The plastic behavior of a material is described by its yield point and its post-yield hardening. The shift from elastic to plastic behavior occurs at a certain point, known as the elastic limit or yield point, on a material's stress-strain curve (see Figure 8–2). The stress at the yield point is called the yield stress. In most metals the initial yield stress is 0.05 to 0.1% of the material's elastic modulus.

The deformation of the metal prior to reaching the yield point creates only elastic strains, which are fully recovered if the applied load is removed. However, once the stress in the metal exceeds the yield stress, permanent (plastic) deformation begins to occur. The strains associated with this permanent deformation are called plastic strains. Both elastic and plastic strains accumulate as the metal deforms in the post-yield region.

The stiffness of a metal typically decreases dramatically once the material yields (see Figure 8–2). A ductile metal that has yielded will recover its initial elastic stiffness when the applied load is removed (see Figure 8–2). Often the plastic deformation of the material increases its yield stress for subsequent loadings: this behavior is called work hardening.

Another important feature of metal plasticity is that the inelastic deformation is associated with nearly incompressible material behavior. Modeling this effect places some severe restrictions on the type of elements that can be used in elastic-plastic simulations.

A metal deforming plastically under a tensile load may experience highly localized extension and thinning, called necking, as the material fails (see Figure 8–2). The engineering stress (force per unit undeformed area) in the metal is known as the nominal stress, with the conjugate nominal strain (length change per unit undeformed length). The nominal stress in the metal as it is necking is much lower than the material's ultimate strength. This material behavior is caused by the geometry of the test specimen, the nature of the test itself, and the stress and strain measures used. For example, testing the same material in compression produces a stress-strain plot that does not have a necking region because the specimen is not going to thin as it deforms under compressive loads. A mathematical model describing the plastic behavior of metals should be able to account for differences in the compressive and tensile behavior independent of the structure's geometry or the nature of the applied loads. This goal can be accomplished if the familiar definitions of nominal stress, , and nominal strain, , where the subscript 0 indicates a value from the undeformed state of the material, are replaced by new measures of stress and strain that account for the change in area during the finite deformations.

Strains in compression and tension are the same only if considered in the limit as ; i.e.,

The stress measure that is the conjugate to the true strain is called the true stress and is defined as

When defining plasticity data in ABAQUS, you must use true stress and true strain. ABAQUS requires these values to interpret the data in the input file correctly.

Quite often material test data are supplied using values of nominal stress and strain. In such situations you must use the expressions presented below to convert the plastic material data from nominal stress/strain values to true stress/strain values.

The relationship between true strain and nominal strain is established by expressing the nominal strain as

The relationship between true stress and nominal stress is formed by considering the incompressible nature of the plastic deformation and assuming the elasticity is also incompressible, so

The *PLASTIC option in ABAQUS defines the post-yield behavior for most metals. ABAQUS approximates the smooth stress-strain behavior of the material with a series of straight lines joining the given data points. Any number of points can be used to approximate the actual material behavior; therefore, it is possible to use a very close approximation of the actual material behavior. The data on the *PLASTIC option define the true yield stress of the material as a function of true plastic strain. The first piece of data given defines the initial yield stress of the material and, therefore, should have a plastic strain value of zero.

The strains provided in material test data used to define the plastic behavior are not likely to be the plastic strains in the material. Instead, they will probably be the total strains in the material. You must decompose these total strain values into the elastic and plastic strain components. The plastic strain is obtained by subtracting the elastic strain, defined as the value of true stress divided by the Young's modulus, from the value of total strain (see Figure 8–3).