Extended Drucker-Prager

You can specify the hardening data for elastic-plastic materials that use the Drucker-Prager plasticity.

For granular materials these models are often used as a failure surface, in the sense that the material can exhibit unlimited flow when the stress reaches yield. This behavior is called perfect plasticity. The models are also provided with isotropic hardening. In this case plastic flow causes the yield surface to change size uniformly with respect to all stress directions. This hardening model is useful for cases involving gross plastic straining or in which the straining at each point is essentially in the same direction in strain space throughout the analysis. Although the model is referred to as an isotropic “hardening” model, strain softening, or hardening followed by softening, can be defined.

The evolution of the yield surface with plastic deformation is described in terms of the equivalent stress $\overline{\sigma }$ , which can be chosen as either the uniaxial compression yield stress, the uniaxial tension yield stress, or the shear (cohesion) yield stress.

• Compression: Define the hardening behavior by giving the uniaxial compression yield stress, ${\sigma }_c$ , as a function of uniaxial compression plastic strain, ${\overline{\epsilon }}^{pl}=\left|{\epsilon }_{11}^{pl}\right|$ .
• Tension: Define the hardening behavior by giving the uniaxial tension yield stress, ${\sigma }_t$ , as a function of uniaxial tension plastic strain, ${\overline{\epsilon }}^{pl}={\epsilon }_{11}^{pl}$ .
• Shear: Define the hardening behavior by giving the cohesion, $d=\frac{\sqrt{3}}{2}\tau (1+\frac{1}{K})$ , as a function of equivalent shear plastic strain, ${\overline{\epsilon }}^{pl}=\frac{{\gamma }^{pl}}{\sqrt{3}}$ , where $\tau$ is the yield stress in shear, $K$ is the ratio of flow stress in triaxial tension to the flow stress in triaxial compression, and ${\gamma }^{pl}$ is the engineering shear plastic strain.

Classical “creep” behavior of materials that exhibit plasticity according to the extended Drucker-Prager models can be defined for implicit analyses.

Creep and plasticity can be active simultaneously, in which case the resulting equations are solved in a coupled manner. To model creep only (without rate-independent plastic deformation), large values for the yield stress must be provided in the Drucker-Prager hardening definition: the result is that the material follows the Drucker-Prager model while it creeps, without ever yielding. When using this technique, a value must also be defined for the eccentricity, $ϵ$ , since both the initial yield stress and eccentricity affect the creep potentials. This capability is limited to the linear model with a von Mises (circular) section in the deviatoric stress plane ( $K=1$ ; that is, no third stress invariant effects are taken into account) and can be combined only with linear elasticity.

Drucker-Prager has four options for defining the creep rate, ${\dot{\overline{\epsilon }}}^{cr}$ , as a function of the creep stress, ${\overline{\sigma }}^{cr}$ , reference stress, $q_0$ ,

• Power law: ${\dot{\overline{\epsilon }}}^{cr}={\dot{\epsilon }}_0{\left[{\left(\frac{{\overline{\sigma }}^{cr}}{q_0}\right)}^n{\left[(m+1){\overline{\epsilon }}^{cr}\right]}^m\right]}^{\frac{1}{m+1}}$ where, ${\overline{\sigma }}^{cr}$ is the creep stress, ${\overline{\epsilon }}^{cr}$ is the creep strain, ${\dot{\epsilon }}_0$ , is a reference strain rate, $q_0$ is a reference stress, and $m$ and $n$ are material parameters.
• Time power law: ${\dot{\overline{\epsilon }}}^{cr}={\dot{\epsilon }}_0{\left(\frac{{\overline{\sigma }}^{cr}}{q_0}\right)}^n{\left({\dot{\epsilon }}_{0}t\right)}^m$ where, ${\overline{\sigma }}^{cr}$ is the creep stress, ${\dot{\epsilon }}_0$ , is a reference strain rate, $q_0$ is a reference stress, $t$ is time, and $m$ and $n$ are material parameters.
• Singh-Mitchell: ${\dot{\overline{\epsilon }}}^{cr}=A{e}^{\left(\alpha {\overline{\sigma }}^{cr}\right)}{\left(\frac{t_1}{t}\right)}^m$ where, ${\overline{\sigma }}^{cr}$ is the creep stress, $t$ is time, $t_1$ is a reference time value (must be small compared to the total time), and $A$ , $\alpha$ , $m$ , and $n$ are material parameters.
• User: To define the creep law using user subroutine CREEP. There is no input data for the user subroutine.