decoupling of the elastic law in part hydrostatic and deviatoric and the additional hypothesis that the shear modulus is constant.

We therefore consider an isotropic elastic law, with a linear deviatoric part and a non-linear volume part:

Deviatory part:

\({\tilde{\epsilon }}^{e}=\frac{s}{2\mu }\) eq 3.3-1

Volume part:

\({\dot{\epsilon }}_{v}^{e}=-\frac{\dot{e}}{1+{e}_{0}}\text{ou}e={e}_{0}-\kappa \text{Ln}(\frac{P}{{K}_{\text{cam}}})\text{si}P<\text{Pconsolidation}\) eq 3.3-2

The law [éq 3.3-2] is in fact derived from an oedometric test where we measure the variation of the void index as a function of loading [Figure 2.2-a]. Recall that a homogeneous oedometric test consists in increasing the effective axial stress while maintaining zero radial deformation on a cylindrical specimen.

Note:

The pressures \(P\) correspond to drained or undrained tests. However, in modeling with the Code_Aster, the constraints manipulated in the laws of behavior are effective, that is to say that one does not take into account the hydrostatic pressure of the fluid that can circulate in the pores, this being calculated in models THM.

Volume loading tests (cf. [Figure 2.2-a]) lead us to the following elastic law:

\({k}_{0}P+{K}_{\text{cam}}=({k}_{0}{P}_{0}+{K}_{\text{cam}})\text{exp}\left[{k}_{0}({\epsilon }_{v}^{e}-{\epsilon }_{\mathrm{v0}}^{e})\right]\text{avec}{k}_{0}=\frac{(1+{e}_{0})}{\kappa }\) eq 3.3-3

Likewise, the growth of the charge surface during the contraction phase, its decrease in expansion, and the experimental results suggest writing:

\({P}_{\text{cr}}={P}_{\text{cr}}^{0}\text{exp}\left[k({\epsilon }_{v}^{p}-{\epsilon }_{\mathrm{v0}}^{p})\right],\text{avec}k=\frac{(1+{e}_{0})}{(\lambda -\kappa )}\) eq 3.3-4

\({\epsilon }_{\mathrm{v0}}^{p}\) and \({e}_{0}\) correspond to the volume deformation and to the initial void index, determined by extrapolation of the curve of the oedometric test at pressure \({K}_{\text{cam}}\) (cf. [Figure 2.2-a]).

3.4. Plastic flow law

The two plastic variables are the volume plastic deformation \({\epsilon }_{v}^{p}\) and the deviatoric tensor of the plastic deformations \({\tilde{\epsilon }}^{p}\). The internal variable is also \({\epsilon }_{v}^{p}\) but associated with the work hardening force \({P}_{\text{cr}}\). The material is non-generalized standard. The flow rule is written as:

:math:`` \({\dot{\epsilon }}^{p}=\dot{\Lambda }\frac{\partial f}{\partial \sigma },,{\dot{\epsilon }}_{v}^{p}=-\dot{\Lambda }\frac{\partial F}{\partial {P}_{\text{cr}}}\), eq 3.4-1

\(\Lambda\) being the plastic multiplier.

By breaking down the first term, we get:

\({\dot{\epsilon }}_{v}^{p}=\dot{\Lambda }\frac{\partial f}{\partial P}{\tilde{\dot{\epsilon }}}^{p}=\dot{\Lambda }\frac{\partial f}{\partial s}{\dot{\epsilon }}_{v}^{p}=-\dot{\Lambda }\frac{\partial F}{\partial {P}_{\text{cr}}}\) eq 3.4-2

knowing that:

\(\underline{\underline{P=-\frac{1}{3}\text{tr}(\sigma )}}\text{et}\underline{\underline{{\epsilon }_{v}=-\text{tr}(\epsilon )+\mathrm{3\alpha }(T-{T}_{0})}}\) eq 3.4-3

\(F\) is the plastic potential associated with the work hardening phenomenon. Note that the third part of [éq 3.4-2] is only formal. Indeed, we know \({\dot{\epsilon }}_{v}^{p}\) by the first relationship so we know the evolution of \({P}_{\text{cr}}\).

3.5. Energy writing and plastic work hardening module

We are therefore in the context of « non-generalized standard materials » (we then use three potentials: the charge surface \(f\), the plastic potential \(F\), and the free energy \(\psi\)). Even in this configuration that is less favorable than the traditional framework of non-generalized standard materials, we are sure to satisfy the second principle of thermodynamics [bib4]. With the aid of the consistency condition (expressing that the point representing the load « follows » the load surface) which is written as follows:

\(\text{df}=\frac{\partial f}{\partial P}\text{dP}+\frac{\partial f}{\partial Q}\text{dQ}+\frac{\partial f}{\partial {P}_{\text{cr}}}{\text{dP}}_{\text{cr}}=0,\) eq 3.5-1

we determine the expression for the plastic multiplier [bib4]:

\(\Lambda =\frac{1}{{H}_{p}}\frac{\partial f}{\partial \sigma }\mathrm{d\sigma }=-\frac{1}{{H}_{p}}\frac{\partial f}{\partial {P}_{\text{cr}}}{\text{dP}}_{\text{cr}}\) eq 3.5-2

with [bib4]:

\({H}_{p}=\frac{\partial f}{\partial {P}_{\text{cr}}}\frac{{\partial }^{2}\psi }{\partial {\epsilon }_{v}^{{p}^{2}}}\frac{\partial F}{\partial {P}_{\text{cr}}},\text{où}{H}_{p}\text{est le module d'écrouissage}\) eq 3.5-3

Identifying the first and third parts of [éq 3.4-2] makes it possible to calculate \(F\) which is written as:

\(F=-\int \frac{\partial f}{\partial P}{\text{dP}}_{\text{cr}}={M}^{2}{P}_{\text{cr}}({P}_{\text{cr}}-\mathrm{2P}+{\mathrm{2P}}_{\text{trac}})\) eq 3.5-4

The concept of work hardening being associated with that of blocked energy:

\({P}_{\text{cr}}=\frac{\partial \psi }{\partial {\epsilon }_{v}^{p}}\text{donc}{\text{dP}}_{\text{cr}}=\frac{{\partial }^{2}\psi }{{\partial }^{2}{\epsilon }_{v}^{p}}{\mathrm{d\epsilon }}_{v}^{p}\) eq 3.5-5

where \(\psi\) is the free energy density:

\(\psi =\frac{3}{2}\mu ({\epsilon }_{\text{eq}}^{e}{)}^{2}+\frac{{P}_{0}}{{k}_{0}}\text{exp}({k}_{0}{\epsilon }_{v}^{e})+\frac{{P}_{\text{cr}}^{0}}{k}\text{exp}(k({\epsilon }_{v}^{p}-{\epsilon }_{\mathrm{v0}}^{p}))\) eq 3.5-6

Using [éq 3.4-2], [éq 3.5-4] and [éq 3.5-6], we can derive from [éq 3.5-3] the expression for the plastic work hardening module:

\({H}_{p}=\frac{\partial f}{\partial {P}_{\text{cr}}}\frac{{\partial }^{2}\psi }{\partial {\epsilon }_{v}^{{p}^{2}}}\frac{\partial F}{\partial {P}_{\text{cr}}}=4{\text{kM}}^{4}(P-{P}_{\text{trac}}){P}_{\text{cr}}(P-{P}_{\text{trac}}-{P}_{\text{cr}})\) eq 3.5-7

The work-hardening module is positive in the \((P-{P}_{\text{trac}}>{P}_{\text{cr}})\) contraction phase and negative in the \((P-{P}_{\text{trac}}<{P}_{\text{cr}})\) expansion phase. For \(P-{P}_{\text{trac}}={P}_{\text{cr}}\), the behavior is perfect plastic and takes place at a constant plastic volume.

3.6. Incremental relationships

The equation [éq 3.4-3] and the consistency condition give the flow relationships:

\({\mathrm{d\epsilon }}_{v}^{p}=\frac{1}{k}\left[(\frac{1}{{P}_{\text{cr}}}-\frac{1}{(P-{P}_{\text{trac}})})\text{dP}+\frac{Q}{{M}^{2}(P-{P}_{\text{trac}}){P}_{\text{cr}}}\text{dQ}\right]\) eq 3.6-1

\({\mathrm{d\epsilon }}_{\text{eq}}^{p}=\frac{1}{k}\left[\frac{Q}{{M}^{2}(P-{P}_{\text{trac}}){P}_{\text{cr}}}\text{dP}+\frac{{Q}^{2}}{{M}^{4}(P-{P}_{\text{trac}}){P}_{\text{cr}}(P-{P}_{\text{trac}}-{P}_{\text{cr}})}\text{dQ}\right]\) eq 3.6-2

\(d{\tilde{\epsilon }}^{p}={\mathrm{d\epsilon }}_{\text{eq}}^{p}\frac{3}{2}\frac{s}{Q}\) eq 3.6-3

The rearrangement of [éq 3.6-1] and [éq 3.6-2] leads to:

\(\frac{{\mathrm{d\epsilon }}_{\text{eq}}^{p}}{{\mathrm{d\epsilon }}_{v}^{p}}=\frac{Q}{{M}^{2}(P-{P}_{\text{trac}}-{P}_{\text{cr}})}\) eq 3.6-4

that is to say with the equation [éq 3.6-3],

\(\frac{d{\tilde{\epsilon }}^{p}}{{\mathrm{d\epsilon }}_{v}^{p}}=\frac{3}{2}\frac{s}{{M}^{2}(P-{P}_{\text{trac}}-{P}_{\text{cr}})}\) eq 3.6-5

Special case of the critical point:

For \(f=0\text{et}P-{P}_{\text{trac}}={P}_{\text{cr}}\): \({\dot{P}}_{\text{cr}}=0\), \({\dot{\epsilon }}_{v}^{p}=0\). We deduce from this, considering the elastic law: \(\dot{P}={k}_{0}P{\dot{\epsilon }}_{v}\). The consistency condition gives us \(\dot{Q}=0\).

3.7. Summary of behavioral relationships

Elasticity

\(s=\mathrm{2\mu }{\tilde{\epsilon }}^{e}\) eq 3.7-1

\(P={P}_{0}\text{exp}({k}_{0}{\mathrm{\Delta \epsilon }}_{v}^{e})+\frac{{K}_{\text{cam}}}{{k}_{0}}(\text{exp}({k}_{0}{\mathrm{\Delta \epsilon }}_{v}^{e})-1)\) eq 3.7-2

Plasticity

The criterion: \(f(\sigma ,{P}_{\text{cr}})={Q}^{2}+{M}^{2}{(P-{P}_{\text{trac}})}^{2}-{\mathrm{2M}}^{2}(P-{P}_{\text{trac}}){P}_{\text{cr}}=0\) with \((Q={\sigma }_{\text{eq}})\)

\(\frac{\partial f}{\partial \sigma }=\underset{}{\underset{\underbrace{}}{(-\frac{1}{3}\frac{\partial f}{\partial P}{I}^{d}+\frac{3}{2}\frac{\partial f}{\partial Q}\frac{s}{Q})}}\) eq 3.7-3

So:

\({\tilde{\dot{\epsilon }}}^{p}=3\dot{\Lambda }s\) eq 3.7-4

\({\dot{\epsilon }}_{v}^{p}=\dot{\Lambda }{\mathrm{2M}}^{2}(P-{P}_{\text{trac}}-{P}_{\text{cr}})\) eq 3.7-5

Crossing

\({P}_{\text{cr}}({\epsilon }_{v}^{p})={P}_{\text{cr}0}\text{exp}(k({\epsilon }_{v}^{p}-{\epsilon }_{v}^{{p}_{0}}))\) eq 3.7-6

Elastic behavior: If \(f<0\) then:

\({\dot{P}}_{\text{cr}}=0\) eq 3.7-7

\({\tilde{\dot{\epsilon }}}_{\text{eq}}^{p}=\mathrm{0,}{\dot{\epsilon }}_{v}^{p}=0\) eq 3.7-8

\(\dot{s}=\mathrm{2\mu }\tilde{\dot{\epsilon }}\) eq 3.7-9

\(\dot{P}=({k}_{0}P+{K}_{\text{cam}}){\dot{\epsilon }}_{v}\) eq 3.7-10

Elasto-plastic behavior: If \(f=0\) and \(\dot{f}=0\) then:

\({\dot{P}}_{\text{cr}}\ne 0;{\dot{P}}_{\text{cr}}=k{\dot{\epsilon }}_{v}^{p}{P}_{\text{cr}}\) eq 3.7-11

\({\tilde{\dot{\epsilon }}}^{p}=3\dot{\Lambda }s\text{si}P-{P}_{\text{trac}}\ne {P}_{\text{cr}}\) eq 3.7-12

\({\dot{\epsilon }}_{v}^{p}=\dot{\Lambda }{\mathrm{2M}}^{2}(P-{P}_{\text{trac}}-{P}_{\text{cr}})\text{si}P-{P}_{\text{trac}}\ne {P}_{\text{cr}}\) eq 3.7-13

\(\dot{s}=\mathrm{2\mu }\tilde{\dot{\epsilon }}\) eq 3.7-14

\(\dot{P}=({k}_{0}P+{K}_{\text{cam}}){\dot{\epsilon }}_{v}\) eq 3.7-15

Notes:

• From the only unknown \({\dot{\epsilon }}_{v}^{p}\) , we can deduce the other unknowns \({\tilde{\dot{\epsilon }}}^{p}\) and \({\dot{P}}_{\text{cr}}\) .

• If \(P-{P}_{\text{trac}}={P}_{\text{cr}}\) : \({\dot{\epsilon }}_{v}^{p}=0\) : , \(\dot{Q}={\dot{P}}_{\text{cr}}=\mathrm{0,}\dot{P}={k}_{0}P{\dot{\epsilon }}_{v}\) .