"Methods of the Classical Theory of Elastodynamics" dealsnot only with classical methods as developed in the pastdecades, but presents also very recent approaches.Applications and solutions to specific problems serve toillustrate the theoretical presentation.Keywords: Smirnov-Sobolev method with further developments;integral transforms; Wiener-Hopf technique; mixedboundary-value problems; time-dependent boundaries;solutions for unisotropic media (Willis method); 3-ddynamical problems for mixed boundary conditions.
"Methods of the Classical Theory of Elastodynamics" dealsnot only with classical methods as developed in the pastdecades, but presents also very recent approaches.Applications and solutions to specific problems serve toillustrate the theoretical presentation.Keywords: Smirnov-Sobolev method with further developments;integral transforms; Wiener-Hopf technique; mixedboundary-value problems; time-dependent boundaries;solutions for unisotropic media (Willis method); 3-ddynamical problems for mixed boundary conditions.
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.
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.
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