Point-like mechanical singularity in the method of boundary states

This paper presents an effective technique for solving some spatial problems in the theory of elasticity, particularly for a bounded set of point-like physical singularities. This technique development involves the advanced energy method of boundary states (MBS) for a class of solutions with singularities. A test problem for a single singularity such as the center of expansion complying with the proposed technique calculations reliability has been solved and analyzed. The use of the method of boundary states demonstrates the efficiency of the technique in solving problems with many centers of expansion and concentrated forces in an unbounded medium. The results of problem solving are displayed in a convenient graphical form.


Introduction
The theory of elasticity gives special attention to the problems, which solutions satisfy all the defining relations but normalize displacements or stresses to infinitely large values in the singular points. The singularities of the concentrated force and the center of expansion (and contraction) need special consideration.
In the classical theory of elasticity, there is a concept that characterizes the combination of three double forces with zero moments acting along the coordinate axes and characterized by P value. Usually, this feature is called the center of contraction (figure 1a), and P with the opposite sign is called the center of expansion (figure 1b). The corresponding point location can be in a cavity inside the body [1,2].
The concept of the concentrated force is an idealization of a force applied to a point in space, by virtue of the fact that a point is defined as a dimensionless and infinitely small unit of space. This concept is often used to solve various problems in continuum mechanics. The singular point on the boundary, to which P force is applied, can be represented as a small and limited surface on which the surface force is distributed (figure 1c). Its resultant corresponds to the concentrated force. In the theory of elasticity, some works were devoted to the plane problems in rigid deformable body mechanics complicated by the presence of concentrated forces. [3,4]. Several works dedicated to the method of boundary states (MBS) development considered the stress-strain state problems in concentrated force presence [5,6].
The other works solved special problems in the spatial theory of elasticity in an unbounded elastic medium for force point-like singularities [3,7,8]. The work [7] solves the problem of concentrated force acting on a spherical body in the three-dimensional case.
Within the framework of the MBS, the concentrated force [9,10] and the center of expansion were considered by the works of [11]. The singularities were taken into account in the classical (regular) way, with the help of special solutions directly included in the initial basis of internal and boundary states, but small neighborhoods of singular force action were excluded from the elastic body. The approach proposed below avoids artificial complications in describing the shape of the body and seeks solutions not in the class of regular functions but of singular ones. This technique is based on the method of boundary states.

Method of boundary states
The fundamental concept of MBS is the condition of the medium, which is a particular solution to the defining equations of the medium, regardless of the conditions determined at the boundary of a body [12]. Defining relations in the mathematical model of a homogeneous elastostatic body are presented in tensor-index notation (a point in the index means differentiation, repetition of indexes means summation) and are enclosed in Cauchy relations generalized Hooke's law 2 equilibrium equation where  -Poisson's ratio, i Bcomponent of an arbitrary harmonic vector. General solutions (4) are an effective tool for building the basis of a state space representation for a body with no singular factors [12].
The concept of the medium condition is transformed into the notions of internal  and boundary  conditions, if it is the case of a specific body A set of all possible conditions   forms isomorphic Hilbert spaces of internal Ξ and boundary Γ conditions with scalar products, which are equal due to the principle of virtual displacements After orthogonalization, the attributes of resulting internal and boundary conditions are presented in Fourier series by elements of orthonormal bases Thus, the solution to the primal problems for any linear media and bodies of arbitrary outlines is reduced to elementary calculation of quadratures. The convergence of a series of factors k c was proven [13].  Let the linear functional be responsible for estimating the value of

Stating problems for a body with a mechanical singularity
Then from its application to the resulting field (6) we get For the given boundary conditions and the reference impact in the localization of the singularity, the values of the functionals on the right side of the equation are calculated, and on the left side they must be equal to p . Then it follows from the equation that By setting 0 p parameter value in the  special solution, we provide the required value in p resulting state.

Test problem for a single mechanical singularity
The MBS approach testing for solving problems complicated by the presence of a physical singularity such as the center of expansion was performed within the first main problem (according to the classification of N.I. Muskhelishvili [3]) for a homogeneous elastostatic medium enclosed inside a hemisphere of  The obtained solution reliability's main criterion is the quadratic residual between the boundary conditions and the on-surface solution results. Figure 5 presents the root-mean-square residual (horizontal axis) dependence on the number of used basis elements (vertical axis) detailed analysis in the form of a bar chart with grouping; the parentheses indicate the maximum degree for independent harmonic polynomials used linearly. To conclude the approach efficiency, 426 orthonormal elements were used in isomorphic spaces of Ξ internal and Γ boundary states (eleventh order of polynomials).

Problems of interaction for concentrated forces in unbounded medium
The first two main problems of the theory of elasticity for a homogeneous elastostatic medium enclosed inside a parallelepiped in equilibrium were solved using the MBS in a non-dimensionalized formulation, within:  The obtained solution responsible for the stress-strain state has a numerical-analytical form. We do not consider it here due to the visual immensity of the internal state components. For brevity, figure 7 shows the display of the saturation of stress fields in the form of isolines for the first problem in the section of 2 z = , and figure 8-9 does the same for the second problem in the section of 1/ 2 z = section. In the figures, a darker layer corresponds to more compression. The figures show the values of two adjacent layers, and we can easily calculate the remaining values in this step. The components of the stress tensors within the region have finite values. When approaching the coordinates of the "source points", they increase and highlight the stress concentration zone, and when moving away from the coordinates, they decrease.

Problem of interaction of three centers of expansion
The first main problem of the theory of elasticity for a homogeneous elastostatic medium enclosed inside a cube in equilibrium was solved using the MBS in a non-dimensionalized formulation, within:  The conclusions on isolines for this problem solution coincide with those of the solved above problem of interaction for concentrated forces in an unbounded medium. The quadratic residual of the boundary conditions with the on-surface results at 102 n = was 0.705.

Conclusions
The main conclusions on the work done are as follows: 1. A technique of numerical and analytical construction of the stress-strain state for spatial problems of solid mechanics, including a finite set of point-like physical singularities, was developed based on the method of boundary states.
2. The proposed technique has shown its effectiveness for solving specific problems. 3. By means of the MBS, the following problems were solved: a test problem for a single mechanical singularity, a problem of interaction for concentrated forces in an unbounded medium in two versions, and a problem of interaction of three centers of expansion.