Asymptotical Mechanics of Composites (eBook)

Igor V. Andrianov was born in the Ukraine (former Soviet Union (fSU)) in 1948. He obtained his Master of Applied Mechanics degree from Dnepropetrovsk State University (DSU, Ukraine) in 1971, his PhD in Structural Mechanics from the DSU in 1975, and his DSc in Mechanics of Solids from the Moscow State Institute of Electronics and Mathematics in 1990. From 1974 to 1977, he was a Research Scientist at the DSU, from 1977 to 1990, an Associate Professor, and from 1990 to 1997, a Full Professor of Mathematics at Dnepropetrovsk Civil Engineering Institute. From 2006 to 2013 he was a Research Scientist at RWTH Aachen University (Germany). Andrianov is the author or co-author of 14 books and more than 300 papers in peer-reviewed journals. He has presented papers at more than 150 international congresses, conferences, and seminars, and also supervised 21 PhD students. Selected as a Soros Professor (1996). His research interests include Mechanics of Solids, Mechanics of Composite Materials, Nonlinear Dynamics, and Asymptotic Methods.Jan Awrejcewicz, Head of the Department of Automation, Biomechanics and Mechatronics, Head of Ph.D. School on 'Mechanics' (since 1996) and of graduate/postgraduate programs on Mechatronics (since 2006). Recipient of multi-doctor honoris causa and the Humboldt Award. Author/co-author of over 730 journal papers and refereed international conference papers and 48 monographs. Editor of 22 books and 26 journal special issues. Principal investigator in 39 research grants by Polish Ministry of Education and Science (21), Committee for Scientific Research (17), National Science Centre (2). Member of Editorial Boards of 74 Journals (15 with IF). Organizer and Head of the series of 14 International Conferences Dynamical Systems-Theory and Application (Lodz, 1992-2017) and of 3 International Conferences “Mechatronics: Ideas for Industrial Applications” (Warsaw 2012; Lodz, 2014; Gdansk, 2015), Chair of the "International Conference of the Polish Society of Biomechanics" (Lodz, 2014). Research interest:  His papers and research cover various disciplines of mathematics, mechanics, biomechanics, automatics, physics and computer oriented sciences, with main focus on nonlinear processes.Vladyslav V. Danishevskyy obtained his Masters (1996), Ph.D. (1999), and Doctor of Sciences (2008) degrees in Structural Mechanics from Prydniprovska State Academy of Civil Engineering and Architecture, Ukraine. He is a Full Professor at this Academy. He has authored 3 monographs and over 50 journal papers refereed by SCOPUS. Among his awards are Soros Post-Graduate Student’s Award (1997), Prize of the National Academy of Sciences of Ukraine for the best academic achievement among young scientists (2000), Alexander von Humboldt Foundation Research Fellowship (2001), NATO Research Fellowship (2003), NATO Reintegration Grant (2005), institutional academic co-operation grant of the Alexander von Humboldt Foundation (2007), research grant of the German Research Foundation (2014), Marie-Curie Research Fellowship by the European Union’s Horizon 2020 research and innovation programme (2015). He has conducted long-term research abroad in RWTH Aachen University, Germany (2001–2002), in the University of Rouen, France (2003–2004), and in Keele University, UK (2015– 2017). His research interests cover mechanics of heterogeneous media and structures, asymptotic methods, nonlinear dynamics.

Preface 6
Sec1 9
Contents 10
1 Introduction 13
References 23
2 Models of Composite Materials and Mathematical Methods of Their Investigation 32
2.1 General Relations of the Linear Theory of Elasticity 32
2.2 General Relations of the Linear Theory of Viscoelasticity 34
2.3 General Relations of the Nonlinear Theory of Elasticity 36
2.4 Elementary Provisions of the Percolation Theory 38
2.5 Integral Transforms 40
2.6 The Method of Multiple Scales 43
2.7 Differential Equations with Periodically Discontinuous Coefficients 46
2.8 Homogenization Approach for Differential Equation with Rapidly Changing Coefficients 50
2.9 Homogenization of Periodically Perforated Media. Schwarz Alternating Method 56
2.10 Boundary Perturbation Method 62
2.11 The Papkovich--Fadle Approach 64
2.12 The Padé Approximants 67
2.13 Two-Point Padé Approximants 71
2.14 Method of Asymptotically Equivalent Functions 72
References 76
3 Conductivity of Fibre Composites: Analytical Homogenization Approach 79
3.1 Mathematical Models of Conductivity-Type Transport Phenomena 79
3.2 Effective Coefficient of Conductivity 80
3.3 Local Fields on the Microlevel 89
3.4 Coated Fibres 91
3.5 Imperfect Bonding Between Fibres and Matrix 96
3.6 Random Composites (Security-Spheres Approach) 98
3.7 Cluster Conductivity of the Fibre Composites 101
3.8 Edge Effects 104
References 109
4 Conductivity of Particle-Reinforced Composites: Analytical Homogenization Approach 110
4.1 Effective Coefficient of Conductivity 110
4.2 Local Fields on the Microlevel 115
4.3 Coated Particles 117
4.4 Imperfect Bonding Between Particles and Matrix 121
4.5 Random Composites 123
4.6 Cluster Conductivity of Particle-Reinforced Composites 124
4.7 Using of Asymptotically Equivalent Functions 127
References 130
5 Elastic and Viscoelastic Properties of Fibre- and Particle-Reinforced Composites 131
5.1 Effective Elastic Characteristics of Composites with Circular Cross-Sectional Fibres 131
5.2 Asymptotic Determination of Effective Elastic Properties of Composites with Fibrous Square-Shaped Inclusions 135
5.2.1 Effective Moduli langleE1rangle, langle?12rangle, langle?13rangle,langleG12rangle, langleG13rangle 135
5.2.2 Effective Moduli langleG23rangle, langleE2rangle, langleE3rangle 139
5.3 Analogy Between Effective Elastic and Transport Properties 150
5.4 Edge Effects in Fibre Composites 153
5.5 Longitudinal Tension of Viscoelastic Composites 158
5.6 Longitudinal Shear of Viscoelastic Composites 161
5.7 Longitudinal Shear of Viscoelastic Composites with Diamond-Shaped Cross-Sectional Fibres 163
5.8 Effective Elastic Characteristics of Particle-Reinforced Composites 168
References 172
6 Local Stresses in Elastic Fibrous Composites 174
6.1 Strong Anisotropy Asymptotics in 2D Case 174
6.2 Elastic Problem for Single Fibre in a 2D Matrix 180
6.3 Periodic Systems of Fibres in 2D Case 185
6.4 Various Types of Fibre Composite Fracture for 2D Case 197
6.5 Load Transfer from Fibre to Half-Space with Elastic Coating 206
6.6 Asymptotic Analysis of Thin Interface in Composite Materials ƒ 215
6.7 Strong Anisotropy Asymptotics for Transversally Isotropic Media 223
6.8 Load Transfer from Fibre to Anisotropic Half-Space for Dilute Composites 225
6.9 Load Transfer from Fibre to Anisotropic Half-Space for Non-dilute Composites 229
6.10 Buckling of Fibres in Fibre Composite 236
References 243
7 Asymptotic Analysis of Perforated Membranes, Plates and Shells 249
7.1 Introduction 249
7.2 Torsion of a Rod with Perforated Cross Section and Deflection of the Perforated Membrane 252
7.3 Bending of a Perforated Plate with Small Holes 259
7.4 Error Estimations 268
7.5 Bending of a Perforated Plate with Large Holes 271
7.6 Homogenization of the Perforated Shallow Shell 278
7.7 Solution of the Plane Stress Unit Cell Problems 283
7.8 The Homogenized Equations for the Perforated Shallow Shell 287
References 290
8 Nonlinear Elastic Problems 293
8.1 1D Case. Physical and Geometrical Nonlinearities 293
8.2 Anti-plane Shear Deformation of Fibre Composite and Structural Nonlinearity 300
8.3 Some Remarks Concerning Nonlinear Deformation and Failure of Fibre Composites 310
References 312
9 Conclusion 314
References 317
Appendix A 319
A.1 Solutions for the infinite domains 319
A.2 Bending plate. Formulae for the coefficients of the Eqs. 7.68, 7.71, 7.72, 7.73 321
A.3 Plane problem. Formulae for the coefficients entering Eqs. 7.172, 7.173, 7.180, 7.186 328
References 333