Preface ...................................................... V
0 Introduction ................................................. 1
0.1 Numerical and asymptotic procedures in the theory of
heterogeneous materials ................................. 1
0.2 Mathematical standpoint ................................. 3
0.3 Physical statements of the homogenization problem ....... 6
1 Definitions, assumptions and theorems in homogenization
problems ..................................................... 7
1.1 Definitions for homogenization problems in solid of
periodic microctructure ................................. 7
1.2 Cell problems and cell solutions for an elastic solid
of periodic microstructure ............................. 10
1.3 Asymptotic series in homogenization problems of
periodic microstructure ................................ 14
2 Application of cell functions for the calculation of
binary composite elastic moduli ............................. 20
2.1 Laminated composite .................................... 20
2.2 Particulate-filled composite ........................... 26
2.2.1 Structural model ................................ 26
2.2.2 Boundary-value problems and a numerical
technique for their solution .................... 29
2.2.3 Elastic properties of a binary composite of
periodic structure with perfectly bonded
components ...................................... 33
2.2.4 Effect of adhesion on the effective elastic
moduli of a binary composite of periodic
structure ....................................... 44
2.2.5 Analysis of micromechanical field
distributions ................................... 48
3 Asymptotic study of linear vibrations of a stretched beam
with concentrated masses and discrete elastic supports ...... 58
3.1 Statement of the problem ............................... 58
3.2 Asymptotic analysis .................................... 61
3.2.1 Empty frequency domains ......................... 61
3.2.2 Low-frequency region, α=0. Long-wave modes ...... 64
3.2.3 Medium-frequency region, α=2. Tooth-like wave
modes ........................................... 67
3.2.4 High-frequency region, α=2.5. Vibrations of
the beam between immobile heavy masses ......... 71
3.2.5 Conclusions .......................................... 73
4 Reinforced plates ........................................... 76
4.1 Rexural vibrations of a rectangular reinforced plate
on an elastic foundation ............................... 76
4.2 Static problem ......................................... 92
4.3 Flexural vibrations and equilibrium state of circular
plates reinforced by radial ribs ....................... 97
4.4 Geometrically nonlinear flexural vibrations of
rectangular reinforced plates ......................... 106
4.5 Account of ribs torsion rigidity ...................... 112
4.6 Account of ribs eccentricity .......................... 116
4.7 Homogenization for plates with wide ribs .............. 122
5 Problems of elasticity theory for reinforced orthotropic
plates ..................................................... 128
5.1 Reinforced orthotropic strip .......................... 128
5.2 Force transfer to a stringer orthotropic strip via
an elastic element .................................... 146
5.3 Plane vibrations of circular cylindrically
orthotropic plates with radial ribs ................... 151
6 Reinforced shells .......................................... 156
6.1 Stringer cylindrical shells ........................... 156
6.2 Shells of revolution with meridional ribs ............. 166
6.3 Cross-reinforced shells ............................... 174
6.4 Waffle- and ring-reinforced shells .................... 176
6.5 Cylindrical shells rarely reinforced using
stringers ............................................. 178
7 Corrugated plates .......................................... 188
7.1 Model problem ......................................... 190
7.2 Transformation of basic equations ..................... 190
7.3 Axisymmetrical deformation of a circular corrugated
plate ................................................. 194
7.4 Rectangular corrugated plate .......................... 203
7.5 Axisymmetrical vibrations of a circular corrugated
plate ................................................. 209
8 Other periodic structures .................................. 212
8.1 Vibrations of a cylindrical shell with a large
number of apparent masses ............................. 212
8.2 Plates on an elastic foundation with strip-shaped
and support-free par .................................. 216
8.3 Laminated structures .................................. 218
8.4 Multisupported plates ................................. 221
8.5 Plates and shells with a periodic system of hinges .... 225
8.6 Simplified nonlinear equations for smooth plates and
shells ................................................ 228
9 Perforated plates and shells ............................... 233
9.1 Bending of rectangular plates with periodic square
perforations .......................................... 233
9.2 Eigenvalue problem for a perforated plate ............. 241
9.3 Analytical approach for a large hole .................. 242
9.4 Matching of asymptotic solutions by means of two-
point Pade approximants ............................... 246
9.5 The plane theory of elasticity in a perforated
domain ................................................ 248
9.6 Perforated shallow shells ............................. 254
Concluding remarks. Perspectives and open problems ......... 254
References ................................................. 255
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