Hindawi Advances in Tribology Volume 2018, Article ID 6498503, 13 pages https://doi.org/10.1155/2018/6498503

Research Article Numerical Analysis of Elastic Contact between Coated Bodies

Sergiu Spinu 1,2

1 Department of Mechanics and Technologies, Stefan cel Mare University of Suceava, 13th University Street, 720229, Romania 2Integrated Center for Research, Development and Innovation in Advanced Materials, Nanotechnologies, and Distributed Systems for Fabrication and Control (MANSiD), Stefan cel Mare University of Suceava, Romania

Correspondence should be addressed to Sergiu Spinu; [email protected]

Received 31 May 2018; Revised 28 September 2018; Accepted 11 October 2018; Published 1 November 2018

Academic Editor: Patrick De Baets

Copyright © 2018 Sergiu Spinu. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Substrate protection by means of a hard coating is an efcient way of extending the service life of various mechanical, electrical, or biomedical elements. Te 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. Te iterative derivation of contact area and contact tractions requires repeated displacement evaluation; therefore the robustness of a contact solver relies on the efciency of the algorithm for displacement calculation. Te fast Fourier transform coupled with the discrete convolution theorem has been widely used in the contact modelling of homogenous bodies, as an efcient computational tool for the rapid evaluation of convolution products that appear in displacements and stresses calculation. Te 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. Te proposed algorithm for displacement calculation is coupled with a state-of-the-art contact solver based on the conjugate gradient method. Te predictions of the newly advanced computer program are validated against existing results derived by a diferent method. Multiple contact cases are simulated aiming to assess the infuence of coating thickness and of its elastic properties on the contact parameters and the strass state. Te performed simulations prove that the advanced algorithm is an efcient 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.

1. Introduction the contact behavior may deviate signifcantly from the framework of the contact between homogenous bodies. Te service life of machine elements undergoing contact load For a multilayered body, the closed-form relationship is expected to beneft from substrate protection by means between the excitation (surface tractions) and the response of a protective hard coating, providing low friction, high (displacements and stresses) may be conveniently derived by wear resistance, and long fatigue life. Te competent design the use of the integral transform theory. Te latter substi- of protective coatings requires the knowledge of the stresses tutes a complicated problem in the spatial domain with its generated in the coated system under contact load, as well as equivalent form in the transformed frequency domain, which the material response to these stresses. Te complex math- may result much simpler and its solution easier to obtain. Te ematical models arising in the analysis of layered systems, mathematical modelling of stress and displacement in layered admitting closed-form solutions only under strong limiting systems pioneered with the work of Burmister [1], involving assumptions, suggest the use of numerical methods, capable Bessel type stress functions in the contact analysis of fully of treating deterministic contacting surfaces resulting from bonded or frictionless interfaces under prescribed axisym- various manufacturing processes. Te classical Hertz model metric normal load. Chen and Engel [2] extended the results is usually employed as a starting point in any nonconforming to nonaxisymmetric normal loading, whereas O’Sullivian¨ contact calculation. However, in case of coated bodies, due to and King [3] obtained the frequency response functions the elastic mismatch between the coating and the substrate, (FRF) for the contact of bilayered materials under normal and 2 Advances in Tribology shear loads. Closed-form expressions were explicitly derived are obtained by superposition of solutions of point forces, in the transform domain, but solutions in the spatial domain also referred to as the Green’s function method. Te Green’s required numerical inverse transformation. Te fast Fourier function describes the half-space response in the time/space transform (FFT) was frst applied to Contact Mechanics by Ju domain, and its counterpart in the frequency domain, and Farris [4], resulting in increased computational efciency describing the spectral response of the half-space to a Dirac compared to the continuous integral transform method. Te delta function, is referred to as the FRF. Tis duality is of contact of rough surfaces with coatings was subsequently particular interest in case of coated bodies, where only the studied by Nogi and Kato [5] using the FFT technique FRFs are known, while the Green’s functions derivation is and the FRFs derived by O’Sullivian¨ and King [3]. Te difcult, if not impossible. For example, the spectral normal conjugate gradient method (CGM) was used for the solution displacement �̃ induced by a unit point force acting normally of the contact problem for both homogenous and layered on the boundary of a half-space of Young modulus � and solids. Polonsky and Keer [6] discussed the periodicity error Poisson’s ratio ] is associated with the application of the FFT to nonperiodic 2 −1 2 2 −1/2 problems and proposed a correction based on the multilevel �̃ (�, �) =2(1−] )� (� +� ) , (1) multisummation (MLMS) technique. A method for analysis � � � � of the stress state developed in layered elastic solids with where and are the angular frequencies in the and cracks was subsequently proposed [7]. Te source of the directions, respectively, both contained in bounding plane. Equation (1) is the spectral counterpart of the well-known periodicity error in the FFT technique was further studied �(�, �) by Liu et al. [8], and a method of discrete convolution and Boussinesq solution (i.e., the Green’s function) that is usually employed to compute the displacement induced by FFT (DCFFT) was advanced, which completely avoids any �(�, �) additional error except for the discretization error. Liu and a prescribed pressure in a homogenous, isotropic and Wang [9] applied the latter method to the study of contact linear elastic half-space: stress felds caused by surface tractions. A similar technique +∞ +∞ was employed by Wang et al. [10] in the study of the partial � (�, �) = ∫ ∫ � (�, �) � (�−�,�−�) �� ��. (2) −∞ −∞ slip contact of coated bodies. For layered materials, the explicit expressions of the FRFs stand at the core of the Te integral in (2) can be performed analytically only FFT-based techniques. Recently, Yu et al. [11] obtained the in a few cases of simple or axisymmetric contact geometry, analytical FRFs of multilayered materials in a recurrence such as the Hertz contact. In the general contact problem, format and studied the elastic contact of layered bodies with the contact area and the pressure distribution are a pri- various confgurations. ori unknown and therefore must be determined by trial- Tis paper advances a numerical simulation technique for and-error. An iterative approach is usually employed, in the contact of bodies with a single layer coating, by coupling a which the initial guess is a domain expected to include the state-of-the-art contact solver [12] with a displacement com- contact area. Te search for the contact solution involves putation technique based on the FRFs derived in the literature repeated computation of displacement for arbitrary pressure [5]. As the contact solver [12] was originally designed for distributions such as the ones resulting during the iterative the purely elastic response, the MLMS method employed process. Such robustness can only be achieved by employing for displacement computation requires the Green’s functions a numerical method, in which the initial guess domain is split for the elastic half-space in the space domain and therefore up into elementary patches over which a prescribed pressure cannot be applied to coated bodies. In this paper, the MLMS distribution is assumed. In this manner, the continuous is substituted by a novel FFT-based technique that can take pressure distribution �(�, �) is substituted by a reunion of advantage of the existing FRFs [5]. It should be noted that pressure elements ��� that are adjusted during the iterative Nogi and Kato [5] have also obtained the displacement in process so that, at convergence, all boundary conditions are a coated body based on the FRFs, but their technique was satisfed. Tis digitization implies that the contact problem is found lacking [6] in precision due to a poorly controlled reported to discrete spatial coordinates (��,��) instead of the periodicity error. In this paper, the convolution products continuous coordinates (�, �). Te method employed in this arising in the stress and displacement calculation as a result paper assumes uniform pressure elements, i.e., a piecewise ofsuperpositionofefectsoffundamentalsolutionsareeval- constant distribution, in which pressure is assumed uniform uated in the frequency domain with improved computational for each set (�, �). Te most important part of the contact efciency and high precision. A comprehensive set of contact solution is the displacement computation, which concen- simulations is performed, aiming to provide insight on the trates most of the required computational resources. Once combined infuence of the coating thickness and of the elastic the displacement can be calculated for arbitrarily prescribed mismatch between the coating and the substrate, on the discretized pressure, using either the Green’s function or contact parameters and on the stress state in the coated body. the corresponding FRF, the solution of the contact between homogenous or in homogenous bodies can be obtained in an 2. Spectral Response of Elastic iterative manner, as detailed in the next section. Equation (2) Coated Half-Spaces is in fact a two-dimensional convolution product that can also be assessed in the frequency domain. In the frame of linear theory of elasticity, stresses and dis- Te study of stresses and displacements in layered mate- placements arising in a half-space due to general loadings rials with application to layered soil deposits was pioneered Advances in Tribology 3 by Burminster [1]. O’Sullivan and King [3] derived the stress the edges of the computational domain, where the infuence felds in a layered half-space due to normal and shear trac- of the neighboring periods is more prominent. Tis assertion tions using Papkovich-Neuber elastic potentials and double also suggest a way of reducing the periodicity error, by Fourier transform. Based on these results, Nogi and Kato [5] moving away the neighboring pressure periods so that their later obtained the explicit form of the FRF for displacement contribution to the target domain is minimized. A larger in the three-dimensional normal contact problem involving period, i.e., a larger than expected computational domain acoatedmaterial: may be considered, and the original pressure must be zero- 2 padded in the extended domain. Tis extension may result −(1−]1)(1+4�ℎ��−��� )�� �̃ (�, �) = , (3) in series with more terms, thus increasing the computational �1 efort. Numerical experimentation may be required to assess the optimum magnitude of the domain extension. √ 2 2 −2�ℎ where ℎ is the coating thickness, �= � +� , �=� , Nogi and Kato [5] advanced a robust algorithm for the �=�1/�2, � = (� − 1)/(� + 3 − 4]1), �=1−4(1−]1)/[1 + contact of homogenous or bilayered bodies, with consider- −2 2 2 2 �(3 − 4]2)],and�=−� /[1+(�+�+4��ℎ )� + ��� ]. ation of microtopography, in which the displacement solu- Te shear moduli and the Poisson’s ratios for the layer and tion was protected against false periodicity by doubling the the substrate are denoted by �� and ]�, respectively, with �=1 number of grids along each direction. Liu et al. [8] discussed for the coating and �=2for the substrate. It should be noted the source of the periodicity error in the convolution theorem that the counterpart of (3) in the time/space domain, i.e., the and advanced the DCFFT method that can efciently handle Green’s function, is not available in closed-form expression. the conversion of linear convolution into cyclic convolution. Te FRF (3) can be used to derive displacement from Teir strategy implies the doubling of the target domain in a superposition of efects of type (2), by employing the each physical dimension, thus obtaining a virtual domain in discrete convolution theorem [13]. Te spectral computation which the techniques of zero padding and wrap-around order of displacement is convenient for coated contact analysis areemployed.Amorein-depthstudyoftheerrorsources, for two reasons: (1) the displacement and stress response such as the aliasing and the Gibbs phenomena, is performed of multilayered materials is only available in the frequency in [9]. Te DCFFT technique requires the knowledge of the domain as the FRFs, and (2) the convolution theorem states Green’s function, and therefore application to coated bodies that, for series with � terms, the computational complexity may not be straightforward. A conversion process of the 2 decreases from �(� ), when the convolution is evaluated in needed infuence coefcients from the FRFs is indicated in the time/space domain, to an improved �(� log �) in the [9], and a correction factor, i.e., the level of aliasing control, frequency domain. Te reduction comes from the fact that is employed. a convolution operation in the time/space domain can be In this paper, the elastic response of the coated body computed as an element-wise product between the Fourier is calculated using an algorithm based on the computation transforms of the convolution members. Te application of of convolution into the frequency domain. Te periodicity this technique implies not only direct but also invers Fourier error is minimized by employing a target domain extension, transform, the latter required to retrieve the convolution in which the excitation is zero-padded. As described in [9], result in the time/space domain. the accuracy of such a technique depends on the intensity Additional difculty arise due to application of the of the aliasing phenomena, which can only be reduced by convolution theorem to a discretized system as opposed to using a small sampling interval in the frequency domain. Te the continuous one. Expression in (2) is a linear continuous latter can be achieved by considering a larger computational convolution, and its discrete counterpart is a discrete cyclic domain in the time/space domain while maintaining the convolution. Te sampling of the convolution members sample interval (in the space domain) constant. Te algo- creates series with a fnite numbers of terms, and the transfer rithm steps are described below. Without losing generality, of these series to the frequency domain is achieved via theequationsarepresentedinonedimensiononly(i.e.,line the FFT algorithm. However, application of FFT to a series contact) for brevity, but extension to the three-dimensional implies that the series is periodical with a period equal to point contact is straightforward. the considered number of terms. In case of nonconforming Te algorithm requires as input the target computational contact problems, this implicit periodization infers that we domain in the spatial dimension, �,andthenumber� of are considering as excitation not the actual pressure, but a pressure elements ��, � = 1⋅⋅⋅�.Itisconvenienttochoose periodical load having as period the pressure distribution � as a power of 2, considering that FFT is computed on a in the initial guess domain. Tis problem periodization vector which is otherwise zero-padded to the next power of may be desirable in case of contact simulation involving 2. A domain extension ratio � is next chosen to minimize the nominally fat surfaces, when the inherent microtopography periodicity error. In this paper, a value �=8(in each spatial can be replicated by the periodic repetition of a characteristic dimension) was imposed, the available RAM memory being asperity. In case of nonperiodic contact problems, an error the main limiting factor. Te resulting extended domain �� is thus introduced in the displacement computation. Te should be discretized by considering �� grids, resulting in latter periodicity error implies that we are computing the uniform pressure patches having the same size as the initial displacement induced not only by the actual pressure distri- ones. By increasing the number of pressure elements, the bution, but by the spurious neighboring periods as well. It is sample interval in the frequency domain is decreased, aiming expected that the periodicity error will be more signifcant at for the reduction of the aliasing phenomena. Indeed, by 4 Advances in Tribology imposing a domain extension of ratio � and by keeping the Te aforementioned algorithm can be used to compute data interval Δ=�/�in the spatial domain constant, the the displacement or stress response of a coated half-space to data interval in the Fourier transform domain is decreased a prescribed, but otherwise arbitrary, set of pressure elements from 2�/� to 2�/(��). acting on the boundary. Special care is required with the Te next step consists in the computation of the discrete evaluation of the FRF at the origin of the frequency domain. frequency samples ��, which are the counterpart of the grids Most FRFs describing the elastic response of a coated body �� from the spatial domain. Te set of �� discrete samples are singular in origin; e.g., �(�,̃ �) in relation (3) cannot should be centered on the origin of the frequency domain: be evaluated for �=�=0. Te discretization in the frequency domain implies that the FRF is assumed piecewise 2� (� − ��/2) constant,andarepresentative pointfromtheseries (4) isused �� = , �=1⋅⋅⋅��. (4) �� for each small interval. Te characteristic function value on the latter interval is derived by a simple evaluation of the A discrete counterpart �̂� of the FRF �̃ is computed next FRF in the chosen point. Steep gradients of the FRF, such by evaluating �̃ in each frequency sample: as those appearing in the vicinity of the origin, where the function is usually singular, may challenge the choice of the �̂ = �̃ (� ) , �=1⋅⋅⋅��. � � (5) representative point. Te FRFs may be singular in origin; however, they are all integrable everywhere [5], including on Te terms of the latter series are subsequently rearranged a vicinity of the origin. Consequently, an average value of the in wrap-around order; i.e., the terms corresponding to the FRF can be computed over the patch centered on origin: negative frequencies are transferred afer the ones corre- �̂ sponding to positive frequencies. A new series of discrete �� �/(��) spectral samples is obtained: �̃�V� = ∫ �̃ (�) ��. (12) 2� −�/(��) �̂ = �̂ ; � �+��/2 Te latter average value is the best available indicator �̂ = �̂�, of the local behavior of the FRF in a vicinity of origin and �+��/2 (6) therefore can be used instead of the noncalculable �(0)̃ .As �� �=1⋅⋅⋅ . discussed in the following section, the value of the FRF in 2 origin may not even be needed for the solution of the contact problem involving coated bodies. A diferent 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 3. The Contact of Coated Bodies � , � = 1⋅⋅⋅� elements � , but otherwise can be arbitrary. In the framework of Contact Mechanics, the nonconforming Te expanded pressure series is obtained by zero padding � contact between idealized geometries is preponderantly ana- the original pressure series in both directions: the original lyzed as its nature is well-defned, easily controlled, repeatable pressure samples are transferred to the middle � terms of the � =0 and relatively insensitive to manufacturing imperfections. extended series so that � everywhere, except for In this case, the contact stresses arise as a local stress concentration that can be assumed independent of stresses ��+(�−1)�/2 ←� � �, �=1⋅⋅⋅�. (7) in the bulk of the bodies. Consequently, bodies of arbitrary Ten, the fast Fourier transform of the pressure series is surface may be considered as linear elastic half-spaces, and computed: the results from the previous section can be directly applied to computation of stresses and displacements generated in the �=̂ FFT (�) . (8) contacting bodies. Te signifcance of the term �(0)̃ is discussed frst, for the According to the discrete convolution theorem, the case when �̃ is the FRF describing the spectral displacement element-wise product of �̂ and �̂ is computed next, producing due to a normal load described by a Dirac delta function the displacement output series �̂ in the frequency domain: (a unit point normal force). Te spectral series �̂�,� = 1⋅⋅⋅�� is implicated in displacement computation afer �̂ = �̂ �̂ , �=1⋅⋅⋅��. � � � (9) rearrangement in wrap-around order according to relation (6). Afer this rearrangement, �(0)̃ becomes the frst term in Te convolution result in the spatial domain is then �̂ , �=1⋅⋅⋅�� the new series � . When inverse FFT is applied, obtained by inverse FFT: as in relation (10), to a series whose frst term is perturbed by a constant value �, each term of the resulting series is �=IFFT (�̂) , (10) perturbed by the quantity �/�,where� is the number of �̂ from which only the middle � terms, corresponding to the terms in the series. Moreover, if the term 1 is misevaluated target domain, are kept as algorithm output: by a constant, only the frst term of the displacement series, �̂1, will be compromised, as �̂ is calculated by element-wise �� ←� � �+(�−1)�/2, �=1⋅⋅⋅�. (11) product. Consequently, afer application of inverse FFT to �̂, Advances in Tribology 5 the displacement series in the space domain, �,willbeknown except for a constant, related to the misevaluation of �̂1. Te situation in which only relative displacements of the W loaded half-space boundary can be computed is well known Body (a) in the Contact Mechanics theory from the framework of the (rigid indenter) 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 [14]. For a three-dimensional  problem (i.e., point contact), the solution is unique, as shown by the displacement solution calculated by Love [15]. However, the strategy designed [14] for the resolution of the hi plane contact problem, making use of relative displacements h u only, can be extended to the three-dimensional contact case. Te model for the frictionless contact used in this paper coating is based on a well-known formulation employed in contact Body (b) scenarios with various constitutive laws: elastic [12, 16, 17], (coated half-space) a elastic-plastic [18], or viscoelastic [19–21]. For nonconform- substrate ing contacts, the initial point of contact evolves into a contact surfaceasnormalloadisapplied,andtheelasticbodies deform as a result of compression. Te equation of the surface Figure 1: Indentation of a coated half-space by a rigid sphere. of separation between the deformed contacting bodies results [14] from comparison of contact geometry before and afer the deformation, along the normal contact axis (i.e., the �- axis, passing through the initial point of contact and normal restarted with a larger guess. An additional assumption is to the plane that separates best the contacting surfaces in the required for the convergence of this iterative process: latter point of contact): ��� ≥0, (�,�)∈�, (14) ℎ�� =ℎ��� +��� −�, (�,�)∈�, (13) where the strict inequality applies to the contact area. Rela- where ℎ denotes the gap between the deformed surfaces, tion (14) implies adhesion is not accounted for. Whereas the ℎ� the initial (i.e., prior to deformation) gap, and � is the latter may play an important role in the contact or rubber- approach between points in the contacting bodies distant to like materials, the real contact area between metallic bodies, the contact region (i.e., the rigid-body motion). A contact established between the higher asperities, is much smaller process schematic is shown in Figure 1. Te boundaries of than the apparent contact surface, so that adhesion may be the two contacting bodies (a) and (b) in underfomed state assumed trivial. are displayed using dashed lines. Te normal approach � is As bodies are assumed impenetrable in the frame of linear indicated as the distance between the centers of the positions theory of elasticity, occupied by the spherical indenter before and afer the half- ℎ ≥0, (�,�)∈�, space deformation. Without loosing generality, body (a) is �� (15) assumed rigid and body (b) is an elastic coated half-space, for where the strict inequality applies to the noncontact region of simplicity. In this manner, the superposition of initial contact � (�) (�) . Te boundary conditions (14) and (15) can be streamlined geometry and of normal displacement, ℎ� = ℎ� +ℎ� (�) (�) (�) as and �=� +� , takes a simpler form, as ℎ� =0and (�) � ℎ =0, (�,�)∈�, � =0. Te contact model is also valid when the indenter �� �� (16) is elastic (coated or not). In this case, the corresponding (�) the latter equation being used to assess, in the iterative displacements � must be computed separately and added �(�) � process, the contact or noncontact status of every patch in to to obtain the composite displacement needed in the computational domain �. Te contact model is completed (13). Te displacement computation described in the previous with the static force equation, relating the normal force � to section assumes that the elastic contacting bodies can be the pressure distribution: approximated with elastic half-spaces.

Te discretization of pressure is extended to all problem �= ∑ ��� . parameters, and two indexes (�, �) are used for spatial local- (�,�)∈� (17) ization inside a rectangular computational domain � divided into small nonintersecting rectangles. Te domain � is the Te model also assumes that the resultant of the pressure initial guess in an iterative process aiming to fnd, by trial-and distribution is aligned with the normal force, so that no error, the contact area �, defned as the reunion of patches tilting moment or rolling is expected to occur. Te con- having associated nonvanishing pressure elements. Te con- tact model (13)–(17) describes any instantaneous contact tact area must be contained in the computational domain state and therefore can be used beyond the framework of at any iteration, otherwise the searching sequence must be linear elasticity, to simulate the contact process of bodies 6 Advances in Tribology

Acquire the input: normal Adopt the initial Compute the Compute the Compute the force, initial gap, elastic guess values for CGM descent displacement CGM residual parameters pressure direction

Adjust pressure Compute the Adjust system Adjust system to balance the CGM descent START size size normal force step No Yes Yes

Adjust pressure Pressure Negative gap? Negative pressure? to minimize the converges? CGM residual STOP No No

Yes

Figure 2: Algorithm fowchart for the frictionless contact problem of coated bodies.

(�) (�) with complicated response (i.e., elastic-plastic, viscoelastic, displacement � and the computed one, � , are then related inhomogeneous materials), provided the displacement � is by properly evaluated. Te difculty in solving the latter system (�) (�) of equations and inequalities stems from the fact that neither ��� =��� +�, (�,�)∈�. (18) the contact area, nor the pressure distribution, are known in advance. Relation (13) written for the contacting patches Te rigid-body approach � may be expressed in terms (�, �) ∈ � yields a system of equations having the pressure of displacement and initial contact geometry from (13), � elements �� asunknowns.Butthesizeofthissystemis considering that ℎ�� =0, (�, �) ∈ �.Consequently, also unknown, because the contact area is unknown. Te solution is to determine the contact area (i.e., the size of the ℎ��� +��� system), and the pressure elements (i.e., the system solution) �= , (�, �) ∈�, (19) simultaneously. Te existence and uniqueness of the solution card (�) of system (13)–(17) was discussed in [22], and convergence is guaranteed for strictly compressive contact tractions, i.e., where card(�) denotesthenumbersofcontactingpatches, {(�, �) : � >0} ��� <0is not allowed. i.e., the cardinal of the set �� .Bycombining Te system resulting from (13) is essentially linear in relations (18) and (19), the relation between the true and the the pressure elements and can be solved using appropriate calculated rigid-body approach results as numerical techniques, such as the conjugate gradient method (�) (�) (CGM). Te advantage of the latter, beside the superlinear � =� +�. (20) rate of convergence, consists in the possibility of adding new constraints during the iterative process. Indeed, it is Considering (13), (18), and (20), it follows that the system (�) (�) convenient not to include the static equilibrium equation (17) residual remains unchanged when � is used instead of � . in the system, but to enforce outside the CGM core, thus Tis algorithm feature suggest that pressure distribution and conserving the symmetrical nature of the system matrix, as the contact area in the three-dimensional contact problem required by the CGM. It should be noted that the numerical can be calculated by making use of relative displacements solution of the CGM sequence is achieved by minimization only. For computational purposes, one may prefer to take ℎ (�, �) ∈ � �̂ of the residual ��, . Te size of the system is a shortcut by setting the value of 1 to zero, thus avoiding iterated based on the boundary conditions (14)–(16). Te potential convergence issues with the numerical integration patches (�, �) ∈ � for which, during system resolution, in (12). Whereas the literature [5, 9] suggests 64-point Gaus- pressure results negative, ��� <0, are excluded from the sian quadrature, we encountered no convergence problem contact area, and the size of the system decreases. On the with the Matlab function “quad2d”. However, the related other hand, patches (�, �) ∈ � − �,forwhichthegapresults computational efort ofen exceeds that for the evaluation of negative, ℎ�� <0, are included in the contact area, and the size all the other terms in the �̂ series. Tis shortcut is nonexistent of the system increases. Te stabilization of this apparently when the rigid-body approach is needed (requiring absolute opposing processes leads to convergence, which is established displacement evaluation), or in the computation of stresses based on the ratio of the change in two subsequent pressure developing in the contacting bodies due to contact load. iterations to the imposed load. Te fowchart of the contact solver employed to derive Apparently, (13) may be compromised by a misevaluation contact area and pressure distribution is presented in of displacement �.Let� be the constant introduced by setting Figure 2. A complete description of the algorithm is beyond �̂ =0 the FRF in origin to zero, i.e., 1 .Tetruevalueof the point of this paper, and an interested reader is directed Advances in Tribology 7 to the literature [12, 18, 21]. By substituting in (13) the dis- each tangential direction, according to the proposed value placement calculated with the numerical technique (4)–(11), of � for the minimization of the periodicity error. Te a numerical model for the contact of coated bodies, bounded calculation of the normal approach and of the stress state by arbitrary rough surfaces, becomes readily available. in the elastic medium requires FRF evaluation at the origin of the frequency domain, and the Matlab function “quad2d” 4. Results and Discussions wasemployedwiththistask.Noconvergenceerrorswere encountered by using the Matlab default absolute and relative Te frictionless, quasistatic contact of a rigid spherical inden- tolerances. Te CGM based contact solver also converged ter pressed against an elastic coated half-space is simulated in all simulations, with an imposed precision chosen with using the proposed numerical technique. Te parameters respect to the number of grids in the target domain, as of the elastic medium, that is, the Young modulus and the suggested in [12]. It should be noted that the derivation Poisson’s ratio, are denoted by �� and ]�,with�=2for of contact parameters requires a two-dimensional surface the semi-infnite substrate and �=1for the dissimilar grid only, whereas the stress computation involves grid elastic coating that is perfectly bonded to the substrate. Te extension in the depth direction. Bases on a convergence reference scenario is the Hertz contact for the homogenous analysis, a 128 × 128 × 128 mesh was found to provide elastic medium, i.e., when �1 =�2 and ]1 = ]2, yielding a reasonable compromise, assuring a computational time of the contact area ��, the central pressure �� and the rigid- less than 10 minutes (for each contact case) on a 4 core body approach ��, which are used as normalizers for the 3.2GHz CPU, with a peak RAM memory utilization of 8 GB. matching contact parameters. Te magnitude of the normal Contact parameters only can be derived on a much fner two- load, the initial gap, the Young modulus of the substrate, dimensional mesh, e.g., with 1024 × 1024 grids in the target �2 = 210 GPa, the Poisson’s ratios of the elastic materials, computational domain, in a matter of seconds. Te stress state ]1 = ]2 =0.3, are kept constant in all simulations, whereas the computation requires the derivation of the nodal values of coating thickness ℎ and the ratio �1/�2 are varied. Contact each of the six stress tensor components, hence the important parameters and stresses are derived numerically based on computational efort. Te propensity of plastic yielding can be a rectangular target computational domain of side lengths assessed based on the relation between the magnitude of the 4�� ×4�� ×2��. Te displacement and stress computation von Mises equivalent stress, ���, and the yield strength of the was performed, however, in a domain 8 times larger in elastic material, ��:

1 2 2 2 � = √ [(� −� ) +(� −� ) +(� −� ) +6(�2 +�2 +�2 )] ≥�. (21) �� 2 �� �� �� �� �� �� �� �� �� �

Te contact simulation program is frst validated against arising in a linearly elastic half-space with a functionally existing results obtained by a diferent method. Te algorithm graded coating subjected to indentation by a punch with a for the computation of elastic displacement in a layered spherical tip. half-space is authorized by the pressure profles depicted Te methodology applied in this paper difers from the in Figure 3, matching the literature results [3]. Te Hertz aforementioned works, and the goals of the paper are difer- semielliptic pressure distribution, corresponding to the case ent. Te newly proposed numerical algorithm can address �1 =�2, matches well the prediction of the numerical the layered contact problem for arbitrary contact geometry, program. Te pressure profles in the coated contact can and the improved computational efciency allows for high deviate signifcantly from semielliptic for specifc coating density meshes that can tackle rough contact scenarios. To confgurations, as shown in Figure 4. Tin coatings harder our best knowledge, the combined infuence of the coating than the substrate lead to a more evenly distributed pressure thickness and of the elastic mismatch between the coating on a receding contact area. and the substrate, on the contact parameters and on the Te understanding and prediction of the indentation of stress feld, has not yet been analyzed. Extrapolation of design a layered half-space was the subject of numerous research recommendations based on the limited number of specifc eforts [23–25], both analytical and numerical. Yang and Ke cases presented in the literature may be difcult, especially [23] investigated the two-dimensional frictionless contact considering the complexity of the involved dependencies. problem of a coating structure consisting of a surface coating, A parametric study is therefore performed using the newly a functionally graded layer and a substrate under a rigid proposed method, aiming to advance coating thickness and cylindrical punch. Te contact problems for thin flms and compliance recommendations, based on the numerically cover plates, in which the loading consists of any one or predicted contact parameters, as well as on the calculated combination of stresses caused by uniform temperature depth and intensity of the von Mises equivalent stress. From a changes and temperature excursions, far feld mechanical mechanical point of view, these parameters are of paramount loading, and residual stresses resulting from flm processing importance for the improvement of the contact resistance. or welding, was considered by these authors [24]. Volkov et al. Te combined infuence of the coating thickness ℎ and [25] obtained the felds of displacements, stresses and strains of the elastic mismatch between the coating and the substrate 8 Advances in Tribology

2.5 1.5

1.2

1

1.6

0.8 （ ；

2 2 （

1.4 Ｊ

1.8

0.6 2.2 1.5 1

1.2 1 1.6

0.8 1 2 1.4 1.8

0.6

Dimensionless pressure, p / pressure, Dimensionless 1.6 0.5 0.5 1.2 1.4 Dimensionless coating thickness, h / coating Dimensionless

1 1.2 0 0 0.5 1 1.5 0.5 1 1.5 2 2.5 3 3.5 4 ； Dimensionless radial coordinate, r / （ Elastic mismatch, ％1 / ％2 ％ ％ 4 ％ ％ 1 / 2= 1 / 2= 1 / 2 Figure 5: Iso-contours of dimensionless maximum pressure �/��. ％1 / ％2= 2 ％1 / ％2= 1 / 4 ％1 / ％2= 1 Hertz framework Figure 3: Radial pressure profles in the spherical indentation of a

（ 2.5 coated half-space, ℎ/�� =1. Ｊ

2

1.2 1.5

1 （

Ｊ 1

0.8 0.5

0.6

Dimensionless maximum pressure, p / pressure, maximum Dimensionless 0 0.5 1 1.5 0.4 Dimensionless coating thickness, h / ；（

％1 / ％2= 1 / 4 ％1 / ％2= 3 Dimensionless pressure, p / pressure, Dimensionless 0.2 ％1 / ％2= 1 / 2 ％1 / ％2= 4 ％1 / ％2= 2 0 0 0.2 0.4 0.6 0.8 1 Figure 6: Infuence of coating thickness on the maximum pressure �/��. Dimensionless radial coordinate, r / ；（

％1 / ％2= 1 ％1 / ％2= 5 ％1 / ％2= 2 Hertz framework ％1 / ％2= 4 Figures 5–8 suggest that the more compliant the coating Figure 4: Spherical indentation of thin hard coatings, ℎ/�� =0.25. with respect to the substrate, the larger the contact radius and the shallower the maximum central pressure. For �1/�2 < 1, the contact radius increases with the coating thickness, whereas the maximum pressure decreases. Te opposite is �1/�2, on the contact parameters, is depicted in Figures 5–10. true for �1/�2 >1. For coatings thick and stif, i.e., To this end, the dimensionless coating thickness ℎ/�� was �1/�2 ≥3.5and ℎ/�� >1, the pressure distribution difers incremented in the interval [0.25, 1.5] with a step of 0.025, signifcantly from the Hertz case, the central pressure being and the ratio �1/�2 was varied between 0.2 and 4 with an at least twice its reference counterpart. Figure 8 exposes a increment of 0.1, resulting in 51 × 39 contact simulations. limitation of the numerical approach: the contact area can Te predicted iso-contours of central maximum pressure, only vary in increments equal to the sampling interval, hence contact radius and rigid-body approach, are depicted in the steps in the plotted data. Figures 5, 7, and 9, respectively. Particular curves showing the Figures 9 and 10 suggest that the rigid-body approach infuence of coating thickness on the contact parameters for increases with the coatings thickness for �1/�2 <1,whereas specifc �1/�2 ratios are presented in Figures 6, 8, and 10. theoppositeistruefor�1/�2 >1. In the latter case, the stifer Advances in Tribology 9

1.5 1.5 1 （ 0.9 0.8 0.7 ；

1 0.9 0.95 0.65 0.7

0.8 1.8 0.85 0.75 1.3

1.2 1.1

1.2 1.1

1.4

1.4 （ ；

1 0.7

1 1 0.7 0.8

1.5 0.9 0.75 0.9 0.8 1 0.95

1.3 0.85

1.2 1.1

1.2 1.1

1.4

1.4

0.8 Dimensionless coating thickness, h / coating Dimensionless 0.5 0.8 0.85

1 0.9 0.9 0.5

1 0.95 1.3 1.2 1.1 Dimensionless coating thickness, h / coating Dimensionless

1.2 1.1 0.9 0.5 1 1.5 2 2.5 3 3.5 4 Elastic mismatch, ％1 / ％2 0.5 1 1.5 2 2.5 3 3.5 4 Elastic mismatch, ％1 / ％2 Figure 7: Iso-contours of dimensionless contact radius �/��. Figure 9: Iso-contours of dimensionless rigid-body approach �/��.

1.6 2 （ ； 1.4 （  /  1.5 1.2

1 1

0.8 approach, Dimensionless Dimensionless contact radius, a / radius, contact Dimensionless

0.5 0.6 0.5 1 1.5 0.5 1 1.5 Dimensionless coating thickness, h / ；（ Dimensionless coating thickness, h / ；（ ％1 / ％2= 1 / 4 ％1 / ％2= 3 ％1 / ％2= 1 / 2 ％1 / ％2= 4 ％1 / ％2= 1 / 4 ％1 / ％2= 3 ％1 / ％2= 2 ％1 / ％2= 1 / 2 ％1 / ％2= 4 ％1 / ％2= 2 Figure 10: Infuence of coating thickness on the dimensionless �/� Figure 8: Infuence of coating thickness on the dimensionless rigid-body approach �. contact radius �/��.

Figures 12 and 13 show that the von Mises equivalent the coating, the more rapid the decrease of �/�� with the stress is discontinuous across the interface between the coat- increase of ℎ/��. ing and the substrate, despite the continuity of displacements Te stress state analysis aims to assess the infuence of and of the stress tensor components ���, ���,and���.Te the input parameters on the dimensionless magnitude and maximumvonMisesstressisalwayslocatedonthecontact depth of the maximum von Mises equivalent stress. Te use axis. Te FRFs used in the stress state computation are given of relation (12) is frst benchmarked against results from the in Appendix. literature [3] and a good agreement is found. Position of Te efect of varying �1/�2 and ℎ/�� on the intensity and maximum equivalent stress is denoted by the symbol “X” in depth of the maximum von Mises stress is also assessed by Figures 11–13. For comparison purposes, it should be noted numerical simulation of the aforementioned contact cases. that in [3], the stress tensor second invariant �2 is plotted, Relevant magnitude and depth iso-contours are depicted with ��� = √3�2. Stress iso-contours in Figure 11 match well in Figures 14 and 15, whereas in Figure 16 the depth of the analytical Hertz solution, giving confdence in the stress the maximum stress is normalized by the layer thickness ℎ calculation method. instead of ��. In the latter plot, a smaller than unity value 10 Advances in Tribology

0 0.173 0.173 0 0.324 0.0983 0.546 0.0983 0.369 0.247 0.397 0.0568 0.0568 0.62 0.413 0.458 （

（ 0.471 0.5 0.5 ； ； 0.546 0.369 0.324 0.471 0.397 1 1 0.369 0.324 0.322 0.235 0.235 1.5 0.19 0.28 1.5 z / depth, Dimensionless 0.101 Dimensionless depth, z / depth, Dimensionless 0.173 0.247 0.146 0.19 0.101 0.0983 0.0983 0.146 0.173 2 2 − − −2 −1 012 2 1 021 ； Dimensionless coordinate, x / ；（ Dimensionless coordinate, x / （

Figure 11: Iso-contours of dimensionless von Mises equivalent stress Figure 13: Iso-contours of dimensionless von Mises equivalent ���/�� �=0 ℎ/�� =1 �1/�2 =1/2 ���/�� in the plane �=0; �1 =�2. stress in the plane , ,and .

1.5 0 0.246 1 0.652 0.3 0.6

0.145 （ 1.3 0.753 ； 0.9 0.854 0.5 1.2 0.145 0.246 （ 0.5 0.449 0.4 ； 0.8 0.246 0.753 1.1 0.652 0.449 0.7 0.55 1 1 0.62 0.449 0.347 0.9 0.145 1.1 1.3 0.246 1.5 0.7 1.7 1.5 Dimensionless depth, z / depth, Dimensionless 0.4 0.5 1 1.2 1.4 1.6

0.145 thickness, h / coating Dimensionless 0.5 0.8 0.62 2 − − 2 1 012 0.5 1 1.5 2 2.5 3 3.5 4 ； Dimensionless coordinate, x / （ Elastic mismatch, ％1 / ％2

Figure 12: Iso-contours of dimensionless von Mises equivalent Figure 14: Iso-contours of dimensionless maximum von Mises (max) stress ���/�� in the plane �=0, ℎ/�� =1,and�1/�2 =2. equivalent stress ��� /��. indicates that the maximum is located in the coating. For of complex initial geometry, may beneft from the use of these cases, the contact performance could greatly beneft numerical techniques. Te latters are expected to provide from the coating process, considering that the coating mate- reliable simulation data for real engineering applications, by � rialcanbechosenwithayieldstrength � much greater than solving models with fewer limiting assumptions. that of the substrate, thus impeding plastic yielding and crack Te numerical method advanced in this paper combines nucleation which eventually lead to contact failure by fretting a state-of-the-art contact solver, based on the conjugate fatigue. Te iso-contours in Figure 16 show that in a domain gradient method, with a novel technique for computation 1<�/� <4 roughly overlapping the rectangle of sides 1 2 of displacement in coated half-spaces undergoing prescribed 0.2 < ℎ/� <0.9 and � ,themaximumvonMisesstressis normal load. In the newly proposed method, the displace- located in the coating, but very close to the interface, i.e., ment computation is performed in the Fourier transform (max) 0.97 < ℎ�� /ℎ < 1. domain for two reasons: (1) at the time of the analysis, the response of the coated half-space to a unit normal force is 5. Conclusions only available in the frequency domain, as the frequency response functions, and (2) calculation of convolution prod- Te competent design of protective coatings, requiring ucts in a discrete system can be performed more rapidly in detailed knowledge of stresses developing in a coated contact the frequency domain than in the time/space domain. Advances in Tribology 11

1.5 advantages of the proposed methodology consist in (1) the ability to treat prescribed, yet arbitrary, contact geometry, 0.6

（ including the inherent microtopography of real contacting ； 0.5 0.4 surfaces; (2) increased computational efciency due to the calculation of convolution products in the frequency domain; and (3) efcient control of periodicity error and of the aliasing 1 0.45 phenomenon. Te predictions of the newly advanced numerical pro- 0.9 0.45 0.8 0.5 gram agree well with results from the literature obtained 0.4 0.7 by other methods. A parametric study is performed, aiming 0.6 to assess the combined infuence of the coating thickness 0.3 and of the elastic mismatch between the coating and the 0.5 0.5 0.45 substrate, on the contact parameters and on the stress state Dimensionless coating thickness, h / coating Dimensionless 0.7 0.4 developed in the coated body. Tese results may provide 0.6 0.5 0.3 basis for the design of highly efcient coatings that improve the contact performance, thus extending the service life of 0.5 1 1.5 2 2.5 3 3.5 4 various machine elements undergoing contact loading. Elastic mismatch, ％1 / ％2 Figure 15: Iso-contours of dimensionless depth of the maximum (max) Appendix von Mises equivalent stress ℎ�� /��. Te stresses and displacements in the coated system can be � � � ,�=1,2 1.5 expressed in Cartezian coordinates , ,and � ,with the origin of the �-axis for each layer located on its top. In this paper, �=1is used for the coating and �=2for the substrate. （ ； 0.3 Te FRFs of the six stress tensor components are functions of 0.4 �, �,and�1 in the coating and of �, �,and�2 in the substrate, where � and � are the coordinates in the frequency domain 0.5 corresponding to � and �, respectively, 1

(�) 2 (�) −��� ��� �̃�� (�, �, ��)=−� (� � + � � ) 0.6 0.7

0.8 (�) −�� (�) �� 2 (�) −�� 0.97 +2�] (� � � − � � � )−�� (� � � (A.1) 1.2 0.9 � � 0.5 1.1 1.5 1

Dimensionless coating thickness, h / coating Dimensionless (�) 1.4 ��� 2 + � � ); 1.7 2.3 0.5 1 1.5 2 2.5 3 3.5 4 �̃�� (�, �, ��)=�̃�� (�, �, ��); (A.2) Elastic mismatch, ％1 / ％2 (�) 2 (�) −��� ��� Figure 16: Iso-contours of dimensionless depth of the maximum �̃�� (�, �, ��)=� (� � + � � )+2�(1 (max) von Mises equivalent stress ℎ�� /ℎ. (�) (�) −��� ��� 2 (�) −��� − ]�)(� � − � � )+��� (� � (A.3)

(�) �� Application of the convolution theorem to contact prob- + � � � ); lems, 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 �� � (A.4) employed, seconded by an increase the resolution in the (�) (�) −��� ��� frequency domain to minimize the aliasing phenomenon. −���� (� � + � � ); Additional difculty arises in the evaluation of the frequency response function at the origin of the frequency domain. �̃�� (�,�,��)=�̃�� (�,�,��); (A.5) In the newly proposed method, this computation was (�) √ (�) −��� ��� performed numerically, given that the frequency response �̃�� (�,�,��)=− −1 [�� (� � − � � ) function is integrable in a neighborhood of the origin, and consequently an average value can be calculated for the latter (�) −�� (�) �� +�(1−2] )(� � � + � � � ) (A.6) 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. Te +���� (� � − � � )] , 12 Advances in Tribology where the needed coefcients are [3] T. C. O’sullivan and R. B. King, “Sliding contact stress feld due to a spherical indenter on a layered elastic half-space,” Journal of Tribology,vol.110,no.2,pp.235–240,1988. �(1) =�{−(1−2] ) [1−(1−2�ℎ) ��] 1 [4] Y. Ju and T. N. Farris, “Spectral Analysis of Two-Dimensional (A.7) Contact Problems,” Journal of Tribology,vol.118,no.2,p.320, 1 + (� − � − 4��2ℎ2)�}; 1996. 2 [5]T.NogiandT.Kato,“Infuenceofahardsurfacelayeronthe (2) (2) limit of elastic contact-Part I: Analysis using a real surface � = � =0; (A.8) model,” Journal of Tribology,vol.119,no.3,pp.493–500,1997.

(1) [6]I.A.PolonskyandL.M.Keer,“Afastandaccuratemethod � =��{(1−2]1)�(1+2�ℎ−��) for numerical analysis of elastic layered contacts,” Journal of (A.9) Tribology,vol.122,no.1,pp.30–35,2000. 1 + (� − � − 4��2ℎ2)} ; [7] I. A. Polonsky and L. M. Keer, “Stress analysis of layered elastic 2 solids with cracks using the fast Fourier transform and conju- 1 gate gradient techniques,” Journal of Applied Mechanics,vol.68, �(2) =− no. 5, pp. 708–714, 2001. 2 [8]S.Liu,Q.Wang,andG.Liu,“Aversatilemethodofdiscrete ⋅�√�{(3−4] ) (1−�) [1−(1−2�ℎ) ��] (A.10) convolution and FFT (DC-FFT) for contact analyses,” Wear,vol. 2 243, no. 1-2, pp. 101–111, 2000. + (�−1)(1+2�ℎ−��)}; [9] S. Liu and Q. Wang, “Studying contact stress felds caused by surface tractions with a discrete convolution and fast fourier (1) � = [1−(1−2�ℎ) ��] ��; (A.11) transform algorithm,” Journal of Tribology,vol.124,no.1,pp. 36–45, 2002. (1) � = (1+2�ℎ−��) ����; (A.12) [10]Z.Wang,W.Wang,H.Wang,D.Zhu,andY.Hu,“PartialSlip Contact Analysis on Tree-Dimensional Elastic Layered Half (2) (1) Space,” Journal of Tribology,vol.132,no.2,p.021403,2010. � = (1−�) √�� . (A.13) [11] C. Yu, Z. Wang, and Q. J. Wang, “Analytical frequency response functions for contact of multilayered materials,” Mechanics of Data Availability Materials,vol.76,pp.102–120,2014. [12] I. A. Polonsky and L. M. Keer, “A numerical method for Te expressions of the frequency response functions are solving rough contact problems based on the multi-level multi- from previously reported studies, which have been cited. Te summation and conjugate gradient techniques,” Wear,vol.231, results of the numerical simulations that support the fndings no. 2, pp. 206–219, 1999. of this study are available from the corresponding author [13] W.H. Press, S. A. Teukolsky, W.T. Vetterling, and B. P.Flannery, upon request. Numerical Recipes: Te Art of Scientifc Computing,Cambridge University Press, Cambridge, UK, 3rd edition, 1992. Conflicts of Interest [14] K. L. Johnson, Contact Mechanics, Cambridge University Press, Cambridge, UK, 1985. Te author declares that there are no conficts of interest [15] A. E. Love, “Te stress produced in a semi-infnite solid by regarding the publication of this paper. pressure on part of the boundary,” Philosophical Transactions of the Royal Society A, vol. 228, pp. 377–420, 1929. [16] L. Gallego, D. N´elias, and S. Deyber, “Afast and efcient contact Acknowledgments algorithm for fretting problems applied to fretting modes I, II Tis work was partially supported from the project “Inte- and III,” Wear, vol. 268, no. 1-2, pp. 208–222, 2010. grated Center for Research, Development and Innova- [17] S. Spinu and M. Glovnea, “Numerical Analysis of Fretting tion in Advanced Materials, Nanotechnologies, and Dis- Contact between Dissimilar Elastic Materials,” Journal of the tributed Systems for Fabrication and Control”, Contract Balkan Tribological Association,vol.18,no.2,pp.195–206,2012. no. 671/09.04.2015, and Sectoral Operational Program for [18] V.Boucly, D. N´elias, and I. Green, “Modeling of the rolling and Increase of the Economic Competitiveness cofunded from sliding contact between two asperities,” Journal of Tribology,vol. 129, no. 2, pp. 235–245, 2007. the European Regional Development Fund. [19] W. W. Chen, Q. J. Wang, Z. Huan, and X. Luo, “Semi-analytical viscoelastic contact modeling of polymer-based materials,” References Journal of Tribology, vol. 133, no. 4, Article ID 041404, 10 pages, 2011. [1] D. M. Burmister, “Te general theory of stresses and displace- [20] S. Spinu, “Numerical simulation of viscoelastic contacts. part I. ments in layered systems. I,” Journal of Applied Physics,vol.16, algorithm overview,” Journal of the Balkan Tribological Associa- no.2,pp.89–94,1945. tion,vol.21,no.2,pp.269–280,2015. [2] W. T. Chen and P. A. Engel, “Impact and contact stress [21] S. Spinu and D. Cerlinca, “Modelling of rough contact between analysis in multilayer media,” International Journal of Solids and linear viscoelastic materials,” Modelling and Simulation in Engi- Structures,vol.8,no.11,pp.1257–1281,1972. neering,vol.2017,ArticleID2521903,11pages,2017. Advances in Tribology 13

[22] J. J. Kalker and Y.A. van Randen, “Aminimum principle for fric- tionless elastic contact with application to non-Hertzian half- space contact problems,” Journal of Engineering Mathematics, vol.6,no.2,pp.193–206,1972. [23] J. Yang and L.-L. Ke, “Two-dimensional contact problem for a coating-graded layer-substrate structure under a rigid cylindri- cal punch,” International Journal of Mechanical Sciences,vol.50, no.6,pp.985–994,2008. [24]M.A.Guler,F.Erdogan,andS.Dag,“Modelingofthinflms and cover plates bonded to graded substrates,” AIP Conference Proceedings,vol.973,pp.790–795,2008. [25] S. S. Volkov, A. S. Vasiliev, S. M. Aizikovich, N. M. Seleznev, and A. V. Leontieva, “Stress-strain state of an elastic sof functionally-graded coating subjected to indentation by a spherical punch,” PNRPU Mechanics Bulletin,no.4,pp.20–34, 2016. International Journal of Rotating Advances in Machinery Multimedia

Journal of The Scientifc Journal of Engineering World Journal Sensors Hindawi Hindawi Publishing Corporation Hindawi Hindawi Hindawi www.hindawi.com Volume 2018 http://www.hindawi.comwww.hindawi.com Volume 20182013 www.hindawi.com Volume 2018 www.hindawi.com Volume 2018 www.hindawi.com Volume 2018

Journal of Control Science and Engineering

Advances in Civil Engineering Hindawi Hindawi www.hindawi.com Volume 2018 www.hindawi.com Volume 2018

Submit your manuscripts at www.hindawi.com

Journal of Journal of Electrical and Computer Robotics Engineering Hindawi Hindawi www.hindawi.com Volume 2018 www.hindawi.com Volume 2018

VLSI Design Advances in OptoElectronics International Journal of Modelling & International Journal of Simulation Aerospace Navigation and in Engineering Observation Engineering

Hindawi Hindawi Hindawi Hindawi Volume 2018 www.hindawi.com Volume 2018 www.hindawi.com Volume 2018 Volume 2018 Hindawi www.hindawi.com www.hindawi.com www.hindawi.com Volume 2018

International Journal of International Journal of Antennas and Active and Passive Advances in Chemical Engineering Propagation Electronic Components Shock and Vibration Acoustics and Vibration

Hindawi Hindawi Hindawi Hindawi Hindawi www.hindawi.com Volume 2018 www.hindawi.com Volume 2018 www.hindawi.com Volume 2018 www.hindawi.com Volume 2018 www.hindawi.com Volume 2018