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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4. Conclusions Constructing a mathematical model for structures made of fiber-composite materials requires taking into account a number of conditions and restrictions on solutions. This includes possible delamination of the reinforcing fiber from the surrounding matrix, interfacial interaction in the fiber-matrix system, as well as the interaction between contacting fibers. An effective method for constructing model problems is modern mathematical methods in the calculus of variations, such as variational inequalities. For the problem under consideration, the variational formulation allows using a finite element apparatus to find a numerical solution that satisfies all specified conditions. In this study, we have constructed a mathematical model of deformation of a two-dimensional elastic body containing delaminated thin inclusions, one of which is elastic and the other semirigid. The model contains a differential formulation with inequality-type conditions on the boundary, as well as an equivalent variational statement. The problem posed is nonlinear. For the numerical solution of the variational inequality, the domain decomposition method is applied, which allows considering a family of linear problems whose solutions converge to the solution of the original nonlinear problem. For the linear problem of a thin semirigid inclusion in one of the subdomains, an algorithm is constructed that allows finding a solution that satisfies all restrictions on solutions in this subdomain. For the plane-stressed problem of junction of a thin semirigid inclusion and a Timoshenko inclusion in an elastic body, the indicated algorithm is numerically implemented and a solution is found with all the stated conditions of the problem. References Gaudiello, A., Monneau, R., Mossino, J., Murat, F., Sili, A., 2002. On the Junction of Elastic Plates and Beams. C. R. Acad. Sci. Paris, Ser. I 335 (8), 717–722. Itou, H., Khludnev, A. M., 2016. On Delaminated Thin Timoshenko Inclusions Inside Elastic Bodies. Math. Methods Appl. Sci. 39:17, 4980– 4993. Kazarinov, N. A., Rudoy, E. M., Slesarenko, V. Yu., Shcherbakov, V. V., 2018. Mathematical and Numerical Simulation of Equilibrium of an Elastic Body Reinforced by a Thin Elastic Inclusion. Comput. Math. Math. Phys. 58:5, 761–774. Khludnev, A.M., 2010. Elasticity Problems in Non-Smooth Domains. Fizmatlit, Moscow, pp. 252 (in Russian). Khludnev, A. M., Leugering, G. R., 2014. Delaminated Thin Elastic Inclusion Inside Elastic Bodies. Math. Mech. Complex Systems 2:1, 1–21. Khludnev, A., Popova, T., 2016. On the Hierarchy of Thin Delaminated Inclusions in Elastic Bodies. Mathematical notes of NEFU 23(1), 87-107. Khludnev, A.M., Popova, T., 2016. Junction Problem for Rigid and Semi-Rigid Inclusions in Elastic Bodies. Arch. Appl. Mech. 86 (9), 1565 1577. Khludnev, A.M., Popova, T.S., 2017. Timoshenko Inclusions in Elastic Bodies Crossing an External Boundary at Zero Angle. Acta Mechanica Solida Sinica 30(3), 327-333. Khludnev, A.M., Popova, T.S., 2017. On the Mechanical Interplay Between Timoshenko and Semirigid Inclusions Embedded in Elastic Bodies. Zeitschrift fur Angewandte Mathematik und Mechanik 97(11), 1406-1417. Khludnev, A., Popova, T., 2019. Semirigid Inclusions in Elastic Bodies: Mechanical Interplay and Optimal Control. Computers and Mathematics with Applications 77(1), 253-262. Lazarev, N. P., 2017. The Derivative of the Energy Functional in an Equilibrium Problem for a Timoshenko Plate with a Crack on the Boundary of an Elastic Inclusion. J. Appl. Industr. Math. 11:2, 252–262. Le Dret, H. 1989. Modeling of the Junction Between Two Rods. J. Math. Pures Appl. 68, 365-397. Neustroeva, N.V., 2008. Contact Problem for Elastic Bodies of Different Dimensions. Vestn. NSU. Ser. Mat., Mech., Inform. 8(4), 60–7. Neustroeva, N. V., 2009. A Rigid Inclusion in the Contact Problems for Elastic Plates. Sib. J. Industr. Math. 12:4, 92–105. Neustroeva, N.V., Lasarev, N.P., 2016. Junction Problem for Euler-Bernoulli and Timoshenko Elastic Beams. Siberian Electronic Mathematical Reports 13, 26-37 (in Russian). Pasternak, I. M., 2012. Plane Problem of Elasticity Theory for Anisotropic Bodies with Thin Elastic Inclusions. J. Math. Sci. 186:1, 31–47. Popova, T.S., 2018. Problems of Thin Inclusions in a Two-Dimensional Viscoelastic Body. J. Appl. Ind. Math. 12, 313-324. Popova, T.S., 2016. A Contact Problem for a Viscoelastic Plate and an Elastic Beam. J. of Appl. and Ind. Math. 10(3), 404–416. Popova, T., Rogerson, G.A., 2016. On the Problem of a Thin Rigid Inclusion Embedded in a Maxwell Material. Z. Angew. Math. Phys. 67(4): paper #105. Rudoy, E.M., Lazarev, N.P., 2018. Domain Decomposition Technique for a Model of an Elastic Body Reinforced by a Timoshenko’s Beam. Journal of Computational and Applied Mathematics 334, 18-26. Rudoy, E. M., Shcherbakov, V. V., 2016. Domain Decomposition Method for a Membrane with a Delaminated Thin Rigid Inclusion. Sib. Èlektron. Mat. Izv. 13, 395–410.