Eight Non-Classical Problems of Fracture Mechanics

Foreword.............................................................................................................................................. 7

Introduction...................................................................................................................................... 11

Part I. G eneral problems

Chapter 1. Divi sion into classical and non-classical problems

of fracture mechanics........................................................................ 15

1.1 Classical problems of fracture mechanics.............................................................................. 15

1.2. Non-classical problems of fracture mechanics..................................................................... 16

1.3. Eight non-classical problems of fracture mechanics........................................................... 17

1.4. Additional discussion of non-classical problems of

fracture mechanics.................................................................................................................... 20

1.4.1. A brief discussion of models and approaches in non-classical

problems of fracture mechanics. Problems 1 to 8 (21). 1.4.2. On

consideration of non-classical problems of fracture mechanics in

terms of classic problems of fracture mechanics (24). 1.4.3. About

some other publications (26).

Chapter 2. Brief statement of foundations of three-dimensional linearized

theory of the deformable bodies stability (TLTDBS).............................. 27

2.1 On the formation of TLTDBS ..................... 27

2.2. Classification of approaches (variants of theory) in TLTDBS ....... 31

2.2.1. Theory of large (fiinite) sub-critical deformations (31).

2.2.2. The first variant of the theory of small sub-critical deformations (33).

2.2.3. The second variant of the theory of small sub-critical deformations (34).

2.2.4. About linearized theory of stability for small deformations

and small averaged angles of rotation (35). 2.2.5. On theory of incremental

deformations (36). 2.2.6. Approximate approach to TLTDBS(37). 2.2.7.

Notes (39).

2.3. On criteria of stability in TLTDBS........................................................................................ 41

2.3.1. Elastic bodies (42). 2.3.2. Plastic Bodies (43) 2.3.3. Bodies with rheological

properties (46).

2.4. General problems of TLTDBS................................................................................................ 49

2.4.1. The general formulation of TLTDBS for different models of

deformed bodies (49). Sufficient conditions for applicability of Euler's method

(statical method) (51). 2.4.3. Sufficient conditions of stability (53).

2.5. On the variational principles of TLTDBS for elastic and plastic bodies......................... 54

2.5.1. Hu-Vashitsu variation principle in TLTDBS for incompressible bodies

under "dead" external load. Unified form for theories 1, 2, and 3 (55).

2.5.2. Variational principle of TLTDBS for compressible bodies under

"following" load. Results for theory 3 (57).

2.6. General solutions for TLTDBS in homogeneous sub-critical conditions........................ 59

2.6.1. General solutions for compressible bodies (60). 2.6.2. General

Solutions of TLTDBS for incompressible bodies (62). 2.6.3. Complex

potentials in the plane problems of TLTDBS. Preliminary discussion (64).

2.6.4. Main relations and general solutions of TLTDBS in coordinates of

initial state (65). 2.6.5. Complex potentials in plane linearized problems

in coordinates of initial state (68). 2.6.6. Complex potentials in dynamic

plane linearized problems in coordinates of initial state for moving cracks

and loads (73).

Part II. Fracture in composite materials under compression

Chapter 3. Problem 1. Fracture in composite materials under compression

along the reinforcing elements.....................79

3.1. General concept and main directions of research................................................................ 79

3.1.1. General concept (79). 3.1.2. The first direction (very approximate

approaches) (82). 3.1.3. The second direction (strict sequential approaches

based on TLTDBS) (84).

3.1.3.1. Internal fracture (loss of stability in the internal structure) (84).

3.1.3.2. Surface fracture (loss of stability in the near-the-surface layers

of composite) (85).

3.2. Analysis of experimental results for compression of composites ......... 87

3.2.1. Experimental results on the loss of stability in the internal structure

of composites under compression (87). 3.2.2. Experimental results on

fracture of composites under compression along the reinforcing elements (90).

3.2.3. About the study of the phenomenon of "kinking" (95).

3.3. Main results of the second direction (strict sequential approaches

on the base of TLTDBS)...................... 97

3.3.1. Introductory information............................................................................................... 97

3.3.2. Continuum theory of fracture....................................................................................... 99

3.3.2.1. Internal fracture (99). 3.3.2.2. Near-the-surface fracture (103).

3.3.3. Layered composites. Model of piece-wise medium............................................... 104

3.3.3.1. Internal fracture (105). 3.3.3.2. Near-the-surface fracture (107).

3.3.3.3. Additional information to the fracture mechanics of layered

composite materials (108).

3.3.3.3.1. Analysis of continuum fracture mechanics of composites

(108). 3.3.3.3.2. On stability of layered composites (112). 3.3.3.3.3.

Conclusions from the sequential analysis of the Daw-Grunfest-Rosen-

Schurtz theory (113).

3.3.4. Fibrous one-directional composites. Model of piece-wise homogeneous

medium......................... 116

3.3.4.1. Internal fracture (117). 3.3.4.2. Near-the-surface fracture (121).

3.3.4.3. On constructing a research method for complex modes of loss of

stability of fibrous unidirectional composites (121).

3.4. Conclusion to chapter 3......................................................................................................... 123

CHAPTER 4. Problem 2. Model of short fibers in the theory of stability and fracture mechanics of composite materials under compression ............ 125

4.1. Experimental results for loss of stability in the internal structure of composites under compression. Case of short fibers..................... 125

4.2. Statement of the problems..................................................................................................... 126

4.3. Classification of design schemes. About Analogies......................................................... 132

4.3.1. Model of infinitely long fibers and layers within the first direction

of research (132). 4.3.2. Model of infinitely long fibers and layers within

the second direction of research (133). 4.3.3. Model of short fibers and

layers within the second direction of research (134).

4.4. Statement of plane problems of mechanics of brittle fracture of

composites with short reinforcing elements under compression.......136

4.4.1. On statement of problems (136). 4.4.2. About the method

of numerical study of problems of section 4.4. (137).

4.5. Results of studies of the plane problems of mechanics of brittle

fracture of composites with short fibers under compression .........138

4.5.1. Asymptotic transition to the model of "infinitely long fibers" (139).

4.5.2. Results for single fiber under compression along fibers (140).

4.5.3. Results for sequentially located two fibers under compression along

the fibers (143).

4.5.4 Results for parallel located two fibers under compression along the fibers (145).

4.5.5. Results for one periodic row of consistently located fibers under

compression along the fibers (146). 4.5.6. Results for one periodic roe of parallel

located fibers under compression along the fibers (149). 4.5.7 Results for

a single fiber located near surface under compression along the fiber

(analysis of the surface instability) (151).

4.6. Conclusion to chapter 4 ........................155

Chapter 5. Problem 3. Fracture in the form of crumpling of ends

under compression of composite materials............. 157

5.1. Introduction.............................................................................................................................. 157

5.2. Experimental researches........................................................................................................ 157

5.3. Theoretical researches............................................................................................................ 159

5.3.1. General concept (159). 5.3.2. Researches within the framework

of the model of piece-wise homogeneous medium (160). 5.3.3. On

researches within the framework of the model of continuum medium

(continuum approach)(164).

Part III. Other nonclassical problems of fracture mechanics

Chapter 6. Problem 4. Brittle fracture of materials with cracks taking

into account the actions of the initial (residual) stresses along the cracks.....168

6.1. Introduction.............................................................................................................................. 168

6.2. Preliminary discussion. Statement of problems................................................................. 170

6.3. Plane and anti-plane statical problems. Criteria of fracture............................................ 173

6.3.1. Order of singularity (173). 6.3.2. Effects of resonant character (174).

6.3.3. Criteria of fracture (175).

6.4. Spatial statical problems........................................................................................................ 177

6.4.1. To statement of spatial static problems of mechanics of brittle fracture

of materials with initial (residual) stresses acting along cracks (177). 6.4.2. To

the method of research of spatial statical problems (178). 6.4.3. Concrete

results (both accurate and using the computer) obtained for 11 design

schemes (178). 6.4.4. On the phenomena of resonance character for spatial

static problems of non-classical Problem 4 of fracture mechanics (179).

6.5. On the dynamical plane and anti-plane problems of mechanics

of brittle fracture of materials with initial (residual) stresses along cracks......179

6.6. Repeating the results............................................................................................................... 180

6.7. On improving the objectivity of citation............................................................................. 184

Chapter 7. Problem 5. Brittle fracture in the form of "brooming" under tension

and compression of composite materials along the reinforcing elements. 186

7.1. Introduction.............................................................................................................................. 186

7.2. Experimental researches........................................................................................................ 187

7.3. Explanation of mechanism of fracture in the form of "brooming"................................ 189

7.4. On development of mechanics of composites with curved structures ...... 192

7.4.1. Introduction (192). 7.4.2. Continuum theories and results on their basis

(193). 7.4.3. Model of piece-wise medium and results on their basis (196).

Chapter 8. Problem 6. Fracture under compression along the parallel cracks 202

8.1. Introduction.............................................................................................................................. 202

8.2. General statement of problems. General concept. General approaches........................ 203

8.2.1. General statement of problems.................................................................................. 203

8.2.2. General concept............................................................................................................ 206

8.2.3. General approaches...................................................................................................... 207

8.2.3.1. First general approach. Beam approximation or beam

approach (207). 8.2.3.2. Second general approach.

Application of TLTDBS (209).

8.3. Results for homogeneous materials with cracks under brittle

and plastic fracture. Second general approach ................ 212

8.3.1. Results for brittle and plastic fracture of homogeneous materials

with cracks, located in one plane. Second general approach. Exact

solutions (212). 8.3.2. Results for brittle and plastic fracture homogeneous

materials with cracks, located in parallel planes. Second general approach (213).

8.3.3. Results for brittle and plastic fracture homogeneous materials with

cracks, located in parallel planes. Second general approach. United approach

for Problems 4 and 6 (215).

8.4. Results for layered composites with microcracks at interface under

brittle and plastic fracture. Second general approach ...............217

8.4.1. Introduction (218). 8.4.2. Results for brittle and plastic fracture of

layered composites with microcracks at interface. Second general approach

(218). 8.4.3. Results for brittle fracture of layered composites with macrocracks

at interface. Second general approach (221).

8.5. Results for brittle fracture of homogeneous materials with cracks

located in the close arranged parallel planes. Passages to the limit.

Second general approach .......................223

8.5.1. Short description of developed method of research (223).

8.5.2. Near-the-surface crack (225).

8.6. On results for viscoelastic fracture....................................................................................... 228

Chapter 9. Problem 7. Brittle fracture of materials with cracks under action

of dynamic loads (with allowance for interaction of crack edges).... 231

9.1. Introduction.............................................................................................................................. 231

9.2. Substantiation of statement of problems. Method of solving.......................................... 232

9.2.1. Substantiation of statement of problems (232).

9.2.2. On method of research (234).

9.3. Concrete results....................................................................................................................... 235

9.3.1. Two-dimensional problems (235).

9.3.2. Three-dimensional (spatial) problems (236).

Chapter 10. Problem 8. Fracture of thin-wall bodies with cracks

under tension in the case of preliminary loss of stability.......... 240

10.1. Introduction............................................................................................................................ 240

10.2. Statement of problems......................................................................................................... 241

10.3. Methods of research and results......................................................................................... 242

General conclusion to the monograph (parts I, II, and III).............................................. 247

References....................................................................................................................................... 248

A.N. Guz. Short biography ........................277