Clay Tabular Plasticity

The Clay Tabular Plasticity model:

describes the inelastic behavior of the material by a yield function that depends on the three stress invariants, an associated flow assumption to define the plastic strain rate, and a strain hardening theory that changes the size of the yield surface according to the inelastic volumetric strain;
can have an isotropic or an anisotropic yield function;
requires that the elastic part of the deformation be defined by using the isotropic or orthotropic linear elastic material model or, in an implicit analysis, the porous elastic material model within the same material definition (porous elasticity is supported only for isotropic yield functions);
allows for the hardening law to be defined by a piecewise linear form;
may optionally include hardening in hydrostatic tension; and
can be used with a regularization scheme for mitigating mesh dependence in situations where the material exhibits strain localization with increasing plastic deformation.

Clay Tabular Plasticity

This option is used to define the yield surface and flow potential parameters for elastic-plastic materials that use the Clay Tabular plasticity model. It must be used with the Clay Hardening option.


Input Data	Description
Stress Ratio	Stress ratio at critical state, $M$ .
Initial Volumetric Plastic Strain	Initial volumetric plastic strain, $ε_{v o l}^{p l} \|_{0}$ , corresponding to $p_{c} \|_{0}$ . A positive value must be entered.
Yield Surface	$β$ , the parameter defining the size of the yield surface on the “wet” side of the critical state.
Ratio Flow Stress	$K$ , the ratio of the flow stress in triaxial compression. If creep material behavior is included, $0.778 \leq K \leq 1$ .
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.

Clay Hardening

This option is used to define piecewise linear hardening/softening of the Cam-clay plasticity yield surface. You define a relationship between the yield stress in hydrostatic compression, $p_{c}$ , and, optionally, the yield stress in hydrostatic tension, $p_{t}$ , to the corresponding volumetric plastic strain, $ε_{v o l}^{p l}$ .

Table 1. Compressive Hardening
Input Data	Description
Yield Stress	Value of the hydrostatic compression stress at yield, $p_{c}$ . $p_{c}$ is given as a positive value and must increase with increasing plastic strain.
Volumetric Plastic Strain	Absolute value of the corresponding compressive volumetric plastic strain, $ε_{v o l}^{p l}$ .
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 2. Tension Clay Hardening
Input Data	Description
Yield Stress	Value of the hydrostatic tension stress at yield, $p_{t}$ . $p_{t}$ can be zero or negative and must decrease with increasing plastic strain.
Volumetric Plastic Strain	Absolute value of the corresponding compressive volumetric plastic strain, $ε_{v o l}^{p l}$ .
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.

Clay Yield Potential Options

This option is used to define the optional Hill's anisotropic yield potential for clay plasticity:

f (σ) = \sqrt{F {(σ_{22} - σ_{33})}^{2} + G {(σ_{33} - σ_{11})}^{2} + H {(σ_{11} - σ_{22})}^{2} + 2 L σ_{23}^{2} + 2 M σ_{31}^{2} + 2 N σ_{12}^{2}}

where:

F = \frac{1}{2} (\frac{1}{R_{22}^{2}} + \frac{1}{R_{33}^{2}} - \frac{1}{R_{11}^{2}}),

G = \frac{1}{2} (\frac{1}{R_{33}^{2}} + \frac{1}{R_{11}^{2}} - \frac{1}{R_{22}^{2}}),

H = \frac{1}{2} (\frac{1}{R_{11}^{2}} + \frac{1}{R_{22}^{2}} - \frac{1}{R_{33}^{2}}),

L = \frac{3}{2 R_{23}^{2}}, M = \frac{3}{2 R_{31}^{2}}, N = \frac{3}{2 R_{12}^{2}},

R_{11} = \frac{{\bar{σ}}_{11}}{σ^{0}}, R_{22} = \frac{{\bar{σ}}_{22}}{σ^{0}}, R_{33} = \frac{{\bar{σ}}_{33}}{σ^{0}}, R_{12} = \frac{{\bar{σ}}_{12}}{τ^{0}}, R_{13} = \frac{{\bar{σ}}_{13}}{τ^{0}}, R_{23} = \frac{{\bar{σ}}_{23}}{τ^{0}} .

Each

{\bar{σ}}_{i j}

is the measured yield stress value when

σ_{i j}

is applied as the only nonzero stress component;

σ^{0}

is the reference yield stress (

p_{c}

p_{t}

) you specified for the Clay Hardening option; and

τ^{0} = σ^{0} / \sqrt{3}


Input Data	Description
R11, R22, R33, R12, R23, and R13	The six yield ratios: $R_{11}$ , $R_{22}$ , $R_{33}$ , $R_{12}$ , $R_{23}$ , and $R_{13}$ , respectively.
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	Specify material parameters that depend on field variables. Field columns appear in the data table for each field variable you add. For more information, see Specifying Material Data as a Function of Temperature and Independent Field Variables.

Softening Regularization

Granular materials often exhibit strain localization with increasing plastic deformation. Post-failure solutions from conventional finite element methods can be strongly mesh dependent. To mitigate the mesh dependency of the solutions, a regularization method is often used to introduce a micro-structural length scale into the constitutive formulation. Let $l_{c}^{(m)}$ denote the characteristic width of a shear band or a crack band, $l_{c}^{(e)}$ denote the characteristic length of the element, and $ε_{v o l, e}^{p l}$ denote the inelastic strain for the element. Then the inelastic strain in the localization band, $ε_{v o l, m}^{p l}$ is defined to be:

ε_{v o l, m}^{p l} = ε_{v o l, e}^{p l} \min ({(\frac{l_{c}^{(e)}}{l_{c}^{(m)}})}^{n_{r}}, f_{\max}),

where $n_{r}$ is a material parameter and $f_{\max}$ is a positive number used for bounding the magnitude of regularization. This strain regularization method is valid only when the characteristic length of the element is greater than the width of the localization band; that is,, $l_{c}^{(e)} \geq l_{c}^{(m)}$ .


Input Data	Description
Crack Band Length	Crack band length, $l_{c}^{(m)}$ . This value must be greater than zero.
Exponent	Exponent, $n_{r}$ . This value must be greater than zero.
Bound	Bound on the magnitude of regularization, $f_{\max}$ . This value must be greater than zero.
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.