Eight Non-Classical Problems of Fracture Mechanics (Advanced Structured Materials, 159)

Foreword
Preface
Introduction
	References
New Mechanical Effects Detected by Prof. Aleksander N. Guz and His Collaborators for the First Time
Contents
About the Author
Part I General Problems
1 Division into Classical and Non-classical Problems of Fracture Mechanics
	1.1 Classical Problems of Fracture Mechanics
	1.2 Non-classical Problems of Fracture Mechanics
	1.3 Eight Non-classical Problems of Fracture Mechanics
	1.4 Additional Discussion of Non-classical Problems of Fracture Mechanics
		1.4.1 Brief Discussion of Models and Approaches in Non-classical Problems of Fracture Mechanics (Problems 1–8, Sect. 1.3)
		1.4.2 On Consideration of Non-classical Problems of Fracture Mechanics from the Point of View of Classical Problems of Fracture Mechanics
		1.4.3 On Some Other Publications
	References
2 Brief Statement of Foundations of Three-Dimensional Linearized Theory of the Deformable Bodies Stability (TLTDBS)
	2.1 On the Formation of TLTDBS
	2.2 Classification of Approaches (Variants of Theory) in the TLTDBS
		2.2.1 Theory of Large (Finite) Subcritical Deformations
		2.2.2 The First Variant of Theory of Small Subcritical Deformations
		2.2.3 The Second Variant of Theory of Small Subcritical Deformations
		2.2.4 On the Linearized Theory of Stability at Small Deformations and Small Averaged Angles of Rotation
		2.2.5 On the Theory of Incremental Deformations
		2.2.6 Approximate Approach in the ILIDBS
		2.2.7 Notes
	2.3 On Stability Criteria in the TLTDBS
		2.3.1 Elastic Bodies
		2.3.2 Plastic Bodies
		2.3.3 Bodies with Rheological Properties
	2.4 General Questions of the TLTDBS
		2.4.1 General Formulation of the TLTDBS Problems for Various Models of Deformable Bodies
		2.4.2 Sufficient Conditions for Applicability of the Euler’s Method (Statical Method)
		2.4.3 Sufficient Conditions of Stability
	2.5 On Variational Principles of the TLTDBS for Elastic and Plastic Bodies
		2.5.1 Variational Principle of Hu–Vashizu Type in the TLTDBS for Incompressible Bodies with “Dead” External Loads. Unified General Form for Theories 2.1, 2.2, and 2.3
		2.5.2 Variational Principle of the TLTDBS for Compressible Bodies Under “Following” Load. Results for Theory 2.3
	2.6 General Solutions of the TLTDBS Under Homogeneous Subcritical States
		2.6.1 General Solutions of the TLTDBS for Compressible Bodies
		2.6.2 The General Solutions of the TLTDBS for Incompressible Bodies
		2.6.3 The Complex Potentials in Plane Problems of the TLTDBS. Preliminary Discussion
		2.6.4 Basic Relations and General Solutions of the TLTDBS in Coordinates of Initial State
		2.6.5 Complex Potentials in Plane Linearized Problems in Coordinates of Initial State
		2.6.6 Complex Potentials in Dynamical Plane Linearized Problems in Coordinates of Initial State for Moving Cracks and Loads
	References
Part II Fracture in Composite Materials Under Compression
3 Problem 1. Fracture in Composite Materials Under Compression Along the Reinforcing Elements
	3.1 General Concept and Key Directions of Research
		3.1.1 General Concept
		3.1.2 The First Direction (Very Approximate Approaches)
		3.1.3 The Second Direction (Strict Sequential Approaches Based on the TLTDBS)
	3.2 Analysis of Experimental Results on Compression of Composites
		3.2.1 Experimental Results on the Loss of Stability in the Internal Structure of Composites Under Compression
		3.2.2 Experimental Results on Fracture of Composites Under Compression Along the Reinforcing Elements
		3.2.3 About the Study of the Phenomenon of “Kinking”
	3.3 Main Results of the Second Direction (Strict Sequential Approaches Based on the TLTDBS)
		3.3.1 Introductory Information
		3.3.2 Continuum Theory of Fracture
		3.3.3 Layered Composites: Model of a Piece-Wise Homogeneous Medium
		3.3.4 Fibrous Unidirectional Composites: Model of Piece-Wise Homogeneous Medium
	3.4 Conclusion
	References
4 Problem 2. Model of Short Fibers in Theory of Stability and Fracture Mechanics of Composite Materials Under Compression
	4.1 Experimental Results on Loss of Stability in the Internal Structure of Composites Under Compression: Case of Short Fibers
	4.2 Statement of Problems
	4.3 Classification of Design Schemes. About Analogies
		4.3.1 Model of Infinitely Long Fibers and Layers in the First Direction of Research
		4.3.2 Model of Infinitely Long Fibers and Layers in the Second Direction of Research
		4.3.3 Model of Short Fibers and Layers in the Framework of the Second Direction of Research
	4.4 Statement of Plane Problems of Brittle Fracture Mechanics of Composites with Short Reinforcing Elements Under Compression
		4.4.1 On Statement of Problems
		4.4.2 On the Method of Numerical Study of Problems of Sect. 4.4
	4.5 Results of Studies of Plane Problems of Brittle Fracture Mechanics of Composites with Short Fibers Under Compression
		4.5.1 Asymptotic Transition to the Model of “Infinitely Long Fibers”
		4.5.2 Results for Single Fiber Under Compression Along Fibers
		4.5.3 Results for Two Sequentially Located Fibers Under Compression Along the Fibers
		4.5.4 Results for Two Parallel Fibers Under Compression Along the Fibers
		4.5.5 Results for One Periodic Row of Sequentially Located Fibers Under Compression Along the Fibers
		4.5.6 Results for One Periodic Row of Parallel Fibers Under Compression Along the Fibers
		4.5.7 Results for Single Fiber Located Close to Surface Under Compression Along the Fiber (Analysis of Near-the-Surface Instability)
	4.6 Conclusion
	References
5 Problem 3. End-Crush Fracture of Composite Materials Under Compression
	5.1 Introduction
	5.2 Experimental Researches
	5.3 Theoretical Researches
		5.3.1 General Concept
		5.3.2 On Studies Within the Model of Piecewise Homogeneous Medium
		5.3.3 On Studies Within the Continuum Medium Model (Continuum Approximation)
	References
Part III Other Non-Classical Problems of Fracture Mechanics
6 Problem 4. Brittle Fracture of Materials with Cracks Taking into Account the Action of Initial (Residual) Stresses Along Cracks
	6.1 Introduction
	6.2 Preliminary Discussion. Statement of Problems
	6.3 Plane and Anti-plane Statical Problems. Criteria of Fracture
		6.3.1 Order of Singularity
		6.3.2 Effects of Resonant Character
		6.3.3 Criteria of Fracture
	6.4 Spatial Statical Problems
		6.4.1 On Statement of Spatial Statical Problems in Mechanics of Brittle Fracture of Materials with Initial (Residual) Stresses Acting Along Cracks
		6.4.2 To the Method of Research of Spatial Statical Problems
		6.4.3 Concrete Results (Using Both Exact Solutions and Computer Methods) Obtained for the Following 11 Design Schemes
		6.4.4 On Phenomena of Resonant Character for Spatial Statical Problems of Non-classical Problem 4 of Fracture Mechanics
	6.5 On Dynamical Plane and Anti-plane Problems in Mechanics of Brittle Fracture of Materials with Initial (Residual) Stresses Along Cracks
	6.6 Repetition of Results
	6.7 On Increasing the Objectivity of Citation
	References
7 Problem 5. Brittle Fracture in the Form of Separation into the Slender Parts of Composite Materials Under Tension or Compression Along Reinforcing Elements
	7.1 Introduction
	7.2 Experimental Researches
	7.3 Explanation of Mechanism of Fracture in the Form of “Separation into the Slender Parts”
	7.4 On Development of Fundamentals of Mechanics of Composites with Curved Structures
		7.4.1 Introduction
		7.4.2 Continuum Theories and Results Based on Them
		7.4.3 Model of a Piece-Wise Homogeneous Medium and Results Based on It
	References
8 Problem 6. Fracture Under Compression Along Parallel Cracks
	8.1 Introduction
	8.2 General Statement of Problems: General Concept and Basic Approaches
		8.2.1 General Statement of Problems
		8.2.2 General Concept
		8.2.3 General Approaches
	8.3 Results for Homogeneous Materials with Cracks Under Brittle and Plastic Fracture: The Second General Approach
		8.3.1 Results for Brittle and Plastic Fracture of Homogeneous Materials with Cracks Located in the Same Plane. The Second General Approach. Exact Solutions
		8.3.2 Results for Brittle and Plastic Fracture of Homogeneous Materials with Cracks Located in Parallel Planes: The Second General Approach
		8.3.3 Results for Brittle and Plastic Fracture of Homogeneous Materials with Cracks Located in Parallel Planes. Second General Approach: The Combined Approach for Problems 4 and 6
	8.4 Results for Layered Composites with Cracks at Interface Under Brittle and Plastic Fracture. The Second General Approach
		8.4.1 Introduction
		8.4.2 Results for Brittle and Plastic Fracture of Layered Composites with Microcracks at Interfaces: The Second General Approach
		8.4.3 Results for the Brittle Fracture of Layered Composites with Macrocracks at the Interfaces: The Second General Approach
	8.5 Results for Brittle Fracture of Homogeneous Materials with Cracks Located in the Close Arranged Parallel Planes. Passage to the Limit. The Second General Approach
		8.5.1 Short Description of Developed Research Method
		8.5.2 Near-the-Surface Crack
	8.6 On Results for Viscoelastic Fracture
	References
9 Problem 7. Brittle Fracture of Materials with Cracks Under Action of Dynamic Loads (with Allowance for Contact Interaction of the Crack Edges)
	9.1 Introduction
	9.2 Substantiation of Statement of Problems. Method of Solving
		9.2.1 Substantiation of the Discussed Problem Statement
		9.2.2 On Research Method
	9.3 Concrete Results
		9.3.1 Two-Dimensional Problems
		9.3.2 Three-Dimensional (Spatial) Problems
	References
10 Problem 8. Fracture of Thin-Wall Bodies with Cracks Under Tension in the Case of Preliminary Loss of Stability
	10.1 Introduction
	10.2 Statement of Problems
	10.3 Research Methods and Results Obtained
	References
General Conclusion to the Monograph (Parts I, II, III)
Bibliography