� A d µ Universit¨ at Stuttgart Institut f¨ ur Analysis, Dynamik und Modellierung Trapped Modes in Elastic Media for Zero Poisson Coefficient Three-dimensional elastic plate with local perturbation Clemens F¨ orster and Timo Weidl, St. Petersburg in July 2010 1
� A d µ Universit¨ at Stuttgart Institut f¨ ur Analysis, Dynamik und Modellierung Trapped modes in physics Trapped modes are modes of oscillation which occur (in an unbounded domain) at discrete frequencies and consist of motion which is restricted to some localized region of the considered medium near some perturbation. Water wave theory: Water waves in perturbed water channels Acoustic theory: Acoustic resonances in waveguides with obstacles Quantum mechanics: Bound states in bent, twisted and coupled waveguides Electromagnetism: Trapped modes in twisted and coupled waveguides Trapped modes correspond to (embedded) eigenvalues for systems with conti- nuous spectrum. Clemens F¨ orster and Timo Weidl, St. Petersburg in July 2010 2
� A d µ Universit¨ at Stuttgart Institut f¨ ur Analysis, Dynamik und Modellierung Trapped modes in mathematics - a tale of two dimensions Models for trapped modes show usually mixed dimensions: a global dimension - often 1 (wires) or 2 (layers) a local dimension d The global dimension determines the low energy behaviour: odinger operator − ∆ − αV ( x ) in L 2 ( R d ) has for � The Schr¨ V dx > 0 has in the limit of α → +0 one negative eigenvalue − λ 1 ( α ) satisfying � = α � λ 1 ( α ) V dx + o ( α ) for d = 1 , 2 1 = α � V dx + o ( α ) for d = 2 . 4 π ln λ − 1 ( α ) The local dimension determines the high energy behaviour: Weyl type asymptotics for large α → + ∞ . Clemens F¨ orster and Timo Weidl, St. Petersburg in July 2010 3
� A d µ Universit¨ at Stuttgart Institut f¨ ur Analysis, Dynamik und Modellierung Trapped modes in elasticity Consider a semi-strip Ω = [0 , + ∞ ) × J with J = ( − π/ 2 , + π/ 2) . Note that − ∆ on this domain has no eigenvalues, neither in the Dirichlet nor in the Neumann case. If you pass to linear elasticity with zero Poisson coefficient � � u 1 ( x 1 , x 2 ) L 2 (Ω , C 2 ) ∋ u = A = − ∆ ⊗ 1 − grad div on u 2 ( x 1 , x 2 ) with stress-free boundary conditions (corresponds to the scalar Neumann case) ∂u 1 ∂x 2 + ∂u 2 ∂u 1 ∂x 2 + ∂u 2 ∂x 1 = 0 ∂x 1 = 0 for x 2 = ± π and for x 1 = 0 , ∂u 2 ∂u 1 ∂x 2 = 0 ∂x 1 = 0 2 then A has at least one positive eigenvalue embedded into the continuous spectrum. Clemens F¨ orster and Timo Weidl, St. Petersburg in July 2010 4
� A d µ Universit¨ at Stuttgart Institut f¨ ur Analysis, Dynamik und Modellierung Shaw (1956): experiments on edge resonance in circular barium titanate disks many attempts to explain edge resonance by approximative analysis or numerical methods Roitberg, Vassiliev and W (1998): first rigorous proof for the existence of trapped modes in the elastic semi-strip Holst, Vassiliev (2000): Edge resonance in an elastic semi-infinite cylinder Gridin, Adamou, Craster (2005): Trapped modes in bent elastic rods and in curved elastic plates Zernov, Pichugin, Kaplunov (2006): Eigenvalue of a semi-infinite elastic plate Main difference between elasticity and other physical systems: Water wave theory, Acoustics, Quantum mechanics, Electromagnetism: trapped modes = eigenvalues of the Laplace operator (scalar) Elasticity theory: trapped modes = eigenvalues of the elastostatic operator (matrix) Clemens F¨ orster and Timo Weidl, St. Petersburg in July 2010 5
� A d µ Universit¨ at Stuttgart Institut f¨ ur Analysis, Dynamik und Modellierung Three Model Problems Elastic strip with local perturbation of material properties J = ( − π 2 , π G 2 = R × J, 2 ) Elastic plate with local perturbation of material properties ������ ������ ������ ������ ������ ������ G 3 = R 2 × J ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ Elastic plate with perturbation by a hole ������ ������ ������ ������ ������ ������ ������ ������ Ω = Ω 0 × J, Ω 0 ⊂ R 2 G 3 \ Ω , ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ General properties: isotropic, linear elastic medium unperturbed part has homogeneous material zero Poisson’s ratio stress-free boundaries Clemens F¨ orster and Timo Weidl, St. Petersburg in July 2010 6
� A d µ Universit¨ at Stuttgart Institut f¨ ur Analysis, Dynamik und Modellierung The (unperturbed) elasticity operator Consider the elastostatic operator with stress-free (Neumann-type) boundary conditions: u ∈ H 2 ( G d ; C d ) , A 0 u = − div σ ( u ) , d = 2 , 3 , σ ( u ) n ∂G d = 0 , on ∂G d . Here we use σ ( u ) = 2 µǫ ( u ) + λ Tr ( ǫ ( u ))I is the stress matrix , ǫ ( u ) = 1 ( ∇ u ) + ( ∇ u ) T � � is the strain matrix , 2 Eν E λ = (1 + ν )(1 − 2 ν ) , µ = are the Lam´ e constants . 2(1 + ν ) Clemens F¨ orster and Timo Weidl, St. Petersburg in July 2010 7
� A d µ Universit¨ at Stuttgart Institut f¨ ur Analysis, Dynamik und Modellierung Special Case: Zero Poisson Coefficient We put E = 2 and ν = 0 and study the self-adjoint operator L 2 ( G d ; C d ) A 0 = − ∆ ⊗ I − grad div in associated with the Hermitian form � u, v ∈ H 1 ( G d ; C d ) . a 0 [ u, v ] = 2 � ǫ ( u ) , ǫ ( v ) � C d × d dx, G d where ǫ ( u ) = 1 ( ∇ u ) + ( ∇ u ) T � � . 2 Clemens F¨ orster and Timo Weidl, St. Petersburg in July 2010 8
� A d µ Universit¨ at Stuttgart Institut f¨ ur Analysis, Dynamik und Modellierung The Case of Local Changes of Material Coefficients: Additive perturbations of Young’s Modulus Let f ∈ L ∞ ( G d ) be compactly supported, independent of x d -coordinate and 0 ≤ f ( x ) ≤ 1 (but sometimes also just f ( x ) ≤ 1 ). For β ∈ (0 , ∞ ) (scaling) and α ∈ (0 , 1) (coupling) we consider � · A α,β u = − div (1 − αf β )( ∇ u + ( ∇ u ) T ) L 2 ( G d ; C d ) , � in f β := f , β with stress-free boundary conditions. This corresponds to � u, v ∈ H 1 ( G d ; C d ) . a α,β [ u, v ] = 2 (1 − αf β ) � ǫ ( u ) , ǫ ( v ) � C d × d dx, G d Clemens F¨ orster and Timo Weidl, St. Petersburg in July 2010 9
� A d µ Universit¨ at Stuttgart Institut f¨ ur Analysis, Dynamik und Modellierung Symmetries We consider the following subspaces of L 2 ( R × J ; C 2 ) := { u ∈ L 2 ( R × J ; C 2 ) | u 1 symmetric in x 2 , u 2 antisymmetric in x 2 } H 1 := { u ∈ H 1 | u 1 ( x 1 , · ) ⊥ 1 in L 2 ( J ; C ) for a.e. x 1 ∈ R } H 4 The subspaces H 4 and H ⊥ 4 reduce A α,β and A 0 . We consider A (4) A (4) α,β := A α,β | D ( A α,β ) ∩ H 4 , := A 0 | D ( A 0 ) ∩ H 4 . 0 While σ ( A 0 ) = [0 , + ∞ ) we have for the reduced operator σ ( A (4) 0 ) = [Λ , + ∞ ) for a certain Λ > 0 . Let us discuss this more in detail for the unperturbed strip: Clemens F¨ orster and Timo Weidl, St. Petersburg in July 2010 10
� A d µ Universit¨ at Stuttgart Institut f¨ ur Analysis, Dynamik und Modellierung Separation of Variables for d = 2 Apply the unitary Fourier transform Φ : H 4 → H 4 in x 1 -direction to � � 2 ∂ x 1 x 1 + ∂ x 2 x 2 ∂ x 1 x 2 A (4) = − ∆ − grad div = − 0 ∂ x 1 x 2 ∂ x 1 x 1 + 2 ∂ x 2 x 2 ∂ x 2 u 1 + ∂ x 1 u 2 = 0 for x 2 = ± π ∂ x 2 u 2 = 0 , 2 , and consider for ξ ∈ R and ˆ u = Φ u − ∂ 2 2 + 2 ξ 2 � � − iξ∂ 2 A (4) ( ξ ) := (Φ A (4) 0 Φ ∗ )( ξ ) = , − 2 ∂ 2 2 + ξ 2 − iξ∂ 2 ( ∂ x 2 ˆ u 2 ) | x 2 = π/ 2 = ( ∂ x 2 ˆ u 1 + iξ ˆ u 2 ) | x 2 = π/ 2 = 0 . The symmetries in x 2 -direction are preserved. For fixed ξ this Sturm-Liouville system has the (ordered) eigenvalues λ j ( ξ ) which depend continuously in ξ : Clemens F¨ orster and Timo Weidl, St. Petersburg in July 2010 11
� A d µ Universit¨ at Stuttgart Institut f¨ ur Analysis, Dynamik und Modellierung 25 λ 1 λ 2 λ 3 20 2 | ξ | 2 lowest three eigenvalue branches 15 of A (4) ( ξ ) λ 10 and 2 | ξ | 2 for comparison 5 0 | ξ | 0 0.5 1 1.5 2 2.5 3 3.5 4 3 global minimum of λ 1 ( ξ ) at λ 1 2.5 κ ≈ 0 . 64 , Λ ≈ 1 . 88 λ 2 minimum is non-degenerated ( κ , Λ) λ 1 ( κ + ε ) = Λ + q 2 ε 2 + O ( ε 3 ) 1.5 for ε → 0 with q ≈ 0 . 84 . 1 | ξ | 0 0.5 1 1.5 2 Clemens F¨ orster and Timo Weidl, St. Petersburg in July 2010 12
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