5.3 Formulating a Boundary-Value Problem for a System of Two
First-Order Ordinary Differential Equations in the Plastic
Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.4 Formulation and Solution of the Boundary-Value Problem
for Second-Order ODE’s in the Elastic Deformation Region . . . 71
5.5 Determining the Critical Pressure . . . . . . . . . . . . . . . . . . . . . . . 74
5.6 An Analytical Solution of the Cavity Expansion Problem
in a Medium with a Linear Shock Adiabat . . . . . . . . . . . . . . . . 78
5.7 Determining Stresses in a Medium with the Mohr-Coulomb
Yield Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.8 Determining Stress in a Medium with Mohr-CoulombTresca’s Yield Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.9 Comparative Analysis of the Results of Analyzing the Cavity
Problem in Media with Tresca’s, Mohr-Coulomb,
and Mohr-Coulomb-Tresca’s Plasticity Conditions . . . . . . . . . . 83
5.10 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6 Construction of the Solutions of Non-stationary Dynamic
Problems for Linear Viscoelastic Bodies with a Constant
Poisson’s Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
Leonid Igumnov, Ekaterina A. Korovaytseva,
and Sergey G. Pshenichnov
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.2 Mathematical Statement of Problem . . . . . . . . . . . . . . . . . . . . . 90
6.3 Representation of the Problem in Transform Domain . . . . . . . . 91
6.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.5 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
7 Features of Subsonic Stage of Contact Interaction of Viscoelastic
Half-Plane and Absolutely Rigid Striker . . . . . . . . . . . . . . . . . . . . . 97
Leonid Igumnov, Ekaterina A. Korovaytseva,
and Dmitrii V. Tarlakovskii
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
7.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
7.3 Green Function Construction . . . . . . . . . . . . . . . . . . . . . . . . . . 100
7.4 Contact Problem Solution Algorithm . . . . . . . . . . . . . . . . . . . . 104
7.5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
Contents
vii
First-Order Ordinary Differential Equations in the Plastic
Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.4 Formulation and Solution of the Boundary-Value Problem
for Second-Order ODE’s in the Elastic Deformation Region . . . 71
5.5 Determining the Critical Pressure . . . . . . . . . . . . . . . . . . . . . . . 74
5.6 An Analytical Solution of the Cavity Expansion Problem
in a Medium with a Linear Shock Adiabat . . . . . . . . . . . . . . . . 78
5.7 Determining Stresses in a Medium with the Mohr-Coulomb
Yield Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.8 Determining Stress in a Medium with Mohr-CoulombTresca’s Yield Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.9 Comparative Analysis of the Results of Analyzing the Cavity
Problem in Media with Tresca’s, Mohr-Coulomb,
and Mohr-Coulomb-Tresca’s Plasticity Conditions . . . . . . . . . . 83
5.10 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6 Construction of the Solutions of Non-stationary Dynamic
Problems for Linear Viscoelastic Bodies with a Constant
Poisson’s Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
Leonid Igumnov, Ekaterina A. Korovaytseva,
and Sergey G. Pshenichnov
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.2 Mathematical Statement of Problem . . . . . . . . . . . . . . . . . . . . . 90
6.3 Representation of the Problem in Transform Domain . . . . . . . . 91
6.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.5 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
7 Features of Subsonic Stage of Contact Interaction of Viscoelastic
Half-Plane and Absolutely Rigid Striker . . . . . . . . . . . . . . . . . . . . . 97
Leonid Igumnov, Ekaterina A. Korovaytseva,
and Dmitrii V. Tarlakovskii
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
7.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
7.3 Green Function Construction . . . . . . . . . . . . . . . . . . . . . . . . . . 100
7.4 Contact Problem Solution Algorithm . . . . . . . . . . . . . . . . . . . . 104
7.5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
Contents
vii