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

Abstract

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

Solving applied problems of elasticity theory in geomechanics using the method of argument functions of a complex variable

“…However, the problem, due to tangential stresses, does not meet the boundary conditions (1). Let's continue the discussion and apply a more general approach that echoes work [28]. To achieve the result, we shall apply formulas (2), at f(φ)=f(ρ)=0.…”

“…This approach is used in [27] when solving the dynamic problem of elasticity theory. The same approach is used in solving the problems of elasticity theory in Cartesian [28] and polar coordinates [29]. With the help of this approach, the closed problem of the theory of plasticity [30] was solved, which enhances the reliability of the method and its use in the problems of continuum mechanics.…”

“…There is a known solution to the flat problem of continuum mechanics using the complex variable function argument method [27,28]: Fig. 1.…”

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

“…One of the first works in this area was a publication on the theory of plasticity [1], which set out the basic approaches to solving the problem. In the future, there were developments using a combination of methods of argument functions and functions of a complex variable [2], invariant differential generalizations in the polar coordinate system [3]. It should be emphasized that the result obtained by simplification in those works is correlated with the results of the solutions by other authors.…”

“…Paper [3] proposes a more complex solution for polar coordinates. The analysis reveals that such generalizations can also take place in Cartesian coordinates.…”