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.- 2.7.1 Lamé Representation.- 2.7.2 The Case of a Separable Solution to the Vector Wave Equation.- 2.7.3 Iacovache’s Representation.- 2.7.4 Representation Employing Papkovich—Neuber Functions.- 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.- 3.2.1 Reflection of Plane Longitudinal and Transverse Waves.- 3.2.2 The Case of Total Internal Reflection.- 3.2.3 Rayleigh Waves.- 3.3 Homogeneous Solutions.- 3.3.1 Construction of Homogeneous Solutions to the Wave Equation.- 3.3.2 Diffraction of a Plane Shear Wave by a Wedge.- 3.4. The Case of an Elastic Half-Plane.- 3.4.1 Reduction of the Dynamic Problem to Superposition of Problems for Symmetric and Antisymmetric Components of the Displacement Vector.- 3.4.2 Homogeneous Solutions.- 3.4.3 Solution of Some Problems.- 3.5 Mixed Boundary-Value Problems for an Elastic Half-Plane. Crack Propagation.- 3.5.1 Representations of Solutions to Mixed Boundary-Value Problems.- 3.5.2 Solution for the n ? 1-Case. Some Examples.- 3.5.3 Solution for the n ? 0-Case.- 3.6 Solution of Analogous Mixed Boundary-Value Problems. Wedge-Shaped Punch.- 3.6.1 Solution for the n ? 1-Case.- 3.6.2 Indentation of a Wedge with a Sub-Rayleigh Contact Speed.- 3.6.3 Indentation of a Wedge with Super-Rayleigh Contact Speed.- 3.6.4 Solution Singularities at the Edges of the Contact Region.- 3.7 Interrelation Between Three- and Two-Dimensional Problems.- 3.8 Application of the Smirnov-Sobolev Method to Solving Axisymmetric Elastodynamic Problems.- 3.8.1 Representation of Axisymmetric Solutions by Employing Analytical Functions.- 3.8.2 Solutions to Axisymmetric Problems.- 3.9 Solutions to Some Axisymmetric Problems with Mixed Boundary Conditions.- 3.9.1 Solution for the n ? 1-Case.- 3.9.2 Circular Crack Expansion and a Conical Punch Indentation.- 3.9.3 Expansion of a Circular Crack Due to a Concentrated Load.- 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.- 4.2.1 Solution to the Problem.- 4.2.2 Cagniard-de Hoop Method.- 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.- 4.5.1 Diffraction of a Plane Acoustic Wave by a Rigid Cone.- 4.5.2 Diffraction of a Spherical Acoustic Wave by a Cone.- 4.6 Diffraction of Elastic Waves by a Smooth Rigid Cone.- 4.6.1 Diffraction of a Plane Longitudinal Wave by a Cone.- 4.6.2 Diffraction of a Spherical Elastic Wave by a Cone.- 4.7 Impact of a Circular Cylinder on a Stationary Obstacle.- 4.7.1 Formulation and Solution of the Problem.- 4.7.2 Analysis of the Solution at the Points of a Cylinder’s Axis.- 5. Solution to Three-Dimensional Elastodynamic Problems with Mixed Boundary Conditions for Wedge-Shaped Domains.- 5.1 Combined Method of Integral Transforms.- 5.1.1 Problem Formulation.- 5.1.2 Problem Solution.- 5.2 Diffraction of a Spherical Elastic Wave by a Smooth Rigid Wedge.- 5.2.1 Solution to the Problem.- 5.2.2 Analysis of the Obtained Solution.- 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.- 6.1.1 A Semi-Infinite Punch.- 6.1.2 Analysis of the Punch Solution.- 6.2 A Finite-Width Punch.- 6.2.1 Solution to the Problem.- 6.2.2 The Acoustic Case.- 6.3 Problems with Moving Boundary Edges.- 6.3.1 Problem Formulation and Application of Integral Transforms.- 6.3.2 Splitting of Fundamental Solutions.- 6.3.3 Solution of the Two-Dimensional Dynamic Problem.- 6.4 Some Crack and Punch Problems.- 6.4.1 The Plane-Strain Problem of a Semi-Infinite Crack Propagation.- 6.4.2 The Antiplane Problem of a Semi-Infinite Crack Propagation.- 6.4.3 The Acoustic Problem for a Punch.- 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.- 7.2.1 Three-Dimensional Case.- 7.2.2 Two-Dimensional Case.- 7.2.3 Radon Transform.- 7.3 Solution to the Second Boundary-Value Problem.- 7.3.1 Three-Dimensional Case.- 7.3.2 Two-Dimensional Case.- 7.4 Lamb’s Problem.- 7.4.1 Three-Dimensional Case.- 7.4.2 Two-Dimensional Case.- 7.4.3 Isotropic Half-Space.- 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.- 7.7.1 Solution to the Problem.- 7.7.2 Isotropic Media.- 7.8 Two-Dimensional Problems.- 7.8.1 A Strip-Shaped Crack.- 7.8.2 The Axisymmetric Case.- References.