Nonlinear Analysis: Viscoelastic Material Analysis

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

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Nonlinear Analysis:

Viscoelastic Material Analysis

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.

Section 3 – Nonlinear Analysis

Module 4 – Viscoelastic Materials

Page 3

Viscoelasticity
• 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.
• 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.

Section 3 – Nonlinear Analysis

Module 4 – Viscoelastic Materials

Page 4

Time Dependent Responses

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.

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

Shear

Tension

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

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

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.

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

Section 3 – Nonlinear Analysis

Module 4 – Viscoelastic Materials

Page 10

Kelvin Model

Derivation of Governing Equation

• 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.

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

Section 3 – Nonlinear Analysis

Module 4 – Viscoelastic Materials

Page 12

Standard Linear Solid – Laplace Domain

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

Section 3 – Nonlinear Analysis

Module 4 – Viscoelastic Materials

Page 13

Standard Linear Solid – Relaxation Equations

Unit Step Function

• 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

Substitution into the governing equation yields

Taking the inverse Laplace transform yields

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.

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

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.

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)

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

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

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

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

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.

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

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.

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

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

Section 3 – Nonlinear Analysis

Module 4 – Viscoelastic Materials

Page 27

Example – Beam Geometry

Portion of the Inventor model of the sandwich beam.

• 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 6061-T6 aluminum.
• The adhesive layer (shown in blue) is 1/32 inch thick.

Section 3 – Nonlinear Analysis

Module 4 – Viscoelastic Materials

Page 28

• 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 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

Mesh absolute element size is 1/64th of an inch.

Section 3 – Nonlinear Analysis

Module 4 – Viscoelastic Materials

Page 30

• 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 31

Example – Material Properties
• Tension relaxation properties for ISR 70-03 adhesive are given in the referenced document.

Sec.

Mpa

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) 2783-2796.

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

Sec.

Mpa

Section 3 – Nonlinear Analysis

Module 4 – Viscoelastic Materials

Page 33

Example - Instantaneous Form

Instantaneous Shear Modulus Relaxation Data

• The shear relaxation data will be entered into the Simulation Prony series table using the instantaneous option.

Sec.

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-Rivlinhyperelastic 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 = 396.4 psi and C01 = 99.1 psi.

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 0.499 will be assumed, which results in a bulk modulus of approximately 496,000 psi.

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.

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 0.002 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 38

Example - Results

Computed displacement history at the tip of the cantilever.

• 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.

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 SimulationMultiphysics 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.