PSI - Issue 30

Tatiana S. Popova et al. / Procedia Structural Integrity 30 (2020) 113–119 Tatiana S. Popova / Structural Integrity Procedia 00 (2020) 000–000

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1. Introduction One of the methods of mathematical modeling of fibrous composites is the application of the theory of thin beams to describe the behavior of inclusions of various nature in an elastic matrix. For example, problems on thin inclusions of various types, including Bernoulli-Euler, Timoshenko, as well as completely rigid thin inclusions are studied by Neustroeva (2009), Pasternak (2012), Khludnev and Leugering (2014), Itou and Khludnev (2016), Rudoy and Shcherbakov (2016), Lazarev (2017), Khludnev and Popova (2017, 2017), Kazarinov et al. (2018), Popova and Rogerson (2016) and others. Mathematical modeling contains a number of difficulties. Experiments show that the reinforcing fiber does not always have perfect adhesion to the surrounding matrix, see Fig. 1. In the presence of delamination of the inclusion from the elastic matrix, a crack appears, on the faces of which, as on a part of the boundary, it is necessary to set conditions. In this case, the zone of fiber delamination from the matrix may be unknown in advance, therefore, when formulating the corresponding problems, it is necessary to correctly provide the conditions in the zone of possible contact or separation. The condition that makes it possible to exclude mutual penetration of opposite crack edges has the form of an inequality and leads to the formulation of the problem in the form of a variational inequality determined by Khludnev, 2010. We also note that the problems of delaminating thin inclusions in a two-dimensional body are related to studies of the contact of bodies of different dimensions, for example, shown by Gaudiello (2002), Neustroeva (2008), Popova (2016). Also, the correct description requires the interaction of several thin inclusions with each other. These studies lead to junction problems such as studied by Neustroeva and Lazarev (2016), junction conditions for rods found by Le Dret (1989).

Fig.1. Interphase boundaries of polymer composite materials based on polytetraftoroethylene and carbon fibers (scanning by electron microscopy)

There are a number of works in recent years by Khludnev and Popova (2016, 2016, 2019) in which problems on thin semirigid inclusions are studied. A semirigid inclusion is characterized by the presence of anisotropic properties: in the direction of one of the coordinate axes, the displacements satisfy the equations of an elastic beam, and in the other direction the displacements are rigid, i.e. the corresponding functions have a predefined structure. The paper Khludnev and Popova (2017) investigates the problem of the equilibrium of a two-dimensional elastic body containing two thin inclusions: one of them is elastic and is modeled within the framework of the Timoshenko beam theory, the other thin inclusion is semirigid. The equations of elasticity are written for horizontal displacements, and vertical displacements are rigid. The formulation of the equilibrium problem for elastic body with two thin inclusions in variational form is given, and an equivalent formulation in the form of a boundary value problem is obtained, and the unique solvability of the problem is proved. A complete system of junction conditions is obtained, which are satisfied at the contact point of two inclusions and have a clear physical interpretation.