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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. Education Community Nonlinear Analysis: Viscoelastic 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. Education Community Objectives The objective of this module is to provide an introduction to the theory and methods used in the analysis of components containing materials described by viscoelastic material models. Topics covered include models based on elastic and viscous mechanical elements; Representation of relaxation data in the form of a Prony series; Instantaneous and long term relaxation moduli; Data required by Autodesk Simulation Multiphysics to perform a viscoelastic analysis; and Results from a Mechanical Event Simulation Analysis with Nonlinear Material Models. Section 3 – Nonlinear Analysis Module 4 – Viscoelastic 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. Education Community Viscoelasticity Section 3 – Nonlinear Analysis Module 4 – Viscoelastic Materials Page 3 Viscoelasticity is concerned with describing elastic materials that exhibit strain rate or time dependent response to applied stress. Viscoelastic materials exhibit hysteresis, creep, and relaxation. Polymers often exhibit viscoelastic properties. Linear Viscoelasticity The relaxation and creep functions are a function only of time. Nonlinear Viscoelasticity The relaxation and creep functions are a function of both time and stress or strain.

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. Education Community Time Dependent Responses Section 3 – Nonlinear Analysis Module 4 – Viscoelastic Materials Page 4 Instantaneous elasticity Creep under constant stress Relaxation under constant strain Instantaneous recovery followed by delayed recovery and permanent set W. N. Findley, Lai, J.S., Onaran, K., Creep and Relaxation of Nonlinear Viscoelastic Materials, Dover, 1989, pp.50. Polymers respond differently to different types of time dependent loading.

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. Education Community Relaxation Modulus When subjected to a constant strain, the stress in polymers will relax (i.e. stress will decrease to a steady state value). In a linear viscoelastic material the relaxation is proportional to the applied strain. The relaxation modulus is defined as: Section 3 – Nonlinear Analysis Module 4 – Viscoelastic Materials Page 5 Relaxation Curves for a Linear Viscoelastic Material 2 times Tension Shear

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. Education Community Creep Compliance When subjected to constant stress, polymers will creep (i.e. strain will continue to increase to a steady state value). If the creep response is proportional to the applied stress, the material is linear. The creep compliance is defined by: Section 3 – Nonlinear Analysis Module 4 – Viscoelastic Materials Page 6 Creep Curves for a Linear Viscoelastic Material 2 times

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. Education Community Sinusoidal Response When subjected to a sinusoidally varying stress there will be a phase angle between the stress and strain. This phase angle creates the hysteresis seen in cyclic stress- strain curves. The phase angle can be related to the damping of the material. Section 3 – Nonlinear Analysis Module 4 – Viscoelastic Materials Page 7 t

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. Education Community Mechanical Element Analogs Section 3 – Nonlinear Analysis Module 4 – Viscoelastic Materials Page 8 Mechanical elements provide a means to construct potential viscoelastic material models. Elastic Element – Stress is proportional to strain. Viscous Element – Stress is proportional to strain rate. The proportionality constant is called viscosity due to its similarity to a Newtonian fluid.

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. Education Community Section 3 – Nonlinear Analysis Module 4 – Viscoelastic Materials Page 9 Maxwell Model The Maxwell model uses a spring and dashpot in series. The Maxwell model doesnt match creep response well. It predicts a linear change in stress versus time for the creep response. Combining yields Units are seconds Derivation of Governing Equation

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. Education Community Kelvin Model Section 3 – Nonlinear Analysis Module 4 – Viscoelastic Materials Page 10 The Kelvin model uses a spring and dashpot in parallel. The Kelvin model doesnt match relaxation data. It doesnt exhibit time dependent relaxation. Derivation of Governing Equation

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. Education Community Standard Linear Solid – Governing Equations Section 3 – Nonlinear Analysis Module 4 – Viscoelastic Materials Page 11 Elastic Arm Maxwell Arm Characteristic Time Derivation of Governing Equation The Standard Linear Solid model is a three- parameter model that contains a Maxwell Arm in parallel with an elastic arm. Laplace transforms will be used to develop relaxation and creep constitutive equations.

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. Education Community Standard Linear Solid – Laplace Domain Laplace Domain Time Domain Governing Equation in Laplace Domain Section 3 – Nonlinear Analysis Module 4 – Viscoelastic Materials Page 12 It is easier to determine the governing equation in the Laplace domain than in the time domain. Elastic Arm Maxwell Arm The overscore indicates the Laplace transform of the variable.

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. Education Community Standard Linear Solid – Relaxation Equations The relaxation behavior is obtained by finding the response to a step change in strain. At time t=0, there is an instantaneous stress response equal to At infinite time the stress relaxes to a steady state value of Unit Step Function Substitution into the governing equation yields Taking the inverse Laplace transform yields Section 3 – Nonlinear Analysis Module 4 – Viscoelastic Materials Page 13

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. Education Community Standard Linear Solid – Relaxation Plot The relaxation modulus, E(t), is shown in the figure. The values chosen for the parameters E r, E m, and are for demonstration purposes only. The stress relaxes to a steady state value controlled by the parameter E r. Section 3 – Nonlinear Analysis Module 4 – Viscoelastic Materials Page 14

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. Education Community Standard Linear Solid – Creep Equations The creep behavior is obtained by finding the response to a step change in stress. At time t=0, there is an instantaneous stress response equal to At infinite time the strain grows to a steady state value of Substitution into the governing equation yields Taking the inverse Laplace transform yields Section 3 – Nonlinear Analysis Module 4 – Viscoelastic Materials Page 15

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. Education Community Standard Linear Solid – Creep Plot Section 3 – Nonlinear Analysis Module 4 – Viscoelastic Materials Page 16 The creep compliance modulus, J(t), is shown in the figure. The values chosen for the parameters E r, E m, and are for demonstration purposes only. The strain creeps to a steady state value controlled by the parameter C r. Since C g is greater than C r the characteristic creep time is slower than that for relaxation.

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. Education Community Standard Linear Solid - Summary The Standard Linear Solid more accurately represents the response of real materials than does the Maxwell or Kelvin models. Instantaneous elastic strain when stress applied; Under constant stress, strain creeps towards a limit; Under constant strain, stress relaxes towards a limit; When stress is removed, instantaneous elastic recovery, followed by gradual recovery to zero strain; Two time constants One for relaxation under constant strain One for creep/recovery under constant stress (Relaxation is quicker than creep) Section 3 – Nonlinear Analysis Module 4 – Viscoelastic Materials Page 17

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. Education Community Wiechert Model The Wiechert model is a generalization of the Standard Linear Solid model and can be used to model the viscoelastic response of many materials. It consists of a linear spring in parallel with a series of springs and dashpots (Maxwell elements). Relaxation Time Section 3 – Nonlinear Analysis Module 4 – Viscoelastic Materials Page 18 Relaxation Modulus The shear relaxation modulus is used from this point forward since Simulation expects data for the shear relaxation modulus to be entered.

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. Education Community Section 3 – Nonlinear Analysis Module 4 – Viscoelastic Materials Page 19 is the value of G(t) at time equal to zero. It is the instantaneous shear modulus. is the value of G(t) at time equal to infinity. It is the final or fully relaxed shear modulus. Relaxation function versus time

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. Education Community Weichert Model – Multiple Relaxation Times The Wiechert model can accurately model the response characteristics of real materials because it can include as many relaxation times and corresponding moduli as needed. In the figure, five Maxwell elements are used to fit the experimental data. Each Maxwell element has a relaxation modulus and corresponding relaxation time constant n Section 3 – Nonlinear Analysis Module 4 – Viscoelastic Materials Page 20 Example Relaxation Data for a Real Material

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. Education Community Prony Series The challenge in describing a material by the Weichert model is to find the coefficients, G i and relaxation times, i, of the Prony Series. Specialized optimization algorithms are used to determine the best set of moduli, G i, and relaxation times, i, that match experimental data. Prony Series Section 3 – Nonlinear Analysis Module 4 – Viscoelastic Materials Page 21

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. Education Community Alternate Forms Section 3 – Nonlinear Analysis Module 4 – Viscoelastic Materials Page 22 This form of the equation is used when the relaxation properties are specified in terms of the instantaneous modulus, G 0. This form of the equation is used when the relaxation properties are specified in terms of the long term modulus,.

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. Education Community Autodesk Simulation Multiphysics Material Data Screen Defines the instantaneous shear modulus First Constant Second Constant The instantaneous form of the relaxation modulus equation is used. Section 3 – Nonlinear Analysis Module 4 – Viscoelastic Materials Page 23 (Mooney-Rivlin)

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. Education Community Volumetric Relaxation Data Unless the Independent Volumetric/Deviatoric Relaxation box is checked, the relaxation data will be applied to both the deviatoric (shear) and volumetric material properties. Many polymers are nearly incompressible and remain so (i.e. no relaxation of the volumetric properties). Zeros have been added for the volumetric Prony series data. Section 3 – Nonlinear Analysis Module 4 – Viscoelastic Materials Page 24

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. Education Community Example - Sandwich Problem Elastomeric adhesives are commonly used as vibration dampers. The hysteresis associated with elastomers provides natural damping. A sandwich type construction where the elastomer is placed between two stiff materials is shown in the figure. Locating the elastomer in the middle exposes it to the highest shear stresses. Section 3 – Nonlinear Analysis Module 4 – Viscoelastic Materials Page 25 Section of Sandwich Beam 6061-T6 Aluminum ISR Industrial Adhesive 1/16 in 1/32 in

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. Education Community Example – 2D Model The beam is modeled using a 2D plane strain representation. A 3D representation would require elements in the thickness direction. The plane strain representation is acceptable since there will be little stress variation through the thickness direction. Section 3 – Nonlinear Analysis Module 4 – Viscoelastic Materials Page 26 Thickness Direction

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. Education Community Example – Beam Geometry The dynamic response of the cantilevered sandwich beam will be computed. The beam is ½ inch wide and 12 inches long. The top and bottom plates are made from 1/16 inch thick T6 aluminum. The adhesive layer (shown in blue) is 1/32 inch thick. Portion of the Inventor model of the sandwich beam. Section 3 – Nonlinear Analysis Module 4 – Viscoelastic Materials Page 27

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. Education Community Loads and Boundary Conditions The displacements at one end of the beam are fixed to simulate a clamped condition. The other end is exposed to a step force of 1 lbs. Displacement Constraints 1 lb divided among 21 nodes Section 3 – Nonlinear Analysis Module 4 – Viscoelastic Materials Page 28

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. Education Community FEA Model A nonlinear dynamic analysis will be performed using the MES with Nonlinear Material Models analysis type. The 2D elements will allow the analysis to run much quicker than if 3D elements were used. Section of Sandwich Beam 6061-T6 Aluminum Simson Industrial Adhesive 1/16 in 1/32 in Mesh absolute element size is 1/64 th of an inch. Section 3 – Nonlinear Analysis Module 4 – Viscoelastic Materials Page 29

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. Education Community Element Definition: Adhesive A viscoelastic Mooney-Rivlin Material is selected. This will give a nonlinear stress- strain relationship with a linear viscoelastic response. The plane strain option is selected. The mid-side nodes option is selected. By default, this is a large displacement analysis. Section 3 – Nonlinear Analysis Module 4 – Viscoelastic Materials Page 30

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. Education Community Example – Material Properties Tension relaxation properties for ISR adhesive are given in the referenced document. Section 3 – Nonlinear Analysis Module 4 – Viscoelastic Materials Page 31 Garcia-Barruetabena, J., et al, Experimental Characterization and Modelization of the Relaxation and Complex Moduli of a Flexible Adhesive, Materials and Design, 32 (2011) Reference Mpa Sec.

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. Education Community Example - Shear Relaxation Properties The relaxation properties given on the previous slide are for tension. Simulation expects shear relaxation properties. Poissons ratio for an incompressible material is 0.5. The shear relaxation data is obtained by dividing the tension data by three. Section 3 – Nonlinear Analysis Module 4 – Viscoelastic Materials Page 32 Mpa Sec. Shear Relaxation Data

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. Education Community Example - Instantaneous Form The shear relaxation data will be entered into the Simulation Prony series table using the instantaneous option. Sec. Instantaneous Shear Modulus Relaxation Data Section 3 – Nonlinear Analysis Module 4 – Viscoelastic Materials Page 33

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. Education Community Example - Mooney-Rivlin Properties The adhesive will be modeled using a hyperelastic material model in conjunction with linear viscoelasticity. The Mooney-Rivlin hyperelastic material model will be used. These constants are normally obtained from the slope and y- intercept of a Mooney curve. As an approximation, the ratio of C 10 /C 01 will be set equal to 4. Section 3 – Nonlinear Analysis Module 4 – Viscoelastic Materials Page 34 These two equations lead to constants of C 10 = psi and C 01 = 99.1 psi.

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. Education Community Example - Bulk Modulus The bulk modulus will be approximated from the equation Section 3 – Nonlinear Analysis Module 4 – Viscoelastic Materials Page 35 For an incompressible material =0.5, and the bulk modulus is infinite. A Poissons ratio of will be assumed, which results in a bulk modulus of approximately 496,000 psi.

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. Education Community Example - Prony Series Data The alpha constants and relaxation times are entered in the Prony series table for the Deviatoric Relaxation data. Note the alpha constants are non- dimensional since they have been normalized by the instantaneous shear modulus, G 0. Assuming that there is no relaxation of the bulk modulus, the volumetric relaxation data will be set to zeros. Section 3 – Nonlinear Analysis Module 4 – Viscoelastic Materials Page 36

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. Education Community Analysis Parameters The response will be computed for 1 second (Event Duration). The response will be captured at 500 time points. This gives an initial time step of seconds. Autodesk Simulation Multiphysics will automatically adjust the time step as needed. The multiplier in the Load Curve table is set to 1 at the beginning and end of the event. This will result in the loads being applied as a step input. Section 3 – Nonlinear Analysis Module 4 – Viscoelastic Materials Page 37

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. Education Community Example - Results The plot shows the computed displacement history for the tip of the cantilever. The peak displacement is approximately twice the steady state response which is consistent with the step response of a linear system. The effect of the damping in the adhesive layer is very evident. Computed displacement history at the tip of the cantilever. Section 3 – Nonlinear Analysis Module 4 – Viscoelastic Materials Page 38

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. Education Community Module Summary An introduction to viscoelastic materials has been provided to help explain the parameters and information required by Autodesk Simulation Multiphysics software. Shear relaxation data is needed to define the deviatoric material properties. Volumetric relaxation data can also be entered and used during the analysis. Autodesk Simulation Multiphysics software provides the ability to couple nonlinear hyperelastic material models with linear viscoelastic models. Although the material is defined in terms of relaxation data, the creep and dynamic response can also be computed. Section 3 – Nonlinear Analysis Module 4 – Viscoelastic Materials Page 39


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