


ARTICLE 

Year : 2012  Volume
: 2
 Issue : 2  Page : 97103 

Evaluation of the Contact Parameters of A Structural Rigid Sphere and A Deformable Flat Contact Model by Considering the Strain Hardening Effect
Veeramalai Chinnasamy Sathish Gandhi^{1}, Sengottuvelu Ramesh^{2}, Radhakrishnan Kumaravelan^{3}
^{1} Department of Mechanical Engineering, Anna University of Technology Tirunelveli (Nagercoil Campus), Konam, Nagercoil, India ^{2} Department of Mechanical Engineering, Sona College of Technology, Salem, India ^{3} Department of Mechanical Engineering, Velalar College of Engineering and Technology, Erode, Tamilnadu, India
Date of Web Publication  4Aug2012 
Correspondence Address: Veeramalai Chinnasamy Sathish Gandhi Department of Mechanical Engineering, Anna University of Technology Tirunelveli (Nagercoil Campus), Konam, Nagercoil India
Source of Support: None, Conflict of Interest: None  Check 
DOI: 10.4103/09768580.99296
Abstract   
The study aims to get an elastic−plastic contact analysis of a rigid sphere with a deformable flat (RSmodel), by considering the effect of a tangent modulus on the contact parameters of a nonadhesive frictionless elastic−plastic contact. The tangent modulus parameter has been considered for the different materials, and analysis is carried out through finite element contact analysis. A set of generalized equations are derived for evaluating the contact parameters, such as, area of contact between two consecutive steps, volume of the bulged material, and the angle at which the bulged material comes out from the flat. These contact parameters are evaluated with the effect of the tangent modulus. The result shows that for a maximum tangent modulus 0.5E of the material E/Y<1000 and 0.3E of material E/Y> 1000, the elastic core is pushing up to the free surface in the flat, up to this maximum tangent modulus.The pedestal on this investigation the volume of material squeezed in the contact region is decreased as well as the angle at which the squeezed material runaway from the contact region is increased. Keywords: Tangent modulus, E/Y ratio, elasticplastic, elastic core
How to cite this article: Gandhi VC, Ramesh S, Kumaravelan R. Evaluation of the Contact Parameters of A Structural Rigid Sphere and A Deformable Flat Contact Model by Considering the Strain Hardening Effect. J Eng Technol 2012;2:97103 
How to cite this URL: Gandhi VC, Ramesh S, Kumaravelan R. Evaluation of the Contact Parameters of A Structural Rigid Sphere and A Deformable Flat Contact Model by Considering the Strain Hardening Effect. J Eng Technol [serial online] 2012 [cited 2019 Oct 17];2:97103. Available from: http://www.onlinejet.net/text.asp?2012/2/2/97/99296 
1. Introduction   
The theory of contact mechanics is concerned with the stresses and deformation that arise when the surfaces of two solid bodies are brought into contact. The two surfaces that fit exactly or closely together without deformation are called 'conforming contacts,' and the surfaces, or one of the two surfaces that deform when there is a contact area in between them are called 'nonconforming contacts'. When two rough solids are brought into contact under a normal preload, the contact junctions are formed at their contacting asperity tips, which may deform elastically, or in a elasticplastic or plastic manner. The stress and deflections arising from the contact between the two solids have a practical application in hardness testing, wear and impact damage of engineering ceramics, the design of dental prostheses, gear teeth, and ball and roller bearings. In nonconforming bodies, the contact area between them is generally small when compared with the dimensions of the bodies themselves; the stresses are highly concentrated in the region close to the contact zone and are not greatly influenced by the shape of the bodies at a distance from the contact area. The existing contact analyses are carried out based on the stress and strain in the contact bodies under loading and unloading conditions. The need of the present study is to investigate the under loading condition; how the strain hardening effect is influenced in the contact parameters, such as, the contact area, volume of material squeezed, and the angle at which the squeezed broken material comes out from the contact region of the deformed body, an understanding of the Tribological phenomenon such as contact fatigue, wear, and damage.
2. Theoretical Background   
The contact of a sphere and a flat is a fundamental problem in contact mechanics, with important scientific and technological aspects. The subject of the normally loaded spherical contact, stems from the classical work of Hertz in 1881, which derived an analytical solution for the frictionless (i.e., perfect slip) contact of two elastic spheres [1]. It is important to analyze either a single asperity contact or the contacting of rough surfaces consisting of multiple asperity contacts. Two fundamental approaches for modeling a single asperity contact are, either by considering a deformable hemisphere in contact with a rigid flat [2] (Flattening approach), or by solving the contact mechanics problem of a rigid spherical indenter penetrating a deformable halfspace [3] (Indentation approach). In the elastic deformation regime these two approaches are based on the Hertz solution [4], and hence, produce identical results. When they are beyond the elastic deformation, these two approaches yield different contact mechanics responses. Bodies that have dissimilar profiles are said to be nonconforming contacts.
3. Literature Review   
Contact analysis can be traced back to 1882, in which Hertz studied the elastic contact between two glass lenses. The Hertz theory is restricted to the normal frictionless contact between an elastic halfspace with small deformation. Abbott and Firestone (AF Model) [5] introduced the basic plastic contact model, known as the surface microgeometry model. In this model, the deformation of a rough surface against a smooth rigid flat is assumed to be equivalent to the truncation of the undeformed rough surface at its intersection with the flat. Tabor [6] proposed that hardness is not a unique material property. Greenwood and Williamson [7] used the Hertz theory and proposed an asperitybased elastic model where asperity heights follow a Gaussian distribution. In order to bridge the two extreme models of GW (Elastic model) and AF (Plastic model). The Chang, Etsion and Bogy (CEB model) [2] developed an elasticplastic contact model, based on the volume conservation of the plastically deformed asperities. Davis [8] defines hardness as, "Resistance of metal to plastic deformation, usually by indentation". Kogut and Etison [9] (KE Model) have used the solution of the finite element method for the elasticplastic contact of a deformable sphere and a rigid flat, by using constitutive laws appropriate to any mode of deformation, which may be elastic or plastic. It also offers a general dimensionless solution, not restricted to a specific material or geometry. Jackson and Green [4] (JG Model) have incorporated a variation of material property on deformed geometry and present some empirical relations of contact area and contact load. Chang et al. [10] introduced the hardness coefficient, which is related to the Poisson's ratio of the sphere. Malayalamurthi and Marappan [11] analyzed the nature of the material dependency of the elasticplastic contact behavior of a deformable sphere and a rigid flat by using the finite element method. Analysis was carried out between the elastic limit and the inception of plasticity for various materials, with different radii. E/Y values less than 300 show strikingly different contact phenomena. Mayuram and Shankar [12] analyzed that an axissymmetrical hemispherical asperity in contact with a rigid flat is modeled for an elastic perfectly plastic material. This analysis shows the critical values in the dimensionless interference ratios ω/ω_{c} for the evolution of the elastic core and the plastic region within the asperity, for different Y/E ratios. The FE Analysis of a single asperity model with the elastic perfectly plastic assumption depends on the Y/E ratio of the material. Jiayong Tian [13] investigated dynamic contact stiffness at the contact interface between a rigid sphere and a semiinfinite cubic solid. Assal T. Hussein [14] stated that an elasticplastic nonlinear analysis, under the dynamic load Adaptive Shifted Integration (ASI) technique, is capable of predicting the behavior of steel frame structures with reasonable accuracy.
According to the literature review, contact analysis of a deformable sphere with a rigid flat using FE Analysis has been conducted by several researchers, and some of these studies consider the effect of material properties. The reserchers have also roughly taken the tangent modulus as 10% of the Young's modulus. [Figure 1] shows that the RSmodel (like the indentation approach). In the Brinell test a hard ball of diameter 'D' is pressed under a load 'W' into the plane surface under test.
4.0 Materials and Methods   
The present study aims to study the effect of strain hardness for single asperity contact parameters, for different materials, under the loading condition of the RS model. In the indentation process the indenter is a rigid member and the plate is a deformable member. In this study, the RS model chosen for analysis is based on the similarity of an indentation processes. The finite element analysis software 'ANSYS' has been used to carry out this analysis, in an axisymmetric condition. Hence, a quarter sphere is considered for the analysis. The finite element contact model of a rigid sphere against a deformable flat is shown in [Figure 2]. For the RSmodel contact analysis the contact pair is created and confirmed between the sphere and flat.
The meshed model is shown in [Figure 3]. For this investigation all the elements used are eightnoded quadrilaterals, plane 82, conta 172, and target 169 [15]. The nodes lying on the axis of the hemisphere are restricted to move in the radial direction.
Also the nodes in the bottom of the hemisphere are restricted in the axial direction. The average sphere size used for this analysis is 50 mm. Here, the frictionless rigid deformable contact analysis is performed for different materials. The material properties are selected based on the Young's modulus, to yield strength ratio [8]. The following are the materials, shown in [Table 1], used for the application of contact problems, such as, cylinder over a flat plate, wheel and rail contact, roller bearings, and meshing of gear teeth.
4.1 Finite element analysis  Effect of tangent modulus
The tangent modulus is the slope of the stressstrain curve. For strain less than the yield point it will be a constant and is the same as in the elastic modulus. For a higher strain it will vary with the strain, and depending on the material it could be any value from zero, up to bigger than the elastic modulus. A wide range of values of the tangent modulus are taken to get a fair idea about the effect of this in different materials, the hardening parameter, and the area of contact. The tangent modulus is taken as %E for linear hardening material. The FE analysis is carried out for different materials, that is, 500≤E/Y≥1750. The stress values with respect to the tangent modulus of different materials are given in [Table 2].
4.2 Penetration depth and width of contact
The rigid sphere and a deformable flat contact model as shown in [Figure 4]. The load is applied on the top of the sphere. The sphere is penetrated into the flat.
In this study an attempt has been made to modify the indentation depth in the new form by incorporating the tangent modulus, in terms of %E. The loading relationship for the penetration depth is given by the relation
ω={9L^{2} /8D0} ^{1/3} [ 2{(1  ν^{2} ) / (E* + E_{T} ) }] ^{2/3}
In Eq. (1), L is the applied load, D is the ball diameter, and the paired material constants ν, E*, and E_{T} are the Poisson's ratio, equivalent Young's modulus, and tangent modulus, respectively. E* is given by
In Eq. (2), 1 and 2 denote the material properties of the ball and plate, respectively.
The projected surface diameter (d) of the residual impressed indentation is shown in [Figure 1]. The following relationship gives the mathematical formula for calculating the diameter:
The material E/Y value of 552.63 is taken for observation of various parameters and it is related to the contact behavior of the sphere with flat (indentation approach), with incorporation of the tangent modulus. This is given in [Table 3].
4.3 Evaluation of contact area
Estimation of the contact area between the two consecutive steps of penetration is very important for analysis of the contact bodies in contact mechanics. The contact area between the two circles of the sphere and the flat, as shown in [Figure 5], is expressed by the double integral method in polar coordinates ςςrdrdθ, with suitable limits. The limits are between π/2 and π/2, and scosθ and 2scosθ. The area lying inside the two circles is calculated from the following expression. Where, A _{1} =inner circle, A _{o} =outer circle, r=position of circle from the center of contact between the bodies, θ=angle at which the circle propagated, and dr=radius between the two circles.
where s=size of the circle (Difference in contact width).
For the estimation of an area between two circles, the outer circle area is considered to be zero. From this reference the remaining area between the two consecutive circles is calculated, as shown in [Table 4].
4.4 Evaluation of volume of squeezed material
The evaluation of the volume of material squeezed between the two contact bodies is very important for contact analysis. This parameter plays a vital role in contact fatigue failure and wear analysis.
The volume of the material between the two circles of the sphere and the flat, as shown in [Figure 6], is expressed by the triple integral method in Cartesian form 8ςςςdxdydz with suitable limits. The limits are between 0 and s, 0 and . 0, The volume lying inside the two circles is calculated from the following expression.
where s=size of the circle (Difference in contact width).
In the estimation of volume between two circles, the outer circle volume is considered to be zero. From this reference the volume between the two consecutive circles is calculated and shown in [Table 5].
4.5 Estimation of angle
The estimation of the angle at which the squeezed material comes out from the contact zone as shown in [Figure 7] is important for contact analysis, to estimate the principle stresses in the sphere and a flat. The angle at which the squeezed material escapes is calculated and shown in [Table 6].
An estimation of the angle, as shown in [Figure 8], at which the squeezed material comes out from the contact region provides a platform for analyzing the transformation of stress in the contact zone.
5. Results and Discussion   
In this finite element analysis the effect of the tangent modulus is studied. The result shows that an increase in the tangent modulus value increases the stress in the material (E/Y<1000) up to 0.5E, after which the stress decreases. With an increase in the tangent modulus value the stress in the material (E/Y>1000) increases up to 0.3E after which the stress decreases, as shown in [Figure 9]. This is due to the effect of strain hardening of the material.
It shows the nonlinear behavior between stress and tangent modulus. It is observed that higher stress is developed in the material E/Y<1000 of hardness H=1 and H=0.43 than for the material having E/Y>1000. It is confirmed that the tangent modulus increases the hardness of the material. The material behavior is dependent on the tangent modulus.
[Figure 10] shows the relation between the difference in area and volume of material between two consecutive circles and the tangent modulus. The tangent modulus of the material increases linearly up to 0.3E, with an increase in the difference in area and volume of the material between two circles. Subsequently the modulus of the material decreases gradually after 0.3E. When it is very close to the contact surface, this area reduces due to the strain hardening effect of the material for any further increase in the tangent modulus. The effect of tangent modulus has a greater influence on the contact parameter.  Figure 10: Difference in area and volume of the material between two circles versus tangent modulus
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[Figure 11] shows the relation between the angle at which the squeezed material escapes from the contact surfaces and tangent modulus. The tangent modulus of the material increases with the angle at which the squeezed material comes out from the contact region.
The angle is a reference from the line of action of the load applied. It is inveterate that the increase in angle reduces the volume of the squeezed material that comes out from the contact zone toward the free surface. The cause of tangent modulus is greatly influenced by the contact parameter.
6. Conclusion   
The tangent modulus of the material is considered for the study of rigid sphere and a flat contact model. The effect of the tangent modulus on the contact parameters is very important for the contact phenomena. A detailed study of the effect of tangent modulus and strain hardening has been carried out by FE Analysis and analytical solutions. The straining action effect is increased by increasing the tangent modulus. This effect is considered for analyzing contact parameters, such as, area between two consecutive steps, volume of squeezed material, and the angle at which the squeezed material comes out from the contact region. This effect causes a decrease in the area between the two consecutive circles near the top surface of the flat, and the volume of material squeezed also decreases. The angle at which the squeezed material comes out increases with an increase in the straining action. It shows that the elastic core in the contact region is pushing up to the free surface due to the effect of the tangent modulus.
References   
1.  K. L. Johnson, Contact Mechanics, Cambridge University Press: Cambridge, 1985. 
2.  W. R. Chang, I. Etsion, and D. B. Bogy, "An elasticplastic model for the contact of rough surfaces", ASME J. Tribol, Vol. 109, pp. 25763, 1987. 
3.  L. Polin and J. F. Lin, "A new method for elasticplastic contact analysis of a deformable sphere and a rigid flat", ASME Tribol., Vol. 128, pp. 2219, 2006. 
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5.  E. J. Abbott, and F. A. Firestone, "Specifying surface quality  A Method Based on Accurate Measurement and Compration", Mech. Eng. (Am. Soc. Mech. Eng.), Vol. 55, pp. 56972, 1993. 
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8.  J. R. Davis, Metals Handbook, 2 ^{nd} ed., ASM International: Metals Park, OH; 1999. 
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10.  W. R. Chang, I. Etsion, and D. B. Bogy, "Static Friction Coefficient Model for metallic rough surfaces", ASME J. Tribol, Vol. 110, pp. 5763, 1988. 
11.  R. Malayalamurthi and R. Marappan, "Finite element study on the residual strain in a sphere after unloading from the elastic plastic state", Int. J.Comput.Methods Eng. Sci. & Mechan., Vol. 10, pp. 27781, 2009. 
12.  S. Shankar and M. M. Mayuram, "A Finite Element Based Study on the Elasticplastic Transition Behavior in a Hemisphere in Contact With a Rigid Flat", ASME Tribol., Vol. 130, pp. 04450216, 2008. 
13.  J. Tian, "Anisotropy Influence of Cubic Solid on Dynamic Hertzian Contact Stiffness for a Vibrating Rigid Indenter", American Journal of Engineering and Applied Sciences, Vol. 3, pp. 5663, 2010. DOI: 10.3844/ajeassp.2010.56.63. 
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15.  Y. Nakasone, T. A. Stolarski, and S. Yoshimoto, Engineering Analysis with ANSYS Software. 1 ^{st} Edn., ButterworthHeinemann, Oxford, Burlington, Mass., ISBN: 075066875X, pp: 456, 2006. 
Authors   
Mr. V. C. SathishGandhi is an Assistant Professor in Department of Mechanical Engineering at Anna University of Technology Tirunelveli, Nagercoil Campus, Nagercoil, Tamilnadu, India. He completed his M.E. Engineering Design at Anna University, Chennai, India in 2004. He has 11 years of teaching experience. He published one paper in international journal. He presented 4 papers in National and one paper in International conferences. He is member in professional bodies like ISTE, IAENG. His current research interest are contact mechanics and fracture mechanics
Dr. S. Ramesh is a Professor in Department of Mechanical Engineering at Sona College of Technology, Salem, Tamilnadu, India. He completed his M.Tech Mechanical Engineering with the Specialization of Manufacturing Engineering from Indian Institute of Technology Madras, Chennai, India in 1997 and Ph.D. in Mechanical Engineering at Anna University, Chennai, India in 2008. He has 22 years of teaching experience. He published 8 papers in international journals and 2 papers in Indian journals. He presented more than 50 papers in National and International conferences. He is member in professional bodies like ISTE, ASME, FIE, IAEME, CII etc. His current research interest are composite material machining, aerospace material machining, material characterization and contact mechanics
Dr. R. Kumaravelan is a Professor and Head, Department of Mechanical Engineering at Velalar College of Engineering and Technology, Erode, Tamilnadu, India. He completed his M.E. Engineering Design at Bharathiar University, Coimbatore, India in 1999 with University First rank and Ph. D. in Mechanical Engineering at Anna University, Chennai, India in 2011 with a specialization in Functionally Graded materials and composite mechanics. He has 18 years of teaching experience. He published 3 papers in international journals and 2 papers in Indian journals. He presented more than 15 papers in National and International conferences. He is member in professional bodies like ISTE, IIPE, IEI, SAE etc. His current research interest are contact mechanics and composite mechanics.
[Figure 1], [Figure 2], [Figure 3], [Figure 4], [Figure 5], [Figure 6], [Figure 7], [Figure 8], [Figure 9], [Figure 10], [Figure 11]
[Table 1], [Table 2], [Table 3], [Table 4], [Table 5], [Table 6]
