Plasticity corrections can be used to obtain postprocessed output requests based
on the evaluation of common plasticity correction rules. The plasticity correction
capabilities are available with linear elasticity to estimate the elastic-plastic
response of the material.
Plasticity corrections:
provide an estimate of the plastic solution for a model analyzed with purely
elastic material response;
can be applied to an isotropic linear elastic material model in general and
static linear perturbation procedures;
can be used with elastic-plastic materials with an isotropic von Mises yield
surface (however, the correction is evaluated only in static perturbation
procedures, where the material response is elastic);
are meaningful only when plastic deformation is localized in small regions of
the structure;
are evaluated using the Neuber or Glinka rules;
are based on either a tabular stress-strain curve or the Ramberg-Osgood
definition of the plastic response of the material; and
have no effect on the solution (additional output is provided only through
postprocessing of a linear elastic solution).
Many types of engineering applications require the solution of nonlinear
elastic-plastic problems. The finite element analysis method provides a robust way
of obtaining such results accurately. However, performing a nonlinear,
elastic-plastic analysis can be computationally expensive.
The Neuber and Glinka plasticity correction rules available in Abaqus/Standard provide an effective method to approximate the elastic-plastic stress and strain
solution by performing only a linear elastic analysis, which can significantly
reduce the computational cost. This is particularly important in concept design
optimization workflows, where the analysis must be performed multiple times. The
plasticity correction capabilities are also supported in static linear perturbation
steps with multiple load cases, which can further substantially decrease the
computational cost of the analysis.
The plasticity corrections provide postprocessed output results based on the
evaluation of the Neuber or Glinka rules applied to a linear elastic response, but
they do not affect the linear solution. To trigger the evaluation of plasticity
corrections, you must request one of the output variables described in Output.
Specifying the Plastic Response
Although the output variables associated with the Neuber and Glinka corrections are
available only when the material response is purely elastic, their evaluation still
requires knowledge of the plastic properties of the material. The plastic properties
are used only to evaluate the additional plasticity corrections output variables;
other solution results, which are based on linear elasticity, are not affected. You
can define the plastic response for the evaluation of plasticity corrections by specifying:
the coefficients of the Ramberg-Osgood model,
the tabular definition of the hardening curve, or
an elastic-plastic material definition with isotropic von Mises
plasticity.
In the latter case, the plasticity corrections are evaluated only in static
perturbation procedures, where the material response is always assumed to be purely
elastic.
Ramberg-Osgood Definition
The Ramberg-Osgood model is based on the observation that the stress versus
plastic strain response is linear when plotted in the logarithmic scale (that
is, it has a power law relation). In this model, the total strain is decomposed
into the elastic and plastic components:
where is the equivalent strain, is the equivalent stress, is the Young's modulus, and and are material parameters.
Tabular Definition
You can specify the hardening curve directly by providing the yield stress as a
function of the equivalent plastic strain in tabular form.
Elastic-Plastic Material Definition
The plasticity corrections can be evaluated using the plastic response specified
in an elastic-plastic material definition with an isotropic von Mises yield
surface. The definition of the hardening curve is provided as a table of yield
stresses versus equivalent plastic strains, similar to the tabular definition
discussed above. However, in this case, output variables for the plasticity
corrections are evaluated only in static linear perturbation procedures, where
the material response is always assumed to be elastic. They are not available in
general procedures, which always compute a fully nonlinear elastic-plastic
solution.
Neuber's Rule
Neuber’s rule is one of the most widely used methods for estimating the
elastic-plastic stress and strain response from purely elastic stress results. It
assumes that the stress-strain product of the elastic solution is equal to the
stress-strain product of the elastic-plastic solution. It can be expressed as
This equation is depicted graphically in Figure 1 and means that the areas of the two triangles shown must
be equal. The dashed line is called the Neuber hyperbola, and the solution to the
problem lies on the intersection of this line with the elastic-plastic curve. To
obtain Neuber's stress and strain ( and ), Abaqus solves the above equation together with the relationship describing the plastic
response.
Glinka's Rule
Glinka’s rule, also known as the equivalent strain energy density method (ESED), is
based on the assumption that the strain energy density distribution in the localized
plastic region near a notch is the same as that predicted from a linear elastic
solution. This approach generally leads to smaller values of stress and strain
compared to Neuber’s rule.
The corrected stress-strain response corresponds to a point on the elastic-plastic
curve such that the area under the curve is equal to the area of the triangle under
the purely elastic response, as shown in Figure 2. The figure also shows the triangle representing the
Neuber response for comparison. Glinka's rule can be expressed as
As in Neuber's method, to obtain Glinka's stress and strain ( and ) Abaqus solves the above equation together with the relationship describing the plastic
response.
Plasticity Corrections in Static Perturbation Procedures
When the static perturbation step follows a general step, Abaqus/Standard calculates the elastic stress () that is used to evaluate the plasticity corrections as the sum of
the base stress and the perturbation stress. In addition, if you request plasticity
corrections for elements that have an elastic-plastic material definition, Abaqus/Standard evaluates the corrections taking into account the fully nonlinear elastic-plastic
state of the material at the end of the general step. In this case, Abaqus/Standard uses the modified Neuber and Glinka rules, graphically depicted in Figure 3 and Figure 4, to compute the corrected stresses and strains. Abaqus/Standard evaluates the yield stress, taking into account the value of equivalent plastic
strain in the base state and shifting the strain to account for the change of the
stress-free configuration.
Material Options
The plasticity correction capabilities are available only with isotropic linearly
elastic materials and elastic-plastic materials with a von Mises yield surface. In
the latter case, Abaqus/Standard computes plasticity corrections only in the static linear perturbation
procedure.
Elements
The plasticity correction capabilities are available with any elements that include
mechanical behavior (elements that have displacement degrees of freedom).
Output
The following output variables can be requested to evaluate plasticity
corrections:
GKEEQ
Glinka equivalent strain,
GKPEEQ
Glinka equivalent plastic strain,
GKSEQ
Glinka equivalent stress,
NBEEQ
Neuber equivalent strain,
NBPEEQ
Neuber equivalent plastic strain,
NBSEQ
Neuber equivalent stress,
References
Molski, K., and G. Glinka, “A Method of Elastic-Plastic Stress and Strain Calculation at a Notch Root,” Materials Science and Engineering, vol. 50, pp. 93–100, 1981.
Neuber, H., “Theory of Stress Concentration for Shear-Strained Prismatical Bodies with Arbitrary Nonlinear Stress-Strain Law,” Journal of Applied Mechanics, vol. 28, pp. 544–550, 1961.