Kaya, Yusuf; Polat, Alper; Şükrü Özşahin, Talat; Bora, Pınar

In the study, the problem of discontinuous contact in two layers, one of which is functionally graded is solved using the theory of elasticity. In this problem functionally graded (FG) layer resting on homogeneous layer and loaded with two different rigid flat blocks. In addition, homogeneous layer is resting on a rigid plane. Frictions on all surfaces are neglected. The heights of the FG layer and the homogeneous layer are h1 and h2, respectively. When solving the problem, displacement and stress equations are substituted in the boundary conditions. Then the problem is demained to singular integral equations. Wherein unknown functions are the contact stresses under the two rigid flat blocks and the slope of the separation. These singular integral equations are solved numerically using the Gauss- Chebyshev integration formulas. The analyzes were performed for different inhomogeneity parameter (βh1), shear modulus ratio (μ2/μ-h1). distance between the rigid blocks ((a3-a2)/h1) and density parameter (γh1). Consequently, the stress distributions and starting and ending points of the separation area between the FG layer and homogeneous layer are determined for several dimensionless quantities.

Polat, Alper; Kaya, Yusuf

In this study, the problem of discontinuous contact in two functionally graded (FG) layers resting on a rigid plane and loaded by two rigid blocks is solved by the finite element method (FEM). Separate analyzes are made for the cases where the top surfaces of the problem layers are metal, the bottom surfaces are ceramic and the top surfaces are ceramic and the bottom surfaces are metal. For the problem, it is accepted that all surfaces are frictionless. A two-dimensional FEM analysis of the problem is made by using a special macro added to the ANSYS package program The solution of this study, which has no analytical solution in the literature, is given with FEM. Analyzes are made by loading different Q and P loads on the blocks. The normal stress (sigma y) distributions at the interfaces of FG layers and between the substrate and the rigid plane interface are obtained. In addition, the starting and ending points of the separations between these surfaces are determined. The normal stresses (sigma x, sigma y) and shear stresses (tau xy) at the point of separation are obtained along the depth. The results obtained are shown in graphics and tables. With this method, effective results are obtained in a very short time. In addition, analytically complex and long problems can be solved with this method.

Kaya, Yusuf; Polat, Alper; Özşahin, Talat Şükrü

In this study, we solve the continuous contact problem of two layers with different material properties, loaded with two rigid flat blocks and resting on a rigid plane, by analytical and finite element methods. All surfaces are assumed to be frictionless. The upper layer is the functionally graded layer (FG), the height h1, the underlying layer is homogeneous, and the height is h2. It is assumed that the FG layer is isotropic and that the shear modulus and density change exponentially. The FG layer is loaded by means of two rigid flat blocks whose external loads are Q and P (per unit thickness in z direction). In the analytical solution of the problem, stress and displacement expressions are substituted in the boundary conditions and the problem is demoted to singular integral equations where the contact stresses are the unknown functions. These singular integral equations are solved numerically using the Gauss–Chebyshev integration formulas. Two-dimensional finite element analysis of the problem is performed using the ANSYS package program. The analyses are performed for different shear modulus ratios (μ2/μ-h1), distances between the blocks ((c − b)/h1) and inhomogeneity parameters (βh1). The contact stresses under the blocks, the normal stresses (σy) between interfaces of the FG layer and the homogeneous layer and between the homogeneous layer and rigid plane are also measured. In addition, the initial separation load and the initial separation distances (χcr, λcr) of the FG layer and homogeneous layer are determined. The results are presented comparatively as graphs and tables.

Polat, Alper; Kaya, Yusuf; Özşahin, Talat Şükrü

In this study, a comparative analysis of the continuous contact problem of a homogeneous layer loaded by two rigid flat blocks and resting on elastic half plane is investigated using the finite element method (FEM). The results obtained from the elasticity solution are compared with FEM solutions. Also in this problem, it is assumed that the all surfaces are frictionless. Two dimensional finite element analysis of the problem is carried out using ANSYS Mechanical APDL. The analyzes are performed for different loading conditions of Q and P loads. Initial separation load and initial separation points (xcr, λcr) and normal stresses (σy) between the homogenous layer and the elastic half plane are determined. The obtained results are shown in graphs and tables.

Kaya, Yusuf; Polat, Alper; Özsahin, Talat Sukru

The continuous and discontinuous contact problem of a homogeneous layer loaded with three rigid flat blocks and resting on the elastic semi-infinite plane was investigated in this study by using the finite element method (FEM). All surfaces are assumed to be frictionless. The homogenous layer was loaded by means of three rigid flat blocks whose external loads are Q and P (per unit thickness in z direction). Two dimensional finite element analysis of the problem was performed using ANSYS package program. Analyzes were performed for loading conditions which are Q=P, Q=2P, Q=4P, Q=8P and Q=10P, and different values of block widths, distances between blocks and shear modulus. The initial separation loads and initial separation distances (lambda(cr), chi(cr)) between the homogeneous layer and the elastic semi-infinite plane were determined for continuous contact condition. The length of the separation zone was obtained for discontinuous contact condition. In addition to these, normal stress distributions between the layer and the semi-infinite plane were determined for continuous and discontinuous contact conditions. As a result of the study, the analytical results, and the results of the finite element method were shown by comparing as graphs and tables.

Polat, Alper; Kaya, Yusuf; Özşahin, Talat Şükrü

A continuous contact problem of functionally graded layer resting on an elastic semi-infinite plane, which is loaded with through two different blocks is addressed in this study. The elasticity theory and integral transformation techniques are used in solution of the problem. The problem is reduced to a system of singular integral equations, and solved numerically by the aid of appropriate Gauss–Chebyshev integration formula. It is assumed that the elastic semi-infinite homogeneous plane is isotropic and all surfaces are frictionless and continuous. The shear modulus and the mass density of the FG layer vary exponentially along the thickness direction.