Products: ABAQUS/Standard ABAQUS/Explicit
Incremental plasticity theory is based on a few fundamental postulates, which means that all of the elastic-plastic response models provided in ABAQUS (except the deformation theory model in ABAQUS/Standard, which is primarily provided for fracture mechanics applications) have the same general form. The basic equations of the models are defined in their general form in this section.
Plasticity models are written as rate-independent models or as rate-dependent models. A rate-independent model is one in which the constitutive response does not depend on the rate of deformation—the response of many metals at low temperatures relative to their melting temperature and at low strain rates is effectively rate independent. In a rate-dependent model the response does depend on the rate at which the material is strained. Examples of such models are the simple metal “creep” models provided in ABAQUS/Standard and the rate-dependent plasticity model that is used to describe the behavior of metals at higher strain rates. Because these models have similar forms, their numerical treatment is based on the same technique.
A basic assumption of elastic-plastic models is that the deformation can be divided into an elastic part and an inelastic (plastic) part. In its most general form this statement is written as
whereThis decomposition can be used directly to formulate the plasticity model. Historically, an additive strain rate decomposition,
has been used in its place. HereIt is shown in The additive strain rate decomposition, Section 1.4.4, that Equation 4.2.1–2 is a consistent approximation to Equation 4.2.1–1 when the elastic strains are infinitesimal (negligible compared to unity) and when the strain rate measure used in Equation 4.2.1–2 is the rate of deformation:
The elastic part of the response is assumed to be derivable from an elastic strain energy density potential, so the stress is defined by
where U is the strain energy density potential. Since we assume that, in the absence of plastic straining, the variation of elastic strain is the same as the variation in the rate of deformation, conjugacy arguments define the stress measureIn some materials the elastic response is essentially incompressible. But this is not usually the case for the materials whose inelastic deformation is considered with the models provided in ABAQUS, so Equation 4.2.1–3 can be taken to define the stress completely. However, the inelastic response is often assumed to be approximately incompressible (in metals, for example, or in soils undergoing large plastic flows), so the user must be careful to ensure that the elements chosen can accommodate incompressible response without exhibiting “locking” problems when the model is three-dimensional, plane strain, or axisymmetric. This requires the use of hybrid or fully or selectively reduced-integration elements.
For several of the plasticity models provided in ABAQUS the elasticity is linear, so the strain energy density potential has a very simple form. For the soils model the volumetric elastic strain is proportional to the logarithm of the equivalent pressure stress. For the concrete model damaged elasticity is used to account for the crack opening after the concrete has cracked: in that case the elasticity model is more complex.
The rate-independent plasticity models in ABAQUS and one of the rate-dependent models all have a region of purely elastic response. The yield function, f, defines the limit to this region of purely elastic response and is written so that
for purely elastic response. HereIn the concrete and the jointed material plasticity models in ABAQUS the yield behavior is modeled with several independent inelastic flow systems. For this case Equation 4.2.1–4 is generalized to
Stress states that cause the yield function to have a positive value cannot occur in rate-independent plasticity models, although this is possible in a rate-dependent model. Thus, in the rate-independent models we have the yield constraints
When the material is flowing inelastically the inelastic part of the deformation is defined by the flow rule, which we can write as
whereThe form in which the flow rule is written above assumes that there is a single flow potential, , in the ith system. More general plasticity models might have several active flow potentials at a point. This is, for instance, the case in the concrete and jointed material models built into ABAQUS.
For some rate-independent plasticity models the direction of flow is the same as the direction of the outward normal to the yield surface:
The rate form of the flow rule is essential to incremental plasticity theory, because it allows the history dependence of the response to be modeled.
The final ingredient in plasticity models is the set of evolution equations for the hardening parameters. We write these equations as
whereEquation 4.2.1–1 to Equation 4.2.1–6 define the general structure of all of the plasticity models in ABAQUS. Since the flow rule and the hardening evolution rules are written in rate form, they must be integrated. The general technique of integration is discussed in Integration of plasticity models, Section 4.2.2. The sections immediately following that discussion describe the details of the specific plasticity models that are provided in ABAQUS.