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Substrate protection by means of a hard coating is an efficient way of extending the service life of various mechanical, electrical, or biomedical elements. The assessment of stresses induced in a layered body under contact load may advance the understanding of the mechanisms underlying coating performance and improve the design of coated systems. The iterative derivation of contact area and contact tractions requires repeated displacement evaluation; therefore the robustness of a contact solver relies on the efficiency of the algorithm for displacement calculation. The fast Fourier transform coupled with the discrete convolution theorem has been widely used in the contact modelling of homogenous bodies, as an efficient computational tool for the rapid evaluation of convolution products that appear in displacements and stresses calculation. The extension of this technique to layered solids is tantalizing given that the closed-form analytical functions describing the response of layered solids to load are only available in the frequency domain. Whereas the false problem periodization can be treated as in the case of homogenous solids, the aliasing phenomenon and the handling of the frequency response function in origin require adapted techniques. The proposed algorithm for displacement calculation is coupled with a state-of-the-art contact solver based on the conjugate gradient method. The predictions of the newly advanced computer program are validated against existing results derived by a different method. Multiple contact cases are simulated aiming to assess the influence of coating thickness and of its elastic properties on the contact parameters and the strass state. The performed simulations prove that the advanced algorithm is an efficient tool for the contact analysis of coated bodies, which can be used to further understand the mechanical behavior of the coated system and to optimize its design.

The service life of machine elements undergoing contact load is expected to benefit from substrate protection by means of a protective hard coating, providing low friction, high wear resistance, and long fatigue life. The competent design of protective coatings requires the knowledge of the stresses generated in the coated system under contact load, as well as the material response to these stresses. The complex mathematical models arising in the analysis of layered systems, admitting closed-form solutions only under strong limiting assumptions, suggest the use of numerical methods, capable of treating deterministic contacting surfaces resulting from various manufacturing processes. The classical Hertz model is usually employed as a starting point in any nonconforming contact calculation. However, in case of coated bodies, due to the elastic mismatch between the coating and the substrate, the contact behavior may deviate significantly from the framework of the contact between homogenous bodies.

For a multilayered body, the closed-form relationship between the excitation (surface tractions) and the response (displacements and stresses) may be conveniently derived by the use of the integral transform theory. The latter substitutes a complicated problem in the spatial domain with its equivalent form in the transformed frequency domain, which may result much simpler and its solution easier to obtain. The mathematical modelling of stress and displacement in layered systems pioneered with the work of Burmister [

This paper advances a numerical simulation technique for the contact of bodies with a single layer coating, by coupling a state-of-the-art contact solver [

In the frame of linear theory of elasticity, stresses and displacements arising in a half-space due to general loadings are obtained by superposition of solutions of point forces, also referred to as the Green’s function method. The Green’s function describes the half-space response in the time/space domain, and its counterpart in the frequency domain, describing the spectral response of the half-space to a Dirac delta function, is referred to as the FRF. This duality is of particular interest in case of coated bodies, where only the FRFs are known, while the Green’s functions derivation is difficult, if not impossible. For example, the spectral normal displacement

The integral in (

The study of stresses and displacements in layered materials with application to layered soil deposits was pioneered by Burminster [

The FRF (

Additional difficulty arise due to application of the convolution theorem to a discretized system as opposed to the continuous one. Expression in (

Nogi and Kato [

In this paper, the elastic response of the coated body is calculated using an algorithm based on the computation of convolution into the frequency domain. The periodicity error is minimized by employing a target domain extension, in which the excitation is zero-padded. As described in [

The algorithm requires as input the target computational domain in the spatial dimension,

The next step consists in the computation of the discrete frequency samples

A discrete counterpart

The terms of the latter series are subsequently rearranged in wrap-around order; i.e., the terms corresponding to the negative frequencies are transferred after the ones corresponding to positive frequencies. A new series

A different treatment is required for the second member of the convolution product, i.e., the excitation. Pressure is assumed known in the target domain as a set of pressure elements

Then, the fast Fourier transform of the pressure series is computed:

According to the discrete convolution theorem, the element-wise product of

The convolution result in the spatial domain is then obtained by inverse FFT:

The aforementioned algorithm can be used to compute the displacement or stress response of a coated half-space to a prescribed, but otherwise arbitrary, set of pressure elements acting on the boundary. Special care is required with the evaluation of the FRF at the origin of the frequency domain. Most FRFs describing the elastic response of a coated body are singular in origin; e.g.,

The latter average value is the best available indicator of the local behavior of the FRF in a vicinity of origin and therefore can be used instead of the noncalculable

In the framework of Contact Mechanics, the nonconforming contact between idealized geometries is preponderantly analyzed as its nature is well-defined, easily controlled, repeatable and relatively insensitive to manufacturing imperfections. In this case, the contact stresses arise as a local stress concentration that can be assumed independent of stresses in the bulk of the bodies. Consequently, bodies of arbitrary surface may be considered as linear elastic half-spaces, and the results from the previous section can be directly applied to computation of stresses and displacements generated in the contacting bodies.

The significance of the term

The situation in which only relative displacements of the loaded half-space boundary can be computed is well known in the Contact Mechanics theory from the framework of the plane contact problem, i.e., the line contact. In the latter case, this model inability is a consequence of the way the plane problem solution is derived [

The model for the frictionless contact used in this paper is based on a well-known formulation employed in contact scenarios with various constitutive laws: elastic [

Indentation of a coated half-space by a rigid sphere.

The discretization of pressure is extended to all problem parameters, and two indexes

As bodies are assumed impenetrable in the frame of linear theory of elasticity,

The model also assumes that the resultant of the pressure distribution is aligned with the normal force, so that no tilting moment or rolling is expected to occur. The contact model (

The system resulting from (

Apparently, (

The rigid-body approach

Considering (

The flowchart of the contact solver employed to derive contact area and pressure distribution is presented in Figure

Algorithm flowchart for the frictionless contact problem of coated bodies.

The frictionless, quasistatic contact of a rigid spherical indenter pressed against an elastic coated half-space is simulated using the proposed numerical technique. The parameters of the elastic medium, that is, the Young modulus and the Poisson’s ratio, are denoted by

The contact simulation program is first validated against existing results obtained by a different method. The algorithm for the computation of elastic displacement in a layered half-space is authorized by the pressure profiles depicted in Figure

Radial pressure profiles in the spherical indentation of a coated half-space,

Spherical indentation of thin hard coatings,

The understanding and prediction of the indentation of a layered half-space was the subject of numerous research efforts [

The methodology applied in this paper differs from the aforementioned works, and the goals of the paper are different. The newly proposed numerical algorithm can address the layered contact problem for arbitrary contact geometry, and the improved computational efficiency allows for high density meshes that can tackle rough contact scenarios. To our best knowledge, the combined influence of the coating thickness and of the elastic mismatch between the coating and the substrate, on the contact parameters and on the stress field, has not yet been analyzed. Extrapolation of design recommendations based on the limited number of specific cases presented in the literature may be difficult, especially considering the complexity of the involved dependencies. A parametric study is therefore performed using the newly proposed method, aiming to advance coating thickness and compliance recommendations, based on the numerically predicted contact parameters, as well as on the calculated depth and intensity of the von Mises equivalent stress. From a mechanical point of view, these parameters are of paramount importance for the improvement of the contact resistance.

The combined influence of the coating thickness

Iso-contours of dimensionless maximum pressure

Influence of coating thickness on the maximum pressure

Iso-contours of dimensionless contact radius

Influence of coating thickness on the dimensionless contact radius

Iso-contours of dimensionless rigid-body approach

Influence of coating thickness on the dimensionless rigid-body approach

Figures

Figures

The stress state analysis aims to assess the influence of the input parameters on the dimensionless magnitude and depth of the maximum von Mises equivalent stress. The use of relation (

Iso-contours of dimensionless von Mises equivalent stress

Iso-contours of dimensionless von Mises equivalent stress

Iso-contours of dimensionless von Mises equivalent stress

Figures

The effect of varying

Iso-contours of dimensionless maximum von Mises equivalent stress

Iso-contours of dimensionless depth of the maximum von Mises equivalent stress

Iso-contours of dimensionless depth of the maximum von Mises equivalent stress

The competent design of protective coatings, requiring detailed knowledge of stresses developing in a coated contact of complex initial geometry, may benefit from the use of numerical techniques. The latters are expected to provide reliable simulation data for real engineering applications, by solving models with fewer limiting assumptions.

The numerical method advanced in this paper combines a state-of-the-art contact solver, based on the conjugate gradient method, with a novel technique for computation of displacement in coated half-spaces undergoing prescribed normal load. In the newly proposed method, the displacement computation is performed in the Fourier transform domain for two reasons: (1) at the time of the analysis, the response of the coated half-space to a unit normal force is only available in the frequency domain, as the frequency response functions, and (2) calculation of convolution products in a discrete system can be performed more rapidly in the frequency domain than in the time/space domain.

Application of the convolution theorem to contact problems, which are essentially nonperiodic unless explicitly considered so, may be challenged by the periodicity error. In this paper, a domain extension in the space domain is employed, seconded by an increase the resolution in the frequency domain to minimize the aliasing phenomenon. Additional difficulty arises in the evaluation of the frequency response function at the origin of the frequency domain. In the newly proposed method, this computation was performed numerically, given that the frequency response function is integrable in a neighborhood of the origin, and consequently an average value can be calculated for the latter vicinity. When required, this mean value was considered instead of the missing sample in the discrete series resulted from digitization of the frequency response function. The advantages of the proposed methodology consist in (1) the ability to treat prescribed, yet arbitrary, contact geometry, including the inherent microtopography of real contacting surfaces; (2) increased computational efficiency due to the calculation of convolution products in the frequency domain; and (3) efficient control of periodicity error and of the aliasing phenomenon.

The predictions of the newly advanced numerical program agree well with results from the literature obtained by other methods. A parametric study is performed, aiming to assess the combined influence of the coating thickness and of the elastic mismatch between the coating and the substrate, on the contact parameters and on the stress state developed in the coated body. These results may provide basis for the design of highly efficient coatings that improve the contact performance, thus extending the service life of various machine elements undergoing contact loading.

The stresses and displacements in the coated system can be expressed in Cartezian coordinates

The expressions of the frequency response functions are from previously reported studies, which have been cited. The results of the numerical simulations that support the findings of this study are available from the corresponding author upon request.

The author declares that there are no conflicts of interest regarding the publication of this paper.

This work was partially supported from the project “Integrated Center for Research, Development and Innovation in Advanced Materials, Nanotechnologies, and Distributed Systems for Fabrication and Control”, Contract no. 671/09.04.2015, and Sectoral Operational Program for Increase of the Economic Competitiveness cofunded from the European Regional Development Fund.