Plastic Models

There are six plasticity models in PISA. They are the von-Mises, Tresca, associated and non-associated Drucker-Prager, and associated and non-associated Mohr-Coulomb models. Modified Cam Clay and Double Yield Surface models are also available. All of the models assume linear elastic perfectly plastic brittle weakening behaviour. In other words, the stress strain response assumes that the material behaves in a linear elastic manner, according to the elastic parameters specified in the model, prior to yielding and in a perfectly plastic behaviour after yielding. The peak and residual strength values can be different. If the material has no weak-ening characteristics, the residual value can be zero or equal to the peak value.

Von-Mises Model: Associated Flow Rule

The von-Mises model is an elastic brittle perfectly plastic model with associated flow rule. Deformation prior to yielding is assumed to be linear elastic, governed by the elastic parameters E and .

The von-Mises criterion can be expressed as:

f = q - k = 0.

• where
  • f is the yield function;
  • q is related to the second stress invariance;
• and
  • k is the peak or residual uniaxial compressive yield strength of the material..

Tresca Model: Associated Flow Rule

The Tresca model is an elastic brittle perfectly plastic model with associated flow rule. Deformation prior to yielding is assumed to be linear elastic governed by the elastic parameters E and .

Yield function of the Tresca criterion is given by:

f =2 q cos() - k = 0.

• where
  • f is the yield function;
  • q is related to the second stress invariance;
  • is the Lode angle;
• and k is the peak or residual uniaxial compressive yield strength of the material.

Drucker-Prager Model: Associated and Nonassociated Flow Rule

The Drucker-Prager model is an elastic brittle perfectly plastic model with associated and non-associated flow rule. Deformation prior to yielding is assumed to be linear elastic governed by the elastic parameters E and .

Yield function f for the Drucker-Prager Model is given by:

.

• where
  • is the mean stress and J2 is the second stress invariant;
  • c and are the peak or residual cohesion and friction angle of the material.

If the associated flow rule is used, the plastic potential is equal to the yield function. For non-associated flow rule, the plastic potential is given by: - after yielding.

• where is the dilation angle;
• and is the first stress invariant.

Mohr Coulomb Model: Associated and Nonassociated Flow Rule

The Mohr-Coulomb model is an elastic brittle perfectly plastic model with associated and non-associated flow rule. Deformation prior to yielding is assumed to be linear elastic governed by the elastic parameters E and .

Yield function f for the Mohr-Coulomb Model is given by:

at yielding;

If the associated flow rule is used, the plastic potential g is equal to the yield function. For non-associated flow rule, the plastic potential is given by:

at yielding;

• where is the dilation angle.