Design of composite structures with extremal elastic properties in the presence of technological constraints J. Awrejcewicz a,⇑ , S.P. Pavlov b , K.S. Bodyagina b , M.V. Zhigalov b , V.A. Krysko b a Lodz University of Technology, Department of Automation, Biomechanics and Mechatronics, 1/15 Stefanowskiego Str., 90-924 Lódz ´ and Warsaw University of Technology, Department of Vehicles, 84 Narbutta Str., 02-524 Warsaw, Poland b Saratov State Technical University, Department of Mathematics and Modelling, Politehnicheskaya 77, 410054 Saratov, Russian Federation article info Article history: Received 22 February 2017 Revised 31 March 2017 Accepted 6 April 2017 Available online 21 April 2017 Keywords: Composites Technological restrictions Homogenization Topology optimization Extreme elastic properties Finite element method abstract In this paper, composites made of periodically repeating micro structures are investigated. The study aims at identifying the optimal spatial distribution of constituents within a composite material to obtain the material of desired/improved functional properties. To find the relationship between micro- and macro-structural properties of the composite material, the method of homogenization is used. The prob- lem of finding optimal microstructures of various materials, with the aim of obtaining maximum rigidity, i.e., maximum volume and shear modules for the base cell of a composite that contains the original instal- lation of technological holes and/or inclusions was first investigated. For illustration and validation of the proposed approach, numerical examples are provided. Ó 2017 Elsevier Ltd. All rights reserved. 1. Introduction It is well known that mechanical properties of composite mate- rials can be improved by modifying the topology of the material microstructure. For instance, an approach based on the structural topology optimization can be employed to find the best space dis- tribution of material phases constituting the composite microstructure. The main idea of designing a microstructure of a composite material having periodic patterns/cells is based on find- ing optimal distribution of periodic stress-strain fields already on a micro-scale, for a periodic elementary cell, called a base cell. A base cell can be studied with the help of the finite element method, and then a procedure of the topological optimization of this elemen- tary, periodically repeated cell can be investigated instead of studying the whole composite structure. Usually, the method of homogenization is applied in order to average the complex micro structural behavior of an elastic medium to determine the macro- scopic properties of a unit cell. The theory of homogenization has been recognized as a rigorous modeling methodology for charac- terizing the mechanical behavior of cellular materials and compos- ites with periodic microstructures [1–3]. However, for complex microstructures of the elastic medium, analytical determination of the stress/strain fields is extremely difficult. Therefore, to find the most effective properties of the elastic medium, the homoge- nization procedure is employed by means of numerical approaches like the finite element method (FEM) [4,5]. The inverse problem is to design a new microstructure of the periodic unit cell so that the resulting material has desirable phys- ical properties. The material design concept based on topology optimization and homogenization has been applied to design elas- tic [6–13] and thermo elastic [14,15] composite materials. A sys- tematic and scientific means of microstructural design is formulated as an optimization problem for the parameters that represent the material properties and topology of the material microstructure. Over the last two decades, various topology optimization algo- rithms and interpolation schemes, e.g. solid isotropic material with penalization (SIMP) [16–19], evolutionary structural optimization (ESO) [20], and level set technique [21,22] have been developed. These topology optimization techniques have been used exten- sively to solve design problems not only for macroscopic struc- tures, but also for microstructures of materials/composites in recent years. Some attempts have been also made to design new materials with extraordinary physical properties, e.g., extreme thermal con- ductivity [23] and maximum stiffness and thermal conductivity http://dx.doi.org/10.1016/j.compstruct.2017.04.008 0263-8223/Ó 2017 Elsevier Ltd. All rights reserved. ⇑ Corresponding author. E-mail addresses: jan.awrejcewicz@p.lodz.pl (J. Awrejcewicz), pspsar@yandex.ru (S.P. Pavlov), bodksen@mail.ru (K.S. Bodyagina), zhigalovm@ya.ru (M.V. Zhigalov), tak@san.ru (V.A. Krysko). Composite Structures 174 (2017) 19–25 Contents lists available at ScienceDirect Composite Structures journal homepage: www.elsevier.com/locate/compstruct