Solution of a Problem Linear Plane Elasticity with Mixed Boundary Conditions by the Method of Boundary Integrals

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

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“…The statement of the flat problem of the theory of elasticity is well known in the literature [4,[16][17][18]. In this case, we obtain:…”

Section: The Study Materials and Methodsmentioning

confidence: 91%

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

Chigirinsky

Naumenko

2021

EEJET

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“…Their knowledge allows one to determine unknown functions. It was shown in [31] how boundary conditions are formed based on certain approaches (the collocation method). The formation of boundary conditions in the method of argument functions makes it possible to define one of the basic trigonometric functions.…”

Section: The Methods Of Argument Functions Of a Complex Variablementioning

confidence: 99%

“…, . Similar approaches in determining normal shear stresses at a shift of the trigonometric function: Signs in (31), (32) in front of square brackets mean that the derivation was performed at different signs in the Cauchy-Riemann relations (12).…”

Section: The Methods Of Argument Functions Of a Complex Variablementioning

confidence: 99%

Invariant differential generalizations in problems of the elasticity theory as applied to polar coordinates

Chigirinsky¹,

Naumenko²

2020

EEJET

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The method of argument functions has become famous for solving problems of continuum mechanics. The solution of problems of the elasticity theory in polar coordinates was the further development of this method. The same approaches are applied to solving problems of the theory of plasticity, the theory of elasticity, and the theory of dynamic processes. If regularities of the solution are determined correctly, then they should be continued in other fields including the problems of the theory of elasticity in polar coordinates. The proposed approach features finding not the solution itself but the conditions for its existence. These conditions may include differential or integral relations which make it possible to close the solution in a general form. This becomes possible when additional functions are introduced into consideration or the argument functions of coordinates of the deformation zone. Basic dependences that satisfy the boundary or edge conditions as well as the functions that simplify the solution of the problem in general should be the carriers of the proposed argument functions. For various reasons, two basic dependences were used in the solution: trigonometric and exponential. Their arguments are two unknown argument functions. In the process of transformations, a mathematical connection was established between them in a form of the Cauchy-Riemann relations which had a stable tendency to be repeated in problems of the continuum mechanics. From these positions, the flat problem was solved in the most detailed way, tested, and compared with the studies of other authors. By reducing the solution to a particular result, a way to classical solutions was found which confirms its reliability. The result obtained is useful and important since it becomes possible to solve an extensive class of axisymmetric applied problems using the method of argument functions of a complex variable

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“…Work [27] demonstrates how the boundary conditions of the problem are mathematically justified based on the collocation method. Their determination is a problem in continuum mechanics.…”

Section: Literature Review and Problem Statementmentioning

confidence: 99%

“…There are greater opportunities to satisfy more complex boundary conditions. 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].…”

Section: Literature Review and Problem Statementmentioning

confidence: 99%

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

Chigirinsky,

Naizabekov,

Lezhnev

et al. 2024

EEJET

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The plane problem of rolling theory is analytically solved using the method argument of functions of a complex variable. The solution to the plane problem has been strengthened from the point of view of the asymmetry of the process, which made it possible to consider the applied problem as the interaction of differently directed zones in the deformation zone. The interaction of lagging and advancing zones is represented as a combination of multidirectional processes in a single deformation zone. With a change in kinematic, power characteristics in local zones, the process parameters change in the entire deformation zone. Stressed states of intermediate loading schemes between stable and unstable rolling are considered. A feature of the interaction of zones with the opposite flow of metal is the analogy with the action of back tension on the deformation zone in literally all parameters - this is the presence of tensile stresses in the lagging zone, a decrease in local specific pressures, a shift in maximum normal stresses towards the exit from the rolls, a change in the length of the advance zone, reduction in rolling force. The studies confirm and repeat the generally accepted provisions of the theory of rolling but reveal the effects of changes in the stress state under different loading models. The results of the work make it possible to determine the modes of rolling processes visually and computationally under conditions of strong and weak interaction of zones with an oppositely directed metal flow. The effects of plastic deformation with a decrease in the total effort in the processes that are within the reach of the limiting focus of crimping under conditions of increasing kinematic load when the gripping angles vary between 0,077…0,168 are given

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Solution of a Problem Linear Plane Elasticity with Mixed Boundary Conditions by the Method of Boundary Integrals

Cited by 3 publications

References 30 publications

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

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

Invariant differential generalizations in problems of the elasticity theory as applied to polar coordinates

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

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