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Friction 7(6): 587–602 (2019) ISSN 2223-7690 https://doi.org/10.1007/s40544-018-0245-3 CN 10-1237/TH RESEARCH ARTICLE

Three-dimensional finite element analysis of shallow indentation of rough strain-hardening surface

Chenghui GAO1, Henry PROUDHON2,*, Ming LIU1,* 1 School of Mechanical Engineering and Automation, Fuzhou University, Fuzhou 350116, China 2 Centre des Matériaux, MINES Paris Tech, Evry Cedex 91003, France Received: 09 November 2017 / Revised: 09 May 2018 / Accepted: 05 September 2018 © The author(s) 2018. This article is published with open access at Springerlink.com

Abstract: Three-dimensional finite element modeling of the contact between a rigid spherical indenter and a rough surface is presented when considering both the loading and unloading phases. The relationships among the indentation load, displacement, contact area, and mean contact pressure for both loading and unloading are established through a curve fitting using sigmoid logistic and power law functions. The contact load is proportional to the contact area, and the mean contact pressure is related to the characteristic stress, which is dependent on the material properties. The residual displacement is proportional to the maximum indentation displacement. A proportional relationship also exists for plastically dissipated energy and work conducted during loading. The surface roughness results in an effective elastic modulus calculated from an initial unloading stiffness several times larger than the true value of elastic modulus. Nonetheless, the calculated modulus under a shallow spherical indentation can still be applied for a relative comparison. Keywords: finite element modeling; surface analysis; contact mechanics; indentation; spherical indenter

1 Introduction

There has been a plethora of research on the surface roughness and its dependence on the machining parameters [1, 2] owing to its influence on the contact heat transfer [3], wear [4], adhesion, stiction, electrical conductivity of the interface, and surface functions (e.g., coating performance, frictional behavior, and fluid load capacity [5]), and because it plays a key role in the physical properties of micro/nano structures. Adhesion between particles and surfaces, which is closely associated with various technological pro- cesses (e.g., pharmaceutical processes and powder painting), depends on the surface roughness because it reduces the true contact area leading to reduced body interaction [6]. Surface roughness also plays a significant role in the dispersion forces, which can generate severe problems such as spontaneous stiction or permanent adhesion between separate elements

(e.g., suspended structures with a very small gap distance) in micro-electromechanical systems [7]. Lyon et al. [8] theoretically studied the effects of roughness at the atomic scale on the surface plasmon excitation, and found visible effects of the surface roughness on the image potential and stopping power. Through a molecular dynamics simulation, Liu et al. [9] found that the surface roughness plays a significant role in the plasticity initiation of silicon nanowires. Nunez and Polycarpou [10] experimentally investigated the effects of the counterpart surface roughness on the formation of a transfer layer from the polymer films for applica- tion to dry or solid lubrication, and determined the dependence of the friction coefficient and wear rate on the surface roughness. Zhu et al. [11] studied the influence of the surface roughness on wheel-rail adhesion and wear through the use of a rolling- sliding tribometer. They found that increasing the surface roughness will increase the wear, and that the

* Corresponding authors: Henry PROUDHON, E-mail: henry.proudhon@mines-paristech.fr; Ming LIU, E-mail: mingliu@fzu.edu.cn

588 Friction 7(6): 587–602 (2019)

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surface roughness influences the adhesion recovery. Dawood et al. [12] studied the effects of the surface roughness during the friction stir welding of 6061 aluminum alloy workpieces. They found perceptible influences of the surface roughness of a workpiece on the quality of the weld surface, as well as the heat flux, grain refinement, tensile properties, micro-hardness, and fracture of the joints. Curry et al. [13] found that the structure, failure, and thermal conductivity of a columnar coating generated by suspension plasma spaying are strongly influenced by the topography of the surface onto which the coating is deposited.

The important effects of the surface roughness have motivated extensive studies on the contact between the rough surfaces of fractal or sinusoidal features [14−19] (e.g., frictionless normal contact between rough surfaces [20], elastic frictionless non-adhesive contact between rough surfaces [21−23], plastic deformation of rough rolling contact [24], contact stiffnesses of rough surfaces [25], and plastic contact with/without strain hardening [26−28]). Greenwood and Williamson [29] assumed a Gaussian distribution in the heights of spherical asperities through the application of Hertz theory [30] to each asperity. In addition, variations in radius, height, and ellipticity were considered by Bush et al. [31]. A scaling approach was used by Persson [32] as a way to consider the roughness of successive length scales owing to the self-affine fractal characteristic of the surfaces. Because the contact pressure is often sufficiently high to induce plastic deformation, statistical models considering elastic-plastic asperities have also been developed [33−35]. These theoretical models ignore the interaction between neighboring asperities as well as the bulk deformation of the base material, however, and suffer from many shortcomings

(e.g., a discontinuous elastic-plastic transition [34]). Because the actual surface topography, interactions among asperities, and the plasticity can be considered, the finite element method is a deterministic approach to investigating the elastic-plastic contact of rough sur- faces. Pei et al. [36] adopted a fully three-dimensional (3D) finite element (FE) modeling for contact of an elastic-plastic rough surface and a rigid flat surface, and found that the results for elastic-plastic solids were in contrast to previous studies on elastic solids [37]. Poulios and Kilt [38] studied the frictionless contact of nominally flat rough surfaces by applying a 3D FE model with a real surface topography and elastic- plastic material behavior. Dong and Cao [39] simulated the deformation behavior of the elastic-linear plastic asperity of a sinusoidal profile using a 3D FE model, and correlated the subsurface stress conditions with a fatigue layer. Liu et al. [40] studied the normal contact of elastic asperities with an elliptical contact area, and investigated the effects of the geometrical and material parameters on contact stiffness. This work is devoted to addressing the contact mechanism of a rough surface through the finite element modeling of a spherical indentation of an experimentally measured rough surface. The contact variables (e.g., normal load and contact area) are quantified, and explicit relations are presented based on the simulation results.

2 Finite element modeling of rough surface contact

The material used herein is gold, and the surface topography is extracted from atomic force microscopy (AFM). Figure 1 shows an AFM image of a gold

Fig. 1 AFM image of surface topography (size 5 μm×5 μm): (a) three-dimensional and (b) projected views.

Friction 7(6): 587–602 (2019) 589

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surface with a scan size 5 μm × 5 μm, data array of 512 × 512, and surface resolution Δ = 9.8 nm, which is sufficient for modeling the contact behavior because a high resolution of about tens of nanometers is required to properly represent the surface topography [41]. The data are visualized and analyzed using Gwyddion 2.41: the arithmetical mean deviation Ra ≈ 6.71 nm, the root-mean-square (rms) roughness Rq ≈ 8.21 nm, the skewness Rsk = –0.0285, the kurtosis Rku = –0.436; and the rms slope 2| / | 0.94z x (the brackets indicate the average of all values, where z is the surface height). The average peak-to-valley and curvature radius of the asperities are 35 nm and 240 nm, respectively [42].

The one-dimensional height-height correlation function in this study is defined as

2 , ,

1 1

1( ) ( ) ( )

N M m

x x n m l n l l n

H z z N M m

(1)

where N = M = 512 (the same as a data array of 512 × 512 for the AFM scan), and x is the distance from point (n + m, l) to point (n, l). As expected, ( )x xH is proportional to 2Hx (H is the Hurst or roughness exponent) when x is smaller than the lateral length (≈ 10Rq for the case shown in Fig. 2(a)), and ( )x xH reaches 2Rq2 when x is larger than the lateral correla- tion length (( 60 )qR in Fig. 2). Figure 2 is obtained by analyzing the experimentally measured surface data with Eq. (1). As expected, 0 < H < 1, and our result, H = 0.77, for the case shown in Fig. 2 are in line with typical experimental values of H, namely, 0.5 0.8H [43−45]. In addition, 2H = 4–2D (where D is the fractal dimension of the surface profile), and D = 1.23 in line

Fig. 2 One-dimensional height-height correlation function extracted from the image.

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