Energy flux and dissipation of inhomogeneous plane waves in hereditary viscoelasticity
aa r X i v : . [ phy s i c s . c l a ss - ph ] A p r Proceedings of the Royal Society A, :20190478 (2019)http://dx.doi.org/10.1098/rspa.2019.0478Published 6 November 2019
Energy flux and dissipation of inhomogeneous planewaves in hereditary viscoelasticity
N. H. Scott ∗ , School of Mathematics, University of East Anglia,Norwich Research Park, Norwich NR4 7TJ. UK.27 April 2020 Dedicated to the memory of Peter Chadwick FRS
Abstract
Inhomogeneous small-amplitude plane waves of (complex) frequency ω are prop-agated through a linear dissipative material which displays hereditary viscoelasticity.The energy density, energy flux and dissipation are quadratic in the small quantities,namely, the displacement gradient, velocity and velocity gradient, each harmonicwith frequency ω , and so give rise to attenuated constant terms as well as to inho-mogeneous plane waves of frequency 2 ω . The quadratic terms are usually removedby time averaging but we retain them here as they are of comparable magnitudewith the time-averaged quantities of frequency ω . A new relationship is derived inhereditary viscoelasticity that connects the amplitudes of the terms of the energydensity, energy flux and dissipation that have frequency 2 ω . It is shown that thecomplex group velocity is related to the amplitudes of the terms with frequency2 ω rather than to the attenuated constant terms as it is for homogeneous waves inconservative materials. Keywords
Complex exponential solutions, group velocity, energy velocity, anisotropicviscoelasticity
MSC (2010) · · PACS · For the most general form of plane wave in a linear continuous medium, the particledisplacement field u ( x , t ) takes the complex exponential form u ( x , t ) = { U exp i( K · x − ωt ) } + (1.1) ∗ Email: [email protected]
N. H. Scottwhere i = √− U = U + + i U − , ω = ω + + i ω − , K = K + + i K − are, respectively, the complex wave amplitude vector, the complex frequency and thecomplex wave vector, all of which are constant. Throughout this paper, the superscripts + and − refer to real and imaginary parts of a complex quantity. The real variables x and t denote position and time, respectively. If the planes of constant phase K + · x = const.and the planes of constant amplitude K − · x = const. are not parallel, then (1.1) is saidto represent an inhomogeneous plane wave.Inhomogeneous plane waves arise in many different areas of mechanics both for conser-vative and for dissipative media, for example, Rayleigh and Stoneley waves in elasticity,electromagnetic radiation in wave guides, and surface and body waves in viscous fluids,viscoelastic solids and thermoelastic solids. Inhomogeneous plane waves are importantbecause all solutions of a linear problem may be written as superpositions, as finite sumsor integrals, of waves of the form (1.1).We show that the energy density, energy flux and dissipation in hereditary viscoelas-ticity are connected by an energy-dissipation equation, see (3.7) below. These quantitiesare quadratic in the small quantities and so each consists of two parts, one an attenuatedharmonic term with frequency 2 ω and the other an attenuated constant. Usually, wetime-average the governing equations by integrating over a cycle of the harmonic termsso that only the attenuated constant terms remain. We find that these terms satisfy therelationship (5.17) below in hereditary viscoelasticity. Here, however, we explore the con-sequences of retaining the terms of frequency 2 ω in the theory of hereditary viscoelasticitybecause these terms are of the same order of magnitude as the attenuated constant terms.We shall see these terms satisfy the relationship (5.8) below. For inhomogeneous waveswe find that the (complex) group velocity is related to the amplitudes of the quadraticterms rather than to those of the (attenuated) constant terms as it is for homogeneouswaves in conservative media, see (6.9) below.The theory of inhomogeneous plane waves propagating through continuous mediahas been given a detailed exposition by Boulanger & Hayes [4]. The idea of retainingthe quadratic terms in the energy density, energy flux and dissipation in the theoriesof thermoelasticity and Kelvin-Voigt viscoelasticity has been explored by Scott [20, 21].Boulanger [3] obtained some of our results in the particular case of incompressible isotropicviscoelastic fluids. Energy density, group velocity and dissipation in homogeneous and in-homogeneous plane waves have been much studied previously in a variety of contexts. Forexample, Chadwick et al. [6] and Borejko [2] consider homogeneous and inhomogeneouswave propagation in a constrained elastic body, e.g. an incompressible or inextensiblebody, and discuss energy propagation and group velocity. Cerveny & Psencik [7] discusstime-averaged and time-dependent energy-related quantities in inhomogeneous harmonicwaves in anisotropic viscoelastic media, especially Kelvin-Voigt viscoelasticity. Declercqet al. [9] discuss the history and properties of ultrasonic inhomogeneous waves, includingcomplex frequency and bounded beams. Deschamps & Huet [11] consider complex surfacenergy flux and dissipation in hereditary viscoelasticity 3waves associated with inhomogeneous skimming and Rayleigh waves in linear elastody-namics. Rodrigues Ferreira & Boulanger [18] extend the theory of damped inhomogeneouswaves to the finite-amplitude case in a deformed Blatz-Ko material. Vashishth & Sukhija[23] extend the theory of inhomogeneous waves to the case of porous piezo-thermoelasticsolids.The paper is constructed as follows. In Section 2 we write down the constitutive equa-tions of viscoelasticity in integral form. Then in Section 3 we use the equations of motionand double integral forms for the strain energy and the dissipation to derive an equation ofenergy balance including dissipation effects. In Section 4 we derive the propagation con-dition for inhomogeneous plane waves in hereditary viscoelasticity. In Section 5 we recallthat the energy density, energy flux and energy dissipation are all quadratic in the smallquantities occurring in the linear theory, e.g. the displacement and velocity gradients,and obtain expressions for them, each containing a constant term and one harmonic withfrequency 2 ω where the linear quantities are harmonic with frequency ω . We obtain someresults valid for all dissipative media and further results proved here only for viscoelasticmedia. Section 6 uses the dispersion relation to obtain a relation between the complexgroup velocity and the complex energy velocity, which is new to hereditary viscoelasticity,and concludes with some examples, namely, linear elasticity, the Newtonian viscous fluidand the Kelvin-Voigt viscoelastic solid. The double integrals for the strain energy anddissipation are evaluated in the Appendix. The particle velocity v ( x , t ) is given by v ( x , t ) = ˙ u ( x , t ) where u denotes particle dis-placement, as at (1.1), and the superposed dot denotes the time partial derivative. Thecomponents of the infinitesimal strain tensor e and the infinitesimal rate-of-strain tensor d = ˙ e are given by e ij = ( u i, j + u j, i ) , d ij = ( v i, j + v j, i ) , (2.1)respectively. The notation ( ) , j denotes the spatial partial derivative ∂ ( ) /∂x j .The constitutive equations of anisotropic linear hereditary viscoelasticity are t ij = Z t −∞ G ijkl ( t − τ ) d kl ( τ ) d τ (2.2)for the Cauchy stress t , see for example [14, Eq. (2.27)], in which twice-occuring romansuffices are summed over. On putting s = t − τ in (2.2) we see that t ij = Z ∞ G ijkl ( s ) d kl ( t − s ) d s. (2.3)We assume that the components G ijkl ( s ) vanish for s < G ijkl ( s ) ≥ , G ′ ijkl ( s ) ≤ , lim s → G ijkl ( s ) = c inst ijkl , lim s →∞ G ijkl ( s ) = c ijkl , (2.4) N. H. Scottwhere prime denotes differentiation with respect to argument. The constants c inst ijkl are theinstantaneous (small t ) elastic moduli and the constants c ijkl are the equilibrium (large t )elastic moduli. If the c ijkl all vanish then the material is a viscoelastic fluid rather thana viscoelastic solid.The tensor components G ijkl ( s ) have the symmetries G ijkl ( s ) = G jikl ( s ) = G ijlk ( s ) , s ≥ , (2.5)because of the symmetries of t and d . We shall need the further symmetry property G ijkl ( s ) = G klij ( s ) , s ≥ . (2.6)If this symmetry is present in the elastic moduli of a purely elastic material it implies theexistence of a strain energy function. Gurtin & Herrera [13] have shown that c inst ijkl and c ijkl each satisfy the symmetry (2.6), that is, the short and long time material behaviourin hereditary viscoelasticity both satisfy the elastic symmetries. Day [8] has gone further.He shows that (2.6) is obeyed for 0 < s < ∞ if and only if a certain work integral, namely, W ( e ) = Z ∞−∞ t ( τ ) · d ( τ ) d τ, is invariant under time reversal; i.e. if W ( e ( − t )) = W ( e ( t )). However, it cannot beclaimed that (2.6) has been proved for all s > For hereditary viscoelasticity, the linearised equations of motion in the absence of bodyforce are t ij, j = ρ ˙ v i , (3.1)where the mass density ρ may be taken to be constant.On multiplying (3.1) by v i we see that˙ k + r i, i + t ij v i, j = 0 , (3.2)in which v i are the components of the particle velocity v = ˙ u and k = ρv i v i , r i = − t ji v j , (3.3)denote the kinetic energy and the components of the energy flux vector r , respectively.By modelling an isotropic viscoelastic material as consisting of springs and dashpotsconnected in series and parallel Bland [1] and Hunter [17] obtained expressions for thenergy flux and dissipation in hereditary viscoelasticity 5strain energy and dissipation in such a material as certain double integrals. When gener-alised to the anisotropic case these double integrals become, for the strain energy, w = 12 Z t −∞ Z t −∞ G ijkl (2 t − τ − τ ) d ij ( τ ) d kl ( τ ) d τ d τ , = 12 Z ∞ Z ∞ G ijkl ( s + s ) d ij ( t − s ) d kl ( t − s ) d s d s (3.4)and for the dissipation d = − Z t −∞ Z t −∞ G ′ ijkl (2 t − τ − τ ) d ij ( τ ) d kl ( τ ) d τ d τ , = − Z ∞ Z ∞ G ′ ijkl ( s + s ) d ij ( t − s ) d kl ( t − s ) d s d s , (3.5)where, as before, prime denotes differentiation with respect to argument. The expres-sion (3.4) for the stored energy is attributed by Del Piero & Desiri [10] to Staverman &Schwarzl [22]. The determination of a suitable expression for w is discussed further byGolden [12] and the references therein. Buchen [5] applied this model to plane waves inlinear isotropic viscoelastic solids and Boulanger [3] applied it to plane waves in incom-pressible isotropic viscoelastic fluids.Differentiating (3.4) under the integral sign with respect to t gives˙ w = 12 Z t G ijkl ( t − τ ) d ij ( t ) d kl ( τ ) d τ + 12 Z t G ijkl ( t − τ ) d ij ( τ ) d kl ( t ) d τ + Z t −∞ Z t −∞ G ′ ijkl (2 t − τ − τ ) d ij ( τ ) d kl ( τ ) d τ d τ . From (3.5) we see that the double integral above is equal to − d . From the symmetryproperty (2.6) the first two integrals above may be combined to give Z t −∞ G ijkl ( t − τ ) d ij ( t ) d kl ( τ ) d τ, which from (2.2) is equal to t ij v i, j . Combining these results leads to˙ w = t ij v i, j − d. (3.6)Eliminating t ij v i, j between this equation and (3.2) gives the energy balance equation inthe form ˙ e + r i, i = − d (3.7)with the total energy e given by e = k + w. (3.8) N. H. Scott We now assume the small disturbance in the viscoelastic body to have the most generalcomplex exponential plane wave form possible, so that the particle velocity v takes theform v = { V e i χ } + (4.1)in which, as before, + denotes the real part of the quantity in braces and the phase factor χ is defined by χ = K · x − ωt = χ + + i χ − , (4.2)where χ + = K + · x − ω + t , χ − = K − · x − ω − t (4.3)are the real and imaginary parts of χ expressed in terms of those of K and ω . From (4.3),we see that the wave amplitudes (4.1) may be written v = { V e i χ + } + e − χ − (4.4)from which it is clear that these wave amplitudes represent a sinusoidal travelling wave offrequency ω + and wave vector K + which is attenuated by the real exponential factor e − χ − .From the component form of the particle velocity (4.1) and its definition (2.1) we seethat the symmetrised velocity gradient may be written d ij = (cid:8) i( V i K j + K i V j )e i χ (cid:9) + (4.5)so that from its definition (2.3) the stress becomes t ij = (cid:8) i H ijkl ( ω ) V k K l e i χ (cid:9) + (4.6)with H ijkl ( ω ) = Z ∞ G ijkl ( s )e i ωs d s (4.7)denoting the half-range Fourier transform.Applying the momentum balance equation (3.1) to the stress (4.6) and the velocity(4.1) leads to the propagation condition { ρωδ ik + i H ijkl ( ω ) K j K l } V k = 0 (4.8)with δ ik denoting the components of the Kronecker delta.nergy flux and dissipation in hereditary viscoelasticity 7 General results
We see from (3.3)–(3.5) and (3.8) that the energy density, energy flux and energy dissi-pation in the energy balance equation (3.7) are quadratic in the small quantities and so,for inhomogeneous plane waves, are expressible as linear combinations of products of theform f ( x , t ) = { A e i χ } + { B e i χ } + (5.1)in which χ continues to be given by (4.2) and A and B are complex constants. Using(4.3), we may evaluate the product (5.1) to obtain f ( x , t ) = { F e χ } + + f e − χ − (5.2)and from (4.4) f ( x , t ) = { F e χ + } + e − χ − + f e − χ − (5.3)where F = 12 AB , f = (cid:26) AB ∗ (cid:27) + (5.4)in which F is a (usually) complex constant and f is a real constant. Here and throughout, ∗ denotes the complex conjugate. The first term of (5.2) represents an inhomogeneousplane wave with phase factor 2 χ , while the second consists of the real constant f at-tenuated by the real exponential factor e − χ − . From (5.3) and (4.4), we see that theinhomogeneous plane wave is attenuated by the same factor and is sinusoidal with fre-quency 2 ω + .To interpret f , we follow [4, Section 11.5] and integrate (5.3) over a cycle of χ + atconstant χ − to show that the mean value of f is f e − χ − , which depends on x and t through χ − . The real constant f is then regarded as a weighted mean of f ( x , t ).We have already observed that the energy density, energy flux, and energy dissipationoccurring in (3.7) may be expressed as linear combinations of products of the form of(5.1) and so, using (5.2) and (5.3), we obtain e = { E e χ + } + e − χ − + e e − χ − (5.5) r j = { R j e χ + } + e − χ − + r j e − χ − (5.6) d = { D e χ + } + e − χ − + d e − χ − (5.7)in which E, R j , D are (usually) complex constants and e, r j , d are real constants. Theselatter constants are the weighted means of e, r j , d as discussed above.Previously, discussion of energy and dissipation has focused on the weighted meansat the expense of terms involving the complex quantities E, R j , D , see for example [4], N. H. Scottoften on the grounds that these terms do not contribute when averaged over a cycle of χ + .However, the energy-dissipation equation (3.7) is valid for all x and t , without averaging,and the neglected terms are of the same order of magnitude (before averaging) as theretained terms.It is our chief purpose here to explore the consequences of retaining the attenuatedharmonic terms on an equal footing with the weighted means.On substituting (5.5)–(5.7) into (3.7) and equating the coefficients of the attenuatedharmonic terms, and those of the purely attenuated terms, we obtain ωE − K · R + i D = 0 (5.8) ω − e − K − · r + d = 0 (5.9)respectively. Equation (5.8) has appeared at [20, Eq. (44)] and [21, Eq. (4.15)]. Equation(5.9) has appeared at [19, Eq. (76)], [20, Eq. (45)], [21, Eq. (4.17) ] as well as at [4, Eq.(11.5.8)], where it was derived by a different method. Equations (5.8) and (5.9) have awide range of validity, not only in viscoelasticity, since they are valid for any system thathas an energy-dissipation equation of the form (3.7) with energy density, energy flux, andenergy dissipation being quadratic in the small quantities and taking the forms given by(5.5)–(5.7).We may conclude from (5.9) that if both ω and K are real, then d = 0 and there isno weighted mean dissipation. Alternatively, if there is dissipation ( d = 0), then we maydraw the conclusion from (5.9) that not both of ω and K can be real. Hereditary viscoelasticity
For the kinetic energy k defined by (3.3) we take v in the form (4.4) and use the formula(5.3) to show that k = ρ n V · V e χ + o + e − χ − + ρ V · V ∗ e − χ − . By adding this equation to (A7) we obtain the equation (5.5) for the total energy e = k + w defined at (3.8) with E and e given by (5.10) and (5.13), respectively.For hereditary viscoelasticity, we may obtain explicit expressions for the quantities E, R j , D and e, r j , d occurring in (5.5)–(5.7) by substituting the inhomogeneous planewave forms of (4.1) into (3.3)–(3.5) and (3.8) for the energy density, energy flux, andenergy dissipation and using (5.2) and (5.4): E = ρ V i V i + i4 H ′ ijkl ( ω ) V i K j V k K l (5.10) R j = − i2 H ijkl ( ω ) V i V k K l (5.11) D = −
12 ( H ijkl ( ω ) + ωH ′ ijkl ( ω )) V i K j V k K l (5.12)nergy flux and dissipation in hereditary viscoelasticity 9 e = ρ V i V ∗ i + 14 ω + H − ijkl ( ω ) V ∗ i K ∗ j V k K l (5.13) r j = (cid:26) − i2 H ijkl ( ω ) V ∗ i V k K l (cid:27) + (5.14) d = − (cid:18) ω − ω + H − ijkl ( ω ) − H + ijkl ( ω ) (cid:19) V i K j V ∗ k K ∗ l . (5.15)We may use the symmetry property (2.6), in the form H ijkl ( ω ) = H klij ( ω ) , (5.16)to verify that e is real. The reality of r j is clear. It also follows from (2.6) that d is real.As in the general case, the non-vanishing of d implies that not both of ω and K can bereal.We may verify the general equation (5.8) in the present case of hereditary viscoelas-ticity by substituting for E, R , and D from (5.10)–(5.12) into (5.8) and observing that itis satisfied. In the same way, we may use (5.13)–(5.15) for the weighted means e, r j , d toderive the identity ωe − K · r + i d = 0 . (5.17)Equations (5.8) and (5.17) have the same form, the first involving the amplitudes of theattenuated harmonic terms and the second involving the weighted means, but it shouldbe remembered that (5.8) has general validity while (5.17) has been demonstrated hereonly for viscoelasticity.Bearing in mind that e, r j , d are real, we may take real and imaginary parts of (5.17)to obtain r · K + = ω + e , r · K − = ω − e + d. (5.18)As might be expected from the absence of d , (5.18) is valid also for conservative mediaand was proved by Hayes [16, Eq. (4.8) ]. Equation (5.18) simply verifies for hereditaryviscoelasticity the general result (5.9). In theories of continuous media, one typically derives from the propagation conditions,such as (4.8), an equation giving the frequency as a function of the wave vector: ω = ω ( K ) (6.1)known as the dispersion relation. In the present case of hereditary viscoelasticity we couldobtain from (4.8) the dispersion relation (6.1) in the implicit formdet { ρωδ ik + i H ijkl ( ω ) K j K l } = 0 . ρωV i + i H ijkl ( ω ) K j V k K l = 0 . (6.2)It follows from (6.1) and (6.2) that ω and V depend on K but not on its complexconjugate K ∗ . Then E, R j , D , defined by (5.10)–(5.12), are functions of K but not K ∗ .Clearly, ω ∗ and V ∗ are functions of K ∗ but not K . It follows that the real quantities e, r j , d , defined by (5.13)–(5.15), are functions of both K and K ∗ .Now (5.8) holds for all possible complex wave vectors K and so may be regarded asan identity in K . Allowing the operator ∂/∂K p to act upon this equation then gives ∂ω∂K p E − R p = − (cid:26) ω ∂E∂K p − K j ∂R j ∂K p + i ∂D∂K p (cid:27) . (6.3)In the general case of a linear dissipative material, we do not have explicit expressions for E, R j , D and so can make no further progress.Equation (5.9) also holds for all possible complex wave vectors K , but since its termsdepend explicitly also on K ∗ , it is to be regarded as an identity in each of the six quantities K + p , K − p , p = 1 , ,
3. Equivalently, (5.9) is an identity in each of the six components of K and K ∗ , with K ∗ now regarded as independent of K . Therefore, we rewrite (5.9) as( ω − ω ∗ ) e − ( K j − K ∗ j ) r j + 2i d = 0and allow ∂/∂K p to act upon it, bearing in mind that ω depends only on K and ω ∗ depends only on K ∗ , to obtain ∂ω∂K p e − r p = − (cid:26) ω − ∂ e∂K p − K − j ∂ r j ∂K p + ∂ d∂K p (cid:27) . (6.4)As with (6.3), we do not have explicit expressions for e, r j , d in the general case of a lineardissipative material and so can make little further progress.There is, however, one deduction we can make from (6.4). In the case of homogeneouswaves in a dissipationless system, the complex wave vector K is replaced by the real one k , the frequency ω also is real, and the dissipation d vanishes, so that ω − = 0 , K − j = 0 , d = 0and (6.4) reduces to r p = e ∂ ω∂K p (6.5)valid for homogeneous waves in a general dissipationless system as proved by Hayes [15,Eq. (20)].We return to our discussion of energy and dissipation in hereditary viscoelasticity andseek a connection between the complex group velocity ∂ω/∂ K and quantities E and R nergy flux and dissipation in hereditary viscoelasticity 11already defined at (5.10) and (5.11). We apply the operator ∂/∂K p to (6.2) and contractthe resulting equation with V i to obtain { ρωV i + i H klij K j V k K l } ∂V i ∂K p + (cid:8) ρV i V i + i H ′ ijkl V i K j V k K l (cid:9) ∂ω∂K p (6.6)+ i H ipkl V i V k K l + i H ijkp V i K j V k = 0 . We may use the symmetry property (5.16) to show from the propagation condition (6.2)that the first term of (6.6) vanishes. Furthermore, we contract (6.2) with V i and use theresult to eliminate ρV i V i from the second term of (6.6). We then use the definitions (5.10)and (5.11) to show that the remaining two terms of (6.6) reduce to R p = E ∂ω∂K p . (6.7)This is an important result in the theory of inhomogeneous waves in dissipative media,here demonstrated for hereditary viscoelasticity. It has previously been demonstrated forthermoelasticity, see Scott [20, Eq. (69)], and for Kelvin-Voigt viscoelasticity, see Scott[21, Eq. (5.1)]. One might expect this result to have a wider validity in the theory ofdissipative media, but this has not been demonstrated.We define a complex energy velocity G associated with the attenuated harmonic termsof the energy-dissipation equation by G = attenuated harmonic energy fluxattenuated harmonic energy density = R E (6.8)provided E = 0, which from (6.7) may be written G = ∂ω∂ K . (6.9)In terms of G , (5.8) becomes G · K = ω + i D/E. (6.10)The energy velocity vector more usually considered in the literature, that associatedwith the weighted mean quantities, is defined by g = weighted mean energy fluxweighted mean energy density = r e (6.11)a purely real vector. In terms of g , (5.17) becomes g · K = ω + i d/e (6.12)comparable with (6.10), and has real and imaginary parts g · K + = ω + , g · K − = ω − + d/e. (6.13)2 N. H. Scott ExamplesLinear elasticity G ijkl ( s ) = c ijkl h ( s ), where h ( s ) is the Heaviside step function: h ( s ) =0 if s < h ( s ) = 1 if s ≥ . The quantities c ijkl are the usual elastic moduli. From(4.7) we see that H ijkl = ( − / i ω ) c ijkl so that the propagation condition (4.8) becomes (cid:8) ρω δ ik − c ijkl K j K l (cid:9) V k = 0 = ⇒ det (cid:8) ρω δ ik − c ijkl K j K l (cid:9) = 0 . (6.14)Thus we see that in the dispersion relation (6.1), ω is homogeneous of degree one in K .Elasticity is a conservative theory, so that there is no dissipation, and the counterpartsin elasticity of the present equations (6.10) and (6.12), with D = d = 0, are simpleconsequences of the fact that ω is homogeneous of degree one in K . Many of the presentresults have been obtained for constrained elastic materials by Chadwick et al. [6, Eqns(4.16), (4.22) and (4.24)], see also Borejko [2, Eqns (3.19), (4.18), (4.20), (4.24), (4.27)and (4.30)]. Newtonian viscous fluid G ijkl ( s ) = η ijkl δ ( s ), where δ ( s ) is the Dirac delta function: δ ( s ) = 0 if s = 0 and R ∞ f ( s ) δ ( s ) d s = f (0) for any function f continuous at x = 0. Forthe Newtonian viscous fluid we have η ijkl = ( κ − µ ) δ ij δ kl + µ ( δ ik δ jl + δ il δ jk ) , (6.15)where κ is the bulk viscosity and µ the shear viscosity. Thus, from (4.7) we see thatfor viscous fluids H ijkl = η ijkl , independent of ω . The propagation condition (4.8) thenbecomes { ρ i ωδ ik − η ijkl K j K l } V k = 0 , where η ijkl is given at (6.15). This is discussed further in [19] in which the presentequations (5.9) and (5.17) are derived for compressible viscous fluids. Kelvin-Voigt viscoelasticity G ijkl ( s ) = c ijkl h ( s ) + η ijkl δ ( s ), which is simply a linearcombination of linear elasticity and the Newtonian viscous fluid described above. Then H ijkl ( ω ) = − ω c ijkl + η ijkl so that the propagation condition (4.8) becomes (cid:8) ρω δ ik − ( c ijkl − i ωη ijkl ) K j K l (cid:9) V k = 0 , the same as [21, Eq. (3.9)] derived directly for a Kelvin-Voigt viscoelastic material. Thepresent equations (5.9) and (5.17), and many others, are derived directly for a Kelvin-Voigt viscoelastic material in [21]. Acknowledgement
The author is much indebted to the referees for helpful suggestions andsupplying a new reference.nergy flux and dissipation in hereditary viscoelasticity 13
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A Appendix
Evaluation of the integral (3 . for w From (4.5) and (5.3) we see that d ij ( t − s ) d kl ( t − s ) = − n ( V i K j + K i V j )( V k K l + K k V l )e χ + − χ − e i ω ( s + s ) o + + 18 n ( V i K j + K i V j )( V ∗ k K ∗ l + K ∗ k V ∗ l )e − χ − e i ωs − i ω ∗ s o + . (A1)We can show from this equation that the integrand of (3.4) may be written G ijkl ( s + s ) d ij ( t − s ) d kl ( t − s ) = − n G ijkl ( s + s ) V i K j V k K l e χ + − χ − e i ω ( s + s ) o + + 12 G ijkl ( s + s ) V i K j V ∗ k K ∗ l e − χ − e − ω − ( s + s ) cos ω + ( s − s ) , (A2)where (2.5) has been used. By taking the complex conjugate of G ijkl ( s + s s ) V i K j V ∗ k K ∗ l and using the symmetry (2.6) we see that this quantity is real leading to the last lineof (A2).nergy flux and dissipation in hereditary viscoelasticity 15Dropping temporarily the suffixes ijkl , we evaluate the integrals I = Z ∞ Z ∞ G ( s + s )e i ω ( s + s ) d s d s , (A3) I = Z ∞ Z ∞ G ( s + s )e − ω − ( s + s ) cos ω + ( s − s ) d s d s , (A4)as these integrals will be needed in performing the double integral in (3.4) with integrand(A2). By means of the substitutions α = s + s and β = − s + s we see that I = 12 Z ∞ Z α − α G ( α )e i ωα d β d α and on performing the β integral I = Z ∞ G ( α ) α e i ωα d α = 1i dd ω Z ∞ G ( α )e i ωα d α = − i H ′ ( ω ) , (A5)where H ( ω ) = R ∞ G ( α )e i ωα d α , see (4.7). Also, we see that I = 12 Z ∞ Z α − α G ( α )e − ω − α cos( ω + β ) d α d β = 1 ω + Z ∞ G ( α )e − ω − α sin( ω + α ) d α = 12i ω + Z ∞ G ( α ) (cid:16) e i ωα − e (i ω ) ∗ α (cid:17) d α = 12i ω + (cid:16) H ( ω ) − { H ( ω ) }∗ (cid:17) = H − ( ω ) ω + , (A6)where H − ( ω ) denotes { H ( ω ) } − , the imaginary part of H ( ω ).We now evaluate w by performing the integrals in (3.4) taking the integrand in theform (A2). The ensuing integrals I and I are given by (A5) and (A6), respectively, sothat we finally obtain w = { W e χ + } + e − χ − + w e − χ − , (A7)where W = i4 H ′ ijkl ( ω ) V i K j V k K l , w = 14 ω + H − ijkl ( ω ) V i K j V ∗ k K ∗ l . Evaluation of the integral (3 . for d To evaluate d by means of (3.5) we need to evaluate the same double integral as in(3.4) except that G ijkl ( s + s ) is replaced by its derivative G ′ ijkl ( s + s ). Therefore theintegrand of (3.5) must be replaced by (A2) except that G ijkl ( s + s ) is replaced byits derivative G ′ ijkl ( s + s ). In place of I and I above we must therefore evaluate theintegrals I = Z ∞ Z ∞ G ′ ( s + s )e i ω ( s + s ) d s d s (A8) I = Z ∞ Z ∞ G ′ ( s + s )e − ω − ( s + s ) cos ω + ( s − s ) d s d s . (A9)Substituting α = s + s and β = − s + s as before, we see that I = 12 Z ∞ Z α − α G ′ ( α )e i ωα d β d α = Z ∞ G ′ ( α ) α e i ωα d α. Integrating by parts gives I = (cid:2) G ( α ) α e i ωα (cid:3) ∞ − Z ∞ G ( α ) (cid:0) e i ωα + i ωα e i ωα (cid:1) d α = − H ( ω ) − ωH ′ ( ω ) , (A10)where the integrated out limits vanish and the integral (A5) has been used.Also, we see that I = 12 Z ∞ Z α − α G ′ ( α )e − ω − α cos( ω + β ) d β d α = 1 ω + Z ∞ G ′ ( α )e − ω − α sin( ω + α ) d α = 1 ω + h G ( α )e − ω − α sin( ω + α ) i ∞ α =0 − ω + Z ∞ G ( α ) n − ω − e − ω − α sin( ω + α ) + ω + e − ω − α cos( ω + α ) o d α = ω − ω + Z ∞ G ( α )e − ω − α sin( ω + α ) d α − Z ∞ G ( α )e − ω − α cos( ω + α ) d α = ω − ω + H − ( ω ) − Z ∞ G ( α ) (cid:16) e i ωα + e (i ω ) ∗ α (cid:17) d α = ω − ω + H − ( ω ) − (cid:16) H ( ω ) + { H ( ω ) }∗ (cid:17) = ω − ω + H − ( ω ) − H + ( ω ) . (A11)We now evaluate d by performing the integrals in (3.5) using the integrals I and I givenby (A10) and (A11), respectively, so that we finally obtain equation (5.7) with D and dd