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On attenuation of free and forced waves in an infinitely long visco-elastic layer of a constant thickness
Wave Motion 68 (2017) 114–127
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On attenuation of free and forced waves in an infinitely long visco-elastic layer of a constant thickness Sergey Sorokin ∗ , Radoslav Darula Department of Mechanical and Manufacturing Engineering, Aalborg University, Fibigerstrœde 16, 9220 Aalborg, Denmark
highlights • A hierarchy of reduced order models is employed to analyse wave propagation in a viscoelastic layer. • The measures of attenuation of free and forced waves are proposed and compared with each other. • A high sensitivity of the attenuation levels to excitation conditions at high frequencies is demonstrated and explained.
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Article history: Received 23 May 2016 Received in revised form 21 August 2016 Accepted 3 September 2016 Available online 17 September 2016 Keywords: Loss factor Wave attenuation Visco-elastic layer Free and forced waves Reduced order models
abstract The conventional concepts of a loss factor and a complex-valued elastic module are used to study wave attenuation in a visco-elastic layer. The hierarchy of reduced order models is employed to assess attenuation levels of free and forced waves in various situations. First, the free waves are considered. In the low frequency limit, the attenuation of these waves is found to be in the excellent agreement with the existing knowledge. At high frequencies, predictions of the reduced order models fully agree with the solutions of exact Rayleigh–Lamb problem. Alternative excitation cases are considered for the forcing problem and a measure of the attenuation level is proposed and validated. The differences between two regimes, the low frequency one, when a waveguide supports only one propagating wave, and the high frequency one, when several waves are supported, are demonstrated and explained. © 2016 Elsevier B.V. All rights reserved.
1. Introduction The assessment of origins and actual levels of damping has always been a challenging task in various applications such as structural intensity analysis and non-destructive structural health monitoring. As a manifestation of uncertainties in this assessment, commercially available finite element software typically leaves to a customer the choice of parameters defining the extent to which the damping matrix is proportional to the stiffness and to the mass ones. On the other hand, the quantification of viscous damping in linear one-degree-of-freedom systems is truly elementary and based on simple energy considerations, which bring to light a concept of the loss factor. Therefore, it has long been attempted to extend this quantification to multi-degree-of-freedom and continuous systems. This paper goes along these lines and is concerned with assessment of the attenuation of time-harmonic waves in an infinitely long visco-elastic layer of a constant thickness. The goal is to demonstrate to what extent the loss factor, generally recognised as a classical measure of the energy dissipation in mechanical systems of finite dimensions, is applicable to estimate spatial and temporal attenuation of waves in visco-elastic structures.
∗
Corresponding author. E-mail address: [email protected] (S. Sorokin).
http://dx.doi.org/10.1016/j.wavemoti.2016.09.001 0165-2125/© 2016 Elsevier B.V. All rights reserved.
S. Sorokin, R. Darula / Wave Motion 68 (2017) 114–127
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For the longitudinal (i.e., the first symmetric) and the flexural (i.e., the first skew-symmetric) waves, this has been done in the classical text [1] by means of the elementary Bernoulli–Euler theory known to be asymptotically consistent in the low frequency — long wave limit with the exact Rayleigh–Lamb solution, see [2]. In fact, the wave attenuation analysis reported in [1] serves as a theoretical background to link the material loss factor and the imaginary part of Young’s module. However, such an assessment has not yet been done for the two-mode (i.e., Timoshenko and Herrmann–Mindlin) models as well as for higher-order models. This paper is aimed to fill in this gap and to give a systematic account of the free waves’ attenuation in the framework of simplified theories verified against the exact solution. Alongside, an assessment of wave attenuation in the case of a forced response of an infinite waveguide, which supports several propagating waves, constitutes the research goal of this paper. To the best of the authors’ knowledge, these issues have not yet been fully addressed in the literature. The paper is structured as follows. Section 2 is concerned with discussion of the concepts of the loss factor and complexvalued Young’s module. In Section 3, the brief exposition of the canonical Rayleigh–Lamb problem is presented and a hierarchy of reduced order models of wave propagation in a layer is described. In Section 4, the alternative measures of attenuation of free waves are discussed. This section is also concerned with comparison of such attenuation predicted by low order theories and in the exact formulation. In Section 5, an attenuation measure for the forced response is proposed and used in the framework of the theories presented in Section 3. Attenuations of free and forced waves are compared with each other in the representative frequency ranges and the influence of excitation conditions is explored. The findings are summarised in conclusions. 2. The loss factor and complex-valued Young’s module As the wave attenuation is caused by the internal material damping in a structure exhibiting steady state cyclic deformation, it is natural to characterise the energy dissipation in terms of energy quantities. The loss factor, defined as N a ratio of the energy dissipated per oscillation cycle ND and the total energy N, see [1,3,4]: η = 2πDN , is such a measure, originally introduced for single-degree-of-freedom systems. Since any linear oscillatory system may be decomposed into a set of uncoupled oscillators, loss factors are commonly used in the modal format, i.e., to assess the level of attenuation at each mode of vibration of a multi-degree-of-freedom or continuous system of finite dimensions. The point to be addressed in this paper is how the concept of a modal loss factor is adjusted in a waveguide theory for calculation of the level of attenuation of otherwise (i.e., in the absence of viscosity) propagating free waves, not necessarily in the low frequency — small wavenumber limit. It should be noted that Refs. [5–7] are concerned with the estimation of loss factors for waves propagating in laminated structures by means of the wave finite element method. The comprehensive literature survey on the damping estimation in the framework of the finite element method is also presented in [6]. However, the modal strain energy method as formulated in Eq. (10), p. 3931 in this reference gives the assessment of attenuation of an individual free wave. In what follows in the present paper, a model, much simpler than those introduced in abovementioned references, is considered to facilitate a rigorous analysis of attenuation of free and forced waves based on the canonical definition of the loss factor. The authors are of the opinion that such an analysis will serve as a novel useful supplement to the findings reported in [5–7]. Steady state harmonic vibrations hereafter are treated with the time-dependence of all state variables chosen in the form exp (−iωt). This modelling brings to light the complex-valued Young’s module [1] with (given the stated above sign convention) the negative imaginary part, Eˆ = E − iEi . It is in common practice to present the imaginary part of the complex-valued Young’s module via a loss factor in the form Ei = ηE. Here the loss factor η is considered as frequencyindependent (see [1, p. 153]). This model is widely used in the conventional harmonic finite element analysis, where the complex-valued stiffness matrices have been introduced and used in many research papers with [6–8] being just a few examples. Furthermore, the commercially available finite element software, such as ANSYS, utilises this concept [9]. Classical Refs. [1,3,10–12] explain pros and cons of the model used hereafter and provide the detailed discussion of damping mechanisms. In the classical theory of linear wave propagation in viscoelastic materials, introduction of the complex-valued Young’s module entails introduction of complex-valued propagation speeds: ‘If one substitutes the modulus of elasticity ¯ = D′ (1 + iη) into the wave equation, one finds that the propagation speed becomes complex’ ([1, p. 155], the third and D fourth lines from the bottom, time dependence is taken as exp (iωt)). The concept of a complex-valued propagation speed has been adopted in several publications [13–17] and implemented in SAFE (semi-analytical finite element) method. We employ this concept in an analytical solution of the canonical Rayleigh–Lamb problem. 3. The low-order models of wave propagation in an elastic straight layer of constant thickness We consider an infinite elastic layer in the plane strain state with the traction-free boundary conditions, as shown in Fig. 1(a). Analysis of the wave propagation in such a layer constitutes the classical Rayleigh–Lamb problem. Due to the symmetry in the boundary conditions at ydim = ± 2h , the symmetric and skew-symmetric modes (see Fig. 1(b)) are decoupled from each other. The conventional spatial dependence is taken as exp (ikdim xdim ), so that the dispersion equation for free symmetric modes is [2, p. 223]: tan tan
q 2p = − 2
4k2 pq
q2 − k2
2 .
(1)
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S. Sorokin, R. Darula / Wave Motion 68 (2017) 114–127
Fig. 1. (a) An elastic layer; (b) Symmetric and skew-symmetric deformation.
In this equation, the non-dimensional parameters scaled to the thickness of an elastic layer h are: k = kdim h, Ω = ωc h , p2 = 1 Ω 2 − k2 and q2 = α 2 Ω 2 − k2 . The notation α = cc1 ≡ 2(1(1−−ν) , where ν is the Poisson ratio, is used hereafter. The speed of 2ν) 2 E (1−ν) dilatation wave is c1 = ρ(1+ν)(1−2ν) and the speed of shear wave is c2 = 2ρ(1E+ν) .
For skew-symmetric waves, the dispersion equation is [2, p. 223]: tan tan
q
2 p 2
=−
q2 − k2 4k2 pq
2 .
(2)
The solutions of dispersion equations (1) and (2) are used as the reference to assess the validity of the low-order models. In the absence of damping, these solutions give the full account of propagating, evanescent and attenuated waves (see [2,18]). Usually, only purely real wavenumbers corresponding to travelling waves are computed. Due to damping, all roots of dispersion equations are complex-valued, and calculations of these wavenumbers from exact transcendental dispersion equations become rather challenging [19]. In this situation, it is particularly practical to employ low-order models, or theories, because dispersion equations derived from Bernoulli–Euler, Herrmann–Mindlin and Timoshenko models are polynomials, and it does not present any difficulty to compute all their roots at once. The same holds true regarding a hierarchy of the finite product method approximations and the Legendre approximations shown to be in the excellent agreement with exact solution in the absence of damping [20,21]. In the remaining part of this section we introduce the reduced-order theories used hereafter for the assessment of wave attenuation in a visco-elastic layer. The Bernoulli–Euler models specialised for the plane strain case are well known. For symmetric waves, the dispersion equation is k2 −
α4 Ω 2 = 0. 4 α2 − 1
(3)
For skew-symmetric waves, it is k4 −
3α 4
α2 − 1
Ω 2 = 0.
(4)
Both these low-frequency models capture only one propagating wave and are asymptotically consistent with the exact equations (1) and (2) in the low frequency limit. The Timoshenko model may be formulated in many alternative ways, see the detailed survey [22]. In what follows, we use this theory as explained in [20]. Then the dispersion equation acquires the form (see [20, p. 5462, Eq. (6)]): k4 −
12α 2
π2
k2 Ω 2 −
3α 4
α2 − 1
Ω2 +
3α 6 Ω 4 = 0. π 2 α2 − 1
(5)
Formulation of the Herrmann–Mindlin model may be found in [18]. The dispersion equation converted to the same notations as in (1)–(5) is
α 2 12α 2 + π 2 2 2 α2 α4 α4 4 2 k + k − Ω2 + Ω 4 = 0. k Ω − 2 2 2 2 2 48 α − 1 48 α − 1 π 4 α −1 4π α 2 − 1
(6)
These models capture two propagating waves. The Timoshenko theory gives an accurate prediction of location of the first branch of the dispersion diagram in a broad frequency range. Its accuracy with respect to the location of the second branch is controlled by a choice of the shear correction factor. If it is chosen as in Eq. (5), then the frequency range, where this theory gives acceptable results for the second branch, is of the same size as the size of the frequency range of applicability of the Bernoulli–Euler theory for the first branch. To advance to high frequencies with reduced order models, various methods may be applied, as discussed in [18,23,24]. The hierarchy of these models for symmetric waves in this waveguide has been formulated in [21] from the exact formulation of elasto-dynamics by means of expansion of displacements on Legendre polynomials in the transverse coordinate y and use of the Hamilton principle. The first terms in series for displacements are polynomials of the zeroth and the first order
S. Sorokin, R. Darula / Wave Motion 68 (2017) 114–127
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in the transverse coordinate. Therefore, the low-order approximations become identical to the conventional elementary theories, which employ the ‘plane cross-section’ hypothesis. Extension of these expansions to higher-order terms allows one to account for an arbitrary large number of branches of dispersion diagram, with this number being controlled by the truncation level in expansions of displacements. l x The kinetic energy per unit width of a layer, extended from x = 0 to x = l ≡ dim in the scaled axial coordinate x = dim , h h is: T =
1 2
ρh
4
1/2
l
−1/2
0
∂ u (x, y, t ) ∂t
2
+
∂v (x, y, t ) ∂t
2
dxdy.
(7a)
The potential energy per unit width is: V =
1 2
ρ h2 c22
+
1/2
−1/2
l 2 ∂ u (x, y, t ) 2 ∂v (x, y, t ) ∂ u (x, y, t ) α + α − 2 ∂x ∂y ∂x 0
2 ∂ u (x, y, t ) ∂v (x, y, t ) α −2 + α2 ∂x ∂y
∂v (x, y, t ) + ∂y
∂v (x, y, t ) ∂x
2
dxdy.
(7b)
The differential equations for symmetric waves in the three-mode theory are obtained with the following ansatz for displacements: u (x, y, t ) = U0 (x, t ) · 1 + U2 (x, t ) · 1 − 12y2 ,
v (x, y, t ) = V1 (x, t ) · y.
Substitution of this ansatz to formulas (7) and standard by parts integration in δ H = δ
(8)
t2 t1
[T − V ] dt = 0 gives the
system of linear differential equations with respect to U0 (x, t ) , V1 (x, t ) and U2 (x, t ). The boundary/loading conditions at the edges x = 0, x = l are obtained as the non-integral terms in the course of by-parts integration. These are concerned with the following pairs of generalised forces and displacements: F0 (x, t ) ≡ F1 (x, t ) ≡ F2 (x, t ) ≡
1/2
−1/2
1/2
σxy (x, y, t ) · ydy =
∂ U0 (x, t ) 2 + α − 2 V1 (x, t ) and U0 (x, t ) ∂x
1 ∂ V1 (x, t )
(9a)
− 2U2 (x, t ) and V1 (x, t )
(9b)
4 ∂ U2 (x, t ) σx (x, y, t ) · 1 − 12y2 dy = α 2 and U2 (x, t ) . 5 ∂x −1/2
(9c)
−1/2
σx (x, y, t ) · 1dy = α 2
12
∂x
1/2
To derive the dispersion equation, the standard substitution U0 (x, t ) = U00 exp (ikx − iωt ) ,
V1 (x, t ) = V10 exp (ikx − iωt ) ,
U2 (x, t ) = U20 exp (ikx − iωt )
to the differential equations of motion with the inertia-correction factors (see [21] for details) is used. The dispersion equation of the sixth order both in the frequency parameter and the wavenumber has the following form:
α 4 15 + 12α 2 + π 2 4 2 α 2 −60 + 120α 2 + π 2 α 4 2 2 α2 α4 k6 + k4 − k Ω k Ω + k2 − 60 2880 α 2 − 1 48 α 2 − 1 240 α 2 − 1 π 2 5π 2 α 4 + 4α 2 15 + π 2 2 4 α4 4 + α2 α4 α6 Ω4 − Ω 6 = 0. Ω2 + + k Ω − 960 α 2 − 1 π 4 4 α2 − 1 16π 2 α 2 − 1 16π 4 α 2 − 1
(10)
The derivation of the model for skew-symmetric waves is similar. The displacements are presented in the truncated series on Legendre polynomials: u (x, y, t ) = U1 (x, t ) · y,
v (x, y, t ) = V0 (x, t ) · 1 + V2 (x, t ) · 1 − 12y2 .
(11)
The boundary/loading conditions are concerned with the following pairs of generalised forces and displacements: F0 (x, t ) ≡ F1 (x, t ) ≡ F2 (x, t ) ≡
1/2
−1/2
1/2
−1/2
σxy (x, y, t ) · 1dy = σx (x, y, t ) · ydy =
∂ V0 (x, t ) + U1 (x, t ) and V0 (x, t ) ∂x α 2 ∂ U1 (x, t ) + 2 α 2 − 2 V2 (x, t ) and U1 (x, t ) 12 ∂x
(12a)
(12b)
1/2
4 ∂ V2 ( x , t ) σxy (x, y, t ) · 1 − 12y2 dy = and V2 (x, t ) . 5 ∂x −1/2
(12c)
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5
4
4 Kr
5
3
3
2
2
1
1
0
0
1
2
3
4
5
0
6
0
1
2
3
4
5
Fig. 2. The dispersion diagrams after exact solution (1) and (2), Mindlin–Herrmann and Timoshenko theories (5) and (6) and three-mode theories (10) and (13): (a) symmetric waves; (b): skew-symmetric waves. Table 1 Models used in the assessment of wave attenuation. Model
No. of waves/Order
Symmetric
Skew-symmetric
Bernoulli–Euler Herrmann–Mindlin Timoshenko Three-mode theory
1 2 2 3
× ×
× × ×
×
The dispersion equation is of the sixth order both in the frequency parameter and the wavenumber:
−
α 2 36 + 20α 4 + 3π 2 α 2 4 2 19π 2 α 2 − 20α 4 3 + π 2 2 2 α2 k6 + k4 − k Ω + k Ω 240 α 2 − 1 720 α 2 − 1 π 2 20 α 2 − 1 π 2 α 4 60α 2 + 9π 2 + 5α 4 π 2 2 4 10α 6 3α 4 α8 2 Ω4 − Ω 6 = 0. + Ω + k Ω − 2 α −1 180 α 2 − 1 π 4 3π 2 α 2 − 1 3π 4 α 2 − 1
(13)
In Fig. 2, the first three branches of dispersion diagrams after exact solution (1) and (2) are plotted. These are purely real wavenumbers in the absence of damping. For comparison, dispersion diagrams after reduced order theories are also plotted. The accuracy of Timoshenko and Herrmann–Mindlin theories has been a subject of the thorough analysis for years. We just note that the Herrmann–Mindlin theory does not perform as good as the Timoshenko one (see also, for instance, [18, p. 150]. As can be seen from Fig. 2, the three mode theories are equally good to approximate in a relatively broad frequency range the first three branches of dispersion diagrams for symmetric and skew-symmetric waves. The overview and classification of the low-order theories introduced in this section are summarised in Table 1. 4. Assessment of attenuation of free waves in a visco-elastic layer We begin with the analysis of free waves. It is confined to the solutions of the dispersion equations presented in Section 3. These transcendental and polynomial equations contain only purely real coefficients and can be written in the generic form: F Ω 2 , k2 = 0.
(14)
The odd powers of a wavenumber are absent due to the natural symmetry of the waveguide. The odd powers of the frequency parameter are also absent due to the choice of the model of material damping. If the complex-valued Young’s module ˜ ≈ Ω 1 + 1 iη . ˜ = √ Ω , or for η ≪ 1, Ω E˜ = E (1 − iη) is introduced, the frequency parameter becomes Ω 2 1−iη 4.1. Temporal and spatial attenuation The attenuation of free waves may be assessed from two viewpoints. The first one is similar to the assessment of damping for free vibrations of a single-degree-of-freedom system. It is the temporal decay over one period of oscillations [1,3]. For an infinite waveguide, it means that a free wave defined by a purely real wavenumber is described by a complex-valued ω frequency ω = ωr − iωi . Then the temporal attenuation Dt is characterised by the ratio ω i , or, in the non-dimensional form, ˜i Ω ˜r Ω
ω
. For a single-degree-of-freedom system, this ratio is simply ω i = r
r
1 2
η. The dispersion equation (14) remains the same
S. Sorokin, R. Darula / Wave Motion 68 (2017) 114–127
119
Fig. 3. The ratio of attenuation levels predicted by the Timoshenko model DTs . (η, Ω ) and the Bernoulli–Euler model DBs .−E . (η) ≡ DTs . (η, 0).
for the non-dimensional frequency Ω regardless of whether elastic or visco-elastic material is considered. Therefore, the temporal attenuation of a free wave with any purely real wavenumber has the same rate Dt = 12 η. The alternative characterisation of attenuation is concerned with the free wave at a purely real frequency ω. In this ˜ becomes complex-valued, and so do all wavenumbers found from the case, the non-dimensional frequency parameter Ω k dispersion equation (14). The spatial attenuation is characterised by the ratio Ds = k i . For consistency, the derivation of this r formula is presented here, although it is completely analogous to the classical derivation of the formula for temporal decay rate (see [3]). We consider a free wave, characterised by the amplitudes of displacements and forces in the form Z (x, t ) = Z0 exp (ikx − iωt ) = Z0 exp (−ki x)· exp (ikr x − iωt ). Here Z0 exp (−ki x) may be the amplitude of any of these state variables at the point x. In the distance equal to the wavelength λ = Z (x,t ) Z (x+λ,t )
= exp
2π ki kr
and ln
Z (x,t ) Z (x+λ,t )
=
2π ki kr
2π kr
, the amplitude decays to Z0 exp −ki x −
2π ki x kr
. Thus,
= 2π Ds .
In the elementary cases of modelling of longitudinal (symmetric) and flexural (skew-symmetric) waves by means of the Bernoulli–Euler theory, the spatial decay is known to be frequency-independent [1, p. 161, Table 4.2]. Specifically, for longitudinal waves Ds = Dt = 12 η and for flexural waves Ds = 21 Dt = 41 η. These results are immediately available from Eqs. (3) and (4).
4.2. Comparison of spatial attenuation predicted by alternative theories The remaining part of this section is concerned with the spatial attenuation of free waves described by a hierarchy of models presented in Table 1. The research questions to be addressed are whether the predictions of Bernoulli–Euler models agree with the advanced models and to what extend the attenuation measure Ds is applicable for higher-order modes in a visco-elastic layer. Comparison between the Bernoulli–Euler and the Timoshenko theories reveals that the elementary formula DsB.−E . (η) = 1 η, strictly speaking, is valid only in the limit Ω → 0. The Timoshenko theory gives in this limit the following formula 4
DTs .
(η, Ω ) ≈ η + 4
. Ω +O Ω
48 α 2 − 1 + π 2 α 2
1
32π 2
2
(15)
3 α2 − 1
The ratio of the attenuation level as predicted by the Timoshenko model DTs . (η, Ω ) to the Bernoulli–Euler model’s attenuation level DsB.−E . (η) ≡ DTs . (η, 0) as a function of the frequency parameter and the Poisson ratio ν is shown in Fig. 3. As seen, the Bernoulli–Euler theory markedly underestimates the attenuation of the free wave. The Bernoulli–Euler is more accurate for the symmetric waves in the same low-frequency limit. The Herrmann–Mindlin model (and, consequently, the three-mode model) gives the following result:
DsH .−M .
(η, Ω ) ≈ η
1 2
−
α2 − 2
2
48 + α 2 π 2 − 48
2
384π 2 α 2 − 1
4 . Ω +O Ω 2
(16)
120
S. Sorokin, R. Darula / Wave Motion 68 (2017) 114–127
.−M . Fig. 4. The ratio of attenuation levels predicted by the Herrmann–Mindlin model DH (η, Ω ) and the Bernoulli–Euler model DBs .−E . (η) ≡ DHs .−M . (η, 0). s
2.5 1
2 1.5 1
0.5
0.5 0
0 0
1
2
3
4
5
6
0
1
2
3
4
5
Fig. 5. The scaled attenuation levels Dηs . (a): Herrmann–Mindlin model (lines) versus exact solution (dots); (b): the Timoshenko model (lines) versus exact solution (dots).
.−M . Similarly to the previous case, the ratio DH (η, Ω ) /DBs .−E . (η) , DsB.−E . (η) ≡ DHs .−M . (η, 0) as a function of the frequency s parameter and the Poisson coefficient ν is shown in Fig. 4. In this case, the Bernoulli–Euler theory also underestimates the attenuation of the free wave, but not to the same extend as for skew-symmetric waves. The comparison of attenuation levels of free waves predicted by approximate theories with the exact solution of the Rayleigh–Lamb problem is illustrated in Fig. 5 (Timoshenko and Herrmann–Mindlin model) and 6 (three-mode theories). As seen, the validity ranges of these theories regarding the attenuation measure Ds remain the same as for purely real wavenumbers for an ideally elastic layer. However, this measure does not seem to be appropriate in vicinity of the cut-on frequencies of higher-order modes. In a waveguide without damping, the cut-on frequency for a given branch of the dispersion diagram is found from the condition ki = 0, which can hold true either when kr = 0 (the conventional cut-on, see, for example, the second branch in Fig. 2(b)) or when kr ̸= 0 (see, for example, the second branch in Fig. 2(a)). In the 3D space (Ω , kr , ki ), this corresponds to the sharp turn of the branch onto the ki = 0 plane. The presence of damping invalidates this simple definition. As discussed in [19], a cut-on frequency range is identified as the frequency range at which the imaginary part of a wavenumber becomes sufficiently small and, more importantly, its decay rate diminishes and becomes to be ‘‘controlled’’ only by the damping coefficient. For small amounts of damping, the cut-on frequency range is centred at the cut-on frequency for a waveguide without material losses. Simultaneously, the attenuation measure Ds becomes very large in this range, as seen in Figs. 5 and 6. Interestingly, the proportionality of this measure to a magnitude of the loss factor η is preserved, see the scaling of the vertical axes in these figures. However, the dominant influence is produced by the frequency. As soon as it deviates from the cut-on value, the attenuation levels of all free waves acquire the same order of magnitude. To assess significance of this result, forcing problems should be solved in these cut-on frequency ranges, see Section 5. To conclude this section, we note that the spatial attenuation of free waves depends not only upon the loss factor as suggested by the elementary theories, but also upon the frequency.
S. Sorokin, R. Darula / Wave Motion 68 (2017) 114–127
121
2.5 2
1
1.5 1
0.5
0.5 0
0
1
2
3
4
5
0
6
0
1
2
3
4
5
Fig. 6. The scaled attenuation levels Dηs . (a): the three-mode theory for symmetric waves (lines) versus exact solution (dots); (b): the three-mode theory for skew-symmetric waves (lines) versus exact solution (dots).
5. Assessment of spatial wave attenuation in a visco-elastic layer for forcing problems As discussed in Section 2, the loss factor is defined as a measure of the energy dissipation in the material of an oscillator. For an infinite waveguide, it is natural to introduce an excitation source, calculate its power input and to consider decay k in the power flow as the observation point departs from a source. The applicability of the damping measure Ds = k i , r used for free waves, needs to be examined. An excitation at the frequencies slightly above the non-zero cut-on ones is of particular interest, because, as shown in the previous section, this measure is much different for co-existing free waves in these frequency ranges. The closely related issue is the sensitivity of attenuation levels to the excitation conditions at high frequencies, when more than just one wave can propagate in an ideally elastic waveguide. Since the attenuation of free waves predicted by reduced-order theories has been checked against the exact Rayleigh–Lamb solution, the forcing problems are considered here for small loss factors (η ≪ 1) by means of the reduced order models. 5.1. Elementary models For consistency, we begin with the elementary cases of attenuation of symmetric (longitudinal) and skew-symmetric (flexural) waves in the Bernoulli–Euler model. For propagation of symmetric waves excited by a point axial force F0 applied at the origin of coordinates, the power flow is defined as
1 N (x) = − Re F (x) V ∗ (x) 2 F0 ω ωη F (x) = − exp i x − x , 2 c0 2c0
(17) V ∗ (x) =
F0 c0 2EA
1+i
η 2
exp −i
ω c0
x−
ωη 2c0
x .
For simplicity, the analysis here is specialised for a uni-axial strain–stress state, c02 = E /ρ . Then formula (17) is reduced as N (x) =
F02 c0 8EA
exp −
ωη c0
x .
The energy injected into the waveguide is gradually dissipated as the wave travels away from the source. To measure this dissipation, a characteristic length should be introduced. The natural choice is the wavelength of the free wave in the absence 2π c of dissipation, λ = 2kπ ≡ ω 0 . For small damping, kr ≈ k0 . Then the spatial energy dissipation rate is 0
D (x0 , Ω , η) =
∆N (x0 ) N (x0 ) − N (x0 + λ) = = 1 − exp (−2π η) ≈ 2π η. N (x0 ) N (x0 )
Alternatively, this formula may be obtained as D (x0 , Ω , η) = ln
N ( x0 ) N (x0 + λ)
= 2π η.
It has the same magnitude regardless of the choice of the reference point x0 , because the theory does not predict evanescent waves and, therefore, the near-field boundary layers do not exist. As seen, this formulation matches exactly the definition of the temporal loss factor for a single-degree-of-freedom oscillatory system with viscous damping.
122
S. Sorokin, R. Darula / Wave Motion 68 (2017) 114–127
Fig. 7. Upper surface — the attenuation level D (x0 , Ω , η) from the Timoshenko theory, lower surface — the attenuation level D (x0 , Ω , η) from the Bernoulli–Euler theory.
In the case of Bernoulli–Euler model of flexural waves (with the conventional beam theory definitions and notation), the power flow is defined as N (x) =
1 2
Re Q (x) V ∗ (x) +
1 2
Re M (x) VΓ∗ (x) .
(18)
Here Q (x) and M (x) are the shear force and the bending moment, while V ∗ (x) and VΓ∗ (x) are complex conjugates of transverse and rotation velocities of a cross-section of the beam. Formulas for these state variables are well known and not presented here. We consider the excitation by a point transverse force at x = 0: Q (0) = 12 sgn (±ε) , ε → 0w ′ (0) = 0. The characteristic length to scale the spatial decay rate of flexural wave equals the length of a propagating wave in the absence of dissipation λ = 2kπ ≡ 2π 0
point x0 , D (x0 , Ω , η) = ln
c0 rg
ω
with rg as the gyration radius. The dependence of the spatial attenuation at the observation
N (x0 ) N (x0 +λ)
upon the loss factor η in the far-field zone (x0 → ∞) coincides with the function
Dfree (Ω , η) = 4π = 4π Ds . As shown in Section 3, the quantity 2π Ds = 2π kri defines the decay rate of the amplitude of a travelling wave over its length. The slopes, bending moments and shear forces have the same decay rate, because the Bernoulli–Euler model is a single-mode one, see Table 1. As follows from formula (18) the rate of the energy dissipation should be twice as large. Analysis of the excitation by a point moment gives same results. As expected, the investigation of the energy dissipation in forcing problems for the elementary theories fully agrees with the classical results [1]. ki kr
k
5.2. Timoshenko and Herrmann–Mindlin models In the Timoshenko theory, the formula (18) for the power flow remains valid provided that shear force, bending moment and angle of rotation of cross-section are defined properly. The attenuation level D (x0 , Ω , η) obtained in the framework of the Bernoulli–Euler theory (lower surface) is compared with its counterpart from the Timoshenko theory (upper surface) as illustrated in Fig. 7 for η = 0.01 and ν = 0.3. Wavelengths λ are calculated accordingly to the theories used. There is a significant difference between predictions of these theories both frequency-wise and the reference pointwise as seen in this figure. Only at very low frequencies the predictions are in a reasonable agreement. The near-field phenomenon, which does not exist for axial waves, is also illustrated in this figure. If the reference point x0 is located in the near field, where an evanescent wave is ‘visible’, then the loss factor exceeds its ‘nominal’ far-field value. As soon as frequency grows and the reference point departs from the source, both theories predict the levels, which differ from each other, but match the attenuations of the free wave Dfree (Ω , η) for each theory. In Fig. 8, the comparison between ‘forced’ attenuation rate D (x0 , Ω , η) (upper surface, the same as the upper surface in Fig. 7) and the attenuation rate of the free wave Dfree (Ω , η) (lower surface) is illustrated for the Timoshenko theory. The differences vanish as the reference point moves away from the source and the frequency grows. The attenuation levels predicted by the Timoshenko theory in two excitation cases: by a point force and by a point moment are compared in Fig. 9. In the near field, the excitation by a moment gives attenuation levels more sensitive to the location of the reference point. In the far field and at higher frequencies there is no difference in the level of attenuation. As known, the Timoshenko theory agrees with the exact elasto-dynamics in predictions of the second branch of dispersion diagram. In Fig. 10, the same diagram as in Fig. 7 after this theory is plotted for higher frequencies. As soon as the second
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Fig. 8. Upper surface — the attenuation level D (x0 , Ω , η), lower surface — the attenuation level Dfree (Ω , η), the Timoshenko theory.
Fig. 9. Upper surface — the attenuation level for a point force excitation, lower surface — the attenuation level for a point moment excitation.
Fig. 10. The attenuation level D (x0 , Ω , η) for a point force excitation around the cut-on frequency, the Timoshenko theory.
wave cuts on, the attenuation level at a given frequency becomes a periodic function of the location of the reference point x0 .
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The two-mode theory for symmetric waves gives results, similar to those obtained from the Timoshenko theory. However, as can be seen from the dispersion diagram in Fig. 2(b), this theory does not predict correctly the shape of the second branch. Therefore, these results are not discussed here, but mentioned, where appropriate, in discussion of the threemode theory for symmetric waves. 5.3. Three mode theories As shown in Section 3, the three-mode theories capture three branches of dispersion diagrams for symmetric and skewsymmetric waves. Therefore, these theories may be used to solve forcing problems at frequencies beyond the validity ranges of Bernoulli–Euler, Timoshenko and Herrmann–Mindlin ones. The power flow is defined as follows: For the symmetric waves Nsymm (x) =
1 ωh 2 c1
Re i F0 (x) · U0∗ (x) + F1 (x) · V1∗ (x) + F2 (x) · U2∗ (x)
(19)
(20)
with the forces and displacements defined as given in (9). For the skew-symmetric waves Nskew (x) =
1 ωh 2 c1
Re i F0 (x) · V0∗ (x) + F1 (x) · U1∗ (x) + F2 (x) · V2∗ (x)
with the forces and displacements defined as given in (12). The forcing conditions are formulated for three alternative excitation cases at x = 0: 1. F0 (0 ± ε) =
1 2
sgn (±ε) ,
ε → 0;
2. Y0 (0) = 0;
F1 (0 ± ε) =
3. F0 (0) = 0;
Z1 (0) = 0;
1 2
Z1 (0) = 0;
sgn (±ε) ,
ε → 0;
F2 (0 ± ε) =
1 2
F2 (0) = 0
(21a)
Y2 (0) = 0
(21b)
sgn (±ε) ,
ε → 0.
(21c)
For symmetric waves, Z1 ≡ V1 , Y0,2 ≡ U0,2 and the forces are defined by (9). For skew-symmetric waves, Z1 ≡ U1 , Y0,2 ≡ V0,2 and the forces are defined by (12). In the low-frequency range, when only one wave propagates in a waveguide, the three-mode theories confirm the assessment of attenuation obtained by means of Bernoulli–Euler (symmetric) and Timoshenko (skew-symmetric) models. The results reported hereafter are obtained in the frequency ranges, where the three-mode theories agree with the exact solutions of the Rayleigh–Lamb problem in describing three propagating waves. The issues of interest are, besides the frequency dependence of attenuation levels, the influence of location of the reference point and the influence of excitation conditions. We begin with the attenuation levels of skew-symmetric waves generated in the loading conditions (21). In Fig. 11(a), (b), the attenuation levels D (x0 , Ω , η) versus excitation frequency are shown for three locations of the reference point: x0 = 5 (green curve), x0 = 50 (blue curve) and x0 = 500 (red curve) for ν = 0.3 and η = 0.01. For reference, the k attenuation level for the first free wave Dfree (Ω , η) ≡ 4π k i is plotted in magenta and it almost coincides with the red r curve. The mid-frequency range covered in this figure captures two regimes: below and above the cut-on frequency of the second wave, and there is a qualitative difference between these regimes. In the frequency range, where only one wave propagates, the decay rates D (x0 , Ω , η) are the same regardless of position of the reference point and regardless of the loading case. In this regime, the measure Dfree (Ω , η) suffices (although it should be calculated in accordance with the threemode model of skew-symmetric waves). Around the cut-on frequency, the attenuation levels become very sensitive to the position of a reference point. If this point is located relatively close to the excitation point, these levels may be much higher than Dfree (Ω , η) for the first propagating wave, as clearly seen in Fig. 11(b). In Fig. 12, the frequency range is shifted higher, to capture the cut-on frequency of the third wave. In this situation, all three loading cases are considered and, qualitatively, the results are the same as in the previous case above the cut-on frequency. This is readily explained by inspection into the components of the power flow (see, for example, [21]). In the near field, the energy injected into the waveguide is distributed between the ‘alternative transmission paths’, i.e., between propagating waves. If a waveguide supports only one wave of this type, this distribution is unique and, in a viscoelastic layer, the far-field decay rate is controlled solely by Dfree (Ω , η) of this wave. As soon as more than one wave propagates, the effective attenuation builds up as a result of superposition of free waves, each of which has its own dissipation rate. It should be noted that the attenuation levels of free waves Dfree (Ω , η) have been calculated in Section 3 with their own wavenumbers, and, therefore, attained extremely large values slightly above cut-on frequencies. The measure D (x0 , Ω , η) for forcing problems is introduced with the shortest wavelength, so that, effectively, the attenuation level of a free wave around its cut-on is re-scaled. Another feature of co-existing travelling waves is that the energy distribution between them
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Fig. 11. The attenuation levels D (x0 , Ω , η) in the mid-frequency range for the skew-symmetric excitation. (a) loading case (21a); (b) loading case (21b). x0 = 5 (green curve), x0 = 50 (blue curve) and x0 = 500 (red curve). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 12. The attenuation levels D (x0 , Ω , η) in the high frequency range for the skew-symmetric excitation. (a) loading case (21a); (b) loading case (21b); (c) loading case (21c). x0 = 5 (green curve), x0 = 50 (blue curve) and x0 = 500 (red curve). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
is dependent on the distance from the excitation point. As these waves have different decay rates, the total attenuation is also dependent upon x0 , as clearly seen from Fig. 12(a)–(c). The three-mode model for symmetric waves gives qualitatively same results as its counterpart for skew-symmetric ones. The point of particular interest is the analysis of the energy dissipation in the high frequency range, where, in the absence of damping, the wave having group and phase velocities of opposite signs, exists, see Fig. 2(a). As known, the wave with the positive group velocity has to be used for solving the forcing problem. Results obtained in three excitation cases in this frequency range are presented in Fig. 13 in the same way as in Fig. 12 also for ν = 0.3 and η = 0.01. The attenuation levels are strongly dependent upon the frequency and the position of a reference point above the cut-on frequency of the second wave, which in the absence of damping should be found from the condition ddkω = 0 rather, than from the conventional condition k = 0. There is also the substantial difference in the attenuation levels D (x0 , Ω , η) between three loading cases above this cut-on frequency, whereas these levels are the same and equal to Dfree (Ω , η) when only one wave propagates in the waveguide. Finally, the difference between dependence of D (x0 , Ω , η) upon x0 below (at Ω = 2.6) and above (at Ω = 3.1) the cuton frequency of the second wave is demonstrated in Fig. 14(a) and (b) for the loading conditions (21a). In the former case,
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S. Sorokin, R. Darula / Wave Motion 68 (2017) 114–127
Fig. 13. The attenuation levels D (x0 , Ω , η) in the high-frequency range for the symmetric excitation. (a) loading case (21a); (b) loading case (21b); (c) loading case (21c). x0 = 5 (green curve), x0 = 50 (blue curve) and x0 = 500 (red curve). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 14. The dependence of attenuation level D (x0 , Ω , η) upon the position of the reference point x0 in the high-frequency range for the symmetric excitation. (a) loading case (21a); (a) −Ω = 2.6; b) Ω = 3.1.
this dependence exists only in the near field, where the evanescent waves are still noticeable. In the latter case, as already discussed, the energy exchanges between co-existing travelling waves are present in the far-field and so the attenuation levelD (x0 , Ω , η) fluctuates. To conclude this section, we note that the spatial energy dissipation rate depends not only upon the loss factor and the excitation frequency as explained in Section 4 for free waves, but also upon the position of the reference point and upon the excitation type. The latter becomes essential as soon as a waveguide supports several propagating waves. To capture this effect, it is necessary to use multi-modal theories summarised in Table 1. 6. Conclusions The reported results suggest that a concept of the loss factor, in principle, is plausible for the analysis of attenuation of free and forced waves in a visco-elastic layer, because both the temporal and the spatial attenuation levels may be expressed via its magnitude η. The difficulty lies in the functional link between the loss factor and the attenuation levels. In the framework described in Section 2, the temporal decay of free waves in a visco-elastic layer is determined in full agreement with [1] by the elementary formula Dt = 21 η. The applicability of the loss factor as a sole parameter,
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which controls the spatial decay of free waves, is case-sensitive simply because of its frequency-dependence illustrated in Section 4. In the low-frequency limit, the results obtained in [1] by means of the classical Bernoulli–Euler models for both symmetric (Ds = Dt = 21 η) and skew-symmetric (Ds = 21 Dt = 14 η) free waves are fully confirmed by comparison with the results obtained from more advanced models. However, these classical formulas underestimate the damping level at higher frequencies. To the leading order, the frequency dependence of spatial attenuation of the first propagating wave is linear in the skew-symmetric case and quadratic in the symmetric case. Assessment of spatial attenuation of high-order free waves shows that it is proportional to the loss factor, but its magnitude is mainly frequency-controlled, especially around cut-on frequencies. In a forcing problem, the loss factor gives a fair estimate of spatial power flow decay only in the low-frequency range, where the waveguide supports one propagating wave of a given (symmetric or skew-symmetric) type. In this case, the attenuation rate Dfree (Ω , η) calculated with the wavelength of the first propagating wave as a reference distance may be reliably used. In the situations, when a waveguide supports more than one wave of the same type, the attenuation level becomes also a function of the position of the point x0 , from which the reference distance is measured, D (x0 , Ω , η). On top of that, the loading conditions influence the attenuation levels. This is explained by partition of the total energy input between free propagating waves, each of which has its own decay rate. On balance, the magnitude of D (x0 , Ω , η) may significantly deviate from Dfree (Ω , η). Therefore, a reliable prediction of the spatial energy dissipation rate in the situation of a high-frequency excitation appears to be very difficult, if based solely on the knowledge of the loss factor. Acknowledgement The financial support for the second author from the Danish Council for Independent Research (the postdoctoral research grant DFF-4005-00348) is gratefully acknowledged. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
L. Cremer, M. Heckl, B.A.T. Petersson, Structure-Borne Sound, Springer-Verlag, Berlin Haidelberg, New York, 2005. J.D. Achenbach, Wave Propagation in Elastic Solids, North-Holland, Amsterdam, 1973. S.S. Rao, Mechanical Vibrations, fourth ed., Prentice Hall, New Jersey, USA, 2004. E.E. Ungar, E.M. Kerwin, Loss factors of viscoelastic systems in terms of energy concepts, J. Acoust. Soc. Am. 34 (1962) 954–957. C.D. Johnson, D.A. Kienholz, Finite element prediction of damping in structures with constrained viscoelastic layers, AIAA J. 20 (1982) 1284–1290. E. Manconi, B.R. Mace, Estimation of the loss factor of viscoelastic laminated panels from finite element analysis, J. Sound Vib. 329 (2010) 3928–3939. E. Manconi, B.R. Mace, R. Garziera, The loss-factor of pre-stressed laminated curved panels and cylinders using a wave and finite element method, J. Sound Vib. 332 (2013) 1704–1711. P.J. Shorter, Wave propagation and damping in linear viscoelastic laminates, J. Acoust. Soc. Am. 115 (2004) 1917–1925. ANSYS Customer Training ‘ANSYS Structural Analysis Guide’ Chapter 4 Located 20th November 2013. S.H. Crandall, The role of damping in vibration theory, J. Sound Vib. 11 (1970) 3–18. G.B. Muravskii, On frequency-independent damping, J. Sound Vib. 274 (2004) 653–668. D. Jones, Handbook of Viscoelastic Vibration Damping, John Wiley & Sons, Chichester, GB, 2001. I. Bartoli, A. Marzani, F.L. di Scalea, E. Viola, Modeling wave propagation in damped waveguides of arbitrary cross-section, J. Sound Vib. 295 (2006) 685–707. A. Bernard, M.J.S. Lowe, M. Deschamps, Guided waves energy velocity in absorbing and non-absorbing plates, J. Acoust. Soc. Am. 110 (2001) 186–196. F. Simonetti, M.J.S. Lowe, On the meaning of lamb mode non-propagating branches, J. Acoust. Soc. Am. 118 (2005) 186–192. F. Simonetti, P. Cawley, A guided wave technique for the characterization of highly attenuative viscoelastic materials, J. Acoust. Soc. Am. 114 (1) (2003) 158–165. A. Marzani, E. Viola, I. Bartoli, F.L. di Scalea, P. Rizzo, A semi-analytical finite element formulation for modeling stress wave propagation in axisymmetric damped waveguides, J. Sound Vib. 318 (2008) 488–505. R.D. Mindlin, in: J. Yang (Ed.), An Introduction to the Mathematical Theory of Vibrations of Elastic Plates, World Scientific, Singapore, 2006. E. Manconi, S.V. Sorokin, On the effect of damping on dispersion curves in plates, Int. J. Solids Struct. 50 (2013) 1966–1973. S.V. Sorokin, C.J. Chapman, A hierarchy of rational Timoshenko dispersion relations, J. Sound Vib. 330 (2011) 5460–5473. S.V. Sorokin, C.J. Chapman, A hierarchy of high-order theories for symmetric modes in an elastic layer, J. Sound Vib. 333 (2014) 3505–3521. I. Elishakoff, Yu. Kaplunov, E. Nolde, Celebrating the centenary of Timoshenko’s study of effects of shear deformation and rotary inertia, Appl. Mech. Rev. 67 (2015) 060802. L.I. Slepyan, Transient Elastic Waves, Sudostroenie, Leningrad, 1973, in Russian. J.F. Doyle, Wave Propagation in Structures, Springer, New York, 1997.
Contents lists available at ScienceDirect
Wave Motion journal homepage: www.elsevier.com/locate/wavemoti
On attenuation of free and forced waves in an infinitely long visco-elastic layer of a constant thickness Sergey Sorokin ∗ , Radoslav Darula Department of Mechanical and Manufacturing Engineering, Aalborg University, Fibigerstrœde 16, 9220 Aalborg, Denmark
highlights • A hierarchy of reduced order models is employed to analyse wave propagation in a viscoelastic layer. • The measures of attenuation of free and forced waves are proposed and compared with each other. • A high sensitivity of the attenuation levels to excitation conditions at high frequencies is demonstrated and explained.
article
info
Article history: Received 23 May 2016 Received in revised form 21 August 2016 Accepted 3 September 2016 Available online 17 September 2016 Keywords: Loss factor Wave attenuation Visco-elastic layer Free and forced waves Reduced order models
abstract The conventional concepts of a loss factor and a complex-valued elastic module are used to study wave attenuation in a visco-elastic layer. The hierarchy of reduced order models is employed to assess attenuation levels of free and forced waves in various situations. First, the free waves are considered. In the low frequency limit, the attenuation of these waves is found to be in the excellent agreement with the existing knowledge. At high frequencies, predictions of the reduced order models fully agree with the solutions of exact Rayleigh–Lamb problem. Alternative excitation cases are considered for the forcing problem and a measure of the attenuation level is proposed and validated. The differences between two regimes, the low frequency one, when a waveguide supports only one propagating wave, and the high frequency one, when several waves are supported, are demonstrated and explained. © 2016 Elsevier B.V. All rights reserved.
1. Introduction The assessment of origins and actual levels of damping has always been a challenging task in various applications such as structural intensity analysis and non-destructive structural health monitoring. As a manifestation of uncertainties in this assessment, commercially available finite element software typically leaves to a customer the choice of parameters defining the extent to which the damping matrix is proportional to the stiffness and to the mass ones. On the other hand, the quantification of viscous damping in linear one-degree-of-freedom systems is truly elementary and based on simple energy considerations, which bring to light a concept of the loss factor. Therefore, it has long been attempted to extend this quantification to multi-degree-of-freedom and continuous systems. This paper goes along these lines and is concerned with assessment of the attenuation of time-harmonic waves in an infinitely long visco-elastic layer of a constant thickness. The goal is to demonstrate to what extent the loss factor, generally recognised as a classical measure of the energy dissipation in mechanical systems of finite dimensions, is applicable to estimate spatial and temporal attenuation of waves in visco-elastic structures.
∗
Corresponding author. E-mail address: [email protected] (S. Sorokin).
http://dx.doi.org/10.1016/j.wavemoti.2016.09.001 0165-2125/© 2016 Elsevier B.V. All rights reserved.
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For the longitudinal (i.e., the first symmetric) and the flexural (i.e., the first skew-symmetric) waves, this has been done in the classical text [1] by means of the elementary Bernoulli–Euler theory known to be asymptotically consistent in the low frequency — long wave limit with the exact Rayleigh–Lamb solution, see [2]. In fact, the wave attenuation analysis reported in [1] serves as a theoretical background to link the material loss factor and the imaginary part of Young’s module. However, such an assessment has not yet been done for the two-mode (i.e., Timoshenko and Herrmann–Mindlin) models as well as for higher-order models. This paper is aimed to fill in this gap and to give a systematic account of the free waves’ attenuation in the framework of simplified theories verified against the exact solution. Alongside, an assessment of wave attenuation in the case of a forced response of an infinite waveguide, which supports several propagating waves, constitutes the research goal of this paper. To the best of the authors’ knowledge, these issues have not yet been fully addressed in the literature. The paper is structured as follows. Section 2 is concerned with discussion of the concepts of the loss factor and complexvalued Young’s module. In Section 3, the brief exposition of the canonical Rayleigh–Lamb problem is presented and a hierarchy of reduced order models of wave propagation in a layer is described. In Section 4, the alternative measures of attenuation of free waves are discussed. This section is also concerned with comparison of such attenuation predicted by low order theories and in the exact formulation. In Section 5, an attenuation measure for the forced response is proposed and used in the framework of the theories presented in Section 3. Attenuations of free and forced waves are compared with each other in the representative frequency ranges and the influence of excitation conditions is explored. The findings are summarised in conclusions. 2. The loss factor and complex-valued Young’s module As the wave attenuation is caused by the internal material damping in a structure exhibiting steady state cyclic deformation, it is natural to characterise the energy dissipation in terms of energy quantities. The loss factor, defined as N a ratio of the energy dissipated per oscillation cycle ND and the total energy N, see [1,3,4]: η = 2πDN , is such a measure, originally introduced for single-degree-of-freedom systems. Since any linear oscillatory system may be decomposed into a set of uncoupled oscillators, loss factors are commonly used in the modal format, i.e., to assess the level of attenuation at each mode of vibration of a multi-degree-of-freedom or continuous system of finite dimensions. The point to be addressed in this paper is how the concept of a modal loss factor is adjusted in a waveguide theory for calculation of the level of attenuation of otherwise (i.e., in the absence of viscosity) propagating free waves, not necessarily in the low frequency — small wavenumber limit. It should be noted that Refs. [5–7] are concerned with the estimation of loss factors for waves propagating in laminated structures by means of the wave finite element method. The comprehensive literature survey on the damping estimation in the framework of the finite element method is also presented in [6]. However, the modal strain energy method as formulated in Eq. (10), p. 3931 in this reference gives the assessment of attenuation of an individual free wave. In what follows in the present paper, a model, much simpler than those introduced in abovementioned references, is considered to facilitate a rigorous analysis of attenuation of free and forced waves based on the canonical definition of the loss factor. The authors are of the opinion that such an analysis will serve as a novel useful supplement to the findings reported in [5–7]. Steady state harmonic vibrations hereafter are treated with the time-dependence of all state variables chosen in the form exp (−iωt). This modelling brings to light the complex-valued Young’s module [1] with (given the stated above sign convention) the negative imaginary part, Eˆ = E − iEi . It is in common practice to present the imaginary part of the complex-valued Young’s module via a loss factor in the form Ei = ηE. Here the loss factor η is considered as frequencyindependent (see [1, p. 153]). This model is widely used in the conventional harmonic finite element analysis, where the complex-valued stiffness matrices have been introduced and used in many research papers with [6–8] being just a few examples. Furthermore, the commercially available finite element software, such as ANSYS, utilises this concept [9]. Classical Refs. [1,3,10–12] explain pros and cons of the model used hereafter and provide the detailed discussion of damping mechanisms. In the classical theory of linear wave propagation in viscoelastic materials, introduction of the complex-valued Young’s module entails introduction of complex-valued propagation speeds: ‘If one substitutes the modulus of elasticity ¯ = D′ (1 + iη) into the wave equation, one finds that the propagation speed becomes complex’ ([1, p. 155], the third and D fourth lines from the bottom, time dependence is taken as exp (iωt)). The concept of a complex-valued propagation speed has been adopted in several publications [13–17] and implemented in SAFE (semi-analytical finite element) method. We employ this concept in an analytical solution of the canonical Rayleigh–Lamb problem. 3. The low-order models of wave propagation in an elastic straight layer of constant thickness We consider an infinite elastic layer in the plane strain state with the traction-free boundary conditions, as shown in Fig. 1(a). Analysis of the wave propagation in such a layer constitutes the classical Rayleigh–Lamb problem. Due to the symmetry in the boundary conditions at ydim = ± 2h , the symmetric and skew-symmetric modes (see Fig. 1(b)) are decoupled from each other. The conventional spatial dependence is taken as exp (ikdim xdim ), so that the dispersion equation for free symmetric modes is [2, p. 223]: tan tan
q 2p = − 2
4k2 pq
q2 − k2
2 .
(1)
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Fig. 1. (a) An elastic layer; (b) Symmetric and skew-symmetric deformation.
In this equation, the non-dimensional parameters scaled to the thickness of an elastic layer h are: k = kdim h, Ω = ωc h , p2 = 1 Ω 2 − k2 and q2 = α 2 Ω 2 − k2 . The notation α = cc1 ≡ 2(1(1−−ν) , where ν is the Poisson ratio, is used hereafter. The speed of 2ν) 2 E (1−ν) dilatation wave is c1 = ρ(1+ν)(1−2ν) and the speed of shear wave is c2 = 2ρ(1E+ν) .
For skew-symmetric waves, the dispersion equation is [2, p. 223]: tan tan
q
2 p 2
=−
q2 − k2 4k2 pq
2 .
(2)
The solutions of dispersion equations (1) and (2) are used as the reference to assess the validity of the low-order models. In the absence of damping, these solutions give the full account of propagating, evanescent and attenuated waves (see [2,18]). Usually, only purely real wavenumbers corresponding to travelling waves are computed. Due to damping, all roots of dispersion equations are complex-valued, and calculations of these wavenumbers from exact transcendental dispersion equations become rather challenging [19]. In this situation, it is particularly practical to employ low-order models, or theories, because dispersion equations derived from Bernoulli–Euler, Herrmann–Mindlin and Timoshenko models are polynomials, and it does not present any difficulty to compute all their roots at once. The same holds true regarding a hierarchy of the finite product method approximations and the Legendre approximations shown to be in the excellent agreement with exact solution in the absence of damping [20,21]. In the remaining part of this section we introduce the reduced-order theories used hereafter for the assessment of wave attenuation in a visco-elastic layer. The Bernoulli–Euler models specialised for the plane strain case are well known. For symmetric waves, the dispersion equation is k2 −
α4 Ω 2 = 0. 4 α2 − 1
(3)
For skew-symmetric waves, it is k4 −
3α 4
α2 − 1
Ω 2 = 0.
(4)
Both these low-frequency models capture only one propagating wave and are asymptotically consistent with the exact equations (1) and (2) in the low frequency limit. The Timoshenko model may be formulated in many alternative ways, see the detailed survey [22]. In what follows, we use this theory as explained in [20]. Then the dispersion equation acquires the form (see [20, p. 5462, Eq. (6)]): k4 −
12α 2
π2
k2 Ω 2 −
3α 4
α2 − 1
Ω2 +
3α 6 Ω 4 = 0. π 2 α2 − 1
(5)
Formulation of the Herrmann–Mindlin model may be found in [18]. The dispersion equation converted to the same notations as in (1)–(5) is
α 2 12α 2 + π 2 2 2 α2 α4 α4 4 2 k + k − Ω2 + Ω 4 = 0. k Ω − 2 2 2 2 2 48 α − 1 48 α − 1 π 4 α −1 4π α 2 − 1
(6)
These models capture two propagating waves. The Timoshenko theory gives an accurate prediction of location of the first branch of the dispersion diagram in a broad frequency range. Its accuracy with respect to the location of the second branch is controlled by a choice of the shear correction factor. If it is chosen as in Eq. (5), then the frequency range, where this theory gives acceptable results for the second branch, is of the same size as the size of the frequency range of applicability of the Bernoulli–Euler theory for the first branch. To advance to high frequencies with reduced order models, various methods may be applied, as discussed in [18,23,24]. The hierarchy of these models for symmetric waves in this waveguide has been formulated in [21] from the exact formulation of elasto-dynamics by means of expansion of displacements on Legendre polynomials in the transverse coordinate y and use of the Hamilton principle. The first terms in series for displacements are polynomials of the zeroth and the first order
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in the transverse coordinate. Therefore, the low-order approximations become identical to the conventional elementary theories, which employ the ‘plane cross-section’ hypothesis. Extension of these expansions to higher-order terms allows one to account for an arbitrary large number of branches of dispersion diagram, with this number being controlled by the truncation level in expansions of displacements. l x The kinetic energy per unit width of a layer, extended from x = 0 to x = l ≡ dim in the scaled axial coordinate x = dim , h h is: T =
1 2
ρh
4
1/2
l
−1/2
0
∂ u (x, y, t ) ∂t
2
+
∂v (x, y, t ) ∂t
2
dxdy.
(7a)
The potential energy per unit width is: V =
1 2
ρ h2 c22
+
1/2
−1/2
l 2 ∂ u (x, y, t ) 2 ∂v (x, y, t ) ∂ u (x, y, t ) α + α − 2 ∂x ∂y ∂x 0
2 ∂ u (x, y, t ) ∂v (x, y, t ) α −2 + α2 ∂x ∂y
∂v (x, y, t ) + ∂y
∂v (x, y, t ) ∂x
2
dxdy.
(7b)
The differential equations for symmetric waves in the three-mode theory are obtained with the following ansatz for displacements: u (x, y, t ) = U0 (x, t ) · 1 + U2 (x, t ) · 1 − 12y2 ,
v (x, y, t ) = V1 (x, t ) · y.
Substitution of this ansatz to formulas (7) and standard by parts integration in δ H = δ
(8)
t2 t1
[T − V ] dt = 0 gives the
system of linear differential equations with respect to U0 (x, t ) , V1 (x, t ) and U2 (x, t ). The boundary/loading conditions at the edges x = 0, x = l are obtained as the non-integral terms in the course of by-parts integration. These are concerned with the following pairs of generalised forces and displacements: F0 (x, t ) ≡ F1 (x, t ) ≡ F2 (x, t ) ≡
1/2
−1/2
1/2
σxy (x, y, t ) · ydy =
∂ U0 (x, t ) 2 + α − 2 V1 (x, t ) and U0 (x, t ) ∂x
1 ∂ V1 (x, t )
(9a)
− 2U2 (x, t ) and V1 (x, t )
(9b)
4 ∂ U2 (x, t ) σx (x, y, t ) · 1 − 12y2 dy = α 2 and U2 (x, t ) . 5 ∂x −1/2
(9c)
−1/2
σx (x, y, t ) · 1dy = α 2
12
∂x
1/2
To derive the dispersion equation, the standard substitution U0 (x, t ) = U00 exp (ikx − iωt ) ,
V1 (x, t ) = V10 exp (ikx − iωt ) ,
U2 (x, t ) = U20 exp (ikx − iωt )
to the differential equations of motion with the inertia-correction factors (see [21] for details) is used. The dispersion equation of the sixth order both in the frequency parameter and the wavenumber has the following form:
α 4 15 + 12α 2 + π 2 4 2 α 2 −60 + 120α 2 + π 2 α 4 2 2 α2 α4 k6 + k4 − k Ω k Ω + k2 − 60 2880 α 2 − 1 48 α 2 − 1 240 α 2 − 1 π 2 5π 2 α 4 + 4α 2 15 + π 2 2 4 α4 4 + α2 α4 α6 Ω4 − Ω 6 = 0. Ω2 + + k Ω − 960 α 2 − 1 π 4 4 α2 − 1 16π 2 α 2 − 1 16π 4 α 2 − 1
(10)
The derivation of the model for skew-symmetric waves is similar. The displacements are presented in the truncated series on Legendre polynomials: u (x, y, t ) = U1 (x, t ) · y,
v (x, y, t ) = V0 (x, t ) · 1 + V2 (x, t ) · 1 − 12y2 .
(11)
The boundary/loading conditions are concerned with the following pairs of generalised forces and displacements: F0 (x, t ) ≡ F1 (x, t ) ≡ F2 (x, t ) ≡
1/2
−1/2
1/2
−1/2
σxy (x, y, t ) · 1dy = σx (x, y, t ) · ydy =
∂ V0 (x, t ) + U1 (x, t ) and V0 (x, t ) ∂x α 2 ∂ U1 (x, t ) + 2 α 2 − 2 V2 (x, t ) and U1 (x, t ) 12 ∂x
(12a)
(12b)
1/2
4 ∂ V2 ( x , t ) σxy (x, y, t ) · 1 − 12y2 dy = and V2 (x, t ) . 5 ∂x −1/2
(12c)
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5
4
4 Kr
5
3
3
2
2
1
1
0
0
1
2
3
4
5
0
6
0
1
2
3
4
5
Fig. 2. The dispersion diagrams after exact solution (1) and (2), Mindlin–Herrmann and Timoshenko theories (5) and (6) and three-mode theories (10) and (13): (a) symmetric waves; (b): skew-symmetric waves. Table 1 Models used in the assessment of wave attenuation. Model
No. of waves/Order
Symmetric
Skew-symmetric
Bernoulli–Euler Herrmann–Mindlin Timoshenko Three-mode theory
1 2 2 3
× ×
× × ×
×
The dispersion equation is of the sixth order both in the frequency parameter and the wavenumber:
−
α 2 36 + 20α 4 + 3π 2 α 2 4 2 19π 2 α 2 − 20α 4 3 + π 2 2 2 α2 k6 + k4 − k Ω + k Ω 240 α 2 − 1 720 α 2 − 1 π 2 20 α 2 − 1 π 2 α 4 60α 2 + 9π 2 + 5α 4 π 2 2 4 10α 6 3α 4 α8 2 Ω4 − Ω 6 = 0. + Ω + k Ω − 2 α −1 180 α 2 − 1 π 4 3π 2 α 2 − 1 3π 4 α 2 − 1
(13)
In Fig. 2, the first three branches of dispersion diagrams after exact solution (1) and (2) are plotted. These are purely real wavenumbers in the absence of damping. For comparison, dispersion diagrams after reduced order theories are also plotted. The accuracy of Timoshenko and Herrmann–Mindlin theories has been a subject of the thorough analysis for years. We just note that the Herrmann–Mindlin theory does not perform as good as the Timoshenko one (see also, for instance, [18, p. 150]. As can be seen from Fig. 2, the three mode theories are equally good to approximate in a relatively broad frequency range the first three branches of dispersion diagrams for symmetric and skew-symmetric waves. The overview and classification of the low-order theories introduced in this section are summarised in Table 1. 4. Assessment of attenuation of free waves in a visco-elastic layer We begin with the analysis of free waves. It is confined to the solutions of the dispersion equations presented in Section 3. These transcendental and polynomial equations contain only purely real coefficients and can be written in the generic form: F Ω 2 , k2 = 0.
(14)
The odd powers of a wavenumber are absent due to the natural symmetry of the waveguide. The odd powers of the frequency parameter are also absent due to the choice of the model of material damping. If the complex-valued Young’s module ˜ ≈ Ω 1 + 1 iη . ˜ = √ Ω , or for η ≪ 1, Ω E˜ = E (1 − iη) is introduced, the frequency parameter becomes Ω 2 1−iη 4.1. Temporal and spatial attenuation The attenuation of free waves may be assessed from two viewpoints. The first one is similar to the assessment of damping for free vibrations of a single-degree-of-freedom system. It is the temporal decay over one period of oscillations [1,3]. For an infinite waveguide, it means that a free wave defined by a purely real wavenumber is described by a complex-valued ω frequency ω = ωr − iωi . Then the temporal attenuation Dt is characterised by the ratio ω i , or, in the non-dimensional form, ˜i Ω ˜r Ω
ω
. For a single-degree-of-freedom system, this ratio is simply ω i = r
r
1 2
η. The dispersion equation (14) remains the same
S. Sorokin, R. Darula / Wave Motion 68 (2017) 114–127
119
Fig. 3. The ratio of attenuation levels predicted by the Timoshenko model DTs . (η, Ω ) and the Bernoulli–Euler model DBs .−E . (η) ≡ DTs . (η, 0).
for the non-dimensional frequency Ω regardless of whether elastic or visco-elastic material is considered. Therefore, the temporal attenuation of a free wave with any purely real wavenumber has the same rate Dt = 12 η. The alternative characterisation of attenuation is concerned with the free wave at a purely real frequency ω. In this ˜ becomes complex-valued, and so do all wavenumbers found from the case, the non-dimensional frequency parameter Ω k dispersion equation (14). The spatial attenuation is characterised by the ratio Ds = k i . For consistency, the derivation of this r formula is presented here, although it is completely analogous to the classical derivation of the formula for temporal decay rate (see [3]). We consider a free wave, characterised by the amplitudes of displacements and forces in the form Z (x, t ) = Z0 exp (ikx − iωt ) = Z0 exp (−ki x)· exp (ikr x − iωt ). Here Z0 exp (−ki x) may be the amplitude of any of these state variables at the point x. In the distance equal to the wavelength λ = Z (x,t ) Z (x+λ,t )
= exp
2π ki kr
and ln
Z (x,t ) Z (x+λ,t )
=
2π ki kr
2π kr
, the amplitude decays to Z0 exp −ki x −
2π ki x kr
. Thus,
= 2π Ds .
In the elementary cases of modelling of longitudinal (symmetric) and flexural (skew-symmetric) waves by means of the Bernoulli–Euler theory, the spatial decay is known to be frequency-independent [1, p. 161, Table 4.2]. Specifically, for longitudinal waves Ds = Dt = 12 η and for flexural waves Ds = 21 Dt = 41 η. These results are immediately available from Eqs. (3) and (4).
4.2. Comparison of spatial attenuation predicted by alternative theories The remaining part of this section is concerned with the spatial attenuation of free waves described by a hierarchy of models presented in Table 1. The research questions to be addressed are whether the predictions of Bernoulli–Euler models agree with the advanced models and to what extend the attenuation measure Ds is applicable for higher-order modes in a visco-elastic layer. Comparison between the Bernoulli–Euler and the Timoshenko theories reveals that the elementary formula DsB.−E . (η) = 1 η, strictly speaking, is valid only in the limit Ω → 0. The Timoshenko theory gives in this limit the following formula 4
DTs .
(η, Ω ) ≈ η + 4
. Ω +O Ω
48 α 2 − 1 + π 2 α 2
1
32π 2
2
(15)
3 α2 − 1
The ratio of the attenuation level as predicted by the Timoshenko model DTs . (η, Ω ) to the Bernoulli–Euler model’s attenuation level DsB.−E . (η) ≡ DTs . (η, 0) as a function of the frequency parameter and the Poisson ratio ν is shown in Fig. 3. As seen, the Bernoulli–Euler theory markedly underestimates the attenuation of the free wave. The Bernoulli–Euler is more accurate for the symmetric waves in the same low-frequency limit. The Herrmann–Mindlin model (and, consequently, the three-mode model) gives the following result:
DsH .−M .
(η, Ω ) ≈ η
1 2
−
α2 − 2
2
48 + α 2 π 2 − 48
2
384π 2 α 2 − 1
4 . Ω +O Ω 2
(16)
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.−M . Fig. 4. The ratio of attenuation levels predicted by the Herrmann–Mindlin model DH (η, Ω ) and the Bernoulli–Euler model DBs .−E . (η) ≡ DHs .−M . (η, 0). s
2.5 1
2 1.5 1
0.5
0.5 0
0 0
1
2
3
4
5
6
0
1
2
3
4
5
Fig. 5. The scaled attenuation levels Dηs . (a): Herrmann–Mindlin model (lines) versus exact solution (dots); (b): the Timoshenko model (lines) versus exact solution (dots).
.−M . Similarly to the previous case, the ratio DH (η, Ω ) /DBs .−E . (η) , DsB.−E . (η) ≡ DHs .−M . (η, 0) as a function of the frequency s parameter and the Poisson coefficient ν is shown in Fig. 4. In this case, the Bernoulli–Euler theory also underestimates the attenuation of the free wave, but not to the same extend as for skew-symmetric waves. The comparison of attenuation levels of free waves predicted by approximate theories with the exact solution of the Rayleigh–Lamb problem is illustrated in Fig. 5 (Timoshenko and Herrmann–Mindlin model) and 6 (three-mode theories). As seen, the validity ranges of these theories regarding the attenuation measure Ds remain the same as for purely real wavenumbers for an ideally elastic layer. However, this measure does not seem to be appropriate in vicinity of the cut-on frequencies of higher-order modes. In a waveguide without damping, the cut-on frequency for a given branch of the dispersion diagram is found from the condition ki = 0, which can hold true either when kr = 0 (the conventional cut-on, see, for example, the second branch in Fig. 2(b)) or when kr ̸= 0 (see, for example, the second branch in Fig. 2(a)). In the 3D space (Ω , kr , ki ), this corresponds to the sharp turn of the branch onto the ki = 0 plane. The presence of damping invalidates this simple definition. As discussed in [19], a cut-on frequency range is identified as the frequency range at which the imaginary part of a wavenumber becomes sufficiently small and, more importantly, its decay rate diminishes and becomes to be ‘‘controlled’’ only by the damping coefficient. For small amounts of damping, the cut-on frequency range is centred at the cut-on frequency for a waveguide without material losses. Simultaneously, the attenuation measure Ds becomes very large in this range, as seen in Figs. 5 and 6. Interestingly, the proportionality of this measure to a magnitude of the loss factor η is preserved, see the scaling of the vertical axes in these figures. However, the dominant influence is produced by the frequency. As soon as it deviates from the cut-on value, the attenuation levels of all free waves acquire the same order of magnitude. To assess significance of this result, forcing problems should be solved in these cut-on frequency ranges, see Section 5. To conclude this section, we note that the spatial attenuation of free waves depends not only upon the loss factor as suggested by the elementary theories, but also upon the frequency.
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121
2.5 2
1
1.5 1
0.5
0.5 0
0
1
2
3
4
5
0
6
0
1
2
3
4
5
Fig. 6. The scaled attenuation levels Dηs . (a): the three-mode theory for symmetric waves (lines) versus exact solution (dots); (b): the three-mode theory for skew-symmetric waves (lines) versus exact solution (dots).
5. Assessment of spatial wave attenuation in a visco-elastic layer for forcing problems As discussed in Section 2, the loss factor is defined as a measure of the energy dissipation in the material of an oscillator. For an infinite waveguide, it is natural to introduce an excitation source, calculate its power input and to consider decay k in the power flow as the observation point departs from a source. The applicability of the damping measure Ds = k i , r used for free waves, needs to be examined. An excitation at the frequencies slightly above the non-zero cut-on ones is of particular interest, because, as shown in the previous section, this measure is much different for co-existing free waves in these frequency ranges. The closely related issue is the sensitivity of attenuation levels to the excitation conditions at high frequencies, when more than just one wave can propagate in an ideally elastic waveguide. Since the attenuation of free waves predicted by reduced-order theories has been checked against the exact Rayleigh–Lamb solution, the forcing problems are considered here for small loss factors (η ≪ 1) by means of the reduced order models. 5.1. Elementary models For consistency, we begin with the elementary cases of attenuation of symmetric (longitudinal) and skew-symmetric (flexural) waves in the Bernoulli–Euler model. For propagation of symmetric waves excited by a point axial force F0 applied at the origin of coordinates, the power flow is defined as
1 N (x) = − Re F (x) V ∗ (x) 2 F0 ω ωη F (x) = − exp i x − x , 2 c0 2c0
(17) V ∗ (x) =
F0 c0 2EA
1+i
η 2
exp −i
ω c0
x−
ωη 2c0
x .
For simplicity, the analysis here is specialised for a uni-axial strain–stress state, c02 = E /ρ . Then formula (17) is reduced as N (x) =
F02 c0 8EA
exp −
ωη c0
x .
The energy injected into the waveguide is gradually dissipated as the wave travels away from the source. To measure this dissipation, a characteristic length should be introduced. The natural choice is the wavelength of the free wave in the absence 2π c of dissipation, λ = 2kπ ≡ ω 0 . For small damping, kr ≈ k0 . Then the spatial energy dissipation rate is 0
D (x0 , Ω , η) =
∆N (x0 ) N (x0 ) − N (x0 + λ) = = 1 − exp (−2π η) ≈ 2π η. N (x0 ) N (x0 )
Alternatively, this formula may be obtained as D (x0 , Ω , η) = ln
N ( x0 ) N (x0 + λ)
= 2π η.
It has the same magnitude regardless of the choice of the reference point x0 , because the theory does not predict evanescent waves and, therefore, the near-field boundary layers do not exist. As seen, this formulation matches exactly the definition of the temporal loss factor for a single-degree-of-freedom oscillatory system with viscous damping.
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Fig. 7. Upper surface — the attenuation level D (x0 , Ω , η) from the Timoshenko theory, lower surface — the attenuation level D (x0 , Ω , η) from the Bernoulli–Euler theory.
In the case of Bernoulli–Euler model of flexural waves (with the conventional beam theory definitions and notation), the power flow is defined as N (x) =
1 2
Re Q (x) V ∗ (x) +
1 2
Re M (x) VΓ∗ (x) .
(18)
Here Q (x) and M (x) are the shear force and the bending moment, while V ∗ (x) and VΓ∗ (x) are complex conjugates of transverse and rotation velocities of a cross-section of the beam. Formulas for these state variables are well known and not presented here. We consider the excitation by a point transverse force at x = 0: Q (0) = 12 sgn (±ε) , ε → 0w ′ (0) = 0. The characteristic length to scale the spatial decay rate of flexural wave equals the length of a propagating wave in the absence of dissipation λ = 2kπ ≡ 2π 0
point x0 , D (x0 , Ω , η) = ln
c0 rg
ω
with rg as the gyration radius. The dependence of the spatial attenuation at the observation
N (x0 ) N (x0 +λ)
upon the loss factor η in the far-field zone (x0 → ∞) coincides with the function
Dfree (Ω , η) = 4π = 4π Ds . As shown in Section 3, the quantity 2π Ds = 2π kri defines the decay rate of the amplitude of a travelling wave over its length. The slopes, bending moments and shear forces have the same decay rate, because the Bernoulli–Euler model is a single-mode one, see Table 1. As follows from formula (18) the rate of the energy dissipation should be twice as large. Analysis of the excitation by a point moment gives same results. As expected, the investigation of the energy dissipation in forcing problems for the elementary theories fully agrees with the classical results [1]. ki kr
k
5.2. Timoshenko and Herrmann–Mindlin models In the Timoshenko theory, the formula (18) for the power flow remains valid provided that shear force, bending moment and angle of rotation of cross-section are defined properly. The attenuation level D (x0 , Ω , η) obtained in the framework of the Bernoulli–Euler theory (lower surface) is compared with its counterpart from the Timoshenko theory (upper surface) as illustrated in Fig. 7 for η = 0.01 and ν = 0.3. Wavelengths λ are calculated accordingly to the theories used. There is a significant difference between predictions of these theories both frequency-wise and the reference pointwise as seen in this figure. Only at very low frequencies the predictions are in a reasonable agreement. The near-field phenomenon, which does not exist for axial waves, is also illustrated in this figure. If the reference point x0 is located in the near field, where an evanescent wave is ‘visible’, then the loss factor exceeds its ‘nominal’ far-field value. As soon as frequency grows and the reference point departs from the source, both theories predict the levels, which differ from each other, but match the attenuations of the free wave Dfree (Ω , η) for each theory. In Fig. 8, the comparison between ‘forced’ attenuation rate D (x0 , Ω , η) (upper surface, the same as the upper surface in Fig. 7) and the attenuation rate of the free wave Dfree (Ω , η) (lower surface) is illustrated for the Timoshenko theory. The differences vanish as the reference point moves away from the source and the frequency grows. The attenuation levels predicted by the Timoshenko theory in two excitation cases: by a point force and by a point moment are compared in Fig. 9. In the near field, the excitation by a moment gives attenuation levels more sensitive to the location of the reference point. In the far field and at higher frequencies there is no difference in the level of attenuation. As known, the Timoshenko theory agrees with the exact elasto-dynamics in predictions of the second branch of dispersion diagram. In Fig. 10, the same diagram as in Fig. 7 after this theory is plotted for higher frequencies. As soon as the second
S. Sorokin, R. Darula / Wave Motion 68 (2017) 114–127
123
Fig. 8. Upper surface — the attenuation level D (x0 , Ω , η), lower surface — the attenuation level Dfree (Ω , η), the Timoshenko theory.
Fig. 9. Upper surface — the attenuation level for a point force excitation, lower surface — the attenuation level for a point moment excitation.
Fig. 10. The attenuation level D (x0 , Ω , η) for a point force excitation around the cut-on frequency, the Timoshenko theory.
wave cuts on, the attenuation level at a given frequency becomes a periodic function of the location of the reference point x0 .
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The two-mode theory for symmetric waves gives results, similar to those obtained from the Timoshenko theory. However, as can be seen from the dispersion diagram in Fig. 2(b), this theory does not predict correctly the shape of the second branch. Therefore, these results are not discussed here, but mentioned, where appropriate, in discussion of the threemode theory for symmetric waves. 5.3. Three mode theories As shown in Section 3, the three-mode theories capture three branches of dispersion diagrams for symmetric and skewsymmetric waves. Therefore, these theories may be used to solve forcing problems at frequencies beyond the validity ranges of Bernoulli–Euler, Timoshenko and Herrmann–Mindlin ones. The power flow is defined as follows: For the symmetric waves Nsymm (x) =
1 ωh 2 c1
Re i F0 (x) · U0∗ (x) + F1 (x) · V1∗ (x) + F2 (x) · U2∗ (x)
(19)
(20)
with the forces and displacements defined as given in (9). For the skew-symmetric waves Nskew (x) =
1 ωh 2 c1
Re i F0 (x) · V0∗ (x) + F1 (x) · U1∗ (x) + F2 (x) · V2∗ (x)
with the forces and displacements defined as given in (12). The forcing conditions are formulated for three alternative excitation cases at x = 0: 1. F0 (0 ± ε) =
1 2
sgn (±ε) ,
ε → 0;
2. Y0 (0) = 0;
F1 (0 ± ε) =
3. F0 (0) = 0;
Z1 (0) = 0;
1 2
Z1 (0) = 0;
sgn (±ε) ,
ε → 0;
F2 (0 ± ε) =
1 2
F2 (0) = 0
(21a)
Y2 (0) = 0
(21b)
sgn (±ε) ,
ε → 0.
(21c)
For symmetric waves, Z1 ≡ V1 , Y0,2 ≡ U0,2 and the forces are defined by (9). For skew-symmetric waves, Z1 ≡ U1 , Y0,2 ≡ V0,2 and the forces are defined by (12). In the low-frequency range, when only one wave propagates in a waveguide, the three-mode theories confirm the assessment of attenuation obtained by means of Bernoulli–Euler (symmetric) and Timoshenko (skew-symmetric) models. The results reported hereafter are obtained in the frequency ranges, where the three-mode theories agree with the exact solutions of the Rayleigh–Lamb problem in describing three propagating waves. The issues of interest are, besides the frequency dependence of attenuation levels, the influence of location of the reference point and the influence of excitation conditions. We begin with the attenuation levels of skew-symmetric waves generated in the loading conditions (21). In Fig. 11(a), (b), the attenuation levels D (x0 , Ω , η) versus excitation frequency are shown for three locations of the reference point: x0 = 5 (green curve), x0 = 50 (blue curve) and x0 = 500 (red curve) for ν = 0.3 and η = 0.01. For reference, the k attenuation level for the first free wave Dfree (Ω , η) ≡ 4π k i is plotted in magenta and it almost coincides with the red r curve. The mid-frequency range covered in this figure captures two regimes: below and above the cut-on frequency of the second wave, and there is a qualitative difference between these regimes. In the frequency range, where only one wave propagates, the decay rates D (x0 , Ω , η) are the same regardless of position of the reference point and regardless of the loading case. In this regime, the measure Dfree (Ω , η) suffices (although it should be calculated in accordance with the threemode model of skew-symmetric waves). Around the cut-on frequency, the attenuation levels become very sensitive to the position of a reference point. If this point is located relatively close to the excitation point, these levels may be much higher than Dfree (Ω , η) for the first propagating wave, as clearly seen in Fig. 11(b). In Fig. 12, the frequency range is shifted higher, to capture the cut-on frequency of the third wave. In this situation, all three loading cases are considered and, qualitatively, the results are the same as in the previous case above the cut-on frequency. This is readily explained by inspection into the components of the power flow (see, for example, [21]). In the near field, the energy injected into the waveguide is distributed between the ‘alternative transmission paths’, i.e., between propagating waves. If a waveguide supports only one wave of this type, this distribution is unique and, in a viscoelastic layer, the far-field decay rate is controlled solely by Dfree (Ω , η) of this wave. As soon as more than one wave propagates, the effective attenuation builds up as a result of superposition of free waves, each of which has its own dissipation rate. It should be noted that the attenuation levels of free waves Dfree (Ω , η) have been calculated in Section 3 with their own wavenumbers, and, therefore, attained extremely large values slightly above cut-on frequencies. The measure D (x0 , Ω , η) for forcing problems is introduced with the shortest wavelength, so that, effectively, the attenuation level of a free wave around its cut-on is re-scaled. Another feature of co-existing travelling waves is that the energy distribution between them
S. Sorokin, R. Darula / Wave Motion 68 (2017) 114–127
125
Fig. 11. The attenuation levels D (x0 , Ω , η) in the mid-frequency range for the skew-symmetric excitation. (a) loading case (21a); (b) loading case (21b). x0 = 5 (green curve), x0 = 50 (blue curve) and x0 = 500 (red curve). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 12. The attenuation levels D (x0 , Ω , η) in the high frequency range for the skew-symmetric excitation. (a) loading case (21a); (b) loading case (21b); (c) loading case (21c). x0 = 5 (green curve), x0 = 50 (blue curve) and x0 = 500 (red curve). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
is dependent on the distance from the excitation point. As these waves have different decay rates, the total attenuation is also dependent upon x0 , as clearly seen from Fig. 12(a)–(c). The three-mode model for symmetric waves gives qualitatively same results as its counterpart for skew-symmetric ones. The point of particular interest is the analysis of the energy dissipation in the high frequency range, where, in the absence of damping, the wave having group and phase velocities of opposite signs, exists, see Fig. 2(a). As known, the wave with the positive group velocity has to be used for solving the forcing problem. Results obtained in three excitation cases in this frequency range are presented in Fig. 13 in the same way as in Fig. 12 also for ν = 0.3 and η = 0.01. The attenuation levels are strongly dependent upon the frequency and the position of a reference point above the cut-on frequency of the second wave, which in the absence of damping should be found from the condition ddkω = 0 rather, than from the conventional condition k = 0. There is also the substantial difference in the attenuation levels D (x0 , Ω , η) between three loading cases above this cut-on frequency, whereas these levels are the same and equal to Dfree (Ω , η) when only one wave propagates in the waveguide. Finally, the difference between dependence of D (x0 , Ω , η) upon x0 below (at Ω = 2.6) and above (at Ω = 3.1) the cuton frequency of the second wave is demonstrated in Fig. 14(a) and (b) for the loading conditions (21a). In the former case,
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Fig. 13. The attenuation levels D (x0 , Ω , η) in the high-frequency range for the symmetric excitation. (a) loading case (21a); (b) loading case (21b); (c) loading case (21c). x0 = 5 (green curve), x0 = 50 (blue curve) and x0 = 500 (red curve). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 14. The dependence of attenuation level D (x0 , Ω , η) upon the position of the reference point x0 in the high-frequency range for the symmetric excitation. (a) loading case (21a); (a) −Ω = 2.6; b) Ω = 3.1.
this dependence exists only in the near field, where the evanescent waves are still noticeable. In the latter case, as already discussed, the energy exchanges between co-existing travelling waves are present in the far-field and so the attenuation levelD (x0 , Ω , η) fluctuates. To conclude this section, we note that the spatial energy dissipation rate depends not only upon the loss factor and the excitation frequency as explained in Section 4 for free waves, but also upon the position of the reference point and upon the excitation type. The latter becomes essential as soon as a waveguide supports several propagating waves. To capture this effect, it is necessary to use multi-modal theories summarised in Table 1. 6. Conclusions The reported results suggest that a concept of the loss factor, in principle, is plausible for the analysis of attenuation of free and forced waves in a visco-elastic layer, because both the temporal and the spatial attenuation levels may be expressed via its magnitude η. The difficulty lies in the functional link between the loss factor and the attenuation levels. In the framework described in Section 2, the temporal decay of free waves in a visco-elastic layer is determined in full agreement with [1] by the elementary formula Dt = 21 η. The applicability of the loss factor as a sole parameter,
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which controls the spatial decay of free waves, is case-sensitive simply because of its frequency-dependence illustrated in Section 4. In the low-frequency limit, the results obtained in [1] by means of the classical Bernoulli–Euler models for both symmetric (Ds = Dt = 21 η) and skew-symmetric (Ds = 21 Dt = 14 η) free waves are fully confirmed by comparison with the results obtained from more advanced models. However, these classical formulas underestimate the damping level at higher frequencies. To the leading order, the frequency dependence of spatial attenuation of the first propagating wave is linear in the skew-symmetric case and quadratic in the symmetric case. Assessment of spatial attenuation of high-order free waves shows that it is proportional to the loss factor, but its magnitude is mainly frequency-controlled, especially around cut-on frequencies. In a forcing problem, the loss factor gives a fair estimate of spatial power flow decay only in the low-frequency range, where the waveguide supports one propagating wave of a given (symmetric or skew-symmetric) type. In this case, the attenuation rate Dfree (Ω , η) calculated with the wavelength of the first propagating wave as a reference distance may be reliably used. In the situations, when a waveguide supports more than one wave of the same type, the attenuation level becomes also a function of the position of the point x0 , from which the reference distance is measured, D (x0 , Ω , η). On top of that, the loading conditions influence the attenuation levels. This is explained by partition of the total energy input between free propagating waves, each of which has its own decay rate. On balance, the magnitude of D (x0 , Ω , η) may significantly deviate from Dfree (Ω , η). Therefore, a reliable prediction of the spatial energy dissipation rate in the situation of a high-frequency excitation appears to be very difficult, if based solely on the knowledge of the loss factor. Acknowledgement The financial support for the second author from the Danish Council for Independent Research (the postdoctoral research grant DFF-4005-00348) is gratefully acknowledged. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
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