INFLUENCE OF IMPERFECT BONDING ON INTERFACE OF MAGNETO- ELECTRO-ELASTIC HETEROSTRUCTURES: SH WAVES DISPERSION RELATIONS

Transcription

1 NFLUENCE OF MPERFECT BONDNG ON NTERFACE OF MAGNETO- ELECTRO-ELASTC HETEROSTRUCTURES: SH WAVES DSPERSON RELATONS J. A. Otero, H. Calas, nstituto e Cibernética, Matemática Física (CMAF), 5 No. 55, entre C D, Veao, Habana 4, CP 0400, Cuba. nstituto e Acustica, Serrano 44, Mari, España. R. Roríguez-Ramos, reinalo@matcom.uh.cu, J. Bravo-Castillero, jbravo@matcom.uh.cu Faculta e Matemática Computación, Universia e La Habana, San Lazaro L, Veao, Habana 4, CP 0400, Cuba. Aair Aguiar, aguiarar@sc.usp.br Department of Structural Engineering, Universit of São Paulo, São Carlos, SP, Brazil. Abstract. n the present work the ispersion relations of stationar SH waves in a heterostructure with magnetoelectro-elastic properties uner imperfect mechanical boning on the interface have been obtaine. The calculations of the ispersion relations were base on the consieration of a smmetric sstem an results are presente b consiering onl the smmetric moes of the ispersion curves. Different limit cases are presente. The first ispersion branch for the smmetric moes is presente for five ifferent values of a parameter use in the moeling of the imperfect boning. Kewors: Piezoelectricit, Piezomagnetism, Magneto-electro-elasticit, Dispersion curve, Wave propagation.. NTRODUCTON The evelopment of smart structures is currentl receiving wiesprea attention owing to potential applications in several branches of engineering, such as in integrate control architecture with highl istribute sensors an actuators. More recentl, applications can be foun in the esign of smart materials, piezoelectric an piezomagnetic properties are involve. Here, the basic iea is to take avantage of each constituent s properties an obtain a composite material that has superior magneto-electric coupling effects as compare to conventional piezoelectric or piezomagnetic materials. The problem of wave propagation in this tpe of material has been stuie b ifferent authors using ifferent geometries. Alshits et al. (994) stuie the eistence of localize acoustic waves on the interface between two piezocrstals of arbitrar anisotrop. Aguiar an Angel (000) investigate the scattering of antiplane waves from a slab region containing a ranom istribution of clinrical cavities. Later, Aguiar an Angel (006) obtaine an integroifferential equation to escribe the propagation of these waves insie the slab an showe that the secon erivative of the corresponing isplacement fiel has a square-root singularit near the bounar of the slab. Pan (00) erive an eact close-form solution for the static eformation of multilaere piezoelectric an piezomagnetic plates base on the quasi-stroh formalism an the propagator matri metho. Pan an Heliger (00) etene the analtical metho of Pan (00) to the free vibration, three-imensional, linear anisotropic, magneto-electro-elastic, simpl supporte, an multilaere rectangular plates. Wang et al. (003) erive the state vector equations for three-imensional, orthotropic, an linearl magneto-electro-elastic meia. The solution of theses equations is base on a mie formulation, the basic unknowns are not onl isplacements, electrical potential, an magnetic potential, but also stresses, electric isplacements, an magnetic inuction. Recentl, Chen et al. (007) presente an analtical treatment for the propagation of harmonic waves in infinitel etene, magneto-electro-elastic multilaere plates base on the state vector approach. This work is motivate b recent contributions reporte b Calas et al. (008), Lombar an Pirau (006) an Melkuman an Mai (008). The work of Melkuman an Mai (008) is use to obtain an etension of the results given b Calas et al. (008) to the case of imperfect contact at the interface. Here, the behavior of stationar SH waves in a heterostructure with magneto-electro-elastic materials with imperfect mechanical boning on the interface is stuie. n particular, the case of permeable interfaces are iscusse in etail. The governing sstem of partial ifferential equations is solve using the smmetr of the sstem an consiering the superficial acoustic wave. Solutions can be separate into smmetric an anti-smmetric parts. The ispersion relations are erive onl for the smmetric part of the solution. Different limit cases are presente. The ispersion curves an the influence of the imperfection at the interface are shown for some cases.. WAVE EQUATON FOR THE SH MODE A material ehibiting magneto-electro-elastic properties with 6mm smmetr with magnetization an polarization in the z -ais irection is consiere. t is well known (Calas et al., 008) that the sstem of equations that governs the namic behavior of a homogeneous magneto-electro-elastic material has five couple partial ifferential equations with three unknown components of the isplacement file, u, u, uz, an two potential functions, ϕ an ψ.

2 Let us consier a plane wave propagating in an infinite homogeneous soli in a stea-state regime. n this case, the solution U = ( u u uz ϕ ψ ) T of the sstem of equations is sought in the comple plane wave form, which is given b 0 ( ikr ωt) U( r, t) = U e k is the wave vector, ω is the circular frequenc an U = ( u ) T u uz ϕ ψ is the amplitue vector. The wave front is in a plane that is perpenicular to the wave vector k = k ˆ ˆ ˆ i + k j+ kzk. Since both magnetic an electric effects appear in the SH moe, we confine the scope of the present work to them, an the equation escribing the phenomena reuce to (Calas et al., 008) u c u+ e + f ψ = ρ, t () e u ε ϕ g ψ = 0, () f u g ϕ μ ψ 0, +, u u z, c c44, e e5, f f5, ε ε, μ μ, g g = (3), an the material parameters ce,, f, ε, μ, g are the elastic, piezoelectric, piezomagnetic, ielectric, magnetic permeabilit, an magnetoelectric coefficients, respectivel. The sstem of equations ()-(3) escribes the motion of a SH wave in an homogeneous material an, therefore, epens onl on (, t, ). This is a couple sstem of equations for the elastic isplacement in the z -ais irection, u, the electric potential ϕ, an the magnetic potential ψ. The sstem ()-(3) can be solve using two auiliar potential functions ϕ an ψ efine b e g f g ϕ = ϕ u + ψ, ψ = ψ u + ϕ, (4) ε ε μ μ respectivel. Replacing (4) into both () an (3), we obtain ϕ = 0, ψ = 0. (5) Solving (4) for the functions ϕ an ψ, we fin ϕ = χϕ + βu αψ, ψ = χψ + β u α ϕ, (6) εμ μ g ε g eμ f g χ =, α =, α =, εμ g εμ g εμ g, f ε eg β = β = εμ g εμ g. (7) Replacing the epressions given b Eq. (6) into Eq. (), the tpical wave motion equation for the mechanical isplacement u is obtaine, being given b u 0, = (8) vs h t v s h is the wave propagation velocit in the meium an is given b c+ eβ + f β vs h =. (9) ρ Noting from above that all variables in this work are harmonic in time, we can rop the term e iω t, an obtain the solution of Eq. (8) in the form + ik ik ik u = ( A e + A e ) e, (0) A + an A enote the amplitues of the waves propagating to the right an to the left of the -ais irection, respectivel. The solution of the sstem in Eq. (5) has the form + k k ik + k k ik ϕ = ( Aes e + Aes e ) e, ψ = ( Ams e + Ams e ) e, () A + es an A es enote the amplitues of the auiliar electric potential relate to the surface waves that are propagating in the positive an negative irections of the -ais, respectivel. On the other han, A + ms an A ms enote the amplitues of the auiliar magnetic potential relate to the surface waves that are propagating in the positive an negative irections of the -ais, respectivel. The amplitues are couple through the electric an magnetic potentials. Replacing the epressions of Eqs. (0) an () into Eq. (6), we finall get + k k + ik ik + k k ik ϕ = χ( Aes e Aese ) β( A e A e ) α ( Ams e Amse ) e, ()

3 + + ik ik + ( ) ( ) ( ) ψ = χ A e + A e + β A e + A e α A e + A e e k k k k ik ms ms es es. (3) 3. SH WAVES N MAGNETO-ELECTRO-ELASTC HETEROSTRUCTURE: DSPERSON RELATON Let us now consier an infinite laer of with bone imperfectl to two semi-infinite meia, one at each sie of the laer. The origin of the coorinate sstem is at the mile of the infinite laer, enote meium, such that the interfaces between the laer an the semi-infinite meia, enote meia an, are locate at =± /. Since we are intereste in the stu of confine moes, the functions u, ϕ, ψ iel evanescent waves in meia an. Consiering that the semi-infinite meia an are of the same material, these functions are smmetric with respect to the z-plane an can be ecouple into smmetric an anti-smmetric moes. We then restrict the application of the bounar conitions on onl one interface. Using Eqs. (0), (), an (3), the functions u, ϕ, ψ can be written in the following form for the smmetric moe onl. k /) Ce, / ik u = e Acos( k ), /, (4) k ( /) Ce, / k /) k /) β u+ χ Φ e α Ψ e, / ik ϕ = e β u+ χ Φ cosh( k) α Ψ cosh( k), /, (5) k( /) k( /) β u+ χ Φ e α Ψ e, / k /) k /) β u α Φ e + χ Ψ e, / ik ψ = e β u α Φ cosh( k ) + χ Ψ cosh( k ), /. (6) k ( /) k ( /) β u α Φ e + χ Ψ e, / The parameters k, k, k, an ω appearing in Eqs. (4)-(6) are relate to each other through the epressions ω sh k + ( k ) = ( v ), /, ω sh k + ( k ) = ( v ), /, (7) j j j j j j j sh v = ( c + e β + f β ) ρ, j =,, is the velocit of the SH wave in meium j. The epression for the stress T4 = c u + e ϕ + f ψ, electric isplacement D = e u ε ϕ g ψ, an magnetic flu B = f u g ϕ μ ψ can be calculate in the form ( ) ( ) k /) k /) C c k e + e Φ + f Ψ k e, < / T4 = Ac k sin( k ) + e Φ + f Ψ ksinh( k / ), < /, k ( /) k ( /) C c k e e f Φ + Ψ k e, > / (8) k /) ε k Φ e, < / D = ε k Φ sinh( k), < /, k ( /) ε k Φ e, > / (9) k /) μ k Ψ e, < / B = μ k Ψ sinh( k), < /, k ( /) μ k Ψ e, > / (0) c = c+ eβ+ f β, e = eχ f α, f = f χ eα. ()

4 Now, we refer to the mathematical statement of the bounar conitions at the interface between the laer an the semi-infinite meia, which is locate at = ± /. n this case, we stu the wave propagation with imperfect bone effect, i.e., electromagneticall permeable boning (Melkuman an Mai, 008), for which T4 = T4 = K( u u ), ϕ = ϕ, D = D, ψ = ψ, B = B, () with si unknown amplitues ( AC,, Φ, Φ, Ψ, Ψ ) that appear in Eqs. (8)-(0). The first epression in Eq. () escribes an elastic interface with elastic, or, spring constant K > 0. With this moel, the interface is allowe to eform an the isplacement at the interface can be iscontinuous. The case K correspons to a perfectl bone interface an the case K 0 correspons to mechanical free bounar with no elastic interaction at the interface. Replacing Eqs. (8)-(0) into the bounar conitions given b Eq. (), the ispersion relations for the smmetric moes can be written as ( tan ( )) sinh ( ε μ K c k c k k + Kk k ) e e N + f f N N + ε μ (3) cc kk tan ( k ) kk sinh ( ) tan ( ε μ + ) 0, c k k e N f N N + = ε μ ε βχ coth(k /)+ βχ + ( β β-ββ ) e k N = ε, (4) μ βχ coth(k/)+ βχ + ( ββ -β β ) f k μ ( ) N = αn / β+ χ/ β sinh(k /), (5) an μ μ β = ( β -β ) K+ β c k, α = β k f K coth(k /, α + α μ μ (6) ε ε β = ( β-β ) K+ β c k, χ = β ke + K χ + χ coth(k/). ε ε 4. LMT CASES The ispersion relations given b Eq. (3) satisf ifferent limit cases, i.e., the stu is reuce to either perfect boning K (permeable interface) or free bounar with no boning 0 K (permeable interface). a) Perfect boning: K. ( tan ( ) ) sinh ( ε μ c k ) 0 c k k + k k e e N + f f N N =, (7) ε μ N an N are given b Eqs. (4) an (5), respectivel, with μ ε β = ( β - β ), β = ( β- β ), α = α + α coth(k /, χ χ + χ coth(k /). = μ ε This case reprouces the same ispersion curves reporte b Calas et al. (008). b) Free bounar with no boning: K 0. tan ( ) sinh ( ) tan ( ε μ cc kk k + kk ) 0 c k k e N f N N + =, (8) ε μ N an N are given b Eqs. (4) an (5), respectivel, with μ ε β = β ck, β = β ck, α = β kf, χ = β ke. μ ε 5. NUMERCAL RESULTS Numerical calculations were performe to show the quantitative an qualitative influence of boning on the namic properties of laere composites. n the following numerical eamples, we use the piezomagnetic material CoFe O 4 an the piezoelectric material PZT-4 to analze the ispersion curves of the composite CoFe O 4 /PZT-4/CoFe O 4. The

5 material parameters are available in Melkuman an Mai (008). n the stu of SH-wave propagation in multi-laer sstems, it is convenient to use the ispersion relation relate to the phase velocit. The relation k = ω vs, v s is the phase velocit (Cheeke, 00), is introuce into Eq. (7) an the components k, k of the wave vectors in meia an, respectivel, are written as = ω ( ) ( ), k ω ( vsh) ( vs ) k v v sh s =. (9) The influence of the imperfect boning at the interface of the composite meium can be observe in Figs. an. Different values of the imperfection material parameter K are consiere in orer to show the influence of this parameter on the ispersion curves. Onl the first ispersion branch is presente for five values of K. The real parts of the solutions v s are foun as functions of the frequenc f = ω /π. The thickness of the intermeiate laer use in the calculations is 0 mm in Fig. an mm in Fig.. These figures suggest that the effect of the imperfection parameter K becomes more noticeable as the laer becomes thinner; especiall, in the low frequenc region. Figure. Dispersion curves for the first moe consiering five ifferent values of K an = 0 mm. Figure. Dispersion curves for the first moe consiering five ifferent values of K an = mm. Figure 3 shows the real part of the velocit with respect to the imperfection parameter K using a thickness laer of mm an frequenc of 0.05MHz. Notice that as K increases, the velocit increases asmptoticall, approaching the velocit of perfect contact.

6 Fig. 3: Depenence of wave velocit on imperfect parameter K in the CoFe O 4 / PZT-4/CoFe O 4 composite. 6. CONCLUSONS n this paper, the ispersion relation for imperfect contact at the interface of a magneto-electro-elastic heterostructure consiering superficial components in magnetic an electric potentials has been erive. Different limit cases ( K an K 0 ) are stuie for the general solution. The first ispersion branch is presente for five values of K. The magneto-electric coupling effect an the imperfect boning have influences on the ispersion curves. n particular, the influence of K becomes more noticeable as the laer becomes thinner. 7. ACKNOWLEDGEMENTS The authors wish to acknowlege the National Council for Scientific an Technological Development (CNPq), Proc. No /009-9 an the National Program of Basic Sciences, CTMA, Cuba, PNCT BMFQC , for their support of this research. 8. REFERENCES Aguiar, A.R. an Angel, Y.C., 006. nterface Effects an Coherent Waves in Porous Elastic Meia, Mathematics an Mechanics of Solis, pp Aguiar, A.R. an Angel, Y.C., 000. Antiplane coherent scattering from a slab containing a ranom istribution of cavities, Proceeings of the Roal Societ of Lonon A 456, pp Alshits, V.., Barnett, D.M., Darinskii, A. N., Lothe, J., 994. On the eistence problem for localize acoustic waves on the interface between two piezocrstals, Wave Motion 0, pp Calas, H., Otero, J. A., Roríguez-Ramos, R., Monsivais, G.an Stern, C, 008. Dispersion relations for SH wave in magneto-electro-elastic heterostructures, nternational Journal of Soli an Structures 45, pp Cheeke, J. Davi N., 00. Funamentals an Applications of Ultrasonic Waves, E. CRC Press LLC, Boca Raton, Floria. Chen, J., Pan, E., Chen, H., 007. Wave propagation in magneto-electro-elastic multilaere plates, nternational Journal of Soli an Structures 44, pp Lombar B., an Pirau, J., 006. Numerical moeling of elastic waves across imperfect contacts, SAM Journal Science Computational 8:, pp Melkuman, A.an Mai Y. W, 008. nfluence of imperfect boning on interface waves guies b piezoelectric/piezomagnetic composites, Philosophical Magazine 88: 3, pp Pan, E., 00. Eact solution for simpl supporte an multilaere magneto-electro-elastic plates, Journal of Applie Mechanics 68, pp Pan, E. an Heliger, P. R., 00. Free vibrations of simpl supporte an multilaere magneto-electro-elastic plates, Journal of Soun an Vibration 5, pp Wang, J., Chen, L., Fang, S., 003. State vector approach to analsis of multilaere magneto-electro-elastic plates, nternational Journal of Soli an Structures 40, pp RESPONSBLTY NOTCE The authors are the onl responsible for the printe material inclue in this paper.