The Cam-Clay material model was developed in the 1970s at the University of Cambridge, and since then it has experienced different modifications. The modified Cam-Clay model is the most commonly used model due to its smooth yield surface, and it is the one implemented in the Geomechanics Module.

The modified Cam-Clay model is a so-called critical state model, where the loading and unloading of the material follows different trajectories in stress space. The model also features hardening and softening of clays. Different formulations can be found in textbooks about the model (see Ref. 13, Ref. 14, and Ref. 15).

This is an ellipse in p-q plane, with a cross section independent of the Lode angle. Note that p, q, and pc are positive variables.

The parameter M > 0 defines the slope of the critical state line in the p-q space. This parameter can be related to the angle of internal friction φ in the Mohr-Coulomb criterion as

In the modified Cam-Clay model, hardening is controlled by the consolidation pressure pc, which depends on the volumetric plastic strain εp,vol as

The evolution of the consolidation pressure depends on the values for the reference void ratio eref, the swelling index κ, and the compression index λ, which are positive parameters that fulfill

The compression index λ is the slope of the virgin isotropic consolidation line, and the swelling index κ is the slope of the rebound-reloading line (also called loading-reloading line) in the e versus ln(p) plane.

Here, σ is the stress tensor, ε is the total strain tensor, εinel is the inelastic strain tensor, σ0 is the initial or external stress tensor, and C is the fourth-order elasticity tensor. Assuming only elastic stresses in a linear isotropic elastic medium, Hooke’s law simplifies to

where K is the bulk modulus and G is the shear modulus. By using the convention that the pressure is the mean stress defined as positive in compression, the trace of the stress tensor is linearly related to the volumetric elastic strain εel,vol (the trace of the elastic strain tensor) by the bulk modulus

here, the pressure p is positive in compression, but the elastic volumetric strain εel,vol is positive in tension.

As opposed to a Linear Elastic Material, the modified Cam-Clay soil model introduces a nonlinear relation for the hydrostatic pressure as a function of the volumetric elastic strain:

here, the reference pressure pref is the pressure at which the reference void ratio eref was measured, and κ is the swelling index.

The initial bulk modulus is then given by the expression K0 = Belpref, and the tangent bulk modulus by the expression Kt = Belpmcc. See also the section Tangent and secant moduli.

The yield surface and the associated flow rule (Qp = Fy) give the rate equation for the plastic strains. The plastic strain tensor εp is calculated from the plastic multiplier λp and the derivatives of the plastic potential Qp with respect to the stress tensor σ

The trace of the plastic strain rate tensor (the volumetric plastic strain rate ) then reads

This relation explains the reason why there is isotropic hardening when the pressure is p > pc/2 and isotropic softening when p < pc/2. As opposed to what happens in J2 plasticity, in the modified Cam-Clay soil model the volumetric plastic strain can either increase or decrease as plastic deformation occurs.

In the modified Cam-Clay model, hardening is controlled by the consolidation pressure pc as a function of volumetric plastic strain, as described in Equation 3-40.

Hardening introduces changes in the shape of the Cam-Clay ellipse, since its major semi-axis depends on the value of the consolidation pressure pc. The initial consolidation pressure pc0 defines the size of the ellipse before plastic deformation occurs.

When an external pore pressure pf is added to the modified Cam-Clay material, the yield function is shifted on the p axis, and the yield function reads:

The quantity p − pf is normally regarded as the effective pressure, or effective stress, which should not be confused with von Mises stress. To add the effect of a fluid pressure in the pores pf to the Modified Cam-Clay material, add an External Stress feature.

The structured Cam-Clay model (SCC) was developed (Ref. 20, Ref. 21) to circumvent the limitations of the Cam-Clay model when applied to structured soils and clays. The SCC model, however, does not considered the influence of the soil structure neither on strength characteristics (especially cohesion) nor in the softening behavior, and it is also not suitable to model cemented clays.

The modified structured Cam-Clay model (MSCC) was further developed to model destructured, naturally structured and artificially structured clays (Ref. 20), and it is the one implemented in the Geomechanics Module.

In the MSCC model, the reduction of mean effective stress due to structure degradation, pb, depends on the shear plastic strain εp,dev. The destructuring mechanism is the process of reducing structure strength due to the degradation and crushing of the structure. The structure degradation is given by

here, pbi is the initial structural strength, pbf the failure structural strength, εp,devc is the effective plastic strain at failure, and ds is the destructuring index due to shear deformation.

where is e void ratio of the structured clay, e* is void ratio of the destructured clay at the same stress state, Δei is the additional void ratio at consolidation pressure, and dv is the destructuring index due to volumetric deformation.

here, ς is a parameter to smooth the shape of the plastic potential.

As opposed to The Modified Cam-Clay Soil Model, the slope of the critical state line, M, depends on both the Lode angle θ and the angle of internal friction

In the modified structured Cam-Clay soil model, hardening is controlled by the consolidation pressure pc as a function of volumetric plastic strain, as described in Equation 3-40 and the contribution to the mean stress due to the structure, pb, as written in Equation 3-41.

The suction value depends on the amount of water in the soil. The BBM uses the concepts of plasticity theory, incorporating the critical state model (Ref. 22). This soil model matches the results obtained with the Modified Cam-Clay Model in fully saturated soils.

The so-called extended Barcelona Basic model (BBMx) was further developed to overcome the numerical limitations of the original BBM model (Ref. 23). The BBMx model presents a smooth yield surface with respect to both stress and suction, and it is the one implemented in the Geomechanics Module.

here, pcs is the consolidation pressure at current suction, b is a dimensionless smoothing parameter, and sy is the yield value at current suction. The tensile strength due to current suction, ps, is linearly related to the suction level as ps = ks, where k is the tension to suction ratio.

As described in the modified Cam-Clay model, hardening is controlled by the consolidation pressure pc, which depends on the volumetric plastic strain εp,vol as

The evolution of the consolidation pressure depends on the values for the reference void ratio eref, the swelling index κ, and the compression index λ, which are positive parameters.

The evolution of the yield value at current suction sy is also governed by the volumetric plastic strain εpl,vol as

where λs is the compression index at current suction and patm is the atmospheric pressure. Note that Equation 3-42 and Equation 3-43 are normally given with opposite sign, but here we use Structural Mechanics convention so the increments in consolidation pressure and suction are positive in compression.

The consolidation pressure at current suction pcs is given by

where λ0 is the compression index at saturation and the compression index at current suction, λs, is given by

where w and m are weighting and soil stiffness parameters.

In the BBMx model, the suction s is linearly related to the volumetric elastic strain εel,vol and the pressure

here K is the bulk modulus, Kc the stiffness to suction, and s0 is the initial value of suction. The pressure p is positive in compression, but the elastic volumetric strain εel,vol is positive in tension.

The Hardening Soil model is a nonlinear elastic material model with stress dependent and stress path dependent stiffness approach. It is a so-called double stiffness model, which has different stiffness during primary loading and unloading/reloading cases (Ref. 24). The yield surface for th Hardening Soil model is combination of a conical surface and a elliptic cap surface in stress space. The failure in shear occurs according to Mohr-Coulomb criterion.

The stiffness modulus for primary loading, denoted by E50, and for unloading/reloading, denoted by Eur, are given by the expressions

here and are reference stiffness moduli at reference pressure pref, c is the cohesion, phi is the angle of internal friction, and m is the stress exponent. From the Mohr-Coulomb criterion, the ultimate deviatoric stress, qf, and the stress to failure qa are defined as

where Rf is failure ratio.

here, γp is the accumulated plastic shear strain, and ψm is the mobilized dilatancy angle.

here, Rc is ellipse aspect ratio.

The evolution of the internal variables pc and γp are governed by the volumetric plastic strain εpl,vol and the plastic strain invariant J2 (εpl) as

here, H is the hardening modulus, which is derived from the bulk modulus in compression (for primary loading) Kc, and the bulk modulus in swelling (for unloading and reloading) Ks.

The dilatancy cut-off is implemented by setting the mobilized dilatancy angle ψm equal to zero when the void ratio reaches the critical void ratio emax.

where C is a function of the stiffness modulus Eur and Poisson’s ratio ν.