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Ingeniería, investigación y tecnología

versión On-line ISSN 2594-0732versión impresa ISSN 1405-7743

Ing. invest. y tecnol. vol.21 no.2 Ciudad de México abr./jun. 2020  Epub 13-Nov-2020

https://doi.org/10.22201/fi.25940732e.2020.21n2.014 

Artículos

Prony series calculation for viscoelastic behavior modeling of structural adhesives from DMA data

Cálculo de Series de Prony a partir de datos de DMA para modelado de comportamiento viscoelástico de adhesivos estructurales

Manuel Alejandro Tapia-Romero1 
http://orcid.org/0000-0003-0366-1165

Mariamne Dehonor-Gómez2 
http://orcid.org/0000-0002-8414-8007

Luis Edmundo Lugo-Uribe3 
http://orcid.org/0000-0003-0573-7098

1Posgrado CIATEQ A.C. Consejo Nacional de Ciencia y Tecnología, Manufactura Avanzada, México. E-mail: aletstapia@gmail.com

2Posgrado CIATEQ A.C. Consejo Nacional de Ciencia y Tecnología, Manufactura Avanzada, México. E-mail: mariamne.dehonor@ciateq.mx

3Posgrado CIATEQ A.C. Consejo Nacional de Ciencia y Tecnología, Manufactura Avanzada, México. E-mail: luis.lugo@ciateq.mx


Abstract

In product design is important to choose the correct material for a specific application. Viscoelastic behavior let us know how much energy the material can dissipate on its internal structure or either return it to the surroundings, and the property that describe this is the Complex Modulus G*, it is a complex quantity that can be separated in a real and an imaginary part called G' storage modulus and iG'' loss modulus respectively. These properties can be measured experimentally from a small material sample easily by performing Dynamical Mechanical Analysis (DMA). In Product Design process there are both, computational and physical validations and there is the need of improving computational studies by understanding the physics of each component. Viscoelastic characteristics of materials can be represented by Prony series, also known as relaxation modulus in function of time. Relaxation modulus can be defined in most of Computer Aided Engineering (CAE) Software. In this article the procedure for calculating Prony Series from DMA data will be explained.

Keywords: Prony series; viscoelasticity; complex modulus; storage modulus; loss modulus; DMA

Resumen

En el diseño de un producto es importante elegir el material correcto para una aplicación específica. El comportamiento viscoelástico nos muestra qué tanta energía puede disipar un material en su estructura interna, o bien, devolver a sus alrededores; la propiedad que describe esto es el módulo complejo G*, que es una cantidad compleja que se puede separar en una parte real y una imaginaria llamada G' módulo de almacenamiento e iG'' módulo de pérdida, respectivamente. Estas propiedades se pueden medir experimentalmente de una pequeña muestra de material, aplicando Análisis Mecánico Dinámico (DMA). En el proceso de diseño de un producto existen las validaciones computacionales y las validaciones físicas, por lo que existe la necesidad de mejorar los estudios computacionales mediante el entendimiento de la física de cada componente. Las características viscoelásticas de los materiales se pueden representar por Series de Prony, también conocidas como el módulo de relajación en función del tiempo. El módulo de relajación se puede definir en la mayoría de los paquetes de Software para Ingeniería Asistida por Computadora (CAE). En este artículo se explicará el procedimiento para calcular las Series de Prony de datos obtenidos con DMA.

Descriptores: Serie de Prony; viscoelasticidad; módulo complejo; módulo de almacenamiento; módulo de pérdida; DMA

Introduction

In this article the method to calculate Prony series from DMA data will be explained with a real example. The Prony series of a piece of structural adhesive used in automotive industry will be obtained showing in detail the considerations and background of each decision taken during the analysis with the purpose of being a good reference for the reader. The motivation for writing this article is the lack of literature available on this subject.

Background on this article analysis is as follows: Automotive body is manufactured from steel or aluminum plates (Davies, 2003). Structural adhesives are used to fill the cavities between these plates and therefore increasing the strength of the structure. Normally they are installed before paint process as it is shown in Figure 1, where they will be cured with the furnace temperature. The installation process consists on place them on the surface of a metal plate in order they adhere by themselves to it. This process is most of the times done manually by an operator. Some of these adhesives when heated liberate gases that will create small bubbles on them, and they will increase their volumes, this is the expansion mechanism that promotes the filling of the closed section in the body structure as shown in Figure 2.

Figure 1: Structural adhesive before curing process 

Figure 2: Structural adhesive after curing process 

Energy during the impact will be dissipated by the structural adhesives. The properties that describe the energy dissipation capacity of materials are known as dynamic properties. The complex modulus G* is a complex quantity that can be divided in a real part and an imaginary part, called storage modulus G', and the loss modulus G'', respectively. Their relationship is indicated in equation 1.

G*=G'+iG'' (1)

The storage modulus G' describes the capacity of a material of storing and returning energy to the surroundings, and the loss modulus G'' describes the capacity of a material of dissipating energy on its internal structure. A graphical explanation of these concepts are often explained with the Figure 3.

Figure 3: Concept explanation of storage & loss moduli (Saba et al., 2016

Dynamical mechanical analysis

The dynamical properties are obtained by an experimental method called dynamical mechanical analysis, DMA from now on. DMA is performed in a laboratory using a Dynamic Mechanical Analyzer, a device commonly used in the study of polymers rheology. Dynamical properties are studied by applying a cyclic stress into a material sample (Sepe, 1998), it can be applied in different modes: tension, bending, cantilever, dual cantilever, torsional, etc. Often the results are shown as frequency response diagrams, also named frequency sweep, where in the vertical axis are shown the values of Storage Modulus G' and loss Modulus G'', and in the horizontal axis are shown the frequencies where the analysis was performed. Most of the times horizontal axis is presented in a logarithmical scale since it is important to show high and low frequency values. An example of a frequency scan can be observed in Figure 4.

Figure 4: Example of a frequency sweep 

The Dynamical Mechanical Analyzer is a device that has movable and fixed clamps. The fixed clamps remain static while the movable clamp will be displaced up and down by a driveshaft deforming the material in an oscillation pattern as illustrated in Figure 5. The amplitude and frequency of the oscillation is programed based on the characteristics of each material. The mathematical background for the calculation of G' and G'' can be found in (Ferry, 1980).

Figure 5: Dynamical Mechanical Analyzer 

In this work a sample of cured structural adhesive (epoxy) with low expansion capability (around 15 % of volume increase) was evaluated. Frequency sweep from 0.1 to 100 Hz was carried out with a Q800 TA Instrument DMA device using a dual cantilever clamp configuration as is shown in Figure 5. An oscillation amplitude of 5 mm was applied at a constant temperature of 30 °C to obtain the experimental viscoelastic properties. Structural adhesive sample dimensions for dual cantilever DMA test were 35.0x13.1x3.7 mm. These samples were prepared cutting the sample of epoxy adhesive with the required dimensions and cured in a convection furnace at 120 °C for 2 hours. Structural adhesives can go from a 15 % to a 300 % expansion ratio depending on the application and performance required. DMA is performed in order to obtain the values of G', G'' and tanδ that is defined as:

tanδ=G''G' (2)

In a DMA experiment where a cyclic force is applied on a material sample, it can be noticed a phase lag between the applied load curve and the strain curve, as shown in Figure 6. This phase lag is an angle named δ. tan δ is a quantity often used in polymers rheology, also known as damping factor and it is related to the energy that the material is dissipating on its structure.

Figure 6: Explanation of tan δ concept: a) shows the plots of an ideal 100 % elastic material, where both curves are in phase, b) shows the plots of a viscoelastic material where there is a phase lag between both curves 

The method used to get the dynamical properties of the structural adhesive was the following: the imposed deformation mode was dual cantilever, and the size of the sample was 35.0x13.1x3.7mm. The analysis was performed in a range of frequencies that goes from 0.1 to 100 Hz, with 10 measurements on each frequency decade. A decade must be understood as an order of magnitude, for example; from 0.1 to 1 we have 1 decade and from 1 to 10 we have another one. This frequencies range was selected based on vibrations commonly detected in automotive bodies due to the input of pavement irregularities through vehicle’s suspension. That input typically goes from 1 to 2 Hz, as mentioned in (Spinola, 2012), nevertheless it was decided to measure in a wider range in order to have the necessary data to make the curve approximation of G' or G'' as functions of the frequency 𝜔 as it will be exposed later in this work.

It is desirable to have a wide range of measurements along 3 or more frequency decades for any material under study to have a good material characterization. Also it is recommended that the common usage input frequency of the product that is under study is included in this range (in this case the structural adhesive used in automotive body). The advantage of perform testing in the frequencies domain is that it requires less time than carry out them on time domain, since the frequency is an inverse quantity with regards of time. This characterization in the frequencies domain will allow a good curve fitting that will let us know the numerical values that will be finally substituted in the Prony series that are functions in the time domain.

The deformation applied to the sample depends on the nature of the material. One method is to impose a small deformation (less than 1 % of the sample size) and keep increasing it until a constant value of the moduli is obtained.

The measured values obtained by performing DMA to this sample are shown in the Table 1.

Table 1: Data obtained from DMA 

Time Temperature Frequency Storage
Modulus
Loss
Modulus
Tan
Delta
Angular
Strain
min °C Hz MPa MPa %
7.49 30.26 0.1 1941 70.08 0.03611 0.08203
16.95 30.19 0.13 1944 70.48 0.03626 0.082
24.06 30 0.16 1953 69.81 0.03574 0.082
25.81 30 0.2 1961 68.64 0.035 0.08202
27.8 30 0.25 1968 68.17 0.03463 0.08202
32.92 30 0.32 1976 67.41 0.03411 0.08201
34.66 30.01 0.4 1983 67.03 0.03379 0.08202
35.74 30 0.5 1990 66.37 0.03335 0.08202
45.19 30 0.63 1998 65.99 0.03302 0.08201
56.9 30 0.79 2007 65.61 0.0327 0.08201
59.52 30 1 2014 65.14 0.03234 0.08203
60.5 30 1.3 2021 64.65 0.03199 0.08202
61.7 30 1.6 2027 64.54 0.03184 0.08202
62.08 30 2 2033 64.26 0.0316 0.08203
62.32 30 2.5 2039 64.04 0.0314 0.08203
63.33 30 3.2 2046 63.56 0.03106 0.08202
63.7 30.01 5 2058 64.16 0.03118 0.08203
64.68 30 6.3 2064 64.53 0.03127 0.08203
65.88 30 7.9 2069 64.9 0.03136 0.08203
66.26 30 10 2076 65.59 0.0316 0.08203
66.78 30 12.6 2082 66.4 0.0319 0.08203
67.4 30 15.8 2087 67.69 0.03243 0.08203
67.65 30 19 2093 68.91 0.03293 0.08203
67.8 30 25 2100 71.17 0.03389 0.08203
68.32 30 31.6 2106 73.72 0.035 0.08203
68.94 30 39.8 2112 76.72 0.03632 0.08203
69.19 30 50 2119 81.4 0.03842 0.08203
69.34 30 63 2121 85.96 0.04053 0.08203
69.59 30 79.5 2137 97.34 0.04556 0.08203
69.76 30 100 2141 102.3 0.04779 0.08203

Models of viscoelastic behavior

Most of all materials have a viscoelastic behavior and in the case of adhesives it is remarkable due to their polymeric nature. Elastic behavior is modeled by Hooke law that can be represented by the following equations:

σ=Eε (3)

In the case of tensile stress, and:

τ=Gγ (4)

In the case of shearing stress; where:

σ =

tensile stress

E =

Young’s modulus

ε =

tensile strain

τ =

shear stress

G =

shear modulus

γ =

strain by shear

Viscous behavior is modeled by the Newton’s law:

τ=ηγ˙ (5)

Where:

τ =

shear stress

η =

viscosity

γ˙ =

strain rate

In the case of viscoelasticity of polymers, a more complex model is needed because it depends on several facts: the size of the molecules, if they are crosslinked or not (Ferry, 1980), and also stress relaxation due to polymeric chains re-ordering with time. By these reasons, the equation that models viscoelasticity uses a modulus that is a function of time, as follows:

τ=Gtγ˙ (6)

Where G(t) is the stress relaxation modulus of the material.

Ferry (1980) shows the same constitutive equation with the following notation:

σijt=-tGt-t'γ˙ijt'dt' (7)

Where:

σij =

stress tensor

γ˙ij =

strain rate tensor

t' =

all the past times were the integration must be carried out over all of them, this must be considered due to the memory of the material during deformation.

Two basic mechanical models that predict viscoelastic properties of materials are Voigt and Maxwell elements, that are represented by springs and dashpots connected between them, as illustrated in Figure 7. The materials behavior can be described as combinations of Maxwell and Voigt elements connected between them. One simple model is the generalized Maxwell model that is just a set of Maxwell elements connected in parallel.

Figure 7: Basic viscoelasticity modeling elements 

In generalized Maxwell model, the contribution in relaxation modulus of each Maxwell element is given by the equation 8:

Git=Gie-t/τi (8)

Where Gi is the spring constant and τi is the relaxation time of the element, defined in equation 9.

τi=ηi/Gi (9)

Where ηi is the dashpot constant. The relaxation modulus of the complete model is the sum of the relaxation moduli of all Maxwell elements connected in parallel, as follows:

Gt=iGie-t/τi (10)

When t→∞ in the case of materials that have a solid-like behavior, G(t) approaches a finite value Ge. For liquid-like materials, G(t) approaches zero (Ferry, 1980), this is why in the case of structural adhesives, that become solid-like materials after the curing process, the equation 4 can be rewritten as:

Gt=Ge+iGie-t/τi (11)

The equation 11 is also known as the Prony Series (Tschoegl, 1989) of the material, and is useful to describe viscoelastic properties in CAE Software (ABAQUS, 1998).

Using DMA data to get Prony series of structural adhesives

As mentioned previously, it is possible to obtain the values of storage modulus G' and loss modulus G'' in function of the frequency by performing DMA to a sample of material. Also, as exposed by Ferry (1980) & Tschoegl (1989), these quantities are represented as functions of frequency in the following equations:

G'ω=Ge+iGiω2τi21+ω2τi2 (12)

G''ω=iGiωτi1+ω2τi2 (13)

This means that is possible to make a curve approximation in order to know the values of the Gi and τi terms, depending on the number of terms desired for the sum. One theoretical method is exposed by Baumgaertel & Winter in (1989). For practical uses, this can be done using software to approximate the curve to these custom equations. The first step is to decide the number of terms. For this work, we will approximate a curve to the first two terms of the sum shown in equation 12.

G'ω=Ge+G1ω2τ121+ω2τ12+G2ω2τ221+ω2τ22 (14)

In equation 14 there are five unknown values: Ge, G1, G2, τ1 and τ2, these values will be approximated by fitting the curve with the form of equation 14 to the data obtained by DMA on the “Storage Modulus” column in Table 1. Software such as Matlab® can be used for a fast procedure capable to meet the Automotive Industry demands on quick analysis. The command used in Matlab® to make the curve approximation is “cftool” (curve fitting tool). This command will display an interactive fitting tool where the column vectors previously created with the information of the columns “Storage Modulus” and “Frequency” in the Table 1 can be loaded as vertical and horizontal axis respectively. In the case of solid-like materials is convenient to use the storage modulus instead of the loss modulus, for example, in this material it was possible to make the approximation of the curve with 30 points over 3 decades, but in the case of loss modulus the experiment should be made in a wider range of frequencies to get a better approximation of loss modulus curve. The loss modulus values are small in comparison to storage modulus values (by a factor of 30 times approximately). This also can be noticed in tanδ values that approach zero. By this reason it is very difficult to get a good approximation with loss modulus curve, since dominant behavior is described by storage modulus. Only storage modulus curve approximation will be discussed in this work.

The obtained values from storage modulus G'(ω) curve approximation are the following: Ge=1947.00, G1=85.79, G2=97.63, τ1=0.0659, τ2=1.6950, replacing these values in equation 14 it is obtained a function of the frequency ω only, as is shown in equation 15:

  G'ω=1947.00+85.79ω20.065921+ω20.06592+ 97.63ω2(1.695)21+ω2(1.695)2 (15)

By plotting the equation 15 as a function of the frequency 𝜔 is obtained the chart in Figure 8:

Figure 8: Curve approximation of storage modulus to data obtained by DMA 

The values obtained in the curve approximation can also be replaced in the equation 11, in this case for the two first terms of the series, obtaining:

Gt=1947.00+85.79e-t/0.0659+97.63e-t/1.695 (16)

The terms of equation 16 are the Prony series of the structural adhesive, also known as the relaxation modulus of the material in function of time. In Figure 9 is showed a plot of equation 16 as a function of time t, it is possible to observe the behavior of the relaxation modulus, where can be noticed that it’s value when a load have been applied for a long time tends to the value of Ge=1947.00 MPa, as mentioned previously.

Figure 9: Relaxation modulus G(t) in function of time 

It can be confirmed the accuracy of the model since the approximated values of G'(ω) calculated with equation 15 have less than 1 % of error with regards of the data obtained by experimental measurement through DMA experiment. Table 2 shows the error calculation for each DMA data point vs. the calculated value with the approximated function.

Table 2: Error calculation of values measured with DMA vs. values calculated with the fitted function G'(ω) (equation 15) 

Frecuency Calculated storage
modulus with equation 15
Measured storage
modulus
Error=measured value-calculated valuemeasured value
Hz MPa MPa %
0.1 1949.7303 1941 0.4498
0.13 1951.5271 1944 0.3872
0.16 1953.6981 1953 0.0357
0.2 1957.0780 1961 0.2000
0.25 1961.882 1968 0.3107
0.32 1969.2311 1976 0.3426
0.4 1977.8047 1983 0.2620
0.5 1987.9030 1990 0.1054
0.63 1999.1612 1998 0.0581
0.79 2009.9052 2007 0.1448
1 2019.7899 2014 0.2875
1.3 2028.5758 2021 0.3749
1.6 2033.8793 2027 0.3394
2 2038.2660 2033 0.2590
2.5 2041.7259 2039 0.1337
3.2 2045.0396 2046 0.0469
5 2051.6141 2058 0.3103
6.3 2056.2783 2064 0.3741
7.9 2062.2139 2069 0.3280
10 2070.0282 2076 0.2877
12.6 2079.1046 2082 0.1391
15.8 2088.7098 2087 0.0819
19 2096.4331 2093 0.1640
25 2106.6909 2100 0.3186
31.6 2113.6681 2106 0.3641
39.8 2118.8206 2112 0.3229
50 2122.4477 2119 0.1627
63 2124.9605 2121 0.1867
79.5 2126.6367 2137 0.4849
100 2127.7134 2141 0.6206

These results allow us to confirm that model showed in equation 15 is a good prediction for G'(ω) since the error values are below 1 %, and this assure that G(t) values will have good correlation with real behavior.

CAE modeling using Prony series

As mentioned previously, the Prony Series of equation 16, are also known as the discrete relaxation modulus in function of time. When equation 16 is substituted in equation 7, the constitutive equation is completely defined, the resulting equation is:

σij=0t1947.00+85.79e-t-t'/0.0659+97.63e-t-t'/1.695γ˙ijt'dt' (17)

CAE makes possible to incorporate the geometry of the product under analysis by dividing a complex geometry in small elements through the creation of a mesh. Equation 17 will be discretized and solved for each one of the elements of this mesh, since σij is the stress tensor of an infinitesimal element as ilustrated in Figure 10.

Figure 10: Stress tensor components 

The definition of Prony Series may be different on each commercial CAE software package. In the case of Abaqus® when the model is viscoelastic and defined in the time domain, the software displays a table similar to Table 3, where the requested parameters are: gi, ki, and τi. The gi terms are the normalized Prony coefficients for shear (deviatoric) behavior, the ki terms are the normalized Prony coefficients for volumetric behavior, and the τi values are the relaxation times of Prony series. Abaqus® assumes that the frequency dependence of gi and ki is independent. In this work the volume changes in the material will not be considered, therefore the column for ki values will be left blank. The changes in shape are described by the deviatoric behavior, described by gi terms, they are defined as follows (Chen, 2000):

gi=GiG0 (18)

Table 3: Dimensionless Prony series coefficients input in Abaqus® 

g1 k1 τi
0.0403 - 0.0659
0.0458 - 1.695

Where the Gi values are the coefficients of the Prony series, and G0 is the relaxation modulus G(t) evaluated in t=0.

Calculating the values of G0, g1 and g2:

G0=G0=2130.42 MPa

g1=G1G0=85.79 [MPa]2130.42 [MPa]=0.0403

g2=G2G0=97.63 [MPa]2130.42 [MPa]=0.0458

g1 and g2 are also known as the dimensionless Prony series coefficients.

Table 3 shows the input of values in Abaqus®. This table can have as many rows as known terms of the Prony series are available. By filling this table, the viscoelastic characteristics of the material under study are completely defined.

The method to calculate the relaxation modulus in function of time presented in this work has many application possibilities. Examples of other applications in automotive industry that used Prony series to describe viscoelastic behavior are : Prediction of damping of railway sandwich type dampers (Merideno et al., 2014), Modeling of viscoelastic parameters for automotive seating applications (Deng et al., 2003), Stress concentration analysis near holes in viscoelastic bodies (Levin et al., 2013), Analysis of adhesion of thermoplastics to steel (Golaz et al., 2011), Finite element analysis of seat cushion and soft-tissue materials focused on passenger-vehicle occupants comfort (Grujicic et al., 2009), Modeling of composites viscoelastic behavior (Machado et al., 2016; Ishikawa et al., 2018).

Conclusions

Viscoelastic properties of materials applied on CAE studies can provide more accurate results since it is a better approximation to real behavior of the material. The relaxation modulus of the analyzed structural adhesive showed a significant dependence with time since modulus values change from the maximum value in G0=2130.42 MPa to the value of Ge=1949.00 MPa due to stress relaxation. This shows that for vehicle CAE impact evaluation that occurs at a high speed the modulus will have also a high value since the stress relaxation will not occur at this speed. In the case of durability CAE evaluation, where the load is applied in long periods of time or in a time dependant load variation the stress relaxation will occur, therefore the modulus value will achieve the Ge value. Viscoelastic properties of structural adhesives can be incorporated to CAE by estimating Relaxation modulus in function of time through Prony series by fitting a model to DMA experimental data in function of frequency.

Model fitting for G'(ω) with respect to experimental DMA data showed an error smaller than 1 % using only two terms of the Prony series.

Through this methodology, parameters required to model viscoelasticity behavior in function of time in CAE software like Abaqus® can be obtained from the information of a single DMA experiment performed to a small material sample.

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Received: June 04, 2019; Accepted: February 28, 2020

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