Fracture of materials under compression along cracks

Preface
Contents
1 Preliminary Consideration: Background, Approaches and Their Analysis
	1.1 Fracture Phenomena Analyzed and Basic Design Diagrams
		1.1.1 Classical and Non-classical Problems of Fracture Mechanics
		1.1.2 Basic Design Diagrams for Problems on the Fracture of Bodies Loaded Along Cracks
	1.2 General Concepts of Investigating Problems of Fracture Mechanics of Materials Under the Action of Forces Directed Along Cracks
		1.2.1 Formulation of the General Concept of and Approaches to Investigating the Problems of the Fracture Mechanics of Bodies Compressed Along Cracks
		1.2.2 General Approaches to Investigating Problems of Fracture Mechanics of Materials with Initial (Residual) Stresses Acting Along Cracks
	1.3 A Brief Review of the Results of Investigating Problems of Fracture Mechanics of Materials Loaded Along Cracks
		1.3.1 On the Results of Investigating Problems of Fracture Mechanics of Materials Compressed Along Cracks
		1.3.2 The Results of Investigating Problems of Fracture Mechanics of Materials with Initial (Residual) Stresses Acting Along Cracks
	References
2 Basic Principles of Fracture Mechanics of Materials Loaded Along Cracks
	2.1 Introductory Notes
		2.1.1 Classes of the Problems Considered
		2.1.2 Principles of Constructing Linearized Relations and Basic Assumptions of the Theory Developed
		2.1.3 Methods Used to Describe the Deformed State and Coordinate Systems
	2.2 Basic Relationships of the Three-Dimensional Linearized Mechanics of Deformable Bodies
		2.2.1 Linearized Geometric and Force-Based Relationships
		2.2.2 Linearized Constitutive Stress-Strain Relationships
		2.2.3 General Formulation of Linearized Problems in Displacements
	2.3 Basic Relationships and General Solutions of Linearized Problems for Uniform Initial States
		2.3.1 Basic Relationships for the Uniform Initial State
		2.3.2 General Formulations of Fracture Mechanics Problems on Materials Loaded Along Cracks
	2.4 Criteria of the Fracture of Materials Loaded Along Cracks
		2.4.1 Description of Fracture Mechanisms Under Compression Along Parallel Cracks
		2.4.2 The Criterion of Material Fracture Under Compression Along Parallel Cracks
		2.4.3 The Criteria of the Brittle Fracture of Materials with Initial (Residual) Stresses
	References
3 Two-Dimensional Problems on the Fracture of Bodies Under Compression Along Cracks
	3.1 Representation of the General Solutions of Two-Dimensional Problems in Terms of Harmonic Potentials
		3.1.1 General Solutions for Compressible Bodies
		3.1.2 General Solutions for Incompressible Bodies
	3.2 Complex Potentials for Plane Linearized Problems
		3.2.1 Complex Potentials for Compressible Bodies
		3.2.2 Complex Potentials for Incompressible Bodies
	3.3 Uniaxial Compression of a Body Along an Array of Coplanar Cracks
		3.3.1 Problem Formulation
		3.3.2 Solving Problems in the Case of Equal Roots
		3.3.3 Solving Problems in the Case of Non-equal Roots
		3.3.4 Critical Compression Parameters for Different Models of Materials
	3.4 Compression of a Body Along a Near-Surface Crack (Near-Surface Delamination)
		3.4.1 Problem Formulation
		3.4.2 Dual Integral Equations
		3.4.3 Resolving Integral Equations
		3.4.4 Critical Compression Parameters
	3.5 Compression of Bodies Along Systems of Parallel Interacting Cracks
		3.5.1 Problems on a Body with Two Parallel Cracks
		3.5.2 Problems on a Body with a Periodic Array of Parallel Cracks
		3.5.3 Critical Compression Parameters for Different Models of Materials
	References
4 Three-Dimensional Problems on Loading of Bodies Along Cracks
	4.1 Representations of the General Solutions of Spatial Linearized Problems in Terms of Potential Harmonic Functions and the General Methodology of Their Investigation
		4.1.1 General Solutions for Equal Roots. Compressible and Incompressible Bodies
		4.1.2 General Solutions for Non-equal Roots. Compressible and Incompressible Bodies
		4.1.3 Examples of Calculating Parameters Involved in Representations of General Solutions in the Natural State Coordinates
		4.1.4 General Solutions of Linearized Problems in the Initial State Coordinates
		4.1.5 General Procedure of Investigating Spatial Problems on the Bodies Loaded Along Cracks
	4.2 Biaxial Uniform Body Compression Along Cracks Located in a Single Plane
		4.2.1 The Problem on the Fracture of a Body Compressed Along an Array of Coplanar Cracks
		4.2.2 Problem on Body Compression Along an Internal Penny-Shaped Crack
		4.2.3 Critical Compression Parameters for Different Models of Materials
	4.3 Biaxial Uniform Compression of Semi-infinite Bodies Along Near-Surface Penny-Shaped Cracks
		4.3.1 Axisymmetric Problem
		4.3.2 Non-axisymmetric Problem
		4.3.3 Critical Parameters Under Body Compression Along a Near-Surface Crack (Near-Surface Delamination Buckling) for Various Material Models
	4.4 Loading of Bodies by Forces Directed Along Systems of Parallel Interacting Cracks
		4.4.1 Problems on a Body with Two Parallel Coaxial Penny-Shaped Cracks
		4.4.2 Problems on a Body with a Periodic Array of Parallel Coaxial Penny-Shaped Cracks
		4.4.3 Critical Compression Parameters for Different Models of Materials
	References
5 Evaluating the Applicability Limits of the Beam Approximation in the Analysis of Problems on the Compression of Bodies Along Cracks
	5.1 Analysis of the Stress-Strain State Near Crack Tips Under Compression
	5.2 Quantitative Evaluations of the Compressive Strength of Bodies with Interacting Cracks by Using the Rigorous Approach and the Beam Approximation
		5.2.1 Highly Elastic Bodies
		5.2.2 Composites
	5.3 Brittle Fracture of Bodies with Closely Spaced Parallel Cracks Under Compression
		5.3.1 General Issues
		5.3.2 Procedure of the Investigation of Resolving Fredholm Integral Equations
	5.4 Evaluations of the Applicability Limits of Beam Approximation for Different Design Diagrams
		5.4.1 Half-Space with a Near-Surface Crack
		5.4.2 Body with Two Parallel Cracks
		5.4.3 Body with a Periodic Array of Parallel Cracks
	References