Solution of contact problems of anisotropic plates bending on an elastic base using the compensating loads method

. In critical structures used elements consisting of anisotropic materials with concentrated force or moment applied to them are often found. In order to equalize the internal force factors, as well as to ensure the required strength and rigidity of the structure, in places where a local load is applied, it is necessary to strengthen the plate with a rigid lining (stamp). There are several ways to solve contact problems. The way seems simpler if it is possible to determine the exact fundamental solution (FS) for the plate. The exact FS significantly reduces the amount of computational work due to the fact that the boundary conditions and the conditions of conjugation of solutions at the boundary of the contact area are met in advance. It remains only to formulate the conditions for the compatibility of movements in the contact area. With this formulation, the integral equation of the contact condition becomes a singular integral equation of the first kind. Such equations are reduced by regularization to Fredholm equations of the second kind and the problem becomes mathematically and physically correct. The paper considers contact problems without taking into account tangential interaction between objects. For such problems, the following way of simplification is possible: replacing the singular core for a half-space with a core for a layer of constant thickness modeled by an elastic base with a single base coefficient. Such a replacement for thin shells is quite justified and gives solutions that are in good agreement with the solutions of the theory of elasticity. The deflection of the plate is sought using the compensating loads method. To work out the methodology, a test contact problem was solved.


Introduction
In critical structures used, for example, in aviation, shipbuilding, and chemical engineering, elements consisting of anisotropic materials on elastic bases with concentrated force or moment applied to them are often found. Studies show that in places where such loads are applied, there is a sharp concentration of stresses, and the rest of the plate practically does not work. In order to equalize the internal force factors, as well as to ensure the required strength and rigidity of the structure, in places where a local load is applied, it is necessary to strengthen the plate with a rigid lining (stamp).
There are two ways to solve contact problems. The first way is to integrate the equilibrium equations of each object in the contact area, outside it, and conjugate solutions at the boundary and the contact surface. This path is associated with significant mathematical difficulties and even for one-dimensional contact problems leads the problem to a large number of equations. The way seems simpler if it is possible to determine the exact fundamental solution (FS) for the plate (for this purpose, the method of distinguishing features was used in the work). The exact FS significantly reduces the amount of computational work due to the fact that the boundary conditions and the conditions of conjugation of solutions at the boundary of the contact area are met in advance. It remains only to formulate the conditions for the compatibility of movements in the contact area.
The correctness of the formulation of contact problems of the theory of plates and shells depends entirely on the accuracy of the construction of the FS of the objects being contacted. For thin-walled elements, the simplest and at the same time rather strict method of constructing the FS obtained in the classical theory of plates, which gives plate movements as a result of bending and stretching, and the FS for the half-space (half-plane), which characterizes the local deformation of the plate, its compressibility in the transverse direction. The fact that in a small neighborhood of a concentrated force applied to a thinwalled element (beam), the stress state is close to that observed in a half-plane under the action of a concentrated force has been confirmed by photoelasticity methods. Taking this into account, we present the influence function for an arbitrary plate in the form [1]:  ). The second term is continuous everywhere and is modulo bounded. Therefore, with this formulation, the integral equation of the contact condition becomes a singular integral equation of the first kind. Such equations are reduced by regularization to Fredholm equations of the second kind, and the problem becomes mathematically and physically correct.
Below we will consider contact problems without taking into account tangential interaction between objects. For such problems, the following way of simplification is possible: replacing the singular core for a half-space with a core for a layer of constant thickness h/2 modeled by an elastic base with a single base coefficient. Such a replacement for thin shells is quite justified and gives solutions that are in good agreement with the solutions of the theory of elasticity.
Thus, the FS has the form: is the Dirac delta function, k is the base coefficient of the thickness h/2. For small h, this coefficient can be obtained from the asymptotic solution for the layer and is equal to [2] Suppose that in some way (no matter under what loads) a rigid flat stamp is given displacements, as a result of which it assumes a position described by a linear function ( , ) = + + . If the origin of the coordinates is taken in the center of the stamp, then α will be the draft of the stamp, and x  and y  are the tangents of the angles of inclination of the stamp relative to this center in the direction of the corresponding axes. The stamp acts on the plate by means of the required normal contact stresses, which we denote ( ) in the S region. By the compensating loads method (the indirect boundary elements method), we look for the distribution of the deflection of the plate in the form: where, for the problem of bending an anisotropic plate, the relations for moments and shearing forces are determined through displacements according to the formulas [3,4] ( , = 1,2,6) are the anisotropy coefficients. The contact condition is set by movements based on the fact that in the contact area the side of the plate on which the rigid stamp is fixed takes the specified shape and position of the stamp. Using (1) and the filtering property of the Dirac delta function, in the contact area we obtain the deflection of the median plane of the plate in the form (contact condition): The term in (6) has the meaning of deflection of the contact surface of the plate due to compression arising from the action of contact stresses. Substituting the expression of the plate deflection (2) into the boundary conditions (3)-(5) and the contact condition (5), we obtain a system of resolving integral equations of the method of compensating loads.
As the analysis of the integral equations kernels has shown, when passing through a smooth contour, the generalized transverse force Vn and the bending moment Mn suffer discontinuities of the first kind (which is similar to the isotropic case): Thus, it can be concluded that the anisotropy of the material properties does not affect the magnitude of the jump when passing through a smooth contour.
Substituting the expression of plate deflection (2) into the boundary conditions (3)-(5) and the contact condition (6), we obtain a system of integral resolving equations of the compensating loads method. So, in the case of a hard pinching of the edge of the plate, we have: In the case of a hinged support of the edge of the plate, we have (equations (7.1) and (7.3) is preserved) instead of (7.2): In the case of a free edge of the edge of the plate, we have (equations (8) and (7.3) is preserved)instead of (7.1) and (7.2): where 2 , 1 ,  i L i are linear differential operators.
In the numerical implementation of the algorithm, the contour is approximated by segments of straight lines or arcs of circles and is divided into boundary elements, within which compensating loads are considered constant. Integrals that do not contain singularities are calculated on the contour elements according to the eight-node quadrature Gauss formula. Singular integrals are calculated analytically or by the sixteen-node quadrature Gauss formula.
If the boundary conditions (3)-(5) and the contact condition (6) are satisfied, after calculating the integrals, the solution of the contact problem is reduced to a system of linear algebraic equations of the form:

Problem statement
The differential equation of a thin anisotropic linear elastic plate bending lying on an elastic base has the form: is the elliptic differential operator; w(x,y) is the deflection points of the median surface of the plate; pz(x,y) -intensity of the normal pressure acting on the plate; p -reactive pressure, which depends on the elastic base on which the plate lies.
Consider an elastic base, which is characterized by the following function: where Ki (i=1,2,3) and gi (i=1,2) are the coefficients of the elastic base. Table 1 shows several possible variants of bases. 1 ≠ 0; 2 = 3 = 0; g 1 = g 2 = g ≠ 0 -Pasternak-Vlasov linear base; 1 ≠ 0; 2 = 3 = 0; g 1 ≠ g 2 ≠ 0 -linear orthotropic base; In the future, we will consider the linear elastic bases of Winkler and Pasternak-Vlasov as the most common: is the differential operator; kz, kt are the parameters of the elastic base (the first and second base coefficients or compression and shear coefficients, respectively);

Determination of the fundamental solution to the problem of bending an anisotropic plate on an elastic base of Winkler and Pasternak-Vlasov
Let be given a system of linear partial differential equations [5]: where is the original linear partial differential operator, given vector function of the right parts, , = 1, . The solution of system (1)   y x U , is represented as a convolution: of a system of linear differential equations;  -the domain of definition of the differential operator .
of the FS is determined from an expression of the form: is a two-dimensional Dirac delta function; I is a unit matrix of dimension N N  . The FS are determined up to the solution of a homogeneous system of equations and, in addition, are generalized functions.
It can be seen from (11)  The elements of the matrix    , t G are determined from the solution of systems of differential equations of the form: where is the Kronecker symbol. The solution of systems (12) for finding FS can be implemented, for example, using a two-dimensional integral Fourier transform, using the method of sequential integration, etc., using the method of distinguishing features [8][9][10][11][12][13][14][15][16]. Without limiting generality and in order to save space, we further assume that the point of application of single concentrated loads, which are modeled by the generalized Dirac δ-function, is at the origin.
In the process of obtaining FS for the problem of bending anisotropic plates, all the authors, following [3], believed that the roots of the characteristic equation of the bending problem for real homogeneous anisotropic materials are complex conjugate. The same applies to anisotropic plates on an elastic base.
To find a new exact FR of the problem of bending anisotropic plates, we will use the Levy method, having previously made a brief description of the method.
Let there be the following elliptic type differential equation: where the operators Λ and Δ have the form: There will be exactly 2n roots and they will all be complex conjugate (for real coefficients of anisotropy of materials  The required FS to the problem of anisotropic plates bending can be obtained after analyzing the features of the function ψ in the form: According to the above formula, the problems of FS an anisotropic plate bending on an elastic base of interest to us were obtained. To verify the correctness of the found FS, two methods were used: using the differentiation of generalized functions formula [2] where ( Ф ) is the usual derivative of the function F, Г is the boundary of the region G (one of the contours inside which the singularity is located) and by checking the equilibrium of a plate bounded by a curve when a unit load is applied to it. The results of the checks confirm the correctness of the FS found. Consider an orthotropic material, the main directions of elasticity (physical axes x / and y / ) which does not coincide with its geometric axes (x and y). The angle φ is between them (Fig. 2).

Comparison of orthotropic and anisotropic material
As a result, the differential operator of the orthotropic material (in case of coincidence of geometric and physical axes) in the above circumstances, after recalculating the stiffness coefficients ( = 1,3) [3], it will take the form similar to an anisotropic material (in case of mismatch of geometric and physical axes)  A rigidly sealed square anisotropic plate (orthotropicplate withφ=30 0 ) with side 1 . 0  a (m) lying on the elastic base of the Winkler is under the action of a square stamp with side = 0.02 (m) (Fig. 3). The following parameter values were adopted for the decision: ℎ = 0.01 (m), k z k  (МN/m 3 ), 1 = 1.2 ⋅ 10 4 (MPa), 2 = 0.6 ⋅ 10 4 (MPa),  Figure 3 shows the deflection distribution for a die with a side = 0.02 (m).
The obtained results of solving the contact problem for an anisotropic plate (orthotropic plate with φ=30 0 ) and orthotropic plate (with φ=0 0 ) turned out to be identical.

Conclusion
In the article, the contact problem of bending an anisotropic plate on an elastic base was solved using MGE. To implement the planned using the Levy method, the FR of the problem of bending an anisotropic plate on an elastic base was determined.
Then the integral equations of the compensating loads method (indirect boundary element method) and the contact conditions of the plate and stamp were formulated. For the formulated integral equations, the analysis of the limit values of potentials was carried out. As a result of boundary-element discretization, systems of linear algebraic equations were obtained for various boundary conditions on the contour of the plate, from which normal contact stresses and compensating loads were determined using the Gauss method, with the help of which the deflection function distributions and contact stresses of the anisotropic plate were obtained.