Advancing a generalized method for solving problems of continuum mechanics as applied to the Cartesian coordinate system

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

Determining the patterns of asymmetric interaction of plastic medium with counter-directional metal flow

“…Works [26,27,29,30] are presented as fundamental for solving plane problems of the theory of elasticity using the method of argument functions. As a continuation of studies [28][29][30], the theory was refined and the applied problem of the action of a transverse force at the end of a console with a rigid embedment was solved [31]. Some difficulties arose in satisfying the stress boundary conditions.…”

“…In accordance with [31,34], for linear partial differential equations we use a fundamental substitution of the form:…”

“…A generalized method for solving problems in continuum mechanics was repeatedly confirmed by publications in highly rated journals in various fields. Including the theory of elasticity [29][30][31], plasticity [15], and the theory of dynamic processes [28]. There are publications in such journals on geomechanics [39].…”

Analytical modelling and design of linear controlled dynamic systems

“…Continuous improvements in the fields of mining technologies [1][2][3][4], transportation and hoisting [5][6][7][8], technological processes [9][10][11][12], and dynamics of technical systems [13][14][15][16] facilitate wider and more thorough development of analytical and computational simulation methods of processes and phenomena occurring within the systems.…”