A Numerical Solution for a Frictional Contact Problem between an Orthotropic Strip and Punch System

Abstract In this study, a numerical solution of elasticity problem is examined. This problem is a plane contact problem. The frictional contact problem for an elastic strip under a rigid punch system is considered. The frictional contact problem is related to infinite length elastic strip in contact with N punches under the influence of horizontal and vertical forces. The lower boundary of the strip is hinged. The solution of contact problems is often reduced to the solution of an integral equation. This integral equation system can be derived from contact problem by using the basic equations of elasticity theory and the given boundary conditions. The singular integral equation system is solved with the help of Gauss Jacobi Quadrature Collocation Method. The frictional contact problem for a homogenous and orthotropic elastic layer are investigated numerically the pressure distribution under the punch system due to the geometrical and mechanical properties of elastic layer are examined and the results are shown in the graphics and tabular form.


Introduction
The elements of many structural and mechanical systems are in contact with each other. The shape of this contact plays an important role in the deformation of the objects and in the behavior of stress distribution along the contact zone. The elementary theory did not meet all needs in the solution of the contact problems. In recent years, the rapid development of computer technology has triggered the development of numerical solution methods. In addition to this, the solution of contact problems has gained speed with the aid of elasticity theory and important studies have been done.
Many problems related to the mechanic of elastic bodies can be converted into the singular integral equations. For this reason, studies on the solution of singular integral equations hold an important place in mathematic. The first important studies on this subject were made by Muskhelishvili. [1][2]. Many methods have been developed to obtain an analytic solution of integral equations. [3][4][5].
The numerical solution methods have been developed since it is difficult to solve integral equations analytically. There are many important studies. Erdogan and Gupta has obtained approximate solution of the system of singular integral equations bu using the properties of the orthogonal polynomials. [6][7]. Plane contact problems with mixed boundary conditions is studied by Alexandrov [8,9]. Chebakov investigated the asymptotic solution of contact problems for a relatively thick elastic layer when there are friction forces in the contact area. [10]. Yusufoglu E. and his friends [11] investigated a plane contact problem for an elastic orthotropic strip by using a functional method for thick strips and an asymptotic method for thin strips. There are many important studies in addition to this studies.
In this study, a numerical solution of elasticity problem is investigated. The frictional contact problem is related to infinite length elastic strip in contact with N punches under the influence of horizontal and vertical forces.

Formulation of the Problem
Upper boundary of the infinite length elastic strip is in contact with a punch system consisting of N punches under the influence of horizontal and vertical forces.
It is assumed that the surface forces do not affect the area outside the contact zone. The lower boundary of the strip is hinged. (Figure 2.1). In addition, when x → ∞ , the stresses will be zero. The stress on the strip will be regarded as the plane stress under these conditions. The solution of the plane contact problem investigated in study [1]. The solution of the contact problem is reduced to the solution of the following integral equation system by using the basic equations of the elasticity theory.

Discretization of the Integral Equation
In this section, the equation (1) will be discretized. Suppose that Where, We assume that the following equations, Thus, Integral equation (6) will be converted the following form.
( ) 1 1 The additional condition (2) may be expressed as

The Numerical Solution Method
In this section, a numerical solution method of the equation system (10) and (11) is developed. We will suppose that the punches are of equal width, for convenience in practice. In this situation, the width of punches are 1 2 The solution of the system is found as; As is known from the singular integral equations theory, index is equal to 34 A Numerical Solution for a Frictional Contact Problem between an Orthotropic Strip and Punch System ( ) The pressures created by these punches must be infinite at the edge, since both ends of the punches are assumed to enter the elastic strip. In this case value of κ is equal to 1, according to index theory.
Substituting Eq. (12) into Eq.(10), integral equation becomes the following form, We will try to find the corresponding Lagrange interpolation polynomials instead of the unknown We choose the interpolation node points , . Lagrange interpolation polynomials is considered as the following form, (16) As is known, the following equation is true. [12] ( ) ( ) ( ) Substituting the Eq.(15) and (17) into Eq.(14), it is obtained Eq.(18) We choose the k t points as roots of ( , ) Eq.(22) can be written more compactly as follows.

Numerical Results and Conclusions
In this section, the system of equations (24) and (25) which are obtained by applying of Gauss Jacobi Quadrature and collocation method are examined. The punch system corresponding to the case and will be examined. We consider the flat-bottomed punches. In this case, according to Eq.(9), the right side of Eq.(1) will be equal to zero.
It is assumed that the width of punches is same and vertical forces acting on the punches are equal to each other.   ε is friction coefficient and λ is the relative thickness of strip.
1. ε is equal to zero in frictionless contact problem.
In this situation, the smallest pressure value occurs in the middle point of the contact area 0 t = . the pressure graph is also a symmetrical curve. 2. In the case of frictional contact where the coefficient of friction is constant, while the thickness of the strip increases, the strip resistance against the pressure of the punch is decreasing. 3.       The pressure distribution curves along the contact zone are symmetrically with respect to the y axis, in the case of frictionless contact. As the relative thickness increases ( 1 λ > ), the minimum pressure value also increases. The minimum pressure increases as the orthotropic property in the x axis direction increases for the strips of the same thickness. 2. In the case of frictional contact, the amount of pressure in the area under the punch on the right is naturally greater than the amount of pressure on the left punch. The above examinations are made for flat-bottomed and equal width punches. It's assumed that the vertical and horizontal forces are equal. But the proposed algorithm is valid for all contact problems involving non-flat punches with different width.