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We consider a non-coercive boundary value problem for two elastic Kirchhoff–Love plates connected to each other by a thin junction. The non-coercivity of the problem is due to the Neumann-type conditions imposed at the external boundaries of the plates. A solution existence is proved for suitable given external forces. Passages to limits are justified as a rigidity parameter of the junction tends to infinity and to zero. We prove that the model corresponding to the first limit case describes an equilibrium of elastic plates with a thin rigid junction; the second limit model fits to the equilibrium state of two elastic plates independent of each other. [ABSTRACT FROM AUTHOR]

The paper addresses analysis of equilibrium state of an elastic body with a thin nonhomogeneous inclusion in a non-coercive case. We assume that a part of the inclusion is located outside the elastic body, and the traction free boundary condition at the external boundary of the elastic body does not provide a coercivity of the problem. The inclusion is delaminated from the surrounding elastic material that implies a presence of the interfacial crack. Constraint boundary conditions at the crack faces describe a mutual nonpenetration and have an inequality type form. We prove a solution existence of the equilibrium problem, analyze a passage to the limit as the rigidity parameter of the inclusion tends to zero, and prove a solution existence of an inverse problem. The inverse problem assumes that along with the displacement field we have to find an elasticity coefficient for the inclusion provided that an additional information is given which can be found from a measurement. [ABSTRACT FROM AUTHOR]

An inverse problem for an elastic body with a thin elastic inclusion is investigated. It is assumed that the inclusion crosses the external boundary of the elastic body. A connection between the inclusion and the elastic body is characterized by the damage parameter. We study a dependence of the solutions on the damage parameter. In particular, passages to infinity and to zero of the damage parameter are investigated. Limit models are analyzed. Assuming that the damage and rigidity parameters of the model are unknown, inverse problems are formulated. Sufficient conditions for the inverse problems to have solutions are found. Estimates concerning solutions of the inverse problem are established. [ABSTRACT FROM AUTHOR]

This article discusses the use of non-smooth variational methods in a wide range of applications, particularly in fields that involve partial differential equations. The authors focus on nonlinear problems in solid and fluid mechanics, as well as inverse and optimal control problems. They aim to develop non-standard analysis and effective numerical methods for solving these problems. The article also highlights the contributions of various authors from different countries and discusses the physical applications of non-smooth variational methods in areas such as elastic and inelastic solids, heat transfer, and fracture mechanics. [Extracted from the article]

This paper is devoted to a rigorous analysis of an equilibrium problem for a two-dimensional homogeneous anisotropic elastic body containing a thin isotropic elastic inclusion. The thin inclusion is modelled within the framework of Euler–Bernoulli beam theory. Partial delamination of the inclusion from the elastic body results in the appearance of an interfacial crack. We deal with nonlinear conditions that do not allow the opposing crack faces to penetrate each other. We derive a formula for the first derivative of the energy functional with respect to the regular crack perturbation along the interface, which is related to energy release rates. It is proved that the energy release rates associated with crack translation and self-similar expansion are represented as path-independent integrals along smooth curves surrounding one or both crack tips. The path-independent integrals consist of regular and singular terms and are analogues of the well-known Eshelby–Cherepanov–Rice J-integral and Knowles–Sternberg M-integral. [ABSTRACT FROM AUTHOR]

The paper is concerned with a control of thin rigid inclusion and crack shapes in elastic bodies. It is assumed that rigid inclusions are delaminated; thus, cracks are located on the boundary of inclusions as well as outside of inclusions. We provide the problem formulations and analyze the shape sensitivity with respect to geometrical perturbations in the frame of free boundary models. Inequality type boundary conditions are considered at the crack faces to guarantee a mutual non-penetration between crack faces. Inclusion and crack shapes are considered as control functions. The cost functional, which is based on the Griffith rupture criterion, characterizes the energy release rate and provides the shape sensitivity with respect to a change of the geometry of the structure. We prove an existence of optimal shapes in the problems considered. [ABSTRACT FROM AUTHOR]

Mathematical methods based on the variational approach are successfully used in a broad range of applications, especially those fields that are oriented on partial differential equations. Our problem area addresses a wide class of nonlinear variational problems described by all kinds of static and evolution equations, inverse and ill-posed problems, non-smooth and non-convex optimization, and optimal control including shape and topology optimization. Within these directions, we focus but are not limited to singular and unilaterally constrained problems arising in mechanics and physics, which are governed by complex systems of generalized variational equations and inequalities. Whereas classical mathematical tools are not applicable here, we aim at a non-standard well-posedness analysis, numerical methods, asymptotic and approximation techniques including homogenization, which are successful within the primal as well as the dual variational formalism. In a broad scope, the theme issue objectives are directed toward advances that are attained in the mathematical theory of non-smooth variational problems, its physical consistency, numerical simulation and application to engineering sciences. This article is part of the theme issue 'Non-smooth variational problems and applications'. [ABSTRACT FROM AUTHOR]