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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]

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]

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'.