Asymptotic Analysis of Fields in Multi-Structures

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1 Asymptotic Analysis of Fields in Multi-Structures VLADIMIR KOZLOV Department of Mathematics, Linkoeping University, Sweden VLADIMIR MAZ'YA Department of Mathematics, Linkoeping University, Sweden ALEXANDER MOVCHAN Department of Mathematical Sciences, University of Liverpool, United Kingdom CLARENDON PRESS OXFORD 1999

2 List of symbols CONTENTS 1 Introduction to compound asymptotic expansions Elementary examples of perturbation problems for ordinary differential equations A one-dimensional singularly perturbed problem Neumann boundary value problem in a domain with small cavity Formulation of the problem The leading order approximation Remainder estimate Complete asymptotic expansion Asymptotic formula for the energy Dirichlet boundary value problem in a domain with small inclusion The leading order approximation The next approximation The complete asymptotic expansion Mixed boundary value problem for the Laplacian in a thin rectangle Formulation of the boundary value problem Two-term approximation The next approximation Higher-order approximation Problem of junction between thin bodies Model problems The leading order approximation The next-order approximation The complete asymptotic expansion The remainder estimate 53 2 A boundary value problem for the Laplacian in a multistructure Formulation of the problem Model problems Limit domains, Model problem in fi Model problem for the junction region Junction layer 70 xiv

3 2.2.5 Model problem for the bottom region Two model problems for a thin cylinder Algebraic system for the skeleton Right-hand sides Local coordinates and limit domains Cut-off functions Asymptotic representations of the right-hand sides The leading term of the asymptotic solution Domain fi Junction layer Thin cylinder Bottom layer (j) Evaluation oi W (i) Evaluation of the constants T 0 and C o Concluding remarks on formal algorithm Complete asymptotic expansion Structure of the asymptotic expansion The asymptotic algorithm Justification of the asymptotic expansion Auxiliary estimates for functions in H 1 (Q e ) Estimate for solutions Estimate for the remainder term A constant right-hand side Application to the asymptotics of the energy integral The case of the right-hand sides concentrated in fi The case of the Dirichlet data at the bases of thin cylinders On a general 1D-3D multi-structure A multi-structure with a thin-walled tube 108 Auxiliary facts from mathematical elasticity Basic formulae of linear elasticity Stress and strain Equations of equilibrium and boundary conditions Two-dimensional problems of linear elasticity Plane strain Anti-plane shear Differential equations for engineering models of elastic rods 120

4 xi 3.4 Classical solutions of linear elasticity for a half-space Boussinesq-Cerruti's solution Mindlin's solution Connection between the Boussinesq-Cerruti and Mindlin solutions Special solutions for a bounded two-dimensional domain The torsion potential The bending potentials Example Special solutions of linear elasticity for an infinite cylinder Representation of differential operators The spectral problem ^-polynomial solutions Biorthogonality conditions The normalised stiffness coefficients Biorthogonality relations for eigenvectors and generalised eigenvectors There are no other polynomials Green's matrix in Q, Definition Asymptotics Korn's inequalities The case of bounded Lipschitz domains Half-space and a cylinder Asymptotics at infinity for solutions to the traction problem for a half-cylinder 151 Elastic multi-structure Multi-structure and boundary value problem Model problems Limit domains Model problem for the body fi Junction layer Model problem for the bottom layer Model problem for a bounded two-dimensional domain Model problems on the axis of an elastic rod Model matrices and the pile structure Special cases Asymptotic expansion of the solution 180

5 4.3.1 Asymptotic representation of the right-hand sides Description of the asymptotic series for the solution Auxiliary solutions of the Lame system in a thin elastic rod Expansions for displacement in a thin rod Junction layer Displacement in ft Bottom layer Functions v ( k m ' j) The recurrent procedure for the asymptotic expansion Justification of the asymptotic expansion Korn's inequality in ft e An estimate for the solution The leading order approximation The term u^- 1 ) The term u< ) Physical interpretation of the results The case Mg + Fi + F 2 1 ^ The case F 2 = F 2 = M3 = Non-degenerate elastic multi-structures Pile structure model Skeleton of the multi-structure The pile structure Mathematical model of the pile structure Solution of the pile structure equations Algebraic system for the pile structure model Non-degenerate and degenerate pile structures Examples Multi-structure and the boundary value problem Description of the multi-structure Formulation of the boundary value problem Model problems Junction layer Remaining model problems Asymptotic expansion of the solution Cut-off functions Asymptotic representation of the right-hand sides for the case of a non-degenerate multi-structure 233

6 xiii Structure of the asymptotic series for the displacement field in ft The junction layer Displacement in ft The bottom layer Functions vj m>i) Evaluation of the lock forces and moments at junction points Algebraic system for a<< m \ /?< m ) The recurrent procedure for the asymptotic expansion Estimate for the remainder of the asymptotic expansion Analysis of the leading term Physical interpretation Spectral analysis for 3D-1D multi-structures An abstract scheme for the asymptotics of eigenvalues Spectral problem for the Laplacian The first eigenvalue The first eigenfunction Asymptotics of first eigenvalues of the Lame operator Spectral problem Korn-type inequalities Spaces X o and H o Asymptotic formula for the eigenvalues Spectral problem for an inhomogeneous elastic multistructure The spectral problem Asymptotic formulae for the eigenvalues 268 Bibliographical remarks 274 Bibliography 276 Index 281

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