Nonlinear Analysis:
Viscoelastic Material Analysis
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Section 3 – Nonlinear Analysis
Objectives
Module 4 – Viscoelastic Materials
Page 2

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.





© 2011 Autodesk
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.
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Section 3 – Nonlinear Analysis
Viscoelasticity
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.
© 2011 Autodesk

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.
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Section 3 – Nonlinear Analysis
Time Dependent Responses
Module 4 – Viscoelastic Materials
Page 4
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.
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Section 3 – Nonlinear Analysis
Relaxation Modulus
Module 4 – Viscoelastic Materials
Page 5

When subjected to a constant
strain, the stress in polymers will
relax (i.e. stress will decrease to a
steady state value).
Relaxation Curves for a Linear
Viscoelastic Material
0.009
0.008
In a linear viscoelastic material
the relaxation is proportional to
the applied strain.
0.007
0.006
Stress

0.005
0.004
2 times
2 times
0.003
0.002
2 times
2 times
0.001

The relaxation modulus is defined
as: Tension
Shear
 t 
o
© 2011 Autodesk
 E t 
 t 
o
0
0
1
2
3
4
5
6
Tim e, sec.
 G t 
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Section 3 – Nonlinear Analysis
Creep Compliance
Module 4 – Viscoelastic Materials
Page 6

When subjected to constant
stress, polymers will creep (i.e.
strain will continue to increase to
a steady state value).
Creep Curves for a Linear
Viscoelastic Material
Creep Curves

If the creep response is
proportional to the applied
stress, the material is “linear”.
The creep compliance is defined
by:
 t 
o
© 2011 Autodesk
0.4
0.35
2 times
0.3
Strain, in/in

0.45
 J t 
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2 times
0.25
0.2
0.15
2 times
2 times
0.1
0.05
0
0
1
2
3
4
5
6
Tim e, sec.
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Section 3 – Nonlinear Analysis
Sinusoidal Response
Module 4 – Viscoelastic Materials
Page 7


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 stressstrain curves.
   o cos  t
   o cos  t   


t

The phase angle can be related to
the damping of the material.

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T 
2

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Section 3 – Nonlinear Analysis
Mechanical Element Analogs
Module 4 – Viscoelastic Materials
Page 8
Mechanical elements provide a means to construct potential
viscoelastic material models.

Elastic Element – Stress is
proportional to strain.
  E
E


Viscous Element – Stress is
proportional to strain rate. The
proportionality constant is called
viscosity due to its similarity to a
Newtonian fluid.
© 2011 Autodesk

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 
d
  
dt

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Section 3 – Nonlinear Analysis
Maxwell Model
Module 4 – Viscoelastic Materials
Page 9
 The Maxwell model uses a
spring and dashpot in series.
Derivation of Governing Equation

 The Maxwell model doesn’t
match creep response well.
1
E
 It predicts a linear change in
stress versus time for the

creep response.
2

  E1
  1   2
    2
  1   2
Combining yields
 
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Units are
seconds
E
 
 
© 2011 Autodesk



1



E
  E 
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Section 3 – Nonlinear Analysis
Kelvin Model
Module 4 – Viscoelastic Materials
Page 10
 The Kelvin model uses a spring
Derivation of Governing Equation

and dashpot in parallel.
1
2
 The Kelvin model doesn’t match
relaxation data.

E
 1  E
 2   
  1  2
 It doesn’t exhibit time dependent
relaxation.
 

E
 
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
1

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 


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Section 3 – Nonlinear Analysis
Standard Linear Solid – Governing Equations
Module 4 – Viscoelastic Materials
Page 11
 The Standard Linear
Solid model is a threeparameter model that
contains a Maxwell Arm
in parallel with an
elastic arm.
 Laplace transforms will
be used to develop
relaxation and creep
constitutive equations.
© 2011 Autodesk
Derivation of Governing Equation

  1  2
2
1
Elastic Arm
 1  E r
Em
Er
Maxwell Arm
 
 2
Em


2

Characteristic Time

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 

Em
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Section 3 – Nonlinear Analysis
Standard Linear Solid – Laplace Domain
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
  1  2
Elastic Arm
Maxwell Arm
 1 s   E r  s 
 1  E r
 
 2
Em

2



s


 s   E r  E m
s 1




© 2011 Autodesk
 s    1 s    2 s 
The overscore
indicates the
Laplace transform
of the variable.




s
  s 
 2 s   E m 
s 1 


 


Governing Equation in

 s 
Laplace Domain


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Section 3 – Nonlinear Analysis
Standard Linear Solid – Relaxation Equations
Module 4 – Viscoelastic Materials
Page 13


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
 0    E r  E m  0

At infinite time the stress
relaxes to a steady state value
of
    E 
r
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0
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 t    0 u  t 
 s  
u t  Unit Step
0
Function
s
Substitution into the
governing equation yields


s

 s    E r  E m
s 1





  0
 s

Taking the inverse Laplace
transform yields
t


 t    E r  E m e 

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
 0

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Section 3 – Nonlinear Analysis
Standard Linear Solid – Relaxation Plot
Module 4 – Viscoelastic Materials
Page 14


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.
© 2011 Autodesk
 t 
0
 E t   E r  E m e

t

25
E r  10
20
Stress

E m  10
15
 1
10
5
0
0
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1
2
3
4
5
6
Tim e, sec.
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Section 3 – Nonlinear Analysis
Standard Linear Solid – Creep Equations
Module 4 – Viscoelastic Materials
Page 15


 t    0 u t 
The creep behavior is obtained
by finding the response to a
step change in stress.
 s  
At time t=0, there is an
instantaneous stress response
equal to
At infinite time the strain
grows to a steady state value
of
    C r  0
© 2011 Autodesk
s
Substitution into the
governing equation yields



s


 Er  Em
 s 



s
s 1 
 


0
 0   C g  0

0
Taking the inverse Laplace
transform yields
t




 t    C g  C r  C g  1  e c



Cg 
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1
Er  Em
Cr 
1
Er
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c 

 
 0

Cr

Cg
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Section 3 – Nonlinear Analysis
Module 4 – Viscoelastic Materials
Standard Linear Solid – Creep Plot
Page 16


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.
t



 t 

 J t    C g  C r  C g  1  e c

0


Cg 
1
Er  Em
Cr 
1
Er
c 
Cr





Cg
0.120000
0.100000
The strain creeps to a steady state
value controlled by the parameter
Cr.
0.080000
in/in

0.060000
0.040000
E r  10
0.020000
E m  10
 1
0.000000

Since Cg is greater than Cr the
characteristic creep time is slower
than that for relaxation.
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0
2
4
6
Tim e, sec.
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Section 3 – Nonlinear Analysis
Standard Linear Solid - Summary
Module 4 – Viscoelastic Materials
Page 17
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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Section 3 – Nonlinear Analysis
Wiechert Model
Module 4 – Viscoelastic Materials
Page 18

The Wiechert model is a
generalization of the Standard
Linear Solid model and can be used
to model the viscoelastic response
of many materials.

G

It consists of a linear spring in
parallel with a series of springs and
dashpots (Maxwell elements).

 t   G t  0
G t   G  
i 
© 2011 Autodesk
i
Ei

n

Gie
t
i
Relaxation Modulus
i 1
Relaxation Time
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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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Section 3 – Nonlinear Analysis
Module 4 – Viscoelastic Materials
G 0 and G 
Page 19
is the value of G(t) at time
equal to zero.
 G0
Relaxation function versus time
25
It is the instantaneous shear
modulus.
is the value of G(t) at time
equal to infinity.
 G
G0
20
Stress

15
G
10
5
0
0

It is the final or fully relaxed shear
modulus.
1
2
3
4
5
6
Tim e, sec.
n
G0  G 
G
i
i 1
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Section 3 – Nonlinear Analysis
Weichert Model – Multiple Relaxation Times
Module 4 – Viscoelastic Materials
Page 20

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.
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Example Relaxation Data for
a Real Material
1
2
3
4
n
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Section 3 – Nonlinear Analysis
Prony Series
Module 4 – Viscoelastic Materials
Page 21


The challenge in describing a
material by the Weichert model is
to find the coefficients, Gi and
relaxation times, i, of the Prony
Series.
G t   G  
n
Ge

t
i
i
i 1
Prony Series
Specialized optimization
algorithms are used to determine
the best set of moduli, Gi, and
relaxation times, i , that match
experimental data.
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Section 3 – Nonlinear Analysis
Alternate Forms
Module 4 – Viscoelastic Materials
Page 22
This form of the equation is used
when the relaxation properties are
specified in terms of the long term
modulus, G  .
G t   G  

n

Gie
n
i 1
G t   G  
n
Ge

G
i
i 1
t
i
i
n
G  G0   Gi
i 1
i 1
t
i




G t    G 0 


G t   G 0 
This form of the equation is used
when the relaxation properties are
specified in terms of the
instantaneous modulus, G0.

n

i 1
n

i 1
© 2011 Autodesk

i
e
i
G0  G 
t
i 1

G t   G  1 

n
Gie
t
i




t


i

Gi 1  e



Gie
t
i
i 1




t
n



i

G t   G 0 1    i 1  e


i 1

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
n




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n

i
1
i 1
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Autodesk Simulation Multiphysics Material
Data Screen
Section 3 – Nonlinear Analysis
Module 4 – Viscoelastic Materials
Page 23
The instantaneous form of the
relaxation modulus equation is used.
t
n




G t   G 0 1    i  1  e i


i 1





(Mooney-Rivlin)
Defines the instantaneous
shear modulus G 0  2 C 10  C 01 
First
Constant
© 2011 Autodesk
Second
Constant
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Section 3 – Nonlinear Analysis
Volumetric Relaxation Data
Module 4 – Viscoelastic Materials
Page 24

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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Section 3 – Nonlinear Analysis
Example - Sandwich Problem
Module 4 – Viscoelastic Materials
Page 25




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.
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Section of Sandwich Beam
6061-T6 Aluminum
1/16 in
1/32 in
1/16 in
6061-T6 Aluminum
ISR 70-03 Industrial Adhesive
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Section 3 – Nonlinear Analysis
Example – 2D Model
Module 4 – Viscoelastic Materials
Page 26

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.
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Thickness Direction
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Section 3 – Nonlinear Analysis
Example – Beam Geometry
Module 4 – Viscoelastic Materials
Page 27

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 6061T6 aluminum.

The adhesive layer (shown in
blue) is 1/32 inch thick.
© 2011 Autodesk
Portion of the Inventor model
of the sandwich beam.
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Section 3 – Nonlinear Analysis
Loads and Boundary Conditions
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
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1 lb divided among
21 nodes
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Section 3 – Nonlinear Analysis
FEA Model
Module 4 – Viscoelastic Materials
Page 29


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 70-03 Industrial Adhesive
Mesh absolute element size is
1/64th of an inch.
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Section 3 – Nonlinear Analysis
Element Definition: Adhesive
Module 4 – Viscoelastic Materials
Page 30





A viscoelastic Mooney-Rivlin
Material is selected.
This will give a nonlinear stressstrain 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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Section 3 – Nonlinear Analysis
Example – Material Properties
Module 4 – Viscoelastic Materials
Page 31
Tension relaxation properties for
ISR 70-03 adhesive are given in
the referenced document.
E   4 . 101 Mpa
1
Mpa
Sec.
0 . 378
1 . 60 x10
2
0 . 427
2 . 91 x10
3
0 . 523
9 . 965
4
0 . 773
4 . 14 x10
5
1 . 014
2 . 01 x10
6
1 . 521
1 . 68 x10
7
2 . 501
2 . 44 x10
8
3 . 364
4 . 82 x10
9
5 . 904
9 . 83 x10
© 2011 Autodesk
3
2
1
2
3
4
5
6
E t   E  

9

Eie
t
i
i 1
Relaxation Modulus
20
Extension Relaxation Moduls [Mpa]

1.E-06
18
16
14
12
10
8
6
4
2
1.E-04
1.E-02
0
1.E+00
1.E+02
Log(time) [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) 2783-2796.
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Section 3 – Nonlinear Analysis
Example - Shear Relaxation Properties
Module 4 – Viscoelastic Materials
Page 32



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.
G 

E
2 1   

E
3
The shear relaxation data is
obtained by dividing the tension
data by three.
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Shear Relaxation Data
G   1 . 367 Mpa
Mpa
Sec.
1
0 . 126
1 . 60 x10
2
0 . 142
2 . 91 x10
3
0 . 174
9 . 965
4
0 . 258
4 . 14 x10
5
0 . 338
2 . 01 x10
6
0 . 507
1 . 68 x10
7
0 . 834
2 . 44 x10
8
1 . 121
4 . 82 x10
9
1 . 968
9 . 83 x10
3
2
1
2
3
4
5
6
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Section 3 – Nonlinear Analysis
Example - Instantaneous Form
Module 4 – Viscoelastic Materials
Page 33

The shear relaxation data will
be entered into the Simulation
Prony series table using the
instantaneous option.
Instantaneous Shear Modulus
Relaxation Data
G 0  6 . 835 Mpa  991 . psi
n
G0  G 
G
i
 6 . 835 MPa

Sec.
1
0 . 01843
1 . 60 x10
2
0 . 02082
2 . 91 x10
3
0 . 02550
9 . 965
4
0 . 03770
4 . 14 x10
5
0 . 04945
2 . 01 x10
6
0 . 07417
1 . 68 x10
7
0 . 1220
2 . 44 x10
8
0 . 1640
4 . 82 x10
9
0 . 2879
9 . 83 x10
3
2
i 1
n




G t   G 0 1    i  1  e i


i 1

t
© 2011 Autodesk




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1
2
3
4
5
6
Education Community
Section 3 – Nonlinear Analysis
Example - Mooney-Rivlin Properties
Module 4 – Viscoelastic Materials
Page 34

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 yintercept of a Mooney curve.

As an approximation, the ratio of
C10/C01 will be set equal to 4.
G 0  2 C 10  C 01   941 ps i
© 2011 Autodesk
C 10
These two equations lead to constants of
C10 = 396.4 psi and C01 = 99.1 psi.
4
C 01
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Section 3 – Nonlinear Analysis
Example - Bulk Modulus
Module 4 – Viscoelastic Materials
Page 35
The bulk modulus will be
approximated from the equation
 
2 G 0 1  
3 1  2


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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Section 3 – Nonlinear Analysis
Example - Prony Series Data
Module 4 – Viscoelastic Materials
Page 36

The alpha constants and
relaxation times are entered in
the Prony series table for the
Deviatoric Relaxation data.

Note the alpha constants are nondimensional 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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Section 3 – Nonlinear Analysis
Analysis Parameters
Module 4 – Viscoelastic Materials
Page 37






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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Section 3 – Nonlinear Analysis
Example - Results
Module 4 – Viscoelastic Materials
Page 38
 The plot shows the computed
displacement history for the tip of
the cantilever.
Computed displacement history at 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.
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Section 3 – Nonlinear Analysis
Module Summary
Module 4 – Viscoelastic Materials
Page 39





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.
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