Rate-Dependent Hardening Options


Type	Description
Hardening Type	The following rate-dependent hardening models are available: Power law: Define yield stress ratios with the Cowper-Symonds overstress law. Yield ratio: Enter yield stress ratios directly in tabular form as a function of equivalent plastic strain rates. Johnson-Cook: Use an analytical Johnson-Cook form to define $R$ . Chaboche: Define yield ratios with the Chaboche rate dependence law. For all four models, the strain rate behavior can be assumed to be separable, so that the stress-strain dependence is similar at all strain rate levels: $\bar{σ} = σ^{0} ({\dot{\bar{ε}}}^{p l}, θ, f_{i}) R ({\bar{ε}}^{p l}, θ, f_{i}),$ where $σ^{0} ({\dot{\bar{ε}}}^{p l}, θ, f_{i})$ (or $B ({\dot{\bar{ε}}}^{p l}, θ, f_{i})$ in the crushable foam model) is the static stress-strain behavior, and $R ({\dot{\bar{ε}}}^{p l}, θ, f_{i})$ is the ratio of the yield stress at nonzero strain rate to the static yield stress (so that $R (0, θ, f_{i}) = 1.0$ ).

Power Law

The Cowper-Symonds overstress power law has the form

{\dot{\bar{ε}}}^{p l} = D {(R - 1)}^{n} f o r \bar{σ} \geq σ^{0} (o r \bar{B} \geq B i n t h e c r u s h a b l e f o a m \mod e l),

where $D (θ, f_{i})$ and $n (θ, f_{i})$ are material parameters that can be functions of temperature.

Power law rate dependence can be used in conjunction with the Johnson-Cook plasticity model, the isotropic hardening metal plasticity models, the extended Drucker-Prager plasticity model, and the crushable foam plasticity model.


Input Data	Description
Multiplier	Material parameter, $D$ .
Exponent	Material parameter, $n$ .
Use temperature-dependent data	Specifies material parameters that depend on temperature. A Temperature field appears in the data table.
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.

Yield Ratio

Alternatively,

R

can be entered directly as a tabular function of the equivalent plastic strain rate (or the axial plastic strain rate in a uniaxial compression test for the crushable foam model),

{\dot{\bar{ε}}}^{p l}

, and temperature,

θ

Yield ratio rate dependence can be used in conjunction with the Johnson-Cook plasticity model, the isotropic hardening metal plasticity models, the extended Drucker-Prager plasticity model, and the crushable foam plasticity model.


Input Data	Description
Yield Stress Ratio	Yield stress ratio, $R = \bar{σ} / σ^{0}$ .
Equivalent Plastic Strain Rate	Equivalent plastic strain rate, ${\dot{\bar{ε}}}^{p l}$ , (or $\| {\dot{\bar{ε}}}_{a x i a l}^{p l} \|$ , the absolute value of the axial plastic strain rate in uniaxial compression, for the crushable foam model).
Use temperature-dependent data	Specifies material parameters that depend on temperature. A Temperature field appears in the data table.
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.

Johnson-Cook

Johnson-Cook rate dependence has the form

{\dot{\bar{ε}}}^{p l} = {\dot{ε}}_{0} \exp [\frac{1}{C} (R - 1)] f o r \bar{σ} \geq σ^{0},

where ${\dot{ε}}_{0}$ and $C$ are material constants that do not depend on temperature.

Johnson-Cook rate dependence can be used in conjunction with the Johnson-Cook plasticity model and the isotropic hardening metal plasticity models, and the extended Drucker-Prager plasticity model, but it cannot be used in conjunction with the crushable foam plasticity model.

If you specify Johnson-Cook rate dependence, you must also specify one of the other two rate dependence models (power law or yield ratio).


Input Data	Description
C	Material constant, $C$ , which is independent of temperature and field variables.
Epsilon dot zero	Material constant, ${\dot{ε}}_{0}$ , which is independent of temperature and field variables.

Chaboche

Chaboche rate dependence has the form

{\dot{\bar{ε}}}^{p l} = {\dot{ε}}_{0} {(\frac{σ^{0}}{K})}^{n} {(R - 1)}^{n}

where ${\dot{ε}}_{0}$ , $K$ , and $n$ are material constants that can be functions of temperature and field variables.

Chaboche rate dependence can be used in conjunction with the Johnson-Cook plasticity model, the isotropic hardening metal plasticity models, the extended Drucker-Prager plasticity model, the Cap Plasticity (Modified Drucker-Prager) model, and the crushable foam plasticity model.


Input Data	Description
Epsilon dot zero	Material constant, ${\dot{ε}}_{0}$ , which can be a function of temperature and field variables.
K	Material constant, $K$ , which can be a function of temperature and field variables.
Exponent	Material constant, $n$ , which can be a function of temperature and field variables.
Use temperature-dependent data	Specifies material parameters that depend on temperature. A Temperature field appears in the data table.
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.