# 15.11. Karafillis Boyce Plasticity Model

    BEGIN PARAMETERS FOR MODEL KARAFILLIS_BOYCE_PLASTICITY
      #
      # Elastic constants
      #
      YOUNGS MODULUS = <real>
      POISSONS RATIO = <real>
      SHEAR MODULUS  = <real>
      BULK MODULUS   = <real>
      LAMBDA         = <real>
      TWO MU         = <real>
      #
      # Yield surface parameters
      #
      YIELD STRESS = <real>
      A            = <real> (4.0)
      C            = <real> (0.0)
      COEFF        = <real> (2.0/3.0)
      ALPHA 1      = <real> (1.0)
      ALPHA 2      = <real> (1.0)
      GAMMA 1      = <real> (1.5)
      GAMMA 2      = <real> (1.5)
      GAMMA 3      = <real> (1.5)
      #
      # Hardening model
      #
      HARDENING MODEL = LINEAR | POWER_LAW | USER_DEFINED |
        CUBIC_HERMITE_SPLINE
      #
      # Linear hardening
      #
      HARDENING MODULUS = <real>
      #
      # Power law hardening
      #
      HARDENING CONSTANT = <real>
      HARDENING EXPONENT = <real> (0.5)
      #
      # User defined hardening
      #
      HARDENING FUNCTION = <string>hardening_function_name
      #
      # Spline based hardening curve
      #
      CUBIC SPLINE TYPE  = <string>
      CARDINAL PARAMETER = <real> val
      KNOT EQPS          = <real_list> vals
      KNOT STRESS        = <real_list> vals
      #
      # Material coordinates system definition
      #
      COORDINATE SYSTEM             = <string> coordinate_system_name
      DIRECTION FOR ROTATION        = <real> 1|2|3
      ALPHA                         = <real> (degrees)
      SECOND DIRECTION FOR ROTATION = <real> 1|2|3
      SECOND ALPHA                  = <real> (degrees)
    END [PARAMETERS FOR MODEL KARAFILLIS_BOYCE_PLASTICITY]

The Karafillis and Boyce model [[1]] is an anisotropic plasticity model. The stress is transformed, based on the anisotropy, and the transformed stress is used in the yield function. The transformed stress, using Voigt notation in the material coordinate system, is given by

\[{\bf s}^{\prime} = {\bf C}:{\boldsymbol{\sigma}}\]
\[\begin{split}\left[ {\bf C} \right] = C \begin{bmatrix} 1 & \beta_{1} & \beta_{2} & 0 & 0 & 0 \\ \beta_{1} & \alpha_{1} & \beta_{3} & 0 & 0 & 0 \\ \beta_{2} & \beta_{3} & \alpha_{2} & 0 & 0 & 0 \\ 0 & 0 & 0 & \gamma_{1} & 0 & 0 \\ 0 & 0 & 0 & 0 & \gamma_{2} & 0 \\ 0 & 0 & 0 & 0 & 0 & \gamma_{3} \\ \end{bmatrix}\end{split}\]

where the terms \(\beta_{k}\) are

\[\beta_{1} = \frac{\alpha_{2} - \alpha_{1} - 1}{2}\]
\[\beta_{2} = \frac{\alpha_{1} - \alpha_{2} - 1}{2}\]
\[\beta_{3} = \frac{1 - \alpha_{1} - \alpha_{2}}{2}\]

The response is isotropic if \(\alpha_{1} = \alpha_{2} = 1\), \(\gamma_{1} = \gamma_{2} = \gamma_{3} = 1.5\), and \(C = 2/3\).

The principal stresses of the transformed stress, \({\bf s}^{\prime}\), are used in the yield function

\[\begin{split}\phi = \left\{ \left( 1-c \right) \phi_{1} + c \phi_{2} \right\}^{1/a} \\ \\ \phi_{1} = \frac{1}{2} \Big( | s_{1} - s_{2} |^{a} + | s_{2} - s_{3} |^{a} + | s_{3} - s_{1} |^{a} \Big) \\ \\ \phi_{2} = \frac{3^{a}}{2^{a}+2} \Big( | s_{1} |^{a} + | s_{2} |^{a} + | s_{3} |^{a} \Big)\end{split}\]

The exponent, \(a\), is similar to the exponent in the Hosford plasticity model and the constant, \(c\) (not to be confused with \(C\) above), is a parameter that provides a mixture of two yield functions.

In the command blocks that define the Hosford plasticity model:

- Consult the Sierra/SM User Manual for more information on elastic constants input.

- The reference nominal yield stress, \(\bar{\sigma}\), is defined with the `YIELD STRESS` command line.

- The exponent for the yield surface description, \(a\), is defined with the `A` command line.

- The coefficient \(C\) in the stress transformation is defined with the `COEFF` command line.

- The term \(\alpha_{1}\) in the stress transformation is defined with the `ALPHA 1` command line.

- The term \(\alpha_{2}\) in the stress transformation is defined with the `ALPHA 2` command line.

- The term \(\gamma_{1}\) in the stress transformation is defined with the `GAMMA 1` command line.

- The term \(\gamma_{2}\) in the stress transformation is defined with the `GAMMA 2` command line.

- The term \(\gamma_{3}\) in the stress transformation is defined with the `GAMMA 3` command line.

- The type of hardening law is defined with the `HARDENING MODEL` command line, other hardening commands then define the specific shape of that hardening curve.

- The hardening modulus for a linear hardening model is defined with the `HARDENING MODULUS` command line.

- The hardening constant for a power law hardening model is defined with the `HARDENING CONSTANT` command line.

- The hardening exponent for a power law hardening model is defined with the `HARDENING EXPONENT` command line.

- The hardening function for a user defined hardening model is defined with the `HARDENING FUNCTION` command line.

- The shape of the spline for the spline based hardening is defined by the `CUBIC SPLINE TYPE`, `CARDINAL PARAMETER`, `KNOT EQPS`, and `KNOT STRESS` command lines.

Output variables available for this model are listed in Table 15.11.1.

Table 15.11.1 State Variables for KARAFILLIS_BOYCE_PLASTICITY Model
Index | Name | Description
--- | --- | ---
1 | `EQPS` | equivalent plastic strain, \(\bar{\varepsilon}^{p}\)

[1]

A. P. Karafillis and M. C. Boyce. A general anisotropic yield criterion using bounds and a transformation weighting tensor. Journal of the Mechanics and Physics of Solids, 41(12):1859–1886, 1993.