Cap Plasticity (Modified Drucker-Prager)

The cap plasticity model:

is intended to model cohesive geological materials that exhibit pressure-dependent yield, such as soils and rocks;
is based on the addition of a cap yield surface to the Drucker-Prager plasticity model that provides an inelastic hardening mechanism to account for plastic compaction and helps to control volume dilatancy when the material yields in shear;
can be used in an implicit analysis to simulate creep in materials exhibiting long-term inelastic deformation through a cohesion creep mechanism in the shear failure region and a consolidation creep mechanism in the cap region;
can be used with either the elastic material mode or, in an implicit analysis if creep is not defined, the porous elastic material model; and
provides a reasonable response to large stress reversals in the cap region; however, in the failure surface region the response is reasonable only for essentially monotonic loading.

Cap Plasticity

The addition of the cap yield surface to the Drucker-Prager model serves two main purposes: it bounds the yield surface in hydrostatic compression, thus providing an inelastic hardening mechanism to represent plastic compaction; and it helps to control volume dilatancy when the material yields in shear by providing softening as a function of the inelastic volume increase created as the material yields on the Drucker-Prager shear failure surface.

The yield surface has two principal segments: a pressure-dependent Drucker-Prager shear failure segment and a compression cap segment. The Drucker-Prager failure segment is a perfectly plastic yield surface (no hardening). Plastic flow on this segment produces an inelastic volume increase (dilation) that causes the cap to soften. On the cap surface plastic flow causes the material to compact.


Input Data	Description
Material Cohesion	Material cohesion, $d$ , in the $p - t$ plane in an implicit analysis and in the $p - q$ plane in an explicit analysis.
Angle of Friction	Material angle of friction, $β$ , in the $p - t$ plane in an implicit analysis and in the $p - q$ plane in an explicit analysis.
Cap Eccentricity Parameter	Cap eccentricity parameter, $R$ . Its value must be greater than zero. Typically $0.0001 \leq R \leq 1000.$
Initial Cap Yield Surface Position	Initial cap yield surface position on the volumetric inelastic strain axis, $ε_{v o l}^{i n} \|_{0}$ .
Transition Surface Radius Parameter	Transition surface radius parameter, $α$ . Its value must be a small number compared to unity. If the material model includes creep properties, $α$ must equal zero.
Flow Stress Ratio	The value of $K$ must be such that $0.778 \leq K \leq 1$ . If creep properties are included in the material model, $K$ must be set to 1.0.
Use temperature-dependent data	Specifies material parameters that depend on temperature. A Temperature field appears in the data table. For more information, see Specifying Material Data as a Function of Temperature and Independent Field Variables.
Number of field variables	Specifies material parameters that depend on one or more independent field variables. A Field column appears in the data table. For more information, see Specifying Material Data as a Function of Temperature and Independent Field Variables.

Cap Hardening

This option is used to specify the hardening part of the material model for elastic-plastic materials that use the Drucker-Prager/Cap yield surface.

You can specify a stress scale factor equal to scale the yield stress.


Input Data	Description
Stress Scale Factor	Yield stress scale factor.

The hardening curve specified for this model interprets yielding in the hydrostatic pressure sense: the hydrostatic pressure yield stress, $p_{b}$ , is defined as a tabular function of the volumetric inelastic strain, and, if required, a function of temperature and other predefined field variables. The range of values for which you define $p_{b}$ must be sufficient to include all values of effective pressure stress subjected to the material during the analysis.


Input Data	Description
Hydrostatic Pressure Yield Stress	Hydrostatic pressure yield stress. (The initial tabular value must be greater than zero, and values must increase with increasing volumetric inelastic strain.)
Volumetric Inelastic Strain	Absolute value of the corresponding plastic strain.
Use temperature-dependent data	Specifies material parameters that depend on temperature. A Temperature field appears in the data table. For more information, see Specifying Material Data as a Function of Temperature and Independent Field Variables.
Number of field variables	Specifies material parameters that depend on one or more independent field variables. A Field column appears in the data table. For more information, see Specifying Material Data as a Function of Temperature and Independent Field Variables.

Cap Creep

Classical “creep” behavior of materials that exhibit plasticity according to the capped Drucker-Prager plasticity model can be defined in implicit analyses. The creep behavior in such materials is intimately tied to the plasticity behavior through the definitions of creep flow potentials and definitions of test data. As a result, you must include cap plasticity and cap hardening in the material definition. If rate-independent plastic behavior is not required in the model, large values for the cohesion, $d$ , as well as large values for the compression yield stress, $p_{b}$ , must be provided in the plasticity definition: as a result the material follows the capped Drucker-Prager model while it creeps, without ever yielding. This capability is limited to cases in which there is no third stress invariant dependence of the yield surface $(K = 1)$ and cases in which the yield surface has no transition region $(α = 0)$ . The elastic behavior must be defined using linear isotropic elasticity.

Creep behavior defined for the cap model is active only during soils consolidation, coupled temperature-displacement, and transient quasi-static procedures.

Cap creep has two possible creep mechanisms that are active in different loading regions: a cohesion mechanism, which follows the type of plasticity active in the shear-failure plasticity region, and a consolidation mechanism, which follows the type of plasticity active in the cap plasticity region. You can choose to activate one or both of the creep mechanisms.

Table 1. Creep Mechanism
Input Data	Description
Cohesion	Use a cohesion mechanism.
Consolidation	Use a consolidation mechanism.
Cohesion and Consolidation	Use both cohesion and consolidation mechanisms.

Cap Plasticity has four options for defining the creep rate, ${\dot{\bar{ε}}}^{c r}$ , as a function of the creep stress, ${\bar{σ}}^{c r}$ , reference stress, $q_{0}$ ,

Power law: ${\dot{\bar{ε}}}^{c r} = {\dot{ε}}_{0} {[{(\frac{{\bar{σ}}^{c r}}{q_{0}})}^{n} {[(m + 1) {\bar{ε}}^{c r}]}^{m}]}^{\frac{1}{m + 1}}$ where, ${\bar{σ}}^{c r}$ is the creep stress, ${\bar{ε}}^{c r}$ is the creep strain, ${\dot{ε}}_{0}$ , is a reference strain rate, $q_{0}$ is a reference stress, and $m$ and $n$ are material parameters.
Time power law: ${\dot{\bar{ε}}}^{c r} = {\dot{ε}}_{0} {(\frac{{\bar{σ}}^{c r}}{q_{0}})}^{n} {({\dot{ε}}_{0} t)}^{m}$ where, ${\bar{σ}}^{c r}$ is the creep stress, ${\dot{ε}}_{0}$ , is a reference strain rate, $q_{0}$ is a reference stress, $t$ is time, and $m$ and $n$ are material parameters.
Singh-Mitchell: ${\dot{\bar{ε}}}^{c r} = A e^{(α {\bar{σ}}^{c r})} {(\frac{t_{1}}{t})}^{m}$ where, ${\bar{σ}}^{c r}$ is the creep stress, $t$ is time, $t_{1}$ is a reference time value (must be small compared to the total time), and $A$ , $α$ , $m$ , and $n$ are material parameters.
User: To define the creep law using user subroutine CREEP. There is no input data for the user subroutine.

Table 2. Law=Power Law
Input Data	Description
q0	Reference stress, $q_{0}$ .
n	Material parameter, $n$ .
m	Material parameter, $m$ .
Strain Rate	${\dot{ε}}_{0}$ .
Use temperature-dependent data	Specifies material parameters that depend on temperature. A Temperature field appears in the data table. For more information, see Specifying Material Data as a Function of Temperature and Independent Field Variables.
Number of field variables	Specifies material parameters that depend on one or more independent field variables. A Field column appears in the data table. For more information, see Specifying Material Data as a Function of Temperature and Independent Field Variables.

Table 3. Law=Time Power Law
Input Data	Description
q0	Reference stress, $q_{0}$ .
n	Material parameter, $n$ .
m	Material parameter, $m$ .
Strain Rate	${\dot{ε}}_{0}$ .
Use temperature-dependent data	Specifies material parameters that depend on temperature. A Temperature field appears in the data table. For more information, see Specifying Material Data as a Function of Temperature and Independent Field Variables.
Number of field variables	Specifies material parameters that depend on one or more independent field variables. A Field column appears in the data table. For more information, see Specifying Material Data as a Function of Temperature and Independent Field Variables.

Table 4. Law=Singh-Mitchell
Input Data	Description
A	Material parameter, $A$ .
Alpha	Material parameter, $α$ .
m	$m$ .
t1	Reference time value, $t_{1}$ .
Use temperature-dependent data	Specifies material parameters that depend on temperature. A Temperature field appears in the data table. For more information, see Specifying Material Data as a Function of Temperature and Independent Field Variables.
Number of field variables	Specifies material parameters that depend on one or more independent field variables. A Field column appears in the data table. For more information, see Specifying Material Data as a Function of Temperature and Independent Field Variables.