Igor V. Andrianov

Asymptotic Homogenization of Composite Materials and Structures

The present paper provides details on the new trends in application of asymptotic homogenization techniques to the analysis of composite materials and thin-walled composite structures and their effective properties. The problems under consideration are important from both fundamental and applied points of view. We review a state-of-the-art in asymptotic homogenization of composites by presenting the variety of existing methods, by pointing out their advantages and shortcomings, and by discussing their applications. In addition to the review of existing results, some new original approaches are also introduced. In particular, we analyze a possibility of analytical solution of the unit cell problems obtained as a result of the homogenization procedure. Asymptotic homogenization of 3D thin-walled composite reinforced structures is considered, and the general homogenization model for a composite shell is introduced. In particular, analytical formulas for the effective stiffness moduli of wafer-reinforced shell and sandwich composite shell with a honeycomb filler are presented. We also consider random composites; use of two-point Pade approximants and asymptotically equivalent functions; correlation between conductivity and elastic properties of composites; and strength, damage, and boundary effects in composites. This article is based on a review of 205 references. DOI: 10.1115/1.3090830

Higher order asymptotic homogenization and wave propagation in periodic composite materials

We present an application of the higher order asymptotic homogenization method (AHM) to the study of wave dispersion in periodic composite materials. When the wavelength of a travelling signal becomes comparable with the size of heterogeneities, successive reflections and refractions of the waves at the component interfaces lead to the formation of a complicated sequence of the pass and stop frequency bands. Application of the AHM provides a long-wave approximation valid in the low-frequency range. Solution for the high frequencies is obtained on the basis of the Floquet–Bloch approach by expanding spatially varying properties of a composite medium in a Fourier series and representing unknown displacement fields by infinite plane-wave expansions. Steady-state elastic longitudinal waves in a composite rod (one-dimensional problem allowing the exact analytical solution) and transverse anti-plane shear waves in a fibre-reinforced composite with a square lattice of cylindrical inclusions (two-dimensional problem) are considered. The dispersion curves are obtained, the pass and stop frequency bands are identified.

Improved Continuous Models for Discrete Media

The paper focuses on continuous models derived from a discrete microstructure. Various continualization procedures that take into account the nonlocal interaction between variables of the discrete media are analysed.

Asymptotical Mechanics of Thin-Walled Structures

1 Asymptotic Approximations.- 2 Regular Perturbations of Parameters.- 3 Singular Perturbation Problems.- 4 Boundary Value Problems of Isotropic Cylindrical Shells.- 5 Boundary Value Problems - Orthotropic Shells.- 6 Composite Boundary Value Problems - Isotropic Shells.- 7 Composite Boundary Value Problems - Orthotropic Shells.- 8 Averaging.- 9 Continualization.- 10 Homogenization.- 11 Intermediate Asymptotics - Dynamical Edge Effect Method.- 12 Localization.- 13 Improvement of Perturbation Series.- 14 Matching of Limiting Asymptotic Expansions.- 15 Complex Variables in Nonlinear Dynamics.- 16 Other Asymptotical Approaches.- Afterword.- References.

Mechanics of periodically heterogeneous structures

0 Introduction.- 1 Definitions, assumptions and theorems in homogenization problems.- 2 Application of cell functions for the calculation of binary composite elastic moduli.- 3 Asymptotic study of linear vibrations of a stretched beam with concentrated masses and discrete elastic supports.- 4 Reinforced plates.- 5 Problems of elasticity theory for reinforced orthotropic plates.- 6 Reinforced shells.- 7 Corrugated plates.- 8 Other periodic structures.- 9 Perforated plates and shells.- Concluding remarks. Perspectives and open problems.- References.

Perforated plates and shells

We consider the biharmonic equation

Homogenization procedure and Pade approximations in the theory of composite materials with parallelepiped inclusions

D{\nabla ^4}W = P(X,Y)

A Little History

(9.1) in the domain G with a large number of square holes which are arranged in a periodic manner (Fig. 9.1).