Abstract
Solving the problem of continuum mechanics has revealed the defining generalizations using the function argument method. The aim of this study was to devise new approaches to solving problems of continuum mechanics using defining generalizations in the Cartesian coordinate system. Additional functions, or the argument of the coordinates function of the deformation site, are introduced into consideration. The carriers of the proposed function arguments should be basic dependences that satisfy the boundary or edge conditions, as well as functions that simplify solving the problem in a general form. However, there are unresolved issues related to how not the solutions themselves should be determined but the conditions for their existence. Such generalized approaches make it possible to predict the result for new applied problems, expand the possibilities of solving them in order to meet a variety of boundary and edge conditions. The proposed approach makes it possible to define a series of function arguments, each of which can be a condition of uniqueness for a specific applied problem. Such generalizations concern determining not the specific functions but the conditions of their existence. From these positions, the flat problem was solved in the most detailed way, was tested, and compared with the studies reported by other authors. Based on the result obtained, a mathematical model of the flat applied problem of the theory of elasticity with complex boundary conditions was built. Expressions that are presented in coordinateless form are convenient for analysis while providing a computationally convenient context. The influence of the beam shape factor on the distribution of stresses in transition zones with different intensity of their attenuation has been shown. By bringing the solution to a particular result, the classical solutions have been obtained, which confirms its reliability. The mathematical substantiation of Saint-Venant's principle has been constructed in relation to the bending of a beam under variable asymmetric loading
Highlights
An option for overcoming such difficulties is the use of a complex variable function argument method, which has demonstrated its capabilities in solving diverse problems of continuum mechanics [15]
The analysis reveals that such generalizations can take place in Cartesian coordinates
One can see that the proposed approach makes it possible to define a series of argument functions, each of which can be a unique condition for a specific applied task
Summary
Underlying the development of the economy of any state is the introduction and use of advanced technical solutions in various areas of production. There are unresolved issues related to how not the solutions themselves should be determined but the conditions for their existence Such generalized approaches make it possible to predict the result for new applied tasks, expand the possibilities of solving them in order to meet a variety of boundary and edge conditions in the problems of continuously changing production. An option for overcoming such difficulties is the use of a complex variable function argument method, which has demonstrated its capabilities in solving diverse problems of continuum mechanics [15] Those general regularities that have been identified make it possible to pose and solve new problems in the theory of elasticity: for example, the study of the stressed state in Cartesian coordinates using argument functions for more complex models of theoretical and applied problems
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