Модели и некоторые свойства треугольных решеток Коссера с хиральной микроструктурой

Abstract

Structural, discrete and continual models of the Cosserat triangular lattice with a chiral microstructure are elaborated. The structural model is constructed on the base of particles of finite size with complex spring bonds. With symmetric diagonal bonds, one has the usual Cosserat triangular lattice, while with different diagonal bonds the Cosserat lattice with a chiral microstructure is realized. The choice of different variants enables one to compare the lattices with the ordinary and chiral microstructures and to select the specific properties caused by the chirality. For this purpose, the problems for the hexagonal cell are solved. In the first problem, the behaviour of the cell under a uniform compression (tension) of the cell is investigated. The ordinary lattice is compressed (elongated) in the direction of the force action. The particles of the chiral cell deviate from the direction of the force action during their displacements. The direction of the deviation is different for compressive (tensile) forces, i. e. the chiral lattice responds qualitatively differently to compression and tension. In the second problem, the top and bottom layers of the cell are rigidly compressed (stretched). The middle layer of the ordinary lattice is then stretched (compressed). This is a typical reaction of ordinary materials to tension (compression). The particles displace along the layer. In the chiral lattice, the middle layer is compressed under compression and is stretched under tension of the cell, i. e. chiral cell possesses the auxeticity property. In this case, the middle layer is inclined. The direction of inclination is different for compression or tension. Thus, the chirality effect is manifested. On the basis of the discrete model, equations of the continuum micropolar model are obtained. For symmetric connections, one has equations of the ordinary micropolar theory. The chiral lattice enables one to obtain equations of the chiral micropolar theory and to select the terms of the equations provided by chirality. Using the continuum model, one can find the values of the parameters, for which the second-order derivatives disappear in the rotational equations for the triangular lattice. Thus, it is possible to select a special case of parameters, for which the lattice is described by the equations of the reduced Cosserat theory.