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 of clays. Different formulations can be found in textbooks, see for instance Ref. 15, Ref. 16, and Ref. 17.

This is an ellipse in the pq-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 pq-plane. This parameter can be matched to the angle of internal friction ϕ in the Mohr–Coulomb criterion as

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 3-27.

The compression index λ is the slope of the virgin isotropic consolidation line, and the swelling index κ is 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 initial void ratio e0 is measured at the initial mean stress pi. The starting value of initial mean stress pi is the reference pressure pref. The equation for the virgin isotropic consolidation line is written as

The void ratio at the initial consolidation pressure pc0 is given by

where e0 is the initial void ratio calculated as

Integrating Equation 3-113, the equation is reformulated as

At zero volumetric elastic strain and with zero contribution from 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 3-116 is used to compute the void ratio which matches exactly with Equation 3-117 which may 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, σ0 is the initial or external stress 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 (see Equation 3-114 and Equation 3-115). This is needed as the MCC model do not have stiffness at zero stress. The reference pressure appears as an additional term in the variational formulation (weak equation).

As opposed to the Linear Elastic Material, the MCC model introduces a nonlinear relation for the hydrostatic pressure as a function of the volumetric elastic strain given by Equation 3-114.

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.

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 J2 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 the consolidation pressure pc as a function of volumetric plastic strain, as described in Equation 3-118.

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, 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 the equivalent 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 (SCC) model was developed (Ref. 22, Ref. 23) 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. 22), 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 equivalent plastic strain at failure, and ds is the destructuring index due to shear deformation.

where is e void ratio of the structured clay, Δe is additional void ratio, e* is void ratio of the destructured clay at the same stress state. The void ratio can be found by using Equation 3-117. Further, 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. The slope of the critical state line, M can be found by using Equation 3-112.

In the modified structured Cam-clay soil model, hardening is controlled by the consolidation pressure pc as a function of volumetric plastic strain, and 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 the same as the one outlined in The Modified Cam-Clay Soil Model,

At zero volumetric strain, and with zero contribution from Initial Stress and Strain or External Stress features, the pressure in the MSCC model is equal to the reference pressure (see Equation 3-120 and Equation 3-115). This is needed as the MSCC model do 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. 24). 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 of the original BBM model (Ref. 25). The model implemented in COMSOL Multiphysics follows Ref. 25 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 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 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 3-112

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 3-122 and Equation 3-123 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 in the total strain tensor for BBMx model, so any changes in suction with keeping pressure constant can cause volumetric strains.

At zero volumetric strain, and with zero contribution from Initial Stress and Strain or External Stress features, the pressure in the BBMx model is equal to the reference pressure (see Equation 3-124 and Equation 3-115). This is needed as 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. 26).

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. 26), failure in shear occurs according to the The Matsuoka–Nakai Criterion. In Ref. 28, the so-called Panteghini and Lagioia version of Matsuoka–Nakai criterion is used.

The 50% failure stiffness for primary loading, E50, and the stiffness for unloading and reloading, Eur, are used to define the plastic potential and failure criterion. 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 angle of internal friction, 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. Consider the stress invariants and , the yield function and plastic potential for the shear hardening are given by

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

Here, Rc is the ellipse aspect ratio, and it can directly entered, or given in terms of the coefficient of earth pressure at rest

The internal variables pc and γp depend on the volumetric plastic strain εpl,vol and the plastic strain invariant J2 (εpl), and their evolution is defined as

where H is the hardening modulus which depends on the bulk modulus in compression Kc and the bulk modulus in swelling Ks

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.

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. 26) 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. 26) for the mobilized dilatancy angle include the Wehnert correction

The Rowe–Li–Dafalias model uses Equation 3-126 to define the mobilized dilatancy angle when , and Equation 3-129 otherwise.

The Hardening Soil Small Strain model is a combination of Small Strain Overlay model and The Hardening Soil Model.

Here is the reference initial shear modulus for primary loading at reference pressure, c is the cohesion, ϕ is the angle of internal friction, 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 The Hardening Soil Model.