Nonlinear Analysis: Viscoelastic Material Analysis

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Nonlinear Analysis:Viscoelastic Material Analysis


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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


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ViscoelasticitySection 3 – Nonlinear Analysis


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 ViscoelasticityThe relaxation and creep functions are a function only of time.

Nonlinear ViscoelasticityThe relaxation and creep functions are a function of both time and stress or strain.


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Time Dependent ResponsesSection 3 – Nonlinear Analysis


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.


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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:



0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0 1 2 3 4 5 6

Time, sec.

Stre

ss

Relaxation Curves for a Linear Viscoelastic Material

2 times2 times

2 times2 times

tEt

o

Tension

tGt

o

Shear


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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

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0 1 2 3 4 5 6

Time, sec.

Stra

in, i

n/in

Creep Curves for a Linear Viscoelastic Material

tJt

o

2 times2 times

2 times2 times


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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

2

T

to cos

to cos


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Mechanical Element AnalogsSection 3 – Nonlinear Analysis


Mechanical elements provide a means to construct potential viscoelastic material models.

E

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.

E

dtd


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E

1

2



Maxwell Model

2

1

E

21

21

E

E

E

1

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.

Combining yields

Units are seconds

Derivation of Governing Equation


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Kelvin ModelSection 3 – Nonlinear Analysis


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.

E

1 2

2

1 E

21

E

1



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Standard Linear Solid – Governing EquationsSection 3 – Nonlinear Analysis


rEmE

1 2Elastic Arm

Maxwell Arm

Characteristic Time


21

rE1

22

mE

mE

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.


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Standard Linear Solid – Laplace Domain

Laplace DomainTime Domain

Governing Equation in Laplace Domain



It is easier to determine the governing equation in the Laplace domain than in the time domain.

21

rE1

22

mE

Elastic Arm

Maxwell Arm

sss 21

sEs r 1

ss

sEs m

12

sssEEs mr

1

The overscore indicates the Laplace transform of the variable.


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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



00 mr EE

0 rE

s

s

tut

0

0

ss

sEEs mr0

1

0

t

mr eEEt

tu


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Standard Linear Solid – Relaxation Plot

The relaxation modulus, E(t), is shown in the figure.

The values chosen for the parameters Er, Em, and are for demonstration purposes only.

The stress relaxes to a steady state value controlled by the parameter Er.

0

5

10

15

20

25

0 1 2 3 4 5 6

Time, sec.

Stre

ss



t

mr eEEtEt

0

11010

m

r

EE


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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



00 gC

0 rC

s

s

tut

0

0

sssEE

s mr

1

0

01

c

t

grg eCCCt

mrg EEC

1

rr EC 1

g

rc CC


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Standard Linear Solid – Creep Plot

0.000000

0.020000

0.040000

0.060000

0.080000

0.100000

0.120000

0 2 4 6

Time, sec.

in/in



11010

m

r

EE

c

t

grg eCCCtJt

1

0

The creep compliance modulus, J(t), is shown in the figure.

The values chosen for the parameters Er, Em, and 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.

mrg EEC

1r

r EC 1

g

rc CC


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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)




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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



0 tGt

i

tn

iieGGtG

1

i

ii E

Relaxation Modulus

G

The shear relaxation modulus is used from this point forward since Simulation expects data for the shear relaxation modulus to be entered.


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GG 0 and

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.

0G

G

0

5

10

15

20

25

0 1 2 3 4 5 6

Time, sec.

Stre

ss

0G

G

Relaxation function versus time

n

iiGGG

10


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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.

3

12

4

n



Example Relaxation Data for a Real Material


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Prony Series

The challenge in describing a material by the Weichert model is to find the coefficients, Gi and relaxation times, i, of the Prony Series.

Specialized optimization algorithms are used to determine the best set of moduli, Gi, and relaxation times, i , that match experimental data.

Prony Series



i

tn

iieGGtG

1


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Alternate FormsSection 3 – Nonlinear Analysis


i

tn

iieGGtG

1

n

iiGGG

10

n

iiGGG

10

i

ii

tn

ii

tn

ii

n

i

t

i

eGGtG

eGeGGtG

11

0

110

i

tn

ii eGtG 11

10 1

1

n

ii

i

i

tn

ii

tn

ii

eGtG

eGGtG

1

1

1

This form of the equation is used when the relaxation properties are specified in terms of the instantaneous modulus, G0.

This form of the equation is used when the relaxation properties are specified in terms of the long term modulus, .G


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Autodesk Simulation Multiphysics Material Data Screen

Defines the instantaneous shear modulus 01100 2 CCG

First Constant

Second Constant

i

tn

ii eGtG 11

10

The instantaneous form of the relaxation modulus equation is used.



(Mooney-Rivlin)


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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.




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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

6061-T6 Aluminum

ISR 70-03 Industrial Adhesive

1/16 in

1/16 in

1/32 in


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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


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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 6061-T6 aluminum.

The adhesive layer (shown in blue) is 1/32 inch thick.

Portion of the Inventor model of the sandwich beam.




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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




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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 Beam6061-T6 Aluminum

6061-T6 Aluminum

Simson 70-03 Industrial Adhesive

1/16 in

1/16 in

1/32 in

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




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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.




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Example – Material Properties

Tension relaxation properties for ISR 70-03 adhesive are given in the referenced document.



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.

Reference

6

5

4

3

2

1

2

3

1083.9904.591082.4364.381044.2501.271068.1521.161001.2014.151014.4773.04

965.9523.031091.2427.021060.1378.01

xxxxxx

xx

Mpa Sec.Mpa 101.4E

1E-05 1E-04 1E-03 1E-02 1E-01 1E+00 1E+01 1E+02 1E+030

2

4

6

8

10

12

14

16

18

20

Relaxation Modulus

Log(time) [sec]

Exte

nsio

n R

elax

atio

n M

odul

s [M

pa]

i

t

iieEEtE

9

1


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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.



312EEG

6

5

4

3

2

1

2

3

1083.9968.191082.4121.181044.2834.071068.1507.061001.2338.051014.4258.04

965.9174.031091.2142.021060.1126.01

xxxxxx

xx

Mpa Sec.Mpa 367.1G

Shear Relaxation Data


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Example - Instantaneous Form

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

i

tn

ii eGtG 11

10

MPa 835.61

0

n

iiGGG

6

5

4

3

2

1

2

3

1083.92879.091082.41640.081044.21220.071068.107417.061001.204945.051014.403770.04

965.902550.031091.202082.021060.101843.01

xxxxxx

xx

Sec.

psi Mpa .G 991 835.60

Instantaneous Shear Modulus Relaxation Data




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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.



i ps9412 01100 CCG 401

10 CC

These two equations lead to constants of C10 = 396.4 psi and C01 = 99.1 psi.


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Example - Bulk Modulus

The bulk modulus will be approximated from the equation



213

12 0

G

For an incompressible material =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.


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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.




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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.




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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.




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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.



Nonlinear Analysis: Viscoelastic Material Analysis

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