18.2.3 Rate-dependent yield

**Products: **ABAQUS/Standard ABAQUS/Explicit ABAQUS/CAE

Rate-dependent yield:

is needed to define a material's yield behavior accurately when the yield strength depends on the rate of straining and the anticipated strain rates are significant;

is available only for the isotropic hardening metal plasticity models (Mises and Johnson-Cook), the isotropic component of the nonlinear isotropic/kinematic plasticity models, the extended Drucker-Prager plasticity model, and the crushable foam plasticity model;

can be conveniently defined on the basis of work hardening parameters and field variables by providing tabular data for the isotropic hardening metal plasticity models, the isotropic component of the nonlinear isotropic/kinematic plasticity models, and the extended Drucker-Prager plasticity model;

can be defined through specification of user-defined overstress power law parameters or yield stress ratios for the isotropic hardening metal plasticity models, the extended Drucker-Prager plasticity model, or the crushable foam plasticity model;

cannot be used with any of the ABAQUS/Standard creep models (metal creep, time-dependent volumetric swelling, Drucker-Prager creep, or cap creep) since creep behavior is already a rate-dependent mechanism; and

in dynamic analysis should be specified such that the yield stress increases with increasing strain rate.

Work hardening dependencies

Generally, a material's yield stress, (or 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, ; the inelastic strain rate, ; temperature, ; and predefined field variables, :

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.

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. The yield stress at a given strain and strain rate is interpolated directly from these tables.

Input File Usage: | Use one of the following options: |

*PLASTIC, HARDENING=ISOTROPIC, RATE= *CYCLIC HARDENING, RATE= *DRUCKER PRAGER HARDENING, RATE= |

ABAQUS/CAE Usage: | Use one of the following models: |

Property module: material editor:
Hardening: Isotropic,
Use strain-rate-dependent data: MechanicalPlasticityDrucker Prager: SuboptionsDrucker Prager HardeningUse strain-rate-dependent dataCyclic hardening is not supported in ABAQUS/CAE. |

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:

where (or in the foam model) is the static stress-strain behavior and is the ratio of the yield stress at nonzero strain rate to the static yield stress (so that ).

Two methods are offered to define *R* in ABAQUS: specifying an overstress power law or defining *R* directly as a tabular function. In addition, in ABAQUS/Explicit an analytical Johnson-Cook form can be specified to define *R*.

The Cowper-Symonds overstress power law has the form

where and are material parameters that can be functions of temperature and, possibly, of other predefined field variables.

Input File Usage: | *RATE DEPENDENT, TYPE=POWER LAW |

ABAQUS/CAE Usage: | Property module: material editor: Hardening: Power Law (available for valid plasticity models) |

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), ; temperature, ; and field variables, .

Input File Usage: | *RATE DEPENDENT, TYPE=YIELD RATIO |

ABAQUS/CAE Usage: | Property module: material editor: Hardening: Yield Ratio (available for valid plasticity models) |

Johnson-Cook rate dependence has the form

where and

Input File Usage: | *RATE DEPENDENT, TYPE=JOHNSON COOK |

ABAQUS/CAE Usage: | Property module: material editor: Hardening: Johnson-Cook (available for valid plasticity models) |