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- Hindawi Publishing Corporation
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- Copyright © 2018 Sergiu Spinu. This 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.
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- 1687-5915
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- 1687-5923
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- 10.1155/2018/9432894
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Abstract
Hindawi Advances in Tribology Volume 2018, Article ID 9432894, 16 pages https://doi.org/10.1155/2018/9432894 Research Article 1,2 Sergiu Spinu Department of Mechanics and Technologies, Stefan cel Mare University of Suceava, 13th University Street, 720229 Suceava, Romania Integrated Center for Research, Development and Innovation in Advanced Materials, Nanotechnologies, and Distributed Systems for Fabrication and Control (MANSiD), Stefan cel Mare University of Suceava, Suceava, Romania Correspondence should be addressed to Sergiu Spinu; sergiu.spinu@fim.usv.ro Received 29 January 2018; Accepted 16 April 2018; Published 20 May 2018 Academic Editor: Huseyin C ¸ imenogl ˇ u Copyright © 2018 Sergiu Spinu. This 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. Characterization of viscoelastic materials from a mechanical point of view is oeft n performed via dynamic mechanical analysis (DMA), consisting in the arousal of a steady-state undulated response in a uniaxial bar specimen, allowing for the experimental measurement of the so-called complex modulus, assessing both the elastic energy storage and the internal energy dissipation in the viscoelastic material. The existing theoretical investigations of the complex modulus’ influence on the contact behavior feature severe limitations due to the employed contact solution inferring a nondecreasing contact radius during the loading program. In case of a harmonic cyclic load, this assumption is verified only if the oscillation indentation depth is negligible compared to that due to the step load. This limitation is released in the present numerical model, which is capable of contact simulation under arbitrary loading profiles, irregular contact geometry, and c omplicated rheological models of linear viscoelastic materials, featuring more than one relaxation time. eTh classical method of deriving viscoelastic solutions for the problems of stress analysis, based on the elastic-viscoelastic correspondence principle, is applied here to derive the displacement response of the viscoelastic material under an arbitrary distribution of surface tractions. The latter solution is further used to construct a sequence of contact problems with boundary conditions that match the ones of the original viscoelastic contact problem at specific time intervals, assuring accurate reproduction of the contact process history. eTh developed computer code is validated against classical contact solutions for universal rheological models and then employed in the simulation of a harmonic cyclic indentation of a polymethyl methacrylate half-space by a rigid sphere. eTh contact process stabilization aer ft the first cycles is demonstrated and the energy loss per cycle is calculated under an extended spectrum of harmonic load frequencies, highlighting the frequency for which the internal energy dissipation reaches its maximum. 1. Introduction viscoelastic problem to a formally identical elastic problem whose solution is easier to achieve. Although application of Important engineering applications involving products like this method is limited as the transient boundary conditions automotive belts and tires, seals, or biomedical devices encountered in most contact problems cannot be directly require accurate prediction of tribological processes between treated by means of Laplace transform, solutions of limited viscoelastic materials such as elastomers or rubbers. Consid- viability were successfully obtained. Lee and Radok [3] ering that a closed-form description of the viscoelastic con- derived the solution of a Hertz-type problem involving linear tact is difficult to achieve due to complexity of the emerging viscoelastic materials for the case when the contact area equations, numerical simulation presents itself as a worthy increases monotonically with time. Hunter [4] solved the substitute, capable of assisting the design of tribologically same problem for monotonic contact radius or when the competent products using viscoelastic materials. radius possesses a single maximum. The contact problem of The classic method for solving the linear viscoelastic viscoelastic bodies was extended by Yang [5] to cover general problems of stress analysis is based on the concept of asso- linear materials and arbitrary quadratic contact geometry. A ciated elastic problem [1, 2]. This approach involves removal more versatile solution, allowing for any number of loadings and unloadings, in which the contact area is a simply of time dimension via Laplace transform, thus reducing the 2 Advances in Tribology connected region, was later achieved by Ting [6]. A further of the emerging linear system of equations, whereas (2) iteration [7] involves multiple connected contact regions and the Discrete Convolution Fast Fourier Transform (DCFFT) contact radii described by arbitrary functions of time. eTh technique [17] is engaged in the rapid computation of discrete contact problem between an axisymmetric indenter and a convolution products. viscoelastic half-space was recently revisited by Greenwood [8]. eTh mathematical complexity of these partially analytical 2. Viscoelastic Constitutive Law and solutions challenges their wide range application, especially Associated Contact Solutions in the case of the contact process under harmonic cyclic load, when up to vfi e different cases [7] have to be considered In the framework of linear theory of viscoelasticity [18], the due to the specifics of contact radius dependence on the material exhibits a linear stress-strain relationship; that is, an history of the loading program. Algorithmization may also increase in stress by a constant factor leads to an increase in be problematic as the resulting implicit solutions require thestrainresponsebythesamefactor.Theresponsefunctions numerical integration and differentiation, as well as the to excitations conveyed by Heaviside step functions are resolution of transcendental equations, which may raise referred to as the material functions of the viscoelastic body, convergence issues. namely, the creep compliance and the relaxation modulus, Recent developments in numerical resolution of elastic which are both functions of time 𝑡. eTh creep compliance contact problems encouraged a new approach to the problem function Φ(𝑡)describes the viscoelastic strain response to of viscoelastic contact. Kozhevnikov et al. [9] advanced a aunitstepchangeinstress, andthe relaxation modulus new algorithm for the indentation of a viscoelastic half-space Ψ(𝑡), conversely, describes the stress response to a unit step based on the Matrix Inversion Method (MIM). Chen et al. change in strain. With these functions, the linear viscoelastic [10] developed a new semi-analytical method (SAM) for con- response to various sequences of stress or strain is assessed, tact modeling of polymer-based materials with complicated according to the Boltzmann hereditary integral, by either of properties and surface topography. es Th e authors [11] studied the two Volterra integral equations: the multi-indentation of a viscoelastic half-space by rigid bodies using a two-scale iterative method (TIM). (𝑡 ) Spinu and Gradinaru [12] advanced a semi-analytical 𝑠 (𝑡 )= ∫ Ψ(𝑡 − 𝑡 ) ; 𝜕𝑡 method forthecomputationofdisplacementinlinearvis- (1) coelastic bodies subjected to arbitrary surface tractions. A (𝑡 ) solution for the Cattaneo-Mindlin problem involving vis- 𝑒 (𝑡 )= ∫ Φ(𝑡 − 𝑡 ) , 𝜕𝑡 coelastic materials was advanced by Spinu and Cerlinca [13], based on the algorithm for the frictionless viscoelastic contact where 𝑠 and 𝑒 are the tensors of deviatoric stress and devia- problemreportedin[14]. Morerecently,thesameauthors toric strain, respectively. Consequently, (1) can be regarded as studied [15] the rough contact of viscoelastic materials by the superposition of a series of loading histories consisting in imposing a simplified manner in which plasticity eeff cts at inn fi itesimal changes in strain or stress, respectively, applied thetip oftheasperitiesareaccountedfor.Whiledealing separately in a window of observation [0, 𝑡],chosensothat with friction or with surface microtopography, these contact initially (i.e., at 𝑡=0 ) the viscoelastic body was undisturbed. simulations employ simple loading programs consisting usu- Analog mechanical models, constructed from linear ally in step or ramped loadings. eTh contact process under springs and dashpots, arranged in series or in parallel, are cyclic harmonic load is investigated in this paper based on convenient tools to model the linear viscoelastic behavior an extended version of the viscoelastic contact algorithm under uniaxial loading. The combination rules for these basic advanced in [14]. New algorithm developments allow for units state that creep compliances add in series and relaxation more general boundary conditions, involving displacement moduli in parallel. driven contact scenarios, as well as the assessment of the post- The ideal spring, also referred to as the Hooke model unloading contact state. or the ideal solid, is the elastic element in which the force From the point of view of algorithmic complexity, the is proportional to the extension. By identifying force with semi-analytical method (SAM) employed herein, derived stress and elongation with strain and according to Hooke’s from the boundary element method (BEM), has significant law, 𝜎(𝑡) = 𝐸𝜀(𝑡),with 𝐸 being the pertinent (i.e., longitudinal computational advantages over the finite element method or shear) elasticity modulus. The dashpot, also referred to as (FEM), as it requires a 2D spatial discretization only (i.e., the the Newton model or the perfect liquid, is the viscous element meshing of the potential contact surface), whereas a FEM in which the force is proportional to the rate of extension. simulation entails the 3D meshing of the entire contacting According to the Newton equation, 𝜀(𝑡) ̇ = 𝜎(𝑡)/𝜂,where bodies. According to this review [16], the computational 𝜀=𝜕 ̇ 𝜀/𝜕𝑡 istherateofstrainand 𝜂 is the viscosity coefficient. efficiency of SAM greatly exceeds that of FEM; for example, Both Hooke and Newton models represent limiting cases of a 3D SAM contact simulation can be conducted with a viscoelastic bodies. computational effort comparable to that of a 2D finite element analysis. In this paper, the algorithmic computational e-ffi Abranchconstitutedbyaspringinparallelwithadashpot ciency is optimized by employing state-of-the-art numerical is known as the Kelvin-Voigt model, whereas a branch techniques: (1) the conjugate gradient method (CGM), with constituted by a spring in series with a dashpot is known as its superlinear rate of convergence, is used for the resolution the Maxwell model. The differential equation for the Maxwell 𝑑𝑡 𝜕𝑠 𝑑𝑡 𝜕𝑒 Advances in Tribology 3 model can be expressed as [18] 𝜀(𝑡) ̇ =𝜎(𝑡)/𝐸 ̇ + 𝜎(𝑡)/𝜂 for yielding the following creep compliance function: 𝑡≥𝑡 , under the assumption that 𝜀(𝑡) = 𝜎(𝑡) = for 0 𝑡<𝑡 . 0 0 1 1 1 −𝑡 The creep compliance function for the viscoelastic half-space Φ 𝑡 = ( + (1 −exp ( ))), () 2 𝐺 𝐺 𝜏 described by a Maxwell model consisting in a spring of shear (5) modulus 𝐺 in series with a dashpot of viscosity 𝜂 results as where 𝜏= . [19] Φ(𝑡) = 1/(2𝐺)(1 +,w 𝑡/𝜏)here 𝜏 denotes the relaxation time: 𝜏=𝜂/𝐺 . Ting’sformalism[6,7] canbeemployedtodescribethe The Zener model exhibits instantaneous elastic strain indentation of the Maxwell viscoelastic half-space by a rigid when stress is instantly applied; if the stress is held constant, spherical punch of radius 𝑅 in a step loading 𝑊(𝑡) = 𝑊 𝐻(𝑡), the strain creeps towards a limit, whereas, under constant where 𝐻(𝑡) denotes the Heaviside step function and 𝑊 is strain, the stress relaxes towards a limit. Moreover, when xfi ed but otherwise arbitrarily chosen. eTh equation for the stress is removed, instantaneous elastic recovery occurs, pressure distribution achieved at time 𝑡 aeft r the rfi st contact, followed by gradual recovery towards vanishing strain. Using at the radial coordinate 𝑟,results as Ting’s formalism [6, 7], the pressure distribution in the step loading of a Zener half-space by a rigid spherical punch of 1/2 8𝐺 2 2 radius 𝑅 results as 𝑝 𝑡, 𝑟 = ((𝑎 𝑡 −𝑟 ) ( ) () (2) 8𝐺 1/2 1 2 2 𝑡 𝑝 (𝑡, 𝑟 )= ((𝑎(𝑡 )−𝑟 ) − 1/2 1 𝑡 −𝑡 2 2 − ∫ exp ( )Re ((𝑎 () 𝑡 −𝑟 ) )𝑑𝑡 ), 𝜏 𝜏 2(𝑡 −𝑡) where 𝑎(𝑡)denotes the contact radius at time 𝑡: (6) ⋅ ∫ exp ( ) 1/3 3𝑅𝑊 Φ (𝑡 ) (3) 𝑎 (𝑡 )=( ) , 1/2 2 ⋅ Re ((𝑎 (𝑡)−𝑟 ) )𝑑𝑡 ), and Re() denotes the real part of its complex argument. This partially analytical solution is a more computationally with 𝑎(𝑡)being the contact radius given by (3). friendly form of the corresponding equation derived in [20]. These basic models, having only one relaxation time, are The Maxwell model accounts well for relaxation but capable of providing only qualitative description of viscoelas- handles badly both creep (model creeps without bound at tic behavior, whereas precise quantitative assessments require constant rate; therefore it is also referred to as the Maxwell more parameters, related to the naturally occurring spectrum fluid) and recovery (only elastic deformation is recovered, of relaxation times of the real viscoelastic material. Such a andthisisdoneinstantaneously). eK Th elvin-Voigt model, goal can be accomplished by using a complex model such on the other hand, handles creep and recovery fairly well but as the generalized Wiechert model, which consists in several does not account for relaxation. Moreover, the latter model Maxwell units and a free spring, connected in parallel. eTh exhibits no instantaneous elastic response; consequently, the shear relaxation modulus function of the Wiechert model can elasticity modulus is formally infinite, and contact pressure be expressed as [18] results to be infinite at the contact boundary in the beginning of the loading process, as demonstrated by Ting’s model −𝑡 Ψ (𝑡 )=𝑔 + ∑𝑔 exp ( ), (7) ∞ 𝑖 [6, 7] implementation reported in [19]. er Th efore, it can be 𝑖=1 asserted that the assumptions of the Kelvin model make it inappropriate for contact analyses. where 𝑔 is the long-term modulus (longitudinal or shear) The Maxwell and Kelvin models are adequate for quali- once the material is totally relaxed and 𝜏 and 𝑔 ,with 𝜏 = 𝑖 𝑖 𝑖 tative or conceptual analyses, but quantitative representation 𝜂 /𝑔 ,aretherelaxationtimeandthespringstiffnessof 𝑖 𝑖 of the behavior of real materials requires an increase in the each Maxwell subunit. eTh naturally occurring spectrum of number of parameters. eTh generalized Maxwell model is relaxation times of a viscoelastic material can be described composed of a number of Maxwell models and an isolated by includingasmanyexponential termsasneeded.Relation spring in parallel, whereas the generalized Kelvin model (7) is also referred to as a Prony series. The Prony series consists in a number of Kelvin units plus an isolated spring ofaviscoelasticmaterialisusually obtained by aone- in series. eTh Standard Linear Solid Model, also known as the dimensional relaxation test, in which the viscoelastic material Zener model, can be represented as a spring of shear modulus is subjected to a sudden strain that is kept constant, while 𝐺 in series with a Kelvin model of parameters 𝐺 and 𝜂 or measuring the stress response over time. eTh initial stress 𝐾 𝐾 asaspringinparallelwithaMaxwellmodel.Byadoptingthe is related to the purely elastic response of the material. former representation, the differential equation for the Zener Later on, the stress relaxes due to the viscous eeff cts in the model results as [18] viscoelastic material. Mathematical description employing the Prony series can be achieved by tfi ting the experimental 𝐺+𝐺 𝐺 𝐾 𝐾 data to (7) by adjusting the model parameters 𝑔 , 𝑔 ,and 𝜎 ̇ (𝑡 )+𝜎 (𝑡 ) =𝐺 (𝜀 ̇(𝑡 )+ 𝜀 (𝑡 )), (4) ∞ 𝑖 𝜂 𝜂 𝐾 𝐾 𝜏 . 𝜋𝑅 𝜋𝑅 4 Advances in Tribology The latter equations express the PMMA creep compliance in terms of shear modulus by using a Poisson’s ratio ] = 0.38 [21]. eTh semi-analytical solution of the contact problem involving linear viscoelastic materials described by complex rheologicalmodelslikethePronyseriesisdiscussed in the following sections. 3. Contact Model eTh contact model employed in this paper is based on the general contact model developed by Johnson [20] for the elastic domain. The contact equations, as well as the imposed assumptions and limitations, are repeated here for clarity, and the newly established dependencies, related to the viscoelastic constitutive law described in the previous section, are then discussed in detail. The set of equations and inequalities governing the 1900 contact process are written in a Cartesian coordinate system 0 200 400 600 800 1000 with 𝑥 and 𝑥 axes laying in the common plane of contact 1 2 Time, t (s) (i.e., the plane passing through the initial point of contact, which separates best the two contacting surfaces). The two Figure 1: Relaxation modulus function of PMMA. contacting solids are subjected to a normal force aligned with 𝑥 axis, compressing the two bounding surfaces together. As opposed to a time-independent purely elastic contact process, in which the final state depends only on loading level, the eTh constitutive law of the real viscoelastic material viscoelastic contact state depends on time, as well as on the considered in this paper is that of the polymethyl methacry- loading history, due to the memory effect of the viscoelastic late (PMMA), a thermoplastic polymer whose mechanical materials, thus adding a third time parameter to the elastic properties were studied extensively by Ramesh Kumar and contact model. Narasimhan [21]. es Th e authors obtained experimentally the PMMA relaxation modulus data under uniaxial compression eTh static force equilibrium relates the normal force 𝑊 in a window of observation of 1000 s. Based on their results, to the pressure distribution 𝑝 at any time in the observation the two-term Prony series of PMMA results as [10] window [0, 𝑡 ]. To keep the number of independent parame- ters to a minimum, the contact is assumed to be frictionless, −𝑡 meaning that shear tractions cannot be sustained at the con- Ψ (𝑡 )= [1973 + 254.776 ⋅ exp ( ) + 263.6628 8.93 tact interface. Moreover, the problem is considered as quasi- (8) −𝑡 static, meaning that the inertia forces due to deformation are ⋅( )] , ( MPa), negligible: 117.96 ∞ ∞ which is the modulus relaxation function of the material 𝑊 (𝑡 )= ∫ ∫ 𝑝 (𝑥 ,𝑥 ,𝑡 ) ,𝑡∈ [0, 𝑡 ]. (11) 1 2 1 2 0 (expressed in terms of longitudinal modulus), depicted in −∞ −∞ Figure 1. The equations of the surface of separation between the In the time domain, the creep compliance and relaxation two contacting bodies yield the geometrical conditions of modulus are not reciprocal (like in the purely elastic case); deformation in the normal direction: that is, Ψ(𝑡)Φ(𝑡)=1 ̸ .However,inthe Laplacetransform domain, the following relation applies [18] to their trans- ℎ(𝑥 ,𝑥 ,𝑡) = ℎ𝑖(𝑥 ,𝑥 )+𝑢(𝑥 ,𝑥 ,𝑡) − 𝜔 (𝑡 ), 1 2 1 2 1 2 forms: (12) 𝑡∈[0,𝑡 ], Ψ (𝑠 )Φ (𝑠 )= , (9) where ℎ𝑖 is the gap between the undeformed (i.e., initial, at time 𝑡=0 ) surfaces, ℎ is the gap between the deformed where 𝑠 isthevariableintheLaplacetransform domain.The surfaces, 𝑢 is therelativenormaldisplacement, and 𝜔 is the latter equation can be used to derive the creep compliance rigid-body approach. function of PMMA by computing first its Laplace transform The contact model is completed with the boundary condi- Φ(𝑠) from (9) and by subsequently applying inverse Laplace tions and constraints, also referred to as the complementarity transform to obtain Φ(𝑡) in the time domain, leading to conditions in the literature of the elastic contact. eTh latter −4 −5 equations are required as the contact area is not known Φ 𝑡 = 7⋅10 − 6.17 ⋅ 10 exp −0.1𝑡 − 8.38 () ( ) a priori and consequently must be found in an iterative (10) −5 −3 manner by a trial-and-error approach. eTh gap ℎ between the ⋅10 exp (−7.47 ⋅ 10 𝑡) , ( ). MPa deformed contacting surfaces vanishes on the contact area, Relaxation modulus function, Ψ(t) (MPa) 𝑑𝑥 𝑑𝑥 Advances in Tribology 5 as no interpenetration of the contacting solids is allowed in discretization is imposed to perform the numerical analysis the frame of elasticity. On the other hand, the gap must be of the contact process. positive outside the contact area, where there is clearance The contact model reviewed herein was also used exten- between the contacting bodies. In the same manner, pressure sively in the simulation of contact scenarios involving is positive on the contact area and vanishes outside the history-dependent processes like plasticity [24], wear [25], or contact area. These boundary conditions and constraints that friction [26] by adding an external loop in which the load was must be satisefi d simultaneously can be expressed as applied incrementally. In the latter contact scenarios, the time parameter does not need to be considered explicitly as long as the history of the contact process is properly simulated (i.e., 𝑝(𝑥 ,𝑥 ,𝑡) ≥ 0, 1 2 the load is applied with sucffi iently small increments). eTh present work attempts to link the contact model to the theory ℎ(𝑥 ,𝑥 ,𝑡) ≥ 0, (13) 1 2 of viscoelastic behavior, in which the material properties 𝑡∈[0,𝑡 ]; depend explicitly on time. Manipulation of the existing semi- analytical solution in the elastic domain, aiming to achieve 𝑝(𝑥 ,𝑥 ,𝑡)ℎ(𝑥,𝑥 , 𝑡) = 0, 𝑡 ∈ [0, 𝑡 ]. (14) 1 2 1 2 0 a viscoelastic contact algorithm, is detailed in the following sections. The assumption of non-negativity of pressure leads to neglectofcontactadhesionandcan be consideredvery 4. Viscoelastic Displacement conservative in the case of viscoelastic materials. Adhesion appears virtually in all contacts between real surfaces, but A surface distribution of normal tractions, such as the the force of adhesion can be often neglected in case of pressure resulting from a mechanical contact process, induces metallic materials, when the actual contact area, established a displacement field whose knowledge is essential in solving between the asperity heights, is much smaller than the the contact problem and in performing stress analysis in the apparent (i.e., between the topographically smooth bodies) contacting bodies. Although the limiting boundary of a real contact area. In the framework proposed in this paper, the solid is intrinsically rough, computational contact mechanics contact solution is achieved using an optimization scheme employ thehalf-spaceassumption,allowingfortheuseof that requires the non-negativity of contact tractions. eTh fundamental solutions (i.e., the Green functions) derived latter are obtained as the solution of a variational problem in the theory of linear elasticity for a semi-infinite body originally formulated in the eld fi of contact mechanics by bounded by a plane surface. For this approximation to remain these authors [22], seeking the minimum of a quadratic form, valid, the slope of the contact geometry must remain small that is, the complementary energy, subjected to constraints, throughout the contact region. eTh normal displacement efi ld that is, the boundary conditions. eTh convergence of this (𝑒) 𝑢 generated in a linear elastic and isotropic solid by a quadratic optimization is guaranteed, but the method fails distribution of normal tractions 𝑝(𝑥 ,𝑥 )is computed by 1 2 when adhesion-like tensile contact tractions are assumed. applying the superposition principle to the Green function It should be noted that adhesion was not considered in (𝑒) 𝐺 (𝑥 ,𝑥 )for the elastic half-space derived by Boussinesq 1 2 either the classic or modern literature [3–15] of viscoelastic [27]: contact. The contact complementarity conditions imposed in the elastic contact model used in this paper match the ∞ ∞ (𝑒) (𝑒) 𝑢 (𝑥 ,𝑥 )=∫ ∫ 𝐺 (𝑥 −𝑥 ,𝑥 −𝑥 ) boundary conditions employed in [7] and lead to a surface 1 2 1 2 1 2 −∞ −∞ (15) displacement compatible with the indenter profile within the contact area. As the integration of adhesion in analytical or ⋅𝑝(𝑥 ,𝑥 )𝑑𝑥 , 1 2 1 2 semi-analytical contact models is still in its early stages [23], no step back is made in the framework proposed in this paper. (𝑒) 2 2 where 𝐺 (𝑥 ,𝑥 ) = (1]−)/(2𝜋𝐺 𝑥 +𝑥 )is the normal 1 2 The difficulty in solving the contact model (11)–(14) stems 1 2 displacement inducedatapointofcoordinates (𝑥 ,𝑥 )by from the fact that neither the contact area nor the pressure 1 2 a unit concentrated force acting in origin along direction distribution is known in advance. An iterative approach → is therefore needed, involving a trial-and-error approach, of 𝑥 ,and ] and 𝐺 are Poisson’s ratio and the shear modulus in which a contact region is assumed, and the pressure of the elastic half-space. distribution is then computed based on this assumption. If all Lee and Radok [3] obtained the contact radius in the constraints in the contact model are veriefi d by the obtained viscoelastic spherical contact problem by applying the hered- solution, the contact problem solution is achieved. This solu- itary integral operator of type (2) to the Hertz (i.e., purely tion is unique based on the theorem of uniqueness of solution elastic) solution in which the elastic compliance 1/(2𝐺)was of the elastostatic problem. Otherwise, the contact area is replaced by the viscoelastic creep compliance Φ(𝑡).This adjusted and a new pressure distribution is computed with the course of action is justiefi d by the classic method for solving new guess. This iterative approach requires that the response the linear viscoelastic problems of stress analysis, which is of the contacting material, that is, the displacement induced basedonthe conceptofassociatedelastic problem[1,2]. by the surface tractions, is computed for arbitrary contact Capitalizing on the fact that basic integral equations for area and pressure distribution. eTh latter computation can stress analysis in viscoelastic materials reduce in the Laplace only be achieved semi-analytically, and therefore a problem transform domain to the type of integral equations describing 𝑑𝑥 6 Advances in Tribology stresses in elastic materials, it has been shown [1, 2] that the viscoelastic solution holds true as long as the contact a viscoelastic problem has an associated elastic problem, to radius increases monotonically with time, but additional which the former reduces aeft r removal of time dependency manipulations are required when the time-dependent contact by application of the Laplace transform. Consequently, if the area is an arbitrary function of time, as shown in [6, 7]. boundary conditions are time-independent, a solution in the In this paper, the same technique of replacing the elastic frequency domain is identical in form to the correspond- contact compliance with the viscoelastic creep compliance ing elastic solution. This technique of deriving viscoelastic function is applied to (15): solutions from their elastic counterpart is also referred to as the correspondence principle. eTh indentation of a 𝜕 (V) (𝑒) (16) 𝑢 (𝑥 ,𝑥 ,𝑡 )=2𝐺 ∫ Φ (𝑡−𝑡 ) 𝑢 (𝑥 ,𝑥 ) , 1 2 1 2 viscoelastic half-space by a rigid indenter cannot generally be 𝜕𝑡 solved in this manner, as the contact problem features time- dependent boundary conditions, which impede transfer to yielding the viscoelastic displacement generated by a known Laplace domain. When applying this technique to the contact history of pressure 𝑝(𝑥 ,𝑥 ,𝑡 )in a window of observation 1 2 radius formula in the associated Hertz elastic problem, [0, 𝑡]: 𝑡 ∞ ∞ (1− ])𝑝(𝑥 ,𝑥 ,𝑡 ) [ 1 2 ] (V) 𝑢 (𝑥 ,𝑥 ,𝑡) = Φ(𝑡 − 𝑡 ) . (17) ∫ [ ∫ ∫ ] 1 2 1 2 2 2 𝜕𝑡 0 −∞ −∞ 𝜋 (𝑥 −𝑥 )+(𝑥 −𝑥 ) 1 2 1 2 [ ] Unlike its counterpart expressing the contact radius in used in conjunction with any history of boundary condi- a Hertz-type viscoelastic contact, the displacement equation tions. Interchanging differentiation and integration in (17) (17) does not require additional manipulations and can be yields 𝑡 ∞ ∞ (1− ])Φ(𝑡 − 𝑡 ) (𝑥 ,𝑥 ,𝑡 ) 1 2 (V) 𝑢 (𝑥 ,𝑥 ,𝑡) =∫ ∫ ∫ . 1 2 (18) 1 2 2 2 𝜕𝑡 0 −∞ −∞ 𝜋 (𝑥 −𝑥 )+(𝑥 −𝑥 ) 1 1 2 2 In a viscoelastic contact problem, the contact area digitized counterpart for each continuous distribution. This and the pressure distribution are not known in advance discretization encourages a simplified notation taking as and, moreover, keep changing during the contact pro- arguments the indexes of the cells rather than the continuous cess, as the response of the viscoelastic material also coordinates. For example, 𝑝(𝑖, 𝑗),with 𝑖 and 𝑗 integers, changes with time. Consequently, integral (18) must be denotes the pressure value computed in the center of the evaluated for various loading histories, implying integration (𝑖) (𝑗) (𝑖) (𝑗) cell (𝑖, 𝑗),and𝑝(𝑖, 𝑗) = 𝑝(𝑥 ,𝑥 ),where𝑥 and 𝑥 1 2 1 2 of arbitrary functions over arbitrary domains. eTh semi- are coordinates of the center of the cell (𝑖, 𝑗).Consequently, analytical treatment of these equations to attain a compu- pressure is assumed to be uniform in any rectangular patch tationally friendly form is detailed in the following sec- and therefore can be factored outside the integral operator tions. (𝑒) in (15). eTh integral of the Green function 𝐺 (𝑥 ,𝑥 ), 1 2 taken over the elementary patch of side lengths Δ and → → 5. Problem Discretization Δ along directions of 𝑥 and 𝑥 , respectively, yields 2 1 2 the influence coefficient [28] for the elastic displacement Neither integral (15) for the elastic case nor (18) for the vis- (𝑒) 𝐾 : coelasticframework canbecomputedanalyticallyforgeneral contact geometry, loading history, or material properties. 𝑥 (ℓ)+Δ /2 𝑥 (𝑘)+Δ /2 2 2 1 1 (𝑒) (𝑒) Important research eo ff rts were dedicated to obtaining the 𝐾 (𝑖 −𝑘,𝑗−ℓ) = ∫ ∫ 𝐺 (𝑥 (𝑖 ) 𝑥 (ℓ)−Δ /2 𝑥 (𝑘)−Δ /2 2 2 1 1 solution of these integrals in a semi-analytical manner [17, (19) 28]. The principle of the semi-analytical method consists −𝑥 ,𝑥 (𝑗) − 𝑥 )𝑑𝑥 , in considering all continuous distributions as piecewise 2 1 2 1 2 constant functions, uniform within each surface element which expresses the normal displacement induced in the in a uniformly spaced rectangular mesh established in the observation cell (𝑖, 𝑗)by a uniform pressure of magni- common plane of contact. Control points must be chosen tude 1/(ΔΔ )Pa acting in the cell (𝑘, ℓ).Theclosed-form for all the elementary cells of the grid (the centers of the 1 2 solution of the double integral (19) was derived by Love cells make good candidates), and all continuous parameters are evaluated in these representing points, resulting in a [29]: 𝑑𝑥 𝑑𝑡 𝑑𝑥 𝑑𝑥 𝜕𝑝 𝑑𝑡 𝑑𝑥 𝑑𝑥 𝑑𝑡 Advances in Tribology 7 𝑘(𝑥 (𝑖 )+Δ /2, 𝑥 (𝑗) + Δ/2) + 𝑘 (𝑥 (𝑖 )−Δ /2, 𝑥 (𝑗) − Δ/2)... 1− ] 1 1 2 2 1 1 2 2 (𝑒) 𝐾 (𝑖, 𝑗) = ( ), 2𝜋𝐺 −𝑘 (𝑥 (𝑖 )+Δ /2, 𝑥 (𝑗) − Δ/2) − 𝑘 (𝑥 (𝑖 )−Δ /2, 𝑥 (𝑗) + Δ/2) 1 1 2 2 1 1 2 2 (20) 2 2 2 2 √ √ where 𝑘(𝑥 ,𝑥 )=𝑥 ln (𝑥 + 𝑥 +𝑥 )+𝑥 ln (𝑥 + 𝑥 +𝑥 ). 1 2 1 2 1 2 2 1 1 2 Within this framework, the semi-analytical counterpart adequate to circumvent the continuous integration in (15). of (15) results as The additional integration over the time span in which the body was loaded in (17) requires an additional temporal 𝑁 𝑁 1 2 mesh capable of simulating the memory effect specific to (𝑒) (𝑒) 𝑢 (𝑖, 𝑗) =∑ ∑𝐾 (𝑖 −𝑘,𝑗−ℓ)𝑝 (𝑘, ℓ ), (21) viscoelastic materials (i.e., the property that the current ℓ=1 𝑘=1 state depends upon all previous states succeeded from the first loading). This temporal discretization should be where 𝑁 and 𝑁 denote the numbers of grids along 1 2 → → chosen so that at 𝑡=0 the body was undisturbed, and the directions of 𝑥 and 𝑥 , respectively. The discrete double 1 2 time increment Δ should be small enough so that, during convolution in (21) can be performed for any imposed 𝑡 each step, the problem parameters can be assumed to be pressure distribution. Optimum algorithmic ecffi iency is constant. A piecewise constant law is thus imposed along achieved using the DCFFT algorithm advanced by Liu et the temporal axis, adding a third parameter to the notation al. [17]. eTh reduction of the order of computation stems implemented in the purely elastic model. For example, from the convolution theorem, which states that the convo- 𝑝(𝑖, 𝑗, 𝑘) is the discrete counterpart of 𝑝(𝑥 ,𝑥 ,𝑡), denoting lution operation reduces to an element-wise product in the 1 2 the pressure in the elementary cell (𝑖, 𝑗)in the spatial mesh, Fourier transform domain. eTh semi-analytical displacement achieved aer ft 𝑘 time increments, where 𝑡=𝑘Δ ,with computation using (21) together with the DCFFT technique is now widely used in computational contact mechanics. In 𝑘 = 1⋅⋅⋅𝑁 . es Th e assumptions regarding the temporal this paper, a generalization of this technique to the case of variation of model parameters authorize the substitution of viscoelastic behavior is proposed. the partial derivative (𝑥 ,𝑥 ,𝑡 )𝑑𝑡 /𝜕𝑡 in (18) with the 1 2 As equations describing the purely elastic model are finite difference 𝑝(𝑖, 𝑗, 𝑘) − 𝑝(𝑖, 𝑗, 𝑘 − 1). Discretization of (18) intrinsically time-independent, spatial discretization is in the time domain yields ∞ ∞ (1− ])Φ (𝑘−𝑛 ) (V) 𝑢 (𝑥 ,𝑥 ,𝑘) = ∑ (∫ ∫ ⋅(𝑝(𝑥 ,𝑥 ,𝑛) − 𝑝(𝑥 ,𝑥 ,𝑛 − 1))𝑑𝑥 ). (22) 1 2 1 2 1 2 1 2 2 2 −∞ −∞ 𝑛=1 √ 𝜋 (𝑥 −𝑥 )+(𝑥 −𝑥 ) 1 2 1 2 It should be noted that, in the latter equation, the reference moment in which the body was undisturbed, with 𝑛≤𝑁 , 𝑘≤𝑁 ,and 𝑛≤𝑘 .Thesemi-analytical counterpart simplified notation related to problem digitization was used 𝑡 𝑡 of (18), discrete in both time and space dimensions, can thus only for the temporal parameter, whereas continuous coor- be expressed as dinates are employed for spatial localization. As pressure is uniform within each elementary patch, it can be factored 𝑁 𝑁 𝑁 𝑡 1 2 outside the spatial integral operator as in the purely elas- (V) (V) 𝑢 (𝑖, 𝑗, 𝑘) = ∑ ∑ ∑ 𝐾 (𝑖 −ℓ,𝑗−𝑚,𝑘−𝑛) tic model, allowing for a viscoelastic influence coefficient 𝑛=1 ℓ=1𝑚=1 (24) (V) (𝑒) 𝐾 defined similarly to its elastic counterpart 𝐾 in (19): ⋅(𝑝 (ℓ, 𝑚, 𝑛 )−𝑝 (ℓ, 𝑚, 𝑛 − 1 )), (V) 𝐾 (𝑖−ℓ,𝑗−𝑚,𝑘−𝑛) where 𝑖 = 1⋅⋅⋅𝑁 , 𝑗 = 1⋅⋅⋅𝑁 ,and 𝑘 = 1⋅⋅⋅𝑁 .This 1 2 𝑡 (23) equation clearly shows that the memory eeff ct is consid- (𝑒) ered explicitly in the displacement computation, as pressure =2Φ𝐺 𝑘−𝑛 𝐾 (𝑖 −ℓ,𝑗−𝑚). ( ) distributions in all previous states (i.e., in all previous time The latter equation employs notation related to dis- increments), together with the current pressure, are needed cretized parameters in both spatial and temporal dimensions. to evaluate the current displacement. It is noteworthy that The influence coefficient (23) expresses the displacement the contribution of the historical pressures can be sepa- observed after 𝑘 time steps in the elementary patch (𝑖, 𝑗)of rated from the contribution of the current pressure, which the spatial mesh, due to a uniform pressure of 1/(ΔΔ )Pa leads to the algorithmic strategy described in the following 1 2 which acted in the patch (ℓ, 𝑚)in the 𝑛th time step aer ft the section. 𝑑𝑥 𝜕𝑝 8 Advances in Tribology 6. Algorithm Overview the descent steps previously computed in the CGM residual minimization process. The viscoelastic contact simulation is achieved by construct- In order to use the CGM, which is proven to converge ing a series of elastic contact problems with boundary only for systems having a symmetric and positive den fi ite conditions that match the ones of the viscoelastic contact matrix, it is not convenient to include the static equilibrium problem at each new time increment in the temporal dis- equation (11) in the system. In this manner, the system matrix cretization. This approach is based on the fact that, provided is in fact the influence coefficients matrix, which is symmetric the compatibility and internal equilibrium equations are and positive definite (as a diagonally dominant matrix). It satisfied instantaneously, any elastic solution to the contact should be noted that the diagonal entries in the influence problem befits an instantaneous viscoelastic solution. The coefficients matrix (for any time 𝑡) are reserved for the coef- set of equations and inequalities (11)–(15) describe in fact a cfi ients expressing the contribution of the pressure located purely elastic contact process, whereas substitution of (15) in each cell to the displacement in the same cell. Moreover, with relation (18) accomplishes the algorithm generalization (20) suggests that the influence coefficients decay rapidly with to the viscoelastic constitutive law. the distance between the excitation (i.e., pressure) and the Due to the robust nature of the semi-analytical formula- effect (i.e., displacement); therefore the influence coefficients tion for the purely elastic contact process, history-dependent matrix is diagonally dominant for ne fi meshes. eTh static processes can also be simulated using this algorithm by equilibrium equation (11) is imposed during each iteration of applying the load in small increments, assuring that the load- the CGM algorithm in an additional correction of the system ingpathisproperlyreproduced.In thecaseofviscoelastic solution outside the CGM core, as depicted in Figure 2. This materials, however, the time parameter appears explicitly, and correction consists in a modification of the nodal pressures the contact parameters are expected to vary even when the proportional to the ratio of the current load to the imposed load level is kept constant. load. Essentially, the nodal pressures result as the solution of The iterative process stops when such a system is found the system of equations arising from digitization of (12). The that its solution in pressure features only positive entries and latter is essentially non-linear due to the presence of the the gap distribution takes only positive values on the entire rigid-body approach 𝜔 but can be linearized by adopting an computational domain. eTh n fi ding of a set of equations of estimate for this parameter based on the current pressure. type (12) whose solution additionally satisefi s both boundary Once initial guess values for pressure and the contact area are constraints and static equilibrium marks the achievement adopted, (12) can be considered for every cell in the contact of simultaneously converging solutions for pressure, contact area, and the resulting set of equations, all having vanishing area,andrigid-bodyapproach.For each newtimeinterval, gap ℎ, provide an estimation of 𝜔. In other words, the rigid- a new set of such contact parameters has to be found, as bodyapproach,thepressuredistribution,andthecontactarea the viscoelastic displacement is updated at each new time areiteratedsimultaneously, andthetrue valueof 𝜔 is found increment to account for the new pressure history. when all contact parameters converge. eTh advantage of this At any iteration and for each newly considered time approach consists in linearization of (12), which can thus interval, the computational domain 𝑃 should fully enclose be solved using appropriate numerical methods for linear the contact area; otherwise, the contact process simulation systems of equations. According to this review [30] on the should be restarted with a larger estimate. As discussed in subject of numerical methods applied to contact mechanics, [28], if the predicted contact area reaches the boundaries the best candidate is the CGM, providing the fastest rate of of 𝑃, spurious pressure concentrators due to end effects residual decrease. compromise the solution precision. This is especially true Equation (12) applied to all cells in the computational for rough contact scenarios, in which the specimen rough- domain generates discrete equations, but only the ones ness sample should be properly handled by extending the related to cells in the contact area form the system to be geometry information with smooth convex shapes. For a solved. Additional difficulty arises as the contact area is not detailed description of the algorithm to solve the fric- known in advance, and therefore both pressure distribution tionless normal contact problem, the reader is referred to and contact area have to be iterated simultaneously. Con- [15, 28]. sequently, the size of the linear system, which is directly eTh contact solver is adapted in this paper to allow more linked to the number of cells in the contact area, may also general boundary conditions, thus addressing an impor- vary during the iterative process. At any iteration, cells with tant part of indentation scenarios, in which the rigid-body negative pressure are excluded from the contact area, while approach (i.e., the displacement history), rather than the load cells with negative gap (i.e., cells where the contacting bodies (i.e., the surface stress history), is imposed. eTh contact solver are predicted to overlap) are (re)included. These adjustments advanced by Polonsky and Keer [28] is load-driven (LD) are requested by the contact complementarity conditions in (i.e., the applied normal force is expected as input) but can (13) and (14) and force the computed nodal pressures to com- be modified for displacement-driven (DD) scenarios (i.e., in ply to the boundary conditions of the contact problem. As which the load is not specified, but the rigid-body approach depicted in Figure 2, these restrictions are imposed outside is imposed). A distinctive feature of the original solver [28] thecoreoftheCGM-type iteration. er Th esizingofthelinear is that the normal approach is not explicitly computed. This system leads to reconsideration of the descent directions and can be achieved as the algorithm [28] minimizes the residual Advances in Tribology 9 START Conjugate Gradient Method (CGM) Acquire the input: LD: Linearize LD, DD: Adopt LD, DD: initial geometry, contact the system by LD, DD: guess values for Compute the compliance, and estimating the Compute the pressure CGM descent LD: force rigid-body CGM residual distribution direction DD: rigid-body approach approach No LD: Adjust LD, DD: Adjust LD, DD: LD, DD: Verify pressure LD, DD: Verify Yes pressure to Compute the pressure according to complementarity minimize the CGM descent convergence static conditions CGM residual step equilibrium No Yes LD, DD: Adjust system STOP size and reset the CGM descent direction Figure 2: Generalized algorithm flowchart for the instantaneous (i.e., with 𝑘 fixed but arbitrary) contact state. 𝑟(𝑖, 𝑗, 𝑘) computed from a modified interference equation of The flowchart for the generalized algorithm, comprising type (12): both common and specific steps pertinent to each type of boundary conditions, is depicted in Figure 2. 𝑟 (𝑖, 𝑗, 𝑘) = ℎ𝑖 (𝑖, 𝑗, 𝑘) + 𝑢 (𝑖, 𝑗, 𝑘) , It was shown in the previous section that the instanta- (25) neous viscoelastic displacement can be derived based on the (𝑖, 𝑗) ∈ ,𝑃 𝑘 = 1 ⋅ ⋅ ⋅ 𝑁 , knowledge of all previous contact states. Capitalizing on this result, the simulation of the viscoelastic contact process can in which the normal approach is not directly considered. As be achieved by solving a series of sequential contact states rigid-body approach is independent of spatial localization related to a temporal discretization n fi e enough so that the (but dependent of time), its contribution is intrinsically memory effect specific to viscoelastic materials is accurately accountedforwhen theresidualiscorrected by itsmean captured. eTh main algorithm steps are detailed as follows: value[28]. In theDDformulation,ontheotherhand, the normal approach is known as input data, so the interference (1) Acquire the input of the viscoelastic contact problem: computationshouldbeconductedasinthetheoreticalmodel. the loading history 𝑊(𝑡), 𝑡∈[0,𝑡 ],theinitialcontact The system residual matches the gap ℎ between the deformed geometry ℎ𝑖(𝑥 ,𝑥 ), and the contact compliance, 1 2 surfaces and vanishes on the contact area: that is, the creep compliance function Φ(𝑡)of the 𝑟 (𝑖, 𝑗, 𝑘) = ℎ𝑖 (𝑖, 𝑗, 𝑘) + 𝑢 (𝑖, 𝑗, 𝑘) − 𝜔 (𝑘 ), viscoelastic material. (26) (2) Adopt the computational domain 𝑃 and the spatial (𝑖, 𝑗) ∈ ,𝑃 𝑘 = 1 ⋅ ⋅ ⋅ 𝑁 . and temporal discretization parameters: 𝑁 , Δ , 𝑁 , 1 1 2 Δ , 𝑁 ,and Δ . As a result, the residual correction by its mean value in 2 𝑡 𝑡 the original algorithm [28] becomes redundant and should (3) Digitize all problems inputs. Obtain ℎ𝑖(𝑖, 𝑗),(𝑖, 𝑗) ∈ be removed. The lack of normal load as input, on the other ,𝑃 𝑊(𝑘), Φ(𝑘) , 𝑘 = 1⋅⋅⋅𝑁 . hand, compromises the adoption of the guess value for the (4) Compute the array of influence coefficients 𝐾(𝑖, 𝑗, 𝑘), pressure distribution, as the mean pressure can no longer (𝑖, 𝑗) ∈ ,𝑃 𝑘 = 1⋅⋅⋅𝑁 ,which canbestoredasathree- be computed. In our numerical simulations, no convergence dimensional array, having two dimensions related to problems occurred with nonvanishing uniform pressure dis- the spatial dimensions and the third dimension for tribution as initial guess. eTh algorithm sequence performing the temporal discretization. In the 𝑘th temporal step, pressure correction with respect to the imposed load also with 𝑘≤𝑁 ,onlythe rfi st 𝑘 layers of this array are becomes redundant and should be eliminated. The fact that used. thecorrectvalueofthe rigid-bodyapproach is knownfrom the start coerces pressure to converge without any additional (5) Solve an initial contact state (𝑘=1 ) with the contact algorithm modicfi ations. eTh speed of convergence for the compliance Φ(1) and obtain the related pressure modiefi d DD version of the contact solver was found to distribution 𝑝(𝑖, 𝑗, 1), (𝑖, 𝑗) ∈ .I 𝑃 nthisparticularcase, be of the same order of magnitude as its LD counterpart. the surface displacement is simply the convolution 10 Advances in Tribology Impose a new Get pressure history: No START Contact opened? time increment: (0) (k) p ,..., p k← k + 1 Yes STOP Compute contribution of pressure Impose vanishing history to displacement: pressure: (0) (k−1) (0) (k) (k) u =u p ,..., p ,K ,..., K p =0 ＢＣＭＮ ＢＣＭＮ No Compute post-unloading Yes End of simulation displacement: LD time interval? LD or DD? (0) (k) (0) (k) u =u p ,..., p ,K ,..., K pu pu DD Solve the load-driven instantaneous Solve the displacement-driven contact problem: instantaneous contact problem: (k) (k) (0) (k) (k) (k) (0) (k) p =p u ,K ,W p =p u ,K , ＢＣＭＮ ＢＣＭＮ Figure 3: Algorithm flowchart for the transient viscoelastic contact. between pressure and the influence coefficients, with theDCFFT technique[17]incomputation of viscoelastic no memory effect to account for: displacement, leading to an improved order of operations of 𝑂(𝑁 𝑁 𝑁 log(𝑁 𝑁 )). Practically, a viscoelastic contact 1 2 1 2 𝑁 𝑁 1 2 scenario with a 256 × 256 spatial discretization and 200 𝑢 (𝑖, 𝑗, 1) = ∑ ∑𝐾(𝑖 − ℓ,𝑗 − ,𝑚 1)𝑝 ℓ, 𝑚, 1 , ( ) time steps can be solved on a 3 GHz quad-core processor in a (27) ℓ=1𝑚=1 matter of minutes. (𝑖, ) 𝑗 ∈ .𝑃 In this paper, an additional conditional branch is added to allow for computation of displacement which persists (6) Apply a new time increment (i.e., increase 𝑘)and (for a limited or an indefinite time period, according to the solve the 𝑘th instantaneous contact state to get imposed rheological model) aer ft contact opening. eTh semi- 𝑝(𝑖, 𝑗, 𝑘), (𝑖, 𝑗) ∈ .Th 𝑃 is canbeachievedasall analytical equation (24) for viscoelastic displacement com- historical pressures ( 𝑝 ,𝑗𝑖 ,1)⋅⋅⋅𝑝(,𝑗𝑖 ,𝑘 − 1),(,𝑗𝑖 ) ∈ putation holds true even after contact opening by assuming 𝑃 are known at this point. The product between vanishing pressure distributions for the time steps subsequent the influence coefficients and the historical pressure to load removal and until the load is resumed. However, an distributions can be included in (12) as an initial additional algorithm conditional branch should be added, state(i.e.,superimposedto ℎ𝑖 ). In this manner, the as the contact solver cannot handle vanishing loads directly. structure of the model (11)–(14) remains unchanged, The flowchart of the extended algorithm, proposed for the and the same type of algorithm can be applied to simulation of contact problems involving linear viscoelastic solve each new time step in the viscoelastic contact materials, with general boundary conditions and arbitrary problem. This technique is similar to the one used loading programs, is presented in Figure 3. In that figure, spa- in the semi-analytical resolution [24] of elastic-plastic tial localization is omitted for brevity and integer superscripts contact problems, in which the residual displacement are used to index the referred time increment. due to the plastic region is added to the contact geometry, forming a modiefi d initial state. 7. Code Validation and Results (7) If the end of the simulation window is reached (i.e., 𝑘=𝑁 ), stop program execution and export the The newly proposed algorithm was benchmarked against computed data. Otherwise, go to step 6. the partially analytical results derived by Ting [6, 7] for the Based on (24), it follows that the computational com- Maxwell and Zener viscoelastic half-spaces indented by a 2 2 2 plexity of the proposed algorithm is of order 𝑂(𝑁 𝑁 𝑁 ) rigid spherical punch of radius 𝑅 in a step loading 𝑊(𝑡) = 1 2 𝑡 when direct multi-summation is applied. eTh algorithmic 𝑊 𝐻(𝑡),where 𝐻(𝑡)denotes the Heaviside step function and efficiency can be increased dramatically by implementing 𝑊 is xfi ed but otherwise arbitrarily chosen. It should be 0 Advances in Tribology 11 1 1.4 t= 0 (Hertz) 1.2 0.8 t = 0.5 ∗ t= 0.6 0.8 t= 2 ∗ 0.6 0.4 0.4 0.2 0.2 01 0.2 0.4 0.6 0.8 01 0.5 1.5 Dimensionless time Dimensionless radial coordinate, r/a loading case 1 Figure 4: Comparison of pressure history in the step loading loading case 2 spherical indentation of a Maxwell half-space. loading case 3 Figure 6: Evolution of contact radius in the indentation of a Maxwell half-space under specific loading programs. t= 0 (Hertz) t= /5 0.8 t= /2 match well the data resulting from numerical computation of relations (2) and (6), respectively, displayed using contin- uous lines. Dimensionless pressure distribution and radial 0.6 coordinate are defined as ratio to Hertz central pressure 𝑝 and contact radius 𝑎 , both corresponding to the maximum loading level 𝑊 .Thegoodagreement betweenthe twodata t= sets legitimizes the novel algorithm for viscoelastic contact 0.4 simulation. t= 5 It should be noted that the contact model for the Maxwell material predicts a contact radius that grows indefinitely 0.2 in indentation under constant load. eTh solution should, however, be rejected for large strain. Three different loading programs are considered in the following simulations concerning the Maxwell half-space: 01 0.5 case 1, instantaneous loading followed by ramp unloading Dimensionless radial coordinate, r/a H 𝑊(𝑡) = 𝑊 (1 − 𝑡/(2𝜏))𝐻(𝑡);case2,instantaneous loading and subsequent polynomial unloading 𝑊(𝑡) = (1 + (2 − Figure 5: Comparison of pressure history in the step loading 𝑡/𝜏) )𝐻(𝑡)𝑊 /9; and case 3, sinusoidal loading 𝑊(𝑡) = spherical indentation of a Zener half-space. 𝑊 sin(𝑡/𝜏)𝐻(𝑡). Dimensionless contact radii depicted in Figure 6 are normalized by the Hertz counterpart 𝑎 cor- responding to the maximum load attained in the loading noted that whereas Ting’s framework requires the punch to program (i.e., 𝑊 in all three cases). eTh time axis uses 2𝜏 as be axisymmetric and leads to tedious algebraic manipulations normalizer for loading programs 1 and 2 and for the third duetothe vfi epossiblealgorithm brancheswhenthecontact case, with 𝜏 being the relaxation time of the Maxwell unit. eTh area does not vary monotonically with time, the algorithm predicted contact radius trends agree well with Ting’s results proposed in this paper can readily handle irregular contact [6], proving the ability of the computer code to handle arbi- geometry, generalized boundary conditions, and complex trary loading histories (i.e., contact radii whose evolution is material properties. described by increasing and/or decreasing functions of time). The pressure distributions predicted by the newly It is noteworthy that the maxima and minima of the contact advanced computer program at several times aer ft the first radii do not necessarily coincide with those of the loading contact, depicted with discrete symbols in Figures 4 and 5, program. Dimensionless pressure, p/p Dimensionless pressure, p/p H H Dimensionless contact radius, a/a 𝜋𝜏 12 Advances in Tribology 2.2 1.6 immediately t= t= 3/4 1.4 aer unloadin ft g 1.8 t= 1.2 t= 2 t= 0 (Hertz) t= 2 1.6 1.4 0.8 1.2 0.6 t= 4 0.8 t= 8 0.4 t= 3 post-unloading 0.6 0.2 t= /2 0.4 t= 0 (Hertz) 0.2 01 0.5 1.5 01 0.5 1.5 Dimensionless radial coordinate, r/a Dimensionless radial coordinate, r/a H H (a) (b) Figure 7: Surface displacement in a loading program consisting in a step loading at 𝑡=0 followed by instantaneous unloading at 𝑡=2𝜏 ;(a) Maxwell half-space and (b) Zener half-space. In order to test the novel DD formulation for the linear DMA methodology employs the so-called complex modulus viscoelastic domain, the simulation of a Zener viscoelastic of the viscoelastic material, consisting in a real part, that is, half-space with 𝐺 =𝐺 , indented by a rigid sphere of radius the storage modulus related to the elastic energy storage, and 𝑅, was conducted following an imposed history of rigid-body an imaginary part, that is, the loss modulus related to the approach, as resulting from Ting’s model applied to a step internal energy dissipation. This complex modulus results by applying the Fourier transform to the Boltzmann hereditary loading, that is, described by the equation 𝜔(𝑡) = 𝑎 (𝑡)/𝑅, integral of type (1).Thephaseangleofthecomplexmodulus with the contact radius 𝑎(𝑡)given by (3). eTh predicted his- is defined as the hysteresis angle of the strain response tory of pressure distribution over a [0, 5𝜏] time range, similar falling behind the stress excitation. In a DMA procedure, the to the one depicted in Figure 5, was compared to that gener- response in amplitude and the phase angle of the viscoelastic ated by the numerical computation of (6). A good agreement specimen to various excitation frequencies are employed to between the two data sets was found, legitimizing the new compute the complex modulus in the frequency domain. displacement-driven contact model. The latter modulus can be further used [32, 33] in the Figure 7(a) depicts the surface displacement profiles in characterization of the contact behavior of the viscoelastic the step loading of a contact between a Maxwell viscoelastic material under harmonic load indentation. half-space and a rigid spherical indenter, followed by instan- An approximate analytical solution in terms of the taneous unloading at time 𝑡=2𝜏 .Theelastic component viscoelastic material complex modulus for the indentation of the post-unloading displacement, that is, that corre- depth amplitude in a frictionless spherical indentation under sponding to the elastic spring from the Maxwell model, is sinusoidal load was derived by Huang et al. [34]. eTh latter recovered instantaneously, while the component related to solution is,however,based on thetentative viscoelastic the dashpot unit is predicted to persist indefinitely. A more contact solution [3] that assumes a monotonically increasing realistic scenario is presented in Figure 7(b), in which a Zener contact area during the loading history. eTh latter limitation viscoelastic contact with 𝐺 =𝐺 is unloaded at time 𝑡=2𝜏 , can be considered too severe unless, in a loading program subsequent to a step loading. In this case, the displacement is recovered gradually and is expected to vanish aeft r a specicfi consisting in a sinusoidal load superposed on a carrier step load, the indentation depth amplitude due to the oscillation time interval. eTh post-unloading displacement profiles are is negligible compared to that due to the step load. eTh displayed using red lines in Figure 7. aforementioned limiting assumptions can be released in the numerical simulation, and the analysis of the hysteresis 8. Dynamic Mechanical Analysis (DMA) energy loss under the cyclic load can be conducted with a precision challenged only by the spatial and temporal The inducement of a steady-state undulated response in a vis- discretization errors. coelastic specimen (e.g., a uniaxial bar) during a steady-state The Zener viscoelastic contact performance under har- oscillation test is oen ft used [31] as a method for assessing the mechanical properties of the viscoelastic material. eTh monicloadisfirstanalyzedinthissection usingthe newly Dimensionless displacement, u/ Dimensionless displacement, u/ H Advances in Tribology 13 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 01 0.5 1.5 01 0.5 Dimensionless rigid-body approach, / Dimensionless rigid-body approach, / H H (a) (b) Figure 8: Contact stabilization under harmonic cyclic load: (a) 𝑓 = 0.1 Hz⋅s, simulation time 𝑡 = 40𝜏 (i.e.,4loadingcycles);(b) 𝑓=1 Hz⋅s, simulation time 𝑡=8𝜏 (i.e., 8 loading cycles). 1.4 proposed semi-analytical model. eTh indentation loading history is sinusoidal, and the phase is chosen so that at 𝑡=0 Hysteretic the load vanishes together with its first derivative: delay 1.2 𝑊 𝑡 =𝑊 /2 ⋅ [1+ sin (2𝜋𝑡/𝜏𝑓 + 3𝜋/2 )]𝐻 𝑡 , () () (28) where 𝑓(Hz ⋅ s)denotes the number of oscillations (cycles) that occur each time interval equal to 𝜏. eTh latter parameter 0.8 and the load amplitude are xfi ed but otherwise can be arbitrarily chosen. By varying 𝑓 (i.e., by varying the excitation frequency with respect to 𝜏), the effect of loading frequency 0.6 on contact parameters can be assessed. Dimensionless time parameter is defined as ratio to 𝜏. 0.4 The numerical simulation predicts that, apart from the first cycles, the harmonic excitation induces a steady-state 0.2 sinusoidal response with the same frequency but out of phase. When 𝑓 is small, that is, in the range of 0.1 Hz ⋅ s, the steady- stateregimeisattainedalmostimmediately,asshown by the 6 6.5 7 7.5 8 force-displacement curve depicted in Figure 8(a). eTh latter Dimensionless time, t/ is similar to that of materials exhibiting elastic hysteresis. In this case,onlythe rfi st loading(i.e.,the ascendingbranchof p/p W/W the rfi st cycle) is unique; subsequent loadings and unloadings / a/a overlaptoanimposedprecision.Withincreasing 𝑓, a specific, Figure 9: Contact stabilization, 7th and 8th loading cycles, 𝑓= but limited, number of cycles is needed for response stabi- 0.1 Hz⋅s, simulation time 𝑡=8𝜏 (i.e., 8 loading cycles). lization, as shown in Figure 8(b). Figure 9 suggests contact parameters stabilization by comparison of responses in the 7th and 8th loading cycles. eTh presence of a hysteresis-type delay between the indentation depth and the excitation load is also demonstrated. that the amplitude of contact radius decays with increasing Further observations can be made based on analysis of excitation frequency, while the amplitude of peak pressure contact response in the steady-state regime in an imposed increases, as shown in Figure 10. Another interesting feature frequency spectrum of the excitation load (with 𝑓 ranging is that, at the same loading level, the contact is more between 0.1 and 1.5 Hz⋅s). eTh numerical simulations suggest conformal on the descending branch of the sinusoid, as Dimensionless load, W/W Dimensionless load, W/W Dimensionless contact parameters 14 Advances in Tribology 1.3 1 1.2 0.8 1.1 0.6 0.4 0.9 0.8 0.2 0.7 01 0.5 1.5 Dimensionless rigid-body approach, / 0.6 H 01 0.5 1.5 f = 0.1 (Hz∗s) f = 0.8 (Hz∗s) f (Hz∗s) f = 0.4 (Hz∗s) f = 1.5 (Hz∗s) p/p Figure 12: Hysteresis loops for various loading frequencies, Zener a/a model. Figure 10: Amplitude of peak contact pressure and contact radius versus excitation frequency. by the hysteresis loop. As this loop is obtained in discrete form, n fi e temporal discretization is required to obtain its 0.7 detailed description. eTh loops presented in Figure 12 were obtained by imposing 100 temporal steps per loading cycle, 0.6 and the simulation time was extended until a steady-state regime was reached (a maximum of 10 cycles were necessary 0.5 for the case 𝑓 = 1.5 Hz⋅s). eTh simulation results suggest that, for the investigated spectrum of excitation frequency and for 0.4 the considered rheological model, the energy loss decays with the increasing of the excitation frequency. A study of the variation of energy dissipation under 0.3 various frequencies of the excitation load is conducted for thePMMAmaterial,involving thesphericalindentation 0.2 of a PMMA half-space in a loading program consisting in a sinusoidal load with an amplitude of 𝑊 ,fixedbut 0.1 otherwise arbitrarily chosen, superposed to a step carrier load of 𝑊 =2𝑊 .Theoscillation wasapplied aeft r thecontact 0 𝐴 stabilization under the step load. Considering the creep 0 0.2 0.4 0.6 0.8 1 1.2 behavior of the contact condition obtained for the rough Dimensionless radial coordinate, r/a contact of a PMMA specimen in [15], 500 s were allowed W= W /2, loading for the contact to reach a steady state: W= W W= W /2, unloading 𝑊 𝑡 =[𝑊 +𝑊 sin (𝜔 𝑡) 𝐻 𝑡 − 500 ]𝐻 𝑡 . (29) () ( ) () 0 𝐴 𝑓 Figure 11: Pressure profiles corresponding to the same loading level, By conducting the numerical simulation under an 𝑓 = 0.6 Hz⋅s. extended frequency spectrum of the excitation load, the stabilized viscoelastic contact behavior and the energy loss per cycle were obtained without any simplifying assumptions. proven by the pressure profiles depicted in Figure 11. The gap The predicted hysteresis loops are similar to those presented is more apparent at smaller loading frequencies. in Figure 12, but the energy loss per cycle, related to the inter- The force-displacement curves depicted in Figure 12 nal viscous friction in the viscoelastic material, was found to prove that the contact process is hysteretic. eTh out-of-phase possess a maximum at a specific harmonic load frequency, response between load and indentation depth is the source as depicted in Figure 13. The numerically computed areas of energy damping in the viscoelastic material. eTh energy of the hysteresis loops enclosed by the force-displacement loss occurring each cycle is quantifiable by the area enclosed curves, normalized by the maximum value, suggest that the Dimensionless pressure, p/p Dimensionless contact parameters Dimensionless load, W/W 𝛿𝐻 Advances in Tribology 15 for the displacement-driven contact is achieved by modifying 1.002 the algorithm for the load-driven contact scenario. Another 0.99 model extension deals with computation of post-unloading 0.98 displacement, allowing for contact openings during the loading program. 0.97 0.998 The predictions of the newly advanced numerical pro- 0.96 gram are benchmarked against existing results for a linear 0.996 viscoelastic half-space described by the Maxwell and Zener 0.95 −3 ×10 rheological models, indented by a spherical punch in step loading. 0.94 The contact simulations under harmonic cyclic load 0.93 suggest that a steady-state, out-of-phase harmonic response is attained aer ft a few cycles. For the Zener theoretical model, 0.92 the amplitude of contact radius, as well as the energy loss per 0.91 cycle, decays with increasing excitation frequency, while the 0 0.05 0.1 0.15 0.2 0.25 0.3 −1 amplitude of peak pressure increases. The contact is found to Harmonic load frequency, (Ｍ ) be more conformal on the descending branch of the loading Figure 13: Energy loss versus harmonic load frequency in the profile. eTh numerical computation of the energy dissipation PMMA contact. in polymethyl methacrylate under an extended frequency spectrum of the excitation load suggests that a maximum is attained at a specicfi frequency. Further increase of the harmonic load frequency leads to an asymptotic value of the energy loss rises to a peak value when 𝜔 increases from −1 area enclosed by the resulting hysteresis loop. 0to0.0031s and then decays with further increase of The performed numerical simulations prove the ability the loading frequency to an asymptotic value. This behavior of the newly proposed algorithm to assist the dynamic may be attributed to the use of two relaxation times in the mechanical analysis in derivation of the complex modulus of PMMA relaxation curve, as opposedtothe singlerelaxation the viscoelastic material, providing an important theoretical time in the Zener rheological model. This result proves the toolforthestudyoftheviscoelasticcontactwithfewer sim- ability of the proposed numerical model to simulate and plifying assumptions than other existing partially analytical predictthecontactperformanceofviscoelasticmaterialswith solutions. complicated mechanical properties, as resulting from actual experimental measurements. Data Availability 9. Conclusions eTh data supporting the PMMA constitutive law are from previously reported studies, which have been cited. eTh A robust algorithm for the resolution of linear viscoelastic results of the numerical simulations that support the findings contact scenarios is advanced in this paper by generalizing of this study are available from the corresponding author an existing well-known numerical solution for the contact upon request. of linear elastic bodies. The method relies on applying the correspondence principle together with the Boltzmann hereditary integral to achieve numerical computation of Conflicts of Interest viscoelastic displacement, based on a temporal discretization eTh author declares that there are no conflicts of interest in which every new state is assessed considering all previous regarding the publication of this paper. time increments, thus simulating the memory effect specific to viscoelastic materials. Compared to other existing meth- ods, the new approach can readily handle arbitrary contact Acknowledgments geometry, arbitrary creep compliance (which can be inputted in discrete form as resulting from experimental data), and This work was partially supported from the project “Inte- arbitrary loading programs, leading to contact radii described grated Center for Research, Development and Innova- by increasing or decreasing functions of time. Whereas the tion in Advanced Materials, Nanotechnologies, and Dis- classical viscoelastic contact solution still requires numerical tributed Systems for Fabrication and Control” (Contract treatments that can oen ft raise convergence issues, the newly no. 671/09.04.2015) and Sectoral Operational Program for proposed algorithm is based on the conjugate gradient Increase of the Economic Competitiveness co-funded from method, whose convergence is guaranteed. eTh computa- the European Regional Development Fund. tional efficiency is increased using a well-established method for rapid calculation of convolution products. References The contact solver assessing the contact parameters from the geometrical condition of deformation is enhanced to [1] E. H. Lee, “Stress analysis in visco-elastic bodies,” Quarterly of allow for more general boundary conditions. The solution Applied Mathematics,vol.13, pp.183–190,1955. 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