DETERMINISTIC CONTACT ANALYSIS OF ROUGH SURFACE USING FINITE ELEMENT METHOD - Diponegoro University | Institutional Repository (UNDIP-IR)

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Chapter 2

Contact of rough surface: a literature survey

2.1.Basics of Contact Mechanics

2.1.1 Introduction

Contact phenomena are abundant in everyday life and play a very important role in engineering structures and systems. They include friction, wear, adhesion and lubrication, among other things; are inherently complex and time dependent; take place on the outer surfaces of parts and components, and involve thermal, physical and chemical processes. Contact Mechanics is the study of relative motion, interactive forces and tribological behavior of two rigid or deformable solid bodies which touch or rub on each other over parts of their boundaries during lapses of time. However, the contact between deformable bodies is very complicated and it is not yet well understood.

The contact theory was originally developed by Hertz [1] and it remains the foundation for most contact problems encountered in engineering. It applies to normal contact between two elastic solids that are smooth and can be described locally with orthogonal radii of curvature such as a toroid. Furthermore, the size of the actual contact area must be small compared to the dimensions of each body and to the radii of curvature (non-conforming contact). Hertz made the assumption based on observations that the contact area is elliptical in shape for such three dimensional bodies. The equations simplify when the contact area is circular such as with spheres in contact. At extremely elliptical contact, the contact area is assumed to have constant width over the length of contact such as between parallel cylinders. The Hertz theory is restricted to frictionless surfaces and perfectly elastic solids. It was not until nearly one hundred years later that Johnson, Kendall, and Roberts [2] found a similar solution for the case of adhesive contact. This theory was rejected by Boris Derjaguin and co-workers [3] who proposed a different theory of adhesion in the 1970s. The Derjaguin model came to be known as the DMT (after Derjaguin, Muller and Toporov) model, and the Johnson et al. model came to be known as the JKR (after Johnson, Kendall and Roberts) model for

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adhesive elastic contact. This rejection proved to be instrumental in the development of the Tabor [4] and later Maugis [5] parameters that quantify which contact model (of the JKR and DMT models) represent adhesive contact better for specific materials.

Further advancement in the field of contact mechanics in the mid-twentieth century may be attributed to names such as Bowden and Tabor [58]. They were the first to emphasize the importance of surface roughness for bodies in contact. Through investigation of the surface roughness, the true contact area between friction partners is found to be less than the apparent contact area. Such understanding also drastically changed the direction of undertakings in tribology. The works of Bowden and Tabor yielded several theories in contact mechanics of rough surfaces.

The contributions of Archard [6] must also be mentioned in discussion of pioneering works in this field. Archard concluded that, even for rough elastic surfaces, the contact area is approximately proportional to the normal force. Further important insights along these lines were provided by Greenwood and Williamson [7], Bush [8], and Persson [9]. The main findings of these works were that the true contact surface in rough materials is generally proportional to the normal force, while the parameters of individual micro-contacts (i.e. pressure, size of the micro-contact) are only weakly dependent upon the load.

2.1.2 Elastic Contact

In the elastic contact surface areaA_e, contact forceP_e, maximum contact pressurep_m, and average contact pressure p_{can be expressed in function of interference .} Derived from Hertz’s theory [1] equation contact surface area for elastic contact is given by A_e for elastic contact:

 R

A_e  ... (2.1)

Contact force P_e derived from equation  _ E ap_m 2 

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3 1 2 2 16 9           RE P_e  2 2 3 16 9   RE P_e  3 2 2 9 16 _ R E

P_e  

2 1 3 2 9 16         _ R E P_e 2 3 2 1 3 4 _ R E Pe 

 ... (2.2)

Maximum contact pressure p_m obtained from equation  _ E R p a m 2  , therefore:

 

12 2 2 R E a E pm          2 1 2         R E

p_m 

 ... (2.3)

Average pressure contact elastic, pis given by: 2 1 3 4 3 2          R E p

p_e _m 

 ... (2.4) In 1951 Tabor [13] stated that the maximum Hertz contact pressure reaches

pm0.6 occurs on beginning of yield so that the average pressure p0.4H. From this relationship can be obtained an average contact pressure p_{with hardness}H at the time of the initial yield point is given by Chang, Etsion and Bogy (CEB model) [10] by equation:

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p ... (2.5) Relationship between maximum contact pressures with hardness at the time of the initial yield point is given by Chang, Etsion and Bogy. It expressed by the equation:

p_m  ... (2.6) By subtituting Equation (2.6) into Equation (2.3), critical interference ₁__CEB is given by: R E kH CEB 2 1

2 

     _  

 ... (2.7)

Kogut dan Etsion (KE model) [12] use value k from [10], so critical interference ₁__KE at the begining of yield:

R E H k_KE KE 2 1

2 

     _  

 ... (2.8)

where k_KE 0.4540.41 and  is Poisson’s ratio.

Zhao, Maietta and Chang (ZMC model) [11] obtain critical interference based from average contact pressure, by subtituting Equation (2.5) in Equation (2.4) obtained critical interference ₁__ZMC:

R E H k _ZMC ZMC 2 1 1 4 3         _  

 ... (2.9)

where k_ZMC 0.4

when₁ contact that occurs is contact elastic. In other hand, when ₁ contact that occurs is contact elastic-plastic or fully plastic.

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2.1.3 Fully Plastic Contact

Fully plastic contact occurs when interference  increased to reach ₂ with average contact pressure p reach value H(Fig 2.1). In fully plastic contact, Zhao, Maietta and Chang using contact plastic model [11].

Figure 2.1: Deformation on asperity [8].

During deformation fully plastic



2ZMC



average contact pressure remains constant at a value H or:

p_p__ZMC  ... (2.10) The contact surface area for the fully plastic contact ZMC model using modeling of plastic contacts [11] is given by:

 R

A_p__ZMC 2 ... (2.11) Contact forces P_p equal with the contact surface area multiplied by the average contact pressure.

H R

P_p__ZMC 2  ... (2.12) Zhao, Maietta and Chang estimated minimum value 2 based on the results

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the fully plastic P_p ( = 2) approximately equal to four hundred times the contact

force at the point of initial yieldPy ( =1) or: 400

 y p P

P ... (2.13) By using equation (2.2) is obtained as follows:

2 3 1 2 1

4 _

R E Py



 ... (2.14)

and

2 3 2 2 1

4 _

R E Pp



 ... (2.15)

By dividing Equation (2.15) with Equation (2.14) is obtained:



₂ ₁



32 P_p P_y 400 ... (2.16)

1 2 54

  ... (2.17)

From the equation above, Zhao, Maietta and Chang (ZMC model) [11] defines the value of interference in the fully plastic 2ZMC limit to the value of critical interference at the yield point ₁__ZMC by the equation:

ZMC ZMC 1_ 2 54

 _  ... (2.18)

While on Kogut and Etsion (KE model) [12] the value of interference in the fully plastic ₂__KE_{limit to the value of critical interference at the yield point}₁__KE_is defined by the equation:

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KE 1_

2 110

 _ 

... ... (2.19)

2.1.4 Elastic-plastic Contact

Contact of elastic-plastic contact is a transition from elastic to fully plastic contact. Elastic-plastic contact occurs when the interference is between ₁_{and (}₁ ₂). At the contact of elastic-plastic, deformation that occurs consists of elastic and plastic deformation. Relations between contact surface area and average contact pressure as a function of the interference is a very complex relationship. Zhao, Maietta and Chang (ZMC model) [11] gives the relationship between the average contact pressure and interference in elastic-plastic contact is given by:





ZMC ZMC

ep H H k

p     _     1 2 2 ln ln ln ln 1     ... (2.20)

Whereas the surface area contact on the elastic-plastic expressed as:

                                  2 1 2 1 3 1 2 1 3 2 1 ZMC ZMC ZMC ZMC ZMC ZMC ZMC ep R A         

 ... (2.21)

By using equation (3.20) and (3.21) the elastic-plastic contact force is the product between the average contact pressures with the contact surface area which yields:

 

              R ZMC k H H P ZMC ZMC ZMC ZMC ZMC ZMC ZMC ZMC ZMC ZMC ep                                                  2 1 2 1 3 1 2 1 1 2

2 ₁ ₂ ₃

ln ln ln ln 1 ...(2.22) Kogut and Etsion [12] give the relationship an average contact pressure against interference in elastic-plastic contact with the equation:

289 . 0 1 19 . 1 _               KE KE ep Y p  

for ₁ ₆

1   KE   ...(2.23)

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117 . 0 1 61 . 1 _               KE KE ep Y p  

for ₆ ₁₁₀

1   KE  

where Y is the yield strength of the material.

While the relationship between surface areas in contact in the interference functions of elastic-plastic contact is given in the equation:

136 . 1 1 93 . 0 _           KE c KE ep A A

for 1 6

1   KE   ...(2.24) 146 . 1 1 94 . 0 _           KE c KE ep A A

for 6 110

1   KE  

Ac-KE is critical contact area when 1KE_.

2.2 Surface Topography: Surface Texture, Roughness, Waviness

Surface topography is the three-dimensional representation of geometric surface irregularities. A surface can be curvy, rough or smooth depending upon the magnitude and spacing of the peaks and valleys and also depending upon how the surface is produced. Surface texture refers to the locally limited deviation of a surface from the ideal intended geometry of the part. A realistic characterization of surface roughness is required to analyze the effect of surface roughness on different tribological parameters. Roughness relates to the closely-spaced irregularities left on a surface from a production process. It is a measure of the fine, closely spaced, random irregularities or surface texture. Waviness is the component of texture upon which roughness is superimposed. It relates to the more widely-spaced irregularities than roughness caused by vibration, chatter, heat treatment, or warping strains [16]. Surface roughness cannot be easily defined by a single parameter. In fact, there are several ways to represent roughness. All rough surfaces having height and wavelength, with the former measured at right angle

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to the surface and the latter in the plane of the surface. The distribution of height is measured from a reference plane (say, mean plane).

The characterization of a surface may be one dimensional (1-D) or two dimensional (2-D) depending upon the machining and finishing process. For 1-D case, height (z) varies with one of the coordinates, whereas in other coordinate there exists a lay where variation of z is comparatively small. But for surfaces made by conventional manufacturing processes, when 1-D characterization is not proper, 2-D surface roughnesses description is required. Atomic Force Microscope (AFM) and 3D surface profilometer are used to improve the resolution and accuracy of the roughness measurement. Typical 1-D and 2-D representation of nominally flat surfaces (polished) are shown in Figure 2.2. (a) and 2.2. (b). Scale used for the height is much larger than that for the wavelength because the height of the asperities from the mean plane is very less compared to their wavelengths [16].

Figure 2.2: Typical representation of a surface: (a) one-dimensional (b) two-dimensional [16].

2.3 Contact Problem of Smooth Surfaces

The study of the contact behavior of deformable bodies has divided into two approaches. The first, regard the bodies are smooth and can be adequately described by

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their nominal geometry, the second admits that all surfaces are comprised of multitude of peaks and valleys which have multiple asperities which is regarded as a rough surface. Hertz [1] was first to analyze the elastic contact between two non-conforming spheres. He gave the analytical solution for the normal contact between two curved bodies for contact pressure and subsurface stress field.

In Hertz analysis, following assumptions were made.

a) Radii of curvature of the contacting bodies are large compared with radius of circle of contact.

b) Contact is frictionless.

c) Surfaces are continuous and non-conforming. d) Strain is small.

e) Each solid can be considered as half space.

Based on these assumptions, the stress fields generated by an indenter contacting an elastic solid can be analyzed [17]. For the case of elastic contact with a spherical indenter of radius R, the radius of the circle of contact a between the indenter and the specimen surface increases with the load. The contact pressure distribution p proposed by Hertz is given by

⁄

The total contact load P can be obtained from the above pressure distribution as,

∫

Where maximum pressure = 3/2 , with denoting the mean pressure. The contact circle radius a is given by,

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is the effective modulus of elasticity and is the effective radius of curvature defined as,

, , are the Young’s modulus, Poisson’s ratio and radius of curvature of the indenter material and , , are the corresponding parameters of the specimen. The depth of indentation h is related to indenter radius by,

When two nominally flat surfaces are brought into contact under load, contact occurs only at discrete spots. The real area of contact is the summation of all individual spots. Real area is only fraction of the nominal contact area. The real area of contact between two solid surfaces has a profound influence on friction and wear of machine parts, thermal and electrical conduction, contact stresses and joint stiffness. Greenwood and Williamson [7] explains how Gaussian distribution can be applied to surface parameters in practice and the reasons for regular occurrence of such a distribution governing each individual effect. For a rough surface in contact with a rigid plane, GW model which assumes constant tip radius for the asperities gives good order of magnitude estimates the number of contacts, real area of contact and nominal pressure at any given separation. Some other important models are available in literature like Greenwood and Tripp model (GT Model) [18], Whitehouse and Archard Model (WA Model) [19], Onions and Archard Model (OA Model) [20], Nayak Model [21], The elliptic Model [22], Hisakado Model [23], Francis Model [24], McCool model [25]. In most of these models, the asperities are assumed to be either spherical, paraboloidal or elliptic paraboloidal in shape and the corresponding Hertzian solution for single contact is used in the analysis.

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2.4 Contact Problem of Rough Surface

Traditionally, surfaces were modeled analytically using assumption and simplifications. Surface produced by any conventional machining/manufacturing processes are never smooth. The surface irregularities are termed as asperities. Asperities were modeled as a variety of geometric shapes. In the past, a number of authors study rough surface contact problem using analytical method (Whitehouse & Archad [19]; Onions & Archad [20]; Bush, Gibson, et al, [8]; Hisokado [23]). Their result was very useful but their application is limited to a relatively small range of loads.

Surface asperity height and contact pattern were treated as probability distributions. Behavior of a single pair of interacting asperities was often extrapolated to describe the behavior of a pair of interacting surfaces covered in asperities [25]. Investigation of the contact itself classically follows two types of approach, either stochastically or deterministically. One of the first models has been proposed by Greenwood and Williamson [7], who assumed that the asperity summits are spherical with a constant radius, the asperities deform elastically and their height follows a Gaussian distribution. Statistical models have had a considerable impact on contact analysis and have been considered by many authors [28-31]. Nevertheless, these models do not take into account the real geometry of the surface and the interactions between the asperities. Deterministic approaches were then developed to introduce a more precise geometric description [41].

Surface roughness can affect the performance of components and system in a wide variety of fields including tribology, fluid sealing, heat transfer, electronic packaging, dentistry, and medicine. Although it is possible to measure the topography of a real surface and incorporate that data into a finite element model [32-34], this practice is still relatively uncommon [35].

In fact, most analysts create probabilistic surfaces based on assumed, known, or desired surface geometry [7, 36-39], in part because the real geometry cannot always be measured. In recent time, researchers work on real surfaces analysis by either experimentally [42] or developed rough surface model in finite element software. Finite element modeling permits contact simulation with complex geometry, boundary condition, material properties, and material models. The finite elements method has

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been used to solve the contact problem for artificial fractal surfaces [46]. Starting from roughness measurements, synthesized fractal surfaces were also used in the studies of Vallet et al. [44-45] where they used a numerical procedure to solve the contact problem.

2.5 Modeling Rough Surface

The classical analysis of rough surface contact problems were based upon statistical models. Their asperities were assumed to have a certain shape and their physical dimensions such as the widths and the heights were assumed to have a certain statistical distribution. The rough surface is represented by a collection of asperities of prescribed shape scattered over a reference plane. The height of the summits has a statistical distribution and is assumed that the contacting asperities deform elastically according to Hertz theory.

Many models describing rough surface have followed the pioneering work of Abbot & Firestone [43]. They attempt to characterize roughness by a series of indicators, such as the arithmetic average of vertical deviation Ra and the mean line m,

the root mean squared Rs or the standard deviation σ. However, the surface can’t be

fully described using only a surface profile in the vertical direction. These were done in order to simplify the problem. With the rapid advances of faster computers within the last decade, the development of more realistic models for contact simulation becomes more feasible.

Webster and Sayles [50] made a computer model for the dry, frictionless contact of real rough surface which uses data recorded directly from stylus measuring instrument. It presented a numerical method of studying the elastic contact of real roughness and topography in order to investigate a wide range of frictionless contact problem. Moreover, they investigated numerical elastic-plastic contact model of a smooth ball on a directionally structured anisotropic rough surface. It’s obtained the contact information such as the extent of plastic deformation and the states of contact over a range of rough surface and result from the analysis have been used to correlate results from friction test. The result is used in Poon and Sayles [50] studied about the effect of surface roughness and waviness upon real contact areas, gasp between contact

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spots, and asperity contact along with their distribution for different surface roughness and effect of σ (height distribution/surface roughness) and β* (spatial structure of a

surface). The result is compared with Bush’s model [8] using stochastic approach. Lee and Cheng [52] also using a computer simulation to made a model for the contact between longitudinally oriented rough surface for simulating contact between purely longitudinal surface. During that time, simulating a model in computer was really taking a lot of time. Several methods were investigated to increase time efficiency and reduce the requirement of the computer memory size. Lee & Ren [53] developed simulating dry contact of three-dimensional rough surface based upon Moving Grid Method (MGM). Its method was able to reduce required RAM (320 Kb from previous 12.8 GB). It method reduce required storage space for deformation matrix to the order of N (Fig 2.3). The computing time to construct the matrix is also proportional to N.

Figure 2.3: A schematic of representation of the three dimensional moving grid method [52].

Chang and Gao [57] determined a new method for contact problem to optimize the efficiency of computer. Two surfaces brought into a contact, a pressure is developed at every surface point inside, and only inside, the true contact area is modeled in algorithm principal. The method is used to calculate the pressures and surface displacements in contacts of rough surfaces. Karpenko & Akav [54] also worked on a computational method for analysis of contact between two rough wavy surfaces for

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which the nominal contact area may be arbitrarily large that both wavy and rough (Fig 2.4).

Figure 2.4: A schematic three dimensional model of Karpenko [54].

Another method to generate rough surface such as successive random midpoint algorithm is used by Pei, et al. [55] when they presented a finite element calculation of frictionless, non-adhesive, contact between a rigid plane and an elasto-plastic solid with a self-affine fractal surface (Fig. 2.5.). All of the rough surface research mentioned above are generated by digitized measured profile of contacting surface and used them for computer simulation.

Figure 2.5: A self-affine fractal surface for L = 256 generated by the successive random midpoint algorithm. Heights are magnified by a factor of 10 to make the roughness visible, and the color varies from dark (blue) to light (red) with increasing height [54].

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Bryan et,al. [47] lately analyze elastic-plastic finite element of line contact between cylinder and rigid plane using ABAQUS. However, they still generated rough surface from measured real surface which imported to ABAQUS using a Phyton script (Fig 2.6). Another finite element research lately from Yastrebov & Durand [48] presented normal frictionless mechanical contact between an elastoplastic material and rigid plane using finite element analysis (FEA) and representative surface element (RSE) approach. Their research also introduced a new reduced model for the analysis of rough surface. Their new model can solve problem in a few second instead of FEA that need a few days. The new model is a series of basic curves obtained by means of elementary finite element computation on a single asperities and phenomenological relations to take into account the interaction between neighboring asperities of the rough surface.

Figure 2.6: Measured rough surface model by Bryant & Evans [46].

In this paper author want to introduce a new way to generate rough surface using numerical method and export its result into finite element commercial software (ABAQUS). Furthermore, analysis to evaluate pressure and contact distribution will be applied on contact problem of rough surface against hard smooth ball. Rough surface is obtained from experimental data and the result is compared from the result from simulation.

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2.6 Three Dimensional Model of Rough Surface in Finite Element Commercial Software

Traditionally, surfaces were modeled analytically using assumptions and simplifications. Asperities were modeled as a variety of geometric shapes. Surface asperities height and contact pattern were treated as probability distributions. The behavior of single pair of interaction asperities was often extrapolated to describe the behavior of a pair of interacting surfaces covered in asperities. These assumptions were not made because they were shown to accurately represent the system of interest, but because they made modeling possible. Therefore, people studied rough surface by mean analyze one single asperity which assumed represent the others asperities of the surface. The topography of surface was taken from measured real surface using optical micro and macro scale surface features and record data digitally.

Schwarzer [56] geometrically construct all sort of rough surfaces by applying mathematical functions. Figure 2.7 shown an example of two surface of equal roughness in a mere mathematical contact situation, yet this model cannot represent the real surface due to the asperities which are homogeny.

Figure 2.7: Model of rough surface from Schwarzer [56].

In recent time, people developed rough surface model in finite element software. Finite element modeling permits contact simulation with complex geometry, boundary condition, material properties, and material models. Bhowmik [16] modeled rough surface with homogeny asperities as shown in Figure 2.8. In his work, the mechanics of contact between a rigid, hard sphere and a surface with a well defined roughness profile

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is studied through experiments and finite element simulation. The well defined roughness profile is made up of a regular array of pyramidal asperities.

Figure 2.8: Rough surface modeled by Bhowmik [16].

David et, al.[49] demonstrated RF MEMS simulation. They used either an optical profilometer (VEECO) or an Atomic Force Microscope (AFM) to capture three dimensional data points of contact surfaces. Then, using Matlab functions they convert the closed surface from a stereo-lithographic format to an ASCII file compatible with ANSYS Parametric Design Language (APDL). In the final step, the rough surface was obtained by creating key points from the imported file. Since the key points are not co-planar, ANSYS uses Coons patches to generate the surface, and then we used a bottom up solid modeling to create the block volume with the rough surface on the top. Figure 2.9 describes the full method developed on ANSYS platform. Meanwhile Figure 2.10. shows rough surface interface in ANSYS.

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Figure 2.10: Methodology for generating rough surface from David et,al [48].

M Kathrine Thompson [26] from mechanical department MIT, presents methods for generating, using, and operating on nonuniform variates for the incorporation of probabilistic rough surfaces in ANSYS (Fig. 2.11). Her work discusses how to decouple the surface from the finite element model by transferring the surface information from arrays to tables.

Figure 2.11: Block with normally distributed rough surface. Mesh created by moving all nodes – actual scale (left), 100x displacement (right) [26].

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Thompson’s model shows asperities with sharp peak instead of smooth. Meanwhile, in the real rough surface, the geometry of the asperities is considered as either hemisphere or ellipsoid as have been proved by previous works on modeling asperities from

measured real surface. Moreover, Thompson’s model is difficult to be meshed because

of its manipulated geometry. This paper will discuss a new way to generate surface in ABAQUS with surface treatment in SolidWorks. Rough surface with smooth asperities is expected as the result of this work. The model behavior from this method will be compared with surface that created normally from ABAQUS.

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The methods are presented for creating solid model geometry from the metrology data. Three example surfaces are imported and used in a contact analysis. For this work, surface metrology data is imported into the finite element program as a two dimensional array. These techniques, combined with the ability to model real surfaces in ANSYS, can be used to help researchers in material science, mechanical and electrical engineering, and beyond to better understand micro scale surface phenomena. However, Thompson’s model shows asperities with sharp peak instead of smooth. Meanwhile, in the real rough surface, the geometry of the asperities is considered as either hemisphere or ellipsoid as have been proved by previous works on modeling asperities from measured real surface. Moreover, Thompson’s model is difficult to be meshed because of its manipulated geometry. This paper will discuss a new way to generate surface in ABAQUS with surface treatment in SolidWorks. Rough surface with smooth asperities is expected as the result of this work. The model behavior from this method will be compared with surface that created normally from ABAQUS.