An Incremental Contact Model for Rough Viscoelastic Solids
Graphical abstract
Introduction
Viscoelastic behaviors are frequently observed regarding various materials and structures, such as rail pads [1], frozen soil [2], cells [3], vitrimers [4], robots [5], and gyroscopes [6], to mention a few. The viscoelastic nature of structural components has a significant effect on their mechanical responses [7], [8], [9] and physical processes [10,11]. Recent progresses in functionally graded materials [12], [13], [14], soft materials [15], and microelectromechanical systems [16] also stimulated research on the role of viscoelasticity. In tribology, it is commonly believed that energy dissipation in viscoelastic contact acts as one of the potential origins of friction [17], [18], [19], [20], which was verified experimentally [21,22]. For these reasons, theories of viscoelastic contact sprang up in the last decades.
Theoretical studies on the rate-dependent contact behavior of linear viscoelastic solids were pioneered by Lee [23] and Radok [24]. They provided a method deducing the viscoelastic spherical contact solution from the elastic Hertz solution, which is applicable as the contact area keeps increasing [25]. Their analysis suggested that viscoelastic contact problems can be tackled by referring to their corresponding linear-elastic cases, providing that the boundary condition is properly handled. Hunter [26] extended the solution to the case that the contact area possesses a single maximum. Graham [27] solved the problem for arbitrary asymmetric indentation and Yang [28] for elliptical contact. The contact problem with arbitrary evolution history of contact area could also be addressed by solving elaborate integral equations [29], [30], [31].
The abovementioned theories provide the basis for addressing various kinds of viscoelastic contact problems. Based on Graham's [27] and Ting's [29] theories, a simple form of the initial unloading stiffness can be obtained [32]. Rodriguez et al. [33,34] extended Ting's model [29] to the viscoelastic contact behavior of orthotropic materials. Yakovenko and Goryacheva [35] presented a model for the contact of an array of rigid spherical indenters against a viscoelastic half-space. The loading-unloading contact of two viscoelastic spheres was studied by Jian et al. [36], and the contact of two viscoelastic thin layers was studied by Argatov and Mishuris [37]. Chen et al. [38,39] derived the Boussinesq solutions for linear elastic substrates coated with viscoelastic layers. Usov [40] examined the effect of viscoelastic coating on the contact of lubricated bodies. Starting from the rolling contact theory of Hunter [41], Zhao et al. [42,43] proposed a method for the elastohydrodynamic lubricated contact of viscoelastic solids. Haiat et al. [44] and Greenwood et al. [18] studied the role of adhesion in viscoelastic contact.
The linear viscoelastic theories were also extended to cover the contact of rough solids. By adopting the principle of Lee and Radok [25], Belyi et al. [45] investigated the time dependence of the real contact area, in which the rough surface is treated as a bunch of cylinders with identical radius and varying heights. Creton and Leibler [46] furthered the well-known GW model [47] to study the rough viscoelastic contact. Chau [48] proposed a framework for the dynamics of viscoelastic solids under step-loading. The viscoelastic contact response of rough surfaces with Cantor structures was modeled by Alabed et al. [49].
Apart from theoretical models, numerical techniques, such as finite element methods [20,[50], [51], [52]], boundary element methods [53], [54], [55], Green function methods [56,57], and bi-conjugate gradient stabilized methods [58], were also developed to simulate contact phenomena. The merits of the numerical techniques lie in dealing with more realistic boundary conditions, geometric nonlinearity, material nonlinearity, etc. For example, the Green function methods, pioneered by Carbone and Mangialardi [59], act as a powerful tool in simulating the steady sliding contact of rigid spheres [60], [61], [62] or rigid rough surfaces [63], [64], [65] against viscoelastic substrates. However, the scale of the problems that can be simulated is usually limited by the computing burden.
Certainly, the principle of Lee and Radok [25] allows for tackling the viscoelastic contact problems in a simplified manner as long as the contact area keeps increasing. Ting [29] improved the method to deal with the case that contact area varies arbitrarily. Nonetheless, the sophisticated and nested integrations and differentiations in Ting's method have limited its applications. That is why most existing studies focus on the contact response to constant excitation, and few studies set foot in the cyclic loading of a viscoelastic substrate. The calculation is largely simplified by Greenwood [66]. His study suggested that the contact response of an axisymmetric rigid indenter against a viscoelastic substrate could be regarded as a ‘stack’ of incremental punch indentations, and the variation of the contact area can be achieved by push and pop operations. Recently, such concept was adopted to model the linear-elastic contact of rough surfaces [67,68].
Herein, the principles of Lee and Radok [25] and Ting [29] as well as the concept of Greenwood [66] are adopted to study the viscoelastic contact of rough surfaces. In Section 2 the viscoelastic contact model is thoroughly described. The contact response for standard linear viscoelastic solids under oscillatory harmonic loading is formulated in Section 3. The evolutions of the real contact area and the average interfacial separation as well as the energy dissipation are discussed in Section 4.
Section snippets
The incremental contact model
By accumulating punch indentation responses, Greenwood [66] showed that the contact solution of a rigid sphere and a viscoelastic half-space could be achieved. Based on this concept, Wang et al. [67] recently proposed a model to predict the contact response of linear elastic solids with rough surfaces. Herein, the concept is employed to deal with the contact of rough viscoelastic solids.
Unlike linear elastic solids which react to excitations immediately, viscoelastic solids tend to react
The contact responses for standard linear solids
In this section, we will apply the theory presented in Section 2 to standard linear solids. For standard linear solids, the creep compliance function and relaxation modulus function are given aswhere the modulus reduction ratio k equals E∞*/E0*, E∞* = Ψ(∞) and E0* = Ψ(0) are the relaxed plane strain modulus and the instantaneous plane strain modulus, respectively, and T is the characteristic time for relation.
Consider a rough viscoelastic solid
Results and discussions
In this section, the contact responses of viscoelastic solids with Gaussian random rough surfaces under harmonic excitation are considered. The evolutions of the real contact area and the average interfacial separation will be presented. Based on the relation between average interfacial separation and load, the critical factors that contribute to the viscoelastic energy dissipation will be discussed.
Conclusions
In the present study, an incremental contact model for linear viscoelastic rough solids is presented based on the principle of Lee and Radok. With knowledge of the total area and number of contact patches for any specific surface separation, the viscoelastic contact responses for contact area fractions within the range of 0 to 15 percent are achieved by superposing the creep and relaxation solutions of Boussinesq problem. Evolutions of the real contact area and the average interfacial
CRediT authorship contribution statement
Xuan-Ming Liang: Methodology, Software, Formal analysis, Writing – original draft. Yue Ding: Methodology, Formal analysis. Cheng-Ya Li: Methodology, Validation. Gang-Feng Wang: Conceptualization, Methodology, Writing – review & editing, Supervision.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
Supports from the National Natural Science Foundation of China (Grant No. 11525209) are acknowledged. Y. Ding acknowledges the support from the National Natural Science Foundation of China (Grant No. 12102322).
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