Product: ABAQUS/Standard
The kinematic hardening models in ABAQUS are intended to simulate the behavior of metals that are subjected to cyclic loading. These models are typically applied to studies of low-cycle fatigue and ratchetting. The basic concept of these models is that the yield surface shifts in stress space so that straining in one direction reduces the yield stress in the opposite direction, thus simulating the Bauschinger effect and anisotropy induced by work hardening.
Two kinematic hardening models are available in ABAQUS. The simplest model provides linear kinematic hardening and is, thus, mainly used for low-cycle fatigue evaluations. This model yields physically reasonable results if the uniaxial behavior is linearized in the plastic range (a constant work-hardening slope). This is usually best accomplished by guessing the strain levels that will be attained in the problem and linearizing the actual material behavior accordingly. It is important to recognize this restriction on the theory's ability to provide reasonable results and to provide material data accordingly. This model is available with the Mises or Hill yield surface.
The combined isotropic/kinematic hardening model is an extension of the linear model. It provides a more accurate approximation to the stress-strain relation than the linear model. It also models other phenomena—such as ratchetting, relaxation of the mean stress, and cyclic hardening—that are typical of materials subjected to cyclic loading. This model is available only with the Mises yield surface.
This section first describes those aspects of the formulation that are common to both models; the specific formulation of each model is presented subsequently.
The total strain rate 
is written in terms of the elastic and plastic strain rates as
The elastic behavior can be modeled only as linear elastic
where
represents the fourth-order elasticity tensor and 
and 
are the second-order stress and strain tensors, respectively.The models are pressure-independent plasticity models. For both models the yield surface is defined by the function
where
is the equivalent Mises stress or Hill's potential with respect to the backstress or “kinematic shift” 
, and 
is the size of the yield surface. For instance, the equivalent Mises stress is defined as where 
is the deviatoric part of the backstress and 
is the deviatoric stress tensor.These models assume associated plastic flow:
where
represents the rate of plastic flow and 
is the equivalent plastic strain rate, 
This model is the simpler of the two kinematic hardening models available in ABAQUS. The size of the yield surface, 
, can be a function of temperature only for this model. The evolution of 
is defined by Ziegler's hardening rule, generalized to the nonisothermal case as
is the hardening parameter ( is the work-hardening slope of the isothermal uniaxial stress-strain response, 
, taken at different temperatures) and 
is the rate of change of C with respect to temperature. This form of evolution law for defines the rate of due to plastic straining to be in the direction of the current radius vector from the center of the yield surface, 
, and the rate due to temperature changes to be toward the origin of stress space. Rice (1975) writes this concept quite generally as The particular identification of 
and 
in Equation 4.3.5–6 above is assumed, so the material behavior is defined by the isothermal, uniaxial work hardening data, only.This model is based on the work of Lemaitre and Chaboche (1990). The size of the yield surface, 
, is defined as a function of equivalent plastic strain, 
; temperature, 
; and field variables, 
. This dependency can be provided directly, can be coded in user subroutine UHARD, or can be modeled with a simple exponential law for materials that either cyclically harden or soften as
is the yield surface size at zero plastic strain, and 
and 
are additional material parameters that must be calibrated from cyclic test data.The evolution of the kinematic component of the model is defined as
where C and
are material parameters, with representing the rate of change of C with respect to temperature and field variables. The rate of change of with respect to temperature and field variables is not accounted for in the model. This equation is the basic Ziegler law, generalized to account for temperature and field variable dependency of C and to which a “recall” term, 
, has been added. The recall term introduces the nonlinearity in the evolution law.The evolution of the backstress and the isotropic hardening are illustrated in Figure 4.3.5–1 for unidirectional loading and in Figure 4.3.5–2 for multiaxial loading.
The center of the yield surface is contained within a cylinder of radius
, which follows directly from Equation 4.3.5–9. Therefore, the yield surface is contained within the limiting surface of radius 
, as shown in the figures. This model can be degenerated into the linear kinematic model described above by setting 
and 
.The physical behavior that can be captured by this model, as well as its limitations, is described in detail in the ABAQUS Analysis User's Manual.
