Bonding a linearly piezoelectric patch on a linearly elastic body

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1 onding a linearly iezoelectric atch on a linearly elastic body Christian Licht, omsak Orankitjaroen, Patcharakorn Viriyasrisuwattana, Thibaut Weller To cite this version: Christian Licht, omsak Orankitjaroen, Patcharakorn Viriyasrisuwattana, Thibaut Weller. onding a linearly iezoelectric atch on a linearly elastic body. Comtes Rendus Mécanique, Elsevier Masson, 14, 34 4), <1.116/j.crme >. <hal-13381> HAL Id: hal htts://hal.archives-ouvertes.fr/hal ubmitted on 4 Nov 15 HAL is a multi-discilinary oen access archive for the deosit and dissemination of scientific research documents, whether they are ublished or not. The documents may come from teaching and research institutions in France or abroad, or from ublic or rivate research centers. L archive ouverte luridiscilinaire HAL, est destinée au déôt et à la diffusion de documents scientifiques de niveau recherche, ubliés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires ublics ou rivés.

2 onding a linearly iezoelectric atch on a linearly elastic body Analyse asymtotique d un atch linéairement iézoélectrique lié à un cors linéairement élastique Christian Licht a,b,c,, omsak Orankitjaroen b,c, Patcharakorn Viriyasrisuwattana b,c, Thibaut Weller a, a LMGC, UMR-CNR 558, Université Montellier-, case courier 48, lace Eugène-ataillon, 3495 Montellier cedex 5, France b Deartment of Mathematics, Faculty of cience, Mahidol University, angkok 14, Thailand c Centre of Excellence in Mathematics, CHE, angkok 14, Thailand abstract A rigorous study of the asymtotic behavior of the system constituted by a very thin linearly iezoelectric late bonded on a linearly elastic body sulies various models for an elastic body monitored by a iezoelectric atch. résumé Une étude rigoureuse du comortement asymtotique du système constitué ar une laque linéairement iézoélectrique collée sur un cors linéairement élastique fournit divers modèlesdecontrôledestructuresélastiquesardesatchesiézoélectriques. 1. Introduction Many studies dealing with the mathematical modeling of iezoelectric devices were devoted to the behavior of the sole atches and rovided various asymtotic models for thin linearly iezoelectric lates see [1] and the references therein). However, the essential technological interest of iezoelectric atches being the monitoring of a deformable body they are bonded to, here we intend to roose various asymtotic models for the behavior of the body through the study of the system constituted by a very thin linearly iezoelectric flat atch erfectly bonded to a linearly elastic three-dimensional body. A reference configuration for the body is an oen set laying in {x 3 < } whose art of its Lischitz-continuous boundary is a non-emty domain in {x 3 = } and such that L, ) is included in for some ositive real number L, while the atch occuies :=, ), being a small real number; let O :=. The body is clamed on a art Γ

3 of \ with a ositive two-dimensional Hausdorff measure H Γ ), and subjected to body forces and surface forces on Γ 1 := \ Γ ) of densities f and F.Moreover,forallδ in R, let δ denote + δe 3, {e 1, e, e 3 } being a basis of the Euclidean hysical sace assimilated to R 3, surface forces of density G acts on whilst the atch is free of mechanical loading and electric charges in and on its lateral boundary, ). Ifu, eu ), σ denote the fields of dislacement, strain and stress in O and ϕ, D stand for the electric otential and the electric dislacement, art of the equations describing the electromechanical equilibrium read as: divσ = f in O, u = onγ, σ n = F on Γ 1, σ n = G on, σ n = on,) div D = in, D n = on,) σ = ae u ) in, σ, D ) = 1 M 1) e u ), ϕ ) in f is the extension of f to by, n is the unit outward normal and a denotes the elasticity tensor which satisfies: a L ; Lin 3)), c; c e ax)e e, e 3, a.e. x ) where Lin N ) is the sace of linear oerators on the sace N of N N symmetric matrices whose inner roduct and norm are noted and as in R 3.IfH:= 3 R 3 is equied with an inner roduct and a norm also denoted as reviously, then M is an element of L R; LinH)) indeendent of x 3 satisfying: [ ] α β M = β T, κ > ; κ e αx)e e, κ g γ x)g g, e, g) H, a.e. x R 3) γ The models will be distinguished according to the additional necessary boundary conditions on and, characterized by an index in {1, }.Case 1 = 1 corresonds to a condition for the electric dislacement on : D n = q on q being a density of electrical charges, while 1 = corresonds to a condition of given electrical otential: ϕ = ϕ on roughly seaking, 1 = 1 deals with atches used as sensors, whereas 1 = concerns actuators see [1,]). Index accounts for the status of the interface between the atch and the body: = 1 corresonds to an electrically imermeable interface, = corresonds to a grounded interface: D n = ϕ = on on It will be convenient to use the following notations: ˆk := ê, ĝ), ê := e αβ ) α,β {1,}, ĝ := g 1, g ), k = e, g) H ẽ 3 ; ẽ αβ = e αβ, 1 α,β, ẽ i3 =, 1 i 3, e kr) = kv,ψ):= ev), ψ ), r = v,ψ) H 1 ; R 3) H 1 ) ev) D ; ) ; ev) )αβ = 1 α v β + β v α ), v D ; R ) where the same symbol e ) stands for the symmetrized gradient in the sense of distributions of D O; R 3 ), O {,, O },ord ; R ). Moreover we introduce some saces, linear and bilinear forms in order to suly a variational formulation of 1) 5). An electromechanical state will be an element r = v,ψ) of V := H 1 Γ O ; R 3) Φ, Φ 1,1) = H 1 ), Φ 1,) = H 1 ), Φ,1) = H 1 ), Φ,) = H 1 ) 7) where, for any domain O of R 3, HΓ 1 O; R3 ) and HΓ 1 O) resectively denote the subsaces of H 1 O; R 3 ) and H 1 O) of all elements with vanishing traces on a art Γ of O. One makes the following assumtions on the data: ϕ denotes the restriction to of an element of H 1/ {x 3 = } ) still denoted by ϕ f, F, G, q) L ; R 3) L Γ 1 ; R 3) L ; R 3) L ), q dˆx = 8) G x + e 3 ) = Gx), q x + e 3 ) = qx), ϕ x + e 3) = ϕ x), a.e. x It is well known that for all ϕ in H 1/ {x 3 = }), there exists an element of H 1 L, )) when = 1, H 1 L L, )) when =, still denoted by ϕ whose trace on is ϕ.hencetheelementϕ o, of Φ defined by ϕ o, x) = ϕ ˆx,x 3 )L/) satisfies: 4) 1 4) 5) 1 5) 6)

4 ϕ o, := ϕ on, 1 ϕ o, dx C 9) Let M and L be defined by: M s, r) := aeu) ev) dx + 1 Mks) kr) dx, s = u,ϕ), r = v,ψ) V L 1, )r) = f v dx + F v dh + G v dˆx + q ψ dˆx, r = v,ψ) V 1, ) Γ 1 L, )r) = f v dx + F v dh + G v dˆx 1 Mk,ϕ ) o, kr) dx, r = v,ψ) V, ) 1) Γ 1 It is straightforward to check that M is continuous and coercive on V actually on HΓ 1 ; R 3 ) Φ /R) when = 1, 1)), L is continuous on V, and seeking an equilibrium state leads to the roblem P ) : Find s = u, ) ϕ in V such that M s, ) r = L r), r V 11) which by the Lax Milgram lemma has a unique solution if one adds the condition ϕ := 1 ϕ dˆx = when = 1, 1)). Then an equilibrium state is s = s +, ϕ o, ).. The asymtotic models We will roose our four models by studying the asymtotic behavior of s, when goes to zero, in three stes: te 1 a riori estimates): ytakingr = s in the formulation of P ) and by due account of ), 3), 8), 9) and of the following Lemma.1, wehave: C > ; e u 1 dx + e u + ϕ ) dx C 1) Lemma.1. There hold: i) v L ;R 3 ) C ev) L ; 3 ) + ev) L, 3 ) ), v H Γ 1 O ; R 3 ), ii) ψ ψ dˆx C ψ dx, ψ H 1 ), iii) ψ dx c[ Γ ψ dˆx + ψ dx], ψ H 1 ), Γ {, }. Proof. Points ii) iii) are standard due to the cylindrical geometry of. To rove i), it suffices to introduce a C [ L, L/3]) cut-off function η such that: [ η = on L, L ] [, η = 1 on L 3 3, L ], η 1, dη 4 [ on L 3 dx 3 L 3, L ] 13) 3 and to aly the Korn inequality in H 1 L+ L +, ); R 3 ) to ηv by noticing that the constant in Korn s inequality does not deend on. Therefore the Cauchy chwarz inequality and 1) make it ossible to define the following element of L ; H): k, s ) = e, u ), g,ϕ )) 1 := which satisfies: k, s ) L ;H) C e u ), ϕ ), x3 ) dx 3 14) thus there exists a nonrelabeled subsequence such that k, s ) converges weakly in L ; H) toward some k.theinterest of introducing k, s ) is that 15)

5 k, s Û ) ) = e, φ ), U,φ ) 1 := which will enable us to identify k. s, x 3) dx 3 16) te Convergence of s )): Proosition.1. Let V := {v H 1 Γ ; R 3 ); ˆv H 1 ; R )}, then there exists ū, φ ) in V H 1 ) such that, when goes to zero, i) the restriction to of u converges weakly in H 1 Γ ; R 3 ) toward ū ; ii) Û,φ ) converges weakly in H 1 ; R R), and consequently strongly in L ; R R) toward ū, φ ), k = e ū ), φ ), and φ = when 1, 1). Proof. The estimate 1) yields that a nonrelabeled subsequence u ) has restriction to which weakly converges in HΓ 1 ; R 3 ) and whose trace on strongly converges in L ; R 3 ). Hence, regarding U, it remains to show lim I =, I := 1 x3 3 u ˆx, t dt ) dx 3 ) dˆx 17) The reeated use of the Cauchy chwarz inequality leads to I u dx, and 1), 13) yield: u dx = C ηu dx L+,) O ηu dx e ηu dx C e u L ; 3 ) + e u L ; 3 )) C Moreover 16) imlies that eu ) converges in D ; ) toward both eû ) and ē, thusū belongs to V and ē = e ū )! Lastly, as φ dx 1 ϕ dx, 1), Lemma.1iii) and ii) with ϕ 1,1) = yield that φ is bounded in L ) and converges strongly toward when 1, 1). In the next ste, we will show that U, ḡ ) is necessarily the unique solution to a variational roblem so that the whole sequences do converge. te 3 Identification of ū, φ )): First, the decomosition H = H 1 H H3 1 = 1: H 1 := { h = e, g) H; e i3 =, g 3 = }, H := { h = e, g) H; ê =, ĝ = }, H 3 := {} 1 = : H 1 = { h = e, g) H; e i3 =, ĝ = }, H = { e, g) H; ê =, g = }, H 3 = { e, g) H; e =, g 3 = } 18) induces a decomosition of M in linear oerators M ij maing H i into H j. The key oint in the identification of ū, φ ) is to establish: M k ) = 19) where h i denotes the rojection on H i of any h of H. As14) and 1) imly: 1 M e u ), ϕ ) θ dx = M e, u ), g,ϕ )) θ dˆx 1 lim M e u ), ϕ ) x) x3 θˆx) dx =, θ L ; H) ) equality 19) is a mere consequence of choosing the following test functions in the formulation of P ):

6 { I r = I v, ), I = 1, ; I v = in, 1 v =, x 3 θ), v = x 3 θ,), θ C ) ; R I, in r =,ψ), ψx) = x 3 θˆx), θ C ) 1) Hence, as 18) and oint ii) in Proosition.1 yield k ) 3 =, we deduce: M k ) 1 = M k ) 1 ; M := M 11 Next, for all v,ψ) in V KL ) Ψ,definedby: ) M1 M 1M 1 ) V KL ) = { v H 1 ; R 3) ; v M, v F ) H 1 ; R ) H ); ˆvx) = v M ˆx) x 3 v F ˆx), v 3 x) = v F ˆx), a.e. x } Ψ = { ψ; ψx) = x 3 ) 1 1 θˆx), θ H 1 ) } if = 1, Ψ ={} if = 3) the coule ẽ ˆv), ψ) belongs to H1 almost everywhere in, then) and ) give: 1 lim M e u ), ϕ ) ) ev), ψ dx = Mk ẽ ˆv), ψ ) dˆx = M k ) 1 ẽ ˆv), ψ ) dˆx 4) while, obviously, lim G v dˆx + q ψ dˆx = G v dˆx + q ψ dˆx. Lastly the very definitions of g, ϕ ) and M imlying g,) ) 3 = ϕ and Mk k κ k for all k in H 1, Jensen s inequality and a standard argument of lower semi-continuity achieve the roof of the following convergence result which sulies our asymtotic models in the form of variational roblems P ): Theorem.1. When tends to zero, u,k, ϕ ))1 ) converges strongly in H 1 ; R 3 ) L R 3 1) toward ū,e ū ), E )) where ū, E ) is the unique solution to: { Find u, E) 1 1) 1),ϕ ) + V E such that P ) M u, E), v, E )) = L v, E ) v, E ) V E where E 1,1) := {E L ; R ) { ;!φ:= ) 1 E H 1m ) := ψ H 1 ); E 1,) ={}, E,1) = L ), E,) ={} M u, E), v, E )) ) := aeu) ev) dx + M eû), E e ˆv), E ) dˆx ψˆx) dˆx = L v, E ) = f v dx + F v dh + G v dˆx + 1 ) q ) 1 E dˆx Γ 1 and any element of H 1 is understood as an element of R 3 1. } } ; E = φ When = 1, the model involves an additional state variable to the sole dislacement, which is the limit of the average in the transverse direction of the efficient comonents of the electrical field in the atch. As it can be eliminated, P ) models the equilibrium of the genuine body subjected to the loading f, F, G) and reinforced along, this reinforcement being nonlocal if = 1, 1). 3. Concluding remarks When = 1, ) or =, 1), the electric data q or ϕ do not have any influence on the limit model which corresonds to a urely elastic surface reinforcement of the body along. However, the characteristics of this reinforcement may deend on the dielectric or iezoelectric coefficients as much as such terms aear in the exression of M see [1], where the systematic influence of crystal symmetries has been carried out). On the contrary, electrical data q or ϕ lays a role in models 1, 1) and, ). More recisely, f, F and G being fixed, the maing ϕ,) ϕ = ū,) is one-to-one, it is theoretically ossible to determine what could be the electrical otential to aly on in order to get a desired dislacement in the range of,). An aroximate rocedure may be done easily by finite elements. Another alication is that the atch may shift the sectrum of the body in an interesting way, that is why we may regard the atch as an

7 actuator. When = 1, 1) the maing q 1,1) q = ū 1,1) is also one to one, thus the measurement of q may suly the knowledge of the state of dislacements: the atch acts as a sensor! y using the same technique of averaging the strain and electrical fields in the transverse direction it is ossible to treat the easier case, from the mathematical oint of view, of iezoelectric atches embedded inside an elastic body. References [1] T. Weller, C. Licht, Asymtotic modeling of thin iezoelectric lates, Ann. olid truct. Mech 1 1) [] J.N. Reddy, On laminated comosite lates with integrated sensors and actuators, Eng. truct )