1. Plasticity Chapter Three

2. Training Manual September 30, 2001 Inventory #001491 3-2 Rate-Independent Plasticity Chapter Overview In this chapter, some of the rate-independent plasticity models not covered in the Basic Structural Nonlinearities seminar will be discussed. –Although some of the basic principles of rate-independent plasticity will be covered, it is assumed that the user is already familiar with common isotropic and kinematic hardening models available in ANSYS (BISO, MISO, BKIN, MKIN/KINH). Most of the discussion will be centered around metal inelasticity. –Many of the concepts can be extended to other materials, if applicable. For example, Drucker-Prager is commonly used for granular materials.

3. Training Manual September 30, 2001 Inventory #001491 3-3 Rate-Independent Plasticity... Chapter Overview In this chapter, we will cover the following topics: A. Background on Rate-Independent Plasticity B. von Mises Yield Criteria C. Anisotropic/Hill Potential (HILL) D. Anisotropic/Generalized Hill Potential (ANISO) E. Voce Nonlinear Isotropic Hardening (NLISO) F. Linear Kinematic Hardening G. Chaboche Nonlinear Kinematic Hardening (CHAB) H. Combined Hardening (CHAB + xISO) I. Cyclic Hardening and Cyclic Softening J. Rachetting and Shakedown K. ANSYS Procedural Considerations for Plasticity

4. Training Manual September 30, 2001 Inventory #001491 3-4 Rate-Independent Plasticity A. Background on Plasticity Review of Elasticity: Before proceeding to a discussion on plasticity, it may be useful to review elasticity of metals. –In elastic response, if the induced stresses are below the material’s yield point, the material can fully recover its original shape upon unloading. –From a standpoint of metals, this behavior is due to the stretching but not breaking of chemical bonds between atoms. Because elasticity is due to this stretching of atomic bonds, it is fully recoverable. Moreover, these elastic strains tend to be small. –Elastic behavior of metals is most commonly described by the stress-strain relationship of Hooke’s Law:

5. Training Manual September 30, 2001 Inventory #001491 3-5 Rate-Independent Plasticity... Background on Plasticity Review of Plasticity: In ductile metals, one may also encounter inelastic, or plastic, response. –Beyond the yield stress is the plastic region, where some permanent deformation remains after load removal. –If one considers what is occurring on the molecular level, plastic deformation results from slip between planes of atoms due to shear stresses (deviatoric stresses). This dislocation motion is essentially atoms in the crystal structure rearranging themselves to have new neighbors, which results in unrecoverable plastic strains. –It is instructive to note that this slipping does not result in any volumetric strains (condition of incompressibility), unlike elasticity.

6. Training Manual September 30, 2001 Inventory #001491 3-6 Rate-Independent Plasticity... Background on Plasticity Review of Plasticity (cont’d): –Because plasticity deals with the loss of energy due to dislocations, it is a non-conservative (path-dependent) process. –Ductile materials can sustain much larger plastic than elastic strains. –Elastic deformation essentially occurs independent of plastic deformation, so induced stresses past the yield point will still produce both elastic and plastic strains. Since plastic strains are assumed to be incompressible, the material response becomes nearly incompressible as strains increase.   Yield Point  y Elastic Plastic Unloading

7. Training Manual September 30, 2001 Inventory #001491 3-7 Rate-Independent Plasticity... Background on Plasticity Rate-Independent Plasticity: If the material response is not dependent on the rate of loading or deformation, the material is said to be rate- independent. –Most metals exhibit rate-independent behavior at low temperatures (< 1/4 or 1/3 melting temperature) and low strain rates. –Ch. 4 & 5 on Creep and Viscoplasticity deal with rate-dependent plasticity in metals.

8. Training Manual September 30, 2001 Inventory #001491 3-8 Rate-Independent Plasticity... Background on Plasticity Engineering vs. True Stress-Strain: Recall that, while engineering stress-strain can be used for small-strain analyses, true stress-strain must be used for plasticity, as they are more representative measures of the state of the material.

9. Training Manual September 30, 2001 Inventory #001491 3-9 Rate-Independent Plasticity... Background on Plasticity Engineering vs. True Stress-Strain (cont’d): If presented with engineering stress-strain data, one can convert these values to true stress-strain with the following approximations: –Up until twice the strain at which yielding occurs: –Up until the point at which necking occurs: Note that, only for stress conversion, the following is assumed: Material is incompressible (acceptable approximation for large strains) Stress distribution across cross-section of specimen is assumed to be uniform.

10. Training Manual September 30, 2001 Inventory #001491 3-10 Rate-Independent Plasticity... Background on Plasticity Engineering vs. True Stress-Strain (cont’d): –Beyond necking: There is no conversion equation relating engineering to true stress-strain at necking. The instantaneous cross-section must be measured.

11. Training Manual September 30, 2001 Inventory #001491 3-11 Rate-Independent Plasticity... Background on Plasticity Example animation of metal extrusion (finite strain plasticity): Element 185 (B-Bar), with isotropic hardening model, rigid-deformable contact with friction

12. Yield Criteria Chapter Three, Sections B-D

13. Training Manual September 30, 2001 Inventory #001491 3-13 Rate-Independent Plasticity Background on Yield Criteria Yield Criterion: The yield criteria is used to relate multiaxial stress state with the uniaxial case. –Tensile testing on specimens provide uniaxial data, which can easily be plotted on one-dimensional stress-strain curves, such as those presented earlier in this section. –The actual structure usually exhibits multiaxial stress state. The yield criterion provides a scalar invariant measure of the stress state of the material which can be compared with the uniaxial case.

14. Training Manual September 30, 2001 Inventory #001491 3-14 Rate-Independent Plasticity B. Mises Yield Criterion A common yield criterion is the von Mises yield criterion (also known as the octahedral shear stress or distortion energy criterion). The von Mises equivalent stress is defined as: In matrix form, this can be expressed as where {s} is the deviatoric stress and  m is hydrostatic stress

15. Training Manual September 30, 2001 Inventory #001491 3-15 Rate-Independent Plasticity... Mises Yield Criterion The stress state can be separated into hydrostatic (dilatational) and deviatoric (distortional) components. –The hydrostatic stress is associated with the energy of volume change whereas the deviatoric stress is associated with shape change. The von Mises yield criterion states that only the deviatoric component {s} causes yielding.

16. Training Manual September 30, 2001 Inventory #001491 3-16 Rate-Independent Plasticity... Mises Yield Criterion If plotted in 3D principal stress space, the von Mises yield surface is a cylinder.          The cylinder is aligned with the axis  1 =  2 =  3. Note that if the stress state is inside the cylinder, no yielding occurs. This means that if the material is under hydrostatic pressure (  1 =  2 =  3 ), no amount of hydrostatic pressure will cause yielding. Another way to view this is that stresses which deviate from the axis (  1 =  2 =  3 ) contribute to the von Mises stress calculation {s}.

17. Training Manual September 30, 2001 Inventory #001491 3-17 Rate-Independent Plasticity... Mises Yield Criterion If viewed normal to the axis  1 =  2 =  3, the von Mises yield criterion will look as shown below. Inside the yield surface, as noted earlier, behavior is elastic. Note that the multiaxial stress state can exist anywhere inside of the cylinder. At the edge of the cylinder (circle), yielding will occur. No stress state can exist outside of the cylinder. Instead, hardening rules will describe how the cylinder changes with respect to yielding.  Elastic Plastic 22 11 33  yy Principal Stress SpaceUniaxial Stress-Strain

18. Training Manual September 30, 2001 Inventory #001491 3-18 Rate-Independent Plasticity... Mises Yield Criterion By default, all of the rate-independent plasticity models use the von Mises yield criteria, unless otherwise noted. –Bilinear isotropic hardening (BISO) –Multilinear isotropic hardening (MISO) –Nonlinear isotropic hardening (NLISO) –Bilinear kinematic hardening (BKIN) –Multilinear kinematic hardening (KINH & MKIN) –Chaboche nonlinear kinematic hardening (CHAB)

19. Training Manual September 30, 2001 Inventory #001491 3-19 Rate-Independent Plasticity... Mises Yield Criterion All of the rate-independent material properties can input via TB commands or be found in the Materials GUI: –Main Menu > Preprocessor > Material Props > Material Models… Structural > Nonlinear > Inelastic > Rate Independent

20. Training Manual September 30, 2001 Inventory #001491 3-20 Rate-Independent Plasticity C. Hill Yield Criterion (HILL) Another useful yield criteria is Hill’s criterion, which is anisotropic (unlike von Mises which is isotropic). Hill’s criterion can be thought of as an extension of the von Mises yield criterion. Hill’s criterion may be expressed as: There are six constants, R xx, R yy, R zz, R xy, R yz, R xz, which characterize the Hill yield criterion:

21. Training Manual September 30, 2001 Inventory #001491 3-21 Rate-Independent Plasticity... Hill Yield Criterion (HILL) Hill’s criterion has three planes of symmetry which are conserved during yielding of the material, so 6 (not 21) constants are required through simple testing. In the previous slide, the constants R xx, R yy, R zz, R xy, R yz, R xz represent that ratio of the yield stress in a given direction to the reference (von Mises) yield stress. 22 11 33      3y  2y Principal Stress SpaceUniaxial Stress-Strain

22. Training Manual September 30, 2001 Inventory #001491 3-22 Rate-Independent Plasticity... Hill Yield Criterion (HILL) Six tests can be performed to determine the yield ratios R xx, R yy, R zz, R xy, R yz, R xz. These are the only parameters required for the Hill potential. For linear elastic material properties, isotropic or orthotropic properties can be specified (EX, EY, EZ, etc.) Hill’s criterion does not describe hardening; it only describes the yield criteria. The Hill potential is combined with isotropic, kinematic, and combined hardening models. –In these models, von Mises is used for the ‘reference’ yield stress. The Hill model is then used to determine the actual yield stress value in six directions.

23. Training Manual September 30, 2001 Inventory #001491 3-23 Rate-Independent Plasticity... Hill Yield Criterion (HILL) Hill’s potential can be input via commands or through the Materials GUI. If using commands, issue TB,HILL to activate the Hill criterion –TB, HILL, mat, ntemp –Input the six yield ratios via TBDATA C1 = R xx, C2 = R yy, etc. –Up to 40 temperature-dependent sets can be input –Don’t forget to input other required properties: Isotropic or orthotropic linear material properties via MP (EX, EY, EZ, PRXY, PRYZ, PRXZ, GXY, GYZ, GXZ) Hardening rule via TB (kinematic, isotropic, or combined)

24. Training Manual September 30, 2001 Inventory #001491 3-24 Rate-Independent Plasticity... Hill Yield Criterion (HILL) After defining the required linear material properties (e.g., EX, PRXY), the 6 constants of a specific hardening model with Hill potential can be input. Since Hill potential only describes the yield criteria, linear material properties and a plasticity hardening rule must also be selected. In the next example, bilinear isotropic hardening will be used, but the procedure is the same for any hardening rule selected.

25. Training Manual September 30, 2001 Inventory #001491 3-25 Rate-Independent Plasticity... Hill Yield Criterion (HILL) If isotropic or orthotropic linear materials have not been input, you will be prompted for this info. Next, the parameters for the hardening rule need to be input (in this case, BISO). Note that the yield stress input here will be the ‘reference’ yield stress used for Hill calculations. TB,BKIN,1,1,,1 TBTEMP,0 TBDATA,1,yield,tang_mod MP,EX,1,ex_value MP,PRXY,1,prxy_value MP,GXY,1,gxy_value

26. Training Manual September 30, 2001 Inventory #001491 3-26 Rate-Independent Plasticity... Hill Yield Criterion (HILL) Lastly, the six yield stress ratios for Hill’s criterion can be specified. –All material properties (linear, hardening, yield criterion) can be temperature-dependent as well. TB,HILL,1,1 TBTEMP,0 TBDATA,1,rxx,ryy,rzz TBDATA,4,rxy,ryz,rxz

27. Training Manual September 30, 2001 Inventory #001491 3-27 Rate-Independent Plasticity... Hill Yield Criterion (HILL) The Hill potential can be used to describe anisotropic viscoplastic and creep behavior, too. –When using commands, similar to rate-independent plastic, simply issue TB,HILL for each viscoplastic (RATE) and creep (implicit CREEP) model. –Through the Materials GUI, the procedure is more automated. Simply select the appropriate potential – either Mises or Hill – when defining the material constants. –More will be discussed on creep and viscoplasticity in Ch. 4 and 5.

28. Training Manual September 30, 2001 Inventory #001491 3-28 Rate-Independent Plasticity... Hill Yield Criterion (HILL) In the situation where R xx =R yy =R zz = R xy =R yz =R xz =1, Hill’s criterion reduces to isotropic von Mises yield criterion Note that yielding in a given direction in tension and compression are the same. A generalized Hill criterion is also available where yield in tension and compression are different (uneven materials), as will be discussed next. For anisotropic materials, please keep in mind that postprocessing equivalent strains (EPxx,EQV) may not necessarily be physically significant. Use caution when reviewing equivalent strains for anisotropic materials.

29. Training Manual September 30, 2001 Inventory #001491 3-29 Rate-Independent Plasticity D. Generalized Hill Potential (ANISO) While the generalized Hill potential is similar to Hill’s potential discussed in Section C, they differ in the following respects: –Generalized Hill allows for uneven materials (tensile and compressive yield ratios may differ). –Yield stress in various directions are input directly (units of stress), not as a yield stress ratio (dimensionless). –The hardening rule is bilinear isotropic hardening. This is built into the material definition, so TB,BISO is not issued. No additional hardening laws can be specified. –No temperature-dependency is assumed. –18x elements are not supported.

30. Training Manual September 30, 2001 Inventory #001491 3-30 Rate-Independent Plasticity... Generalized Hill Potential (ANISO) The yield surface of the generalization of Hill’s potential theory can be thought of as a distorted cylinder initially shifted in principal stress space. –The cylindrical yield surface is distorted because of anisotropy (Hill criterion), where yielding is different in different directions. –The cylindrical yield surface is initially shifted because yielding in tension vs. compression can be specified to be different. 22 11 33     3yt Principal Stress SpaceUniaxial Stress-Strain  3yc

31. Training Manual September 30, 2001 Inventory #001491 3-31 Rate-Independent Plasticity... Generalized Hill Potential (ANISO) The yield criterion is as follows: –The [M] matrix contains the information on different yield strengths in different directions. –The [L] matrix accounts for differences of yield in tension and compression. –K is the current yield stress in a given direction. Recall that this is based on bilinear isotropic hardening. –More details can be found in Ch. 4.1.13 of the ANSYS Theory Manual.

32. Training Manual September 30, 2001 Inventory #001491 3-32 Rate-Independent Plasticity... Generalized Hill Potential (ANISO) The generalized Hill potential model requires 18 constants -- yield stress in tension, compression, and shear (9) and tangent modulus in those directions (9). Two problems commonly arise in the definition of ANISO: –The yield surface must always represent a closed yield surface throughout loading. Otherwise, the yield surface would not make physical sense (elastic domain would not be definable). –The consistency equation must always be satisfied. This is a requirement of plastic incompressibility – recall that plastic strains are incompressible and do not contribute to volumetric change. –This means that the anisotropic yield stresses and slopes are not independent. The user must ensure the above criteria, otherwise a warning/error message will be printed in ANSYS. Please refer to the Theory manual for more details.

33. Training Manual September 30, 2001 Inventory #001491 3-33 Rate-Independent Plasticity... Generalized Hill Potential (ANISO) Unlike the other material properties discussed so far in this chapter, the generalized Hill potential cannot be used with 18x elements. Supported elements include: –Core elements: PLANE42, SOLID45, SOLID92, SOLID95 –Other elements: LINK1, PLANE2, LINK8, PIPE20, BEAM23, BEAM24, SHELL43, SHELL51, PIPE60, SOLID62, SOLID65, PLANE82, SHELL91, SHELL93, and SHELL143

34. Training Manual September 30, 2001 Inventory #001491 3-34 Rate-Independent Plasticity... Generalized Hill Potential (ANISO) The material properties can be input via commands or through the Materials GUI. Use TB,ANISO to input the generalized Hill potential material parameters. –TB,ANISO,mat –Use TBDATA to input 18 parameters. C1-C3 Tensile yield stresses in the material x, y, and z directions C4-C6 Corresponding tangent moduli C7-C9 Compressive yield stresses in the material x, y, and z directions C10-C12 Corresponding tangent moduli C13-C15 Shear yield stresses in the material xy, yz, and xz directions C16-C18 Corresponding tangent moduli –No temperature-dependency allowed –Input linear material properties (e.g., orthotropic) via MP. –Input all values as positive constants

35. Training Manual September 30, 2001 Inventory #001491 3-35 Rate-Independent Plasticity... Generalized Hill Potential (ANISO) After defining the required linear orthotropic or isotropic material properties (e.g., EX, PRXY), the 18 constants of the Generalized Hill anisotropic model can be input. –All values need to be input, as no defaults exist. TB,ANISO,1 TBDATA, 1,sxt,syt,szt,mod_xt,mod_yt,mod_zt TBDATA, 7,sxc,syc,szc,mod_xc,mod_yc,mod_zc TBDATE,13,sxy,syz,sxz,mod_xy,mod_yz,mod_xz

36. Training Manual September 30, 2001 Inventory #001491 3-36 Rate-Independent Plasticity... Example of Hill Potential Contour animation of von Mises stress of anisotropic sheet. SHELL181, with bilinear isotropic hardening and Hill yield criterion, rigid-deformable contact with friction

37. Training Manual September 30, 2001 Inventory #001491 3-37 Rate-Independent Plasticity... Workshop Exercise Please refer to your Workshop Supplement: Workshop 3: Hill Yield Criterion

38. Hardening Rules Sections E-H

39. Training Manual September 30, 2001 Inventory #001491 3-39 Rate-Independent Plasticity Background on Flow Rule Plastic Flow Rule: The plastic flow rule defines the relationship between the plastic strain increments and the stress. The flow rule prescribes the direction of the plastic straining when yielding occurs. –That is, it defines how the individual plastic strain components (  x pl,  y pl, etc.) develop with yielding. –For metals and other materials exhibiting incompressible inelastic behavior, the plastic flow develops in a direction normal to the yield surface. Otherwise (as in the case of DP material model), there is some increase in material volume when yielding – i.e., inelastic strains are not fully incompressible. Plastic strains develop in a direction normal to yield surface

40. Training Manual September 30, 2001 Inventory #001491 3-40 Rate-Independent Plasticity... Background on Flow Rule Associative Flow: –The direction of plastic flow is the same as the direction of the outward normal to the yield surface. Non-Associative Flow: –For frictional materials, non-associative flow rules are often needed (in Drucker-Prager model, the dilatation angle often differs from internal friction angle). Direction of plastic flow and normal of yield surface are the same Yield Surface p q Direction of plastic flow and normal of yield surface are not the same

41. Training Manual September 30, 2001 Inventory #001491 3-41 Rate-Independent Plasticity Background on Hardening Rules Hardening Rules: The hardening rule describes how the yield surface changes (size, center,shape) as the result of plastic deformation. The hardening rule determines when the material will yield again if the loading is continued or reversed. –This is in contrast to elastic-perfectly-plastic materials which exhibit no hardening -- i.e., the yield surface remains fixed. Elastic Plastic Yield Surface after Loading Initial Yield Surface

42. Training Manual September 30, 2001 Inventory #001491 3-42 Rate-Independent Plasticity E. Isotropic Hardening Isotropic hardening states that the yield surface expands uniformly during plastic flow. The term ‘isotropic’ refers to the uniform dilatation of the yield surface and is different from an ‘isotropic’ yield criterion (i.e., material orientation).  22 11 33   ' yy '' Initial Yield Surface Subsequent Yield Surface

43. Training Manual September 30, 2001 Inventory #001491 3-43 Rate-Independent Plasticity... Isotropic Hardening The yield criterion can therefore be written as: where {s} is the deviatoric stress and  k is the current yield stress. Isotropic hardening is appropriate for large strain, proportional loading situations. It is not suitable for cyclic loading applications.

44. Training Manual September 30, 2001 Inventory #001491 3-44 Rate-Independent Plasticity... Voce Nonlinear Isotropic Hardening In the Basic Structural Nonlinearities seminar, bilinear and multilinear isotropic hardening were covered (BISO, MISO). The third isotropic hardening law available in ANSYS is Voce’s nonlinear isotropic hardening (NLISO), where a smooth function using 4 material constants [k, R o, R , b] describe the material behavior. Plastic Strain Stress NLISO

45. Training Manual September 30, 2001 Inventory #001491 3-45 Rate-Independent Plasticity... Voce Nonlinear Isotropic Hardening The Voce hardening law is meant for materials which exhibit a smooth transition from the linear elastic region (E) to a final, constant linear strain-hardening slope (R o ). –As shown in the previous slide, the material constant k describes the elastic limit (  0 ). –If b=0, this would reduce to bilinear isotropic hardening (BISO). –If b=0 and R o =0, this would lead to elastic-perfectly-plastic behavior. The material can be entered in the Materials GUI or via TB,NLISO command. –TB,NLISO,mat,ntemp,4 –4 material constants input via TBDATA –Constants can be temperature dependent (TBTEMP) Up to 20 temperature-dependent sets can be input

46. Training Manual September 30, 2001 Inventory #001491 3-46 Rate-Independent Plasticity... Voce Nonlinear Isotropic Hardening After defining the required linear material properties (e.g., EX, PRXY), the 4 constants of the Voce Nonlinear Isotropic Hardening model can be input. –Note that constants can be temperature-dependent as well. TB,NLISO,1,1,4 TBTEMP,0 TBDATA,1,sigy0,r0,rinf,b

47. Training Manual September 30, 2001 Inventory #001491 3-47 Rate-Independent Plasticity... Voce Nonlinear Isotropic Hardening Based on uniaxial data, a curve fit can be performed to determine the Voce Hardening constants. Recall that this is meant for materials which exhibit a smooth transition between the elastic slope (Young’s modulus) and large-strain plastic region (tangent modulus).

48. Training Manual September 30, 2001 Inventory #001491 3-48 Rate-Independent Plasticity F. Linear Kinematic Hardening For linear kinematic hardening, the yield surface translates as a rigid body during plastic flow. –An initially isotropic plastic behavior is no longer isotropic after yielding (kinematic hardening is a form of anisotropic hardening) –The elastic region is equal to twice the initial yield stress. This is called the Bauschinger effect.  22 11 33   y yy '' Initial Yield Surface Subsequent Yield Surface 

49. Training Manual September 30, 2001 Inventory #001491 3-49 Rate-Independent Plasticity... Linear Kinematic Hardening The yield criterion can therefore be expressed as: where {s} is the deviatoric stress,  y is the uniaxial yield stress, and {  } is the back stress (location of the center of the yield surface). –Note in the previous plot that the center of the yield surface has translated {  }. Hence, based on the location {  }, the yield in reversal is still 2  y. –The back stress is linearly related to plastic strain via:

50. Training Manual September 30, 2001 Inventory #001491 3-50 Rate-Independent Plasticity... Linear Kinematic Hardening Bilinear kinematic hardening (BKIN) is an example of linear kinematic hardening. Can be used for cyclic loading since it includes the Bauschinger effect (elastic region equal to twice the initial yield stress). However, linear kinematic hardening is recommended for situations where the strain levels are relatively small (less than 5-10 % true strain). –Because there is only one plastic slope (tangent modulus), this is not representative of true metals as the hardening is constant. Hence, this is not realistic for large strain applications.

51. Training Manual September 30, 2001 Inventory #001491 3-51 Rate-Independent Plasticity G. Nonlinear Kinematic Hardening Nonlinear kinematic hardening is similar to linear kinematic hardening except for the fact that the evolution law has a nonlinear term (the “recall” term  dp): where  pl is equivalent plastic strain while p is accumulated plastic strain. The yield criterion is expressed as: where R is a constant defining the yield stress, similar to linear kinematic hardening.

52. Training Manual September 30, 2001 Inventory #001491 3-52 Rate-Independent Plasticity... Nonlinear Kinematic Hardening The yield surface can be described graphically as shown below: –The current yield surface shifts in principal stress space –There is a limiting yield surface, as explained in the next slide. In other words, the behavior approaches perfectly plastic, unlike linear kinematic hardening, which does not change slope. 22 11 33   R C/   Limiting Yield Surface Current Yield Surface Limiting value of {  

53. Training Manual September 30, 2001 Inventory #001491 3-53 Rate-Independent Plasticity... Nonlinear Kinematic Hardening Nonlinear kinematic hardening has the following characteristics: –Nonlinear kinematic hardening does not have a linear relationship between hardening and plastic strain. –The nonlinear kinematic hardening term is associated with the translation of yield surface. A non-zero value of  results in a ‘limiting value of {  }.’ This means that, unlike linear kinematic hardening, the yield surface cannot translate forever in principal stress space. The translation is limited within a specific region. –The constant R (yield stress), which describes the size of the elastic domain, is added on the response. If a limiting value of {  } exists, then a ‘limiting yield surface’ will also exist. –Nonlinear kinematic hardening is suitable for large strains and cyclic loading, as it can simulate the Bauschinger effect. It can model ratchetting and shakedown (discussed later).

54. Training Manual September 30, 2001 Inventory #001491 3-54 Rate-Independent Plasticity... Chaboche Kinematic Hardening The Chaboche Kinematic Hardening model (CHAB) is an example of nonlinear kinematic hardening. As noted previously, the yield function is 22 11 33   R C/   Limiting Yield Surface Current Yield Surface Limiting value of {  

55. Training Manual September 30, 2001 Inventory #001491 3-55 Rate-Independent Plasticity... Chaboche Kinematic Hardening The back stress {  } is a superposition of up to five kinematic models: where n is the number of kinematic models to use and C i and  i are material constants. –Recall that the evolution of the back stress is nonlinear, hence the term ‘nonlinear’ kinematic hardening. –There is also a temperature-dependency term (last term in above equation) –Note that if n=1 and  1 =0, CHAB will reduce to BKIN (no limiting value for  1 ).

56. Training Manual September 30, 2001 Inventory #001491 3-56 Rate-Independent Plasticity... Chaboche Kinematic Hardening The figure shown on the bottom is an example of the usage of Chaboche model: –n is 3, the number of kinematic models combined together. –R is the yield stress (constant value) –Values  1 -  3 are the back stresses calculated from the previous equation. Constants C 1 -C 3 and  1 -  3 are associated with these values. –R describes the yield surface whereas  describes the shifting of the center of the yield surface. –Note that  3 =0, so there is no limiting surface for . 11 33 22  =  1 +  2 +  3  +R R R=160 C 1 =80000,  1 =2000 C 2 =10000,  2 =200 C 3 =2500,  3 =0

57. Training Manual September 30, 2001 Inventory #001491 3-57 Rate-Independent Plasticity... Chaboche Kinematic Hardening The material can be entered in the Materials GUI or via TB,CHAB command. –TB,CHAB,mat,ntemp,npts npts = 2n+1, where n=number of kinematic models ntemp = number of temperature-dependent sets to input –2n+1 material constants are input via TBDATA C1 is yield stress C2 is C 1 constant for first kinematic model C3 is  1 constant for first kinematic model C4 is C 2 constant for second kinematic model C5 is  2 constant for second kinematic model …etc… up until C11 –Constants can be temperature dependent (TBTEMP)

58. Training Manual September 30, 2001 Inventory #001491 3-58 Rate-Independent Plasticity... Chaboche Kinematic Hardening After defining the required linear material properties (e.g., EX, PRXY), the 2n+1 constants of the Chaboche Nonlinear Kinematic Hardening model can be input. –Note that constants can be temperature-dependent as well. TB,CHAB,1,1,1 TBTEMP,0 TBDATA,1,yield,cons1,gamma1

59. Training Manual September 30, 2001 Inventory #001491 3-59 Rate-Independent Plasticity... Chaboche Kinematic Hardening As with many other models, Chaboche can be temperature- dependent. Use “Add Temperature” to add columns. Chaboche requires 2n+1 constants for n kinematic models. Use “Add Row” to add more rows for material parameter definition. TB,CHAB,1,2,2 TBTEMP,temp1 TBDATA,1,yield TBDATA,2,cons1a,gamma1a TBDATA,4,cons2a,gamma2a TBTEMP,temp2 TBDATA,1,yield TBDATA,2,cons1b,gamma1b TBDATA,4,cons2b,gamma2b

60. Training Manual September 30, 2001 Inventory #001491 3-60 Rate-Independent Plasticity H. Combined Hardening In the case of Chaboche kinematic hardening, the yield criterion is: We can have combined hardening by defining R as an isotropic hardening variable rather than a constant. For example, if we use Voce’s hardening law (NLISO), we would redefine R as: Note that we may use any isotropic hardening law to define R, be it BISO, MISO, or NLISO. This results in the translation and expansion of the yield surface.

61. Training Manual September 30, 2001 Inventory #001491 3-61 Rate-Independent Plasticity... Combined Hardening Combined hardening can be used for large strain and cyclic loading applications. The combined hardening model can be used for cyclic loading applications to model ratchetting, shakedown, cyclic hardening/softening (discussed later). To define combined hardening, one can use TB commands or the Materials GUI (next slide): –Define Chaboche via TB,CHAB and TBDATA commands, discussed in Section G. –Define isotropic hardening law via TB,BISO/MISO/NLISO with appropriate TBDATA or TBPT commands.

62. Training Manual September 30, 2001 Inventory #001491 3-62 Rate-Independent Plasticity... Combined Hardening If using the menus, the combined hardening laws are shown in the Materials GUI on the bottom-right. BISO, MISO, and NLISO can be combined with Chaboche. –First, define linear elastic properties, Chaboche constants, then bilinear isotropic hardening parameters. TB,CHAB,1,1,1 TBTEMP,0 TBDATA,1,yield,cons1,gamma1 TB,BISO,1,1 TBTEMP,0 TBDATA,1,yield,mod

63. Training Manual September 30, 2001 Inventory #001491 3-63 Rate-Independent Plasticity... Combined Hardening Combined Chaboche Kinematic and Isotropic Hardening constants can be temperature-dependent. Ensure that the initial yield stress parameter is the same for both models. –“C1 (R)” constant for CHAB, “Yield Stss” constant for BISO, first stress-strain point for MISO, “Sigy0” constant for NLISO. –Note that, if the yield stress parameters are not the same for CHAB and xISO (BISO/MISO/NLISO), the xISO yield stress will override. Since isotropic hardening is being used for the definition of the yield stress, this is consistent with expectations.

64. Training Manual September 30, 2001 Inventory #001491 3-64 Rate-Independent Plasticity Summary Yield Criteria vs. Hardening Rule Previous discussion centered around yield criteria and hardening (i.e., evolution of the criterion). –“Isotropic” and “anisotropic” can be used to describe both, whereas “kinematic” is used to describe the latter only. –Kinematic hardening is a type of anisotropic hardening, although a distinction will be made where kinematic hardening is used for the translation of the yield surface only. –Below is a summary of yield criteria, hardening rules, and the corresponding ANSYS material models:

65. Training Manual September 30, 2001 Inventory #001491 3-65 Rate-Independent Plasticity... Workshop Exercise Please refer to your Workshop Supplement: Workshop 4: Voce Nonlinear Isotropic Hardening

66. Cyclic Loading Sections I-J

67. Training Manual September 30, 2001 Inventory #001491 3-67 Rate-Independent Plasticity Proportional Loading Behavior Proportional Loading: In principal stress space, any loading path which is a straight line through the origin is termed proportional loading. –In other words, if the principal stress ratios remain constant, this is proportional loading. 22 11 33 Yield Surface Proportional Loading Path The principal stress ratios 2 and 3 can be vary at different integration point locations in the model, but, under proportional loading, they will remain constant.

68. Training Manual September 30, 2001 Inventory #001491 3-68 Rate-Independent Plasticity Cyclic Loading Behavior Monotonic Loading: Monotonic loading simply refers to situations where no unloading occurs. Cyclic Loading: Cyclic loading refers to situations where loading reverses direction. –In tension-compression cyclic loadings, metals may exhibit hardening or softening, depending on the material, temperature, and initial state. –In nonsymmetric stress-controlled situations, ratchetting or shakedown may occur.

69. Training Manual September 30, 2001 Inventory #001491 3-69 Rate-Independent Plasticity... Cyclic Loading Behavior Symmetric Loading: Symmetric loading means that min and max values are the same. For example, a cyclic load of +/- 1200 MPa is an example of a symmetric loading cycle. Nonsymmetric Loading: Nonsymmetric loading cycles occur when the max and min loads are not equal. For example, cyclic load of +10 ksi and -6 ksi is a nonsymmetric load.

70. Training Manual September 30, 2001 Inventory #001491 3-70 Rate-Independent Plasticity I. Cyclic Hardening Cyclic hardening occurs under symmetric loading. –Under a strain-controlled test, the stress range will increase. –Under a stress-controlled test, the strain range will decrease.

71. Training Manual September 30, 2001 Inventory #001491 3-71 Rate-Independent Plasticity... Cyclic Hardening Controlled Stress Controlled Strain t  t  TB,CHAB,1 TBDATA,1,980,224000,400 TB,NLISO TBDATA,1,980,0,400,5 TB,CHAB,1,,2 TBDATA,1,980,224000,400,20000 TB,NLISO TBDATA,1,980,0,400,5

72. Training Manual September 30, 2001 Inventory #001491 3-72 Rate-Independent Plasticity... Cyclic Hardening Considerations for Cyclic Hardening: Combined hardening can be used to model cyclic hardening. –Chaboche (CHAB) plus any isotropic hardening law (BISO, MISO, NLISO) is used to model cyclic hardening. The isotropic hardening laws will increase the yield stress with cyclic strain.

73. Training Manual September 30, 2001 Inventory #001491 3-73 Rate-Independent Plasticity... Cyclic Softening Cyclic softening occurs under symmetric loading. –Under a strain-controlled test, the stress range will decrease. –Under a stress-controlled test, the strain range will increase.

74. Training Manual September 30, 2001 Inventory #001491 3-74 Rate-Independent Plasticity... Cyclic Softening Controlled Stress Controlled Strain t  t  TB,CHAB,1 TBDATA,1,980,224000,400 TB,NLISO TBDATA,1,980,0,-400,5 TB,CHAB,1,2 TBDATA,1,980,224000,400,20000 TB,NLISO TBDATA,1,980,0,-400,5

75. Training Manual September 30, 2001 Inventory #001491 3-75 Rate-Independent Plasticity... Cyclic Softening Considerations for Cyclic Softening: Chaboche Model (CHAB) plus Voce’s nonlinear isotropic hardening model (NLISO) should be used to model cyclic softening. –Only Voce nonlinear isotropic hardening (NLISO) allows for a negative plastic slope. –Note that, for most materials such as metals, a negative stress- strain slope is unphysical since it implies material instability (stress decreases with increasing plastic strain). –However, as long as CHAB + NLISO results in positive hardening with increasing plastic flow, this will allow for proper material modeling. The NLISO isotropic hardening law will decrease the yield stress with cyclic strain (cyclic softening).

76. Training Manual September 30, 2001 Inventory #001491 3-76 Rate-Independent Plasticity J. Ratchetting & Shakedown Ratchetting occurs under a nonsymmetric stress-controlled loading –There is a progressive increase in strain at each cycle. Shakedown occurs under a nonsymmetric stress-controlled loading –There is a progressive stabilization of strain at each cycle.

77. Training Manual September 30, 2001 Inventory #001491 3-77 Rate-Independent Plasticity... Ratchetting & Shakedown Loading Controlled Stress Unsymmetry t  Ratchetting TB,CHAB,1 TBDATA,1,980,224000,400 Shakedown TB,CHAB,1,,2 TBDATA,1,980,224000,400,20000

78. Training Manual September 30, 2001 Inventory #001491 3-78 Rate-Independent Plasticity... Ratchetting Considerations for Ratchetting: –Ratchetting is the accumulation of plastic strain under an unsymmetric stress-controlled cyclic loading. –Linear kinematic models cannot capture ratchetting, as shown below (bilinear kinematic hardening BKIN example):

79. Training Manual September 30, 2001 Inventory #001491 3-79 Rate-Independent Plasticity... Ratchetting Considerations for Ratchetting (cont’d): –On the other hand, a single nonlinear kinematic model (n=1) for the Chaboche model CHAB can capture ratchetting, as shown below. In the figure on the left, the blue lines indicate a symmetric loading sequence. Note that no ratchetting occurs, and it is a stable cycle. The red lines indicate a nonsymmetric loading sequence. Because the plastic slopes are different (due to value of back stresses), plastic strains continue to accumulate.

80. Training Manual September 30, 2001 Inventory #001491 3-80 Rate-Independent Plasticity... Ratchetting Considerations for Ratchetting (cont’d): –Ratchetting occurs due to the fact that the initial slope in compression (A-B) is different from the slope in tension (C-D). Since the loading is unsymmetric, C-D is nearer to the ‘limiting yield surface’, so its slope is more asymptotic. C D A B

81. Training Manual September 30, 2001 Inventory #001491 3-81 Rate-Independent Plasticity... Shakedown Considerations for Shakedown: –Shakedown is similar to ratchetting. However, instead of the plastic strain steadily accumulating under nonsymmetric loading, it comes to a standstill in shakedown.

82. Training Manual September 30, 2001 Inventory #001491 3-82 Rate-Independent Plasticity... Shakedown Considerations for Shakedown (cont’d): One way to model shakedown is with at least two kinematic models in Chaboche (n=2). One of the two should have  =0. –One kinematic model will have  i  0, which will provide a ratchetting effect, as shown previously. –On the other hand, another model will have  i =0 to provide a stabilization effect. Recall that  i =0 is equivalent to bilinear kinematic hardening, so there is no ratchetting for BKIN. –Together, the two models will provide ratchetting with stabilization after a certain number of cycles. This is called shakedown.

83. Training Manual September 30, 2001 Inventory #001491 3-83 Rate-Independent Plasticity Chaboche Model Determination of material parameters involves a series of different tests. Complex material behavior is isolated through these specific tests. –The first and third references on the next page are good starting points for information on material characterization.

84. Training Manual September 30, 2001 Inventory #001491 3-84 Rate-Independent Plasticity... Chaboche Model References for Further Reading: Lemaitre, J. and Chaboche, J.L., Mechanics of Solid Materials, Cambridge University Press, 1990. Chaboche, J.L., “Constitutive Equations for Cyclic Plasticity and Cyclic Viscoplasticity”, International Journal of Plasticity, Vol. 5, pp. 247-302, 1989. Chaboche, J.L., “On Some Modifications of Kinematic Hardening to Improve the Description of Ratchetting Effects”, International Journal of Plasticity, Vol. 7, pp. 661-678, 1991.

85. ANSYS Procedure Section K

86. Training Manual September 30, 2001 Inventory #001491 3-86 Rate-Independent Plasticity K. Procedure for Plasticity Problems In this section, we will cover areas concerning the analysis of models with plasticity. This includes: –Element Selection –Solution Options –Postprocessing The following general considerations are applicable to any hardening model (isotropic, kinematic, or combined) or loading conditions (proportional or cyclic).

87. Training Manual September 30, 2001 Inventory #001491 3-87 Rate-Independent Plasticity... Element Selection for Plasticity Two main considerations exist when selecting elements: –Ensure that the selected elements support the plasticity model of interest –If large plastic strains are expected, verify that the selected elements can handle nearly-incompressible material behavior with appropriate element technology. For large strain problems, lower-order elements tend to behave more robustly because inverted midside nodes are not an issue. Also for large strain problems, grading/biasing the mesh to anticipate areas of large strains is recommended, such that element quality will not deteriorate during the solution. –If the model is bending-dominated, ensure that shear locking will not be a problem by use of appropriate element formulation.

88. Training Manual September 30, 2001 Inventory #001491 3-88 Rate-Independent Plasticity... Element Selection for Plasticity Make sure that the appropriate elements are used which support your material model: –NLISO, CHAB, and HILL are supported the ‘core’ and 18x family of elements (this also includes BISO, MISO, BKIN, KINH/MKIN). The core elements include PLANE42, SOLID45, PLANE82, SOLID92, SOLID95 The 18x elements include LINK180, SHELL181, PLANE182- 183, SOLID185-187, BEAM 188-189. –The Generalized Hill Potential (ANISO) does not support 18x elements but supports the ‘core’ and the following elements: LINK1, PLANE2, LINK8, PIPE20, BEAM23, BEAM24, SHELL43, SHELL51, PIPE60, SOLID62, SOLID65, SHELL91, SHELL93, and SHELL143

89. Training Manual September 30, 2001 Inventory #001491 3-89 Rate-Independent Plasticity... Element Selection for Plasticity Large-strain plasticity requires ability to handle nearly- incompressible situations. –B-Bar, Enhanced Strain, URI, and Mixed U-P can be used. –If the problem is bending-dominated, prevention of shear locking through Enhanced Strain should also be considered. It is recommended to use the 18x elements (please refer to Ch. 2 of this seminar) because of the advanced element technology available. –For PLANE182 and SOLID185, it is recommended to use B-Bar (default) first. If the problem is clearly bending-dominated, use Enhanced Strain formulation. On the other hand, if it is not obvious whether shear locking may be an issue, use B-Bar first, then switch to Enhanced Strain if accuracy is a concern. –Use of higher-order elements PLANE183 and SOLID186-187 is also acceptable.

90. Training Manual September 30, 2001 Inventory #001491 3-90 Rate-Independent Plasticity... Solution Options for Plasticity The solution of models containing plasticity is similar to other nonlinear problems, but there are some special considerations when solving models with plasticity. Main Menu > Solution > -Analysis Type- Sol’n Control… Solution Controls > -Basic Tab- Analysis Options –Specify large displacement solution (NLGEOM,ON), as needed. –The default Solution Control (SOLCONTROL) settings are recommended. By default, Solution Control is on.

91. Training Manual September 30, 2001 Inventory #001491 3-91 Rate-Independent Plasticity... Solution Options for Plasticity Remember that plasticity is a path-dependent, or nonconservative, phenomenon since the energy due to plastic strains is dissipated. –Path-dependent problems are dependent on the load history. The load needs to be applied gradually to ensure accuracy of capturing the load history. –Consequently, nonconservative systems require many substeps to capture this path dependency. –The number of substeps (NSUBST) and cutback controls (CUTCON) help us achieve the level of accuracy we need. –The Monitor file (jobname.mntr) is also a useful device to ensure that no substep has too much plastic strain (i.e., too few substeps, and the path dependency is not accurately captured).

92. Training Manual September 30, 2001 Inventory #001491 3-92 Rate-Independent Plasticity... Solution Options for Plasticity An appropriate number of substeps (NSUBST) should be specified to ensure accuracy of plastic strain calculations. Solution Controls > -Basic Tab- Time Control –Initial, min, and max substeps (NSUBST) should be used to accomplish this. –Be sure to specify a large enough maximum number of substeps in case ANSYS needs to bisect the solution (discussed next). –Also, make sure that the minimum number of substeps is reasonable. A value greater than the default value of 1 should generally be specified.

93. Training Manual September 30, 2001 Inventory #001491 3-93 Rate-Independent Plasticity... Solution Options for Plasticity Use Cutback Control (CUTCONTROL) to specify a maximum equivalent plastic strain increment. Solution Controls > -Nonlinear Tab- Cutback Control –CUTCONTROL,PLSLIMIT,plvalue will impose a maximum equivalent plastic strain increment of plvalue –By default, plvalue is 15%. –If, during a timestep, ANSYS calculates a plastic strain increment larger than plvalue, then the solution is automatically bisected until the plastic strain increment limit is satisfied or the minimum time step is reached. –This command helps ensure that the plastic response is adequately captured. Please remember that this is plastic strain increment, not the actual value of plastic strain itself.

94. Training Manual September 30, 2001 Inventory #001491 3-94 Rate-Independent Plasticity... Solution Options for Plasticity During solution, the Monitor file (jobname.mntr) provides a summary of maximum equivalent plastic strain increment of the entire model. –One can verify at which substeps more plastic straining occurs. SOLUTION HISTORY INFORMATION FOR JOB: extrude.mntr ANSYS RELEASE 6.0 15:43:08 08/17/2001 LOAD SUB- NO. NO. TOTL INCREMENT TOTAL VARIAB 1 VARIAB 2 VARIAB 3 STEP STEP ATTMP ITER ITER TIME/LFACT TIME/LFACT MONITOR MONITOR MONITOR CPU MxDs MxPl 1 1 3 8 16 0.24500E-02 0.24500E-02 36.859 0.98000E-01 0.78886E-30 1 2 1 1 17 0.24500E-02 0.49000E-02 41.984 0.19600 0.78886E-30 1 3 1 1 18 0.36750E-02 0.85750E-02 46.344 0.34300 0.78886E-30 1 4 1 3 21 0.55125E-02 0.14087E-01 55.141 0.56840 0.78886E-30 1 5 1 4 25 0.55125E-02 0.19600E-01 66.188 0.81004 0.10024E-01 1 6 1 3 28 0.82687E-02 0.27869E-01 75.156 1.2191 0.46599E-01 1 7 1 1 29 0.12403E-01 0.40272E-01 79.781 1.8342 0.71383E-01 1 8 1 3 32 0.18605E-01 0.58877E-01 89.000 2.8069 0.11811 1 9 1 3 35 0.18605E-01 0.77481E-01 97.953 3.6952 0.68734E-01 1 10 2 2 47 0.13954E-01 0.91435E-01 126.02 4.3586 0.62924E-01 1 11 1 2 49 0.13954E-01 0.10539 133.14 5.0205 0.65091E-01 1 12 1 3 52 0.20930E-01 0.12632 142.38 6.0058 0.68586E-01 1 13 1 4 56 0.31395E-01 0.15771 154.02 7.5288 0.13620 1 14 2 3 69 0.12102E-01 0.16982 184.55 8.1099 0.37285E-01 1 15 1 2 71 0.12102E-01 0.18192 191.72 8.6983 0.50149E-01 1 16 1 1 72 0.18152E-01 0.20007 196.80 9.5843 0.80046E-01 1 17 1 4 76 0.27229E-01 0.22730 207.89 10.907 0.79968E-01 1 18 1 2 78 0.27229E-01 0.25453 215.42 12.231 0.11344 1 19 2 3 88 0.12601E-01 0.26713 239.25 12.841 0.35480E-01

95. Training Manual September 30, 2001 Inventory #001491 3-95 Rate-Independent Plasticity... Reviewing Results for Plasticity There are various quantities one can review for plasticity, some of which are listed below: –EPPL - plastic strains EPPL,EQV is equivalent plastic strain based on current plastic strain components. At 6.0, EPPL,EQV is calculated and stored in RST results file for solid/plane/shell elements, so no AVPRIN,,effnu needs to be issued. Recall that care needs to be taken when evaluating equivalent plastic strains for anisotropic materials or compressible inelastic strains (plastic flow not normal to yield surface: DP). –NL - nonlinear items NL,HPRES is hydrostatic pressure. Check NL,HPRES to verify that volumetric locking does not occur. If it does, Mixed U-P formulation may be required. NL,EPEQ is accumulated equivalent plastic strain (summation of EPPL,EQV increments) –SEND - strain energy density SEND,PLAS is plastic strain energy density

96. Training Manual September 30, 2001 Inventory #001491 3-96 Rate-Independent Plasticity... Reviewing Results for Plasticity One can use PLESOL, PLNSOL commands to plot these results. The GUI can also be used as shown below: Main Menu > General Postproc > Plot Results > Nodal Solu … Main Menu > General Postproc > Plot Results > Element Solu … In the dialog box shown on the right, the plastic strain category is selected on the left. Components, principal, and effective plastic strains can be selected on the right- hand choices. Note that, at v6.0, Eff Nu is not required. Actual equivalent plastic strains are calculated and stored.

97. Training Manual September 30, 2001 Inventory #001491 3-97 Rate-Independent Plasticity... Reviewing Results for Plasticity Example of plotting plastic strain energy density (SEND,PLAS)

98. Training Manual September 30, 2001 Inventory #001491 3-98 Rate-Independent Plasticity... Workshop Exercise Please refer to your Workshop Supplement: Workshop 5: Chaboche Nonlinear Kinematic Hardening