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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/edcommunity Education Community Section 3 – Nonlinear Analysis Objectives Module 3 – Hyperelastic Materials Page 2 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. © 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 Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 3 Hyperelastic material models are used with materials that respond elastically when subjected to large deformations. Elastomers 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 © 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. Biological Materials Foams www.autodesk.com/edcommunity Education Community Hyperelastic Models: Definition Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 4 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. © 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. Nonlinear response of a typical polymer www.autodesk.com/edcommunity Education Community Hyperelastic Models: Variety of Models Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 5 • Each material model contains constants that must be determined experimentally. Autodesk Simulation Hyperelastic Material Models • 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 © 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. •Neo-Hookian •Mooney-Rivlin •Ogden •Yeoh •Arruda-Boyce •Vander Waals •Blatz - Ko 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. © 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. Strain Invariant Based Models Neo-Hookean Mooney-Rivlin Yeoh Arruda-Boyce polymers, moderate stretch levels Mooney-Rivlin is a commonly used model for polymers. Blatz-Ko polyurethane foams Stretch Based Models Ogden Vander Waals www.autodesk.com/edcommunity High stretch levels Education Community Hyperelastic Models: Limitations Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 7 • 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. © 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. 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. www.autodesk.com/edcommunity Education Community Hyperelastic Models: No Viscous Effects Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 8 Hyperelastic Models are strain-rate independent (i.e. it doesn’t matter how fast or slow the load is applied). 0.009 0.008 0.007 0.006 Stress In addition to the Mullins Effect, creep, relaxation, and losses due to a sinusoidal input cannot be modeled using a hyperelastic material model. Relaxation Curves for a Linear Viscoelastic Material 0.005 0.004 2 times 0.002 Viscoelastic material models covered in Module 4 of this Section can be used to capture some of these phenomena. © 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. 2 times 0.003 2 times 2 times 0.001 0 0 1 2 3 4 5 6 Tim e, sec. www.autodesk.com/edcommunity Education Community 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). © 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. Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 9 Stresses are determined from the derivatives of the strain-energy density functions W W Sij 2 Eij Cij www.autodesk.com/edcommunity Education Community Section 3 – Nonlinear Analysis Finite Deformation Theory: Material Configurations Module 3 – Hyperelastic Materials Page 10 Consider an arbitrary line element defined by points P & Q in the undeformed configuration. Q* P* y,y* 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. © 2011 Autodesk Deformed Configuration Q Undeformed Configuration P x,x* z,z* 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. x * f ( x, y , z ) y * g ( x, y , z ) z * h ( x, y , z ) www.autodesk.com/edcommunity Education Community Section 3 – Nonlinear Analysis Finite Deformation Theory: Deformation Gradient Module 3 – Hyperelastic Materials Page 11 The differential changes in the coordinates of the deformed and undeformed configurations are: f f f dx* dx dy dz x y z g g g dy* dx dy dz x y z h h h dz* dx dy dz x y z The deformation gradient is defined as © 2011 Autodesk f x dx * g dy * dz * x h x f x g F x h x f z dx g dy z h dz z f y g y h y f y g y h y 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. dx* F dx f z g z h z www.autodesk.com/edcommunity Education Community Finite Deformation Theory: Mapping Functions 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. x* x u ( x, y, z ) y* y v( x, y, z ) z* z w( x, y, z ) © 2011 Autodesk f x u ( x, y , z ) g y v ( x, y , z ) h z w( x, y, z ) 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. Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 12 Using these functions, the deformation gradient becomes Deformation Gradient u u u 1 x y z v v v F 1 x y z w w w 1 y z x www.autodesk.com/edcommunity Education Community Section 3 – Nonlinear Analysis Finite Deformation Theory: Stretch Tensor The Deformation Gradient can be broken down into a product of two matrices. Module 3 – Hyperelastic Materials Page 13 U F RU V R The matrix [R] is an orthogonal rotation matrix, and [U] and [V] are symmetric matrices that are called the right and left stretch tensors. © 2011 Autodesk V 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. 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. www.autodesk.com/edcommunity Education Community Finite Deformation Theory: Right Cauchy-Green Deformation Tensor The change in length squared of the line element PQ in the deformed configuration is dx dx F F dx * T dS dx 2 T * T dS dx Cdx 2 T Where [C] is the right CauchyGreen deformation tensor given by Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 14 As shown below, the right Cauchy-Green deformation tensor is equal to the square of the right stretch tensor. C U R RU T T C U U U T 2 C F F T © 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 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. © 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. Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 15 Principal Stretch Ratios 1 , 2 , 3 Equation used to define [U] and [V] F RU V R Equation used to find the square of the principal stretch ratios. C U U U T www.autodesk.com/edcommunity 2 Education Community Section 3 – Nonlinear Analysis Finite Deformation Theory: Stretch Ratio Invariants Module 3 – Hyperelastic Materials Page 16 The square of the principal stretch ratios can be found from the equation C11 2 det C12 C13 C C12 2 22 C23 C23 0 C33 2 C13 The coefficients of the characteristic equation are invariants of [C] and can be written in terms of the principal stretch ratios as which results in the characteristic equation I 2 3 2 2 1 © 2011 Autodesk I 2 I3 0 2 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. Principal Stretch Invariants I1 2 1 2 2 2 3 I2 2 2 1 2 2 2 2 3 2 2 3 1 I3 2 2 2 1 2 3 www.autodesk.com/edcommunity Education Community Section 3 – Nonlinear Analysis Finite Deformation Theory: Volumetric Strain Module 3 – Hyperelastic Materials Page 17 Elastomers are nearly incompressible while undergoing moderate stretch (i.e. there is no volume change). Deformed Volume V dx* dy* dz* Original Volume Vo dx dy dz Cube of material in the deformed configuration dy dz Volume Ratio dx dV dx dy dz * x y z 123 * * dVo dx dy dz Incompressible material constraint 1 1 2 3 2 Incompressible © 2011 Autodesk dV 1 1 2 3 dVo 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. 2 1 2 2 2 3 I 3 det( F ) www.autodesk.com/edcommunity Education Community Section 3 – Nonlinear Analysis Finite Deformation Theory: Multiplicative Decomposition of F Module 3 – Hyperelastic Materials Page 18 The principal stretch invariants can be used to describe the strain energy functions of materials that are incompressible. ~ F Fvol F 1 3 Fvol J I For materials that are nearly incompressible, it is necessary to define volumetric and deviatoric portions of the deformation gradient. 1 3 ~ FJ F Volumetric Contribution Deviatoric Contribution 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. © 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 Section 3 – Nonlinear Analysis 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. © 2011 Autodesk Module 3 – Hyperelastic Materials Page 19 V ~ det F det Fvol det F V0 det Fvol J 13 0 0 0 J 1 0 3 0 3 1 J 3 1 1 J 3 ~ det F det F J 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 Section 3 – Nonlinear Analysis Finite Deformation Theory: Relationships Module 3 – Hyperelastic Materials Page 20 ~ ~ ~ The principal invariants I1 , I 2 and I 3of the deviatoric Cauchy~ ~T ~ Green deformation tensor C F F are related to the principal invariants I1, I2 and I3 of the Cauchy-Green deformation tensor C by the following relationships. ~ I1 J 1 3 I1 ~ I 2 J 4 3 I 2 ~ I3 1 ~ ~ ~ Similarly, the deviatoric principal stretches 1 , 2 and 3 are related to the principal stretches 1, 2, and 3 by the equations: ~ 1 J 1 31 ~ 2 J 1 32 ~ 3 J © 2011 Autodesk 1 3 3 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. I. Doghri, Mechanics of Deformable Solids: Linear, Nonlinear, Analytical and Computational Aspects, Springer-Verlag, 2000, p. 374. 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 m i ~ j ~ 2m W Cij I1 3 I 2 3 Dm J 1 i , j 0 M 1 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/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 i=1, j=0 and M=1 i=1, j=0 and i=0, j=1 and M=1 ~ W C10 I1 3 D1 J 1 ~ ~ W C10 I1 3 C01 I 2 3 D1 J 1 Yeoh i=1,2 3, j=0 and M=1 2 3 ~ ~ ~ W C10 I1 3 C20 I1 3 C30 I1 3 D1 J 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 Hyperelastic Material Models: Common Mooney-Rivlin Constants Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 23 The constants C10 and C01 are related to the instantaneous shear modulus for a two parameter Mooney-Rivlin model. G0 2C10 C01 Two Parameter MooneyRivlin Model i=1, j=0 and i=0, j=1 and M=1 ~ ~ W C10 I1 3 C01 I 2 3 D1 J 1 The constant D1 is related to the instantaneous bulk modulus 0 2 D 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 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 E G or 21 E 2G 1 If the material is incompressible, =0.5, and the above relationships become E G 3 © 2011 Autodesk E 3G 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. In terms of the MooneyRivlin constants, G0 2C10 C01 and E0 6C10 C01 These expressions show that increasing either C10 or C01 will increase the stiffness of the material since G and E will increase. www.autodesk.com/edcommunity Education Community Hyperelastic Material Models: Mooney Plot Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 25 The relationship between the stress and stretch for a uniaxial test specimen can be written as 1 C2 2 2 C1 Plot of uniaxial stress-stretch data for a two parameter incompressible MooneyRivlin Model 1 2 2 This equation can be rewritten as 1 2 2 C1 C2 o o o o o Slope C2 C1 1 which is the equation for a straight C10 C1 line. C01 C2 © 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. o www.autodesk.com/edcommunity Education Community Example Problem: Problem Definition Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 26 The objective is to determine the deformation in a rubber oring used to prevent leakage of a 500 psi fluid between the Rubber O-ring housing and shaft. Cut-away View Close-up View Shaft Housing © 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: 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. © 2011 Autodesk Y-Z Plane An axisymmetric model requires that the mesh be on the y-z plane (y is the radial direction). 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. © 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 When an axisymmetric analysis is performed, the axis of symmetry of the part must be the z-axis as shown in the figure. Coordinate System Required for an Axisymmetric Analysis 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/edcommunity Education Community Example Problem: 2D Axisymmetric Mesh Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 30 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 zdirection. © 2011 Autodesk 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. 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 Section 3 – Nonlinear Analysis Module 3 – Hyperelastic Materials Page 31 The 2D element type is selected. 2D element types can be used to model plane stress, plane strain, and axisymmetric problems. © 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. © 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 MooneyRivlin 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. © 2011 Autodesk Strain Energy Density Function ~ ~ 2 W C10 I1 3 C01 I 2 3 2 J 1 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 The bulk modulus can be related to the shear modulus and Poisson’s ratio through the equation 2G 1 31 2 For an incompressible material =0.5, and the bulk modulus is infinite. © 2011 Autodesk Strain Energy Density Function ~ ~ 2 W C10 I1 3 C01 I 2 3 2 J 1 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 MooneyRivlin material can be approximated using the following procedure. 2G 1 G is taken to be 0.5 in the numerator. 31 2 1 2 The shear modulus for a Mooney-Rivlin material is given by G 2C10 C01 2C10 C01 1 2 © 2011 Autodesk For C10 = 297 psi, C01=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. 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. © 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 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. Von Mises stress superimposed on the deformed shape at 500 psi pressure 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/edcommunity Education Community 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. Shear Stress, yz , superimposed on the deformed shape at 500 psi Strain Energy Density Function ~ ~ 2 W C10 I1 3 C01 I 2 3 2 J 1 Deviatoric © 2011 Autodesk Volumetric 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 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. Von Mises Strain superimposed on deformed shape at 500 psi. • 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/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. © 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