The Cam-clay material model was developed in the 1970s at the University of Cambridge, and since then it has experienced many modifications. The modified Cam-clay (MCC) model is the most commonly used model due to its smooth yield surface, and it is the one implemented in COMSOL Multiphysics.

The modified Cam-clay model is a so-called critical state model, where the loading and unloading of the material follow different paths in stress space. The model also features hardening and softening. Different formulations can be found in textbooks, see for instance Ref. 68, Ref. 69, and Ref. 70.

The yield function was originally written in terms of the Invariants of the Stress Tensor

This is an ellipse in the Meridional Plane, with a circular section in the Octahedral Plane.

If we write the Cam-clay yield function in terms of the invariants described in Pressure-Dependent and Soil Plasticity:

Here, σm is the mean stress, σmises is the von Mises equivalent stress, θ is the Lode angle, and Γ(θ) is a function of θ that defines the shape of the yield function in the Octahedral Plane.

The shape of the yield function in the Octahedral Plane, Γ(θ), can be made a function of the Lode angle θ by using any of the functions described in Octahedral Section. The most common choice, Γ(θ) = 1, results in a circular section that recovers the original Cam-clay model.

The parameter pc > 0 is the consolidation pressure, and the parameter M > 0 defines the slope of the critical state line in the Meridional Plane which also represents the eccentricity of the ellipse. This parameter can be matched to the friction angle ϕ in the Mohr–Coulomb Criterion criterion as

The parameter MQ represents the slope of the critical state line for the plastic potential, which a can be matched to the Mohr–Coulomb Criterion criterion at the compressive meridian by the dilatation angle ψ

The soil response to isotropic compression is described by the curve of the void ratio (or specific volume or volumetric strain) versus the logarithm of pressure as shown in Figure 3-30.

The compression index λ corresponds to the slope of the virgin isotropic consolidation line, and the swelling index κ corresponds to the slope of the swelling line (also called unloading-reloading line) in the e versus ln(p) plane. The reference void ratio eref is measured at the reference pressure pref and the initial void ratio e0 is measured at the initial mean stress p0. The starting value of the initial mean stress p0 is the reference pressure pref. The equation for the virgin isotropic consolidation line reads

The void ratio at the preconsolidation pressure pc0 is given by

where e0 is the initial void ratio calculated as

Integrating Equation 3-131, the equation is reformulated as

At zero volumetric elastic strain and with zero contribution from the Initial Stress and Strain or External Stress features, the initial mean stress is nonzero and equal to the reference pressure pref. The reference pressure acts as an in situ stress.

The tangent bulk modulus Kt is written as

The secant bulk modulus Ks (or bulk modulus K) is written as

Equation 3-134 is used to compute the void ratio, which matches the analytical expression in Equation 3-135. These two equations can be used for verification purposes.

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 initial void ratio e0, the swelling index κ, and the compression index λ, which are positive parameters that fulfill

Here, σ is the stress tensor, ε is the total strain tensor, εel is the elastic strain tensor, and G is the shear modulus.

At zero volumetric strain, and with zero contribution from Initial Stress and Strain or External Stress features, the pressure in the MCC model is equal to the reference pressure pref (see Equation 3-132 and Equation 3-133). This is needed since the MCC model does not have stiffness at zero stress. The reference pressure appears as an additional term in the variational formulation (weak equation).

The modified Cam-clay model introduces a nonlinear relation for the hydrostatic pressure as a function of the volumetric elastic strain given by Equation 3-132.

The plastic strain tensor εp is computed from the flow rule

where λp is the plastic multiplier and the derivatives of the plastic potential Qp with respect to the stress tensor σ. An associated flow rule is used such that Qp = Fy.

where the plastic strain rate tensor includes both deviatoric and volumetric parts. The trace of the plastic strain rate tensor (the volumetric plastic strain rate ) then reads

This relation explains 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 Metal Plasticity, in the modified Cam-clay soil model the volumetric plastic strain can either increase or decrease as plastic deformation occurs.

In the MCC model, hardening is controlled by changes in the consolidation pressure pc as a function of volumetric plastic strain.

Hardening introduces changes in the shape of the Cam-clay ellipse, since its major semiaxis 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 MCC material, the yield function is shifted along the p axis.

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

The structured Cam-clay (SCC) model (Ref. 76) was developed to circumvent the limitations of the Cam-clay model when applied to structured soils and clays. The SCC model, however, does not consider 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 (MSCC) model was further developed to model destructured, naturally structured and artificially structured clays (Ref. 75), and it is the one implemented in COMSOL Multiphysics.

In the MSCC model, the reduction of mean effective stress due to structure degradation, pb, depends on the equivalent plastic strain εpe. 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, pb0 is the initial structural strength, pbf the failure structural strength, εfp,dev is the plastic deviatoric strain at failure, and ds is the destructuring index due to shear deformation.

where is e is the void ratio of the structured clay, Δe is the additional void ratio, and e* is the void ratio of the destructured clay at the same stress state. The void ratio is computed by using Equation 3-135. Further, Δec0 is the additional void ratio at preconsolidation pressure, and dv is the destructuring index due to volumetric deformation.

Here, θ is the Lode angle, and Γ(θ) is a function of θ that defines the shape of the yield function in the Octahedral Plane.

The shape of the yield function in the Octahedral Plane, Γ(θ), can be made a function of the Lode angle θ by using any of the functions described in Octahedral Section. The most common choice, Γ(θ) = 1, results in a circular section that recovers the original criterion.

The parameter MQ represents the slope of the critical state line for the plastic potential, which a can be matched to the Mohr–Coulomb Criterion criterion at the compressive meridian by the dilatation angle ψ

Here, ς is a parameter to smooth the shape of the plastic potential. The slope of the critical state line, MQ, is found by using Equation 3-130.

In the modified structured Cam-clay soil model, hardening is controlled by the consolidation pressure pc as a function of volumetric plastic strain, and it is described as

Here, λ* is the compression index for destructured clay, κ is the swelling index, and η is ratio of shear stress to mean stress.

The relationship between hydrostatic pressure and volumetric elastic strain is outlined in The Modified Cam-Clay Soil Model,

At zero volumetric strain, and with zero contribution from the Initial Stress and Strain or External Stress features, the pressure in the MSCC model is equal to the reference pressure pref (see Equation 3-138 and Equation 3-133). This is needed since the MSCC model does not have stiffness at zero stress. The reference pressure appears as an additional term in the variational formulation (weak equation).

The Barcelona basic model (BBM) was developed to simulate the loading of unsaturated and partially saturated soils, by incorporating an extra state variable for the pore suction. The suction value depends on the amount of water in the soil, and it affects the flow in porous soils as well as the deformation and stress distribution.

The BBM model uses the concepts of plasticity theory, incorporating the critical state model (Ref. 77). 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 numerical limitations in the original BBM model (Ref. 78). The model implemented in COMSOL Multiphysics follows Ref. 78 with certain modifications described in this section.

Here, p and q are stress invariants as defined in The Modified Cam-Clay Soil Model, pcs is the consolidation pressure at current suction, ps is the tensile strength due to current suction, s is the current suction, b is a dimensionless smoothing parameter, sy is the yield value at current suction, and pref is the reference pressure at which the reference void ratio eref was measured.

The shape of the yield function in the Octahedral Plane, Γ(θ), can be made a function of the Lode angle θ by using any of the functions described in Octahedral Section. The most common choice, Γ(θ) = 1, results in a circular section that recovers the original criterion.

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.

The parameter MQ represents the slope of the critical state line for the plastic potential, which a can be matched to the Mohr–Coulomb Criterion criterion at the compressive meridian by using the dilatation angle ψ

Here, ς is a parameter to smooth the shape of the plastic potential. The slope of the critical state line, MQ, is found by using Equation 3-130.

The consolidation pressure at current suction pcs is calculated from

where λ(s) is the compression index at current suction, λ0 is the compression index at saturation, and κ is the swelling index. The compression index at current suction, λ(s), is given by

where w and m are weighting and soil stiffness parameters.

The slope of the critical state line M can be computed from the Matsuoka–Nakai criterion, in which case it depends on both the Lode angle θ and the angle of internal friction ϕ

At the tensile or compressive meridians, where the Lode angle is θ = 0 or θ = π/3, the slope of the critical state line achieves the same expression as when matched to the Mohr–Coulomb Criterion, see Equation 3-130

It can also be matched to the Matsuoka–Nakai Criterion.

As in The Modified Cam-Clay Soil Model, hardening is controlled by the evolution of the consolidation pressure pc, which depends on the volumetric plastic strain εp,vol.

The initial void ratio e0, the swelling index κ, and the compression index at saturation λ0, are positive parameters.

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

Here, λs is the compression index for changes in suction, κs is the swelling index for changes in suction, and patm is the atmospheric pressure, all positive parameters. Note that Equation 3-140 and Equation 3-141 are normally given with opposite sign. Here, however, the structural mechanics convention is used, so the increments in consolidation pressure and suction are positive in compression.

Here K is the bulk modulus, and Kcs the stiffness to suction. Note that the pressure p in compression and suction s is positive variables, but the elastic volumetric strain εel,vol is negative in compression. The volumetric elastic response due to suction is given by

Here, s0 is the initial suction. The volumetric elastic response due to suction is accounted as an inelastic strains in the total strain tensor, any changes in suction while keeping a constant pressure can cause volumetric strains.

At zero volumetric strain, and with zero contribution from the Initial Stress and Strain or External Stress feature, the pressure in the BBMx model is equal to the reference pressure (see Equation 3-142 and Equation 3-133). This is needed since the BBMx model does not have stiffness at zero stress. The reference pressure appears as an additional term in the variational formulation (weak equation form).

The hardening soil model is an elastoplastic material model with a stress and stress path-dependent stiffness. It is a so-called double stiffness model, meaning that the soil stiffness is different during primary loading and unloading/reloading load paths (Ref. 79).

In the original hardening soil model, failure in shear occurs according to the Mohr–Coulomb Criterion. In the so-called hardening soil smooth model (Ref. 79), failure in shear occurs according to Matsuoka–Nakai Criterion.

In Ref. 81, the shear hardening and failure surfaces are combined, and the failure surface imposes the final limit on shear stresses. COMSOL Multiphysics implements the latter version.

The 50% failure stiffness for primary loading, E50, and the stiffness for unloading and reloading, Eur, are used to define the elastic relation and hardening variable. These are given by

Here is the reference failure stiffness for primary loading at reference pressure, is the reference stiffness for unloading and reloading at reference pressure, c is the cohesion, ϕ is the friction angle, and m is the stress exponent.

The ultimate deviatoric stress qf and the stress to failure qa are derived from the Mohr–Coulomb Criterion

where Rf is the failure ratio.

Here, θ is the Lode angle, and Γ(θ) is a function of θ that defines the shape of the yield function in the Octahedral Plane, and ψm is the mobilized dilatancy angle.

The shape of the yield function in the Octahedral Plane, Γ(θ), can be made a function of the Lode angle θ by using any of the functions described in Octahedral Section. The most common choice, Γ(θ) = 1, results in a circular section.

The yield function for the elliptic cap is described in the Compression Cap section.

For linear hardening, the internal variable for the preconsolidation pressure, pc, depends on the volumetric plastic strain εpl,vol as

where Kc is the hardening modulus and

The dilatancy cutoff 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.

If required, add a Compression Cap or a Tension Cutoff criterion to the hardening soil model.

The mobilized dilatancy angle ψm is explicitly used in the definition for the plastic potential Qp. Different authors define this variable using different expressions. The original expression for the mobilized dilatancy derived by Rowe (Ref. 79) is

here, ϕc is the critical friction angle and ϕm is the mobilized friction angle, and these are derived from

where ψ is the dilatancy angle, σi are the principal stresses, c is the cohesion, and ϕ is the angle of internal friction.

Other expressions (Ref. 79) for the mobilized dilatancy angle include the Wehnert correction

The hardening soil small strain model is a combination of Small-Strain Overlay model and The Hardening Soil Model.

Here is the reference small strain shear modulus for primary loading at reference pressure, c is the cohesion, ϕ is the friction angle, and m is the stress exponent.

The critical shear strain, defined as

Here, Gur is the shear modulus for unloading/reloading, as defined in The Hardening Soil Model.

The yield function for the elliptic cap is described in the Compression Cap section. If required, add a Tension Cutoff criterion to the hardening soil small strain model.