1. Viscoelastic Material Analysis
Nonlinear Analysis:
Viscoelastic Material Analysis

2. Section 3 – Nonlinear Analysis
Module 4 – Viscoelastic Materials
Page 2
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.

3. Viscoelastic materials exhibit hysteresis, creep, and relaxation.
Section 3 – Nonlinear Analysis
Module 4 – Viscoelastic Materials
Page 3
Viscoelasticity
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. Time Dependent Responses
Section 3 – Nonlinear Analysis
Module 4 – Viscoelastic Materials
Page 4
Time Dependent Responses
Polymers respond differently to different types of time dependent loading.
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.

5. Relaxation Curves for a Linear Viscoelastic Material
Section 3 – Nonlinear Analysis
Module 4 – Viscoelastic Materials
Page 5
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:
Relaxation Curves for a Linear Viscoelastic Material
2 times
2 times
2 times
2 times
Tension
Shear

6. Creep Curves for a Linear Viscoelastic Material
Section 3 – Nonlinear Analysis
Module 4 – Viscoelastic Materials
Page 6
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:
Creep Curves for a Linear Viscoelastic Material
2 times
2 times
2 times
2 times

7. The phase angle can be related to the damping of the material.
Section 3 – Nonlinear Analysis
Module 4 – Viscoelastic Materials
Page 7
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. t

8. Mechanical Element Analogs
Section 3 – Nonlinear Analysis
Module 4 – Viscoelastic Materials
Page 8
Mechanical Element Analogs
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. The Maxwell model uses a spring and dashpot in series.
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 doesn’t match creep response well.
It predicts a linear change in stress versus time for the creep response.
Derivation of Governing Equation
Combining yields
Units are seconds

10. The Kelvin model uses a spring and dashpot in parallel.
Section 3 – Nonlinear Analysis
Module 4 – Viscoelastic Materials
Page 10
Kelvin Model
The Kelvin model uses a spring and dashpot in parallel.
The Kelvin model doesn’t match relaxation data.
It doesn’t exhibit time dependent relaxation.
Derivation of Governing Equation

11. Standard Linear Solid – Governing Equations
Section 3 – Nonlinear Analysis
Module 4 – Viscoelastic Materials
Page 11
Standard Linear Solid – Governing Equations
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.
Derivation of Governing Equation
Elastic Arm
Maxwell Arm
Characteristic Time

12. Standard Linear Solid – 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.
Laplace Domain
Time Domain
The overscore indicates the Laplace transform of the variable.
Elastic Arm
Maxwell Arm
Governing Equation in Laplace Domain

13. Standard Linear Solid – Relaxation Equations
Section 3 – Nonlinear Analysis
Module 4 – Viscoelastic Materials
Page 13
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

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

15. Standard Linear Solid – Creep Equations
Section 3 – Nonlinear Analysis
Module 4 – Viscoelastic Materials
Page 15
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

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

17. Standard Linear Solid - Summary
Section 3 – Nonlinear Analysis
Module 4 – Viscoelastic Materials
Page 17
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)

18. Section 3 – Nonlinear Analysis
Module 4 – Viscoelastic Materials
Page 18
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).
The shear relaxation modulus is used from this point forward since Simulation expects data for the shear relaxation modulus to be entered.
Relaxation Modulus
Relaxation Time

19. is the value of G(t) at time equal to zero.
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. Weichert Model – Multiple Relaxation Times
Section 3 – Nonlinear Analysis
Module 4 – Viscoelastic Materials
Page 20
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.
Example Relaxation Data for a Real Material t1 t2 t3 t4 tn

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

22. Section 3 – Nonlinear Analysis
Module 4 – Viscoelastic Materials
Page 22
Alternate Forms
This form of the equation is used when the relaxation properties are specified in terms of the long term modulus,
This form of the equation is used when the relaxation properties are specified in terms of the instantaneous modulus, G0.

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

24. Volumetric Relaxation Data
Section 3 – Nonlinear Analysis
Module 4 – Viscoelastic Materials
Page 24
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.

25. Example - Sandwich Problem
Section 3 – Nonlinear Analysis
Module 4 – Viscoelastic Materials
Page 25
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 of Sandwich Beam
6061-T6 Aluminum
1/16 in
1/32 in
1/16 in
6061-T6 Aluminum
ISR Industrial Adhesive

26. The beam is modeled using a 2D plane strain representation.
Section 3 – Nonlinear Analysis
Module 4 – Viscoelastic Materials
Page 26
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.
Thickness Direction

27. Example – Beam Geometry
Section 3 – Nonlinear Analysis
Module 4 – Viscoelastic Materials
Page 27
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.

28. Loads and Boundary Conditions
Section 3 – Nonlinear Analysis
Module 4 – Viscoelastic Materials
Page 28
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

29. Section 3 – Nonlinear Analysis
Module 4 – Viscoelastic Materials
Page 29
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
1/16 in
1/32 in
1/16 in
6061-T6 Aluminum
Simson Industrial Adhesive
Mesh absolute element size is 1/64th of an inch.

30. Element Definition: Adhesive
Section 3 – Nonlinear Analysis
Module 4 – Viscoelastic Materials
Page 30
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.

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

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

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

34. Example - Mooney-Rivlin Properties
Section 3 – Nonlinear Analysis
Module 4 – Viscoelastic Materials
Page 34
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 C10/C01 will be set equal to 4.
These two equations lead to constants of C10 = psi and C01 = 99.1 psi.

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

36. Example - Prony Series Data
Section 3 – Nonlinear Analysis
Module 4 – Viscoelastic Materials
Page 36
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, G0.
Assuming that there is no relaxation of the bulk modulus, the volumetric relaxation data will be set to zeros.

37. Section 3 – Nonlinear Analysis
Module 4 – Viscoelastic Materials
Page 37
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.

38. Computed displacement history at the tip of the cantilever.
Section 3 – Nonlinear Analysis
Module 4 – Viscoelastic Materials
Page 38
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.

39. Section 3 – Nonlinear Analysis
Module 4 – Viscoelastic Materials
Page 39
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.