COEFFICIENTS OF SINGULARITIES FOR A SIMPLY SUPPORTED PLATE PROBLEMS IN PLANE SECTORS

Transcription

1 Electronic Journal of Differential Equations, Vol. 213 (213), No. 238, pp ISSN: URL: or ftp ejde.math.txstate.edu COEFFICIENTS OF SINGULARITIES FOR A SIMPLY SUPPORTED PLATE PROBLEMS IN PLANE SECTORS BOUBAKEUR MEROUANI, RAZIKA BOUFENOUCHE Abstract. This article represents the solution to a plate problem in a plane sector that is simple supported, as a series. By using appropriate Green s functions, we establish a biorthogonality relation between the terms of the series, which allows us to calculate the coefficients. 1. Introduction Let S be the truncated plane sector of angle ω 2π, and radius ρ (ρ is positive and fixed) defined by: S = {(r cos θ, r sin θ) R 2, < r < ρ, < θ < ω} (1.1) and Σ the circular boundary part Σ = {(ρ cos θ, ρ sin θ) R 2, < θ < ω}. (1.2) We are interested in the study of a function u, belonging to the Sobolev space H 2 (S), and being the solution of 2 u =, in S u = Mu =, for θ =, ω, where the operator M represents the bending moment and is defined as (1.3) Mu = ν u + (1 ν)( 2 1un un 1 n un 2 2). (1.4) Here ν is a real number ( < ν < 1/2) called Poisson coefficient and n = (n 1, n 2 ) is the unit outward normal vector to Γ and Γ 1 (See Figure 1). The boundary conditions u = and Mu =, for θ =, θ = ω mean that the plate is simply supported. This type of boundary conditions arises in problems of linear or non linear vibrations of thin imperfect plates. See for example [2, pages 5,6] and the references therein. 2 Mathematics Subject Classification. 35B4, 35B65, 35C2. Key words and phrases. Crack sector; singularity; bilaplacian; solution series. c 213 Texas State University - San Marcos. Submitted May 13, 213. Published October 24,

2 2 B. MEROUANI, R. BOUFENOUCHE EJDE-213/238 Figure 1. We show that the solutions u of this problems can be written as a series of the form u(r, θ) = α E c α r α φ α (θ). (1.5) Here E stands for the set of solutions of the equation in a complex variable α: sin 2 (α 1)ω = sin 2 ω, Re α > 1 (1.6) For further studies of the set E, see for example Blum and Rannacher [1], and Grisvard [4]. We will compute the coefficients c α in (1.5). This sort of calculations have already been done by Tcha-Kondor [5] for the Dirichlet s boundary conditions, and by Chikouche-Aibeche [3] for the Neumann s boundary conditions. These authors have established, thanks to the Green s formula, a relation of biorthorgonality between any two functions φ α, which allows them to calculate the coefficients c α. We follow the same approach. Thus we need to write the appropriate Green formula for the domain S. Using this formula, we establish a relation of biorthorgonality between the functions φ α. In the case of a crack domain (ω = 2π) this relation reduces to the simple one obtained by Tcha-Kondor. This enables us, in this particular situation, to find an explicit formula for the coefficients c β. 2. Separation of variables Replacing u by r α φ α (θ) in problem (1.3) leads us to the boundary value problem φ (4) α (θ) + (2α 2 4α + 4)φ (2) α (θ) + α 2 (α 2) 2 φ α (θ) =, (2.1) φ (2) α (θ) + [να 2 + (1 ν)α]φ α (θ) =, θ =, θ = ω (2.2) φ α (θ) =, θ =, θ = ω (2.3)

3 EJDE-213/238 COEFFICIENTS OF SINGULARITIES 3 A relation similar to the orthogonality obtained for the biharmonic operator is given, in the following theorem. Theorem 2.1. Let φ α and φ β be solutions of (2.1) with α and β solutions of (1.6). Then, for α β, one has [φ α, φ β ] = [ (α 2 ν(α + β) + (3 ν) 2α 2α)φ α φ α] φβ α β + [ (β 2 ν(α + β) + (3 ν) 2β 2β)φ β φ α β β] φα dθ =. (2.4) Proof. We use the Green formula (v 2 u u 2 [ u v v)dx = (unv + Mv) (vnu + n n Mu)] d, (2.5) where S Γ Nu = u n + (1 ν)( 2 1un 1 n 2 12u(n n 2 2) + 2un 2 1 n 2 ), and Γ is the boundary of S. For two functions u, v which are solutions of (1.3), using the Green s formula we obtain [ u v (unv + Mv) (vnu + n n Mu)d] = (2.6) Σ On Σ, for the function u α = r α φ α, we have u α n = u α r = αrα 1 φ α, Mu α = r α 2{ [α 2 (1 ν)α]φ α + νφ α}, (2.7) Nu α = r α 3{ α 2 (α 2)φ α + [(ν 2)α + (3 ν)]φ } α. The results follow from the application of formula (2.6) to the biharmonic functions u α = r α φ α and u β = r β φ β, and by using relations (2.7). Remark 2.2. The relation (2.4) between the functions φ α and φ β is similar to the relation of biorthorgonality obtained when the functions φ α and φ β satisfying (2.1) with the Dirichlet boundary conditions φ α = φ α = φ β = φ β = for θ = and θ = ω. In this latter case, the relation is given by φ α φ βdθ = which is obtained by a double integration by parts: φ α φ βdθ = and using the Dirichlet s boundary conditions. φ αφ β dθ (2.8) φ αφ β dθ + [φ α, φ β] ω [φ α, φ β ] ω, (2.9) The following corollary is an immediate consequence of remark 2.2. Corollary 2.3. Let φ α and φ β be solutions of (2.1) with α and β solutions of (2.6). Suppose in addition that [φ α, φ β] ω [φ α, φ β ] ω =, (2.1)

4 4 B. MEROUANI, R. BOUFENOUCHE EJDE-213/238 and α β, then [φ α, φ β ] = { [(α 2 2α)φ α + φ α]φ β + [(β 2 2β)φ β + φ β ]φ α} dθ = (2.11) Remark 2.4. For u α = r α φ α we have u α 2 r u α r = rα 2 [(α 2 2α)φ α + φ α]. (2.12) Let P be the operator P = 2 r r. From the corollary 2.3 and Remark 2.4, we deduce the following result. Corollary 2.5. Under the hypotheses of corollary 2.3, if α β, we have (P u α u β + u α P u β )d =. (2.13) Σ 3. Formula for the coefficients in the crack case The crack case (ω = 2π) is an important one, among singular domains, in the applications. Moreover in this case the solutions of (2.6) are explicitly known and we have E = { k, k N, k > 2} 2 and these roots are of multiplicity 2. In this framework we assume that the solution u admits the representation u = α E(c α u α + d α v α ), E = { k, k N, k > 2}, 2 u α = r α φ α (θ), v α = r α ψ α (θ) (3.1) the solutions φ α and ψ α, in terms of θ, are the odd functions: and since α = k/2, we obtain and thus φ α (θ) = sin(α 2)θ, (3.2) ψ α (θ) = sin αθ. (3.3) φ α () = φ α (ω) = ψ α () = ψ α (ω) = (3.4) [φ α, φ β] ω = [φ α, φ β ] ω =, [ψ α, ψ β] ω = [ψ α, ψ β ] ω =, [φ α, ψ β] ω = [φ α, ψ β ] ω =. From here comes the idea of decomposing the solution u of (1.3) into two parts as follows: u = w 1 + w 2, w i = α E i (c α u α + d α v α ), i = 1, 2, (3.5) E 1 = {2m, m > 1}, E 2 = {2m + 1, 2m > 1}.

5 EJDE-213/238 COEFFICIENTS OF SINGULARITIES 5 Calculation of c β and d β. From the expressions of φ α, ψ α one easily sees that: if α E 1, then φ α() = φ α(ω) and ψ α() = ψ α(ω), if α E 2, then φ α() = φ α(ω) and ψ α() = ψ α(ω). (3.6) Equations (3.4) and (3.6) allow us to apply corollary 2.5 to functions u α and u β (resp. u α, v β and v α, v β ) and get the relations: (P w i u β + w i P u β )d = 2c β (u β P u β )d + d β (P v β.u β + v β P u β )d, (P w i v β + w i P v β )d = c β (P u β v β + u β P v β )d + 2d β (P v β v β )d. By direct calculations we obtain (P u β v β + u β P v β )d =, (u β P u β )d = (β 2)ωρ 2β 1 and from this we get our main the result. (P v β.v β )d = βωρ 2β 1 Theorem 3.1. Let u be a the solution of (1.3) written in the form where (3.7) (3.8) u = w 1 + w 2 (3.9) w i = α E i (c α u α + d α v α ), i = 1, 2 (3.1) Suppose that the series that gives w i is uniformly convergent in S. Then for any α E i, i = 1, 2 we have c α = ρ1 2α (P w i u α + w i P u α )d 2(α 2)ω (3.11) d α = ρ1 2α (P u i v α + w i P v α )d 2αω Σ Remark 3.2. Let ζ H 3/2 (Σ) H 1 (Σ) be the trace of u on Σ and χ H 1/2 (Σ) the trace of P u on Σ. If u is regular in order that ζ H 4 (], 2π[) and χ H 2 (], 2π[), then we have a uniform convergence of the series in S ρ for all ρ < ρ, see [5] Independence of the coefficients. Proposition 3.3. The coefficients c β (resp d β ) are independent of ρ. Proof. Let us prove that the derivative of c β with respect to ρ is zero. Observing the expression of c β in Theorem 3.1, we just have to prove that the derivative, with respect to ρ, of γ β = ρ 1 2β (P w i u β + w i P u β )d. (3.12)

6 6 B. MEROUANI, R. BOUFENOUCHE EJDE-213/238 vanishes. By derivation with respect to r we have γ β = On Σ, we have and { r ( w i)r 2 β φ β + [ (2 β) w i 2 2 w i r 2 + w i r r β φ β βw i r 1 β[ (β 2 2β)φ β + φ β] } dθ. (2 β) w i 2 2 w i r 2 + (β2 2) 1 r r ( w i) = Nw i + (1 υ) [ 1 r 3 2 w i θ 2 1 r 2 3 w i r θ 2 ], w i ] r 1 β φ β r (3.13) = βmw i + [2 (1 υ)β][ 1 w i r r w i r 2 ]. (3.14) θ2 Using these formulas in the expression of γ β we obtain γ β = + + (βmw i r 1 β φ β + Nw i r 2 β φ β )dθ { [(β 2 (1 υ)β)φ β + φ β] u i r (1 υ) 3 u i r θ 2 φ } β r β dθ { [2 (1 υ)(β 1)] 2 w i θ 2 φ β βw i [(β 2 2β)φ β + φ β] } r 1 β dθ. By a double integration by parts, we verify that 2 w i θ 2 φ βdθ = 3 w i r θ 2 φ βdθ = (3.15) w i φ βdθ (3.16) w i r φ βdθ (3.17) Using (3.15) (3.17) in the expression of γ β and putting the ρ1 2β, we obtain γ β = ρ 1 2β + ρ 1 2β + ρ 1 2β ω(βmw i r β 1 φ β + Nw i r β φ β )ρdθ [(β 2 (1 υ)β)φ β + υφ β]r β 2 w i r ρdθ { [ β 2 (β 2)]φ β + [ (2 υ)β + (3 υ)]φ β} r β 3 w i ρdθ. (3.18) Taking into account of (2.7), whose expressions appear explicitly in γ β, we obtain γ β = ρ 1 2β Σ [( u β Nw i + u β n Mw ) ( i ui Nu β + w i n Mu )] β d =. (3.19) We follow the same analysis to prove the independence of d β with respect to ρ.

7 EJDE-213/238 COEFFICIENTS OF SINGULARITIES Convergence of the series. We first write c α and d α in the form c α = I i ρ α, d α = J i ρ α (3.2) with ρ I i = (P w i φ α + w i ρ 2 [(α 2 2α)φ α + φ 2ω(α 2) α])d, J i = ρ (P w i ψ α + w i ρ 2 [(α 2 2α)ψ α + ψ 2ωα α])d. The solution u of (1.3) is then written as and we have the following result. (3.21) u = w 1 + w 2 (3.22) w i = α E i [( r ρ )α I i φ α + ( r ρ )α J i ψ α ], i = 1, 2 (3.23) Theorem 3.4. The series (3.23) converges uniformly in S ρ for all ρ < ρ. Proof. Set H i,α = = (P u i φ α + u i ρ 2 [(α 2 2α)φ α + φ α])dθ P u i φ α dθ + (α 2 2α)ρ 2 u i φ α dθ + ρ 2 u i φ αdθ (3.24) We show that H i,α is 1/α times by bounded term, for α large enough. According to (3.17), we have u i φ αdθ = u i φ α dθ (3.25) Replacing φ α by its expression and integrating by parts we obtain u i φ α dθ = 1 [ α ω ] u i cos(α 2)θdθ α (α 2) On the other hand, by a triple integration by parts, we have (α 2 2α) u i φ α dθ = 1 [ α 2 ] α (α 2) 2 u i cos(α 2)θdθ Also, integrating by parts, we obtain ( u i 2 u i r r )φ αdθ = 1 [ α α (α 2) intω ( θ ( u i) 2 r Then, we deduce the existence of a constant C such that: (3.26) (3.27) 2 u ] i r θ ) cos(α 2)θdθ (3.28) H i,α C (3.29) α Using this last inequality and the fact that φ α is bounded as well as the term 1/(2ω(α 2)) for large α we deduce the existence of a constant C such that c α r α φ α C α ( r ρ )α (3.3) α Ei α E i which converges uniformly in S ρ proved by the same way. for ρ < ρ. Convergence of α E i d α r α ψ α is

8 8 B. MEROUANI, R. BOUFENOUCHE EJDE-213/238 References [1] H. Blum, R. Rannacher; On the boundary value problem of the biharmonic operator on domains with angular corners, Maths. Methods Appl. Sci. 2 (198), no. 4, [2] Cédric Camier; Modélisation et étude numérique de vibrations non-linéaire de plaques circulaires minces imparfaites, Application aux cymbales. Thèse Présentée et soutenue publiquement le 2 février 29 pour l otention du Docteur de l Ecole Polytechnique. [3] W. Chikouche, A. Aibeche; Coefficients of singularities of the biharmonic problem of Neumann type: case of the crack. IJMMS 23: 5, , Hindawi Publishing Corp. [4] P. Grisvard; Singularities in boundary values problems, Recherches en Mathématiques Appliquées, Vol. 22, Masson, Paris, 1992 (in French). [5] O. Tcha-Kondor; Nouvelles séries trigonométriques adapt ées à l étude de problèmes aux limites pour l équation biharmonique.étude du cas de la fissure [New trigonometric series adapted to the study of boundary value problems for the biharmonic equation. The cas of the crack], C. R. Acad. Sci. Paris sér. I, Math. 315(1992), no. 5, (in French). Boubakeur Merouani Department of Mathemaics, Univ. F. Abbas-Sétif 1, Algeria address: mermathsb@hotmail.fr Razika Boufenouche Department of Mathemaics, Univ. Jijel, Algeria address: r boufenouche@yahoo.fr