This model is an improvement on the Mohr-Coulomb model, generated by combining nonlinear elastic models and elasto-plastic models to make a suitable model for the behavioral characteristics of silt or sand based ground.  The Modified Mohr-Coulomb model can simulate the Double stiffening behavior, which is not affected by the shear failure or compressive yield.

The axial strain and decrease in material stiffness caused by the initial deviatory stress is similar to the Hyperbolic (nonlinear elastic) model, but it is closer to the plastic theory than elastic theory and has differences in dilatancy angle consideration and yield cap application.

The main nonlinear parameters are as follows.

[Elasticity modulus : Three types of reference stiffness modulus]

This model is a more detailed material model than the Mohr-Coulomb model and the Elasticity modulus can be set at different values for loading and unloading. In most cases, the Elasticity modulus for unloading is set larger to prevent uplift (bulging phenomenon) on the cutting surface due to stress release during excavation modeling. For a rough approximation, in case of hard soil (sand, OC clay), unloading stiffness is set equal to 3 times of secant stiffness in standard drained triaxial test. In case of soft soil, based on the relationship between compression and swelling index, unloading stiffness is approximately set equal to 10 times of secant stiffness.

[Failure Ratio]

The ultimate deviatoric stress is derived from the Mohr-Coulomb failure criterion. As soon as the deviatoric stress reaches to the ultimate value, the failure criterion is satisfied and perfectly plastic yielding occurs. The ratio between the ultimate and the quantity in deviatoric stress is given by the failure ratio which must be smaller than1.

[Reference Pressure]

The reference stress used in the triaxial test of specific strengths on the nonlinear elastic curve. This can represent in-situ horizontal stress at mid-level of soil layer depending on OCR(Over Consolidation Ratio).

[Power Law nonlinear elastic model coefficient]

Hardening soil model is the stress dependency of soil stiffness. The reference stiffness modulus which is used in this model has the relationship with the confining stress dependent stiffness. In this relation, the amount of stress dependency is given by the power m. In case of soft clays, the power is recommended to be taken equal to 1. The value of m around 0.5 for hard soil like sand and OC clays can be used to simulate a logarithmic compression behavior. In general, m is in the range of 0.5 to 1.

[Porosity]

The void ratio is the volume ratio between voids and soil particles. Here, the porosity is the volume ratio between voids and the total soil including water. Hence, unlike the void ratio, the porosity cannot have a value larger than 1 and has a value of 0.6 in general. When soil experience shearing, dilating materials reach to the state of critical density. In order to include this soil behavior by means of dilatancy cut-off, the maximum porosity must be entered as advanced parameters. When the soil is subject to shear hardening, solver recalculates dilatancy angle.

[KNC]

KNC is the percentage of K (Coefficient of earth pressure) in a normally consolidated ground. In other words, it is the effective horizontal stress ratio during maximum vertical stress. This can be expressed as 1-sin (Interior friction angle) and because general clays have an interior friction angle of nearly zero, the value is close to 1. However, it cannot be smaller than '0(zero)'

[Friction angle at shear / Ultimate dilatancy angle / Cohesion]

Same as the friction angle, dilatancy angle, cohesion parameters of the Mohr coulomb material model.

Shear hardening can be defined by equivalent plastic strain related to the mobilized shear resistance automatically. Shear yield surface can expand up to the Mohr-Coulomb failure surface.

Increment of Cohesion with depth can be defined with additional options as in Mohr-Coulomb model. The reference height must be inputted based on the Global coordinates.

[Cap : Compression Hardening]

Compressive yield can happen when excessive compression forces occur on the ground. Normally, the compressive forces that cause yield are very large and the Mohr-Coulomb model has no problems in simulating real soils with it omitted. However, in order to simulate the compressive behavior more accurately, the model considers circles or ellipses when considering compressive failure, hence the name 'Cap'.

[OCR / Pc (Pre- overburden pressure]

Yield surface of hardening soil model in p-q plane has the length of preconsolidation stress and its magnitude is determined by Cap shape factor and cap hardening parameter. The smaller value of cap shape factor(α) lead to steep caps underneath the Mohr-coulomb failure line. For this, the preconsolidation stress can be calculated from either the OCR(Over consolidation ratio) or the Pc(Pre-overburden pressure).

The user can input the pressure at which compressive failure occurs.

[Cap shape factor / Cap Hardening Parameter]

Use this variable to change the shape of the Cap, a yield function shape. Refer to Ch.4 of the Analysis manual for a more detailed algorithm. These are considered automatically based on the relation between KNC and Eoedref rather than directly inputting parameters.

Following is the summary of parameters for the hardening soil model.