Eight Non-Classical Problems of Fracture Mechanics

ISBN-13: 9783030775032 / Angielski / Miękka / 2022

Aleksander N. Guz

Opis

This book presents an analysis of eight non-classical problems of fracture and failure mechanics mainly obtained by research in the department of dynamics and stability of continuum of the S. P. Timoshenko Institute of Mechanics of the National Academy of Sciences of Ukraine (NAS of Ukraine). It focusses on the application of the 3D (three-dimensional) theories of stability, dynamics, and statics of solid mechanics to the investigation of non-classical problems of fracture and failure mechanics.  

Spis treści

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

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

 

Part I. General problems

Chapter 1. Division 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

O autorze

A.N.Guz is Director of the S. P. Timoshenko Institute of Mechanics of the NASU, Ukraine. His principal scientific results have been obtained in mechanics of deformable solids and related problems of continuum mechanics: the three-dimensional theory of stability of deformable bodies, the theory of propagation and diffraction of elastic waves in multi-connected bodies and bodies with initial stresses, stress concentration around holes in shells, mechanics of composites materials and structural members utilizing them, aerohydroelasticity, non-classical problems of fracture mechanics, rock mechanics, dynamics of viscous compressible liquid, contact problems, and mechanics of nanocomposites and non-destructive methods of stress determination.