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/edcommunity Education Community Nonlinear Analysis: Hyperelastic Material Analysis

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/edcommunity Education Community Objectives  The 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 Analysis Module 3 – Hyperelastic Materials Page 2

3. © 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/edcommunity Education Community Hyperelastic Materials: Applications  Hyperelastic 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 materials Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 3 Foams Biological Materials Elastomers

4. © 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/edcommunity Education Community Hyperelastic 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 Analysis Module 3 – Hyperelastic Materials Page 4 Nonlinear response of a typical polymer

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/edcommunity Education Community Hyperelastic Models: Variety of Models Each 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.comwww.axelproducts.com Autodesk Simulation Hyperelastic Material Models Neo-Hookian Mooney-Rivlin Ogden Yeoh Arruda-Boyce Vander Waals Blatz - Ko Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 5

6. © 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/edcommunity Education Community Hyperelastic Models: Typical Applications Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 6  Deciding 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 Models Neo-Hookean Mooney-Rivlin Yeoh Arruda-Boyce polymers, moderate stretch levels Stretch Based Models Ogden Vander Waals High stretch levels Mooney-Rivlin is a commonly used model for polymers. Blatz-Ko polyurethane foams

7. © 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/edcommunity Education Community Hyperelastic Models: Limitations Hyperelastic 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 Analysis Module 3 – Hyperelastic Materials Page 7 Load Unload From: 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.

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/edcommunity Education Community Hyperelastic Models: No Viscous Effects  Hyperelastic Models are strain-rate independent (i.e. it doesn’t 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 Material 2 times Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 8

9. © 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/edcommunity Education Community Hyperelastic Models: Strain Energy Density Functions Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 9 The 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. Green’s strain, invariants of the Cauchy-Green deformation tensor, or principal stretch ratios). Stresses are determined from the derivatives of the strain-energy density functions

10. © 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/edcommunity Education Community Finite Deformation Theory: Material Configurations Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 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 Configuration Deformed Configuration P P* Q Q*

11. © 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/edcommunity Education Community The differential changes in the coordinates of the deformed and undeformed configurations are: Finite Deformation Theory: Deformation Gradient Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 11 The deformation gradient is defined as

12. © 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/edcommunity Education Community Deformation Gradient Finite Deformation Theory: Mapping Functions Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 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

13. © 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/edcommunity Education Community Finite Deformation Theory: Stretch Tensor Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 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.

14. © 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/edcommunity Education Community Finite Deformation Theory: Right Cauchy-Green Deformation Tensor Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 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.

15. © 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/edcommunity Education Community Finite Deformation Theory: Principal Stretch Ratios Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 15 The 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

16. © 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/edcommunity Education Community Principal Stretch Invariants Finite Deformation Theory: Stretch Ratio Invariants Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 16 The 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 equation which results in the characteristic equation

17. © 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/edcommunity Education Community Finite Deformation Theory: Volumetric Strain Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 17 Elastomers are nearly incompressible while undergoing moderate stretch (i.e. there is no volume change). Original Volume Deformed Volume Volume Ratio Incompressible Cube of material in the deformed configuration  Incompressible material constraint

18. © 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/edcommunity Education Community Finite Deformation Theory: Multiplicative Decomposition of F Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 18 The 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 Contribution Deviatoric Contribution

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/edcommunity Education Community The 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 Constraint Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 19

20. © 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/edcommunity Education Community Finite Deformation Theory: Relationships Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 20 The principal invariants, and of the deviatoric Cauchy- Green deformation tensor are related to the principal invariants I 1, I 2 and I 3 of the Cauchy-Green deformation tensor C by the following relationships. Similarly, the deviatoric principal stretches, and are related to the principal stretches 1, 2, and 3 by the equations: I. Doghri, Mechanics of Deformable Solids: Linear, Nonlinear, Analytical and Computational Aspects, Springer-Verlag, 2000, p. 374.

21. © 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/edcommunity Education Community Hyperelastic Material Models: Rivlin Polynomial Model Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 21 A general and widely used form of strain energy density function, W, was proposed by Rivlin where C ij and D m 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.

22. © 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/edcommunity Education Community Hyperelastic Material Models: Rivlin Polynomial Model Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 22 Several 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-Rivlin Yeoh i=1, j=0 and M=1 i=1, j=0 and i=0, j=1 and M=1 i=1,2 3, j=0 and M=1

23. © 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/edcommunity Education Community Hyperelastic Material Models: Common Mooney-Rivlin Constants Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 23 Two Parameter Mooney- Rivlin Model i=1, j=0 and i=0, j=1 and M=1 The constants C 10 and C 01 are related to the instantaneous shear modulus for a two parameter Mooney-Rivlin model. The constant D1 is related to the instantaneous bulk modulus

24. © 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/edcommunity Education Community Hyperelastic Material Models: Effects of Changing Constants Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 24  For small strains, the shear modulus and Young’s modulus are related by the relationship or  If the material is incompressible, =0.5, and the above relationships become  In terms of the Mooney- Rivlin constants, and These expressions show that increasing either C 10 or C 01 will increase the stiffness of the material since G and E will increase.

25. © 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/edcommunity Education Community Hyperelastic Material Models: Mooney Plot Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 25 Plot of uniaxial stress-stretch data for a two parameter incompressible Mooney- Rivlin Model o o o o o o The relationship between the stress and stretch for a uniaxial test specimen can be written as This equation can be rewritten as which is the equation for a straight line.

26. © 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/edcommunity Education Community Example Problem: Problem Definition Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 26 Rubber O-ring Housing Shaft 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 View Close-up View

27. © 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/edcommunity Education Community Y-Z Plane Example Problem: Wedge Geometry Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 27  Since 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).

28. © 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/edcommunity Education Community Example Problem: Analysis Type Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 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.

29. © 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/edcommunity Education Community Example Problem: Axisymmetric Coordinate System Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 29 Coordinate System Required for an Axisymmetric Analysis  When 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.

30. © 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/edcommunity Education Community Example Problem: 2D Axisymmetric Mesh  There 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 Analysis Module 3 – Hyperelastic Materials Page 30 Z-displacement constraint Pressure Load The y-direction does not have to be constrained because the circumferential strain will limit the motion in this direction. Axisymmetric 2D Mesh

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/edcommunity Education Community Example Problem: Element Type  The 2D element type is selected.  2D element types can be used to model plane stress, plane strain, and axisymmetric problems. Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 31

32. © 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/edcommunity Education Community Example Problem: Element Definition Data Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 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.

33. © 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/edcommunity Education Community Example Problem: Hyperelastic Material Definition Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 33  A 2-constant standard Mooney- Rivlin material is selected. This material is good for moderate stretch levels.  The First Constant (C 10 ) and Second Constant (C 01 ) material coefficients are taken from the Simulation library. Strain Energy Density Function

34. © 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/edcommunity Education Community Example Problem: Bulk Modulus (Incompressible) Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 34 Strain Energy Density Function The bulk modulus can be related to the shear modulus and Poisson’s ratio through the equation For an incompressible material =0.5, and the bulk modulus is infinite.

35. © 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/edcommunity Education Community Example Problem: Bulk Modulus (Nearly Incompressible) Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 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 C 10 = 297 psi, C 01 =172 psi and =0.499, the bulk modulus is computed to be 496,000 psi. Small changes in can cause large changes in   i.e.  0.4988   400,000 psi . is taken to be 0.5 in the numerator.

36. © 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/edcommunity Education Community Example Problem: Analysis Parameters Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 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.

37. © 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/edcommunity Education Community Von Mises stress superimposed on the deformed shape at 500 psi pressure Example Problem: Analysis Results Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 37  The 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.

38. © 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/edcommunity Education Community Shear Stress,  yz, superimposed on the deformed shape at 500 psi Example Problem: Analysis Results Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 38 The 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 Function Deviatoric Volumetric

39. © 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/edcommunity Education Community Von Mises Strain superimposed on deformed shape at 500 psi. Example Problem: Analysis Results Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 39 The 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%.

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/edcommunity Education Community Summary Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 40  This 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.