Modelling and solution of contact problem for infinite plate and cross-shaped embedment.

UDC 539.3:517.518.12+624.044

Introduction. Development of efficient methods of determination of an intense-strained state of thin-walled constructional designs with inclusions, reinforcements and other stress raisers is an important problem both with theoretical, and from the practical point of view, considering their wide practical application. Plates, reinforced by a different inclusions and ribs, are widely used in practice as components of different constructions. In this study we proposed a method of analytical solution for boundary value problem of stress-strain state of the bending of an infinite plate with a rigid cross-shaped embedment. The exact solution of this boundary value problem is obtained by reduction to the Riemann problem and by its subsequent solution.

The boundary element method is an effective way of solving boundary value problems for systems of differential equations. Methods based on the boundary integral equations, are a powerful tool in many fields of science and technology, including the calculation of plates and shells [1 ... 5].

However, due to the singularity of the fundamental solutions, a problem associated with irregular borders (corners, edges, etc ...) arises. So, the question to use of special techniques for solving the problems with non-smooth boundary is actual.

One of methods for solving numerous classes of integral equations and systems, arising on the basis of the method of boundary integral equations, is an analytical method of reducing these equations and systems to the Riemann problem with their subsequent solution [6.13].

This method was further developed in solving the problem of bending isotropic [6] and orthotropic plates [7, 8] with linear irregularities oriented arbitrarily.

Contact problem of bandpass orthotropic plate Kirchhoff model with a thin semi-infinite rigid reinforcement were studied and solved in [9] by present method, as well as with reinforcement in the form of elastic rib [10].

In [11], an exact solution of the antisymmetric contact problem of bending bandpass orthotropic semi-infinite plate and a rigid support was constructed by reduction to the Riemann problem. The asymptotic behavior of the contact forces at the end of this support has been investigated.

Exact solution of the boundary value problem of bending bandpass shallow shell, which is supported by intermediate thin semi-infinite rib, type Winkler foundation was obtained in [12]; and supported by intermediate thin semi-infinite rigid support, was obtained in [13].

The aim of this research is to develop the analytical mathematical method of studying of an intense-strained state of infinite plate with cross-shaped embedment at a bend. It is also necessary to investigate the asymptotic behavior of the contact forces at the ends of this embedment.

Materials and Methods. We consider the problem of the bending of an infinite plate (-[infinity] < x,y < [infinity]), containing a cross-shaped, thin, absolutely rigid embedment ([absolute value of (x) [less than or equal to] a,x = 0).

The force P applied to the embedment in point x = 0,y = 0 . P is an applied transverse load. The plate is simply supported in 4N points

[M.sub.k] = (bcos([pi]k / (2N) + [pi] / (4N)),bsin([pi]k / (2N) + [pi] / (4N)) = ([x.sub.k], [y.sub.k]), (b > a).

It is necessary to find the deflection of embedment [W.sub.0] and the contact forces of interactions [[psi].sub.1]([xi]),[[psi].sub.2]([xi]) between embedment and plate.

Using the results of [6], we give mathematical formulation of the boundary problem described above. Equation, governing the deflection of mid-surface of plate w(x,y) can be approximated as:

D[[DELTA].sup.2]w(x,y)=[[psi].sub.1](x)[delta](y) + [[psi].sub.2](y)[delta](x) - P/4N [4N.summation over (k=1)] [delta](x - [x.sub.k])[delta](y - [y.sub.k]); (1)

The boundary conditions are the following:

[partial derivative]w(x,y)/[partial derivative][x.sup.i][partial derivative][y.sup.3-i] [right arrow] 0, i = [bar.0,3] as [x.sup.2] + [y.sup.2] [right arrow] [infinity]; (2)

Moreover:

w(x,0) = [W.sub.0], ([absolute value of (x)] [less than or equal to] a); w(0,y) = [W.sub.0], ([absolute value of (y)] [less than or equal to] a;); (3)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (4)

where [delta](x), [delta](y) -- Dirac delta functions.

Using the fundamental solution of the biharmonic equation [OMEGA](x,y) = [(8[pi]).sup.-1] ([x.sup.2] + [y.sup.2])ln[square root of ([x.sup.2] + [y.sup.2])]), we obtain the solution of equation (1), satisfying (2).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (5)

Substituting (5) in (3), we obtain a system of two integral equations for [[psi].sub.1]([xi]) and [[psi].sub.2]([eta]). Posed problem is symmetrical relative to the variables x and y.

Therefore [[psi].sub.1]([xi]) = [[psi].sub.2]([xi]). [[psi].sub.1]([xi])--is even, and eventually we come to an integral equation of the first kind with a smooth kernel:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; (6)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Performing the differentiation (6) three times with respect to x and introducing the notation [tau] = [a.sup.-1][xi], t = [a.sup.-1]x, [[psi].sub.1]([xi]) = [psi]([tau]), we come to a singular equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; (7)

where g(y) = 1/1-y - 1/1+y - 6y/1+[y.sup.2] + 4[y.sup.3]/[(1+[y.sup.2]).sup.2], [f.sub.3](t)=f"'(x)

It is important to note that the solution of equation (7), when substituted into the left side of the equation (6), in general, can give a function that is different from [W.sub.0] + f(x) on an even polynomial of the second order A + B[x.sup.2]. The necessary and sufficient conditions for the equality to zero of this polynomial will be equalities

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (8)

The first is obtained by substituting of x = 0 into (6). The second--by double differentiation of ([d.sup.2] / d[x.sup.2]) (6) with respect to x and subsequent substituting x = 0 into result. To satisfy (8), we should be seeking [psi]([tau]) in such class of functions in which the homogeneous equation, corresponding (7), has two linearly independent solutions. As will show below, it is necessary to search [psi]([tau]) in class of functions with non-integrable singularity at the point [tau] = 1, and the corresponding integrals are understood in regularized sense [14].

We extend the definition of right-hand side of equation (7) as 1 [less than or equal to] t < [infinity], using the unknown function [f.sub.+](t). Introducing the notation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

let rewrite (7) in the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (9)

We applying Mellin transform to the (9)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Then we calculate M[g] = G(p) using the formula 3.241 [15]. As a result, we have the Riemann problem

[[PSI].sup.-](p)G(p) = [F.sup.-](p) + [F.sup.+](p), -[infinity] < Im p < [infinity], (10)

where G(p) = -[pi]tg [pi]p/2 (1 - p - 2/sin [pi]/2), (max([alpha], 0) < Re p = [sigma] < 1).

[alpha] is determined by the asymptotic behavior of function [[psi].sub.-]([tau]) = O([[tau].sup.-a]) as [tau] [right arrow] 0.

Problem (10) is solved by the factorization method [16] with the use of representations

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

As a result, (10) is transformed into

[[PSI].sup.-](p)[G.sup.-](p) + [H.sup.-](p) = [F.sup.+](p)/[G.sup.+](p) + [H.sup.+](p) = Q(p). (11)

To obtain two constants satisfying the conditions (8), it is necessary to have Q(p) = [c.sub.0] + [c.sub.1]p . Results. Thus, exact solutions of equations (7) and (6) have the next form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (12)

where [[psi].sub.1]([xi]) = [psi]([xi][a.sup.-1]) = 2[pi]D[[c.sub.0][[phi].sub.0]([xi] + [c.sub.1][[phi].sub.1]([xi]) + [[phi].sub.*]([xi])];

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Here the constants [c.sub.0], [c.sub.1], [W.sub.0] are found from the equations (8) and (4):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Conclusions. Mathematical formulation of the boundary value problem is done. An integral equation for stress-strain analysis of thin supported elastic plate with rigid cross-shaped embedment is obtained. The exact solution of this boundary value problem is obtained by reduction to the Riemann problem and by its subsequent solution.

The behavior of the function [[psi].sub.1]([xi]) when [xi] [right arrow] a - 0 (defined by the asymptotic behavior [[PSI].sup.-](p) at [absolute value of (p)] [right arrow] [infinity]) is of great interest. Since p[[G(p)].sup.-1] = O([p.sup.1/2]) then, according to [17] [[psi].sub.1]([xi]) = O([(a -- [xi]).sup.-3/2]), i.e., the contact forces have a non-integrable singularity at the ends of the cross-shaped embedment. This singularity coincides with the result in [9].

DOI 10.15276/opu.2.49.2016.14

References

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Accepted July 10, 2016

O.B. Kozin (1), PhD, Assoc.Prof., O.B. Papkovskaya (2), PhD, Assoc.Prof., M.O. Kozina (2), PhD

(1) National University "Odesa Law Academy", 23 Fontans'ka Rd., 65009 Odessa, Ukraine; e-mail: kozindre@rambler.ru

(2) Odessa National Polytechnic University, 1 Shevchenko Ave., 65044 Odessa, Ukraine
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