Mathematical Theory in Periodic Plane Elasticity

by Cai; Hai-Tao

• ISBN13: 9789056992422
• ISBN10: 9056992422
• Format: Hardcover
• Copyright: 2000-07-06
• Publisher: CRC Press

Summary

Presenting the mathematical theory of period problems in plane elasticity by methods of complex variables. The most general formulations of such problems are proposed under the assumption that the stresses are periodic and the displacements are quasi-periodic. The general expression of complex displacements are illustrated. Periodic welding problems are studied by reducing them to periodic Riemann boundary value problems. Various periodic problems of the elastic half-plane (fundamental problems, contact problems) are treated and solved by reduction to Riemann-Hilbert boundary value problems with discontinuous coefficient. Periodic crack problems are investigated which are transferred to singular integral equations whose unique solvability is guaranteed.

Table of Contents

Introduction to the Series	ix
Preface	xi
Periodic Boundary Value Problems For Analytic Functions
Periodic Riemann Boundary Value Problems: Case of Closed Contours	1	(7)
Formulation of the problems	1	(1)
Transfer to classical Riemann boundary value problems	2	(2)
Discussion on homogeneous problem P01	4	(2)
Discussion on non-homogeneous problem P1	6	(1)
A particular case	7	(1)
Periodic Riemann Boundary Value Problems: Case of Open Arcs or Discontinuous Coefficients	8	(4)
Case of open arcs	8	(2)
An important special case	10	(1)
Case of discontinuous coefficients	11	(1)
Periodic Riemann-Hilbert Boundary Value Problems of the Half-plane	12	(7)
Formulation of the problem	12	(1)
Sketch of the method of solution	13	(1)
An important particular case	14	(5)
Some Integral Formulas for Hilbert Kernel in the Half-plane	19	(2)
Periodic Problems For Isotropic Material in Plane Elastic Theory
Stress Functions	21	(11)
General expression of stress functions	21	(6)
The converse of Theorem 2.1	27	(2)
Formulation of the fundamental problem	29	(1)
Stress functions for elastic half-plane	30	(2)
Periodic Welding Problems	32	(5)
The case of uniform material welded	32	(2)
The case of welded material with the same shearing modulus	34	(3)
Periodic Fundamental Problems of Elastic Half-plane	37	(16)
The first fundamental problem	37	(5)
The second fundamental problem	42	(3)
The mixed fundamental problem	45	(8)
Periodic Contact Problems	53	(19)
The case without friction	53	(9)
The case with friction	62	(10)
Periodic Problems for Anisotropic Medium
The Stress Functions	72	(2)
Basic assumptions	72	(1)
Periodicity of the stress functions for anisotropic medium	73	(1)
Periodic Fundamental Problems of Anisotropic Half-plane	74	(10)
The first fundamental problem	74	(7)
The second fundamental problem	81	(3)
Periodic Contact Problem for Anisotropic Medium	84	(9)
Stress functions expressed in terms of boundary values of the stress components	85	(1)
Formulation of the problem	86	(2)
Solution of the problem	88	(1)
Periodic condition for displacements and equilibrium condition at z = - ∞i	89	(3)
The pressure beneath the stamps	92	(1)
Problems with Periodic Moving Loads on Isotropic Elastic Half-Plane
Stress Functions and Fundamental Problems	93	(5)
Periodicity of the stress functions for isotropic medium with moving loads	93	(1)
Formulation and solution of the problems	94	(1)
Periodic condition for displacements and equilibrium condition at z = -∞i	95	(1)
A special case	96	(2)
Periodic Contact Problems with Moving Stamps	98	(5)
Periodic boundary condition and solution of the problems	98	(2)
Periodic condition for displacements and equilibrium condition at z = - ∞i	100	(1)
The pressure beneath the stamps	101	(2)
Periodic Crack Problems in Plane Elasticity
Fundamental Problems of Isotropic Plane with Periodic Collinear Cracks	103	(7)
General comments	103	(2)
The first fundamental problem	105	(4)
The second fundamental problem	109	(1)
Fundamental Problems of Isotropic Elastic Plane with Periodic Arbitrary Cracks	110	(11)
General discussions	110	(2)
The stresses at infinity	112	(1)
The first fundamental problem	113	(5)
The second fundamental problem	118	(3)
Fundamental Problems of Anisotropic Elastic Plane with Periodic Collinear Cracks	121	(11)
General comments	121	(1)
Symmetric loads	122	(5)
Anti-symmetric loads	127	(3)
Stress intensity factors	130	(2)
Doubly-Periodic Problems in Plane Elasticity
Preliminaries	132	(3)
General notion	132	(1)
Weierstrass functions	133	(2)
General Expressions of Complex Stress Functions	135	(5)
General explanations	135	(1)
The case of region with holes	136	(2)
The case of region with cracks	138	(2)
Doubly-periodic Fundamental Problems	140	(7)
Relations among the addenda of related functions	140	(4)
Formulation of the fundamental problems	144	(3)
Bibliography	147	(5)
Index	152