Grobbelaar-Van Dalsen, Marié

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Institute of Mathematics AS CR, v. v. i.

EQUADIFF 10Marié Grobbelaar-Van DalsenThermal effects in an elastic plate-beam structureIn: Jaromír Kuben and Jaromír Vosmanský (eds.): Equadiff 10, CzechoslovakInternational Conference on Differential Equations and Their Applications, Prague,August 27-31, 2001, [Part 2] Papers. Masaryk University, Brno, 2002. CD-ROM; alimited number of printed issues has been issued. pp. 177--181.Persistent URL: http://dml.cz/dmlcz/700351Terms of use:© Institute of Mathematics AS CR, 2002Institute of Mathematics of the Academy of Sciences of the Czech Republic providesaccess to digitized documents strictly for personal use. Each copy of any part of thisdocument must contain these Terms of use.This paper has been digitized, optimized for electronic delivery andstamped with digital signature within the project DML-CZ: The CzechDigital Mathematics Library http://project.dml.czEquadiff 10, August 27–31, 2001Prague, Czech RepublicEquadiff 10 CD ROMPapers, pp. 177–181Thermal Effects in an ElasticPlate-beam StructureMarié Grobbelaar-Van DalsenDepartment of Mathematics and Applied Mathematics,University of Pretoria,0002 South Africa,Email: mgrob@scientia.up.ac.zaAbstract. We consider a linear model for a 2−D hybrid elastic structureconsisting of a thermo-elastic plate which has a beam attached to its freeend. We show that the interplay of parabolic dynamics and hyperbolicdynamics in the model yields analyticity for the entire system. This resultprovides an easy route to uniform stability.MSC 2000. 35Q72,73D35Keywords. thermo-elastic, plate, plate-beam, coupled, dynamic bound-ary conditions.1 Introduction and Statement of the ProblemWe consider well-posedness of the following model, Pr (P ), for the transversalvibrations of a hybrid structure consisting of a thin rectangular thermo-elasticplate which is clamped along three edges, while to its free edge a thin beam withends clamped to the adjoining clamped edges of the plate, is attached:wtt +∆2w + α∆θ = 0 in ΩTw = 0 =∂w∂non ∂ΩT − ΓTβθt − η∆θ − α∆wt = 0 in ΩTθ = 0 on ∂ΩT − ΓTThis is an overview article.178 M. Grobbelaarwtt − [wxxx + (2− ν)wxyy ] + wyyyy − α∂θ∂n+ bθyy = 0 on ΓT∂w∂n= 0 on ΓTw = 0 = wy at ∂ΓTβθt + η∂θ∂n− κθyy − bwyyt = 0 on ΓTθ = 0 at ∂ΓT ,w(x, y, 0) = w0(x, y), wt(x, y, 0) = w1(x, y), θ(x, y, 0) = θ0(x, y) in Ωw(a, y, 0) = µ0(y), wt(a, y, 0) = µ1(y), θ(a, y, 0) = θ1(y) on Γ.Here Ω denotes the interior of the plate with corner points (0, 0), (a, 0), (a, ^) and(0, ^), while Γ is the line joining (a, 0) and (a, ^) and ∂Γ its end-points.The constitutive equations in Pr (P ) are “contact” equations in the sense thatthe deflections as well as the temperatures of the plate and the beam match at theinterface for t > 0, but not necessarily initially. Thus the 1−D biharmonic equationand heat equation along Γ form a system of dynamic boundary conditions for thethermo-elastic plate equations. By allowing for interaction between the plate andthe beam, the partial differential equations along Γ contain additional terms: thethird order space derivatives of the displacement variable w in the beam equationrepresent the combined shear force and twisting moment exerted by the plate onthe beam, while the conormal derivative of the thermal variable θ in the heatequation along Γ reflects the flux of heat from the plate to the beam across theinterface Γ.2 Implicit Evolution Equation for Pr (P )We formulate Pr (P ) as an implicit evolution problem, Pr (AEP ), of the formFind U such thatddt(BU(t)) +AU(t) = 0, U ∈ D ⊂ X, t > 0limt→0+BU(t) = y ∈ Ywith A and B operators from a Banach space X to a second Banach space Y.The construction of a unique solution of Pr (AEP ) with representation U(t) =S(t)y entails the construction of a double family of evolution operators [4], viz.〈{S(t), E(t)}〉 = 〈{S(t) : Y → X |t > 0}, {E(t) : Y → Y |t > 0}〉, with E(t) =: BS(t)a semigroup in Y. The evolution from an initial state in Y to a solution in the spaceX , is generated by the jointly closed operator pair 〈−A,B〉 : D → Y ×Y in whichR(B) is dense in Y.3 Mathematical Setting for Pr (P )We define the following spaces and operators:Thermo-elastic Hybrid Structure 179X0 =: L2(Ω) with inner product (, )0 and norm ‖.‖0.Hm(Ω) = Hm,2(Ω) denotes the usual Sobolev spaces with inner products (, )mand norms ‖.‖m when m > 0 and the Hilbert space L2(Ω) when m = 0. (, )m,Γand ‖.‖m,Γ denote the inner products and norms in Hm(Γ ).For u ∈ Hm(Ω) we denote the trace of u on Γ by γu.We define the following subspaces of X0:X1 =: {w ∈ H1(Ω)∣∣∣w = 0 on ∂Ω − Γ, γw ∈ H10 (Γ )}.X2 =: {w ∈ H2(Ω)∣∣∣w = 0 = ∂w∂n on ∂Ω − Γ, ∂w∂n = 0 on Γ, γw ∈ H20 (Γ )}.The spaces Xi, i = 0, 1, 2 are endowed with the inner products (, )i and the norms‖.‖i. For X2 we also use the equivalent inner product ((, ))2 given bya(w, z) = (wxx, zxx)0+2(1−ν)(wxy, zxy)0+(wyy, zyy)0+ν(wxx, zyy)0+ν(wyy, zxx)0.The associated norm will be denoted by |||.|||2.Y0 =: X0 × L2(Γ ). The (usual) inner product and norm are denoted by (, )Y0 and‖.‖Y0 .The domains D1 and D2 are defined byD1 =: {w ∈ H4(Ω)∣∣∣w = 0 = ∂w∂n on ∂Ω − Γ, ∂w∂n = 0 on Γ, γw ∈ H4(Γ )∩H20 (Γ )}.D2 =: {θ ∈ H2(Ω)∣∣∣θ = 0 on ∂Ω − Γ, γθ ∈ H2(Γ ) ∩H10 (Γ )}.The operators A, B and Cj , j = 1, 2, 3 from X0 into Y0 are defined byAw =:〈∆2w,−[γ(wxxx + (2 − ν)wxyy)] + (γw)yyyy〉,Bw =: 〈w, γw〉 , w ∈ D1 = D(A).C1θ =:〈α∆θ,−αγ ∂θ∂n + b(γθ)yy〉, θ ∈ D2,C2ẇ =: 1β 〈−α∆ẇ,−b(γẇ)yy〉 , ẇ ∈ X2,C3θ =: 1β〈−η∆θ, ηγ ∂θ∂n − κ(γθ)yy〉, θ ∈ D2.Observing that R(B) = {〈w, γw〉 , w ∈ D1} is a proper subset of Y0, we define thefollowing subsets of X1 ×H10 (Γ ) and X2 ×H20 (Γ ):Y1 = C^(B[X1]), Y2 = C^(B[X2]), with closures taken in Y0.Y1 will be endowed with the norm ‖Bw‖Y1 = (‖∇w‖20+‖(γw)y‖20,Γ )12 and Y2 withthe norm |||Bw|||Y2 ≡ ((Bw,Bw))Y2 = (a(w,w) + ‖(γw)yy‖20,Γ )12 .To cast Pr (P ) in the abstract form Pr (AEP ), we define product spaces equippedwith product space inner products and norms, viz. the “finite energy” space XEand its accompanying space YE and a weaker space X , with accompanying spaceY :180 M. GrobbelaarXE =: X2 × (X0)2,YE =: Y2 × (Y0)2.In XE we define the domainDE =: {UE = (w, ẇ, θ), w ∈ D1, ẇ ∈ X2, θ ∈ D2}The linear operators A and B on the common domain DE are now defined byAUE =: 0 −B 0A 0 C10 C2 C3UE , BUE =:B 0 00 B 00 0 BUE , U ∈ DE .To define the weaker spaces X ,Y we first defineH =: {w ∈ X0∣∣∣A 12w = 0}, WX0 = H⊥. [1][2]Z =: {y = 〈y1, y2〉 ∈ Y0∣∣∣A 12 y1 = 0}, WY0 = Z⊥.X =: (WX0 ∩X0)× (X0)2, Y =: (WY0 ∩ Y0)× (Y0)2.To define a domain D in X , we introduce the variable U =: (u, ẇ, θ), Bu =: −A 12w.D =: {U = (u, ẇ, θ), u ∈ (WX0 ∩H2(Ω)), ẇ ∈ X2, θ ∈ D2.}The linear operators L andM from X to Y are now defined byLU =: 0 A 12 0−A 12 0 C10 C2 C3U, MU =:B 0 00 B 00 0 BU, U ∈ D.P r (P )may now be cast in the form of implicit evolution problems, viz. Pr (AEP )I :ddt(BUE(t)) +AUE(t) = 0, UE ∈ DE , t > 0,limt→0+BUE(t) = G ∈ YEor Pr (AEP )II :ddt(MU) + LU = 0, U ∈ D, t > 0,limt→0+MU(t) = F ∈ Y.4 Main ResultsTh e reader is referred to [3] for the detailed proofs.Lemma 1.Re {(AUE ,BUE)YE}= Re {−((Bẇ,Bw))Y2 + (Aw,Bẇ)Y0 + (C1θ,Bẇ)Y0 + (C2ẇ + C3θ,Bθ)Y0}= η‖∇θ‖20 + κ‖(γθ)y‖20,Γ ,Re{(LU,MU)Y} = η‖∇θ‖20 + κ‖(γθ)y‖20,Γ .Thermo-elastic Hybrid Structure 181We proveTheorem 2. The operator pair 〈−A,B〉 generates a unique uniformly boundeddouble family 〈S, E〉 = 〈{S(t) : YE → XE |t > 0}, {E(t) : Y → Y|t > 0}〉 of evolu-tion operators. Thus Pr (AEP )I has unique solution UE ∈ C((0,∞);DE ) withrepresentation UE(t) = S(t)G for any G ∈ R(B) and each t ∈ (0,∞).Corollary 3. Pr (P ) in (w,wt, θ) can be associated with a uniformly boundedevolution operator S(t) : YE → DE ⊂ XE in the sense that S(t)G = UE(t) = (w, ẇ, θ)solves Pr (AEP )I for any G = (G1, G2, G3) = (〈g1, γg1〉 , 〈g2, γg2〉 , 〈g3, γg3〉) suchthat g1 ∈ D1, γg1 ∈ H4(Γ ) ∩H20 (Γ )g2 ∈ X2, γg2 ∈ H20 (Γ )g3 ∈ D2, γg3 ∈ H2(Γ ) ∩H10 (Γ ).The restriction that each Gi, i = 1, 2, 3, of G is of the form 〈gi, γgi〉 , may beinterpreted as meaning that the initial displacement, velocity and temperature inthe plate and the beam should match along Γ.Theorem 4. The operator pair 〈−L,M〉 generates a unique analytic uniformlybounded double family 〈S, E〉 = 〈{S(t) : Y → X|t > 0}, {E(t) : Y → Y|t > 0}〉 ofevolution operators. Thus Pr (AEP )II has unique solution U ∈ C((0,∞);D) withrepresentation U(t) = S(t)F for any F ∈ Y and each t ∈ (0,∞).Corollary 5. Pr (P ) in (w,wt, θ) can be associated with an analytic evolutionoperator S(t) : Y → D ⊂ X in the sense that S(t)F = U(t) = (u, ẇ, θ) solvesPr (AEP )II for any F = (〈f1, f2〉 , 〈g1, g2〉 , 〈h1, h2〉) such thatf1 ∈ X0, 0 = E(〈f1, γf1〉), f2 ∈ L2(Γ )g1 ∈ X0, g2 ∈ L2(Γ )h1 ∈ X0, h2 ∈ L2(Γ )with 2E(〈f1, γf1〉) = a(f1, f1) + ‖(γf1)yy‖20,Γ the elastic potential energy.With the aid of Lemma 1 we obtain uniform stability for Pr (AEP )II :Theorem 6. There exist constants M,σ > 0 such that for t > 0, the uniquesolution U ∈ C((0,∞);D) of Pr (AEP )II , represented as U(t) = S(t)F for anyF ∈ Y, satisfies‖S(t)F‖X ≤M exp(−σt)‖F‖Y .References1. Grobbelaar-Van Dalsen, M., Fractional Powers of a closed Pair of Operators, Proc.Roy. Soc. Edinburgh 102A 1986, 149–158.2. Grobbelaar-Van Dalsen, M., On Fractional Powers of a Pair of Matrices and a Plate-beam Problem, Appl. Anal. 72(3–4) 1999, 369–390.3. Grobbelaar-Van Dalsen, M., Thermal Effects in a two-dimensional Hybrid ElasticStructure, J. Math. Anal. Appl., to appear.4. Sauer, N., Empathy Theory and the Laplace Transform, Linear Operators BanachCenter Publications 38 1997, 325–338.

Thermal effects in an elastic plate-beam structure

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Thermal effects in an elastic plate-beam structure

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Institute of Mathematics AS CR, v. v. i.