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Nonlinear Analysis: Hyperelastic Material Analysis

Nonlinear Analysis: Hyperelastic Material Analysis. Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 2. Objectives. The objectives of this module are to: Provide an introduction to the theory and methods used to perform analyses using hyperelastic materials.

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Nonlinear Analysis: Hyperelastic Material Analysis

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  1. Nonlinear Analysis: Hyperelastic Material Analysis

  2. Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 2 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.

  3. 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 Elastomers Biological Materials Foams

  4. Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 4 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. Nonlinear response of a typical polymer

  5. Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 5 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.com Autodesk Simulation Hyperelastic Material Models • Neo-Hookian • Mooney-Rivlin • Ogden • Yeoh • Arruda-Boyce • Vander Waals • Blatz - Ko

  6. Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 6 Hyperelastic Models: Typical Applications • 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 Mooney-Rivlin is a commonly used model for polymers. polyurethane foams Blatz-Ko Stretch Based Models High stretch levels Ogden Vander Waals

  7. Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 7 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. Unload Load 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. Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 8 HyperelasticModels: NoViscous 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 2 times 2 times 2 times

  9. Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 9 Hyperelastic Models: Strain Energy Density Functions • 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. Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 10 Finite Deformation Theory: Material Configurations • 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. Q* Deformed Configuration P* y,y* Q Undeformed Configuration P x,x* z,z*

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

  12. Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 12 Finite Deformation Theory: Mapping Functions Using these functions, the deformation gradient becomes 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. Deformation Gradient 

  13. Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 13 Finite Deformation Theory: Stretch Tensor • 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 Right Stretch Tensor because it appears on the right of the rotation matrix. The Left Stretch Tensor because it appears on the left of the rotation matrix.

  14. Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 14 Finite Deformation Theory: Right Cauchy-Green Deformation Tensor The change in length squared of the line element PQ in the deformed configuration is As shown below, the right Cauchy-Green deformation tensor is equal to the square of the right stretch tensor. Where [C] is the right Cauchy-Green deformation tensor given by

  15. Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 15 Finite Deformation Theory: Principal Stretch Ratios 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. Principal Stretch Ratios Equation used to define [U] and [V] Equation used to find the square of the principal stretch ratios.

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

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

  18. Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 18 Finite Deformation Theory: Multiplicative Decomposition of F 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. Volumetric Contribution Deviatoric Contribution R.J. Flory, Thermodynamic Relations for High Elastic Materials, Trans. Faraday Soc., 57, (1961) 829-838.

  19. Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 19 Finite Deformation Theory: Incompressibility Constraint 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.

  20. Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 20 Finite Deformation Theory: Relationships The 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.

  21. Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 21 Hyperelastic Material Models: Rivlin Polynomial Model A general and widely used form of strain energy density function, W, was proposed by Rivlin where Cijand Dmare 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. Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 22 Hyperelastic Material Models: Rivlin Polynomial Model 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 i=1, j=0 and M=1 i=1, j=0 and i=0, j=1 and M=1 Yeoh i=1,2 3, j=0 and M=1

  23. Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 23 Hyperelastic Material Models: Common Mooney-Rivlin Constants Two Parameter Mooney-Rivlin Model The constants C10 and C01 are related to the instantaneous shear modulus for a two parameter Mooney-Rivlin model. i=1, j=0 and i=0, j=1 and M=1 The constant D1 is related to the instantaneous bulk modulus

  24. Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 24 Hyperelastic Material Models: Effects of Changing Constants • For small strains, the shear modulus and Young’s modulus are related by the relationship • In terms of the Mooney-Rivlin constants, or and • If the material is incompressible, n=0.5, and the above relationships become These expressions show that increasing either C10 or C01 will increase the stiffness of the material since G and E will increase.

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

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

  27. Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 27 Example Problem: Wedge Geometry • 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. Y-Z Plane An axisymmetric model requires that the mesh be on the y-z plane (y is the radial direction).

  28. Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 28 Example Problem: Analysis Type • 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. Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 29 Example Problem: Axisymmetric Coordinate System • 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. Coordinate System Required for an Axisymmetric Analysis

  30. Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 30 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. Axisymmetric 2D Mesh Pressure Load Z-displacement constraint The y-direction does not have to be constrained because the circumferential strain will limit the motion in this direction.

  31. Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 31 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.

  32. Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 32 Example Problem: Element Definition Data • 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. Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 33 Example Problem: Hyperelastic Material Definition • 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

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

  35. Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 35 Example Problem: Bulk Modulus (Nearly Incompressible) The bulk modulus for a nearly incompressible Mooney-Rivlin material can be approximated using the following procedure. n is taken to be 0.5 in the numerator. 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).

  36. Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 36 Example Problem: Analysis Parameters • 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. Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 37 Example Problem: Analysis Results • 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. Von Mises stress superimposed on the deformed shape at 500 psi pressure

  38. Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 38 Example Problem: Analysis Results • 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. Shear Stress, syz, superimposed on the deformed shape at 500 psi Strain Energy Density Function Volumetric Deviatoric

  39. Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 39 Example Problem: Analysis Results Von Mises Strain superimposed on deformed shape at 500 psi. • 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. Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 40 Summary • 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.

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