About Plasticity

von Mises Yield Surface

The von Mises yield surface is used to define isotropic yielding. It is defined by giving the value of the uniaxial yield stress as a function of uniaxial equivalent plastic strain, temperature, and/or field variables.

Hill Yield Surface

The Hill yield surface allows anisotropic yielding to be modeled. You must specify a reference yield stress, ${\sigma }^0$, for the metal plasticity model and define a set of yield ratios, $R_{ij}$, separately. These data define the yield stress corresponding to each stress component as $R_{ij}{\sigma }^0$. Hill's potential function is discussed in detail in Hill Anisotropic Yield/Creep. Yield ratios can be used to define three common forms of anisotropy associated with sheet metal forming: transverse anisotropy, planar anisotropy, and general anisotropy.

The plasticity model using the Hill yield surface is applicable to strains up to about 25%-30%. It is not recommended for analyses in which these values are exceeded.

Perfect Plasticity

Perfect plasticity means that the yield stress does not change with plastic strain. It can be defined in tabular form for a range of temperatures and/or field variables; a single yield stress value per temperature and/or field variable specifies the onset of yield.

Isotropic Hardening

Isotropic hardening means that the yield surface changes size uniformly in all directions such that the yield stress increases (or decreases) in all stress directions as plastic straining occurs. Abaqus provides an isotropic hardening model, which is useful for cases involving gross plastic straining or in cases where the straining at each point is essentially in the same direction in strain space throughout the analysis. Although the model is referred to as a "hardening" model, strain softening or hardening followed by softening can be defined.

If isotropic hardening is defined, the yield stress, ${\sigma }^0$, can be given as a tabular function of plastic strain and, if required, of temperature and/or other predefined field variables. The yield stress at a given state is simply interpolated from this table of data, and it remains constant for plastic strains exceeding the last value given as tabular data.

Abaqus/Explicit will regularize the data into tables that are defined in terms of even intervals of the independent variables. In some cases where the yield stress is defined at uneven intervals of the independent variable (plastic strain) and the range of the independent variable is large compared to the smallest interval, Abaqus/Explicit may fail to obtain an accurate regularization of your data in a reasonable number of intervals. In this case the program will stop after all data are processed with an error message that you must redefine the material data. See Material Data Definition for a more detailed discussion of data regularization.

Johnston-Cook Isotropic Hardening

Johnson-Cook hardening is a particular type of isotropic hardening where the yield stress is given as an analytical function of equivalent plastic strain, strain rate, and temperature. This hardening law is suited for modeling high-rate deformation of many materials including most metals. Hill's potential function (see Hill Anisotropic Yield/Creep) cannot be used with Johnson-Cook hardening. For more details, see Johnson-Cook Plasticity.

Kinematic Hardening

Three kinematic hardening models are provided in Abaqus to model the cyclic loading of metals. The linear kinematic model approximates the hardening behavior with a constant rate of hardening. The more general nonlinear isotropic/kinematic model will give better predictions but requires more detailed calibration. The multilinear kinematic model combines several piecewise linear hardening curves to predict the complex response of metals under thermomechanical load cycles. This model is based on Besseling (1958) and is available only in Abaqus/Standard. For more details, see Models for Metals Subjected to Cyclic Loading.

Direct Tabular Data

Test data can be provided as tables of yield stress values versus equivalent plastic strain at different equivalent plastic strain rates (${\stackrel{\dot{-}}{\epsilon }}^{pl}$); one table per strain rate. Direct tabular data cannot be used with Johnson-Cook hardening. The guidelines that govern the entry of this data are provided in Rate-Dependent Yield.

Yield Stress Ratios

Alternatively, you can specify the strain rate dependence by means of a scaling function. In this case you enter only one hardening curve, the static hardening curve, and then express the rate-dependent hardening curves in terms of the static relation; that is, we assume thatwhere ${\sigma }^0$ is the static yield stress, ${\stackrel{-}{\epsilon }}^{pl}$ is the equivalent plastic strain, ${\stackrel{\dot{-}}{\epsilon }}^{pl}$ is the equivalent plastic strain rate, and $R$ is a ratio, defined as $R=1.0$ at ${\stackrel{\dot{-}}{\epsilon }}^{pl}=\mathrm{0.0.}$. This method is described further in Rate-Dependent Yield.