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Nonlinear Analysis:Hyperelastic Material Analysis 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk.www.autodesk.com/edcommunityEducation Community1ObjectivesThe objectives of this module are to:

Provide an introduction to the theory and methods used to perform analyses using hyperelastic materials.

Discuss the characteristics and limitations of hyperelastic material models.

Develop the finite deformation quantities used to define the strain energy density functions of hyperelastic materials.

Learn from an example: The deformation in an o-ring subjected to a pressure shows how the theory relates to the input data required by Autodesk Simulation Multiphysics.Section 3 Nonlinear AnalysisModule 3 Hyperelastic MaterialsPage 2 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk.www.autodesk.com/edcommunityEducation CommunityHyperelastic Materials:ApplicationsHyperelastic material models are used with materials that respond elastically when subjected to large deformations.

Some of the most common applications to model are:(i) the rubbery behavior of a polymeric material(ii) polymeric foams that can be subjected to large reversible shape changes (e.g. a sponge)(iii) biological materialsSection 3 Nonlinear AnalysisModule 3 Hyperelastic MaterialsPage 3

FoamsBiological MaterialsElastomers 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk.www.autodesk.com/edcommunityEducation CommunityFinite Element Analysis in Practice Version 22.00 7/28/2008 33Hyperelastic Models:Definition

A hyperelastic material model derives its stress-strain relationship from a strain-energy density function.

Hyperelastic material models are non-linearly elastic, isotropic, and strain-rate independent.

Many polymers are nearly incompressible over small to moderate stretch values. Section 3 Nonlinear AnalysisModule 3 Hyperelastic MaterialsPage 4

Nonlinear response of a typical polymer 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk.www.autodesk.com/edcommunityEducation CommunityHyperelastic Models:Variety of ModelsEach material model contains constants that must be determined experimentally.

Which material model to use depends on which one best matches the behavior of the material in the stretch range of interest.

A good discussion of material tests needed to define hyperelastic material parameters may be found at www.axelproducts.com Autodesk Simulation Hyperelastic Material ModelsNeo-Hookian

Mooney-Rivlin

Ogden

Yeoh

Arruda-Boyce

Vander Waals

Blatz - KoSection 3 Nonlinear AnalysisModule 3 Hyperelastic MaterialsPage 5 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk.www.autodesk.com/edcommunityEducation CommunityHyperelastic Models:Typical ApplicationsSection 3 Nonlinear AnalysisModule 3 Hyperelastic MaterialsPage 6Deciding which hyperelastic material model to use is not easy.

Each model contains coefficients which must be determined by fitting the model to experimental data.

The best model is the one that best matches the experimental data over the stretch range of interest.

Multiaxial tests are generally required to obtain a good match between a particular material model and the experimental data.Strain Invariant Based ModelsNeo-HookeanMooney-RivlinYeohArruda-Boycepolymers, moderate stretch levelsStretch Based ModelsOgdenVander WaalsHigh stretch levelsMooney-Rivlin is a commonly used model for polymers.Blatz-Kopolyurethane foams 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk.www.autodesk.com/edcommunityEducation CommunityHyperelastic Models:LimitationsHyperelastic models are reversible meaning that there is no difference between load and unload response.

Hyperelastic models assume stable behavior (i.e. there is no difference in response between the first and any other load event).

They are perfectly elastic and do not develop a residual strain.

Section 3 Nonlinear AnalysisModule 3 Hyperelastic MaterialsPage 7

LoadUnloadFrom: Dorfmann A., Ogden R.W., A Constitutive Model for the Mullins Effect with Permanent Set in Particle-Filled Rubber, Int. J. Solids Structures, 41, 1855-1878, 3004.The Mullins Effect is a type of response not covered by hyperelastic material models. 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk.www.autodesk.com/edcommunityEducation CommunityHyperelastic Models:No Viscous EffectsHyperelastic Models are strain-rate independent (i.e. it doesnt matter how fast or slow the load is applied).

In addition to the Mullins Effect, creep, relaxation, and losses due to a sinusoidal input cannot be modeled using a hyperelastic material model.

Viscoelastic material models covered in Module 4 of this Section can be used to capture some of these phenomena.

Relaxation Curves for a Linear Viscoelastic Material2 times2 times2 times2 timesSection 3 Nonlinear AnalysisModule 3 Hyperelastic MaterialsPage 8 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk.www.autodesk.com/edcommunityEducation CommunityHyperelastic Models:Strain Energy Density FunctionsSection 3 Nonlinear AnalysisModule 3 Hyperelastic MaterialsPage 9The stress-strain relationship for a hyperelastic material is derived from a strain-energy density function, W.

The strain-energy density functions for hyperelastic materials are defined in terms of finite deformation quantities (i.e. Greens strain, invariants of the Cauchy-Green deformation tensor, or principal stretch ratios).

Stresses are determined from the derivatives of the strain-energy density functions 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk.www.autodesk.com/edcommunityEducation CommunityFinite Deformation Theory:Material ConfigurationsSection 3 Nonlinear AnalysisModule 3 Hyperelastic MaterialsPage 10 Consider an arbitrary line element defined by points P & Q in the undeformed configuration.

The same points are defined by P* and Q* in the deformed configuration.

f,g & h are functions that define the relationship between coordinates in the deformed and undeformed configurations.

x,x*y,y*z,z*Undeformed ConfigurationDeformed ConfigurationPP*QQ*

2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk.www.autodesk.com/edcommunityEducation Community

The differential changes in the coordinates of the deformed and undeformed configurations are:

Finite Deformation Theory:Deformation GradientSection 3 Nonlinear AnalysisModule 3 Hyperelastic MaterialsPage 11The deformation gradient is defined as

2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk.www.autodesk.com/edcommunityEducation CommunityDeformation GradientFinite Deformation Theory:Mapping FunctionsSection 3 Nonlinear AnalysisModule 3 Hyperelastic MaterialsPage 12

The displacements u,v and w in the x, y and z directions can be used to determine the mapping functions f, g and h.

Using these functions, the deformation gradient becomes 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk.www.autodesk.com/edcommunityEducation CommunityFinite Deformation Theory:Stretch TensorSection 3 Nonlinear AnalysisModule 3 Hyperelastic MaterialsPage 13

The Deformation Gradient can be broken down into a product of two matrices.

The matrix [R] is an orthogonal rotation matrix, and [U] and [V] are symmetric matrices that are called the right and left stretch tensors.

The Left Stretch Tensor because it appears on the left of the rotation matrix.

The Right Stretch Tensor because it appears on the right of the rotation matrix. 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk.www.autodesk.com/edcommunityEducation CommunityFinite Deformation Theory:Right Cauchy-Green Deformation TensorSection 3 Nonlinear AnalysisModule 3 Hyperelastic MaterialsPage 14

The change in length squared of the line element PQ in the deformed configuration is

Where [C] is the right Cauchy-Green deformation tensor given by

As shown below, the right Cauchy-Green deformation tensor is equal to the square of the right stretch tensor. 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk.www.autodesk.com/edcommunityEducation CommunityFinite Deformation Theory:Principal Stretch RatiosSection 3 Nonlinear AnalysisModule 3 Hyperelastic MaterialsPage 15The principal stretch ratios could be found by extracting the eigenvalues of [U] or [V].

This is typically not done since it would require that the rotation matrix [R] be found.

Instead it is more customary to find the square of the principal stretch ratios by extracting the eigenvalues of the Cauchy-Green deformation tensor.

Equation used to define [U] and [V]

Equation used to find the square of the principal stretch ratios.

Principal Stretch Ratios 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk.www.autodesk.com/edcommunityEducation CommunityPrincipal Stretch InvariantsFinite Deformation Theory:Stretch Ratio InvariantsSection 3 Nonlinear AnalysisModule 3 Hyperelastic MaterialsPage 16The coefficients of the characteristic equation are invariants of [C] and can be written in terms of the principal stretch ratios as

The square of the principal stretch ratios can be found from the equationwhich results in the characteristic equation

2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk.www.autodesk.com/edcommunityEducation CommunityFinite Deformation Theory:Volumetric StrainSection 3 Nonlinear AnalysisModule 3 Hyperelastic MaterialsPage 17Elastomers are nearly incompressible while undergoing moderate stretch (i.e. there is no volume change).Original VolumeDeformed VolumeVolume RatioIncompressible

Cube of material in the deformed configuration

Incompressible material constraint 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk.www.autodesk.com/edcommunityEducation CommunityFinite Deformation Theory:Multiplicative Decomposition of FSection 3 Nonlinear AnalysisModule 3 Hyperelastic MaterialsPage 18The principal stretch invariants can be used to describe the strain energy functions of materials that are incompressible.

For materials that are nearly incompressible, it is necessary to define volumetric and deviatoric portions of the deformation gradient.

The determinant of F is equal to the ratio of the deformed and undeformed configurations.

R.J. Flory, Thermodynamic Relations for High Elastic Materials, Trans. Faraday Soc., 57, (1961) 829-838.Volumetric ContributionDeviatoricContribution 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk.www.autodesk.com/edcommunityEducation CommunityThe determinant of F is equal to the ratio of the deformed and undeformed configurations.

The determinant of the volumetric contribution is equal to one.

The determinant of the deviatoric contribution is equal determinant of the deformation gradient, J.

Finite Deformation Theory:Incompressibility ConstraintSection 3 Nonlinear AnalysisModule 3 Hyperelastic MaterialsPage 19 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk.www.autodesk.com/edcommunityEducation CommunityFinite Deformation Theory:RelationshipsSection 3 Nonlinear AnalysisModule 3 Hyperelastic MaterialsPage 20The principal invariants , and of the deviatoric Cauchy-Green deformation tensor are related to the principal invariants I1, I2 and I3 of the Cauchy-Green deformation tensor C by the following relationships.

Similarly, the deviatoric principal stretches , and are related to the principal stretches l1, l2, and l3 by the equations:

I. Doghri, Mechanics of Deformable Solids: Linear, Nonlinear, Analytical and Computational Aspects, Springer-Verlag, 2000, p. 374. 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk.www.autodesk.com/edcommunityEducation CommunityHyperelastic Material Models:Rivlin Polynomial ModelSection 3 Nonlinear AnalysisModule 3 Hyperelastic MaterialsPage 21A general and widely used form of strain energy density function, W, was proposed by Rivlin

where Cij and Dm are material constants.

The left hand term controls the distortional response of the material while the right hand term controls the volumetric response.Note the left hand term is written in terms of the principal invariants of the deviatoric portion of the Cauchy-Green deformation tensor. 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk.www.autodesk.com/edcommunityEducation CommunityHyperelastic Material Models:Rivlin Polynomial ModelSection 3 Nonlinear AnalysisModule 3 Hyperelastic MaterialsPage 22Several hyperelastic material models in Autodesk Simulation Multiphysics are obtained from the Rivlin polynomial model by selecting different values for i, j and M. Examples are shown below.

Neo-Hookian

Mooney-RivlinYeoh

i=1, j=0 and M=1i=1, j=0 and i=0, j=1 and M=1i=1,2 3, j=0 and M=1 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk.www.autodesk.com/edcommunityEducation CommunityHyperelastic Material Models:Common Mooney-Rivlin ConstantsSection 3 Nonlinear AnalysisModule 3 Hyperelastic MaterialsPage 23

Two Parameter Mooney-Rivlin Modeli=1, j=0 and i=0, j=1 and M=1

The constants C10 and C01 are related to the instantaneous shear modulus for a two parameter Mooney-Rivlin model.The constant D1 is related to the instantaneous bulk modulus 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk.www.autodesk.com/edcommunityEducation Community23Hyperelastic Material Models:Effects of Changing ConstantsSection 3 Nonlinear AnalysisModule 3 Hyperelastic MaterialsPage 24For small strains, the shear modulus and Youngs modulus are related by the relationship

or

If the material is incompressible, n=0.5, and the above relationships become

In terms of the Mooney-Rivlin constants,

and

These expressions show that increasing either C10 or C01 will increase the stiffness of the material since G and E will increase. 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk.www.autodesk.com/edcommunityEducation CommunityHyperelastic Material Models:Mooney PlotSection 3 Nonlinear AnalysisModule 3 Hyperelastic MaterialsPage 25

Plot of uniaxial stress-stretch data for a two parameter incompressible Mooney-Rivlin ModelooooooThe relationship between the stress and stretch for a uniaxial test specimen can be written asThis equation can be rewritten as

which is the equation for a straight line.

2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk.www.autodesk.com/edcommunityEducation CommunityExample Problem:Problem DefinitionSection 3 Nonlinear AnalysisModule 3 Hyperelastic MaterialsPage 26

Rubber O-ringHousingShaft

The objective is to determine the deformation in a rubber o-ring used to prevent leakage of a 500 psi fluid between the housing and shaft.Cut-away ViewClose-up View 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk.www.autodesk.com/edcommunityEducation Community

Y-Z PlaneExample Problem:Wedge GeometrySection 3 Nonlinear AnalysisModule 3 Hyperelastic MaterialsPage 27Since nothing will vary around the circumferential direction an axisymmetric analysis will be performed.

This will reduce the size of the problem without losing any of the desired information.

A 5-degree wedged shape portion of the model is created in Autodesk Inventor.

This wedge will be used in Autodesk Simulation Multiphysics to create the axisymmetric model.An axisymmetric model requires that the mesh be on the y-z plane (y is the radial direction). 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk.www.autodesk.com/edcommunityEducation CommunityExample Problem:Analysis TypeSection 3 Nonlinear AnalysisModule 3 Hyperelastic MaterialsPage 28

The analysis type is set to Static Stress with Nonlinear Material Models.

This analysis type will allow the selection of one of the hyperelastic material models when the element data is defined.

The analysis type can be selected at startup or changed by editing the Analysis Type in the FEA Editor. 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk.www.autodesk.com/edcommunityEducation CommunityExample Problem:Axisymmetric Coordinate SystemSection 3 Nonlinear AnalysisModule 3 Hyperelastic MaterialsPage 29

Coordinate System Required for an Axisymmetric AnalysisWhen an axisymmetric analysis is performed, the axis of symmetry of the part must be the z-axis as shown in the figure.

The radial direction corresponds to the +y direction.

The radius is equal to zero when y is equal to zero. 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk.www.autodesk.com/edcommunityEducation CommunityExample Problem:2D Axisymmetric MeshThere are several steps involved in creating this mesh and the details are contained in the first of the two videos for this module.

A 500 psi pressure load is applied to one face of the o-ring and the model is constrained in the z-direction.Section 3 Nonlinear AnalysisModule 3 Hyperelastic MaterialsPage 30

Z-displacement constraintPressure LoadThe y-direction does not have to be constrained because the circumferential strain will limit the motion in this direction.Axisymmetric 2D Mesh 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk.www.autodesk.com/edcommunityEducation Community30

Example Problem:Element TypeThe 2D element type is selected.

2D element types can be used to model plane stress, plane strain, and axisymmetric problems.Section 3 Nonlinear AnalysisModule 3 Hyperelastic MaterialsPage 31 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk.www.autodesk.com/edcommunityEducation CommunityExample Problem:Element Definition DataSection 3 Nonlinear AnalysisModule 3 Hyperelastic MaterialsPage 32

The Mooney-Rivlin hyperelastic material model is selected for the O-ring material and isotropic linear elastic materials are selected for the housing and shaft parts.

All parts use the axisymmetric geometry type. 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk.www.autodesk.com/edcommunityEducation CommunityExample Problem:Hyperelastic Material DefinitionSection 3 Nonlinear AnalysisModule 3 Hyperelastic MaterialsPage 33

A 2-constant standard Mooney-Rivlin material is selected. This material is good for moderate stretch levels.

The First Constant (C10) and Second Constant (C01) material coefficients are taken from the Simulation library.

Strain Energy Density Function 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk.www.autodesk.com/edcommunityEducation CommunityExample Problem:Bulk Modulus (Incompressible)Section 3 Nonlinear AnalysisModule 3 Hyperelastic MaterialsPage 34

Strain Energy Density FunctionThe bulk modulus can be related to the shear modulus and Poissons ratio through the equation

For an incompressible material n=0.5, and the bulk modulus is infinite. 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk.www.autodesk.com/edcommunityEducation CommunityExample Problem:Bulk Modulus (Nearly Incompressible)Section 3 Nonlinear AnalysisModule 3 Hyperelastic MaterialsPage 35

The bulk modulus for a nearly incompressible Mooney-Rivlin material can be approximated using the following procedure.The shear modulus for a Mooney-Rivlin material is given by

For C10 = 297 psi, C01=172 psi and n=0.499, the bulk modulus is computed to be 496,000 psi.

Small changes in n can cause large changes in k (i.e. n = 0.4988 k=400,000 psi).n is taken to be 0.5 in the numerator. 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk.www.autodesk.com/edcommunityEducation CommunityExample Problem:Analysis ParametersSection 3 Nonlinear AnalysisModule 3 Hyperelastic MaterialsPage 36

The event duration is set to 1 second.

Time is used only as an interpolation parameter to determine the percentage of load being applied.

The pressure will be applied in 1000 time steps or load increments. Hyperelastic materials are very nonlinear and small time steps are required to achieve converged solutions.

The load curve will increase the pressure linearly from 0 to 1 seconds. 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk.www.autodesk.com/edcommunityEducation CommunityVon Mises stress superimposed on the deformed shape at 500 psi pressureExample Problem:Analysis ResultsSection 3 Nonlinear AnalysisModule 3 Hyperelastic MaterialsPage 37The von Mises stress is invariant to hydrostatic stress states. This can be seen in the figure.

The top half of the o-ring has retained its circular shape due to the contact constraints and the pressure acting on the surface. In this distortion free area the von Mises stress is very low.

The bottom half of the o-ring is not constrained by the pressure and distorts as it is squeezed into the gland. This distorted area has larger von Mises stresses.

2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk.www.autodesk.com/edcommunityEducation Community37

Shear Stress, syz , superimposed on the deformed shape at 500 psiExample Problem:Analysis ResultsSection 3 Nonlinear AnalysisModule 3 Hyperelastic MaterialsPage 38The shear stress in the upper half of the o-ring is nearly zero. This is due to the volumetric response in this area. There is little contribution from the deviatoric portion of the constitutive equation.

The shear stress is greater in the lower half where there is more distortion. The deviatoric portion of the constitutive equation is playing a larger role in this area.

Strain Energy Density FunctionDeviatoricVolumetric 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk.www.autodesk.com/edcommunityEducation Community38Von Mises Strain superimposed on deformed shape at 500 psi.Example Problem:Analysis ResultsSection 3 Nonlinear AnalysisModule 3 Hyperelastic MaterialsPage 39The von Mises strain is also invariant to hydrostatic loading as seen in the figure.

The upper half of the o-ring has very small strains.

The strains in the lower half where the distortion is taking place has a max von Mises strain of 40%.

2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk.www.autodesk.com/edcommunityEducation Community39SummarySection 3 Nonlinear AnalysisModule 3 Hyperelastic MaterialsPage 40This module has provided an introduction to hyperelastic materials and has demonstrated how to perform an analysis with Autodesk Simulation Multiphysics using these material models.

Hyperelastic materials are used to compute the large deformation response of nonlinear elastic materials (i.e. polymers, foams, and biological materials).

The stress-strain response of hyperelastic materials are defined by strain energy density functions expressed in terms of finite deformation variables.

Autodesk Simulation Multiphysics provides a library of hyperelastic material models capable of modeling the behavior of materials over a wide range of stretch. 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk.www.autodesk.com/edcommunityEducation CommunityChart20.0020.0040.0080.00190483740.00380967480.00761934970.00160653070.00321306130.00642612260.00136787940.00273575890.00547151780.00122313020.00244626030.00489252060.00113533530.00227067060.00454134110.0010820850.002164170.004328340.00104978710.00209957410.00419914830.00103019740.00206039480.00412078950.00101831560.00203663130.00407326260.0010111090.0020222180.0040444360.00100673790.00201347590.0040269518

Time, sec.Stress

Sheet1ErEmnt1010101wwtE1E2dEE1/EE2/E112050.244978663120.61552812810.66666711110.1666667778222640.152649328426.30589287590.86666724440.1333334222332830.106735672628.16025568070.93333395560.10000006674428.82352941182.35294117650.081452044328.91940836070.96078495420.07843142485529.23076923081.92307692310.065694801629.29396003740.97435962390.06410260687729.61.40.04726207629.63308961280.98666732440.0466666978101029.8019801980.99009900990.033210376429.81842248970.99340000220.0330033223202029.95012468830.49875311720.016651250229.95427721570.99833815520.016625115303029.97780244170.3329633740.011106540729.97965149640.99926074760.0110987865404029.98750780760.24984384760.008331404829.98854858880.999584260.0083281338505029.99200319870.1999200320.006665679229.99266950260.99973410640.0066640055606029.99444598720.16662038320.005554984129.99490877520.99981553280.0055540165707029.99591920020.14282799430.004761544929.99625924180.99986463990.0047609363808029.99687548820.12498047180.004166425629.99713584950.99989651620.0041660185909029.9975311690.11109739540.003703534429.99773689570.99991837220.00370324910010029.99800020.0999900010.003333209929.9981668440.99993400660.003333002320020029.99950001250.049998750.001666651229.99954167770.99998400040.001666626130030029.99977778020.0333329630.001111106529.99979629850.99999325930.001111099540040029.99987500080.02499984380.000833331429.99988541740.99999650.000833328750050029.99992000030.019999920.000666665729.9999266670.9999980.000666664460060029.99994444460.01666662040.00055555529.99994907420.99999881480.000555554470070029.99995918380.01428568510.000476190129.99996258510.99999930610.000476189880080029.999968750.01249998050.000416666429.99997135420.9999996250.000416666390090029.99997530870.01111109740.000370370229.99997736630.99999984360.00037037021000100029.999980.009999990.000333333229.999981666710.0003333332CgCr10.050.12TimeJ(t)Curve 1Curve 2Curve 300.0500000.050.10.20.10.0524390.05243852880.10487705750.20975411510.50.0610600.06105996080.12211992170.244239843410.0696730.0696734670.1393469340.27869386811.50.0763820.07638167240.15276334470.305526689520.0816060.08160602790.16321205590.32642411182.50.0856750.08567476020.17134952030.342699040630.0888430.0888434920.1776869840.3553739683.50.0913110.09131130280.18262260570.365245211340.0932330.09323323580.18646647170.37293294344.50.0947300.09473003880.18946007750.378920155150.0958960.09589575010.19179150010.38358300030.0001TimeG(t)Curve 1Curve 2Curve 30200.0020.0040.0080.119.04837418040.00190483740.00380967480.00761934970.516.06530659710.00160653070.00321306130.0064261226113.67879441170.00136787940.00273575890.00547151781.512.23130160150.00122313020.00244626030.0048925206211.35335283240.00113533530.00227067060.00454134112.510.82084998620.0010820850.002164170.00432834310.49787068370.00104978710.00209957410.00419914833.510.30197383420.00103019740.00206039480.0041207895410.18315638890.00101831560.00203663130.00407326264.510.11108996540.0010111090.0020222180.004044436510.067379470.00100673790.00201347590.0040269518

Sheet1

Storage ModulusLoss Modulustw

Sheet2

Storage ModulusLoss ModulustwNormalized Moduli

Sheet3

Time, sec.Strain, in/inCreep Curves

Time, sec.Stress