The reduced modulus of elasticity of a layered half-space

A layered half-space is represented as the sum of parts of homogeneous half-spaces of coating materials and the base material, the rigidity of which corresponds to the rigidities of the coating and the base. Using a rigid model of a layered elastic body loaded with an axisymmetric distributed load, a new solution is obtained for determining its reduced elasticity modulus and the Poisson's ratio depending on the elastic characteristics of the base material, the coating material and the relative thickness of the coating. Expressions are given for the reduced elastic modulus and the Poisson's ratio of a layered body with a two-layer coating.


Introduction
To date, the possibility of increasing the life of joints of machine parts due to a change in the design or improvement of materials by optimizing their microstructure is almost exhausted.In this regard, one of the promising areas of increasing operational performance of joints of machine parts, including sealing joints and friction units, is the coating on their working surfaces, or the formation of modified layers based on metals, ceramics, and polymers [1].Experience in operating friction units and seals with such coatings shows that their antifriction properties and sealing capacity are determined not only by the properties of the coating material but also by its thickness [2].Known recommendations for choosing the thickness of the coating are based on experimental data, often contradictory.The absence of the theory of contact interaction of rough surfaces through the coating layer does not allow to develop reliable methods for predicting the friction characteristics of triboconjugations and seal tightness at the design stage, which requires expensive and laborintensive experimental tests.One of the directions for solving this problem is to determine the reduced elastic characteristics of solid objects with thin coatings -the elastic modulus and Poisson's ratio.

State of the problem
The presence of a coating involves taking into account the change in mechanical properties as a function of the distance to the surface.Within the framework of the theory of elasticity, this means that we must consider an elastic body with varying values of the elastic modulus and Poisson's ratio [3].Contact problems for bodies with mechanical properties that vary in depth have been examined by many researchers [2-7, etc.].According to the author [3], research methods can be divided into three groups: analytical, numerical and numerically-analytical.However, it is not possible to apply the results obtained to solve practical problems of friction, wear and tightness.We should also note the paper [8], in which an approximate solution of the axisymmetric contact problem is given for an elastic layer of finite thickness, which may be of interest when using polymer coatings.
Engineering methods for solving contact problems on the basis of simplifying hypotheses, for example, the representation of a layered body as a topocomposite-constructions with special mechanical properties, depending on the mechanical properties of the base and coating materials, the thickness of the coating, should be included in a separate group.In [9,10], it was proposed to use the Hertz theory for this purpose.On the basis of reliable results for the extreme thickness values of the coating, an expression is obtained for the dimensionless elastic geometric parameter Φ connecting the displacement of the surface of the topocomposite with the displacement of the surface of a homogeneous half-space from the base material.
The authors [11][12][13], with the development of the method on the basis of the rigid model of a layered body, determined the reduced elasticity modulus and the Poisson's ratio for any values of the coating thickness for an axisymmetric loading of a layered half-space.

Moving points inside a homogeneous half-space
Consider a homogeneous half-space under loading by an axisymmetric load of the form . Following the classical approach based on the application of Boussinescu's potential functions [14], for the displacement of any point along the symmetry axis into a homogeneous half-space for the case of its loading by a distributed load (1): Taking into account Eq. ( 1) and the fact that a r = ρ и a z z = , аfter integration, we have where ( ) is the Gaussian hypergeometric function Substituting Eq. (3) into Eq.( 2) and taking into account that z ad where ( ) We simplify the notation by adopting ( ) ( ) As the analysis showed, the function ( ) to a small extent depends on the values of the Poisson ratio.In this case ( )

Elastic characteristics of a layered half-space
Consider a layered elastic half-space (Fig. 1), which consists of a coating of thickness 1 Using the rigid model of a layered body for the reduced elasticity modulus and Poisson's ratio of the topocomposite, the authors of [12,13] obtained: ( ) where . Similar expressions are obtained for the indentation of a rigid spherical indenter into a layered body.As calculations have shown, the results obtained when the layered body is loaded by an axially symmetric distributed load and when a spherical indenter is indented into it differ by not more than 1%, therefore, in the future it is recommended to use δ 1 F .In the derivation of Eqs. ( 8) and ( 9), the equality of displacements for homogeneous bodies at point A was used [13].Retaining the notation of [13], we consider a different approach.
The scheme (Fig. 1) can be represented in the form of Fig. 2a.Then the movements where s1, s0 are the rigidity of the layer and the base material, ( ) . We consider two homogeneous half-spaces with elastic characteristics ν1, Е1 и ν0, Е0, loaded respectively by forces P1 и P0 (рис.2b и 2c).Forces P1 и P0 and accordingly the maximum pressures р01 and р00 are chosen from the condition of equality of displacements: . For the scheme (Fig. 2b) where a . For the scheme (Fig. 2c) ( ) For the scheme (Fig. 2а) From the conditions for determining the forces of 1 P and 0 P it follows: The value of р01 is determined from the conditions for the equality of the compression of a coating of thickness δ1 for a layered body under load P and a homogeneous material under load P1: The value of р00 is determined from the condition of equality of displacements for z = δ1 of a layered body under load р0 and a homogeneous material at z = δ1 under load р00: Taking Eq. ( 4) into account, we express the Eq. ( 10) in the form Substituting the Eqs.( 12) -( 14) into Eq.( 11), we obtain ( ) Consequently, by analogy with Eq. ( 8) we have ( ) For the Poisson ratio, the Eq. ( 9) should be used.
Comparison of the dependences of the ( ) 1 δ F with the Eqs.( 8) and ( 16) obtained when the layered body was loaded with the axisymmetric distributed load, showed their insignificant divergence (not more than 3%).Therefore, it is recommended to use Eq. ( 16) for engineering calculations.
For a two-layer coating with thicknesses 1 δ and 2 δ , we obtain: ( ) For the case of a contact between a smooth rigid sphere and a layered half-space, the approach of the bodies, the contact radius and the maximum pressure are determined by the expressions: When contacting a rough surface with an elastic layered half-space for a single asperity, the parameter i 1 δ can be represented in the form  -relative contact area for a particular unevenness; c a -radius of the area per single asperity (parameter of discrete model of roughness).