2.1 Introduction

Viscoelastic damping treatments have been extensively used in various structural applications to control undesirable vibrations and associated noise radiation in a simple and reliable manner (Nashif et al. 1985; Sun and Lu 1995). In this chapter, particular emphasis is placed on studying the dynamic characteristics of such damping treatments and outlining the different mathematical models used to describe the behavior of these treatments over a wide range of operating frequencies and temperatures. Particular focus is given to ascertain the merits and drawbacks of the classical models by Maxwell, Kelvin–Voigt, and Zener (Zener 1948; Flugge 1967; Christensen 1982; Haddad 1995; Lakes 1999, 2009) both in the time and frequency domains.

2.2 Classical Models of Viscoelastic Materials

These models include the of Maxwell, Kelvin–Voigt, and Poynting–Thomson models (Haddad 1995; Lakes 1999, 2009). In these models, the dynamics of ViscoElastic Materials (VEMs) are described in terms of series and/or parallel combinations of viscous dampers and elastic springs as shown in Figure 2.1. The dampers are included to capture the viscous behavior of the VEM, whereas the springs are used to simulate the elastic behavior of the VEM.

2.2.1 Characteristics in the Time Domain

The dynamic characteristics of Maxwell and Kelvin–Voigt models in the time domain are summarized in Table 2.1.

Model	Maxwell model	Kelvin–Voigt model
Stresses and strains of components
Equilibrium and kinematic equations	Stress σ is same for spring and damper Strain ε is sum of strains of spring and damper: σ = σ s = σ d   (2.1) and ε = ε s  + ε d   (2.3)	Strain ε is same for spring and damper Stress σ is sum of stresses of spring and damper: σ = σ s  + σ d   (2.2) and ε = ε s = ε d   (2.4)
Constitutive equations	Spring: σ = E s ε s   (2.5) Damper:   (2.7)	Spring: σ s = E s ε (2.6) Damper:   (2.8)
Model equation	Substituting Eqs. (2.5) and (2.7) into Eq. (2.3) gives:   (2.9) where λ = c d /E s	Substituting Eqs. (2.6) and (2.8) into Eq. (2.2) gives:   (2.10)

(E _s = Young's modulus of elastic element, c _d = damping coefficient of dissipative element)

One can note that the stress–strain equations of the Maxwell and Kelvin–Voigt models can generally be written as follows:

(2.11)

where P and Q are differential operators given by:

(2.12)

Hence, for Maxwell model, p = 1, q = 1, α ₀ = 1, α ₁ = λ, β ₀ = 0 and β ₁ = c _d while for the Kelvin–Voigt model, p = 0, q = 1, α ₀ = 1, β ₀ = E _s and β ₁ = c _d .

The ability of both the Maxwell and the Kelvin–Voigt models to predict the characteristics of realistic VEM will be determined by considering the behavior under creep and relaxation loading conditions.

2.2.2 Basics for Time Domain Analysis

The initial and final value theorems of the Laplace transform are essential to the complete understanding of the behavior of viscoelastic models in the time domain. Appendix 2.A summarizes the two theorems and presents the necessary proofs.

Application of these two theorems to Maxwell and Kelvin–Voigt models is summarized in Tables 2.2 and 2.3 when these models are subjected to creep and relaxation loading, respectively. These two theorems provide the means for determining the initial and final limits of the VEM response under different loading conditions. This feature enables the correct calculation of the time response, between these two limits, when the differential equations describing these models are solved as will be demonstrated later.

Model	Maxwell model	Kelvin–Voigt model
Model
Loading conditions	The stress is constant σ = σ 0 and the initial and final values of the strain ε are predicted.
The strain in the Laplace s domain
Initial value
Final value

Model	Maxwell model	Kelvin–Voigt model
Model
Loading conditions	The strain is constant ε = ε 0 and the initial and final values of the stress σ are predicted.
The stress in the Laplace s domain
Initial value
Final value

Table 2.2 indicates that the Maxwell model experiences an initial strain when the creep load is applied, which is typical in VEMs. However, this strain tends to become unbounded as time grows. This feature is not observed or supported experimentally. As for the Kelvin–Voigt model, the initial value theorem indicates zero initial strain, which is rather unrealistic and a bounded final strain of σ ₀/E _s that is observed in a realistic VEM.

Table 2.3 indicates that the Maxwell model experiences an initial stress when the relaxation strain is applied and that stress is completely relieved as time progresses. Both of these characteristics are typical in VEM. As for the Kelvin–Voigt model, the initial and the final values remain constant E _s ε ₀, which is rather unrealistic behavior of a VEM.

2.2.3 Detailed Time Response of Maxwell and Kelvin–Voigt Models

Tables 2.4 and 2.5 summarize the detailed behavior characteristics of Maxwell and Kelvin–Voigt models in the time domain between the initial and final values predicted in Tables 2.2 and 2.3.

Model	Maxwell model	Kelvin–Voigt model
Loading conditions	The stress is constant σ = σ 0 and the time history of the strain is predicted
Response	As the initial strain ε = ε 0 a at time t = 0, Hence: (2.13) Unbounded strain for bounded stress	As the initial strain ε = 0a at time t = 0, Hence: (2.14) Bounded strain for bounded stress
Unloading Conditions	The stress is reduced back to zero at time t = t 1 and the time history of the strain
Response	At t = t 1 , , Hence, when σ = 0: with solution ε = ε 1= constant (2.15) No contraction after stress removal	At t = t 1 , , Hence, when σ = 0: or with solution (2.16) Complete strain relief after stress removal

^a Table 2.2 (using initial value theorem).

Model	Maxwell model	Kelvin–Voigt model
Loading conditions	The strain is constant ε = ε 0 and determines the time history of the stress
Response	As the initial stress σ = σ 0 a at time t = 0: Hence: with solution σ = E s e −t/λ ε 0 (2.17) stress decays to zero without any residual stress	As the initial strain ε = ε 0 a at time t = 0: Hence: σ = E s ε 0 = constant (2.18) stress remains constant, that is, VEM exhibits no relaxation

^a Table 2.3 (using initial value theorem).

Tables 2.4 and 2.5 indicate that the Maxwell model predicts unrealistic creep characteristics as the strain tends to be unbounded even for finite stress levels or the strain tends to remain constant when the stress is removed. The Kelvin–Voigt model also yields unrealistic relaxation characteristics with the stress remaining constant with time, indicating that the VEM does not exhibit any stress relaxation. Therefore, neither the Maxwell nor the Kelvin–Voigt model replicates the behavior of realistic VEM.

Note that these predictions, particularly at t = 0, are in agreement with the predictions of the initial and final value theorems listed in Tables 2.2 and 2.3.

In order to avoid the drawbacks and limitations of both the Maxwell and Kelvin–Voigt models, several other spring‐damper arrangements have been considered. For example, a damper with series and parallel springs is considered to combine the attractive attributes and compensate for the deficiencies of both the Maxwell and Kelvin–Voigt models. The resulting model is the Poynting–Thomson model, shown in Figures 2.1c and 2.2a. Other common models are also displayed in Figure 2.2 such the “three‐parameter model” and the “standard solid model” (Zener 1948).

Figure 2.3a,b show the most widely used spring‐mass configurations of VEM models that are employed extensively, particularly, in commercial finite element packages. These two configurations are, namely, the generalized Maxwell model and the generalized Kelvin–Voigt model.

These two generalized n classical models are assembled in parallel or series to model the complex behavior of realistic VEMs. These models are augmented with additional springs E ₀ , either in parallel or series, to eliminate the drawbacks associated with the classical models as outlined in Tables 2.2–2.5.

2.2.4 Detailed Time Response of the Poynting–Thomson Model

The stress σ across the series spring of the Poynting–Thomson model, shown in Figure 2.4, is given by:

(2.19)

and the stress σ across the damper and the parallel spring is given by:

(2.20)

Using the Laplace transformation, yields

(2.21)

Hence, the total strain ε across the Poynting–Thomson model is

(2.22)

In the time domain, this equation becomes

(2.23)

From Eqs. (2.11), (2.12), and (2.23), p = 1, q = 1, α ₀ = (E _s  + E _p ), α ₁ = c _d , β ₀ = E _s E _p , and β ₁ = E _s c _d .

1. The creep characteristics of the Poynting–Thomson model are obtained as follows:
   1. Determine the initial and final values strain:

      For stress σ = σ ₀, then Eq. (2.22) reduces to:

      

      Then,

      

      and

      

      where .

   2. Determine the time history of the strain:

      The time history of the strain is determined by solving Eq. (2.23) such that at t = 0, σ = σ ₀, and the initial strain ε ₀ = σ ₀/E _s . Hence, Eq. (2.23) reduces to:

      

      This equation has a solution:

      (2.24)

      where λ = c _d /E _p and E _∞ = E _s E _p /(E _s  + E _p ). Note that Eq. (2.24) has the initial and final values ε ₀ and ε _∞ at t = 0 and t = ∞.

      Figure 2.5 shows the strain–time characteristics as predicted by Eq. (2.24).

2. The Relaxation characteristics of the Poynting–Thomson model are obtained as follows:
   1. Determining the initial and final values stress:

      For strain ε = ε ₀, then Eq. (2.22) reduces to:

      

      Then,

      

      and

      

      where .

   2. Determining the time history of the stress:

      The time history of the stress can be determined by solving Eq. (2.23) such that at t = 0, ε = ε ₀, and the initial stress σ ₀ = E _s ε ₀. Hence, Eq. (2.23) reduces to:

      

      This equation has the following solution:

      (2.25)

      where .

Figure 2.6 shows the stress–time characteristics as predicted by Eq. (2.25).

Table 2.6 summarizes the main characteristics of Maxwell, Kelvin–Voigt, and Poynting–Thomson models.

Parameter	Maxwell	Kelvin–Voigt	Poynting–Thomson
Model
Dynamic equations
Creep characteristics
Comments	Unrealistic	Realistic	Realistic
Relaxation characteristics
Comments	Realistic	Unrealistic	Realistic

The characteristics summarized in Table 2.6 ascertain the ability of the Poynting–Thomson model to simulate a realistic behavior of VEMs. However, several combinations of Poynting–Thomson models are necessary to replicate the behavior of realistic VEMs.

Solution

Table 2.7 lists the solutions of the constitutive equations of Maxwell and Kelvin–Voigt models for the given loading and unloading cycle.

Figure 2.8a,b displays the stress–strain characteristics of the Maxwell and Kelvin–Voigt models. The figures indicate that, according to the Maxwell model, the VEM is stiffer and dissipates less energy, as represented by the enclosed area, than that predicted by the Kelvin–Voigt model.

Parameter	Maxwell	Kelvin–Voigt
Constitutive equation
Equation during loading
Initial condition (ε 0)	ε 0 = 0	ε 0 = 0
Response		ε = (ε 0 + 1)e −t  − 1 + t
Equation during unloading
Initial condition (ε 1)	ε 1 = 1.5	ε 1 = e −1
Response		ε = 3 − t + e −t  − 2e 1 − t

2.3 Creep Compliance and Relaxation Modulus

In Section 2.2, time domain relationships are derived for the different classical VEM models when these models are subjected to creep or relaxation loading. These relationships are obtained by solving the constitutive equations that describe the dynamics of the VEM models subject to initial conditions determined by applying the initial value theorem.

Table 2.8 lists these relationships such that the ratio between the strain and amplitude of the creep stress ε/σ ₀ is denoted by the symbol J(t) and called “the creep compliance” and the ratio between the stress and amplitude of the relaxation strain σ/ε ₀ is designated by the symbol E(t) and called “the relaxation modulus.”

Model	Maxwell model	Kelvin–Voigt model	Poynting–Thomson model
Creep compliance (J)	a	a
Relaxation modulus (E)		b	where ,

^a λ = c _d /E _s .

^b dirac(t) = ∞ at t = 0 and 0 at t = 0+.

It is important to note that these two characteristic properties of VEM are time‐dependent, unlike the corresponding properties for solids, which are constants.

Note also that the process involved in deriving these properties has been tedious and cumbersome as it consists of applying the initial and final value theorems followed by the exhaustive procedure for solving the constitutive equations subject to the initial values and then validating the solution against the obtained final values.

In this section, two other approaches are presented. In the first approach, the direct Laplace and inverse Laplace transformations are applied one after the other to the constitutive equations of the VEM. In the second approach, the topology of the VEM model is translated into a linear set of equations that can be reduced by using the Gauss elimination to determine both J(t) and E(t) simultaneously. The two approaches are implemented in a MATLAB environment to enhance their practicality and utility.

2.3.1 Direct Laplace Transformation Approach

The constitutive equations of the VEM are transformed into the Laplace domain to assume one of the following transfer function forms:

(2.26a)

and

(2.26b)

When the VEM is subjected to creep loading, the stress σ is replaced by its Laplace transform σ ₀/s and Eq. (2.26a) reduces to:

(2.27a)

Similarly, when the VEM is subjected to relaxation loading, the strain ε is replaced by its Laplace transform ε ₀/s and Eq. (2.26b) reduces to:

(2.27b)

The inverse Laplace transform is then used to transform J(s) and E(s) into the time domain creep compliance J(t) and relaxation modulus E(t).

Table 2.9 lists the corresponding creep compliance J(t) and relaxation modulus E(t) for the Maxwell, Kelvin–Voigt, and Poynting–Thomson models.

Operation	Maxwell	Kelvin–Voigt	Poynting–Thomson
Dynamic equations
Laplace transform of the strain due to creep loading σ 0
Inverse Laplace transform of ε/σ 0 = J using MATLAB	>> syms L cd s t >> ilaplace ((L*s + 1)/(cd*s^2),s,t) J = λ/cd + t/cd	>> syms L E s t >> ilaplace(1/(E*s*(L*s + 1)),s,t) J = 1/E − 1/(E * exp(t/λ))	>> syms Es Ep cd s t >> ilaplace (((Es + Ep) + cd*s)/ (s*(Es*Ep+Es*cd*s)),s,t) J = (Ep + Es)/(Ep*Es) ‐ 1/(Ep*exp((Ep*t)/cd))
Laplace transform of the stress due to relaxation loading ε 0
Inverse Laplace transform of σ/ε 0 = R using MATLAB	>> syms L cd s t >> ilaplace ((cd)/(L*s+1),s,t) R = cd/(λ * exp(t/λ))	>> syms L E s t >> ilaplace(E*(L*s + 1)/s,s,t) R = E + E*L*dirac(t)	>> syms Es Ep cd s t >> ilaplace((Es*Ep + Es*cd*s)/ (s*(Es + Ep + cd*s)),s,t) R = (Ep*Es)/(Ep + Es) + Es^2/ (exp((t*(Ep + Es))/cd)*(Ep + Es))

2.3.2 Approach of Simultaneous Solution of a Linear Set of Equilibrium, Kinematic, and Constitutive Equations

This approach was developed by Vondřejc (2009) and translates the topology of the VEM model into a linear set of equilibrium, kinematic, and constitutive equations which can be reduced by using the Gauss elimination to determine both J and R simultaneously. The approach is implemented in a MATLAB environment to enhance its practicality and utility.

In this approach, the topology of the viscoelastic model is described by N serial strings extending between P points. Each string can consist of a spring and a damper in series. For example, Figure 2.9 shows the description of the topology of the Maxwell and Poynting–Thomson VEM models. In Figure 2.9a, the Maxwell model is described by one string and two points whereas in Figure 2.9b, the Poynting–Thomson model is defined by two strings and three points.

In a symbolic MATLAB environment, the topology description of the two models is given by the vectors B such that:

Maxwell Model: B = [1, 2, E ₁ , cd ₁ ],

Poynting–Thomson Model: B = [1, 2, E ₁ , inf; 2, 3, E ₂ , inf; 2, 3, inf, cd ₂ ];

Note that in this description of any string, the component value of a spring or a damper that is missing in the string is set equal to “inf.”

The mathematical formulation of the VEM model, in the Laplace domain, is described as follows:

Constitutive equations	(2.28)
Equilibrium equations	(2.29)
	(2.30)
	(2.31)
Kinematic equations	(2.32)

Note that b _j and e _j denote the beginning and end points of the jth string.

In a matrix form, Eqs. (2.28) through (2.32) take the following form:

(2.33)

where x is a vector of the strains and stresses given by:

(2.34)

where ε _{j, j + 1} = strain between points j and j + 1, σ _i = stress in the ith string, σ = stress applied to the entire VEM topology, and ε = total strain of the entire VEM topology.

Solution

From Eqs. (2.28) through (2.32), the system of equations describing the dynamics of the Maxwell model is given by:

(2.35)

where .

Applying the Gauss elimination method to Eq. (2.35), it reduces to:

(2.36)

Expanding the last row of Eq. (2.36) gives:

(2.37)

Hence, if the VEM is subjected to creep loading such that σ = σ ₀, then Eq. (2.37) reduces to:

(2.38)

Using MATLAB symbolic manipulation gives:

Note that the obtained creep compliance J matches that listed in Table 2.9.

Also, if the VEM is subjected to relaxation strain such that ε = ε ₀, then Eq. (2.37) reduces to:

(2.39)

Using MATLAB symbolic manipulation gives:

The obtained relaxation modulus E matches that listed in Table 2.9.

2.5.1 Hysteresis Characteristics

Consider a VEM subjected to sinusoidal stress σ and strain ε given by

(2.43)

with the stress and strain related by the following constitutive equation

(2.44)

Combining Eqs. (2.43) and (2.44) gives

(2.45)

where σ _e = E′ε ₀ sin(ωt) and σ _d = ηE′ε ₀ cos(ωt) denote the elastic and dissipative components of the applied stress σ.

Then σ _d can be written as:

(2.46)

Rearranging this equation reduces it to

(2.47)

which is an equation of an ellipse as shown in Figure 2.12a. Figure 2.12b shows a plot of the elastic stress versus the strain and Figure 2.12c combines the elastic and dissipative stress components. Figure 2.12c displays accordingly the total stress σ acting on the VEM versus the strain ε.

Note that the dissipative component takes the form of a hysteresis loop. The area inside the loop quantifies the amount of energy D dissipated during the cyclic deformation of the VEM.

2.5.2 Energy Dissipation

The energy dissipated during a full vibration cycle of the VEM, at a frequency ω, per unit volume can be determined from

(2.48)

But as σ _d = ηE′ε ₀ cos(ωt) and ε = ε ₀ sin(ωt), then Eq. (2.48) reduces to

(2.49)

2.5.3 Loss Factor

Two methods can be used to extract the loss factor from the hysteresis characteristics of the VEM. These methods are based on the following:

2.5.3.1 Relationship Between Dissipation and Stored Elastic Energies

Consider now the energy W stored in the elastic component, during one‐quarter of a vibration cycle, which can be determined from:

With σ _e = E′ε ₀ sin(ωt), the equation reduces to

(2.50)

From Eqs. (2.49) and (2.50), the loss factor η can be determined from

(2.51)

Hence, Eq. (2.51) defines the physical meaning of the loss factor as the ratio between the dissipated energy and stored energy. Also, Figure 2.12 shows the graphical representation and physical meaning of both the dissipated and stored energies.

2.5.3.2 Relationship Between Different Strains

Equations (2.45) and (2.46) can be rewritten as:

(2.52)

When the stress σ is set = 0, the corresponding strain ε _sf can be obtained from:

or,

(2.53)

The σ − ε relationship of the upper branch of hysteresis characteristics can be expressed form Eq. (2.45) as:

(2.54)

The maximum stress is attained when:

at ε = ε _maxσ given by:

(2.55)

Figure 2.13 displays the graphical interpretation of the strains ε _sf and ε _maxσ .

From Eqs. (2.53) and (2.55),

(2.56)

Hence, the loss factor can be computed as the ratio between the two strains ε _sf and ε _maxσ as measured from the hysteresis characteristics.

2.5.4 Storage Modulus

The storage modulus can be determined by considering the stress under a strain‐free condition σ _stf . This value can be obtained by setting ε = 0 in Eq. (2.54), giving:

(2.57)

Once η and ε ₀ are determined from Eqs. (2.56) and (2.53), then Eq. (2.57) can be used to compute the storage modulus E′.

Solution

The constitutive equation for the Poynting–Thomson model is:

For sinusoidal stress: σ = sin t, this equation reduces to:

This equation is integrated numerically, with respect to time, using MATLAB to extract the time history of the strain as function of the time history of the input stress. Then, the strain is plotted against the stress to yield the stress–strain characteristics shown in Figure 2.14.

1. From Figure 2.14,
   
   

   Then,

   

   This yields ε ₀ = 1.581.

   Also, from Figure 2.14, σ_stf =  ± ηE′ε ₀ =  ± 0.3148, or

   
2. From Table 2.10,
   

   Then,

   

   and

   

Hence, the two methods yield exactly the same results.

2.6.1 Basic Building Block of Fractional Derivative Models

The basic concepts of fractional calculus are summarized in Appendix 2.B.

In this section, the basic building block of FD models is the “spring‐pot” that replaces the spring and dashpot elements used in the classical models. The spring‐pot element is employed to simplify, improve the applicability, and reduce the number of parameters used to model the complex behavior of viscoelastic polymers.

The spring‐pot element is a nonlinear FD element that has the following constitutive equation:

(2.58)

Note that the stress σ(t) applied to the element is dependent on the FD of order α of the strain ε(t) where α ranging between 0 and 1. When α = 0, the spring‐pot element reduces to a linear spring and when α = 1, the spring‐pot element becomes a linear dashpot (damper) as shown in Figure 2.16.

From Eq. (2.58), the storage and loss moduli of the spring‐pot can be determined as follows:

(2.59)

where and .

The relaxation modulus E(t) of the spring‐pot can be obtained by applying the inverse Fourier transform to Eq. (2.58) knowing that E(s) = E^*/s, as indicated in Eq. (2.27b). This yields the following expressions:

Relaxation modulus

(2.60)

and

Creep compliance

(2.61)

2.6.2 Basic Fractional Derivative Models

The basic FD models presented in this section are the FD Maxwell, FD Kelvin–Voigt, and FD Poynting–Thomson models.

For example, consider the classical Maxwell model shown in Figure 2.17a is transformed to a FD Maxwell model shown in Figure 2.17b by replacing each component by a spring‐pot with different parameters.

For the FD model and using the equivalent spring‐pot Eq. (2.58), the strains ε ₁ and ε ₂ can be written as:

(2.62)

But as, ε = ε ₁ + ε ₂, then:

(2.63)

where , , and α ₁ ≥ α ₂.

Fourier‐transforming Eq. (2.63) gives:

(2.64)

Following the same approach adopted in Section 2.6.1, it can be easily shown that the relaxation modulus and creep compliance of the FD Maxwell model are given by:

(2.65)

Table 2.11 summarizes the complex moduli for the FD Maxwell, Kelvin–Voigt, and Zener models.

Parameter	Maxwell	Kelvin–Voigt	Zener
Model
Storage modulus	a		b

^a where , , and α ₁ ≥ α ₂.

^b .

Solution

From Eq. (2.58), the complex modulus as predicted by the four‐parameter FD model can be obtained by using Eq. (2.B.10), to give:

It can be easily shown that the storage and loss moduli of the four‐parameter FD model are given by:

and

Accordingly, the loss factor η is given by:

This yields a value of α that can be estimated from (see Problem 2.9):

Solution

The storage modulus and the loss factor of the FD and Poynting–Thomson VEM models are plotted versus the frequency ω as shown in Figure 2.18. The plots are obtained for the Poynting–Thomson model with α = 1, β = 20, λ = 3, and E _s = 500 psi while the FD model is given by:

The figure indicates clearly that the four‐parameter FD model adequately replicates the physical behavior of Dyad 606 unlike the linear Poynting–Thomson model.

2.6.3 Other Common Fractional Derivative Models

In this section, some of the commonly used FD models are introduced. The main characteristics of these models are presented in the frequency ω domain as indicated in Table 2.12. These models vary in complexity by increasing the number of the included parameters from three (E _∞ , Δ, τ) as in the Debye model to five parameters (E _∞ , Δ, τ, α, and β) as in the Havriliak–Negami model (Pritz 2003, Ciambella et al. 2011).

Model	E * (ω)	Parameters
Debye	a	3
Cole–Coleb		4
Cole–Davidson		4
Havriliak–Negami		5

^awhere Δ = (E ₀ − E _∞)/E _∞ = relaxation strength, = relaxed elastic modulus, = unrelaxed elastic modulus.

^bFriedrich and Braun (1992), Pritz (2003), and Ciambella et al. (2011).

Solution

According to the set parameters, the system equation reduces to:

or

where,

where A _j + 1 = Grunwald coefficients.

Figure 2.20a,b displays the Grunwald coefficients and the time response of the spring‐mass system with a FD damper, respectively. It is evident, from Figure 2.20a, that the Grunwald coefficients vanish as the number of terms included in the summation of Eq. (2B.14) increases. This demonstrates clearly the “fading memory” characteristics of FDs. Figure 2.20b indicates that the system reaches, with an error of 3.14%, the desired reference command after 15 s.

Note that the approach adopted in Example 2.7 establishes the basis for time domain analysis of finite element models of structures treated with VEMs that are described by FD models.

The initial and final values of a function x(t) are given by the following theorems (Nise 2015):

• Initial value theorem:
• Final value theorem:

Proof

From the definition of the Laplace transform L:

Consider the following two extreme cases:

1. when s → 0, this equation becomes
   
   that is,
2. when s → ∞, we have
   
   that is,

2.B.1 Fractional Integration

The fractional integral of a function f(x) with n‐folds is defined as:

(2.B.1)

Let

(2.B.2)

Then, integrating one more time gives:

(2.B.3)

Integrating another time gives:

(2.B.4)

After n integrations, yields the following expression, which is called “Reimann–Liouville Fractional Integral”:

(2.B.5)

But, as Γ(n) = (n‐1)! = gamma function, then this equation reduces to:

(2.B.6)

2.B.2 Convolution Theorem

As the Laplace inverse of a product G(s) F(s) is given by the following convolution integral (Weber and Arfken 2003)¹:

(2.B.7)

If

Then,

or

(2.B.8)

Equation (2.B.8) indicates that the fractional integration of f(x) is the convolution integral resulting from the inverse Laplace transformation of F(s) operated on by the n‐fold integral operator s ⁻ⁿ .

Solution

With n = 1/2, Eq. (2.B.6) becomes:

which can be solved symbolically using MATLAB as follows:

2.B.3 Fractional Derivatives

Let m = p − l, then: D ^m f(x) = D ^p D ^−l f(x).

From Eq. (2.B.6),

If 0 < m < 1, then p = 1 or

Let n = 1 – l, then:

(2.B.9)

which is called the “Reimann–Liouville Fractional Derivative.”

Solution

With n = 1/2, Eq. (2.B.9) becomes:

which can be solved symbolically using MATLAB as follows:

2.B.4 Laplace Transform of Fractional Derivatives

Let n = 1 − l, then D ⁿ f(t) = D ¹ D ^−l f(t).

Performing the Laplace transform, gives:

Integration by parts gives:

or

But as n = 1 − l, then:

(2.B.10)

2.B.5 Grunwald–Letnikov Definition of Fractional Derivatives

1. Integer Derivatives

   Using the definition of integer derivatives in terms of a backward finite difference quotient, the following expressions can be extracted:

   
   

   and

   

   Hence, in general, the nth derivative can be written as:

   (2B.11)

   Replacing Δt by t/N, then Eq. (2B.11) reduces to:

   (2B.12)

   where the binomial coefficient for j > n.

2. Fractional Derivatives

   To extend Eq. (2B.12) to any fractional order derivative, the following extended definition of the binomial coefficient is used:

   

   or

   (2B.13)

   Substituting Eq. (2B.12) into Eq. (2B.13) gives:

   (2B.14)

   where = Grunwald coefficients. These coefficients can be given also by the following recursive relationships:

   
   

   and

   (2B.15)

Note that the Grunwald coefficients in Eq. (2B.15) decay to zero as the number of terms N increases. This feature describes the “fading memory” phenomenon which indicates that the behavior of the VEM is governed primarily by the recent time history rather than by the most remote history. Due to this property, it is possible to truncate higher order terms of summation in Eq. (2B.15) in order to simplify and speed the computational effort.

Other numerical algorithms to approach the FDs are summarized by Oldham and Spanier (1974), Padovan (1987), Podlubny (1999), and Galucio et al. (2006).

Problems

1. 2.1 Consider the Poynting–Thomson model of the VEM shown in Figure P2.1a. Derive the constitutive equation of the model that relates the applied stress σ to the resulting strain ε. Determine the creep behavior, during loading and unloading of the model according the cycle shown in Figure P2.1b, by solving the constitutive equation. Assume that σ = σ ₀ and ε = σ ₀/(E _p  + E _s ) at t = 0 while σ = 0 and ε = ε ₁ at t = t ₁ .

   Check the initial and final of the strain, during loading and unloading, using the initial and final value theorem (outlined in Appendix 2.A).

   

2. 2.2 If a VEM is represented by the Poynting–Thomson model and is subjected to a step strain ε ₀, show that the initial stress σ ₀ experienced by the VEM is given by:
   

   Use the initial value theorem.

3. 2.3 Consider the three‐parameter Jeffery model of the VEM shown in Figure P2.2a. Derive the constitutive equation of the model that relates the applied stress σ to the resulting strain ε. Determine the creep behavior, during loading and unloading of the model according the cycle shown in Figure P2.1b, by solving the constitutive equation. Assume that σ = σ ₀ and ε = 0 at t = 0 while σ = 0 and ε = ε ₁ at t = t ₁ . Discuss the results.

   Determine also the relaxation behavior when subjected to the strain loading shown in Figure P2.2b, by solving the constitutive equation. Assume that ε = ε ₀ at t = 0. Discuss the results.

   

4. 2.4 Consider the three‐parameter models of a VEM shown in Figure P2.3.

   Derive the constitutive equations of the models that relate the applied stress σ to the resulting strain ε. For sinusoidal excitations, at a frequency ω, such that σ = σ ₀ e ^iωt and ε = ε ₀ e ^iωt , use the derived constitutive equations to extract expressions for the complex modulus E, storage modulus E′, and loss factor η for each model [E = E′(1 + iη)].

   If E _s = 1, E _p = 1, and c = 1, plot the effect of the frequency ω on the storage modulus E′and loss factor η for each model. Comment on the results.

   

5. 2.5 Derive the governing constitutive equations of Burgers model of a VEM, which is shown in Figure P2.4. Note that the model is combination between Maxwell and Kelvin–Voigt models.

   For sinusoidal excitations at a frequency ω, such that σ = σ ₀ e ^iωt and ε = ε ₀ e ^iωt , extract expressions for the complex modulus E, storage modulus E′ and loss factor η for each model [E = E′(1 + iη)].

   If E _s = 1, E _p = 1, and c = 1, plot the effect of the frequency ω on the storage modulus E′ and loss factor η for each model. Compare the results with those of the three‐parameter models of Problem 2.4.

   

6. 2.6 For the viscoelastic hysteresis loop shown in Figure P2.5 show that:
   1. the strain ε _max = ε ₀,
   2. the stress σ _maxε = E′ε ₀, and
   3. the loss factor η = σ _stf /σ _maxε .

   

7. 2.7 Show that the complex modulus of the Jeffery model of VEMs with an added inertia element I, as shown in Figure P2.6, is given by:
   

   

8. 2.8 Show that the complex modulus E ^* of the hierarchical viscoelastic model of Figure P2.7, satisfies the following:
   

   Show that this equation implies that: with λ = c _d /E, that is, the hierarchical viscoelastic model is actually a FD model (Heymans 1996).

   

9. 2.9 The complex modulus as predicted by the four‐parameter FD model is given by:
   

   Show that:

   1. the storage and loss moduli of the model are given by:
      

      and

      
   2. the loss factor η is given by:
      
   3. the value of α that can be estimated from:
      

   (Hint: , and )

10. 2.10 Determine the parameters of the following two fractional models:
    1. (Parameters are: G ₀, G _∞, τ, α)
    2. (Parameters are: G _m , τ, α)

    that best fit the actual experimental behavior of Dyad606 VEM of Soundcoat operating at 38°C/100°F. Note that G, τ _σ , and γ denote shear modulus, shear stress, and shear strain respectively. For these two models, derive the expressions for the:

    1. The storage modulus G′ and
    2. The loss factor η.

    The storage modulus and loss factor of Dyad606 are given in the table here. Compare and comment on the results.

    Frequency (Hz)	0.001	0.01	0.040	0.10	0.20	0.40	0.70	1.0	3.0	5.0	10
    G′ (kpsi)	0.040	0.050	0.065	0.100	0.150	0.200	0.250	0.300	0.500	0.58	0.67
    Loss Factor η	0.40	0.50	0.60	0.70	0.75	0.80	0.90	0.92	0.93	0.94	0.95
    Frequency (Hz)	20	30	40	50	60	70	80	90	100	200	300
    G′ (kpsi)	0.75	1.1	1.3	1.6	1.8	2	2.2	2.3	2.6	4	5.2
    Loss Factor η	0.95	0.96	0.98	0.99	1.	1.	1.	1.	1.	0.99	0.95
    Frequency (Hz)	400	500	600	700	800	900	1000	2000	3000	4000	5000
    G′ (kpsi)	6.3	7.2	8.5	9.5	11	12	13	19	23	2.6	3
    Loss Factor η	0.91	0.88	0.85	0.82	0.79	0.77	0.75	0.63	0.56	0.51	0.48

11. 2.11 Show that the plots of loss modulus/E _s versus the storage modulus/E _s of the Maxwell, Kelvin–Voigt, and Poynting–Thomson models are as displayed in Figure P2.8. Note that these plots are called the “Nyquist diagrams” or “Cole–Cole” plots.

    

12. 2.12 Show that the plots of loss factor versus the storage modulus/E _s of the Maxwell, Kelvin–Voigt, and Poynting–Thomson models are as displayed in Figure P2.9. Note that the plot is called the “Wicket” plot.

    

13. 2.13 If the relaxation modulus of a VEM is given by:

    E(t) = 1 + e ^−t in GPa, where t = time in seconds.

    Determine its creep compliance J(t).

14. 2.14 Show that the complex modulus E ^* of the Ladder Viscoelastic Model in Figure P2.10, satisfies the following equation:
    

    Also show this equation implies that: with λ = η/E ₀.

    

15. 2.15 Consider the dynamic system shown in Figure P2.11 that simulates the dynamics of VEMs. Derive the following:
    1. relationship between the force F and the net deflection (x ₁ ) in the following Laplace domain form:
       
       where K ^* = complex stiffness = K′(1 + i η) with K′ and η denoting the storage modulus and loss factor, respectively.
    2. storage modulus of the VEM K′ in terms of the frequency
    3. loss factor η of the VEM in terms of the frequency

    

16. 2.16 Show that the mechanical system with coulomb damping, which is displayed in Figure P2.12a, has the electrical analog circuit represented schematically in Figure P2.12b. Show also that the system has the hysteresis characteristics illustrated in Figure P2.12c.

    Relate the parameters of the mechanical elements of Figure P2.12a to those of their electrical analog components of Figure P2.12b. Note that q denotes the charge (Iwan, 1964).