Methods of the Classical Theory of Elastodynamics

ISBN-13: 9783642771019 / Angielski / Miękka / 2011 / 319 str.

Vladimir B. Poruchikov; V. a. Khokhryakov; G. P. Groshev

Opis

"Methods of the Classical Theory of Elastodynamics" deals not only with classical methods as developed in the past decades, but presents also very recent approaches. Applications and solutions to specific problems serve to illustrate the theoretical presentation. Keywords: Smirnov-Sobolev method with further developments; integral transforms; Wiener-Hopf technique; mixed boundary-value problems; time-dependent boundaries; solutions for unisotropic media (Willis method); 3-d dynamical problems for mixed boundary conditions.

Spis treści

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.