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ME565 Advanced Finite Element Analysis Spring 2003

2003, Hormoz Zareh 1 Portland State University

Material Nonlinearity

This type of nonlinearity arises when the material exhibits non-linear stress-strain relationship.Recall that for linear elastic FE analysis the only stress-strain relationship was defined via

modulus of elasticity, E. Now, in the case of non-linear material analysis, the modulus of

elasticity is only the first definition point of an overall behavior. The typical definition andanalysis in the non-linear material domain involves one of post-yield (plastic) behavior. Typical

elasto-plastic material characteristic under tension is shown in figure 1. The unloading line

determines the residual (plastic) strain remaining in the system.

Note that figure 1 also represents a structure which exhibits a softening behavior after yielding.

The numerical solution of this type of non-linear problem involves approximating the non-linearsegment of stress-strain curve with a series of piece-wise linear segments. Each linear segment

is approximated by a tangent modulus (ET) which is computed as the ratio of stress over strain

for that particular line segment (see figure below).

Fig. 1 Typical elasto-plastic behavior

E

ET

Strain

Stress

ySE

Unloading


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For a complete FE analysis three important concepts must be well understood. The first one is

the Yield Criterion. It determines how the applied stresses (on the component) are related to the

yield strength specified in FE study, and determines the onset of yielding. The most commonlyused criterion is the von Mises (or octahedral shear) theory. Therefore, when the von Mises stress

reaches the yield strength, the component is assumed to have yielded, and the plastic regime

begins. Other yield criteria include Tresca (Maximum shear stress theory) andDrucker-Prager(Coulomb-Mohr theory).

The second concept relates the progression of yielding in the plastic domain. This is referred toas theflow rule. The most commonly used is the Prandtl-Reuss relation which relates the strain

increments to the stress increments of the common metals.

The final concept describes the mechanism for the growth of the yield surface. It is called the

hardening rule. It determines how the yield point changes as a result of accumulation of plastic

strain, and depends on the type of material. Metals are usually in the category described by

kinematic hardening. There are a variety of kinematic hardening rules (see I-DEAS hardening

rule section). The isotropic hardening rule assumes the center of yield region remains stationaryin the stress space while the size of yield surface expands as a result of strain hardening. This

theory is best suited for problems in which the plastic strain is considerably more than the onsetof yield, such as manufacturing processes (forming, cold working) and large motion dynamic

problems.

References:

Cook, R. D., Malkus, D. S. and Plesha, M. E., Concepts and Applications of FiniteElement Analysis, 4th Edition, John Wiley & Sons, New York, 2002, pp 530-587.

Crisfield, M. A.,Non-linear Finite Element Analysis of Solids and Structures, Volume 1:Essentials, John Wiley & Sons, Chichester, 1991, pp 77-80,131-132, 211-220.

EDS-PLM solutions, I-DEAS version 9 Simulation help files, 2002.


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Overview of the I-DEAS Hardening Rules for Plasticity Models

Select hardening rules in the physical property tables for the elements. The following hardeningoptions are available:

Isotropic hardening (option 0) Prager kinematic hardening (option 1) Ziegler-Prager kinematic hardening (option 2) Prager combined hardening (option 3) Ziegler-Prager combined hardening (option 4)

Isotropic Hardening

When you select isotropic hardening, the software uses a piece-wise linear stress-strain curve.The isotropic hardening assumption isn't very realistic for most materials subjected to cyclic

loading. However, it's relatively simple and efficient.

Isotropic hardening assumes that the yield surface expands uniformly as a result of plastic

straining. This assumption is achieved by making the yield stress a function of the integrated

effective plastic strain increments, which for a von Mises material is:

The slope of the stress plastic strain curve, Ep, is called the plastic modulus. It can be obtained

from the uniaxial stress-strain curve and is defined by:

For a von Mises material, the effective stress is given by:


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Kinematic Hardening

When you select kinematic hardening, the software assumes a bilinear stress-strain curve. If the

material database contains a multilinear representation, only the yield point and the tangent

modulus of the first segment beyond it are used to characterize the stress-strain behavior.

Kinematic hardening assumes that the yield surface translates in the stress space but doesn't

change size or shape. The yield stress, y, doesn't change, but the back stress, ij, is a function ofplastic straining.

Prager hardening and Ziegler-Prager hardening are the two most widely used models forkinematic hardening.

Prager Kinematic Hardening

The Prager kinematic hardening model assumes that, during plastic deformation, the back stress

increment is in the same direction as the plastic strain increment.

The constant c can be obtained from the uniaxial stress-strain curve and is related to the plastic

modulus, Ep.

Inconsistencies arise in Prager's model for certain stress subspaces such as plane stress. The

assumption of yield surface translation only in the stress space is violated. Prager's model is,

therefore, not recommended for plane stress, shell, beam, or rod elements.

Ziegler-Prager Kinematic Hardening

The Ziegler-Prager kinematic hardening model assumes that the back stress increment is in the

direction of the stress minus the back stress.

The factor, d , depends on the plastic strain history.

If a kinematic hardening model is required, the Ziegler-Prager model is recommended over the

Prager model.


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Prager and Ziegler-Prager Combined Kinematic Isotropic Hardening

When you select combined kinematic isotropic hardening, the software assumes a bilinear stress-

strain curve. If the material database contains a multilinear representation, only the yield point

and the tangent modulus of the first segment beyond it are used to characterize the stress-strain

behavior.

You must also enter the combined hardening parameter M, through the I-DEAS Material DataSystem.

Combined hardening assumes that the yield surface both expands and translates in the stressspace. The plastic strain increment is composed of two components shown in the following

equations:

Where

The reduced effective plastic strain associated with isotropic hardening is related to the effective

plastic strain by the following:

The back stress increment for Prager combined hardening is:

The back stress increment for Ziegler-Prager hardening is:

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