Rate-Dependent Yield

Work Hardening Dependencies

Generally, a material's yield stress, $\bar{σ}$ (or $\bar{B}$ for the crushable foam model), is dependent on work hardening, which for isotropic hardening models is usually represented by a suitable measure of equivalent plastic strain, ${\bar{ε}}^{p l}$ ; the inelastic strain rate, ${\dot{\bar{ε}}}^{p l}$ ; temperature, $θ$ ; and predefined field variables, $f_{i}$ :

\bar{σ} = \bar{σ} ({\bar{ε}}^{p l}, {\dot{\bar{ε}}}^{p l}, θ, f_{i}) .

Many materials show an increase in their yield strength as strain rates increase; this effect becomes important in many metals and polymers when the strain rates range between 0.1 and 1 per second, and it can be very important for strain rates ranging between 10 and 100 per second, which are characteristic of high-energy dynamic events or manufacturing processes.

Defining Hardening Dependencies for Various Material Models

Strain rate dependence can be defined by entering hardening curves at different strain rates directly or by defining yield stress ratios to specify the rate dependence independently.

Direct Entry of Test Data

Work hardening dependencies can be given quite generally as tabular data for the isotropic hardening Mises plasticity model, the isotropic component of the nonlinear isotropic/kinematic hardening model, and the extended Drucker-Prager plasticity model. The test data are entered as tables of yield stress values versus equivalent plastic strain at different equivalent plastic strain rates. The yield stress must be given as a function of the equivalent plastic strain and, if required, of temperature and of other predefined field variables. In defining this dependence at finite strains, “true” (Cauchy) stress and log strain values should be used. The hardening curve at each temperature must always start at zero plastic strain. For perfect plasticity only one yield stress, with zero plastic strain, should be defined at each temperature. It is possible to define the material to be strain softening as well as strain hardening. The work hardening data are repeated as often as needed to define stress-strain curves at different strain rates, starting with the static stress-strain curve and followed by curves corresponding to increasing values of strain rate. The yield stress at a given strain and strain rate is interpolated directly from these tables.

Using Yield Stress Ratios

Alternatively, and as the only means of defining rate-dependent yield stress for the Johnson-Cook and the crushable foam plasticity 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} ({\bar{ε}}^{p l}, θ, f_{i}) R ({\dot{\bar{ε}}}^{p l}, θ, f_{i}),

where $σ^{0} ({\bar{ε}}^{p l}, θ, f_{i})$ (or $B ({\bar{ε}}^{p l}, θ, f_{i})$ in the 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 $R (0, θ, f_{i}) = 1.0$ ).

Three methods are offered to define R in Abaqus: specifying an overstress power law, defining R directly as a tabular function, or specifying an analytical Johnson-Cook form to define R.

Overstress Power Law

The Cowper-Symonds overstress power law has the form

{\dot{\bar{ε}}}^{p l} = D {(R - 1)}^{n} for \bar{σ} \geq σ^{0} (or \bar{B} \geq B in the crushable
	 foam
	 model),

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

Chaboche Rate Dependence

Chaboche rate dependence has the form

{\dot{\bar{ε}}}^{p l} = {{\dot{ε}}_{0} 〈 \frac{\bar{σ} - σ^{0}}{K} 〉}^{n},

where ${\dot{ε}}_{0} (θ, f_{i})$ , $K (θ, f_{i})$ , and $n (θ, f_{i})$ are material parameters that can be functions of temperature and, possibly, of other predefined field variables.

The above relation can be rewritten as

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

If Chaboche rate dependence is used with the crushable foam model, $\bar{σ}$ and $σ$ must be replaced with $\bar{B}$ and $B$ , respectively, in the above relations. In the perfectly plastic case when $σ^{0}$ or $B$ does not depend on plastic strain, this law and the overstress power law become identical if $D = {\dot{ε}}_{0} {(\frac{σ^{0}}{K})}^{n}$ . However, in general, if hardening is defined these laws produce different results.

Tabular Function

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), ${\dot{\bar{ε}}}^{p l}$ ; temperature, $θ$ ; and field variables, $f_{i}$ .

Johnson-Cook Rate Dependence

Johnson-Cook rate dependence has the form

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

where ${\dot{ε}}_{0}$ and C are material constants that do not depend on temperature and are assumed not to depend on predefined field variables. Johnson-Cook rate dependence can be used in conjunction with the Johnson-Cook plasticity model, the isotropic hardening metal plasticity models, and the extended Drucker-Prager plasticity model (it cannot be used in conjunction with the crushable foam plasticity model).

This is the only form of rate dependence available for the Johnson-Cook plasticity model. For more details, see Johnson-Cook Plasticity.

Elements

Rate-dependent yield can be used with all elements that include mechanical behavior (elements that have displacement degrees of freedom).