Hyperelasticity Chapter Six In future:

Hyperelasticity Chapter Six In future:
Add an appendix describing deformation gradient, stress-strain conjugacy, etc. Table of hyperelasticity Gent model Ogden Hyperfoam & Blatz-Ko (18x)

Hyperelasticity Chapter Overview
This chapter will cover the wide range of rate-independent hyperelastic constitutive models available in ANSYS for modeling rubber materials. In this chapter, we will cover the following topics: A. Background on Physics of Rubber B. Background on Hyperelastic Theory C. Particular Forms of the Strain Energy Potential (18x Elements) D. Considerations for HYPERxx Elements E. Solving Hyperelasticity Models F. Material Testing and Curve-Fitting September 30, 2001 Inventory #001491 6-2

Hyperelasticity A. Background on Elastomers
Elastomers are a class of polymers with the following properties Elastomers involve natural and synthetic rubbers, which are amorphous and are comprised of long molecular chains The molecular chains are highly twisted, coiled, and randomly oriented in an undeformed state These chains become partially straightened and untwisted under a tensile load Upon removal of the load, the chains revert back to its original configuration Strengthening of the rubber is achieved by forming crosslinks between molecular chains through a vulcanization process September 30, 2001 Inventory #001491 6-3

Hyperelasticity ... Background on Elastomers
On a macroscopic level, rubber behavior exhibits certain characteristics They can undergo large elastic (recoverable) deformations, anywhere on the order of %. As noted previously, this is due to the untwisting of cross-linked molecular chains. There is little volume change under applied stress since the deformation is related to straightening of chains. Hence, elastomers are nearly incompressible. Their stress-strain relationship can be highly nonlinear Usually, in tension, the material softens then stiffens again. On the other hand, in compression, the response becomes quite stiff. F u Tension Compression Add picture of finite strain September 30, 2001 Inventory #001491 6-4

Hyperelasticity ... Background on Elastomers
An example of an analysis of a rubber boot Element 185 (B-Bar), with nearly-incompressible Neo-Hookean hyperelastic material, rigid-deformable contact & self-contact September 30, 2001 Inventory #001491 6-5

Hyperelasticity B. Background on Hyperelasticity
There are some key assumptions related to the hyperelastic constitutive models in ANSYS Material response is isotropic, isothermal, and elastic Thermal expansion is isotropic Deformations are fully recoverable (conservative) Material is fully or nearly incompressible Requires element formulations discussed earlier such as B-Bar or Mixed U-P to handle incompressibility condition The constitutive hyperelastic models are defined through a strain energy density function Unlike plasticity, hyperelasticity is not defined as a rate formulation Instead, total-stress vs. total-strain relationship is defined through a strain energy potential (W) September 30, 2001 Inventory #001491 6-6

Hyperelasticity ... Definition of Stretch Ratio
Before proceeding to a detailed discussion on different forms of the strain energy potential, some terms will be defined: The stretch ratio (or simply ‘stretch’) is defined as The above is an example of stretch ratio as defined for uniaxial tension of a rubber specimen, where eE is engineering strain. There are three principal stretch ratios l1, l2, and l3 which will provide a measure of the deformation. These will also be used in defining the strain energy potential. September 30, 2001 Inventory #001491 6-7

Hyperelasticity ... Definition of Stretch Ratio
To illustrate the definition of the principal stretch ratios by an example, consider a thin square rubber sheet in biaxial tension. The principal stretch ratios l1 and l2 characterize in- plane deformation. On the other hand, l3 defines the thickness variation (t/to). Additionally, if the material is assumed to be fully incompressible, then l3 will equal l-2. In future, maybe discuss in more detail: Green-Lagrange strain, Second Piola-Kirchhoff stress Deformation gradient det|C-li2|=0 J=det|F| September 30, 2001 Inventory #001491 6-8

Hyperelasticity ... Definition of Strain Invariants
The three strain invariants are commonly used to define the strain energy density function If a material is fully incompressible, I3 = 1. Because we assume that the material is isotropic, some forms of the strain energy potential are expressed as a function of these scalar invariants. In other words, strain invariants are measures of strain which are independent of the coordinate system used to measure the strains. September 30, 2001 Inventory #001491 6-9

Hyperelasticity ... Definition of Volume Ratio
The volume ratio J is defined as As shown above, J can be thought of as the ratio of deformed to undeformed volume of the material. In the case of thermal expansion, the thermal volumetric deformation is The elastic volumetric deformation is related to the total and thermal volumetric deformation by the following: September 30, 2001 Inventory #001491 6-10

Hyperelasticity ... Definition of Strain Energy Potential
The strain energy potential (or strain energy function) is usually denoted as W Strain energy potential can either be a direct function of the principal stretch ratios or a function of the strain invariants The particular forms of the strain energy potential will be discussed shortly. These forms determine whether stretch ratios or strain invariants are used. Based on W, second Piola-Kirchoff stresses (and Green-Lagrange strains) are determined: September 30, 2001 Inventory #001491 6-11

Hyperelasticity ... Definition of Strain Energy Potential
Because of material incompressibility, we split the deviatoric (subscript d or with ‘bar’) and volumetric (subscript b) terms of the strain energy function. As a result, the volumetric term is a function of volume ratio J only where the deviatoric principal stretches and deviatoric invariants are defined as (for p=1,2,3 ): Note that I3=J2, so I3 is not used in the definition of W. September 30, 2001 Inventory #001491 6-12

Strain Energy Potential (18x Elements Only)
Section C Strain Energy Potential (18x Elements Only)

Hyperelasticity C. Particular Forms of W
In this section, the different hyperelastic models for the 18x series of elements will be presented. Each is a particular form of W, based either on the strain invariants or on the principal stretch ratios directly. Polynomial Neo-Hookean Mooney-Rivlin Arruda-Boyce Ogden September 30, 2001 Inventory #001491 6-14

Hyperelasticity ... Particular Forms of W
The strain energy potential W will require certain types of parameters input as material constants. The number of material constants will differ, depending on the strain energy function W chosen. The choice of W will depend on the user, although some very general guidelines will be presented to aid the user in selection of W. From the selection of W and material constants which are input, stress and strain behavior are calculated by ANSYS. The next slides discuss the different forms of strain energy potential W available in ANSYS. September 30, 2001 Inventory #001491 6-15

Hyperelasticity ... Polynomial Form
The polynomial form is based on the first and second strain invariants. It is a phenomenological model of the form where the initial bulk modulus and initial shear modulus are This option is defined via TB,HYPER,,,N,POLY. cij and di are input via TBDATA. Usually, values of N greater than 3 are rarely used. It may be applicable for strains up to 300%. Note to instructors: Values of N greater than 2 or 3 are rarely used because of numerical difficulty in fitting constants. Moreover, the higher order terms may result in unstable energy functions outside the range of experimental data, so using N > 2 will require enough data to cover the range of deformation interested -- otherwise, nonphysical results may occur. The polynomial form is most general -- Neo-Hookean, Mooney-Rivlin are particular forms of this, but they can be made to be equivalent. Other specific functions are Yeoh or James-Green-Simpson, both of which can be modeled using the polynomial form. September 30, 2001 Inventory #001491 6-16

Hyperelasticity ... Polynomial Form
Sample definition of 2-term Polynomial form shown below. Constants c10, c01, c20, c11, c02, d1, d2 to be defined. TB,HYPER,1,1,N,POLY TBTEMP,0 TBDATA,1,c10,c01,c20,c11,c02 TBDATA,6,d_1,d_2 September 30, 2001 Inventory #001491 6-17

Hyperelasticity ... Neo-Hookean Form
The Neo-Hookean form can be thought of as a subset of the polynomial form for N=1, c01=0, and c10=m/2: where the initial bulk modulus is defined as This option is defined via TB,HYPER,,,,NEO. The constants m and d are input via TBDATA. This is the simplest hyperelastic model which can serve as a good starting point, using a constant shear modulus. However, it is limited to strains up to 30-40% in uniaxial tension and up to 80-90% in pure shear (these are general guidelines). September 30, 2001 Inventory #001491 6-18

Hyperelasticity ... Neo-Hookean Form
Sample definition of Neo-Hookean form shown below. Constants m and d to be defined. TB,HYPER,1,1,2,NEO TBTEMP,0 TBDATA,1,mu,d September 30, 2001 Inventory #001491 6-19

Hyperelasticity ... Mooney-Rivlin Form
There are two-, three-, five-, and nine-term Mooney Rivlin models available in ANSYS. These can also be thought of as particular cases of the polynomial form. The two-term Mooney-Rivlin model is equivalent to the polynomial form when N=1: The three-term Mooney-Rivlin model is similar to the polynomial form when N=2 and c20=c02=0: September 30, 2001 Inventory #001491 6-20

The five-term Mooney-Rivlin model is equivalent to the polynomial form when N=2: The nine-term Mooney-Rivlin model can also be thought of as the polynomial form when N=3: September 30, 2001 Inventory #001491 6-21

For all of the preceding Mooney-Rivlin forms, the initial shear and initial bulk moduli are defined as: For the 18x series of elements, this option is defined via TB,HYPER,,,N,MOONEY. Constants cij and d are input via TBDATA. September 30, 2001 Inventory #001491 6-22

Sample definition of 3-term Mooney-Rivlin form shown below. Constants c10, c01, c11, d to be defined for 18x elements. TB,HYPER,1,1,3,MOONEY TBTEMP,0 TBDATA,1,c10,c01,c11,d September 30, 2001 Inventory #001491 6-23

Comments on the different Mooney-Rivlin (MR) models: As a very general guideline, the two-term MR form may be valid up to % tensile strains, although it will not account for stiffening effects of the material, usually present at large strains. Compression behavior may also not be characterized well with only two-term MR. As noted in the figure below, more terms may capture any inflection points in the engineering stress-strain curve. As with the polynomial form, the user must ensure that enough data is supplied with inclusion of higher-order terms. Five- or Nine-term MR may be used up to % strains (general guideline). Note to instructors: The 2-term Mooney Rivlin can be viewed as an extension of the Neo-Hookean model which includes the second invariant, so it may lead to better data fitting. However, similar to the Neo-Hookean form, the 2-term Mooney-Rivlin form only uses linear functions of the strain invariants, so neither case will capture the stiffening (‘inflection point’) at higher strain levels. Two-Term MR Five-Term MR Nine-Term MR September 30, 2001 Inventory #001491 6-24

Hyperelasticity ... Others Based on Polynomial
It was noted in the previous slides that the Neo-Hookean and the Mooney-Rivlin models can be viewed as special cases of the general Polynomial form. There are some other common special cases of the Polynomial form which can be modeled. Reduced polynomial form (similar to polynomial form but with j=0; i.e., omit second invariant dependency): The Yeoh model (use reduced polynomial form, N=3): The James-Green-Simpson model (polynomial form, N=3): September 30, 2001 Inventory #001491 6-25

Hyperelasticity ... Arruda-Boyce Form
The Arruda-Boyce form, also known as the eight-chain model, is a statistical mechanics-based model. This means that the form was developed as a statistical treatment of non- Gaussian chains emanating from the center of the element to its corners (eight-chain network) where the constants Ci are defined as Note to instructors: The model was first proposed by Arruda and Boyce in 1993 for modeling incompressible materials and later extended by Bergstrom and Boyce in 1998 to compressible materials. Bergstrom-Boyce includes rate-dependent behavior, but only its rate-independent form is currently in ANSYS. September 30, 2001 Inventory #001491 6-26

This option is defined via TB,HYPER,,,,BOYCE. Constants m, lL, and d are input via TBDATA. The initial shear modulus is m The limiting network stretch lL is the stretch at which stress starts to increase without limit. Note that as lL becomes infinite, the Arruda-Boyce form becomes the Neo-Hookean form. Generally limited to 300% strain at most. September 30, 2001 Inventory #001491 6-27

Sample definition of Arruda-Boyce form shown below. Constants m, lL, and, d to be defined. TB,HYPER,1,1,3,BOYCE TBTEMP,0 TBDATA,1,mu,lambda_L,d September 30, 2001 Inventory #001491 6-28

One may note from the previous slides that the Arruda-Boyce model is dependent on the first invariant I1 only. The justification of this comes from the observation that the strain energy potential is less sensitive to changes in the second invariant than the first. Also, if only uniaxial data is available, it has been shown that ignoring the second invariant leads to better prediction of general deformation states. From a physical standpoint, this means that the eight chains are equally stretched under any deformation state, i.e., I1= l12+ l22+ l32 represents this chain elongation. Additional usefulness of the Arruda-Boyce model stem from the fact that the material behavior can be characterized well even with limited test data (uniaxial test), and fewer material parameters are required. However, this is a fixed formulation, which may limit its applicability. Note to instructors: Gent model is available at 6.0 as an undocumented feature. September 30, 2001 Inventory #001491 6-29

Hyperelasticity ... Ogden Form
The Ogden form, another phenomenological model, is directly based on the principal stretch ratios rather than the strain invariants: where the initial bulk and shear moduli are defined as September 30, 2001 Inventory #001491 6-30

This option is defined via TB,HYPER,,,N,OGDEN. mi, ai, and di need to be supplied via TBDATA. The model degenerates to the Neo-Hookean form when N=1 m1=m a1=2 The model is equivalent to the two-term Mooney-Rivlin form if N=2 m1=2c10 a1=2 m2=-2c01 a2=-2 Since Ogden is based on principal stretch ratios directly, it may be more accurate and may provide better data fitting. However, it may also be more computationally expensive. In general, Ogden form may be applicable for strains up to 700%. September 30, 2001 Inventory #001491 6-31

Sample definition of 2-term Ogden form shown below. Constants m1, a1, m2, a2, d1, d2 to be defined. TB,HYPER,1,1,2,OGDEN TBTEMP,0 TBDATA,1,mu_1,a_1 TBDATA,3,mu_2,a_2 TBDATA,5,d_1,d_2 September 30, 2001 Inventory #001491 6-32

Hyperelasticity ... Incompressibility Considerations
Considerations for Incompressibility: All rubber-like materials have some very small compressibility. However, assuming full incompressibility is usually a very good approximation. The choice of treatment of material as nearly or fully incompressible is decided by the user and data available. For 18x lower-order elements, use B-Bar as first choice for nearly incompressible problems If shear locking exists, switch to Enhanced Strain. If volumetric locking exists, add Mixed U-P formulation. September 30, 2001 Inventory #001491 6-33

Hyperelasticity ... Incompressibility Considerations
Considerations for Incompressibility (cont’d): If the material is fully incompressible, 18x elements with Mixed U-P must be used. Set d=0 and KEYOPT(6) > 0 for fully incompressible problems. ANSYS will also automatically set KEYOPT(6) > 0 if it is not set by user in fully incompressible case (for all di=0). Note that plane stress problems can handle fully incompressible problems without difficulty (refer to Ch. 2). Do not use Mixed U-P in plane stress case. ANSYS will switch KEYOPT(6)=0 if it is incorrectly set for plane stress. Although ANSYS will automatically handle KEYOPT(6) settings as shown above, it is a good idea to manually set these options, especially in case of SOLID187 since, for SOLID187, KEYOPT(6)=1 or 2. September 30, 2001 Inventory #001491 6-34

Hyperelasticity ... Summary of Choice of W
There are several factors which affect the choice of the strain energy potential used for a particular application: The Neo-Hookean and 2-term Mooney-Rivlin models are simple models which may be used as a starting point for analyses. Note that these models may not predict large tensile strains (material stiffening effect) or compressive modes well. The Arruda-Boyce model may be able to predict multiple modes of deformation well, based only on uniaxial data. The polynomial (higher-term Mooney Rivlin) and Ogden forms may be able to fit experimental data more accurately, especially for larger strain applications. Ogden is especially well-suited since it is based directly on principal stretch ratios, but it may be more computationally expensive. September 30, 2001 Inventory #001491 6-35

Hyperelasticity ... Summary of Choice of W
The preceding slide provides rules of thumb for hyperelastic model selection based on strain range of interest. However, please keep in mind that a good experimental data fitting is the best way to determine which hyperelastic model to use. Make sure that your experimental data covers the expected strain range. If your data is for 50% strain, do not expect it to correlate well for strains of 200% Make sure that your experimental data covers the expected modes of deformation. Data fit of uniaxial tension only may not suffice for complex material response. Correlating your experimental data for the strain range and deformation modes of interest is the best way to ensure that you have selected an appropriate hyperelastic model. Some additional information will be covered in Subsection F. September 30, 2001 Inventory #001491 6-36

Hyperelasticity ... Workshop Exercise
Please refer to your Workshop Supplement: Workshop 10: Hyperelastic Keyboard September 30, 2001 Inventory #001491 6-37

Section D HYPERxx Elements

Hyperelasticity D. Options for HYPERxx Elements
There are two older sets of hyperelastic elements: HYPER5x consists of HYPER56, 58, 74, and 158 These elements are used for modeling nearly-incompressible Mooney-Rivlin materials only. HYPER8x consists of HYPER84 and 86. These elements are used for modeling Blatz-Ko compressible foam-type materials. The material inputs will be discussed in the next pages for HYPER5x and HYPER8x, respectively. September 30, 2001 Inventory #001491 6-39

For HYPER5x family of elements, Mooney-Rivlin is input via TB,MOONEY instead. Constants cij are input via TBDATA. C1 1st strain energy constant (c10) C2 2nd strain energy constant (c01) C3 3rd strain energy constant (c20) C4 4th strain energy constant (c11) C5 5th strain energy constant (c02) C6 6th strain energy constant (c30) C7 7th strain energy constant (c21) C8 8th strain energy constant (c12) C9 9th strain energy constant (c03) Compressibility is not input via “d” constant, but, instead, Poisson’s ratio is input with MP,NUXY with the relation Note to instructors: Recall that k=E/[3(1-2n)] and m=E/[2(1+n)]. Substiuting for E, we arrive at k=[2m(1+n)]/[3(1-2n)]=[2(2)(c10+c01)(1.5)]/[3(1-2n)]=2(c10+c01)/(1-2n) September 30, 2001 Inventory #001491 6-40

Mooney-Rivlin form (TB,MOONEY) for HYPER5x elements. See previous slide for constants. Note that Poisson’s ratio must also be defined separately under “Linear” properties. September 30, 2001 Inventory #001491 6-41

Hyperelasticity ... Blatz-Ko Foam
Blatz-Ko is used for the modeling of compressible foam-type rubbers: where the shear modulus is defined as The parameters involve Young’s modulus and Poisson’s ratio, input via MP,EX and MP,NUXY (or MP,PRXY). The Blatz-Ko model is commonly used to model compressible foam-type polyurethane rubbers. This is limited to HYPER84 and HYPER86 element types only. Blatz-Ko is a special case of the Ogden compressible foam (Hyperfoam) model with N=1 a1=2 b1=0.5 September 30, 2001 Inventory #001491 6-42

Hyperelasticity ... Blatz-Ko Foam
Sample definition of Blatz-Ko foam shown below for HYPER8x Young’s Modulus and Poisson’s Ratio to be defined. September 30, 2001 Inventory #001491 6-43

Section E Solution Procedure

Hyperelasticity E. ANSYS Procedure
When performing an analysis of hyperelastic materials in ANSYS, the following must be kept in mind: Select appropriate element type(s) Define hyperelastic constitutive model constants Check output during solution for specific warnings/errors Verify results in postprocessing September 30, 2001 Inventory #001491 6-45

Hyperelasticity ... Selecting Appropriate Element Type
The selection of the appropriate element type(s) depend upon the hyperelastic model used and degree of incompressibility of the material. 18x Elements refer to SHELL181, PLANE182/183, SOLID B-bar is preferred, although URI, Enhanced Strain also available HYPER5x Elements refer to HYPER56, 58, 74, 158 HYPER8x Elements refer to HYPER84, 86 Note to instructors: Although HYPER84/86 also support TB,MOONEY, these elements are not recommended due to performance reasons and the fact that only 2-term Mooney-Rivlin is supported for HYPER84/86. If TB,MOONEY is to be used, HYPER5x elements are preferred. September 30, 2001 Inventory #001491 6-46

Hyperelasticity ... Selecting Appropriate Element Type
Define Element Types in usual manner Main Menu > Preprocessor > Element Type > Add/Edit/Delete ... For 18x elements, Select under “Hyperelastic” category For HYPERxx elements, Select under “Mooney-Rivlin” category September 30, 2001 Inventory #001491 6-47

Hyperelasticity ... Define Hyperelastic Material
All hyperelastic models can be selected in the Materials GUI under: Structural > Nonlinear > Elastic > Hyperelastic Recall that material “Mooney-Rivlin” is for 18x elements only and “Mooney-Rivlin (TB,MOON)” is for HYPER5x elements. All hyperelastic parameters may also be temperature-dependent. September 30, 2001 Inventory #001491 6-48

Hyperelasticity ... Define Hyperelastic Material
After selecting the appropriate hyperelastic model, a separate dialog box will appear with the required input. In the example below, a 2-term Mooney-Rivlin model has been selected. Constants c10, c01, and d have been input. September 30, 2001 Inventory #001491 6-49

Hyperelasticity ... Running the Solution
Solving models which include hyperelastic materials have similar considerations as other nonlinear analyses Because these usually are finite strain problems, large deformation effects (NLGEOM,ON) should be activated Hyperelastic materials are conservative (path-independent). If the loading is proportional and the stress state corresponds to one of the six typical stress paths, the problem will converge easily (few substeps). If the hyperelastic stress state and loading path are complex, a small enough timestep should be specified to aid convergence. If plasticity, friction, or any other source of path-dependency also exist in the model, the considerations for nonconservative systems will dictate solution behavior (adequate number of substeps to capture path-dependent response) The default behavior of Solution Control (SOLCON,ON) should suffice for most situations. September 30, 2001 Inventory #001491 6-50

The recommended choice of solver for hyperelastic problems is dependent on the element type used: Because hyperelastic problems usually result in ill- conditioned matrices, the sparse direct solver is usually preferred. However, in the case of Mixed U-P 18x elements (KEYOPT(6)>0), because of the presence of Lagrange multipliers, the frontal solver is preferred. Note to instructors: Sparse solver, because of pivoting, may require excessive memory for Mixed U-P 18x elements. Hence, they are not recommended for efficiency reasons. This is an issue which we are hoping to address in the near future. September 30, 2001 Inventory #001491 6-51

During the solution, either physical or numerical instabilities may be experienced, resulting in a ‘negative or small pivot’ warnings. Physical instabilities are usually due to local or global geometric instabilities, such as buckling or wrinkling Numerical instabilities are due to non-positive definite strain energy density function, usually resulting from strains outside the expected range of interest. This stems from insufficient experimental data when curve-fitting material constants. As noted earlier, hyperelastic materials are defined through a strain energy density function, which defines the stress- strain relationship. To ensure that these modes of deformation are realistic (i.e., numerically stable), the Drucker Stability condition must be satisfied during solution. September 30, 2001 Inventory #001491 6-52

The Drucker Stability criterion is defined as the following: In other words, the tangent material stiffness matrix should always be positive definite. To ensure this, ANSYS does a preliminary check of the stretch ratio in the range of 0.1 to for the six typical stress paths. The above condition is checked for uniaxial, equibiaxial, and planar cases, both in tension and compression. (Compression means that the stretch < 1.0, tension is when the stretch > 1.0) This check is automatically done at the beginning of the solution phase September 30, 2001 Inventory #001491 6-53

If the material is stable for the given range of 0.1 to 10.0 for the six typical stress paths, no message will be displayed. Otherwise, a warning message will be printed in the initial solution phase, such as the one shown below: In the above case, uniaxial compression and equibiaxial tension were satisfied; however, the other four cases were not, so the limits are printed in the warning message. *** WARNING *** CP= TIME= 12:50:52 Hyper-elastic material may become unstable, material number 1 at temperature 0. The nominal-strain limits where the material becomes unstable are: UNIAXIAL TENSION E+01 EQUIBIAXIAL COMPRESSION E+00 PLANAR TENSION E+01 PLANAR COMPRESSION E+00 September 30, 2001 Inventory #001491 6-54

For the HYPER5x elements, a secondary check can be performed during each equilibrium iteration. This is activated by setting KEYOPT(8)=1 for the HYPER5x elements (HYPER56, HYPER58, HYPER74, and HYPER158). During each equilibrium iteration, every Gauss integration point is checked for stability violations. If some Gauss points fail a stability check, a note indicating the total number of Gauss points will be printed in the Output Window/File. An example is shown below where 16 Gauss points did not pass the check: *** LOAD STEP SUBSTEP COMPLETED. CUM ITER = *** TIME = TIME INC = *** AUTO STEP TIME: NEXT TIME INC = UNCHANGED FORCE CONVERGENCE VALUE = CRITERION= E-01 >>> Gauss points have exceeded the material stability limit DISP CONVERGENCE VALUE = E-01 CRITERION= E-02 EQUIL ITER 1 COMPLETED. NEW TRIANG MATRIX. MAX DOF INC= E-01 September 30, 2001 Inventory #001491 6-55

Please note that these checks performed at the beginning of the analysis are used to help diagnose problems if the solution fails to converge A stability check is often an indication that there may be convergence difficulty in that strain range. A material failing the stability check does not necessarily mean that the solution is invalid in that region. When the equilibrium iteration checks are activated for the HYPER5x elements, if any Gauss point per element does not pass the stability check, it will be saved in the results. A non-zero STFLAG (SMISC record) value will indicate which elements failed a stability check during that substep. This can help diagnose possible convergence problems. September 30, 2001 Inventory #001491 6-56

Hyperelasticity ... Verifying Results
The output of stress and strain results for hyperelastic materials differ, depending on the element types used: 18x series of elements report Cauchy (true) stress and Logarithmic (true) strain in the global coordinate system. HYPERxx elements save Cauchy (true) stress in the global coordinate system while Logarithmic strain is in the original element coordinate system. September 30, 2001 Inventory #001491 6-57

The 18x series of elements and HYPER5x elements support postprocessing hydrostatic pressure via NL,HPRES Main Menu > General Postproc > Plot Results > Element Solu... To ensure that volumetric locking has not occurred, plot hydrostatic pressure and verify that a checkboard pattern does not exist. Otherwise, a different element formulation may need to be selected. Please note that for 18x elements, HPRES is defined as +p whereas, for HYPER5x, HPRES is defined as -p September 30, 2001 Inventory #001491 6-58

Some troubleshooting tips: Check the mesh distortion when postprocessing. Poorly shaped elements at any point in the solution can lead to convergence difficulties. Check for poor aspect ratios and angles between element edges approaching 0 or 180 degrees. For Mixed U-P elements, recall that when the number of pressure DOF (Np) is greater than the number of active (unconstrained) displacement DOF (Nd), this is an over-constrained model, which results in locking. Ideally, the ratio of Nd/Np should be 2/1 for 2D problems or 3/1 for 3D problems. Over-constrained models can be overcome by mesh refinement, especially in areas without displacement b.c. September 30, 2001 Inventory #001491 6-59

Additional troubleshooting tips: For a fully incompressible problem (18x elements with Mixed U-P and material constant d=0), no unique solution may exist if all boundary nodes have prescribed displacements. This is due to the fact that hydrostatic pressure (internal DOF) is independent of deformation. Hydrostatic pressure needs to be determined by a force/pressure boundary condition. Without this, the hydrostatic pressure cannot be calculated -- i.e., there is no unique solution. For these situations where this occurs, having at least one node without applied boundary condition will remedy this situation. September 30, 2001 Inventory #001491 6-60

Hyperelasticity ... Workshop Exercise
Please refer to your Workshop Supplement: Workshop 11: Hyperelastic O-ring September 30, 2001 Inventory #001491 6-61

Section F Curve-Fitting

Hyperelasticity F. Material Testing and Curve-Fitting
The strain energy function W provides a relationship of stress-strain used by ANSYS to calculate the material response. Material parameters/constants required in the description of a specific choice of W. These parameters can be taken from testing of rubber specimens, based on curve-fitting routines. ANSYS provides curve-fitting for Mooney-Rivlin materials based on experimental data. The GUI procedure will be discussed. Arruda-Boyce curve-fitting will be covered in the last part of the section. September 30, 2001 Inventory #001491 6-63

Hyperelasticity ... Material Testing and Curve-Fitting
Assuming incompressibility, the following modes of deformation are identical: 1. Uniaxial Tension and Equibiaxial Compression 2. Uniaxial Compression and Equibiaxial Tension 3. Planar Tension and Planar Compression September 30, 2001 Inventory #001491 6-64

Test data typically comes from one or more of the following six tests: Uniaxial Tension Uniaxial Compression Biaxial Tension (Circular or rectangular specimen) Planar Shear Simple Shear Volumetric Test (Button specimen) The test data is collected as engineering stress/strain. Engineering stress/strain is used for the curve fitting (recall that stretch ratio l=1+eE). This is different from plasticity curve-fitting, where collected data is converted to true stress/strain. September 30, 2001 Inventory #001491 6-65

Collected data may need to be adjusted to account for effects such as hysteresis and stress-softening behavior. A typical engineering stress-strain curve for a rubber sample under cyclic loading is shown on the right. Note that hysteresis is present (behavior in loading vs. unloading is different). Stress-softening effects (such as Mullins effect) are also present. A stabilized curve (loading path) is then shifted to the origin (zero stress for zero strain) and used for curve-fitting procedures. September 30, 2001 Inventory #001491 6-66

Hyperelasticity ... Mooney-Rivlin Curve-Fitting
This subsection will cover ANSYS’s built-in curve-fitting routine for Mooney-Rivlin constants. To determine how many Mooney-Rivlin terms one should include, the following provides a general guideline: September 30, 2001 Inventory #001491 6-67

You should use at least twice as many data points as the desired number of constants to be calculated. Using more terms usually improves the statistical quality of the curve fit (more tightly fitted through the data points), but the overall shape of the curve may be worse. As a practical matter, you should probably select the two, five, or nine constant function which results in the best combination of tight data fit and satisfactory curve shape.   Fitted Curve Actual Material Response Data Points September 30, 2001 Inventory #001491 6-68

Ideally, one should have data for the three modes of deformation (e.g., uniaxial tension, biaxial tension, planar tension) in order to fully characterize the material. Ensure that the test data represents all modes of expected deformation For example, if you only have uniaxial tension data, do not create a model which experiences significant shear deformations. Verify that the test data also covers the complete strain range of interest. If your test data extends between 0% and 100% strain, do not make a model that experiences 150% strain. Using strains beyond the range of the test data can produce erroneous results. September 30, 2001 Inventory #001491 6-69

Note that where the supplied data points are available, the derived constants match the actual data well. Outside the range the constants produce a response that does not match the physical response. This can cause convergence difficulties.   Data Range Curve from Fitted Constants Actual Material Response September 30, 2001 Inventory #001491 6-70

The GUI procedure for determining and applying the Mooney- Rivlin constants consists of four main steps: 1. Dimension the Stress and Strain Data Arrays 2. Fill in the Stress and Strain Data Arrays 3. Calculate the Mooney-Rivlin Constants 4. Evaluate the Quality of the Mooney-Rivlin Constants September 30, 2001 Inventory #001491 6-71

Dimension the Stress and Strain Data Arrays Utility Menu > Parameters > Array Parameters > Define/Edit Array Parameters Dialog Box > Add The maximum number of data points from any one test. Always 1 Always 3 (for 3 test types) September 30, 2001 Inventory #001491 6-72

Fill in the Stress and Strain Data Arrays Utility Menu > Parameters > Array Parameters > Define/Edit Array Parameters Dialog Box > Edit Note, you can hit the down arrow to scroll down to fill in the rest of the array data. Uniaxial (+/-) Biaxial (+/-) Planar (+/-) September 30, 2001 Inventory #001491 6-73

Fill in the Stress and Strain Data Arrays The stress and strain information needs to be input as engineering stress and engineering strain. Each column of the array refers to a specific test type as shown in the table below: If you do not have data for all three columns of each STRESS and STRAIN arrays, you must leave the missing columns blank. September 30, 2001 Inventory #001491 6-74

Calculate the Mooney-Rivlin Constants Main Menu > Preprocessor > Material Props > Mooney-Rivlin > Define Table… Main Menu > Preprocessor > Material Props > Mooney-Rivlin > Calculate Const … This will execute the GUI function to calculate the Mooney-Rivlin constants (shown on the next slide). Enter material number and number of temperatures. September 30, 2001 Inventory #001491 6-75

The *MOONEY function will create the Mooney-Rivlin constants and store them in three places; the database, the array CONST, and the file specified (defaults to jobname.tb.) Number of Constants Names of Data Arrays Arrays created automatically from the GUI. September 30, 2001 Inventory #001491 6-76

Please note that this procedure will create the Mooney-Rivlin table for HYPER5x elements. (TB,MOONEY) To use this for 18x elements, the appropriate Mooney-Rivlin (TB,HYPER,,,,MOONEY) material model must be created, and the constants input manually. September 30, 2001 Inventory #001491 6-77

As noted earlier, besides creating a Mooney-Rivlin material table for the HYPER5x elements, this curve-fitting procedure will also generate a text file jobname.tb with the appropriate commands to create this material model (sample jobname.tb file shown below). This jobname.tb input file is suitable for HYPER5x elements. To use this material with 18x elements, the TB,MOONEY command must be replaced with an appropriate TB,HYPER,,,npts,MOONEY, where npts=2, 3, 5, or 9-term Mooney-Rivlin models. /COM,ANSYS RELEASE UP :17: /22/2001 /COM, hyper.tb TB,MOONEY, TBDAT,1, E+02 TBDAT,2, E+02 *DEL,CONST *DIM,CONST ,, 2 CONST (1)= E+02 CONST (2)= E+02 September 30, 2001 Inventory #001491 6-78

Evaluate the Quality of the Mooney-Rivlin Constants: The Output Window will contain the following: FOLLOWING TEST DATA TYPES HAVE BEEN SPECIFIED FOR COMPUTING 5 TERM MOONEY-RIVLIN SERIES TYPE OF TEST DATA # OF DATA POINTS UNIAXIAL EQUIBIAXIAL SHEAR ROOT MEAN SQUARE (RMS) ERROR AND COEFFICIENT OF DETERMINATION (COD) FOR 2, 5 AND 9 TERM SERIES # OF TERMS COEFFICIENTS COMPUTED RMS ERROR (%) COD C10,C E C10,C01,C20,C11,C E C10,C01,C20,C11,C02, E C30,C21,C12,C03 The user has chosen 5 term series for which the following constants have been computed C10 = E+00 C01 = E+00 C20 = E+01 C11 = E+01 C02 = E+00 *** WARNING *** C02 IS NON-NEGATIVE. *** CHECK RESULTS CAREFULLY *** September 30, 2001 Inventory #001491 6-79

Evaluate the Quality of the Mooney-Rivlin Constants: The root mean square (RMS error) and the coefficient of determination (COD) are statistical measures of the quality of the curve fit. The RMS error expressed as a percentage, should be “close” to zero. The coefficient of determination will be less than 1.0, but should be “close” to 1.0 (typically 0.99 or better). The output file will also contain any warning messages if any of the constraints were not satisfied for the calculated Mooney-Rivlin constants. In addition to checking the RMS error and the COD, you should also plot the calculated stress-strain data versus the experimental stress-strain data (next slide). September 30, 2001 Inventory #001491 6-80

Evaluating and Plotting Calculated M-R Constants: There are two steps to this process - Calculate Stress and Strain Values Graph the Calculated versus the Experimental Values Main Menu > Preprocessor > Material Props > Mooney-Rivlin > Evaluate Const… Number of calculated stress-strain points Type of test for calculated data Strain Range Calculated Stress and Strain Arrays September 30, 2001 Inventory #001491 6-81

Graph Calculated Values versus Experimental Data Main Menu > Preprocessor > Material Props > Mooney-Rivlin > Graph ... Controls the strain range of the graph, useful for showing how the Mooney-Rivlin model performs outside of the experimental range. September 30, 2001 Inventory #001491 6-82

Graph Calculated Values versus Experimental Data The graph shows the calculated values versus the experimental values for the range of the experimental data. Use of /NOERASE, /XRANGE, /YRANGE commands may help visualize super-imposed graphs of experimental and calculated data. Utility Menu > Erase Options > Erase Between Plots Utility Menu > PlotCtrls > Style > Graphs > Modify Axes… September 30, 2001 Inventory #001491 6-83

Note on Extrapolated Data Evaluating the Mooney-Rivlin constants can be used to evaluate the Mooney-Rivlin model outside of the range of the experimental data. However, use extrapolated data with caution! If you do not have enough test data to cover the full range of stains for your model, get more test data! September 30, 2001 Inventory #001491 6-84

Hyperelasticity ... Arruda-Boyce Curve-Fitting
Unlike Mooney-Rivlin, polynomial form, or Ogden models, Arruda-Boyce can provide accurate results based on uniaxial data only. The equation for the Arruda-Boyce model is repeated below: where the constants C1-C5 are defined as: Three parameters are required: m, lL, and d. September 30, 2001 Inventory #001491 6-85

The limit network stretch lL can be defined as follows: Take uniaxial compression data (true stress vs. stretch ratio) along the equilibrium curve. The stretch at which stress starts to increase without limit is llimit. The limit network stretch can then be calculated from the relationship: Note to instructors: The “equilibrium curve” is an estimation of the response. Recall that, for polymers, we often have hysteresis (loading vs. unloading curve), as noted in the beginning of this section. Instead of taking the uploading curve, we estimate a response in between the loading/unloading response. This is termed the ‘equilibrium curve’ and will be used in the subsequent discussion. In this particular case shown on right, llimit is about Based on the equation shown above, lL is calculated to be This agrees well with the actual value of 2.82 used for this model. September 30, 2001 Inventory #001491 6-86

The initial shear modulus m can be determined as follows: Choose a point on the true stress vs. stretch ratio curve, the values of which are s and l. The initial modulus may be calculated from the following equation: L(x) = coth(x) - 1/x is the Langevin function, and L-1(x) is the inverse Langevin function. A sample macro to calculate the inverse Langevin is shown on the next slide. September 30, 2001 Inventory #001491 6-87

! L_inv.inp ! Obtain the inverse Langevin numerically. ! First, fill a table with Langevin(X) & X, where: ! column 0 (index column) contains Y=Langevin(X) ! column 1 contains X ! If Y is known, the inverse Langevin of Y is X. ! To obtain X (the inverse Langevin of Y), use the ! interpolation feature of table arrays to define a ! scalar (X) in terms of the table array: X=L_inv(Y,1) *dim,L_inv,table,100,1 *vfill,L_inv(1,1),ramp,.1, *do,i,1,100,1 y=1/(tanh(.1*i))-1/(.1*i) L_inv(i,0)=y *enddo ! i /axlab,x,Y /axlab,y,L_inv(Y) /gcolumn,1,X *vplot,L_inv(1,0),L_inv(1,1) /com,Representative inverse Langevin values x1=L_inv(.4,1) x2=L_inv(.5,1) x3=L_inv((2/3),1) x4=L_inv(.8,1) Sample input file to calculate inverse Langevin Supplied by John Dulis of the ANSYS Technical Support Group September 30, 2001 Inventory #001491 6-88

The compressibility parameter d is based on the initial bulk modulus k. The bulk modulus can be determined by a volumetric test. However, if that test data is not available, k can be taken to be sufficiently large, such as k=500m. From that, d can be determined: Note that, as with all rubber analyses, the bulk modulus usually has little effect on the results except for situations where the rubber is highly confined. Assuming d=0 means that the rubber will be fully incompressible (Mixed U-P must be used, except for plane stress cases). September 30, 2001 Inventory #001491 6-89

Hyperelasticity ... Example Animation
Example animation of a rubber bushing. Element 185 (B-Bar and Mixed U/P), with fully-incompressible Mooney-Rivlin hyperelastic material and rigid-deformable contact September 30, 2001 Inventory #001491 6-90

Some general references on Arruda-Boyce model: 1. “A Three-dimensional Constitutive Model for the Large Stretch Behavior of Rubber Elastic Materials”, E.M. Arruda and M.C. Boyce, Journal of the Mechanics and Physics of Solids, Vol. 41 (2), pp (1993). 2. “Constitutive Modeling of the Large Strain Time-dependent Behavior of Elastomers”, J.S. Bergstrom and M.C. Boyce, Journal of the Mechanics and Physics of Solids, Vol. 45 (5), pp (1998). 3. “Direct Comparison of the Gent and the Arruda-Boyce Constitutive Models of Rubber Elasticity”, M.C. Boyce, Rubber Chemistry and Technology, Vol. 69, pp (1997). September 30, 2001 Inventory #001491 6-91

Hyperelasticity References for Further Reading
Some general references on hyperelasticity: 1. Non-Linear Finite Element Analysis of Solids and Structures Vol.1 and 2, M.A. Crisfield, John Wiley & Sons, 1996 & 1997. 2. Nonlinear Elastic Deformations, R.W. Ogden, Dover Publications, Inc., 1984 3. “A Theory of Large Elastic Deformation”, M. Mooney, Journal of Applied Physics, Vol. 6, pp (1940). September 30, 2001 Inventory #001491 6-92

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