Contact Problem for an Elastic Layer on an Elastic Half Plane Loaded by Means of Three Rigid Flat Punches

Abstract

The frictionless contact problem for an elastic layer resting on an elastic half plane is considered. The problem is solved by using the theory of elasticity and integral transformation technique. The compressive loads P and Q (per unit thickness in 𝑧 direction) are applied to the layer through three rigid flat punches. The elastic layer is also subjected to uniform vertical body force due to effect of gravity. The contact along the interface between elastic layer and half plane is continuous, if the value of the load factor, λ, is less than a critical value, 𝜆_cr. In this case, initial separation loads, 𝜆_cr and initial separation points, 𝑥_cr are determined. Also the required distance between the punches to avoid any separation between the punches and the elastic layer is studied and the limit distance between punches that ends interaction of punches is investigated for various dimensionless quantities. However, if tensile tractions are not allowed on the interface, for 𝜆>𝜆_cr the layer separates from the interface along a certain finite region. Numerical results for distance determining the separation area, vertical displacement in the separation zone, contact stress distribution along the interface between elastic layer and half plane are given for this discontinuous contact case.

1. Introduction

Contact between deformable bodies abounds in industry and everyday life. Because of the industrial importance of the physical processes that take place during contact, a considerable effort has been made in their modeling, analysis, and numerical simulations.

The range of application in contact mechanics starts with problems like foundations in civil engineering, where the lift off the foundation from soil due to eccentric forces acting on a building, railway ballasts, foundation grillages, continuous foundation beams, runaways, liquid tanks resting on the ground, and grain silos is considered. Furthermore, foundation including piles as supporting members or the driving of piles into the soil is of interest. Also classical bearing problem of steel constructions and the connection of structural members by bolts or screws are areas in which contact analysis enters the design process in civil engineering [1].

A complete analysis of the interaction problem for elastic bodies generally requires the determination of stresses and strain within the individual bodies in contact, together with information regarding the distribution of displacements and stresses at the contact regions.

The contact problem for an elastic layer has attracted considerable attention in the past because of its possible application to a variety of structures of practical interest. The layer usually rests on a foundation which may be either elastic or rigid and the body force due to gravity may be neglected [2–27] or effect of gravity may be taken into account [28–35]. Studies regarding the frictionless contact along the interface may be found in [2–15]. Also contact region may exhibit frictional characteristics, giving rise to normal and shear traction at the contact surface [16–27].

While initial contact is determined by the geometric features of the bodies, the extent of the contact generally changes not only by the particular loads applied to the bodies but also with the elastic constants of the materials. Due to the bending of the layer under local compressive loads, in the absence of gravity effects, the contact area would decrease to a finite size which is independent of the magnitude of the applied load [2–7]. If the effect of gravity was taken into account, the normal stress along the layer subspace interface will be compressive and the contact is maintained through the frictionless interface unless the compressive applied load exceeds a certain critical value. When the magnitude of the compressive external load exceeds a certain value, a separation will take place between the layer and the foundation. The length of separation region along the interface is unknown, and the problem becomes a discontinuous contact problem [28–35].

Interaction between an elastic medium and a rigid punch forms another group of contact problems. Rigid punches may be structural elements such as foundations, beams, and plates of finite or infinite extent resting on idealized linearly deformable elastic media. Here, the shape of the contact region may be known a priori and remains constant, or contact region may be changed due to the shape of punch profile [8–27]. The problem of flat-ended rigid punch has important applications in soil mechanics, particularly in estimating the safety of foundations. The application of the three punches for an elastic layer resting on an elastic half plane in soil mechanics is obvious; for example, the punches can be taken as foundations placed on layered soil. When the foundations are placed on soil, there is a possibility of pressure isobars of adjacent foundations overlapping each other. The soil is highly stressed in the zones of overlapping, or the difference of settlement between two adjacent foundations, commonly referred to as differential settlement may cause damage to the structure. It is possible to avoid overlapping of pressures or differential settlement by installing the foundations at considerable distance apart from each other.

In this study, contact problem of the three punches for an elastic layer resting on an elastic half plane is considered according to the theory of elasticity with integral transformation technique. The compressive loads 𝑃 and 𝑄 (per unit thickness in 𝑧 direction) are applied to the layer through three rigid flat punches. The width of midmost punch can be different from the other two punches and thickness of the layer is constant, ℎ. The layer is subjected to homogeneous vertical body force due to gravity, 𝜌₁𝑔. All surfaces are frictionless. The layer remains in contact with the elastic half plane where the magnitude of the load factor 𝜆 is less than a critical value, 𝜆_cr(𝜆=𝑃/𝜌₁𝑔ℎ²). If 𝜆>𝜆_cr, the contact is discontinuous and a separation takes place between the layer and the half plane. A numerical integration procedure is performed for the solution of the problems, and different parameters are researched for various dimensionless quantities for both continuous and discontinuous contact cases. Finally, numerical results are analyzed and conclusions are drawn.

2. General Expressions for Stresses and Displacements

Consider a frictionless elastic layer of thickness h lying on an elastic half plane. The geometry and coordinate system are shown in Figure 1. The governing equations are 𝜇_𝑘∇²𝑢_𝑘+2𝜇_𝑘𝜅_𝑘𝜕−1𝜕𝑥𝜕𝑢_𝑘+𝜕𝑥𝜕𝑣_𝑘𝜇𝜕𝑦=0,_𝑘∇²𝑣_𝑘+2𝜇_𝑘𝜅_𝑘𝜕−1𝜕𝑦𝜕𝑢_𝑘+𝜕𝑥𝜕𝑣_𝑘𝜕𝑦=𝜌_𝑘()𝑔,𝑘=1,2 (1a)(1b)where 𝜌_𝑘𝑔 is the intensity of the body force acting vertically in which 𝜌_𝑘 and 𝑔 are mass density and gravity acceleration. 𝑢_𝑘 and 𝑣_𝑘 are the 𝑥 and 𝑦 components of the displacement vector, 𝜇_𝑘 and 𝜅_𝑘 represent shear modulus and elastic constant of the layer and the half plane, respectively.𝜅_𝑘=(3−𝛾_𝑘)/(1+𝛾_𝑘) for plane stress and 𝜅_𝑘=(3−4𝛾_𝑘) for plane strain. 𝛾_𝑘 is the Poisson ratio (𝑘=1,2). Subscript 1 indicates the elastic layer and subscript 2 indicates the elastic half plane.

Figure 1 

Elastic layer resting on an elastic half plane loaded by means of three rigid flat punches.

𝑢_𝑝 and 𝑣_𝑝 represent the displacements for the case in which gravity forces are considered. 𝑢_ℎ and 𝑣_ℎ are the displacements when the gravity forces are ignored, and total field of displacements may be expressed as𝑢=𝑢_𝑝+𝑢_ℎ,𝑣=𝑣_𝑝+𝑣_ℎ. (2a)(2b)

Observing that 𝑥=0 is a plane of symmetry, it is sufficient to consider the problem in the region 0≤𝑥≤∞ only. Using the symmetry consideration, the following expressions may be written: 𝑢₁𝑥,𝑦=−𝑢₁,𝑣−𝑥,𝑦₁𝑥,𝑦=𝑣₁,𝑢−𝑥,𝑦₁=2𝑥,𝑦𝜋^∞₀𝜙₁()𝑣𝛼,𝑦sin𝛼𝑥𝑑𝛼,₁=2𝑥,𝑦𝜋^∞₀Ψ₁𝛼,𝑦cos(𝛼𝑥)𝑑𝛼, (3a)(3b)(3c)(3d)where 𝜙₁ and Ψ₁ functions are inverse Fourier transforms of 𝑢₁and 𝑣₁ respectively. Taking necessary derivatives of (3c) and (3d), substituting them into (1a) and (1b), and solving the second-order differential equations, the following equations may be obtained for displacements: 𝑢_1ℎ=2𝑥,𝑦𝜋^∞₀𝐴+𝐵𝑦𝑒^−𝛼𝑦+𝐶+𝐷𝑦𝑒^𝛼𝑦()𝑣sin𝛼𝑥𝑑𝛼,_1ℎ=2𝑥,𝑦𝜋^∞₀𝜅𝐴+¹𝛼𝜅+𝑦𝐵𝑒^−𝛼𝑦+𝜅−𝐶+¹𝛼−𝑦𝐷𝑒^𝛼𝑦cos(𝛼𝑥)𝑑𝛼. (4a)(4b)

Using Hooke’s law and (4a) and (4b), stress components which do not include the gravity force may be expressed as follows: 𝜎_𝑥1ℎ𝑥,𝑦=4𝜇₁𝜋^∞₀𝛼𝐴+𝐵𝑦−3−𝜅₁2𝐵𝑒^−𝛼𝑦+𝛼𝐶+𝐷𝑦+3−𝜅_𝑖2𝐷𝑒^𝛼𝑦()𝜎×cos𝛼𝑥𝑑𝛼,_𝑦1ℎ𝑥,𝑦=4𝜇₁𝜋^∞₀−𝛼+𝜅𝐴+𝐵𝑦¹+12𝐵𝑒^−𝛼𝑦++𝜅−𝛼𝐶+𝐷𝑦¹+12𝐷𝑒^𝛼𝑦()𝜏×cos𝛼𝑥𝑑𝛼,_𝑥𝑦1ℎ𝑥,𝑦=4𝜇₁𝜋^∞₀−𝛼+𝜅𝐴+𝐵𝑦^𝑖−12𝐵𝑒^−𝛼𝑦+𝛼−𝜅𝐶+𝐷𝑦^𝑖−12𝐷𝑒^𝛼𝑦×sin(𝛼𝑥)𝑑𝛼. (5a)(5b)(5c)

For the case in which gravity force exist, particular part of the displacement components corresponding to 𝜌₁𝑔, the following expressions are obtained, that is, special solution of the Navier equations for a layer with a height ℎ: 𝑢_1𝑝=3−𝜅₁8𝜇₁𝜌₁𝑔ℎ2𝑣𝑥,_1𝑝=𝜌₁𝑔𝑦2𝜇₁𝜅₁−1𝜅₁−𝜅+1𝑦−ℎ¹+18ℎ,𝜎_𝑦1𝑝=𝜌₁𝑔,𝜎𝑦−ℎ_𝑥1𝑝=𝜌₁𝑔ℎ𝑦−21+𝜅₂1+𝜅₂+2𝜇₂1+𝜅₁,𝜏_{𝑥𝑦1𝑝}=0. (6a)(6b)(6c)(6d)(6e)

Considering the orthogonal axes shown in Figure 1, displacements will be zero for 𝑦=−∞, and if 𝜇₂, 𝜈₂ are the elastic constants of the half plane, then the homogenous field of displacements and stresses of the elastic half plane may be obtained as 𝑢_2ℎ=2𝑥,𝑦𝜋^∞₀𝐸+𝐹𝑦𝑒^𝛼𝑦()𝑣sin𝛼𝑥𝑑𝛼,_2ℎ=2𝑥,𝑦𝜋^∞₀𝜅−𝐸+²𝛼−𝑦𝐹𝑒^𝛼𝑦()𝜎cos𝛼𝑥𝑑𝛼,_𝑥2ℎ𝑥,𝑦=4𝜇₂𝜋^∞₀𝛼𝐸+𝐹𝑦+3−𝜅₂2𝐹𝑒^𝛼𝑦()𝜎×cos𝛼𝑥𝑑𝛼,_𝑦2ℎ𝑥,𝑦=4𝜇₂𝜋^∞₀+𝜅−𝛼𝐸+𝐹𝑦₂+12𝐹𝑒^𝛼𝑦()𝜏cos𝛼𝑥𝑑𝛼,_𝑥𝑦2ℎ𝑥,𝑦=4𝜇₂𝜋^∞₀𝛼−𝜅𝐸+𝐹𝑦²−12𝐹𝑒^𝛼𝑦×sin(𝛼𝑥)𝑑𝛼, (7a)(7b)(7c)(7d)(7e)subscript 2 indicates the elastic half plane. Note that the body force acting in the foundation is neglected since it does not disturb the contact pressure distribution. A, B, C, D, E, and F are the unknown constants, which will be determined from the boundary and continuity conditions at 𝑦=0 and 𝑦=ℎ.

3. Case of Continuous Contact

An elastic layer with a height of ℎ resting on an elastic half plane, shown in Figure 1, is analyzed for unit thickness in 𝑧 direction. Widths of punches at both sides are similar, (𝑐−𝑏)/ℎ, and each of these punches transmits a concentrated load of 𝑄 to the elastic layer. The width of the midmost punch is different, 2𝑎/ℎ and it is subjected to a concentrated load, 𝑃. All surfaces are frictionless. Particularly, the initial separation load (𝜆_cr) and point (𝑥_cr) where the layer separated from the elastic half plane and the variation of the stress distribution between elastic layer and elastic half plane is examined depending on material properties, width of punches, and magnitude of the external loads, 𝑃 and 𝑄. Due to the different settlement of punches, a separation takes place between punch I and elastic layer, if punches are close enough. Therefore, the critical distance between the punches indicating the initiation of separation between the punch I and the elastic layer is researched and also limit distance between the punches where the interaction of punches ends is investigated.

If load factor (𝜆) is sufficiently small, then the contact along the layer-subspace, 𝑦=0,0<𝑥<∞, will be continuous, and A, B, C, D, E, and F must be determined from the following boundary and continuity conditions:𝜎_𝑦1()=(𝑥)(𝑥)𝜏𝑥,ℎ−𝑃,0<𝑥<𝑎−𝑄,𝑏<𝑥<𝑐0,𝑎<𝑥<𝑏,𝑐<𝑥<∞,_𝑥𝑦1()𝜏𝑥,ℎ=0,0<𝑥<∞,_𝑥𝑦1()𝜏𝑥,0=0,0<𝑥<∞,_𝑥𝑦2()𝜎𝑥,0=0,0<𝑥<∞,_𝑦1(𝑥,0)=𝜎_𝑦2()𝜕𝑥,0,0<𝑥<∞,𝑣𝜕𝑥²(𝑥,0)−𝑣₁(𝜕𝑥,0)=0,0<𝑥<∞,𝑣𝜕𝑥¹(𝜕𝑥,ℎ)=0,0<𝑥<𝑎,𝑣𝜕𝑥¹(𝑥,ℎ)=0,𝑏<𝑥<𝑐, (8a)(8b)(8c)(8d)(8e)(8f)(8g)(8h)

in which subscripts 1 and 2 indicate relation to the elastic layer and the elastic half plane, respectively. 𝑃(𝑥) is the unknown contact pressure under punch I and 𝑄(𝑥) is the unknown contact pressure under punch II, which have not been determined yet. If a separation occurs between the elastic layer and elastic half plane, this will give rise to a discontinuous contact position and the following results for former solution will no longer be valid and new solution will be attained for the latter case.

Equilibrium conditions of the problem may be expressed as^𝑎₀𝑃(𝑥)𝑃𝑑𝑥=2_𝑐𝑏,𝑄(𝑥)𝑑𝑥=𝑄. (9a)(9b)

Displacement and stress expressions (4a), (4b), (5a)–(5c), (6a)–(6e), and (7a)–(7e) are substituted into boundary conditions (8a)–(8f), and unknown constants A, B, C, D, E, and F are determined in terms of unknown functions 𝑃(𝑥) and 𝑄(𝑥). By making use of (8g) and (8h), after some simple manipulations, one may obtain the following singular integral equations for 𝑃(𝑥) and 𝑄(𝑥) [36, 37]: −1𝜋𝜇₁^𝑎₀𝑘₁(𝑥,𝑡)+1+𝜅₁41+1𝑡+𝑥𝑃(𝑡)−1𝑡−𝑥𝑑𝑡𝜋𝜇₁^𝑐_𝑏𝑘₁(𝑥,𝑡)+1+𝜅₁41+1𝑡+𝑥(𝑡)−1𝑡−𝑥×𝑄𝑑𝑡=0,0<𝑥<𝑎,𝜋𝜇₁^𝑎₀𝑘₁(𝑥,𝑡)+1+𝜅₁41+1𝑡+𝑥𝑃(𝑡)−1𝑡−𝑥𝑑𝑡𝜋𝜇₁^𝑐_𝑏𝑘₁(𝑥,𝑡)+1+𝜅₁41+1𝑡+𝑥(𝑡)𝑡−𝑥×𝑄𝑑𝑡=0,𝑏<𝑥<𝑐, (10a)(10b)where𝑘₁(𝑥,𝑡)=^∞₀𝛼³81+𝜅₁×1+𝜅₂+𝜇₂𝜇₁1+𝜅₁+𝑒^−2𝛼ℎ4𝛼ℎ1+𝜅₂−2𝜇₂𝜇₁1+𝜅₁+𝑒^−4𝛼ℎ−1−𝜅₂+𝜇₂𝜇₁1+𝜅₁(Δ)⁻¹−1+𝜅₁4×{sin𝛼(𝑡+𝑥)−sin𝛼(𝑡−𝑥)}𝑑𝛼, (11) in which Δ=𝛼³2−1−𝜅₂−𝜇₂𝜇₁1+𝜅₁+𝑒^−4𝛼ℎ−1−𝜅₂+𝜇₂𝜇₁1+𝜅₁+𝑒^−2𝛼ℎ1+𝜅₂2+4𝛼²ℎ²+4𝛼ℎ𝜇₂𝜇₁−1−𝜅₁⁻¹. (12)

If evaluated values of A, B, C, and D in terms of 𝑃(𝑥) and 𝑄(𝑥) are substituted into (5b), the expression of the contact stress between elastic layer and half plane may be obtained as 𝜎_𝑦1(𝑥,0)=−𝜌₁1𝑔ℎ−𝜋^𝑎₀𝑘₂()𝑃(𝑡)−1𝑥,𝑡𝑑𝑡𝜋^𝑐_𝑏𝑘₂(𝑥,𝑡)𝑄(𝑡)𝑑𝑡,0<𝑥<∞, (13) in which 𝜌₁ and 𝑔 are mass density and gravity acceleration, respectively, where 𝑘₂(𝑥,𝑡)=^∞₀𝛼³𝜇₂𝜇₁1+𝜅₁×𝑒^−3𝛼ℎ(−1+𝛼ℎ)+𝑒^−𝛼ℎ(1+𝛼ℎ)(Δ)⁻¹×{cos𝛼(𝑡+𝑥)+cos𝛼(𝑡−𝑥)}𝑑𝛼. (14)

To simplify the numerical analysis, the following dimensionless quantities are introduced:𝑥₁=𝑎𝑟₁,𝑡₁=𝑎𝑠₁,𝑥₂=𝑐−𝑏2𝑟₂+𝑐+𝑏2,𝑡₂=𝑐−𝑏2𝑠₂+𝑐+𝑏2,𝑔₁𝑠₁=𝑃𝑎𝑠₁,𝑔𝑃/ℎ₂𝑠₂=𝑄((𝑐−𝑏)/2)𝑠₂+(𝑐+𝑏)/2,𝑃𝑃/ℎ𝛼=𝑤ℎ,𝜆=𝜌₁𝑔ℎ². (15a)(15b)(15c)(15d)(15e)(15f)(15g)(15h)Substituting from (15a)–(15h), (9a), (9b) and (10a), (10b) may be expressed as¹₀𝑔₁𝑠₁𝑎ℎ𝑑𝑠₁=12₁₋₁,𝑔₂𝑠₂𝑐−𝑏2ℎ𝑑𝑠₂=𝑄𝑃,−1𝜋¹₀𝑔₁𝑠₁𝑎ℎ𝑑𝑠₁𝑚₁𝑟₁,𝑠₁+1+𝜅₁4×1𝑎𝑠₁+𝑟₁−1𝑎𝑠₁−𝑟₁−1𝜋¹₋₁𝑔₂𝑠₂𝑐−𝑏2ℎ𝑑𝑠₂×𝑚₂𝑟₁,𝑠₂+1+𝜅₁4×1((𝑐−𝑏)/2)𝑠₂+(𝑐+𝑏)/2+𝑎𝑟₁−1((𝑐−𝑏)/2)𝑠₂+(𝑐+𝑏)/2−𝑎𝑟₁=0,0<𝑟₁−1<1,𝜋¹₀𝑔₁𝑠₁𝑎ℎ𝑑𝑠₁×𝑚₃𝑟₂,𝑠₁+1+𝜅₁4×1𝑎𝑠₁+((𝑐−𝑏)/2)𝑟₂+()−1𝑐+𝑏/2𝑎𝑠₁−((𝑐−𝑏)/2)𝑟₂+()−1𝑐+𝑏/2𝜋¹₋₁𝑔₂𝑠₂𝑐−𝑏2𝑑𝑠₂×𝑚₄𝑟₂,𝑠₂+1+𝜅₁4×1((𝑐−𝑏)/2)𝑠₂+𝑟₂+()−1𝑐+𝑏((𝑐−𝑏)/2)𝑠₂−𝑟₂=0,−1<𝑟₂𝜎<1,_𝑦1(𝑥,0)1𝑃/ℎ=−𝜆−1𝜋¹₀𝑚₅𝑟₁,𝑠₁𝑔₁𝑠₁𝑎ℎ𝑑𝑠₁−1𝜋¹₋₁𝑚₆𝑟₂,𝑠₂𝑔₂𝑠₂𝑐−𝑏2ℎ𝑑𝑠₂, (16a)(16b)(16c)(16d)(16e)where𝑚₁𝑟₁,𝑠₁=𝑘₁𝑥₁,𝑡₁,𝑚₂𝑟₁,𝑠₂=𝑘₁𝑥₁,𝑡₂,𝑚₃𝑟₂,𝑠₁=𝑘₁𝑥₂,𝑡₁,𝑚₄𝑟₂,𝑠₂=𝑘₁𝑥₂,𝑡₂,𝑚₅𝑟₁,𝑠₁=𝑘₂𝑥₁,𝑡₁,𝑚₆𝑟₂,𝑠₂=𝑘₂𝑥₂,𝑡₂. (17a)(17b)(17c)(17d)(17e)(17f)

The index of the integral equations in (16a)–(16e) is +1; so the functions 𝑔₁(𝑠₁) and 𝑔₂(𝑠₂) may be expressed in the following forms:𝑔₁𝑠₁=𝐺₁𝑠₁1−𝑠²₁^−1/2,0<𝑠₁,𝑔<1₂𝑠₂=𝐺₂𝑠₂1−𝑠²₂^−1/2,−1<𝑠₂<1, (18a)(18b)where 𝐺₁(𝑠₁) is bounded in [0,1] and 𝐺₂(𝑠₂) is bounded in [−1,1]. Then using appropriate Gauss-Chebyshev integration formula given in [36], (16a)–(16d) are replaced by the following algebraic equations:^𝑛_𝑖=1𝜋𝑊_𝑖𝑎ℎ𝐺₁𝑠_1𝑖=12_𝑛𝑖=1,𝜋𝑊_𝑖𝑐−𝑏𝐺2ℎ²𝑠_2𝑖=𝑄𝑃,−^𝑛_𝑖=1𝑊_𝑖𝐺₁𝑠_1𝑖𝑎ℎ𝑚₁𝑟_1𝑗,𝑠_1𝑖+1+𝜅₁4×1𝑎𝑠_1𝑖+𝑟_1𝑗−1𝑎𝑠_1𝑖−𝑟_1𝑗−^𝑛_𝑖=1𝑊_𝑖𝐺₂𝑠_2𝑖𝑐−𝑏×𝑚2ℎ₂𝑟_1𝑗,𝑠_2𝑖+1+𝜅₁4×1((𝑐−𝑏)/2)𝑠_2𝑖+(𝑐+𝑏)/2+𝑎𝑟_1𝑗−1((𝑐−𝑏)/2)𝑠_2𝑖+(𝑐+𝑏)/2−𝑎𝑟_1𝑗,−=0,𝑗=1,…,𝑛−1_𝑛𝑖=1𝑊_𝑖𝐺₁𝑠_1𝑖𝑎ℎ×𝑚₃𝑟_2𝑗,𝑠_1𝑖+1+𝜅₁4×1𝑎𝑠_1𝑖+((𝑐−𝑏)/2)𝑟_2𝑗+()−1𝑐+𝑏/2𝑎𝑠_1𝑖−((𝑐−𝑏)/2)𝑟_2𝑗+()−𝑐+𝑏/2_𝑛𝑖=1𝑊_𝑖𝐺₂𝑠_2𝑖𝑐−𝑏×𝑚2ℎ₄𝑟_2𝑗,𝑠_2𝑖+1+𝜅₁4×1((𝑐−𝑏)/2)𝑠_2𝑖+𝑟_2𝑗+()−1𝑐+𝑏((𝑐−𝑏)/2)𝑠_2𝑖−𝑟_2𝑗,𝑊=0,𝑗=1,…,𝑛−1₁=𝑊_𝑛=12𝑛−2,𝑊_𝑖=1,(),𝑠𝑛−1𝑖=2,…,𝑛−1_1𝑖=cos𝑖−1𝜋2𝑛−1,𝑠_2𝑖=cos𝑖−1𝜋,(),𝑟𝑛−1𝑖=1,…,𝑛_1𝑗=cos2𝑗−1𝜋4𝑛−2,𝑟_2𝑗=cos2𝑗−1𝜋,.2𝑛−2𝑗=1,…,𝑛−1 (19a)(19b)(19c)(19d)(19e)(19f)(19g)

The unknowns 𝐺₁(𝑠_1𝑖) and 𝐺₂(𝑠_2𝑖)(𝑖=1,…,𝑛) are determined from the system (19a)–(19d). By using (18a), (18b), substituting the results into (16e), and using Gauss-Chebyshev integration formula, the contact stress 𝜎_𝑦1(𝑥,0)ℎ/𝑃 is evaluated. It should be observed that the integral equations (16c) and (16d) are valid provided the contact stress 𝜎_𝑦1(𝑥,0)ℎ/𝑃 is compressive everywhere; that is, 0<𝑥<∞. The critical load factor, 𝜆_cr and the corresponding location of interface separation, 𝑥_cr can be determined through the use of the following condition for various dimensionless quantities: 𝜎_𝑦1(𝑥,0)𝑃/ℎ=0. (20)

4. Case of Discontinuous Contact

Since the interface cannot carry tensile tractions for 𝜆 >𝜆_cr, there will be separation between the elastic layer and the elastic half plane in the neighborhood of 𝑥=𝑥_cr on the contact plane 𝑦=0, as shown in Figure 1. Assuming that the separation region is described by 𝑑<𝑥<𝑒, 𝑦=0, where 𝑑 and 𝑒 are unknowns and functions of 𝜆, boundary and continuity conditions for the discontinuous contact case are defined as follows:𝜎_𝑦1()=(𝑥)(𝑥)𝜏𝑥,ℎ−𝑃,0<𝑥<𝑎−𝑄,𝑏<𝑥<𝑐0,𝑎<𝑥<𝑏,𝑐<𝑥<∞,_𝑥𝑦1()𝜏𝑥,ℎ=0,0<𝑥<∞,_𝑥𝑦1()𝜏𝑥,0=0,0<𝑥<∞,_𝑥𝑦2()𝜎𝑥,0=0,0<𝑥<∞,_𝑦1(𝑥,0)=𝜎_𝑦2()𝜕𝑥,0,0<𝑥<∞,𝑣𝜕𝑥²(𝑥,0)−𝑣₁(=𝜑(𝑥)𝜎𝑥,0),𝑑<𝑥<𝑒0,0<𝑥<𝑑,𝑒<𝑥<∞,_𝑦1(𝑥,0)=𝜎_𝑦2()𝜕𝑥,0=0,𝑑<𝑥<𝑒,𝑣𝜕𝑥¹(𝜕𝑥,ℎ)=0,0<𝑥<𝑎,𝑣𝜕𝑥¹(𝑥,ℎ)=0,𝑐<𝑥<𝑏. (21a)(21b)(21c)(21d)(21e)(21f)(21g)(21h)(21i)

After utilizing the boundary and continuity conditions defined in (21a)–(21f), new values for the constants A, B, C, D, E, and F which appear in (4a), (4b), (5a)–(5c), and (7a)–(7e) may be obtained in terms of new unknown functions 𝑃(𝑥), 𝑄(𝑥), and 𝜑(𝑥). Unknown functions are then determined from the conditions (21h)-(21i) which have not yet been satisfied. These conditions give the following system of singular integral equations:−1𝜋𝜇₁^𝑎₀𝑘₁(𝑥,𝑡)+1+𝜅₁41−1𝑡+𝑥𝑃(𝑡)−1𝑡−𝑥𝑑𝑡𝜋𝜇₁^𝑐_𝑏𝑘₁(𝑥,𝑡)+1+𝜅₁41−1𝑡+𝑥𝑄(𝑡)+1𝑡−𝑥𝑑𝑡𝜋^𝑒_𝑑𝑘₂()𝜑(𝑡)−1𝑥,𝑡𝑑𝑡=0,0<𝑥<𝑎,𝜋𝜇₁^𝑎₀𝑘₁(𝑥,𝑡)+1+𝜅₁41−1𝑡+𝑥𝑃(𝑡)−1𝑡−𝑥𝑑𝑡𝜋𝜇₁^𝑐_𝑏𝑘₁(𝑥,𝑡)+1+𝜅₁41−1𝑡+𝑥𝑄(𝑡)+1𝑡−𝑥𝑑𝑡𝜋^𝑒_𝑑𝑘₂()𝜑(𝑡)1𝑥,𝑡𝑑𝑡=0,𝑏<𝑥<𝑐,𝜋^𝑎₀𝑘₂()𝑃(𝑡)1𝑥,𝑡𝑑𝑡+𝜋^𝑐_𝑏𝑘₂()𝑄(𝑡)−𝜇𝑥,𝑡𝑑𝑡₁𝜋^𝑒_𝑑𝑘₃(𝑥,𝑡)−4𝜇₂/𝜇₁1+𝜅₂+𝜇₂/𝜇₁1+𝜅₁×1+1𝑡+𝑥𝜑(𝑡)𝑡−𝑥𝑑𝑡−𝜌₁𝑔ℎ=0,𝑑<𝑥<𝑒, (22a)(22b)(22c)where kernels 𝑘₁(𝑥,𝑡) and 𝑘₂(𝑥,𝑡) are given by (11) and (14) and 𝑘₃(𝑥,𝑡)=^∞₀−2𝛼³𝜇₂𝜇₁𝑒^−2𝛼ℎ2+4𝛼²ℎ²−𝑒^−4𝛼ℎ×(Δ)−1₋₁+4𝜇₂/𝜇₁1+𝜅₂+𝜇₂/𝜇₁1+𝜅₁×{sin𝛼(𝑡+𝑥)+sin𝛼(𝑡−𝑥)}𝑑𝛼, (23) in which Δ is given by (12).

The index of integral equations (22a) and (22b) is +1. On the other hand, the index of the singular integral equation (22c) is −1 due to the physical requirement of smooth contact at the end points 𝑑 and 𝑒 [37]. Thus, in solving the problem the two conditions which would account for the unknowns 𝑑 and 𝑒 are the consistency condition of integral equation (22c) and the single-valuedness condition ^𝑒_𝑑𝜑(𝑥)𝑑𝑥=0. (24)

Defining the following dimensionless quantities𝑥₃=𝑒−𝑑2𝑟₃+𝑒+𝑑2,𝑡₃=𝑒−𝑑2𝑠₃+𝑒+𝑑2,𝑔₃𝑠₃=𝜇₁𝜑((𝑒−𝑑)/2)𝑠₃+(𝑒+𝑑)/2;𝑃/ℎ (25a)(25b)by making use of (15a)–(15h), the integral equations (22a)–(22c) may be expressed as follows:−1𝜋¹₀𝑔₁𝑠₁𝑎ℎ𝑑𝑠₁𝑚^∗₁𝑟₁,𝑠₁+1+𝜅₁4×1𝑎𝑠₁+𝑟₁−1𝑎𝑠₁−𝑟₁−1𝜋¹₋₁𝑔₂𝑠₂𝑐−𝑏2ℎ𝑑𝑠₂×𝑚^∗₂𝑟₁,𝑠₂+1+𝜅₁4×1((𝑐−𝑏)/2)𝑠₂+(𝑐+𝑏)/2+𝑎𝑟₁−1((𝑐−𝑏)/2)𝑠₂+(𝑐+𝑏)/2−𝑎𝑟₁+1𝜋¹₋₁𝑔₃𝑠₃𝑚^∗₃𝑟₁,𝑠₃𝑒−𝑑2ℎ𝑑𝑠₃=0,0<𝑟₁−1<1,𝜋¹₋₁𝑔₁𝑠₁𝑎ℎ𝑑𝑠₁×𝑚^∗₄𝑟₂,𝑠₁+1+𝜅₁4×1𝑎𝑠₁+((𝑐−𝑏)/2)𝑟₂+()−1𝑐+𝑏/2𝑎𝑠₁−((𝑐−𝑏)/2)𝑟₂+()−1𝑐+𝑏/2𝜋¹₋₁𝑔₂𝑠₂𝑐−𝑏2ℎ𝑑𝑠₂𝑚^∗₅𝑟₂,𝑠₂+1+𝜅₁4×1((𝑐−𝑏)/2)𝑠₂+𝑟₂+()−1𝑐+𝑏((𝑐−𝑏)/2)𝑠₂−𝑟₂+1𝜋¹₋₁𝑔₃𝑠₃𝑚^∗₆𝑟₂,𝑠₃𝑒−𝑑2ℎ𝑑𝑠₃=0,−1<𝑟₂−1<1,𝜋¹₋₁𝑔₁𝑠₁𝑚^∗₇𝑟₃,𝑠₁𝑎ℎ𝑑𝑠₁−1𝜋¹₋₁𝑔₂𝑠₂𝑚^∗₈𝑟₃,𝑠₂𝑐−𝑏2ℎ𝑑𝑠₂−1𝜋¹₋₁𝑔₃𝑠₃𝑒−𝑑2ℎ𝑑𝑠₃×𝑚^∗₉𝑟₃,𝑠₃−4𝜇₂/𝜇₁1+𝜅₂+𝜇₂/𝜇₁1+𝜅₁×1((𝑒−𝑑)/2)𝑠₃+𝑟₃+()+1𝑒+𝑑((𝑒−𝑑)/2)𝑠₃−𝑟₃−1𝜆=0,−1<𝑟₃<1, (26a)(26b)(26c)where𝑚^∗₁𝑟₁,𝑠₁=𝑘₁𝑥₁,𝑡₁,𝑚^∗₂𝑟₁,𝑠₂=𝑘₁𝑥₁,𝑡₂,𝑚^∗₃𝑟₁,𝑠₃=𝑘₂𝑥₁,𝑡₃,𝑚^∗₄𝑟₂,𝑠₁=𝑘₁𝑥₂,𝑡₁,𝑚^∗₅𝑟₂,𝑠₂=𝑘₁𝑥₂,𝑡₂,𝑚^∗₆𝑟₂,𝑠₃=𝑘₂𝑥₂,𝑡₃,𝑚^∗₇𝑟₃,𝑠₁=𝑘₂𝑥₃,𝑡₁,𝑚^∗₈𝑟₃,𝑠₂=𝑘₂𝑥₃,𝑡₂,𝑚^∗₉𝑟₃,𝑠₃=𝑘₃𝑥₃,𝑡₃. (27a)(27b)(27c)(27d)(27e)(27f)(27g)(27h)(27i)

Similar to (16a), (16b), additional condition (24) may be expressed as ¹₋₁𝑔₃𝑠₃𝑑𝑠₃=0. (28)

To solve the system of integral equations, it is found to be more convenient to assume that (26c) as well as (26a) and (26b) has an index +1 [29]; consequently, the function 𝑔_𝑖(𝑠_𝑖)  (𝑖=1,…,3) may be expressed in the form 𝑔₁𝑠₁=𝐺₁𝑠₁1−𝑠²₁^−1/2,0<𝑠₁𝑔<1,_𝑖𝑠_𝑖=𝐺_𝑖𝑠_𝑖1−𝑠²_𝑖^−1/2,−1<𝑠_𝑖<1,(𝑖=2,3), (29a)(29b)where 𝐺_𝑖(𝑠_𝑖) is a bounded function. In order to insure smooth contact at the end points of the separation region, we then impose the following conditions on 𝐺₃(𝑠₃): 𝐺₃(−1)=0,𝐺₃(1)=0. (30)Equations (26a)–(26c), (16a), (16b), and (28) can easily be reduced to the following system of linear algebraic equations by employing the appropriate Gauss-Chebyshev integration formula [36]: −^𝑛_𝑖=1𝑊_𝑖𝐺₁𝑠_1𝑖𝑎ℎ𝑚^∗₁𝑟_1𝑗,𝑠_1𝑖+1+𝜅₁4×1𝑎𝑠_1𝑖+𝑟_1𝑗−1𝑎𝑠_1𝑖−𝑟_1𝑗−^𝑛_𝑖=1𝑊_𝑖𝐺₂𝑠_2𝑖𝑐−𝑏×𝑚2ℎ_∗2𝑟_1𝑗,𝑠_2𝑖+1+𝜅₁4×1((𝑐−𝑏)/2)𝑠_2𝑖+(𝑐+𝑏)/2+𝑎𝑟_1𝑗−1((𝑐−𝑏)/2)𝑠_2𝑖+(𝑐+𝑏)/2−𝑎𝑟_1𝑗+^𝑛−1_𝑖=2𝑊_𝑖𝐺₃𝑠_3𝑖𝑒−𝑑𝑚2ℎ^∗3𝑟_1𝑗,𝑠_3𝑖,−=0,𝑗=1,…,𝑛−1_𝑛𝑖=1𝑊_𝑖𝐺₁𝑠_1𝑖𝑎ℎ×𝑚^∗₄𝑟_2𝑗,𝑠_1𝑖+1+𝜅₁4×1𝑎𝑠_1𝑖+((𝑐−𝑏)/2)𝑟_2𝑗+()−1𝑐+𝑏/2𝑎𝑠_1𝑖−((𝑐−𝑏)/2)𝑟_2𝑗+()−𝑐+𝑏/2_𝑛𝑖=1𝑊_𝑖𝐺₂𝑠_2𝑖𝑐−𝑏𝑚2ℎ^∗5𝑟_2𝑗,𝑠_2𝑖+1+𝜅₁4×1((𝑐−𝑏)/2)𝑠_2𝑖−𝑟_2𝑗+()±1𝑐+𝑏((𝑐−𝑏)/2)𝑠_2𝑖−𝑟_2𝑗+^𝑛−1_𝑖=2𝑊_𝑖𝐺₃𝑠_3𝑖𝑒−𝑑𝑚2ℎ^∗6𝑟_2𝑗,𝑠_3𝑖,−=0𝑗=1,…,𝑛−1_𝑛𝑖=1𝑊_𝑖𝐺₁𝑠_1𝑖𝑎ℎ𝑚^∗₇𝑟_3𝑗,𝑠_1𝑖−^𝑛_𝑖=1𝑊_𝑖𝐺₂𝑠_2𝑖𝑐−𝑏𝑚2ℎ^∗8𝑟_3𝑗,𝑠_2𝑖−^𝑛−1_𝑖=2𝑊_𝑖𝐺₃𝑠_3𝑖𝑒−𝑑×𝑚2ℎ_∗9𝑟_3𝑗,𝑠_3𝑖−4𝜇₂/𝜇₁1+𝜅₂+𝜇₂/𝜇₁1+𝜅₁×1((𝑒−𝑑)/2)𝑠_3𝑖−𝑟_3𝑗+()+1𝑒+𝑑((𝑒−𝑑)/2)𝑠_3𝑖−𝑟_3𝑗−1𝜆,=0,𝑗=1,…,𝑛−1_𝑛𝑖=1𝜋𝑊_𝑖𝑎ℎ𝐺₁𝑠_1𝑖=12_𝑛𝑖=1,𝜋𝑊_𝑖𝑐−𝑏𝐺2ℎ²𝑠_2𝑖=𝑄𝑃_{𝑛−1𝑖=2},𝜋𝑊_𝑖𝐺₃𝑠_3𝑖=0, (31a)(31b)(31c)(31d)(31e)(31f)where 𝑊_𝑖, 𝑠_𝑖, and 𝑟_𝑗 are given by (19e)–(19g) (𝑟₂=𝑟₃,𝑠₂=𝑠₃). It was shown in [36] that the consistency condition is automatically satisfied if the Gauss-Chebyshev integration formula is used for solving integral equations. Thus, (31a)–(31d) and (31e)-(31f) give 3𝑛 equation for 3𝑛 unknowns 𝐺₁(𝑠_𝑖), 𝐺₂(𝑠_𝑖), 𝐺₃(𝑠_𝑗)  (𝑖=1,…,𝑛),  (𝑗=2,…,𝑛−1), 𝑑 and 𝑒. The equation system is nonlinear in 𝑑 and 𝑒; so an interpolation scheme is required for the solution. Selected values of 𝑑 and 𝑒 are substituted into (31a)–(31d), and 𝐺₁(𝑠), 𝐺₂(𝑠), and 𝐺₃(𝑠) are obtained which must satisfy (31e), (31f) at the same time for known 𝜆>𝜆_cr. If (31e), (31f) are not satisfied, then solution must be repeated with new values of 𝑑 and 𝑒 until the (31e), (31f) are satisfied at the same time.

It should be noted that (22c) gives the 𝜎_𝑦1(𝑥,0)ℎ/𝑃 outside as well as inside the separation region (𝑒,𝑓). Thus, once the functions 𝐺₁(𝑠), 𝐺₂(𝑠), and 𝐺₃(𝑠) and the constants 𝑑 and 𝑒 are determined, contact stress 𝜎_𝑦1(𝑥,0)ℎ/𝑃 may be easily evaluated. The displacement component 𝑣^∗(𝑥,0) in the separation region (𝑑,𝑒), referring to (21f) and (25b), may be obtained from𝑣^∗(𝑥,0)=𝑣₂(𝑥,0)−𝑣₁(𝑥,0)=^𝑥_𝑑𝑔₃(𝑡)𝑑𝑡,𝑑<𝑥<𝑒 (32a) or 𝜇₁𝑣𝑃/ℎ^∗(𝑥,0)=𝑒−𝑑2ℎ^𝑟3₋₁𝐺₃𝑠₃𝑑𝑠₃,−1<𝑟₃<1, (32b) where 𝑥=𝑒−𝑑2𝑟₃+𝑒+𝑑2. (32c)

Also using appropriate Gauss-Chebyshev integration formula and taking +1 as the index of (32b), the following expression may be written for the vertical displacement in the separation region: 𝜇₁𝑣𝑃/ℎ^∗(𝑥,0)=𝑒−𝑑2ℎ^𝑘−1_𝑖=2𝑊_𝑖𝐺₃𝑠_3𝑖,(𝑘=2,…,𝑛−1), (33) where 𝑊_𝑖 and 𝑠_𝑖 are given by (19e)–(19g).

5. Results and Discussion

Some of the calculated results obtained from the solution of the continuous and discontinuous contact problems for various dimensionless quantities such as 𝜇₂/𝜇₁, 𝑎/ℎ, (𝑐−𝑏)/ℎ, 𝑄/𝑃, 𝜆, and (𝑏−𝑎)/ℎ are presented in Figures 2, 3, 4, 5, and 6 and Tables 1, 2, 3, 4, and 5. First separation point between elastic layer and elastic half plane is determined and first separation region is investigated. Contact pressure 𝜎_𝑦1(𝑥,0)ℎ/𝑃 is presented. Depending on load factor 𝜆, possibilities of other separation regions between elastic layer and elastic half plane are determined. Also possibility of separation between first punch and elastic layer is researched. Besides, the distance (𝑏−𝑎)/ℎ, that ends the interaction of punches is examined. It is assumed that 𝑄/ℎ≥𝑃/ℎ.

Table 1 

Variation of minimum value of distance between two punches (𝑏−𝑎)/ℎ to avoid separation under first punch with 𝑎/ℎ and (𝑐−𝑏)/ℎ for various values of 𝑄/𝑃 (𝜇₂/𝜇₁=6.48).

Table 2 

Variation of distance between two punches (𝑏−𝑎)/ℎ that ends interaction of punches with 𝑄/𝑃 (𝜇₂/𝜇₁=2.75, 𝑎/ℎ=0.25, and (𝑐−𝑏)/ℎ=0.5).

Table 3 

Variation of distance between two punches (𝑏−𝑎)/ℎ that ends interaction of punches with 𝜇₂/𝜇₁ (𝑄=6𝑃, 𝑎/ℎ=0.25, and (𝑐−𝑏)/ℎ=0.5).

Table 4 

Variation of distance between two punches (𝑏−𝑎)/ℎ that ends interaction of punches with 𝑎/ℎ and (𝑐−𝑏)/ℎ (𝑄=4𝑃, 𝜇₂/𝜇₁=0.36).

Table 5 

Variation of load factor values with distance between two punches (𝑏−𝑎)/ℎ (𝑄=2𝑃, 𝜇₂/𝜇₁=1.65, 𝑎/ℎ=0.5, and (𝑐−𝑏)/ℎ=1).

Figure 2 

Variation of minimum value of distance between two punches (𝑏−𝑎)/ℎ to avoid separation under first punch with 𝑄/𝑃 for various values of 𝜇2/𝜇1 (𝑎/ℎ=0.5, (𝑐−𝑏)/ℎ=1).

Figure 3 

Contact stress distribution between elastic layer and elastic half plane for the cases of continuous (𝜆<𝜆cr) and discontinuous contact (𝜆>𝜆cr) (𝑄=2𝑃, 𝜇2/𝜇1=2.75, 𝑎/ℎ=0.25, (𝑏−𝑎)/ℎ=1.5, and (𝑐−𝑏)/ℎ=0.5).

Figure 4 

Contact stress distribution between elastic layer and elastic half plane for the case of discontinuous contact (𝜇2/𝜇1=0.36, 𝑎/ℎ=0.125, (𝑏−𝑎)/ℎ=0.5, (𝑐−𝑏)/ℎ=0.5, and 𝜆=75>𝜆cr).

Figure 5 

Contact stress distribution between elastic layer and elastic half plane for the case of discontinuous contact (𝑄=2𝑃, 𝑎/ℎ=0.5, (𝑏−𝑎)/ℎ=2, (𝑐−𝑏)/ℎ=1, and 𝜆=100>𝜆cr).

Figure 6 

Separation displacement 𝑣∗(𝑥,0) between elastic layer and elastic half space as a function of 𝑥 for various values of second punch load 𝑄(𝜇2/𝜇1=6.48, 𝑎/ℎ=0.75, (𝑏−𝑎)/ℎ=1, (𝑐−𝑏)/ℎ=1.5, and 𝜆=50>𝜆cr), (*: [35]).

Figure 2 shows limit distance between punches that initiates separation under first punch for various values of material constant, 𝜇₂/𝜇₁ with 𝑄/𝑃. The distance (𝑏−𝑎)/ℎ increases, maintaining continuous contact between first punch and elastic layer with 𝑄/𝑃. Besides, for bigger values of 𝜇₂/𝜇₁, limit distance that initiates separation under first punch decreases. In such a case, elastic half plane gets stiffer and it becomes easy to separate first punch from the elastic layer.

Variation of critical distance between punches with 𝑎/ℎ and (𝑐−𝑏)/ℎ for various values of 𝑄/𝑃 is presented in Table 1. For fixed values of second punch width, (𝑐−𝑏)/ℎ, an increase in first punch width requires longer distance between punches to avoid separation under first punch. On the contrary, for fixed values of first punch width, 𝑎/ℎ, an increase in second punch width decreases (𝑏−𝑎)/ℎ and punches can be placed closer to each other without separation.

Interaction between punches ends for a definite value of (𝑏−𝑎)/ℎ. Tables 2–4 show the critical value of the distance that ends interaction between punches with dimensionless quantities 𝜇₂/𝜇₁, 𝑎/ℎ, (𝑐−𝑏)/ℎ, and 𝑄/𝑃. In such a case, there is no need to consider punches together. Also these tables show the values of the load factor that cause separation between elastic layer and elastic half plane, 𝜆_cr. For 𝜆=𝜆_cr, 𝜎_𝑦1(𝑥,0)ℎ/𝑃 is zero. Contact between punches and elastic layer is continuous.

Table 3 shows the critical distance between punches that ends interaction of punches with elastic constant, 𝜇₂/𝜇₁. For small values of 𝜇₂/𝜇₁; that is, it is easy to bend elastic layer, interaction between punches ends in a longer distance. Initial separation point 𝑥_cr between elastic layer and elastic half plane from the origin 𝑥=0 decreases with an increase in 𝜇₂/𝜇₁. Critical load factor also decreases in this situation.

Distance (𝑏−𝑎)/ℎ that ends interaction between punches increases with a decrease in second punch width while first punch width is fixed. If both first and second widths are increased, interaction of punches ends in a shorter distance. This situation is presented in Table 4. In this case, critical load factor, 𝜆_cr increases but initial separation point, 𝑥_cr decreases with increment in punch widths. Separation also occurs at the right-hand side of the second punch.

For fixed values of 𝑎/ℎ=0.5, (𝑐−𝑏)/ℎ=1, 𝑄=2𝑃, and 𝜇₂/𝜇₁=1.65, variations of critical load factor, 𝜆_cr and initial separation point, 𝑥_cr are given in Table 5. For small values of (𝑏−𝑎)/ℎ, initial separation point between elastic layer and elastic half plane appears at the right-hand side of second punch. If (𝑏−𝑎)/ℎ increases, in this case separation takes place between two punches. Keeping on increasing, the distance (𝑏−𝑎)/ℎ ends the interaction of punches. For (𝑏−𝑎)/ℎ=7.9446, there is no need to consider punches together.

In Figures 3–5, the normalized contact stress distribution 𝜎_𝑦1(𝑥,0)ℎ/𝑃 at the interface of elastic layer and elastic half plane is given for the problems described in Section 3, and Section 4. Different scales have been used for continuous and discontinuous contact cases in order to include the entire pressure distribution and to give sufficient details in compact forms.

Variation of the contact stress distribution 𝜎_𝑦1(𝑥,0)ℎ/𝑃 with load factor 𝜆=𝑃/𝜌₁𝑔ℎ² for fixed values of 𝑄=2𝑃, 𝜇₂/𝜇₁=2.75, 𝑎/ℎ=0.25, (𝑏−𝑎)/ℎ=1.5, and (𝑐−𝑏)/ℎ=0.5 is presented in Figure 3, 𝜆<𝜆_cr and 𝜆>𝜆_cr show contact stress distribution for the continuous and discontinuous contact cases, respectively. It can be seen from the figure that another separation region is possible if distance between punches (𝑏−𝑎)/ℎ or load factor 𝜆 is increased. Contact pressure has peaks around the edges of the rigid punches.

In Figure 4, the variation of the normalized contact stress 𝜎_𝑦1(𝑥,0)ℎ/𝑃 with 𝑄/𝑃 is given for the discontinuous contact case (𝜆=75>𝜆_cr). Figure 4 shows that both contact stress and separation zone (𝑒−𝑑)/ℎ increase with an increase in 𝑄/𝑃. Variation of the normalized contact stress 𝜎_𝑦1(𝑥,0)ℎ/𝑃 for the discontinuous contact case shows three different regions between elastic layer and elastic half plane. These are continuous contact region, separation zone, and also continuous contact region, where the effects of external load 𝑄/ℎ and 𝑃/ℎ decrease and disappear infinitely.

Variation of contact stress 𝜎_𝑦1(𝑥,0)ℎ/𝑃 with 𝜇₂/𝜇₁=1.65 is shown in Figure 5. Separation zone increases as elastic half plane gets stiffer than the elastic layer. In this case, peak value of the contact stress also increases.

Results calculated from (33) giving the displacement 𝑣^∗(𝑥,0) in the separation region 𝑑<𝑥<𝑒 as a function of 𝑥 are shown in Figure 6. Also results are compared with those of [35]. It is seen that separation region 𝑑<𝑥<𝑒 and displacement values increase with an increase in 𝑄/𝑃 ratio as this is the case in [35]. Separation zone is nearly the same, but displacement values are smaller than those of [35] for fixed values of 𝜇₂/𝜇₁=6.48, 𝑎/ℎ=0.75, (𝑏−𝑎)/ℎ=1, and (𝑐−𝑏)/ℎ=1.5.

6. Conclusion

In this paper, continuous and discontinuous contact problems for an elastic layer resting on an elastic half plane loaded by means of three rigid flat punches are considered. Numerical procedures developed in this study can be used to find approximate solutions to problems of engineering interest.

Numerical results show that the punch widths, elastic constants, external loads, distance between punches play a very important role in the formation of the continuous and discontinuous contact areas, initial separation point and load, separation displacement, limit distance that ends the interaction of punches, critical distance that causes separation under first punch, and the contact pressure distribution.

From the study, the following conclusions may be drawn:(i)it is easy to separate first punch from elastic layer if 𝑄/𝑃 or 𝑎/ℎ increases. This results also decrease in 𝜇₂/𝜇₁ or (𝑐−𝑏)/ℎ.(ii)Increment in 𝑄/𝑃, 𝑎/ℎ or (𝑐−𝑏)/ℎ ends interaction of punches in a shorter distance. Also an increase in 𝜇₂/𝜇₁ causes the same result.(iii)First separation between elastic layer and elastic half plane occurs between punches or in the region 𝑐<𝑥<∞ depending on 𝜆. If (𝑏−𝑎)/ℎ is big enough, interaction of punches disappears. (iv)Size of separation region is not affected much with 𝑄/𝑃, but separation displacement is increased with an increase in 𝑄/𝑃.

Nomenclature

𝑃∶	Compressive load per unit thickness in 𝑧 direction on punch I, (N/m)
𝑄:	Compressive load per unit thickness in 𝑧 direction on punch II and punch III, (N/m)
ℎ:	Thickness of the layer, (m)
𝜆:	Load factor
𝑢:	𝑥-component of the displacement, (m)
𝑣:	𝑦-component of the displacement, (m)
𝜌𝑘:	Mass density, (kg/m3)
𝑔:	Gravity acceleration, (m/sn2)
𝜇:	Shear modulus, (Pa)
𝜅:	Elastic constant
𝛾:	Poisson’s ratio
(𝑐−𝑏):	Width of punch II and punch III, (m)
𝑎:	Half width of punch I, (m)
𝑃(𝑥):	Contact pressure under punch I, (Pa)
𝑄(𝑥):	Contact pressure under punch II and punch III, (Pa)
(𝑒−𝑑):	Separation length, (m).