Analysis of the stress field in a wedge using the fast expansions with pointwise determined coefficients

The stress problem for the elastic wedge-shaped cutter of finite dimensions with mixed boundary conditions is considered. The differential problem is reduced to the system of linear algebraic equations by applying twice the fast expansions with respect to the angular and radial coordinate. In order to determine the unknown coefficients of fast expansions, the pointwise method is utilized. The problem solution derived has explicit analytical form and it’s valid for the entire domain including its boundary. The computed profiles of the displacements and stresses in a cross-section of the cutter are provided. The stress field is investigated for various values of opening angle and cusp’s radius.


Introduction
A lot of research is devoted to consideration of elastic infinite wedge-shaped domains. The stress field for the antiplane deformation of elastic wedge is examined in [1][2][3]. The plane deformation of the wedge is discussed in [4][5][6]. Some particular results are obtained by applying Mellin transforms [1,5,6] and using functions of a complex variable [2,3,7]. The exact solutions to the problems of plane deformation of the elastic wedge under zero loading on its lateral edges can be derived using Wiener-Hopf method [4]. The wedge has straight cracks on its axis of symmetry. An intrusion of the wedge into the plastic half-space has been investigated in the monograph [8]. The loaded wedge with smooth edges was considered in [9]. The particular solutions for the truncated circular sector are given in [10]. Some studies are devoted to three-dimensional problems for an elastic wedge [11,12]. In [11] the explicit matrix algorithm for three-dimensional wedge problem solution is proposed. In [12] one of the wedge's surfaces is reinforced with a coating of Winkler type. On the other surface some arbitrary boundary conditions are set. In this case, the methods of nonlinear boundary integral equations and of successive approximations are used.
In this work the new analytical method of fast expansions [13] will be applied. It allows to obtain the high accuracy solution to the problem under study in an explicit analytical form. The method of fast expansions is applicable for solution to the problems associated with partial differential [14], integro-differential [13] and ordinary differential [15] equations.

Materials and methods
The equilibrium equations in terms of displacements in cylindrical coordinates for the wedge-shaped 2 1234567890 ''"" The displacements U and V are defined at the rear edge r R = : The rest edges are loaded by external forces Relying on a physical meaning of the problem, the stresses and strains should be bounded throughout the wedge-shaped domain Ω (otherwise the cutter would be broken). For this reason, the additional conditions must be imposed The formulation of the elastic problem (1)-(5) is given in a classical form, but it is not complete. The requirement to the functions U and V of being smooth and bounded is essential, since it imposes specific restrictions on the boundary conditions formulation. For the case of mixed boundary conditions (2)-(4), the relations (5) lead to the necessity of the "consistency" conditions formulation. If these conditions aren't satisfied, a discontinuity will occur at the angular points ( ) , that is, the conditions (5) will be broken. Hence, the accuracy of the solution will be essentially affected.
The boundary conditions (2) and (4) The functions ( ) ( ) where the constants * * are assumed to be unknown in advance and they are to be determined throughout the solution process. Further, the boundary functions of sixth order [13] will be used to determine the displacements U and V , so the functions up to sixth order derivatives as it is required by the condition (5). Assuming fixed condition at the cutter's rear edge r R = , the relations (7) reduce Thus, under the "fixed boundary" we will imply the equality of the displacement components to zero only for 0 ε θ θ ε ≤ ≤ − , and inside the ε − neighborhoods of the angular points the displacements are determined by some additional relations satisfying the conditions (5) and (8).
Assuming the smallness of ε − neighborhoods the concrete relations for are not given. This is sufficient to derive the solution to the problem in a finite form.
At the cusp of the cutter at points ( ) ( ) 0 0 0 ,0 , , r r θ some "consistency" conditions are also to be satisfied. In order to derive these conditions we will make use of the following considerations. For the boundary conditions defined in (2)-(4) the unique solution to the boundary problem exists. However, discontinuities of the stresses might occur at the angular points if this circumstance isn't envisaged in advance when the boundary conditions are being specified. That is, the stresses will depend on the direction of approach to the angular points for arbitrarily chosen functions ( ) These conditions impose the restrictions on a choice of the function ( ) The remaining two conditions are derived by formally substituting the radial loading ( ) where the constants 11 12 , F F * * are assumed to be determined throughout the solution process.
The lower edge ( 0 θ = ) is assumed to be zero loaded, that is Under the conditions (9)-(11), the loadings ( ) ( ) The constant coefficients 01 02 04 , , F Φ Φ in relations (11), (12) depend on pressing force applied to the sample being processed and they are determined experimentally.
From the previous considerations, it follows that instead of relations (4), the consistent boundary conditions should be used where the function  1  2  1  2  11  12 , , , , , to be determined throughout the solution process.
According with the method of fast expansions [13] the displacements ( ) depending only on the radial coordinate r . Following the same approach the unknown functions (18) can also be represented in a fast expansion form: , 5040 720 2160 15120 Substituting the relations (16), (17) As the resultant expressions are quite cumbersome, they are not given here and further referred to as double fast expansion form of the original problem.
In order to determine the unknown constant coefficients (23), the pointwise method, which was developed and tested in [15] will be employed. Following this technique it is necessary to discretize the wedge-shaped domain Ω in a uniform grid with ( )( )  All in all, we derive the consistent system of ( )( ) from the consistency conditions. This system then has been solved numerically in Maple software.

Results and discussion
As an example the numerical results for solution to the boundary value problem (1)    It is apparent that the accuracy achieved (refer to figures 1, 2) is acceptable for the most technical purposes. It should be noted that the similar high accuracy approximate solutions were accomplished in [13][14][15] where the method of fast expansions has also been involved.
The computed profiles of the displacements ( ) , U r θ , ( ) , V r θ and stress components r σ , θ σ , rθ σ in a cross-section Ω of the cutter are shown in figure 3 and figure 4, respectively.