Date December 16, 2001 Memo Number STI:01/11Subject Sheldon's ANSYS Tips and TricksSheldon's ANSYS Tips and TricksSheldon's ANSYS Tips and TricksSheldon's ANSYS Tips and Tricks: : : : Plasticity Hardening RulesPlasticity Hardening RulesPlasticity Hardening RulesPlasticity Hardening RulesKeywords Isotropic, Kinematic Hardening

1.1.1.1. Introduction:Introduction:Introduction:Introduction:ANSYS has a wide range of constitutive models and element technology available to the user. In

this memo, the basics of isotropic and kinematic hardening for plasticity will be discussed,specifically related to proportional/non-proportional, monotonic/cyclic loading, and finite strainapplications.

This memo assumes that the user is familiar with the basics of plasticity, including:• Selection of yield criterion (von Mises or Hill potential)• Selection of hardening rule (isotropic, kinematic, nonlinear kinematic, combined)• Selection of strain-rate-independent vs. rate-dependent models

Although future Tips & Tricks memos may address some of these topics, for more detailedinformation, it is suggested that the reader attend the “ANSYS 6.0 Advanced StructuralNonlinearities” training seminar, which your local ASDs and ANSYS, Inc. may hold periodically.

2.2.2.2. Background Discussion:Background Discussion:Background Discussion:Background Discussion:Hardening rules may best be illustrated when yield surfaces are plotted in principal stress space,

as shown below. Recall that, for the Mises yield criterion, the yield surface is a cylinder, the axis ofwhich lies along the value σ 1= σ 2=σ3.

For most yield criterion in ANSYS, we assume that inelastic strains are incompressible, so theyhave no dependence on hydrostatic pressure. That is why we are usually concerned with deviatoricstresses, which are stress values which deviate from the axis σ 1= σ 2=σ3. The figures represent theview in principal stress space, when looking directly at this “hydrostatic pressure axis.”

Inside of the yield surface is the elastic domain, whereas plasticity is described by the surfaceitself; no stress state can exist outside of the cylinder. Hardening rules describe how this yieldsurface changes in shape as yielding occurs.

Isotropic hardening describes a dilation or isotropicexpansion of the yield surface. This is expressed by thetop figure on the right. As yielding occurs, the yieldsurface expands uniformly. This means that the elasticdomain (inside the cylinder) grows, so, if loading isreversed, yielding occurs at a value of 2σ’. Thehardening can be described as bilinear (BISO),multilinear (MISO), or by the Voce equation (NLISO).

Kinematic hardening (bottom figure on right)represents a translation of the yield surface. This meansthat the elastic domain is always the same size, althoughthe yield surface moves in principal stress space. Hence,if loading is reversed, yielding occurs in compressing at avalue of 2σy. This is an approximation of theBauschinger effect, a behavior seen by most metals. Thehardening can be expressed as bilinear (BKIN),multilinear (KINH/MKIN), or nonlinear (CHAB). TheChaboche model is actually a bit more sophisticated, asthis nonlinear kinematic hardening law can combinethe effects of up to five kinematic models with a limiting yield surface. The Chaboche model can becombined with any isotropic hardening law to also describe combined hardening, which is atranslation and expansion of the yield surface.1

1 For more details on nonlinear kinematic hardening and combined hardening laws, please refer to the “ANSYS 6.0 Advanced StructuralNonlinearities” Seminar Lecture notes or the ANSYS Theory Manual, Ch. 4.1.

3.3.3.3. General Recommendations for Isotropic and Kinematic Hardening:General Recommendations for Isotropic and Kinematic Hardening:General Recommendations for Isotropic and Kinematic Hardening:General Recommendations for Isotropic and Kinematic Hardening:In this memo, only simple isotropic and kinematic hardening will be discussed. There has

sometimes been confusion on when these hardening rules are to be used, so some generalcharacteristics will be discussed.

Isotropic hardening can be used for large-strain analyses of metals (> 5-10% true strain).Isotropic hardening is not meant for cyclic loading applications because it does not account for theBauschinger effect. Moreover, applicability of isotropic hardening for non-proportional loading isleft up to the user, although, generally speaking, it is meant for proportional loading only.2

On the other hand, kinematic hardening is usually meant for non-proportional, cyclic loadingsince the Bauschinger effect is approximated with this model. However, it is generally meant forsmall-strain applications.

Combined hardening (and Chaboche nonlinear kinematic hardening), though not discussed indetail in this memo, can be utilized to model complex, large-strain cyclic behavior such as cyclichardening/softening and rachetting/shakedown.

4.4.4.4. Bilinear and Multilinear Kinematic Hardening:Bilinear and Multilinear Kinematic Hardening:Bilinear and Multilinear Kinematic Hardening:Bilinear and Multilinear Kinematic Hardening:The bilinear kinematic hardening model (BKIN) usually cannot represent large-strain effects

well because of the constant tangent modulus. The true stress-strain slope of most metals usuallychanges as the strains increase, but the bilinear model fails to account for this due to its simplerepresentation. This means that the yield surface can translate forever in principal stress space, evenallowing for the unrealistic possibility of passing through the origin.3

There are two multilinear kinematic hardening models available in ANSYS, namely MKIN andKINH. Both models use the sublayer model, which can be thought of as a weighted response ofmultiple elasto-perfectly-plastic ‘layers.’ A simplified view of this is that, as a layer ‘yields,’ itbecomes perfectly plastic, so it provides no stiffness response; this allows for the modeling of apiecewise linear curve.

The author recommends using KINH over MKIN due to the following reasons:• KINH allows up to 20 points per stress-strain curve, whereas MKIN only allows up to 5

points.• For KINH, input is done via TBPT commands, which is more consistent with other piecewise-

linear models such as MISO and MELAS, but MKIN relies on TBDATA input.• KINH allows up to 40 temperature-dependent curves, whereas MKIN allows only 5

temperature-dependent curves. Furthermore, in the case of temperature-dependent curves,MKIN requires each curve to have the same strain values, whereas KINH does not.

KINH is the same as MKIN with TBOPT=2, or use of Rice’s model for temperature-dependency. As aresult, KINH behaves the same as MKIN (TBOPT=2), so, due to the reasons mentioned above, theuser should consider using KINH.

5.5.5.5. Proportional and Non-proportional Loading:Proportional and Non-proportional Loading:Proportional and Non-proportional Loading:Proportional and Non-proportional Loading:The difference in cyclic loading behavior (inclusion of the Bauschinger effect) has been

discussed above for isotropic and kinematic hardening. For proportional and non-proportionalloading, however, the difference between isotropic and kinematic hardening may not be clear.

The author believes that for monotonic, proportional loading, MISO and KINH should providesimilar results, even for most large-strain problems. This is because the response under proportionalloading is similar, regardless of whether the yield surface expands or translates. This assumption isdependent on the stress-strain curve not having a maximum stress greater than 1.5-2 times the yieldstrength.4 This is a trivial case since, if the loading were proportional, isotropic hardening would bepreferred, anyway.

2 Proportional loading is when, under a given load, the principal stresses maintain constant ratios. Another way to view this is that,during proportional loading, the stress state goes through a straight line through the origin in principal stress space.3 This scenario would mean that, during unloading, the material would yield.4 Although, in Section 3, it was mentioned that bilinear/multilinear kinematic hardening models are meant for small-strain applications,this special case of large-strain, proportional loading is being considered.

For non-proportional loading in large-strain applications, the choice ofisotropic or kinematic hardening is less clear. An example of this is shownon the right, as plotted in 2D principal stress space. Two loads are appliedto a model, but they are non-proportional (load sequence 2 does not passthrough the origin of the yield surface). As noted in Section 3, isotropichardening is generally meant for proportional loading only, as the yieldsurface expands uniformly in principal stress space, which wouldn’tnecessarily account for the change in direction of load 2. Also, kinematichardening is meant for small-strain applications due to the translation of theyield surface, as one could imagine unexpected behavior occurring, if theyield surface translated by a very large amount.

To understand better the difference between these hardening rulesunder non-proportional loading, a simple tensile specimen model was used,as shown on the right. The two specimens had the same mesh and samestress-strain definition, as shown in the third figure on the right, althoughone mesh used MISO, the other used KINH.

A torsional load was applied as load step 1, then an axial load wasimposed as load step 2. This was accomplished with rigid-deformablecontact, and the problem was displacement-controlled.

The results of the simulation are shown in the last two figures on theright. The second-to-the-bottom figure is a plot of principal stresses at anode in the midspan of the specimen. Under load 1 (proportional loading),the response between MISO (blue) and KINH (red) are almost exactly thesame. However, as the second load is applied, because it is non-proportional, the resulting stresses differ between the two models.

Likewise, if the stress-strain response is plotted for the same nodelocations, the difference between MISO and KINH can be readily seen. Upuntil the end of load step 1, both MISO and KINH follow the original stress-strain curve well. During load step 2, however, both models do not fit theoriginal stress-strain curve exactly, although MISO follows the originalstress-strain curve more than KINH, for this particular problem. Thechange in KINH stress-strain response is due to the “reshifting” of the yieldsurface under the second load step.5

6.6.6.6. Conclusion:Conclusion:Conclusion:Conclusion:The choice of hardening law, yield criterion, and stress-strain curve

representation is dependent on the material used and expected loadingconditions. As long as the constitutive model adequately describes thematerial within the strain range of interest, that constitutive modelshould provide useful results in simulation.

This memo is not meant as a comparison of which hardening law is‘better,’ as application dictates the selection of the hardening rule. Instead,the author hopes that this memo serves as an illustration of what happensas the yield surface expands or translates during yielding, in order toprovide a better understanding of hardening rules and their usefulness.The more complex and powerful combined hardening laws (Chaboche andisotropic hardening) can also be used to model a much wider range ofmaterial behavior, although it requires a more thorough understanding ofmaterials’ cyclic response than can be covered in this short memo.

__________________________Sheldon ImaokaSheldon ImaokaSheldon ImaokaSheldon ImaokaANSYS, Inc.ANSYS, Inc.ANSYS, Inc.ANSYS, Inc.This document is not being provided in my capacity as an ANSYS employee. I am solely responsible for the content.This document is not being provided in my capacity as an ANSYS employee. I am solely responsible for the content.This document is not being provided in my capacity as an ANSYS employee. I am solely responsible for the content.This document is not being provided in my capacity as an ANSYS employee. I am solely responsible for the content.

5 The reader needs to consider what happens in the physical material response under non-proportional loading and how this relates to ascalar yield criterion when determining the applicability of MISO vs. KINH for a given problem.

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