Methods of the Classical Theory of Elastodynamics

1. Introduction.- 2. Formulation of Elastodynamic Problems. Some General Results.- 2.1 Fundamental Equations of Elastodynamics.- 2.2 Initial and Boundary Conditions. Interfaces.- 2.3 Constraints Imposed on the Solution Behavior in the Neighborhood of Singular Points/Curves.- 2.4 Continuous and Discontinuous Solutions.- 2.5 Uniqueness Theorem for Solutions to Elastodynamic Problems with Strong Discontinuities.- 2.6 The Green-Volterra Formula.- 2.7 Various Representations of Solutions to the Equations of Motion of a Homogeneous Isotropic Medium.- 2.8 On the Relationships Between Solutions of Transient Dynamic Problems and Those of Static, Steady-State and Stationary Dynamic Problems.- 3. The Method of Functionally Invariant Solutions (the Smirnov-Sobolev Method).- 3.1 Functionally Invariant Solutions to the Wave Equation.- 3.2 Plane and Complex Waves.- 3.3 Homogeneous Solutions.- 3.4. The Case of an Elastic Half-Plane.- 3.5 Mixed Boundary-Value Problems for an Elastic Half-Plane. Crack Propagation.- 3.6 Solution of Analogous Mixed Boundary-Value Problems. Wedge-Shaped Punch.- 3.7 Interrelation Between Three- and Two-Dimensional Problems.- 3.8 Application of the Smirnov-Sobolev Method to Solving Axisymmetric Elastodynamic Problems.- 3.9 Solutions to Some Axisymmetric Problems with Mixed Boundary Conditions.- 3.10 An Alternative Derivation of the Smirnov-Sobolev Representations.- 4. Integral Transforms in Elastodynamics.- 4.1 Application of Integral Transforms to Solving Elastodynamic Problems.- 4.2 Lamb's Problem for a Half-Plane.- 4.3 Diffraction of an Acoustic Wave by a Rigid Sphere.- 4.4 Expansion of an Acoustic Wave Solution for a Sphere Over a Time-Dependent Interval.- 4.5 Diffraction of Acoustic Waves by a Rigid Cone.- 4.6 Diffraction of Elastic Waves by a SmoothRigid Cone.- 4.7 Impact of a Circular Cylinder on a Stationary Obstacle.- 5. Solution to Three-Dimensional Elastodynamic Problems with Mixed Boundary Conditions for Wedge-Shaped Domains.- 5.1 Combined Method of Integral Transforms.- 5.2 Diffraction of a Spherical Elastic Wave by a Smooth Rigid Wedge.- 5.3 Diffraction of an Arbitrary Incident Plane Elastic Wave by a Rigid Smooth Wedge.- 6. Wiener-Hopf Method in Elastodynamics.- 6.1 Problems with a Stationary Boundary.- 6.2 A Finite-Width Punch.- 6.3 Problems with Moving Boundary Edges.- 6.4 Some Crack and Punch Problems.- 7. Homogeneous Solutions to Dynamic Problems for Anisotropic Elastic Media (Willis' Method).- 7.1 Studies in Elastodynamics for Anisotropic Media.- 7.2 Solution to the First Boundary Value Problem.- 7.3 Solution to the Second Boundary-Value Problem.- 7.4 Lamb's Problem.- 7.5 The Wedge-Shaped Punch Problem.- 7.6 Representing the Solutions for an Anisotropic Space in Terms of Displacement/Stress Discontinuities Across a Plane.- 7.7 Expansion of an Elliptic Crack.- 7.8 Two-Dimensional Problems.- References.