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
List of symbols CONTENTS 1 Introduction to compound asymptotic expansions 1 1.1 Elementary examples of perturbation problems for ordinary differential equations 1 1.2 A one-dimensional singularly perturbed problem 5 1.3 Neumann boundary value problem in a domain with small cavity 10 1.3.1 Formulation of the problem 10 1.3.2 The leading order approximation 12 1.3.3 Remainder estimate 13 1.3.4 Complete asymptotic expansion 16 1.3.5 Asymptotic formula for the energy 20 1.4 Dirichlet boundary value problem in a domain with small inclusion 22 1.4.1 The leading order approximation 22 1.4.2 The next approximation 24 1.4.3 The complete asymptotic expansion 25 1.5 Mixed boundary value problem for the Laplacian in a thin rectangle 30 1.5.1 Formulation of the boundary value problem 30 1.5.2 Two-term approximation 31 1.5.3 The next approximation 34 1.5.4 Higher-order approximation 36 1.6 Problem of junction between thin bodies 38 1.6.1 Model problems 40 1.6.2 The leading order approximation 47 1.6.3 The next-order approximation 49 1.6.4 The complete asymptotic expansion 51 1.6.5 The remainder estimate 53 2 A boundary value problem for the Laplacian in a multistructure 56 2.1 Formulation of the problem 58 2.2 Model problems 59 2.2.1 Limit domains, 59 2.2.2 Model problem in fi 60 2.2.3 Model problem for the junction region 63 2.2.4 Junction layer 70 xiv
2.2.5 Model problem for the bottom region 71 2.2.6 Two model problems for a thin cylinder 75 2.2.7 Algebraic system for the skeleton 76 2.3 Right-hand sides 77 2.3.1 Local coordinates and limit domains 78 2.3.2 Cut-off functions 78 2.3.3 Asymptotic representations of the right-hand sides 79 2.4 The leading term of the asymptotic solution 81 2.4.1 Domain fi 81 2.4.2 Junction layer 83 2.4.3 Thin cylinder 85 2.4.4 Bottom layer 87 2.4.5 (j) Evaluation oi W 0 88 2.4.6 (i) Evaluation of the constants T 0 and C o 88 2.4.7 Concluding remarks on formal algorithm 89 2.5 Complete asymptotic expansion 89 2.5.1 Structure of the asymptotic expansion 89 2.5.2 The asymptotic algorithm 91 2.6 Justification of the asymptotic expansion 92 2.6.1 Auxiliary estimates for functions in H 1 (Q e ) 92 2.6.2 Estimate for solutions 94 2.6.3 Estimate for the remainder term 97 2.7 A constant right-hand side 98 2.8 Application to the asymptotics of the energy integral 100 2.8.1 The case of the right-hand sides concentrated in fi 101 2.8.2 The case of the Dirichlet data at the bases of thin cylinders 103 2.9 On a general 1D-3D multi-structure 105 2.10 A multi-structure with a thin-walled tube 108 Auxiliary facts from mathematical elasticity 115 3.1 Basic formulae of linear elasticity 115 3.1.1 Stress and strain 115 3.1.2 Equations of equilibrium and boundary conditions 116 3.2 Two-dimensional problems of linear elasticity 118 3.2.1 Plane strain 118 3.2.2 Anti-plane shear 119 3.3 Differential equations for engineering models of elastic rods 120
xi 3.4 Classical solutions of linear elasticity for a half-space 121 3.4.1 Boussinesq-Cerruti's solution 121 3.4.2 Mindlin's solution 123 3.4.3 Connection between the Boussinesq-Cerruti and Mindlin solutions 124 3.5 Special solutions for a bounded two-dimensional domain 126 3.5.1 The torsion potential 127 3.5.2 The bending potentials 128 3.5.3 Example 128 3.6 Special solutions of linear elasticity for an infinite cylinder 129 3.6.1 Representation of differential operators 129 3.6.2 The spectral problem 130 3.6.3.^-polynomial solutions 132 3.6.4 Biorthogonality conditions 133 3.6.5 The normalised stiffness coefficients 135 3.6.6 Biorthogonality relations for eigenvectors and generalised eigenvectors 136 3.6.7 There are no other polynomials 138 3.7 Green's matrix in Q, 141 3.7.1 Definition 141 3.7.2 Asymptotics 141 3.8 Korn's inequalities 143 3.8.1 The case of bounded Lipschitz domains 143 3.8.2 Half-space and a cylinder 146 3.9 Asymptotics at infinity for solutions to the traction problem for a half-cylinder 151 Elastic multi-structure 155 4.1 Multi-structure and boundary value problem 156 4.2 Model problems 158 4.2.1 Limit domains 158 4.2.2 Model problem for the body fi 158 4.2.3 Junction layer 160 4.2.4 Model problem for the bottom layer 167 4.2.5 Model problem for a bounded two-dimensional domain 171 4.2.6 Model problems on the axis of an elastic rod 172 4.2.7 Model matrices and the pile structure 174 4.2.8 Special cases 179 4.3 Asymptotic expansion of the solution 180
4.3.1 Asymptotic representation of the right-hand sides 180 4.3.2 Description of the asymptotic series for the solution 182 4.3.3 Auxiliary solutions of the Lame system in a thin elastic rod 184 4.3.4 Expansions for displacement in a thin rod 187 4.3.5 Junction layer 189 4.3.6 Displacement in ft 190 4.3.7 Bottom layer 191 4.3.8 Functions v ( k m ' j). 193 4.3.9 The recurrent procedure for the asymptotic expansion 197 4.4 Justification of the asymptotic expansion 198 4.4.1 Korn's inequality in ft e 198 4.4.2 An estimate for the solution 199 4.5 The leading order approximation 203 4.5.1 The term u^- 1 ) 203 4.5.2 The term u< ) 204 4.6 Physical interpretation of the results 208 4.6.1 The case Mg + Fi + F 2 1 ^ 0 208 4.6.2 The case F 2 = F 2 = M3 = 0 210 Non-degenerate elastic multi-structures 213 5.1 Pile structure model 214 5.1.1 Skeleton of the multi-structure 214 5.1.2 The pile structure 216 5.1.3 Mathematical model of the pile structure 216 5.1.4 Solution of the pile structure equations 217 5.1.5 Algebraic system for the pile structure model 219 5.1.6 Non-degenerate and degenerate pile structures 221 5.1.7 Examples 222 5.2 Multi-structure and the boundary value problem 224 5.2.1 Description of the multi-structure 224 5.2.2 Formulation of the boundary value problem 226 5.3 Model problems 227 5.3.1 Junction layer 227 5.3.2 Remaining model problems 230 5.4 Asymptotic expansion of the solution 231 5.4.1 Cut-off functions 231 5.4.2 Asymptotic representation of the right-hand sides for the case of a non-degenerate multi-structure 233
xiii 5.4.3 Structure of the asymptotic series for the displacement field in ft 233 5.4.4 The junction layer 235 5.4.5 Displacement in ft 237 5.4.6 The bottom layer 238 5.4.7 Functions vj m>i) 239 5.4.8 Evaluation of the lock forces and moments at junction points 241 5.4.9 Algebraic system for a<< m \ /?< m ) 242 5.4.10 The recurrent procedure for the asymptotic expansion 243 5.5 Estimate for the remainder of the asymptotic expansion 244 5.6 Analysis of the leading term 245 5.7 Physical interpretation 246 6 Spectral analysis for 3D-1D multi-structures 248 6.1 An abstract scheme for the asymptotics of eigenvalues 249 6.2 Spectral problem for the Laplacian 253 6.2.1 The first eigenvalue 253 6.2.2 The first eigenfunction 255 6.3 Asymptotics of first eigenvalues of the Lame operator 257 6.3.1 Spectral problem 258 6.3.2 Korn-type inequalities 258 6.3.3 Spaces X o and H o 263 6.3.4 Asymptotic formula for the eigenvalues 263 6.4 Spectral problem for an inhomogeneous elastic multistructure 267 6.4.1 The spectral problem 267 6.4.2 Asymptotic formulae for the eigenvalues 268 Bibliographical remarks 274 Bibliography 276 Index 281