Exact Solutions for Simply Supported and Multilayered Piezothermoelastic Plates with Imperfect Interfaces

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1 he Open Mechanics Journal Exac Soluions for Simply Suppored and Mulilayered Piezohermoelasic Plaes wih Imperfec Inerfaces X. Wang * and E. Pan Dep. of Civil Engineering and Dep. of Applied Mahemaics Universiy of Aron Aron OH USA Absrac: Exac soluions are derived for hree-dimensional orhoropic linearly piezohermoelasic simply-suppored and mulilayered recangular plaes wih imperfec inerfaces under saic hermo-elecro-mechanical loadings. In his research he imperfec inerface is described as hermally wealy or highly) conducing mechanically complian and dielecrically wealy or highly) conducing. While he homogeneous soluions for one layer are obained in erms of he socalled pseudo-sroh formalism soluions for mulilayered plaes are expressed in erms of he ransfer marices for boh he layer and he imperfec inerface. Due o he fac ha he hermal effec is incorporaed we adop a special form of he ransfer marix resuling in a very concise soluion srucure for piezohermoelasic mulilayered plaes. Numerical resuls are presened o validae he developed formulas and o demonsrae he influence of he inerface imperfecion on he disribuions of he field variables. 1. INRODUCION hree-dimensional analyical soluions for simplysuppored plaes coninue o arac invesigaors aenion [1-9]. Various echniques including he asympoic expansion scheme [7] he sae space formulaion [47] he Sroh formalism [6] he pseudo-sroh formalism [89] and he ransfer marix or propagaor marix) [7-9] have been proposed in hese sudies. he maerials sudied encompass purely elasic [1] piezoelecric [67] piezohermoelasic [45] and muliferroic [89] maerials. One common assumpion in mos of he aforemenioned wors is ha he exended displacemen and racion vecors see [8] for a deailed definiion) are coninuous across he inerface beween wo adjacen layers. his ind of simplificaion of he inerface is no enough o reflec various damage occurring on he inerface e.g. debonding sliding and/or cracing across he inerface and as a resul he concep of imperfec inerface should be incorporaed. Up o now various imperfec inerface models have been proposed in he conex of hea conducion [10-1] dielecriciy [114] and elasiciy [15-0]. he main focus of his research is he incorporaion of imperfec inerfaces in he hree-dimensional analysis of a simply-suppored and mulilayered piezohermoelasic recangular plae. he imperfec inerface sudied here is mechanically complian and hermally or dielecrically) wealy or highly) conducing. For a mechanically complian inerface we adop he linear spring model for he imperfec inerface [15-19]. In his model racions are coninuous bu displacemens are disconinuous across he imperfec inerface jumps in he displacemen componens are furher assumed o be proporional in erms of he spring-facor-ype inerface parameers o heir respecive inerface racion componens. For a hermally or dielecrically) wealy conducing inerface [10-1] he normal hea flux or he normal elecric displacemen) is coninuous bu he emperaure or elecric poenial) is disconinuous across he inerface he jump in emperaure or elecric poenial) is proporional o he normal hea flux or normal elecric displacemen). For a hermally or dielecrically) highly conducing inerface [11118] he emperaure or elecric poenial) is coninuous across he inerface as he normal hea flux or normal elecric displacemen) has a disconinuiy across he inerface which is proporional o a cerain differenial expression of he emperaure or elecric poenial). I is no a easy as o address his problem since he inerface is imperfec in hea conducion elasiciy and dielecriciy. Under isohermal condiions Chen e al. [1] derived exac soluions for simply-suppored and mulilayered orhoropic piezoelecric recangular plaes wih mechanically complian and dielecrically wealy conducing inerface by means of a sae space formulaion. I should be noiced ha in he discussion of Chen e al. [1] here exiss a sign error in he descripion of he dielecrically wealy conducing inerface see Eq. 11) 4 in [1]).. PROBLEM DESCRIPION.1. Basic Equaions In a fixed Caresian coordinae sysem x he consiuive equaions including he Fourier s law of hea conducion for an orhoropic piezohermoelasic maerial of crysal class mm wih poling in he x -direcion can be wrien in he marix form as q 1 = 11 1 q = q = 1) *Address correspondence o his auhor a he Dep. of Civil Engineering and Dep. of Applied Mahemaics Universiy of Aron Aron OH USA; xuwang@uaron.edu / Benham Science Publishers Ld.

2 he Open Mechanics Journal 007 Volume 1 Wang and Pan 11 c 11 c 1 c e 1 S 11 * 11 c 1 c c e S * c 1 c c e S * c e 4 0 S 0 1 = c 55 0 e S c S 1 0 D e 15 0 ) E 1 0 D e ) 0 E 0 D e 1 e e ) E p ) q i σ ij and D i are he hea flux sress and elecric displacemen; is he emperaure change; S ij and E i are he srain and elecric field; ii is he hermal conduciviy coefficien; c ij e ij and ε ij are he elasic piezoelecric and dielecric coefficiens; β ii and p are he sress-emperaure and pyroelecric coefficiens. he srains and elecric fields can be expressed in erms of he displacemens u i i=1 ) and he elecric poenial φ as S ij = 1 u i j u ji E i = i. ) In addiion he seady sae energy equaion in he absence of hea source and he saic equilibrium equaions in he absence of body force and elecric charge can be expressed as q ii = 0 4) ij j = 0 D ii = 0. 5).. he Boundary Value Problem Le us consider an anisoropic piezohermoelasic and N-layered recangular plae wih horizonal dimensions L x L y in he x 1 - and x -direcions) and he oal hicness H in he x -direcion). Each layer of he plae is orhoropic wih poling in he x -direcion. he Caresian coordinae sysem is esablished in such a way ha is origin is a one of he four corners on he boom surface and he plae is in he posiive x region. Le layer be bonded by he lower inerface x = and he upper inerface x =1 wih is hicness h =1. I is obvious ha z 1 =0 and z N1 =H. he boundary and coninuiy condiions o be saisfied are hose a he four edges of he mulilayered recangular plae as well as hose on he horizonal surfaces and differen inerfaces of he mulilayered recangular plae. hese boundary and coninuiy condiions are specifically lised as follows. i) Simply-suppored elecrically grounded and zero emperaure edge boundary condiions for each layer [4-8 1] u = u = 11 = = = 0 a x 1 = 0 L x for each layer 6) u 1 = u = = = = 0 a x = 0 L y ii) hermo-elecro-mechanical boundary condiions on he boom surface of he plae 1 = 0 1 cos px 1 = 0 sin px 1 )cosqx = 0 sin px 1 D = D 0 sin px 1 ) or = 0 sin px 1 h a = h a a * sin px 1 x = 0) 7) h a is he surface hea ransfer coefficien a x = 0 and D 0 0 * a are nown values and p = n L x q = m L y 8) wih n and m being wo posiive inegers. iii) hermo-elecro-mechanical boundary condiions on he op surface of he plae 1 = 1 H cos px 1 = H sin px 1 )cosqx = H sin px 1 D = D H sin px 1 ) or = H sin px 1 h b = h b b * sin px 1 ) x = H ) h b is he surface hea ransfer coefficien a x = H and H 1 H H D H H * b are also nown values. iv) 9) Imperfec bonding condiions beween wo adjacen piezohermoelasic layers In he following we will discuss in deail he imperfec inerface in hea conducion elasiciy and dielecriciy one by one. Imperfec Inerface in Hea Conducion If he inerface x =z j j=...n) is hermally wealy conducing hen i follows from he definiions oulined in he inroducion ha he following relaionships hold

3 Mulilayered Piezohermo-elasic Plaes wih Imperfec Inerfaces he Open Mechanics Journal 007 Volume 1 q z j ) = q z j z j ) z j ) = j) q z j j) 10) is a nonnegaive consan and he superscrips and denoe he limi values from he upper and lower sides of he inerface z = z j. he case j) = 0 corresponds o a hermally perfec inerface as j) sands for adiabaic conac. On he oher hand if he inerface x =z j is hermally highly conducing hen i follows from he definiions oulined in he inroducion ha he following relaionships hold z j ) = z j q z j ) q z j ) = j) s z j s = x 1 x 11) is he operaor of surface Laplacian [18] j) is a nonnegaive consan. he case j) = 0 corresponds o a hermally perfec inerface as j) describes conac wih a medium of infinie conduciviy. Imperfec Inerface in Elasiciy According o he definiions oulined in he inroducion he boundary condiions on a mechanically complian inerface x =z j can be expressed as 1 z j ) = 1 z j z j ) = z j z j ) = z j u 1 z j ) u 1 z j ) = 1 j) 1 z j u z j ) u z j ) = j) z j u z j ) u z j ) = j) z j 1 j) j) j) 1) are hree nonnegaive consans. he case 1 j) j) j) = 0 corresponds o a mechanically perfec inerface as 1 j) j) j) describes a compleely debonded inerface. Imperfec Inerface in Dielecriciy If he inerface x =z j is dielecrically wealy conducing hen he following relaionships hold D z j ) = D z j z j ) z j ) = D j) D z j D j) 1) is a nonnegaive consan. he case D j) = 0 corresponds o a dielecrically perfec inerface as D j) describes a charge-free insulaing) inerface. On he oher hand if he inerface x =z j is dielecrically highly conducing hen he following relaionships hold z j ) = z j D z j ) D z j ) = D j) s z j D j) 14) is a nonnegaive consan. he case D j) = 0 corresponds o a dielecrically perfec inerface as D j) describes an equipoenial inerface.. HE PSEUDO-SROH FORMALISM.1. Hea Conducion he emperaure saisfying he zero emperaure edge condiions is chosen o be = fe x sin px 1 ). 15) f and η are unnowns. hen i follows from Eq. 1) ha he hree hea flux componens can be expressed as q 1 = h 1 e x cos px 1 q = h e x sin px 1 )cosqx q = ge x sin px 1 h 1 = 11 pf h = qf g = f. 16) Subsiuion of Eq. 15) ino Eq. 1 hen he resuls ino Eq. 4) for he energy equaion we obain he following eigenrelaion 11 p q f = 0. 17) I is apparen ha for a nonrivial soluion of f he wo eigenvalues of should be 1 = 11 p q > 0 = 1 < 0. 18) Wih he aid of Eq. 16 Eq. 17) can be recas ino he following sandard eigenvalue problem M f g = f g 19) 0 1 M = 11 p q. 0) 0 If we disinguish he wo eigenvecors of Eq. 19) by aaching a subscrip o f and g hen he general soluion for he emperaure and normal hea flux of he x -dependen facor) can be expressed as

4 4 he Open Mechanics Journal 007 Volume 1 Wang and Pan q = f 1 f e 1 x 0 g 1 g 0 e 1 x ) 1 ) 1) 1 and are wo unnown consans o be deermined. he wo in-plane hea fluxes q 1 and q of he x - dependen facor) can be expressed in erms of he emperaure and normal hea flux as q 1 = 11 p 0 q q 0 q. ) he eigenvecors of Eq. 19) are acually righ ones. he lef eigenvecors of Eq. 19) are found by solving he following eigenvalue problem M = ) he superscrip denoes marix ranspose. If and f g [ ] are he eigenvalue and eigenvecor of [ ] are he corresponding Eq. 19 hen = and = g f soluions of Eq. ). Since he lef and righ eigenvecors are orhogonal o each oher we hen come o he following normalized orhogonal relaionship g f f 1 f g 1 f 1 g 1 g = ) he choice of f 1 f g 1 and g is no unique as long as hey saisfy Eq. 4) or equivalenly f 1 f = 1 1 ). For example we can choose f 1 = 1 f = 1 1 g 1 = 1 g = 1 which obviously saisfy he above normalized orhogonal relaionship... he Elecroelasic Field In view of he simply suppored and elecrically grounded edge boundary condiions in Eq. 6 he generalized displacemen vecor can ae he following forms u 1 a 1 cos px 1 ) u u = u = e sx a sin px 1 )cosqx ) a sin px 1 ) a 4 sin px 1 ) c 1 cos px 1 ) fe x c sin px 1 )cosqx ) c sin px 1 ) c 4 sin px 1 ) 5) he firs erm on he righ hand side of Eq. 5) gives he homogeneous general soluion while he second erm on he righ hand side of Eq. 5) presens he paricular soluion due o he hermal effec from Eq. 15). Subsiuion of Eq. 5) ino Eq. and hen he resuls ino he consiuive relaions ) yields he generalized racion vecor as follows 1 b 1 cos px 1 ) = = e sx b sin px 1 )cosqx ) b sin px 1 ) D b 4 sin px 1 ) 6) d 1 cos px 1 ) fe x d sin px 1 )cosqx ) d sin px 1 ). d 4 sin px 1 ) Inroducing four vecors a b c d of dimension 4 [ ] b = [ b 1 b b b 4 ] [ ] d = [ d 1 d d d 4 ] a = a 1 a a a 4 7) c = c 1 c c c 4 hen we can find ha he vecor b is relaed o a and d o c by b = R s)a = 1 Q sr)a 8) s d = R )c = 1 Q R)c ) R = R and he hree 4 4 real marices Q R are defined by c = 0 c = c e 0 0 e c 11 p c 66 q ) pqc 1 c 66 ) 0 0 Q = Q pqc = 1 c 66 ) c 66 p c q ) c 55 p c 44 q ) e 15 p e 4 q ) 0 0 e 15 p e 4 q ) 11 p q 0 0 pc 1 pe qc R = qe pc 55 qc pe 15 qe and he wo real vecors 1 are defined as 1 = p 11 q 0 0 0) 0 0 = 1) ) p In addiion he in-plane sresses and he in-plane elecric displacemens can be expressed as 11 q 11 sin px 1 ) q 1 sin px 1 ) 1 q 1 cos px 1 )cosqx ) q cos px 1 )cosqx ) = e sx q 1 sin px 1 ) fe x q sin px 1 ) D 1 q 41 cos px 1 ) q 4 cos px 1 ) D q 51 sin px 1 )cosqx ) q 5 sin px 1 )cosqx ) q 11 q 1 q 1 q 41 q 51 c 11 p c 1 q c 1 s e 1 s c 66 q c 66 p 0 0 = c 1 p c q c s e s e 15 s 0 e 15 p 11 p 0 e 4 s e 4 q q a 1 a a a 4 ) a)

5 Mulilayered Piezohermo-elasic Plaes wih Imperfec Inerfaces he Open Mechanics Journal 007 Volume 1 5 q 1 c 11 p c 1 q c 1 e 1 * c q c 66 q c 66 p q = c 1 p c q c e c 0 c * b) q 4 e 15 0 e 15 p ) 11 p 0 c q 5 0 e 4 e 4 q ) q 4 0 Now insering Eq. 5) ino Eq. hen ino Eq. and finally ino he saic equilibrium equaions 5 we finally arrive a he following eigenrelaions Q sr R ) s a = 0 4) Q R R ) c = 1. 5) I can be easily verified ha if s is an eigenvalue of 4 hen s is also an eigenvalue of he eigenequaion 4). Wih aid of Eqs. 8) and 9 Eqs. 4) and 5) can be recas ino he following sandard eigenrelaions N a b = s a b 6) N c d = c d 7) 1 R 1 N = Q R 1 R R 1 = I R 1. 8) he general soluions Eqs. 6) and 7) can be ermed he pseudo-sroh formalism [8] for a homogeneous piezohermoelasic layer. Assume ha he ih i=1 4) and i4)h eigenvalues of Eq. 6 denoed by s i and s i4 saisfy he relaion s i s i4 = 0. We disinguish he eigh eigenvecors of Eq. 6) by aaching a subscrip o a and b. Also c 1 d 1 is he vecor obained from Eq. 7) for = 1 c d is he vecor obained from Eq. 7) for =. hen he general soluion for he generalized displacemen and racion vecors of he x -dependen facor) can be concisely expressed as u = A 1 A e s x ) K 1 c 1 c e* 1 x 0 f B 1 B K d 1 d 0 e * 1 x 0 f 9) K 1 and K are wo consan vecors o be deermined and A 1 = [ a 1 a a a 4 ] A = [ a 5 a 6 a 7 a 8 ] B 1 = [ b 1 b b b 4 ] B = [ b 5 b 6 b 7 b 8 ] 40) e s x = diag e s 1 x e s x e s x e s 4 x e s 1 x e s x e s x e s 4 x. In view of Eqs. 1) and 4 Eq. 9) can also be furher expressed as u = A 1 A e s x ) K 1 B 1 B K c 1 c * f 1 g f 1 f 41) d 1 d f g 1 * f 1 f q. he eigenvecors of Eq. 6) are acually righ ones. he lef eigenvecors of Eq. 6) are found by solving he following eigenvalue problem N =. 4) I can be shown ha if s and [ ab] are he eigenvalue and eigenvecor of Eq. 6 hen = s and = [ba] are he corresponding soluions of Eq. 4). Since he lef and righ eigenvecors are orhogonal o each oher we hen come o he following imporan orhogonal relaionship [8] B B 1 A A 1 A 1 A B 1 B = I 0 0 I. 4) hus he orhogonal relaionship Eq. 4) provides us wih a simple way of invering he eigenvecor marix which is required in forming he ransfer marix. Furhermore afer edious derivaions he in-plane sresses elecric displacemens and hea fluxes of he x - dependen facor) can be expressed in erms of he generalized displacemen vecor racion vecor emperaure and normal hea flux as 11 p 11 q q 1 p p 1 q u D 1 = p D q q q p 69 0 q q ) Μ ij are only relaed o he maerial properies given by 11 = c 1 c 1 e e 1 c e 1 c 11 c e 1 = c 1 c c 1 e e c e e 1 c e 1 e c 1 c e 17 = c 1 e 1 e c e = c1e ce1 18 c e 19 = c 1 e p ) e 1 e c p ) 11 c e 1 = c 66 = c c e e c e c e 8 = c e c e c e c 7 = c e e c e = c e p ) e e c p ) 9 c e 44 = c e 15 c 45 = e c 54 = c 44 e 4 55 c 56 = e 4 44 c 69 = = ) he above expression Eq. 44) demonsraes ha once he generalized displacemen vecor generalized racion vecor emperaure and normal hea flux are nown all he oher inplane field componens can be easily obained hrough an algebraic operaion. 4. RANSFER MARIX AND SOLUION OF LAY- ERED SRUCURE For a cerain layer wih is lower surface a x = = 1 N i follows from Eqs. 1) and 4) ha 1 and can be expressed in erms of emperaure and normal hea flux a is lower surface x = z as

6 6 he Open Mechanics Journal 007 Volume 1 Wang and Pan 1 = e) 1 0 f 1 f 0 e ) 1 z g 1 g = e) 1 0 g f 0 e ) 1 z g 1 f 1 q 1 q. 46) hen he emperaure and normal hea flux a any posiion wihin his layer are relaed o hose a is lower surface x = z as follows q = P x z ) q 47) P x ) = f g 1 e 1 x f 1 g e 1 x f 1 f e 1 x e 1 x ) 48) g 1 g e 1 x e 1 x ) f g 1 e 1 x f 1 g e 1 x is he ransfer marix for hea conducion of layer. Apparenly P 0) is a ideniy marix. Similarly for a cerain layer wih is lower surface a x = = 1 N i follows from Eqs. 41) and 4) ha he unnown vecors K 1 and K can be expressed in erms of he generalized displacemen and racion vecors as well as he emperaure and normal hea flux a is lower surface x = as K 1 = e s ) z * B K B 1 B B 1 A u A 1 e s ) * A c 1 c f 1 g f 1 f A 1 d 1 d f g 1 f 1 f q. 49) hen he generalized displacemen and racion vecors a any posiion wihin his layer are relaed o he generalized displacemen and racion vecors as well as he emperaure and normal hea flux a he lower surface x = as follows u = E x z ) u S x ) q E x ) = A 1 A e s x ) *B B 1 B B 1 A *A 1 S x ) = c 1 c * f 1 g e 1 x f 1 f e 1 x d 1 d f g 1 e * 1 x * f 1 f e * 1 x *E x ) c 1 c * f 1 g f 1 f d 1 d f g 1 * f 1 f 50) 51) Apparenly E 0) is an 8 8 ideniy marix and S 0) is an 8 zero marix. Now ha Eqs. 47) and 50) can be concisely wrien ogeher as follows u u = Y x ) q q 5) Y x ) = E x ) S x ). 5) 0 8 P x ) is called he layer ransfer marix for he piezohermoelasic problem of layer. o handle he imperfec inerface we now inroduce he inerface ransfer marix. Acually by using he boundary condiions Eqs. 10)-14) on he imperfec inerface he soluion a x = z can be relaed o ha a x = z hrough he following propagaing relaion u q u = Z q 54) he marix Ζ defined below is he so-called inerface ransfer marix for inerface x = z I 44 Z Z = Z 1 I ) Z wih Ζ 1 and Ζ 1 wo 4 4 marices and Ζ a marix. More specifically if he inerface is hermally wealy conducing hen Z = 1 ). 56) 0 1 On he oher hand if he inerface is hermally highly conducing hen 1 0 Z = ) p q ) 1. 57) If he inerface is mechanically complian and dielecrically wealy conducing hen Z 1 = diag ) 1 ) ) D ) Z 1 = ) On he oher hand if he inerface is mechanically complian and dielecrically highly conducing hen Z 1 = diag ) ) ) 1 0 Z 1 = diag ) D p q ). 59) I is observed ha when he inerface x = is hermally or dielecrically highly conducing he inerface ransfer marix Z is dependen on he values of p and q. I can be easily checed ha Z is an ideniy marix when he inerface x = is perfec. Consequenly he soluion a he op

7 Mulilayered Piezohermo-elasic Plaes wih Imperfec Inerfaces he Open Mechanics Journal 007 Volume 1 7 surface x = H of he layered plae can be expressed by ha a he boom surface x = 0 of he layered plae as u q H u = 60) q 0 he marix Ω is deermined by = 0 8 * ) = Y N h N ) Z N Y N 1 h N 1 ) Z N 1 Y h ) Z Y 1 h 1 61) wih Ξ being an 8 8 marix Ψ an 8 marix Φ a marix. here are weny unnowns in Eq. 60). Once he en hermo-elecro-mechanical boundary condiions Eqs. 7) and 9) on he boom and op surfaces of he layered plae are imposed all he unnowns in Eq. 60) can be uniquely deermined. o demonsrae his more clearly Eq. 60) can be equivalenly rewrien as follows uh ) H ) = u0) 0) 0) q 0) H ) q H ) = ) 0) 6) q 0). Ξ ij ij=1) are 4 4 bloc marices of Ξ; Ψ ij ij=1) are 4 1 bloc marices of Ψ. he elecric poenial is given on he wo surfaces of he layered plae: 65) u = [ u 1 u u D ] 66) and 67) he normal elecric displacemen is given on he op surface while he elecric poenial is prescribed on he boom surface: By imposing he hermal boundary condiions Eqs. 7) 5 and 9) 5 he emperaure and normal hea flux a he wo surfaces of he layered plae can be uniquely deermined as 0) 1) h a q 0) N 0 0 ) = h b 1 H ) q H ) ) * h a a N ) * h b b 0 0 6) he superscrips 1) and N) refer o he hermal conduciviy coefficiens of he boom and op layers respecively; Φ ij ij=1) are he componens of Φ. Consequenly by imposing he elecromechanical boundary condiions Eqs. 7) 1-4 and 9) 1-4 he generalized displacemen and racion vecors a he wo surfaces of he layered plae can be uniquely deermined. We presen he resuls below for four differen combinaions of he elecric loads on he wo surfaces of he layered plae. he normal elecric displacemen is given on he wo surfaces of he layered plae: wih u J 1 and J being defined in Eqs. 66) and 67). 68) 69) he normal elecric displacemen is given on he boom surface while he elecric poenial is prescribed on he op surface: 0) = H ) = H D H H H D u0) = 1 1 H ) 1 1 0) ) 1 1 q 0 64) 70) uh ) = H ) )0) ) 0) )q 0 71)

8 8 he Open Mechanics Journal 007 Volume 1 Wang and Pan wih u J 1 and J being defined in Eqs. 66) and 67). Consequenly he generalized displacemen and racion vecors as well as he emperaure and normal hea flux a any posiion wihin he layered plae can be deermined by using Eqs. 5) and 54). In addiion he in-plane field componens can also be easily deermined by using Eq. 44) once he generalized displacemen and racion vecors as well as he emperaure and normal hea flux are nown. 5. NUMERICAL SUDIES Here we consider a nine-layered plae of square shape wih L=L x =L y =1m and H=0.1m. he boom eigh layers are made of graphie-epoxy composie wih he fiber orienaion 90 /0 /90 /0 /0 /90 /0 /90 wih respec o he x 1 -axis and he op layer is made of a PZ-5A piezoelecric maerial. All he layers have he same hicness. he maerial properies of he graphie-epoxy [46] and PZ-5A [46] piezoelecric layers are given in able 1. As in [6] we rea he graphieepoxy layer as a piezoelecric maerial wih he piezoelecric moduli se equal o zero. Furhermore we focus on he hermal loads by seing = sin px 1 ) C x =0 = 0 x =H 0 1 = 0 = 0 = 0 = H 1 = H = H = H = 0 wih n=m=1. his boundary value problem was numerically sudied by Xu e al. [4] using he sae-space formulaion wih he inerface being assumed o be perfec. When he inerface is perfec our resuls are in complee agreemen wih hose calculaed by Xu [4]. For example we demonsrae in Fig. 1) he hrough-he-hicness variaion of he shear sress σ 1 when he inerface is assumed o be perfec. hus he developed formulas based on he pseudo-sroh formalism and ransfer marix are validaed. Here we are more ineresed in he influence of he imperfecion of he inerfaces on he variaions of he field variables. More specifically we consider he following mechanically complian and hermo-elecrically wealy conducing inerfaces described by j) H = j) 1 1) 1 = j) = j) H = j) H 1) D = j= 9) 1) c δ 1 δ and δ are hree dimensionless nonnegaive parameers. Fig. ) shows he disribuions of he normal sress σ along he hicness direcion for differen values of he dimensionless imperfec inerface parameer = 1 = =. he horizonal variables are fixed a he cener x 1 =x =L/. I is observed from Fig. ) ha he magniude of normal sress is very small for δ=0.5. On he oher hand he magniude is very high when δ=10. As a resul he imperfec inerface parameers can be properly designed o reduce he hermal sress level. We also noice ha he imperfecion of he inerface can change he naure of he normal sress from compression o ension. able 1. Maerial Properies of he Graphie-Epoxy and PZ-5A 0 Graphie-Epoxy PZ-5A c 11 GPa) c GPa) c GPa) c 1 GPa) c 1 GPa) c GPa) c 44 GPa) c 55 GPa) c 66 GPa) e 1 Cm ) e Cm ) e Cm ) e 4 Cm ) 0 1. e 15 Cm ) 0 1. ε Fm 1 ) ε Fm 1 ) ε Fm 1 ) β NK 1 m 1 ) β 10 5 NK 1 m 1 ) β 10 5 NK 1 m 1 ) p 10-6 CK 1 m ) WK 1 m 1 ) WK 1 m 1 ) WK 1 m 1 ) CONCLUSIONS We have derived exac soluions for hree-dimensional orhoropic piezohermoelasic simply-suppored and mulilayered recangular plaes wih imperfec inerfaces under

9 Mulilayered Piezohermo-elasic Plaes wih Imperfec Inerfaces he Open Mechanics Journal 007 Volume 1 9 Fig. 1). hrough-he-hicness variaion of shear sress σ 1 when he inerface is assumed o be perfec. Hybrid mulilayered plae subjeced o emperaure change. L=L x =L y =1m and H=0.1m. Fig. ). Disribuion of normal sress σ along he hicness direcion of he plae for differen values of he dimensionless imperfec inerface parameer = 1 = =. Hybrid mulilayered plae subjeced o emperaure change. L=L x =L y =1m and H=0.1m. he horizonal variables are fixed a he cener x 1 =x =L/.

10 10 he Open Mechanics Journal 007 Volume 1 Wang and Pan saic hermo-elecro-mechanical loads. We developed a new and simple formalism ha resembles he Sroh formalism so ha he soluions in a homogeneous piezohermoelasic layer can be obained in a concise and elegan form. We also inroduced he inerface and layer ransfer marices in order o rea he mulilayered case wih imperfec inerface condiions and wih he hermal influence. I is found ha hrough inroducion of he inerface ransfer marix for he general imperfec inerface condiions all imperfec inerfaces e.g. wealy or highly conducing inerface) can be discussed wihin a common framewor. Our soluions can also provide benchmars for various plae heories and numerical mehods. ACKNOWLEDGEMENS his wor was parially suppored by ARL/ARO AFRL/AFOSR. he reviewers commens on he iniial manuscrip are highly appreciaed. REFERENCES [1] Pagano NJ. Exac soluions for composies in cylindrical bending. J Compos Maer 1969; : [] Pan E. An exac soluion for ransversely isoropic simply suppored and layered recangular plaes. J Elas 1991; 5: [] Ray MC Rao KM Samana B. Exac soluion for saic analysis of inelligen srucures under cylindrical bending. Compu Sruc 199; 47: [4] Xu KM Noor AK ang YY. hree-dimensional soluions for coupled hermoelecroelasic response of mulilayered plaes. Compu Mehods Appl Mech Engrg 1995; 16: [5] Dube GP Kapuria S Dumir PC. Exac piezohermoelasic soluion of simply-suppored orhoropic fla panel in cylindrical bending. In J Mech Sci 1996; 8: [6] Vel SS Bara RC. hree-dimensional analyical soluion for hybrid mulilayered piezoelecric plaes. J Appl Mech 000; 67: [7] Reddy JN Cheng ZQ. hree-dimensional soluions of smar funcionally graded plaes. J Appl Mech 001; 68: [8] Pan E. Exac soluion for simply suppored and mulilayered magneo-elecro-elasic plaes. J Appl Mech 001; 68: [9] Pan E Han F. Exac soluion for funcionally graded and layered magneo-elecro-elasic plaes. In J Engng Sci 005; 4: 1-9. [10] Benvenise Y Miloh. he effecive conduciviy of composie wih imperfec conac a consiuen inerfaces. In J Engng Sci 1986; 4: [11] Benvenise Y. On he decay of end effecs in conducion phenomena: a sandwich srip wih imperfec inerfaces of low or high conduciviy. J Appl Phys 1999; 86: [1] Chen. hermal conducion of a circular inclusion wih variable inerface parameer. In J Solids Sruc 001; 8: [1] Fan H Sze KY. A micro-mechanics model for imperfec inerface in dielecric maerials. Mech Maer 001; : [14] Wang X Pan E. A moving screw dislocaion ineracing wih an imperfec piezoelecric bimaerial inerface. Phys Sa Sol b) 007; 4: [15] Ru CQ Schiavone P. A circular inclusion wih circumferenially inhomogeneous inerface in aniplane shear. Proc R Soc Lond A 1997; 45: [16] Fan H Wang GF. Screw dislocaion ineracing wih imperfec inerface. Mech Maer 00; 5: [17] Wang X Zhong Z. hree-dimensional soluion of smar laminaed anisoropic circular cylindrical shells wih imperfec bonding. In J Solids Sruc 00; 40: [18] Benvenise Y. A general inerface model for a hree-dimensional curved hin anisoropic inerphase beween wo anisoropic media. J Mech Phys Solids 006; 54: [19] Wang X Pan E Roy AK. New phenomena concerning a screw dislocaion ineracing wih wo imperfec inerfaces. J Mech Phys Solids 007 available from: doi: /j.jmps [0] Kais MA Mavroyannis G. Feeble inerfaces in bimaerials. Aca Mech 006; 185: [1] Chen WQ Cai JB Ye GR Wang YF. Exac hree-dimensional soluions of laminaed orhoropic piezoelecric recangular plaes feauring inerlaminar bonding imperfecions modeled by a general spring layer. In J Solids Sruc 004; 41: Received: May Revised: May Acceped: May 9 007