Evaluations of the solution to the homogeneous plane problem of the theory of elasticity in the neighborhood of an irregular boundary point

The most complex stress-strain state (SSS) occurs in the domain of stress concentration due to the shape of the boundary ("the geometric factor") and the finite discontinuities of the specified forced deformations, the mechanical properties, emerging at the irregular point of the boundary of the domain. In this paper, there is a review of the methods for analyzing the particular qualities of the solution to the problem of the theory of elasticity, due to the shape of the boundary or "the geometric factor". The features of the stress-strain state of constructions and structures, possessing "constructive heterogeneity" under the action of discontinuous forced deformations, are stress concentrators, determined on the polymer models of the photoelastic method. For interpreting and decoding of experimentally obtained local SSS in stress concentration domains of structures there are given estimations of the solution to homogeneous plane problem of the elasticity theory in neighborhood of an irregular boundary point.


Introduction
The stress-strain state (SSS) of complex structures is characterized by a significant concentration of stresses in the domains of conjugation of elements with different options for the constructive design of the boundary: singular lines, incoming angle, etc.
The research of the stress state of complex structures in the domains of conjugation of elements made from materials with different mechanical properties under the action of forced deformations, that are discontinuous along the contact line (surface) of the elements, is an actual problem in the practice of engineering design. The research of the stress-strain state of complex structures in zones of constructive heterogeneity is focused on the consideration of the problem of elasticity theory in the neighborhood of the irregular point of the domain boundary, which includes the discontinuity (jump) of forced deformations. The fundamental work of V.A. Kondratiev [1] proves that the solution to the general elliptic boundary value problem in the neighborhood of irregular points of the domain boundary is represented in the form of an asymptotic expansion and an infinitely differentiable function. The terms of this expansion contain solutions to homogeneous boundary value problems for model domains: a wedge or a cone. They depend on the local characteristics -the magnitude of the solid or flat angle and the type of boundary conditions, the mechanical characteristics for piecewise homogeneous bodies. The magnitudes of the coefficients to the solution expansion in the neighborhood of a singular point are unknown and depend on the problem. The methods for determining these coefficients of decomposition are complex and difficult to be realized in the practical determination of stresses of complex structures having a complex shape of the boundary.
In the scientific papers of I.T. Denisyuk is given the asymptotics of the elastic solution to a plane complex domain with angular points on the separation lines [2]. The work of V.D. Kuliev [3] shows the advantage of a fundamental research of the singular solution to the boundary value problem in the possibility of applying its results to the investigation of problems on residual stresses and crack propagation across the connection boundary of dissimilar materials. It is shown that the order of the stress singularity depends on the approach degree of the crack tip to the connection domain of materials with different Young's modulus.
Mostly referred works of M.L. Williams show that stresses, deformations, the Erie stress function near the vertex of the sector with rectilinear sides have got a polynomial form [4]. The works of A.I. Kalandii, K.S. Chobanjan, L.A. Bagirov, O.K. Aksentjan, G.P. Cherepanov, V.P. Netrebko and many others represent the solution to a homogeneous boundary value problem in a neighborhood of an irregular boundary point in a polynomial form.

2.3.
In [6,7,8], similarity theory is used to research the solution to an elastic problem in displacements in the neighborhood of an irregular point on a singular boundary line of the domain.
We consider a small neighborhood of an irregular point on a singular line -the line of discontinuity, for example, boundary conditions or the first derivatives of the surface function of the domain. In a small neighborhood of the irregular point of the surface boundary, there is applied the following similarity group: 1 1 1 ; ; ; 0 , = = = > x tx y ty z z t when t is the group parameter. Writing the Lame equation in a small neighborhood of a point on a singular line and passing to the limit for → +∞ t , the solution to the elastic problem comes to solving two homogeneous plane problems, as in case 2.2: a plane deformation and an antiplane deformation.
In [6], the concept of a canonical singular problem characterizing the singularity of the SSS in the neighborhood of an irregular boundary is defined, for which the following two theorems are valid: 1) Any canonical singular problem corresponds to a transcendental equation, to each root of which there is corresponded a homogeneous solution, the number of arbitrary real constants in this solution is equal to the multiplicity of the root.
2) In an infinitesimal neighborhood of a singular point the solution to the correct boundary value problem of the theory of elasticity behaves as an asymptotically largest in absolute value eigenfunction of the corresponding canonical singular problem.

Statement of the problem and estimations of the solution
We consider the solution to the problem of elasticity theory for a homogeneous or piecewise homogeneous body in the neighborhood of an irregular point of the boundary of a plane domain into which a finite discontinuity (jump) of the forced deformations is included.
According to the theoretical analysis [8], the solution to the problem of elasticity theory in the neighborhood of an irregular point of the boundary of a plane domain can be written in the following form: The parameter 1 = t c is great enough and parameter c is small enough so as not to take into account the stresses of the common field σ S ij determined by the given forced deformations. The stress function of a homogeneous plane elastic problem in a polar coordinate system is written in the following form: 0 1 ( ) We apply the binomial expansion in the following form: The stress function (3) takes into account the expansion (4) and is rewritten in the following way:   Therefore, with the necessary accuracy for the stress function, it suffices to take the first two (three) terms of the expansion series for n = 0, 1 (2), i.e.