Abstract
Relevance. The progressive development of views on the Saint-Venant formulated principles and methods underlying the deformable body mechanics, the growth of the mathematical analysis branch, which is used for calculation and accumulation of rules of thumb obtained by the mathematical results interpretation, lead to the fact that the existing principles are being replaced with new, more general ones, their number is decreasing, and this field is brought into an increasingly closer relationship with other branches of science and technology. Most differential equations of mechanics have solutions where there are gaps, quick transitions, inhomogeneities or other irregularities arising out of an approximate description. On the other hand, it is necessary to construct equation solutions with preservation of the order of the differential equation in conjunction with satisfying all the boundary conditions. Thus, the following aims of the work were determined: 1) to complete the familiar Saint-Venants principle for the case of displacements specified on a small area; 2) to generalize the semi-inverse Saint-Venants method by finding the complement to the classical local rapidly decaying solutions; 3) to construct on the basis of the semi-inverse method a modernized method, which completes the solutions obtained by the classical semi-inverse method by rapidly varying decaying solutions, and to rationalize asymptotic convergence of the solutions and clarify the classical theory for a better understanding of the classic theory itself. To achieve these goals, we used such methods , as: 1) strict mathematical separation of decaying and non-decaying components of the solution out of the plane elasticity equations by the methods of complex variable theory function; 2) construction of the asymptotic solution without any hypotheses and satisfaction of all boundary conditions; 3) evaluation of convergence. Results. A generalized formulation of the Saint-Venants principle is proposed for the displacements specified on a small area of a body. A method of constructing asymptotic analytical solutions of the elasticity theory equations is found, which allows to satisfy all boundary conditions.
Highlights
The main factor for refusing a researcher in recognition of his works becomes fairly meaningful and reasonable lack of trust for something overly unconventional and provocative
The progressive development of views on the Saint-Venant formulated principles and methods underlying the deformable body mechanics, the growth of the mathematical analysis branch, which is used for calculation and accumulation of rules of thumb obtained by the mathematical results interpretation, lead to the fact that the existing principles are being replaced with new, more general ones, their number is decreasing, and this field is brought into an increasingly closer relationship with other branches of science and technology
Progressive development of views on the principles underlying deformable body mechanics, growth of the mathematical analysis field, which is used for computation and accumulation of rules of thumb, obtained by interpreting mathematical results, lead to Зверяев Евгений Михайлович, доктор технических наук, профессор, ведущий научный сотрудник Института прикладной математики имени М.В
Summary
The main factor for refusing a researcher in recognition of his works becomes fairly meaningful and reasonable lack of trust for something overly unconventional and provocative. The problem of applying semi-inverse Saint-Venant’s method to mechanics of composite materials was brought to the forefront Such problems require reconsideration of the accumulated practice and its generalization in order to obtain new possibilities of expanded application of classic ideas to new problems and materials on the basis of extended and generalized formulations. Gregory and Wan [5] applied a general method developed by them for obtaining the proper boundary condition series for arbitrarily defined allowed boundary conditions (without an explicit solution of internal or boundary layer) for a number of special cases of general interest, including cases with defined boundary displacements Their overall results demonstrate that in order to be strictly correct, the Saint-Venant’s principle can be used only to the leading terms of the external solution, i.e. classical plate theory. The solution comes down to finding Airy’s stress function φ x, z , which satisfies biharmonic equation
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callMore From: Structural Mechanics of Engineering Constructions and Buildings
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
