Proceedings of Meetings on Acoustics Volume 19, 2013 http://acousticalsociety.org/ ICA 2013 Montreal Montreal, Canada 2-7 June 2013 Engineering Acoustics Session 1aEA: Thermoacoustics I 1aEA7. On discontinuity waves and vibrations in thermo-piezoelectric bodies Adriano Montanaro* *Corresponding author's address: Department of Mathematics, University of Padua, via Trieste, 63, Padova, 35121, Padova, Italy, montanar@math.unipd.it With regard to a body composed of a linear thermo-piezoelectric medium, lying in a natural state, we consider processes constituted by small displacements, thermal deviations and small electric fields superposed to such state. Any discontinuity surface of order r 2 for the above processes is characteristic for the linear thermo-piezoelectric partial differential equations. We find the ordinary di fferential equations of propagation of general standing waves and plane progressive waves. We characterize the regular waves that are reversible in time. Published by the Acoustical Society of America through the American Institute of Physics 2013 Acoustical Society of America [DOI: 10.1121/1.4799922] Received 22 Jan 2013; published 2 Jun 2013 Proceedings of Meetings on Acoustics, Vol. 19, 030007 2013 Page 1
INTRODUCTION The solid body B under consideration is composed of a linear thermo-piezoelectric medium, i.e., a non-magnetizable linearly elastic dielectric medium that is heat conducting and not electric conducting; B has a natural configuration, say a placement κ[b] of the three-dimensional Euclidean space that B can occupy with zero stress, uniform temperature and uniform electric field. Such natural configuration and state will be used as reference. We consider processes of B constituted by small displacements, thermal deviations and small electric fields u, T, E superposed to κ[b]. A smooth singular surface or discontinuity surface of order r in the triple of fields u, T, E is referred to as a weak thermo-piezoelectric wave if r 2. Any singular surface of order r 2 is characteristic for the linear thermo-piezoelectric partial differential equations. Then smooth waves are considered. i It is shown that the wavefront of a plane progressive wave is characteristic if and only if the wave is isothermal. ii The differential equations are characterized for standing waves of a general type and for the standing waves which are sinusoidal. The latter are isothermal, isentropic, have wavefronts which are characteristic, and their directions of polarization and of propagation satisfy certain constitutive conditions. iii The differential equations for plane progressive waves which are reversible in time are characterized. The results presented here are proved in [1]. LINEAR THERMO-PIEZOELECTRICITY Notations We adopt the linearized theory for thermo-piezoelectricity that is developed in [2], [3]; such a general framing contains more particular theories as, for example, the theory in [4] A unique system of coordinates x 1, x 2, x 3 for both the reference configuration and the ambient space will be used, and the notations in [2], [3] are in force: t mechanical Cauchy stress tensor; E electric vector φ electrostatic potential; T incremental absolute temperature; D electric displacement vector. The linear constitutive equations are specified in terms of the constitutive quantities listed below. σ klij = elastic moduli, e ikl = piezoelectric moduli, β kl = thermal stress moduli, κ E kl = dielectric susceptibility, ω k = pyroelectric polarizability, ɛ kl = permittivity moduli, κ kl = Fourier coefficients, γ = heat capacity, η o = entropy at the natural state, T o = absolute temperature at the natural state, ρ o = mass-density at the natural state. Constitutive Equations and Balance Laws The following constitutive equations are assumed respectively for the Cauchy stress, electric displacement vector, heat flux vector and specific entropy: t kl = σ klij u i, j e ikl E i β kl T, 1 Proceedings of Meetings on Acoustics, Vol. 19, 030007 2013 Page 2
D k = e kij u i, j + ɛ ki E i + ω k T, 2 q k = κ kl T, l + κ E kl E l, 3 η = η 0 + γ T 0 T + 1 ρ o β ij u i, j + ω i E i, 4 where E i = φ, i 5 and the following symmetries hold: σ klij = σ ijkl = σ lkij = σ kl ji, 6 e kij = e kji, β ij = β ji, 7 κ kl = κ lk, κ E kl = κe lk. 8 In the absence of external fields the field equations corresponding to the i balance law of linear momentum, ii Maxwell s equation, and iii balance law of conservation of energy, write as t kl, k ρ 0 ü l = 0, 9 D k, k = 0, ρ o θ η q k, k = 0. 10 Field Equations The linearized field equations, that are obtained by replacing the constitutive equations in the balance laws and neglecting the higher order terms, in the homogeneous case write as σ klij u i, jk + e ijl φ, ij β kl T, k = ρ o ü l, l = 1, 2, 3, 11 e kji u j, ik ɛ kj φ, jk + ω k T, k = 0, 12 T, jk + κ E jk φ, jk + T 0 β kj u k, j + ρ o γ Ṫ T 0 ω k φ,k = 0. 13 Characteristic Hypersurfaces of the Linear Thermo-Piezoelectric Equations A characteristic manifold of a system of partial differential equations is a surface in IR 4 that is exceptional for the assignment of data in the appropriate Cauchy initial value problem. The characteristic equation of the linear differential equations 11-13 is the equation obtained by equating to zero the determinant of the coefficients of the system of five equations σ klij n j n k ρ o V 2 δ li λ i + e il j n i n j ϕ = 0 e kji n i n k λ j ɛ kj n j n k ϕ = 0 n j n k τ + T 0 V β ij n j λ i + κ E jk n jn k T 0 n k ω k V ϕ = 0 14 in the five scalar unknowns τ, λ i, ϕ see [5], where this result is shown to hold also for a non homogeneous body. Proceedings of Meetings on Acoustics, Vol. 19, 030007 2013 Page 3
DISCONTINUITY WAVES Compatibility Conditions for Jumps of Partial Derivatives Let E 3 denote the three-dimensional Euclidean ambient space, I = [t o, t 1 ] a time interval and E = I E 3. We consider a smooth hypersurface S in E that admits a suitably regular representation x i = ψ i t, ξ 1, ξ 2, i = 1, 2, 3, 15 with the parameter pair belonging to an open subset of IR 2. For any value of t equation 15 defines a surface S t in E 3, referred to the curvilinear coordinates ξ 1, ξ 2. The totality of surfaces S t for t I is a moving surface in E 3. Thus S can be interpreted as both the hypersurface of E of equations 15 and the associated moving surface in E 3. The speed V of the surface S at time t has x components and the speed of S t in direction of n is V i = ψ i t 16 V = V i n i. 17 Now let f : N IR be a real scalar-valued function, where N = I N with N open subset of E 3 having, for all t I, non-empty intersection with S t. Since the results below refer only to the part of S contained in N, we replace S N by S and S t N by S t. Let f / n denote the derivative of f in the direction of n on S t, where n is distance measured from S t. Hence / n n i / x i. If S is a singular hypersurface in E of order r 2 for the function f = f x 1, x 2, x 3, t, then the compatibility conditions Hadamard [6], pp.103-104 [ r f ] x i x j... x l t r s = V r s [ r f n r ] n i n j...n l 0 s r, 18 hold on S, where r f n r = r f x p... x q n p...n q r indexes, 19 and V is the local speed of propagation with respect to the medium, apply to the derivatives of f. Weak Discontinuity Waves The l.p.d.e.s 14 are of second order; thus the name weak thermo-piezoelectric wave, briefly weak wave, is applied to a singular hypersurface S E := I IR 3 for the dependent variables u i, φ, T of order r 2. Proposition 3.1 Assume a. Then weak waves are characteristic for the l.p.d.e.s 14. This result generalizes to heat-conducting piezoelectric bodies the one in [7] that hold for piezoelectric bodies that are not heat-conducting. SMOOTH WAVES Time-Reversible Processes A possible thermo-piezoelectric process of B is defined as a solution u, φ, T = u, φ, T x, t Proceedings of Meetings on Acoustics, Vol. 19, 030007 2013 Page 4
of the field equations 11-13. We say that a possible process of B p := u, T, φ :B [t o, t 1 ] IR 3 IR IR, p = px, t is time-reversible, orreversible in time, if the process p = u, φ, T : B [0, t 1 t o ] IR 3 IR IR, p := px, r, r := t + t 1 is also possible for B. Proposition 4.1 Let p := u, φ, T be a thermo-piezoelectric process that is possible for B. Then p is possible for B if and only if p is solution of the field equations σ klij u i, jk + e ijl φ, ij β kl T, k = ρ o ü l l = 1, 2, 3, 20 e kji u j, ik ɛ kj φ, jk + ω k T, k = 0, 21 T, jk + κ E jk φ, jk = 0, 22 T 0 β kj u k, j + ρ o γ Ṫ T 0 ω k φ,k = 0. 23 Proof. The above definition of time-reversed process p associated to p implies that if both p and p are possible processes, then they solve Eq.s 11-12, p solves Eq.13, whereas p solves the equation T, jk + κ E jk φ, jk T 0 β kj u k, j + ρ o γ Ṫ T 0 ω k φ,k = 0 24 since p = p t t, 2 p t 2 = p2 t 2. The thesis follows since Eq.s 13 and 24 are equivalent to Eq.s 22 and 23. Q.E.D. Standing Waves In considering standing waves or vibrations propagating in a not heat-conducting piezoelectric body one seeks solutions of the type ux, t = un xcosω t, φx, t = φn xcosω t, 25 where ω is the circular frequency and n is a unit vector, that may be called the direction of vibration. Here, by analogy, for a heat-conducting piezoelectric body we seek solutions u, φ, T having the form ux, t = un xcosω t φx, t = φn xcosω t 26 Tx, t = Tn xcosω t for some ω IR and unit vector n. The next proposition shows that the solutions of vibration type solve of a system of ordinary differential equations. Proceedings of Meetings on Acoustics, Vol. 19, 030007 2013 Page 5
Proposition 4.2 Let the triple of functions u, φ, T be of the form 26 for some ω IR and unit vector n. Then u, φ, T is a solution of the field equations 11-13 if and only if the functions u, φ, T in Eq.s 26 satisfy the system of ordinary differential equations σ klij u i n j n k + ρ o δ li u i ω 2 + e ijl φ n i n j β kl T n k = 0 l = 1, 2, 3, 27 e kji u j n i n k ɛ kj φ n j n k + ω k T n k = 0, 28 T + κ E jk φ n j n k = 0, 29 T 0 u k β kj n j + ρ o γ T T 0 φ ω k n k = 0. 30 Plane Progressive Waves A plane progressive thermo-piezoelectric wave propagating in the direction of a unit vector n with speed or phase velocity V may be represented by u = un x + Vt φ = φn x + Vt T = Tn x + Vt. 31 Definition 4.1 The wavefronts of the plane progressive wave 31 are the hyperplanes of equation n x + Vt= c, with c constant. Definition 4.2 We say that a wavefront π of 31 is characteristic if at each x, t π the 5 tuple λ i, ϕ, τ:= u i, φ, T does not vanish and is a solution of the equations 14 1 3. Proposition 4.3 Let the triple of functions u, φ, T have the form 31 for some V IR and unit vector n. Then u, φ, T is a solution of the field equations 11-13 if and only if σ klij u i n jn k ρ o δ li u i V 2 + e ijl φ n i n j β kl T n k = 0, l = 1, 2, 3, 32 e kji u j n in k ɛ kj φ n j n k + ω k T n k = 0, 33 T n j n k + κ E jk φ n j n k + V T 0 β kj u k n j + ρ o γ T T 0 ω k φ n k = 0. 34 Proposition 4.4 For some V IR and unit vector n let the triple of functions u, φ, T in 31 be a solution of the field equations 32-34. Then the wavefronts of u, φ, T are characteristic if and only if u, φ, T is an isothermal wave. Proceedings of Meetings on Acoustics, Vol. 19, 030007 2013 Page 6
Let u = un x + Vt, φ = φn x + Vt, T = Tn x + Vt 35 u = un x Vt, φ = φn x Vt, T = Tn x Vt 36 Proposition 4.5 Let n be a unit vector and V > 0. Then both p := u, φ, T and p := u, φ, T are solutions to the field equations 11-13 if and only if the equations 37-40 below hold. σ klij n j n k ρ o δ il V 2 u i + e ijlφ n i n j β kl T n k = 0, l = 1, 2, 3, 37 e kji u j n in k ɛ kj φ n j n k + ω k T n k = 0, 38 T n j n k + κ E jk φ n j n k = 0, 39 T 0 β kj u k n j + ρ o γ T T 0 ω k φ n k = 0. 40 Proof. Let p and p be solutions of Eq.s 11-13; then by Proposition 4.3 Eq.s 32-34 hold and furthermore the latter equation also holds with V replaced with V ; hence Eq. 34 splits in Eq.s 39-40. Conversely, if Eq.s 37-40 hold, then also Eq.s 32-34 hold. Q.E.D. Corollary 4.1 A possible plane progressive wave u, φ, T is reversible in time if and only if both the waves u, φ, T and u, φ, T are possible processes. Proof. Eq.s 37-40 are equivalent to Eq.s 20-23 under the substitution 35 or 36. Hence Proposition 4.1 yields the thesis. Q.E.D. Isentropic Plane Waves Now we consider isentropic plane waves, that is, solutions p = u, φ, T u = un x + Vt φ = φn x + Vt T = Tn x + Vt 41 of the field equations such that see 4 T o ρ o η = ρ o γ Ṫ + T o β ij u i, j ω i φ,i = 0. 42 By replacing u = u n x + VtV, φ = φ n x + VtV, Ṫ = T n x + VtV 43 into 42 we obtain Hence T o ρ o η = ρ o γ T + T o β ij u i n j ω i φ n i +V = 0. 44 η = 0 ρ o γ T + T o β ij u i n j ω i φ n i = 0. 45 In particular, if β ij n j = 0 = ω i n i, then the plane wave is isentropic if and only if it is isothermal η = 0 T = 0. Proceedings of Meetings on Acoustics, Vol. 19, 030007 2013 Page 7
Isentropic Standing Waves We can repeat the procedure above for standing waves too. Let u, φ, T be of the form 26 for some ω IR and unit vector n. Hence u = un x [ ωsinωt ] φ = φn x [ ωsinωt ] Ṫ = Tn x [ ωsinωt ] 46 By replacing 46 into 42 we obtain T o ρ o η = [ ] ρ o γ T + T o β ij u i n j ω k φn k ωsinωt 47 Hence η = 0 ρ o γ T + T o β ij u i n j ω k φn k = 0. 48 In particular, if β ij n j = 0 = ω k n k, then the standing wave is isentropic if and only if it is isothermal η = 0 T = 0. Sinusoidal Standing Waves Next we study a standing wave p := u, φ, T that is sinusoidal, in the sense that its form 26 is further specified by ux, t = λsink n xcosω t for some λ IR 3, φ, τ IR, unit vector n, k > 0, ω > 0. Note that by prostaferesi formulae we have φx, t = φsink n xcosω t 49 Tx, t = τsink n xcosω t sin u cos v = 1 sin p + sin q 2 where 2u = p + q, 2v = p q and p = u + v, q = u v. Hence p = p + + p 50 is the sum of the two sinusoidal progressive waves that are defined by p + := u +, φ +, T +, p := u, φ, T, 51 u ± x, t = λ 2 sink n x ± ω t, φ± x, t = ϕ 2 sink n x ± ω t, T± x, t = τ sink n x ± ω t. 52 2 Thw wave p can propagate if and only if both p + and p can propagate. Proposition 4.6 The sinusoidal standing wave p, having the form 49, can propagate if and only if the two sinusoidal plane progressive waves p + and p in 51, 52 can propagate. Proceedings of Meetings on Acoustics, Vol. 19, 030007 2013 Page 8
Corollary 4.2 Any possible sinusoidal wave is reversible in time. Lastly, we note that possible sinusoidal vibrations are characteristic, isothermal, isentropic, and can propagate only along material certain material directions. REFERENCES [1] A. Montanaro, On discontinuity waves and smooth waves in thermo-piezoelectric bodies, Preprint, http://http://paduaresearch.cab.unipd.it/5403/ 1 12 2013. [2] A.C.Eringen, Mechanics of Continua Robert E. Krieger Publishing Company, Inc., second edition 1963. [3] A.C.Eringen and G. Maugin, Electrodynamics of Continua Springer-Verlag New York Inc. 1990. [4] K. Majorkowska-Knap, Dynamical problems of thermo-piezoelectricity, Bullettin de l Academie Polonaise des Sciences, Series des sciences techniques 27, 97 105 1979. [5] A. Montanaro, On propagation of discontinuity waves in thermo-piezoelectric bodies, WSEAS Transactions on Mathematics 6, 648 654 2007. [6] J. Hadamard, Lecons sur la Propagation des Ondes et les Equations de l Hydrodynamique Herman, Paris 1903. [7] A. Montanaro, On discontinuity waves in linear piezoelectricity, J. Elasticity 65, 49 60 2001. Proceedings of Meetings on Acoustics, Vol. 19, 030007 2013 Page 9