The goal of present article is to simulate the behavior of viscoelastic materials. By using ABAQUS software, we can simulate the behavior of viscoelastic materials, conditional to calculate storing shear module and wasting module of viscoelastic materials by the test. First, we give descriptions required parameters such as shear module and loss factor in order to anticipate the behavior of viscoelastic materials. Then, the measurement of each above variable will be illustrated by test. To make use of the result from test, related theories should be explained in brief. Finally, resultant variables are used to simulate in ABAQUS software. The results of test are compared with software results. This comparison indicates that the results are well consistent

Mehrdad Nasirshoaibi

Nader Mohammadi

English

Mathematical Models in Engineering

  © JVE INTERNATIONAL LTD. ISSN PRINT 2351-5279, ISSN ONLINE 2424-4627, KAUNAS, LITHUANIA 67 
9. Simulation of viscoelastic materials by ABAQUS 
Nader Mohammadi1, Mehrdad Nasirshoaibi2 
Department of Mechanical Engineering, Parand Branch, Islamic Azad University, Tehran, Iran 
1Corresponding author 
E-mail: 1nmohamady@ut.ac.ir, 2mehrdadnasirshoaibi@gmail.com 
(Received 11 May 2015; received in revised form 23 July 2015; accepted 11 August 2015) 
Abstract. The goal of present article is to simulate the behavior of viscoelastic materials. By using 
ABAQUS software, we can simulate the behavior of viscoelastic materials, conditional to 
calculate storing shear module and wasting module of viscoelastic materials by the test. First, we 
give descriptions required parameters such as shear module and loss factor in order to anticipate 
the behavior of viscoelastic materials. Then, the measurement of each above variable will be 
illustrated by test. To make use of the result from test, related theories should be explained in brief. 
Finally, resultant variables are used to simulate in ABAQUS software. The results of test are 
compared with software results. This comparison indicates that the results are well consistent. 
Keywords: simulation, viscoelastic materials, ABAQUS software, vibration level. 
1. Introduction 
Noise and vibrations are an environmental topic we are always considered to decrease or 
remove them. In cars, the most common method is adding passive dissipative materials to the 
body in order to reduce noise and vibration level. Anyway, prediction of noise and vibration is 
substantial.  
Body structures for, e.g., automotive, space, aeronautical and naval applications are, to a large 
extent, consisting of thin walled structures and corresponding finite element models are most often 
built from shell elements. Although modelling with three dimensional solid elements is becoming 
more and more common for thin-walled structures, thanks to the extra-ordinary development of 
computer hardware capacity during a number of years, engineers will never have unlimited 
resources in terms of degrees-of-freedom to spend on a finite element model. Therefore, there will 
always be a tradeoff between the simplicity in modelling using solid modelling and the extra 
number of degrees-of-freedom it will cause. 
It is found from a continuous cooperation over several years with the large and well known 
car manufacturers, that today, within the automotive industry, shell modelling is the totally 
dominating finite element modelling (FEM) technique on the car “body-in white” level. Thus, any 
investigation on the effect of “applied damping material” (ADM) to the structure will have to cope 
with the fact that the existing base structure FE model is made up of shell elements. Also, almost 
the whole community of car manufacturers reports large difficulties in how to accurately enough 
model viscoelastic materials, and to get correlation with measurements. An easy-to-use and 
accurate material model which can be communicated using only a few parameters is therefore key 
to an improvement within this field. 
One of influential way to reduce vibrational energy is using viscoelastic materials as 
constrained layer [1, 2]. Loss factor of viscoelastic materials and also the stiffness of constrictive 
layer attached to the structure are very effective to reduce vibrational energy. Thus, when the main 
structure is vibrated abundantly, viscoelastic layer is subjected to a huge transformation and its 
kinetic energy will be converted to thermal.  
Several ways have been presented to show the behavior of viscoelastic materials in the 
ambience of vibration absorbent layers since 1950. One of the ways to simulate the behavior of 
viscoelastic materials is modeling shear modules by a series of small damping oscillators. 
According to the same idea, two scientists performed transient and steady state analysis of 
sandwich panel. The intersection of all recent studies is that parameters in the selected model are 
accurately known or estimated.  
There are two basic information in linear viscoelastic, especially in dynamic behavior of semi-
9. SIMULATION OF VISCOELASTIC MATERIALS BY ABAQUS.  
NADER MOHAMMADI, MEHRDAD NASIRSHOAIBI 
68 MATHEMATICAL MODELS IN ENGINEERING. DECEMBER 2015, VOLUME 1, ISSUE 2  
elastics: real and imaginary part of shear module. These variables are severely count on 
temperature and frequency. Nashif found some information about how to use one-end involved 
beam for extracting mixed and frequency-depended shear module for different temperatures of a 
viscoelastic material [3, 4]. The results can be found in Standard ASTM E756-98. The results of 
these standards have been used in the article [5]. 
2. Viscoelasticity as reduced fractions 
Rubber with nitrile (Acrylic nitrile butadiene) base is one of the most common rubbers. The 
rubber studied in this example contains 35 % of Black Carbon. Rubber has been stock to the steel 
plate. In the process of sticking them, temperature and pressure must be high enough; rubber must 
be also already attached to the steel by a thin layer of glue. 
Rubber with nitrile base is basically a viscoelastic and quite incompressible material. 
Viscoelastic behavior is observed when shear module of material is a function of frequency as 
Fig. 1. Poisson’s coefficient of this material has been achieved with high accuracy of 0.4993. It 
has been assumed in this model that strains are little and linear. Thin layer of rubber has been 
constrained by metal plates which are good thermal conductors. The assumption of being 
isothermal is also valid for the rubber. Elastic behavior (in isothermal and isotropic conditions) 
has been modeled by linear viscoelastic model. However, using classic model of viscoelastic 
materials in which internal variables are first grade according to the variations law, many internal 
variables and consequently many parameters are needed to describe viscoelastic behavior. The 
numbers of required parameters are significantly reduced by using reduced fractions. Linear 
reduced fractions are appropriately consistent with empirical results for a wide range of 
frequencies and polymers. But this consistency is in single-axis state. 
 
Fig. 1. Simple transformations of materials 
Shear stress equation ߬(ݐ) can be written as two Couple equations: 
߬(ݐ) = ܩ଴ߛ(ݐ) − (ܩ଴ − ܩஶ)ߛ௏, ܦఈ +
1
߬∗ఈ ߛ
௏ = 1߬∗ఈ ߛ
௏, ߛ௏ = 0, (1)
where ߛ is the (macroscopic) shear while ߛ௏ is an internal variable of shear type representing a 
distribution of irreversible microstructural processes in the material, ܩ଴ > 0 and ܩஶ > 0  are 
elastic constants which can be identified as instantaneous and longtime shear modulus: 
ܦఈݕ(ݐ) = 1Γ(1 − ߙ)
݀
݀ݐ ቈන
ݕ(ݏ)
(ݐ − ݏ)ఈ ݀ݏ
௧
଴
቉, (2)
9. SIMULATION OF VISCOELASTIC MATERIALS BY ABAQUS.  
NADER MOHAMMADI, MEHRDAD NASIRSHOAIBI 
 © JVE INTERNATIONAL LTD. ISSN PRINT 2351-5279, ISSN ONLINE 2424-4627, KAUNAS, LITHUANIA 69 
where ܦఈ  is the fractional derivative operator of order ߙ ∈ (0, 1) . Using Fourier transform, 
operator ܦ is: 
ܨ[ܦఈݕ(ݐ)](߱) = (݅߱)ఈܨ[ݕ(ݐ)](߱). (3)
Using Eqs. (1) and (3), the Eq. (4) will be obtained: 
ݐ(߱) = ܩ∗(߱)ߛ(߱), ܩ∗(߱) = [ܩ௦(߱) + ݅ܩ௟(߱)] = ܩ଴
ܩ଴ − ܩஶ
1 + [ݐ∗(݅߱)]ఈ, (4)
where ݐ∗ relaxation constant and ܩ∗ is denoted the dynamic shear modulus. 
Using ASTM E756-98 standard, real and imaginary part of shear module can be obtained [5]. 
Real and imaginary part of shear module is a function of temperature and frequency shown 
practically by a one-end involved beam in Fig. 2. A non-contact electromagnetic stimulation has 
been placed at the free-end of beam. The how to perform the test is presented in ASTM E756-98 
[5]. A piezoelectric crystal collects system response at the tingly end. Temperature is increased 
from –30 to 180 degrees within 10 degrees’ steps. When the sample temperature reached to steady 
state, all tests will be performed. The required time to achieve desired temperature at each step is 
about 30 minutes. The length of the beam used for each test is 15 cm and it width 25.4 mm. 
 
Fig. 2. The beam used for test 
Shear module, obtained from test results, has two real and imaginary parts. Real and imaginary 
parts have to be separated. Real and imaginary parts are separated in the Eq. (5): 
ܩ଴
ቂ1 + cos ቀߨߙ2 ቁቃ (ܩ଴ − ܩஶ)(߱ݐ)ఈ
2 ቂ1 + cos ቀߨߙ2 ቁ (߱ݐ)ఈቃ
+ ݅
sin ቀߨߙ2 ቁ (ܩ଴ − ܩஶ)(߱ݐ)ఈ
2 ቂ1 + cos ቀߨߙ2 ቁ (߱ݐ)ఈቃ
⋅ (5)
In order to achieve coefficients of the curve easily, it is necessary to define a variable like ܤ 
instead of cos(ߨߙ/2) in real part of Eq. (5), and then Eq. (6) is given: 
ܩ଴
[1 + ܤ](ܩ଴ − ܩஶ)(߱ݐ)ఈ
2[1 + ܤ(߱ݐ)ఈ] . (6)
Fig. 1 shows the results from test. To achieve the parameters of Eq. (6), minimum squares 
method must be used to have the least error for passing the curve. To do so, MATLAB software 
was used. Finally, the following values were obtained: 
ܩ଴ = 414 MPa,   ܩஶ = 4 MPa, ݐ∗ = 3.31 × 10ି଺s, ߙ = 0.6. (7)
9. SIMULATION OF VISCOELASTIC MATERIALS BY ABAQUS.  
NADER MOHAMMADI, MEHRDAD NASIRSHOAIBI 
70 MATHEMATICAL MODELS IN ENGINEERING. DECEMBER 2015, VOLUME 1, ISSUE 2  
3. Viscoelastic materials and ABAQUS 
ABAQUS software investigates viscoelastic materials’ behavior in two main fields: time and 
frequency. In time field, several transformations of these materials are investigated, and in 
frequency field, the behavior of materials depending on the frequency. 
3.1. Viscoelastic theory in ABAQUS 
Consider a shear test with small strain; this shear strain varies frequently by time: 
ߛ(ݐ) = ߛ଴exp(݅߱ݐ). (8)
It is assumed that this viscoelastic piece has been under oscillation for a long time, and reached 
to a steady state: 
߬(ݐ) = [ܩ௦(߱) + ݅ܩ௟(߱)]ߛ଴exp(݅߱ݐ). (9)
This module (mixed) can be dimensionless first by the help of the Eq. (1), and then Fourier 
transform: 
݃(ݐ) = ܩோ(ݐ)ܩஶ − 1, (10)
and: 
ܩ௦(߱) = ܩஶ[1 − ߱ℑ݃∗(߱)], ܩ௟(߱) = ܩஶ[߱ℜ݃∗(߱)]. (11)
When there is a multi-axis stress in the model, ABAQUS assumes that shear module and 
volume module are independent. The behavior of volume module is defined by two parts of storing 
volume module ܭ௦(߱) and wasting volume module. A shear module similar to this model’s 
module can be dimensionless by the help of Fourier transform ݇∗(߱) and time-dependent volume 
module ܭோ(௧): 
ܭ௦(߱) = ܭஶ[1 − ߱ℑ݇∗(߱)], ܭ௟(߱) = ܭஶ[߱ℜ݇∗(߱)]. (12)
3.2. Defining viscoelastic materials properties in ABAQUS 
There are 4 ways to define viscoelastic materials’ properties in frequency field, including: 
1 – Determining the coefficients of Peron series; 
2 – Results of creeping test; 
3 – Results of releasing test; 
4 – Determining the frequency-dependent coefficients. 
Determining the frequency-dependent coefficients is categorized into two parts of Formula 
and Tabular methods. 
4. Relations of formula method 
The relations of this part are as follows: 
݃∗ = ଵ݃∗݂ି௔,   ݇∗ = ݇ଵ∗݂ି௔. (13)
5. Relations of tabular method 
The relations of this part are as follows: 
9. SIMULATION OF VISCOELASTIC MATERIALS BY ABAQUS.  
NADER MOHAMMADI, MEHRDAD NASIRSHOAIBI 
 © JVE INTERNATIONAL LTD. ISSN PRINT 2351-5279, ISSN ONLINE 2424-4627, KAUNAS, LITHUANIA 71 
߱ℜ݃∗(߱) = ܩ௟ܩஶ ,   ߱ℑ݃
∗(߱) = 1 − ܩ௦ܩஶ , ߱ℜ݇
∗(߱) = ܭ௟ܭஶ , ߱ℑ݇
∗(߱) = 1 − ܭ௦ܭஶ. (14)
6. Simulation of a real example in ABAQUS software 
In this section, we simulate the beam of Fig. 2. In Fig. 2, steel plates are placed in up and down, 
each of them with thickness of 0.5 mm. They are drawn in Part module of ABAQUS software as 
Shell. There is viscoelastic material between two mentioned plates, which is made of NBR with 
thickness of 0.1 mm. In Part module, this part is drawn as Solid. 
In Table 1, the properties of elastic section for steel and viscoelastic layer are presented.  
In load module, we apply 1 N load on the beam end, and close all freedom degrees at the other end. 
Table 1. The properties of elastic part 
 Poison’s coefficient Yang module (GPa) Density (kg/m3) 
Steel 0.3 210 7800 
Layer NBR 0.4939 0.002757904 1250 
7. Finding 
In the following, the results of test are compared with software results. Fig. 3 indicates that the 
results are well consistent. 
 
Fig. 3. Analytical solution results (dotted mode of answer for tingly support) 
8. Conclusion 
By using ABAQUS software, we can simulate the behavior of viscoelastic materials, 
conditional to calculate storing shear module and wasting module of viscoelastic materials by the 
test. Table method is the best choice among them. The results of test which are compared with 
software results indicates that the results are well consistent. 
References 
[1] Alvelid M., Enelund M. Modeling of constrained thin rubber layer with emphasis on damping. Journal 
of Sound and Vibration, Vol. 300, Issues 3-5, 2007, p. 662-675. 
[2] Meunier M., Shenoi R. A. Forced response of FRP sandwich panels with viscoelastic materials. 
Journal of Sound and Vibration, Vol. 263, Issue 1, 2003, p. 131-151. 
[3] Nashif A. D., Jones D. I. G., Henderson J. P. Vibration Damping. Wiley, New York. 
[4] Jones D. I. G. Handbook of Viscoelastic Vibration Damping. John Wiley and Sons. 
[5] ASME Standard, ASME Manual MSnc, An ASME Paper. The American Society of Mechanical 
Engineers, New York, http://www.asme.org/pubs/MSc.html. 


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Mathematical Models in Engineering