1,720,982 research outputs found
Existence theorems in the linear theory of micropolar shells
No full textTheorems regarding existence and uniqueness of weak solutions to mixed boundary value problems in the linear theory of micropolar shells in statics and dynamics are proved. Convergence of FEM for the static mixed problems is established. Eigenvalue problems for micropolar shells are studied and properties of the spectrum and eigenmodes are formulated
Existence of weak solutions in elasticity
No full textSolvability and uniqueness of solutions to the problems of equilibrium, vibration and dynamics in a weak setup for classical and nonclassical models of linear elasticity are established in a unified framework sufficiently flexible to accommodate new elastic models
Mathematical study of boundary-value problems within the framework of Steigmann–Ogden model of surface elasticity
No full textMathematical questions pertaining to linear problems of equilibrium dynamics and vibrations of elastic bodies with surface stresses are studied. We extend our earlier results on existence of weak solutions within the Gurtin–Murdoch model to the Steigmann–Ogden model of surface elasticity using techniques from the theory of Sobolev’s spaces and methods of functional analysis. The Steigmann–Ogden model accounts for the bending stiffness of the surface film; it is a generalization of the Gurtin–Murdoch model. Weak setups of the problems, based on variational principles formulated, are employed. Some uniqueness-existence theorems for weak solutions of static and dynamic problems are proved in energy spaces via functional analytic methods. On the boundary surface, solutions to the problems under consideration are smoother than those for the corresponding problems of classical linear elasticity and those described by the Gurtin–Murdoch model. The weak setups of eigenvalue problems for elastic bodies with surface stresses are based on the Rayleigh and Courant variational principles. For the problems based on the Steigmann–Ogden model, certain spectral properties are established. In particular, bounds are placed on the eigenfrequencies of an elastic body with surface stresses; these demonstrate the increase in the body rigidity and the eigenfrequencies compared with the situation where the surface stresses are neglected
How did generalized solutions arise in solid mechanics?
Full text linkIn recent decades, engineers and physicists have shown an increasing interest in functional analysis and its applications. As many of these practitioners lack special training in mathematics, they sometimes run into trouble when trying to use the tools of this powerful branch of knowledge. Our purpose is to outline the connection between the traditional ideas of mechanics and the newer mathematical concepts of generalized solution and distribution.Fil: Lebedev, L. P.. Universidad Nacional de Salta. Facultad de Ingeniería; ArgentinaFil: Grossi, Ricardo Oscar. Universidad Nacional de Salta. Facultad de Ingeniería; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Salta; Argentin
Advanced engineering analysis: The calculus of variations and functional analysis with applications in mechanics
No full textAdvanced Engineering Analysis is a textbook on modern engineering analysis, covering the calculus of variations, functional analysis, and control theory, as well as applications of these disciplines to mechanics. The book offers a brief and concise, yet complete explanation of essential theory and applications. It contains exercises with hints and solutions, ideal for self-study
Acceleration waves in the nonlinear micromorphic continuum
No full textWithin the framework of the nonlinear elastic theory of micromorphic continua we derive the conditions for propagation of acceleration waves. An acceleration wave, also called a wave of weak discontinuity of order two, can be treated as a propagating nonmaterial surface across which the second derivatives of the placement vector and micro-distortion tensor may undergo jump discontinuities. Here we obtain the acoustic tensor for the micromorphic medium and formulate the conditions for existence of acceleration waves. As examples we consider these conditions for the linear micromorphic medium and for the relaxed micromorphic model
On the existence of solution in the linear elasticity with surface stresses
No full textThe mathematical investigation of the initial-boundary and boundary value problems in the linear elasticity considering surface stresses is presented. Weak setup of the problems based on mechanical variational principles is studied. Theorems of uniqueness and existence of the weak solution in energy spaces of static and dynamic problems are formulated and proved. Some properties of the spectrum of the problems under consideration are established. The studies are performed applying the functional analysis techniques. Finally, the Rayleigh principle for eigenfrequencies is constructed
On weak solutions of boundary value problems within the surface elasticity of Nth order
No full textA study of existence and uniqueness of weak solutions to boundary value problems describing an elastic body with weakly nonlocal surface elasticity is presented. The chosen model incorporates the surface strain energy as a quadratic function of the surface strain tensor and the surface deformation gradients up to Nth order. The virtual work principle, extended for higher-order strain gradient media, serves as a basis for defining the weak solution. In order to characterize the smoothness of such solutions, certain energy functional spaces of Sobolev type are introduced. Compared with the solutions obtained in classical linear elasticity, weak solutions for solids with surface stresses are smoother on the boundary; more precisely, a weak solution belongs to (Formula presented.) where (Formula presented.) and (Formula presented.)
On solvability of initial boundary-value problems of micropolar elastic shells with rigid inclusions
No full textThe problem of dynamics of a linear micropolar shell with a finite set of rigid inclusions is considered. The equations of motion consist of the system of partial differential equations (PDEs) describing small deformations of an elastic shell and ordinary differential equations (ODEs) describing the motions of inclusions. Few types of the contact of the shell with inclusions are considered. The weak setup of the problem is formulated and studied. It is proved a theorem of existence and uniqueness of a weak solution for the problem under consideration
- …
