Impact of a Fixed-Length Rigid Cylinder on an Elastic-Plastic Homogeneous Body

Transcription

1 Raja R. Katta Andreas A. Polycarpou 1 polycarp@illinois.edu Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, Urbana, IL Impact of a Fixed-Length Rigid Cylinder on an Elastic-Plastic Homogeneous Body A contact mechanics (CM) based model of a fixed-length rigid cylinder impacting a homogeneous elastic-plastic homogeneous body was developed and includes an improved method of estimating the residual depth after impact. The nonlinear elastic behavior during unloading was accounted for to develop an improved coefficient of restitution model. The impact model was applied to study a practical case of a cylindrical feature on the slider of a magnetic storage hard disk drive impacting the disk to predict various critical impact contact parameters. The CM model was validated using a plane strain finite element model and it was found that a cylindrical feature with a longer length results in a substantial alleviation of impact damage. DOI: / Keywords: contact mechanics, oblique impact, elastic-plastic impact, coefficient of restitution, magnetic storage 1 Introduction Impact between two solids occurs in numerous mechanical devices and could result in unwanted damage to the bodies involved. During impact between two solids, the initial kinetic energy is converted to recoverable elastic strain energy and unrecoverable plastic deformation, stress wave propagation, material damping, and other forms of energy such as frictional heat and sound 1. Hertz elastic contact theory can be applied to calculate normal elastic impact between spherical bodies or a sphere impacting a plane when the deformations due to the resulting impact are quasistatic in nature,3. The impact process can be captured by quasi-static contact theory as long as the energy dissipation due to the resulting stress waves is negligible. Streator 4 numerically examined the dynamic contact of a rigid sphere with an elastic half-space for typical impact velocities greater than tens of m/s and showed that during the initial moments of impact the resulting stresses can be orders of magnitude higher than those predicted using quasi-static theories. However, the time scale during which this happens is extremely small tens of femtoseconds, and once characteristic time scales are reached; the stress relaxes to magnitudes agreeing with quasi-static theories. These high stresses are relaxed or relieved through stress wave propagation. For most impacts encountered in micromechanical applications, e.g., magnetic storage slider-disk impact, typical impact velocities are less than 3 m/s with characteristic time scales of the order of microseconds, resulting in negligible stress wave energy dissipation 3. Material damping and acoustic losses could also be considered minimal. When the impacts result in elastic-plastic deformation around the contact region, plastic deformation can be a significant source of energy dissipation. In these cases, instead of purely elastic Hertz contact models, elastic-plastic contact constitutive models such as Johnson s model 5, which was developed based on analytical approaches, and Kogut and Etsion 6, Jackson and Green 7, Ye and Komvopoulos 8, and Kogut and Komvopoulos 9 models, which were based on finite element calculations, can be used for modeling impact 10,11. The effect of Poisson s 1 Corresponding author. Present address: 140 Mechanical Engineering Building, MC-44, 106 West Green Street, Urbana, IL Contributed by the Tribology Division of ASME for publication in the JOURNAL OF TRIBOLOGY. Manuscript received November 18, 009; final manuscript received July 30, 010; published online October 1, 010. Assoc. Editor: Shuangbiao Jordan Liu. ratio in the contact behavior of cylindrical solids was investigated by Green 1, where it was shown that the critical stress decreases with increasing Poisson s ratio. Lim and Stronge 13 developed an elastic-plastic cylinder contact model based on a similar approach as Johnson s model 5. In applications where the impact is oblique, frictional heat generated at the contacting interface can result in energy loss. Frictional heat dissipation is considerable only when gross slip occurs 14. An oblique impact process for spherical or cylindrical body impacts can be captured by using Coulomb friction along with contact deformation models 15,16. An important observation in oblique impacts is that as long as the tangential tractions do not significantly affect the normal motion in the contacting bodies, the plastic deformation losses remain insensitive to friction. The normal coefficient of restitution e y is a parameter used to capture the energy loss that occurs due to impact in the normal direction to the plane of impact. In general, it can be used to provide a measure of the elastic stress wave losses, e.g., Ref. 16, and plastic deformation, e.g., Refs.,17 19, during normal impact. In oblique impact, the tangential motions can be described using a tangential coefficient of restitution e x. This is obtained by defining an impulse ratio ratio of tangential impulse over normal impulse, which is a generalization of the coefficient of friction 1. This method of coefficient of restitution modeling approach was successfully applied to magnetic storage oblique impact 0. Due to recent advancements in micromanufacturing techniques, micromechanical devices are increasingly used in consumer applications. In some cases, impact between two components in such devices could be a critical issue. For example, in a magnetic storage hard disk drive, there is a slider flying at a height of few nanometers over the spinning disk. If shock is imparted to the hard disk drive, a feature on the edge of the slider or on the air-bearing surface of the slider can impact the disk and cause physical or magnetic damage 1. Typically, this feature can be approximated using either a spherical profile slider corners 3,10 or a cylindrical profile of a finite length and radius of curvature chamfered edges on the slider. The mass and size of the impacting slider are significantly smaller than that of the disk. The slider feature is usually harder than the disk, so the impact can be modeled by a rigid cylinder impacting an elastic-plastic deformable body. In this work, a dynamic impact model of a rigid cylinder impacting an elastic-plastic deformable body was developed based on an elastic-plastic contact model for two-dimensional plane Journal of Tribology Copyright 010 by ASME OCTOBER 010, Vol. 13 /

2 strain cylinders proposed by Lim and Stronge 13. The dynamic model also accounts for the finite length of the cylinder instead of directly using the plane strain formulation. In this work, Lim and Stronge s plane strain contact model was extended to account for finite length of a cylindrical contact. Also, an improvement of the method of estimation of the residual contact depth and the rebound elastic work compared with Lim and Stronge s model is proposed and used to obtain the normal coefficient of restitution. A parametric study to examine the effect of the cylinder radius, cylinder length, and impact velocity on critical contact parameters was performed. A finite element analysis FEA -based model of the above mentioned elastic-plastic impact was used to compare with the impact model results. The effect of tangential tractions on the normal motion was also examined using the FEA model. Elastic-Plastic Contact Mechanics (CM) Model.1 Contact Model. A constitutive model for plane strain rigid cylinder contacting a deformable elastic-plastic half-space was developed by Lim and Stronge 13. The model was developed for plane strain condition, which means the length of the contacting bodies is infinitely long. To consider the effect of the finite length of a cylinder L, we introduce a parameter n=l/r, where for the plane strain model to be applicable n 1 it is suggested that n 10. For n 10, effects due to the finite size of the cylinder may become dominant, and thus plane strain solutions will be invalid. Hence, Lim and Stronge s model can be rewritten, as shown in Eqs. 1 3 all symbols are explained in the Nomenclature. For purely elastic contact: p m /Y 1.5, P = nr E r ln R n E r 1 P 1 For elastic-plastic contact: 1.5 p m /Y.4, 1 = Y + a Y R a a Y For fully plastic contact: p m /Y =.4, = Y + a Y R a a Y 3 where p m is the mean contact pressure, Y is the yield strength, is the penetration depth, P is the contact force, R is the radius of the cylinder, E r is the reduced elastic modulus, is the ratio of the thickness of the deformable body d and R, is the Poisson s ratio, and a is the half-contact width. The parameters with subscript Y correspond to their values when yield initiates in the body and are given by a Y = 1YR E r Y = R 6Y E r ln +ln E r 6Y 1 P Y = 36Y nr 6 E r To obtain the transition between elastic and fully plastic contacts, Lim and Stronge derived an analytical equation to estimate the mean contact pressure p m based on Johnson s spherical cavity model 5. In Ref. 5, during the transition between perfectly elastic and fully plastic behavior, which is termed as the elasticplastic regime, it is assumed that the mode of deformation results in radial expansion of the material immediately below the cylindrical contact due to hydrostatic stress. By ensuring compatibility between the material displaced during the contact and the radial 4 5 expansion of the cylindrical cavity in the hydrostatic state, the mean contact pressure for the cylindrical cavity model is obtained as p m Y = ln 4 E r a 7 3 YR Note that in this work, the upper limit of the p m /Y is always constant at.4. Some researchers have shown that this ratio, for a deformable sphere contacting a rigid flat, can vary with the materials used and the evolving contact geometry 7,8. A limit to the maximum penetration max up to which the p m /Y ratio is valid has not been defined in the contact model. However, in the current analysis, the evolving contact geometry remains mostly in the regime where this ratio will be a constant; hence, it is safe to use a constant ratio. In this work, the maximum penetrations involved are / Y Residual Contact Depth Model. Lim and Stronge 13 estimated the residual contact depth r to be r = a r R a max 8 R where a r is the residual contact width, a max is the maximum contact width, and R is the radius of the cylindrical crater on the deformable body after elastic-plastic contact. In Ref. 13, it was assumed that a r a max, i.e., the contact width at maximum compression is assumed to be equal to the residual width after the impact. Thus, it does not account for the recovered elastic contact half-width a el. To consider elastic recovery, a el also needs to be accounted, so that a r = a max a el 9 For normal elastic contact of a rigid cylindrical surface onto an elastic half-space, quasi-static Hertz theory gives a in terms of P as a = 4R E r P nr a R = Therefore during loading, at first yield a Y 4P E r nr 10 R = 4P Y 11 E r nr Following the same idea as in Ref. 13, after the first loadunload contact cycle where loading results in elastic-plastic contact, let us assume that a second loading of the plastically deformed area is performed until the resulting contact width is equivalent to a max of the first loading. The elastic limit of the second loading will be at maximum contact depth when the contact radius is a max. The second loading up to a max is elastic since first unloading is elastic. The residual contact area of the unloaded body is assumed to be a concave cylindrical surface with a radius R. Figure 1 shows the cylindrical surface of radius R on a concave cylindrical surface of radius R, Fig. 1 Unloaded contact region with a crater of half-contact width a r and radius R assuming no pile-up or sink-in / Vol. 13, OCTOBER 010 Transactions of the ASME

3 a max 1 R R 1 = 4P max E r nr Equating Eqs. 11 and 1, we obtain Therefore, a max P max 1 R 1 R = a Y a R = R max a max a Y P Y R The elastically unloaded contact width a el is equivalent to the contact width obtained after the elastic loading of a cylinder with radius R on a cylindrical crater of radius R up to P max and can be obtained using Hertz theory as max a el = 4P 1 E r nr R R For elastic-plastic contact: 1.5 p m /Y.4 Based on similar triangles OAB and OCB in Fig. 1 tan = r = a r R r Since, it is elastic-plastic contact r R R r R a r r a r 18 R For fully plastic contact: p m /Y =.4 From triangle OAB Fig. 1, R = R r + a r r R r + a r =0 19 Finding the roots of the quadratic Eq. 19, one of the roots is r = R R a r 0 Thus, residual contact depth r is obtained by using Eq. 18 for elastic-plastic contact and Eq. 0 for fully plastic contact. Note that Eq. 0 is also applicable to elastic-plastic contact and can be used in place of Eq. 19. However, Eq. 19 provides a simple analytical solution for quick estimation of r since most impact applications are such that they are purely elastic or elastic-plastic. 3 Contact Mechanics-Based Dynamic Impact Model The forces considered in this model were the contact forces arising from the impact. At such high velocities, adhesion and damping forces are negligible and the impact contact motion in the normal direction could be described by m = P,R,L, 0 =0, 0 = V y 1 where m is the mass of the cylinder, is the acceleration, 0 is the initial displacement, and 0 is the initial velocity. The primary focus of this work is to estimate the contact damage that occurs due to impact and the mechanical contact damage parameters considered are, p m, and a. The elastic-plastic cylinder contact equations are nonlinear. For elastic contact, P cannot be expressed analytically in terms of. Hence, the values were obtained over a practical range of P/L and R values a priori using Eq. 1 and stored in a table matrix of values. Thus, the right hand side of Eq. 1, i.e., P, can be obtained for a given, R, and L from the table while solving the differential equation as long as the impact is purely elastic. When the contact exceeds the elastic limit, was calculated a priori for a range of a and R using Eq. for elastic-plastic deformation and Eq. 3 for fully plastic deformation. However, to determine if contact is in the elastic-plastic or fully plastic regimes, the mean contact pressure p m needs to be calculated. Hence, beyond initial yield, p m is calculated using Eq. 6 for a range of a and R values and is used to determine if the contact is in the elastic-plastic 1.5 p m /Y.4 or fully plastic p m /Y =.4 regimes. Once this is determined, both and p m are stored in tables, and when contact exceeds the elastic limit, Eq. is solved instead of Eq. 1, where for a given, R, and L, p m and a are obtained from the aforementioned tables. Thus, m = p m a,r nra,r, 0 =0, 0 = V y The above dynamic model Eqs. 1 and is solved numerically to obtain the penetration depth for a given cylinder radius R, cylinder length L, and impact velocity V y. A simple step-by-step simulation process for a cylinder with given L and R impacting an elastic-plastic deformable body is shown below. Step 1: Numerically solve the differential equation Eq. 1 or Eq. with given initial conditions 0 =0, 0 =V y.at the current time step i, the current depth i is calculated using an explicit numerical integration scheme such as the Runge Kutta method. Step : If i Y, then the contact is purely elastic, and P i for the corresponding i, R, and L is obtained from the look-up table and is used as the right hand side of the differential Eq. 1. Step 3: If i Y, then p mi and a i are obtained for the corresponding i, R, and L from the look-up tables and used in the right hand side of the differential Eq.. Step 4: Repeat steps 1 3 by marching ahead in time using the explicit time integrating scheme until the right hand side of the differential equation becomes 0 i.e., the cylinder is no longer in contact with the deformable body. 4 Typical Impact Simulation Results The motivation of this work was to apply the cylinder impact model to a practical case of a cylindrical feature of the slider, impacting a disk in a magnetic storage hard disk drive, and to investigate the resulting contact parameters. The mass m =0.159 mg and size of the slider are significantly smaller than that of the disk. The recording slider is primarily made of Al O 3 TiC designated as AlTiC and is usually covered by a thin 4 nm diamondlike carbon layer. Thus, in the impact model, the cylinder can be treated as rigid body by accounting for the elastic modulus of AlTiC in the reduced elastic modulus E r of the deformable body. The magnetic storage thin-film disk is made up of several thin-film layers but in this work only the substrate mechanical properties were used and thus, treat the disk as a homogeneous material. This assumption is reasonable since the mechanical properties of the thin-film layer are similar to that of the glass-based substrate material 10. The thickness of the deformable body d was taken as 100 m and E r =83.96 GPa, which is a combined modulus of glass-based substrate with E=100 GPa and =0.3, and AlTiC with E=390 GPa and =0.. The yield strength of the glass-based substrate is Y =.41 GPa. Figure a shows typical impact simulation results when R = m, L=0 m, and V y =1 m/s, whereas in Fig. b, R =10 m, L=1000 m, and V y =5 m/s. The impact solution based on the pure elastic contact is also plotted in Fig. for comparison. Figure a clearly shows that the elastic-plastic solution is different than the elastic solution and the deformable body incurred plastic deformation, with plastic deformation being less pronounced in Fig. b. The maximum penetration in Fig. a is higher than that in Fig. b though V y is lower since L is smaller in the case represented by Fig. a. Referring to Fig. a in Ref. 10, which deals with spherical slider corner impacts, max for R= m and V y =1 m/s was 7 m, whereas for the same parameters and cylinder edge with L=0 m, max is lower at Journal of Tribology OCTOBER 010, Vol. 13 /

4 Fig. Comparison between elastic and elastic-plastic CM impact models. Penetration of cylinder into homogeneous disk during impact Table 1 material properties : a R= m, L/R =10, and V y =1 m/s and b R=10 m, L/R=100, and V y =5 m/s. m Fig. a. This result is in agreement with the experimental results presented in Fig. 3 in Ref. 10, where it was measured that at higher impact velocities, the corner spherical impact quickly turns into a cylindrical edge impact, thus resulting in elongated imprints whose depths were lower than those predicted from the spherical impact model, indicating a cylindrical feature contact. In these cases, it is more appropriate to use a cylinder impact model where L is equal to the length of the imprinted feature. The above CM-based model was used to investigate the effect of R 1 10 m, L L/R=10, and V y m/s on the critical contact parameters. Figure 3 a depicts the maximum penetration max for the chosen parametric range of impact velocities and corner radii of curvature. The dependence of max on R is small at low V y up to 0.5 m/s but increases at higher values. Figure 3 b shows residual depth r after the end of the impact where it is clearly seen that below the contour line indicating r =0.05 m, the impact results in purely elastic deformation, which is completely recovered after the impact. The impact duration is shown in Fig. 3 c, where it highly depends on R and is almost independent of V y. The mean contact pressure p m contour is shown in Fig. 3 d, which can be used to determine the deformation regime of an impact via p m /Y Y =.4 GPa in this case. Figure 4 shows contour plots for the same parametric range as in Fig. 3 but with L/R=100. The max values, as seen in Fig. 4 a, are significantly lower than those in Fig. 3 a for a given R and V y. Similarly, the r values in Fig. 4 b are also lower than those in Fig. 3 b. This is because the contact force is distributed over a larger contact area resulting in a lower contact pressure, which can clearly be observed in Fig. 4 d, where the p m values are lower than those in Fig. 3 d. Most of the impact in Fig. 4 is in the elastic and elastic-plastic regimes. Fig. 3 Impact contact parameters using the CM model at various impact velocities and cylinder radii when L/ R=10: a maximum penetration, b residual depth, c impact duration, and d maximum mean contact pressure / Vol. 13, OCTOBER 010 Transactions of the ASME

5 Fig. 4 Impact contact parameters using the CM model at a larger range of impact velocities and cylinder radii when L/ R=100: a maximum penetration, b residual depth, c impact duration, and d maximum mean contact pressure 5 Coefficient of Restitution Model 5.1 Work Done During Impact Phase. The normal coefficient of restitution e y is defined as the ratio of the magnitude of the rebound normal velocity V yr over the impact normal velocity V yi, e y = V yr 3 V yi Since the cylinder is impacting a fixed-length body, it will rebound after the impact. If the impact results in loss of energy due to the contact, it will be manifested in the form of reduced kinetic energy of the cylinder while rebounding. It is reasonable to assume that since the cylinder is rigid, it will not lose any mass during the impact, thus e y = V yr V = yi 1 mv yr 1 mv yi 4 where e y is the ratio of the impact kinetic energy over the rebound kinetic energy of the cylinder. This can also be defined as the magnitude of the work done during the impact phase W tot over the elastic work recovered during the rebound or the restitution phase W reb of the impact, e y = W reb W tot 5 The work done during the impact can be obtained by integrating the load over the penetration depth. Lim and Stronge 13 obtained the work done over the regimes of elastic W el, elasticplastic W ep, and fully plastic W fp deformation, as follows: W el = Y P Y P max C 1 P Y 1 4C where C= ln + ln E r /6Y /1 YR a Y W ep = P Y a Y 3 R ln YR a Y W fp = P Ya Y YR a Y max Y ln P max 6 C P Y / max Y YR a Y 3/ max Y 1 +1 R pl Y / 8 where pl = Y a Y /R Alternatively, instead of using the above equations to obtain the work done during impact, it can be directly calculated using numerical integration of the load-penetration P- values during the loading phase of the impact up to P max. The proposed method to estimate the work recovered during rebound W reb is described below. 5. Rebound Work Done During Impact. The rebound or the restitution phase is assumed to be elastic. To obtain the coefficient of restitution, the work recovered during restitution or the rebound work W reb is needed to be estimated during the impact. Since an analytical solution to obtain W reb is unavailable, Lim and Stronge 13 approximated the rebound P- to be linear and obtained the work done between max and r as Journal of Tribology OCTOBER 010, Vol. 13 /

6 Fig. 5 Typical P- curve obtained during cylinder contact. The dashed line indicates the unloading rebound curve shifted to the origin. W reb = 1 P max max r 9 A more accurate model to estimate the rebound work was sought in this work. The characteristic load-unload behavior during the cylinder impact process and can be represented as shown schematically in Fig. 5. The curve represented by AB is the loading part occurring during the impact phase, and curve BC represents the unloading part during the restitution phase. Hence, area BCD will provide an estimate of the amount of work done during the restitution phase. Note that in Ref. 13, this area was approximated as triangle BCD dash-dotted line in Fig. 5. For a more accurate model, we need to obtain the area under curve BC to obtain an accurate estimate of the rebound work. Thus, the expression for unloading work can be derived as follows. Referring to Fig. 5, rebound work W reb =area BCD=area B C D BCD with origin shifted. The area of region B C D is basically the area of the region encompassed by the rectangle C EB D the area of the region to the left of the loading curve C EB W reb = area C EB D area C EB The area of rectangle C EB D is P max max r, and the area C EB is the area to the left of the curve C B, where W reb = P max max r 0P max n P dp 30 P max P max n P dp P r dp 31 0 = 0 Since the unloading behavior is elastic, it implies that the -P relationship follows Eq. 1, but here, the effective radius will be 1/R 1/R 1 instead of R. This is because unloading is equivalent to a second loading of the cylinder on a cylindrical crater of radius R up to the new yield point P. Y Similarly, the updated quantities Y and P Y from Eqs. 5 and 6, respectively and C based on the new reduced radius can be calculated and introduced into Eq. 1 to yield P = P Y P Y 1 ln P/P Y C 3 Substituting Eq. 3 in Eq. 31 and integrating, simplifies Eq. 30 to W reb = P max P Y C ln P max/p Y 1 4C Y P Y 33 Note that P Y = P max and Y = max r. Thus, Eq. 33 simplifies to Eq. 34 below, which has a similar form as Eq. 40 in Ref. 13 W reb = P max max r C 1 4C 34 Fig. 6 CM-based coefficient of restitution for normal impact e y : a L/R=10, b L/R=10 larger range, and c L/R= Normal Coefficient of Restitution. The normal coefficient of restitution e y can be obtained as follows: for purely elastic impact, e y =1 assuming negligible stress wave losses for elastic-plastic impact, W e y = reb W el + W ep for fully plastic impact, e y = W reb W el + W ep + W fp The coefficient of restitution e y is obtained for the parametric cases discussed earlier and is plotted in Fig. 6. In Fig. 6 a, where L/R=10, R=1 10 m, and V y =0.1 1 m/s, the region to the right of the contour line of e y =1 is where there is no plastic deformation, and thus the elastic strain energy is recovered fully by the cylinder. To examine the effect of cylinder length L on e y, L/R was increased to 100 with R=1 10 m and V y =0.1 1 m/s and shown in Fig. 6 b. Compared with Fig. 6 a, e y values are higher in Fig. 6 b since L/R is higher, resulting in lower contact pressures. Note that in the current analysis, the effect of elastic stress wave energy dissipation has not been presented. This is because for the velocity range and impacting body sizes involved in this work, it was found that the elastic stress wave energy loss is several orders of magnitude lower and is thus negligible / Vol. 13, OCTOBER 010 Transactions of the ASME

7 Fig. 7 Finite element model showing the mesh configuration used for the cylinder impact analysis 6 Finite Element Model To verify the proposed CM model, dynamic FEA was performed using ABAQUS/explicit solver 3. Explicit solvers are particularly efficient for analysis with very short dynamic response times where an explicit central difference time integration scheme is used. In this solver, the accelerations are calculated from dynamic equilibrium equations at time t. The accelerations at time t are used to advance the velocity solution at time t+ t/ and displacement solution to t+ t, where t is the time-step. Plane strain models of a cylinder vertically normal direction and obliquely impacting a homogeneous deformable body were simulated using FEA. The deformable body mesh is made of four-node bilinear plane strain quadrilateral elements with full integration. The dimensions of the deformable body are m length height with the finest mesh of m on and near the surface contact region and coarsest mesh away from the surface. The current mesh configuration as shown in Fig. 7 was arrived at by performing a convergence study of cylinder contact using different mesh and element sizes and comparing the results with analytical cylinder contact solutions. The body size was such that any stress waves generated during the impact do not reflect back to the contact region. Hence, there are no artificial effects due to stress wave reflection. The total number of nodes and elements are 5,551 and 5,000, respectively. Only the bottom half of the rigid cylinder was modeled using an analytical rigid surface of R=10 m since only a small region of the cylinder will come in contact with the deformable body. Since the model is in plane strain, a mass per unit length should be assigned to the cylinder m =m/l where m=0.159 mg. Thus, through m, the finite length of the cylinder can be accounted for in the plane strain simulation. For the current model typical time-step t is 5 ns, whereas the total time taken for a simulation is 3 s, which is in agreement with impact duration observed in experiments. For simplicity, material damping effects were not considered in the analysis. Contact surfaces were defined on the analytical rigid surface and the top surface of the deformable mesh, which ensures that appropriate contact pressures were applied during contact. A constant friction coefficient model was employed to account for friction at the contacting interfaces. In this friction model, as long as the lateral surface traction is less than the product of the friction coefficient and contact pressure p, the contacting interfaces will stick. Gross slip will occur once in the whole contact region increases beyond this limit. For the current analysis, =0.15, which is typical of the contact of the materials used in this work 4. For normal impact simulations, the rigid cylinder was constrained from rotating in any direction and also constrained from moving horizontally X-direction. The horizontal constraint on the cylinder is removed for oblique impact simulations. The bottom boundary of the deformable body is constrained from moving vertically Y-direction. Similarly, the horizontal motion Xdirection of the side-boundaries of the deformable body is constrained. V y ranges from 0. m/s to 1 m/s for cylinders with L/R=10 R=10 m and L=100 m and V y =1 to 5 m/s when L/R=100 R=10 m and L=1000 m. For oblique impacts, the horizontal velocity V x =10 m/s, which is a typical linear velocity for a spinning magnetic storage disk. For the above cases, Fig. 8 FEA maximum von Mises stress for R=10 m, V y =1 m/s during a normal impact and b oblique impact V x =10 m/s. Units in GPa. Journal of Tribology OCTOBER 010, Vol. 13 /

8 Fig. 9 FEA penetration depths for R=10 m, V y =1 m/s during normal impact: a maximum penetration and c residual penetration; during oblique impact V x =10 m/s : b maximum penetration and d residual penetration. Units in m the angle of impact would be within the range of deg, which is sufficiently high to cause gross slip throughout the impact 0. The maximum von Mises stress contours during the course of the impact are shown in Figs. 8 a and 8 b, for normal impact and oblique impact, respectively, for the case when V y =1 m/s, R=10 m, and L/R=10. In both cases, the impact has resulted in plastic deformation since the von Mises stress exceeds Y.41 GPa. Thus, when gross slip occurs in this range, friction has a negligible effect on the energy loss due to plastic deformation. The frictional energy dissipation for the current case will be a very Fig. 10 FEA versus CM impact model penetration depth comparison for L/ R=10: a maximum penetration and c residual depth; for L/ R=100: b maximum penetration and d residual depth / Vol. 13, OCTOBER 010 Transactions of the ASME

9 Table 1 Maximum and average percentage difference of the depths estimated by impact model compared with the FEA model results Maximum % difference Average % difference max Lim and Stronge, r, Eq. 8 Proposed r, Eqs. 17 and 18 max Lim and Stronge, r, Eq. 8 Proposed r, Eqs. 17 and 18 L/ R= L/ R= small component of the total energy dissipation 14. Figure 9 plots the penetration depths from the FEA model for the same cases as discussed in Fig. 8. The max U values for normal and oblique impacts are shown in Figs. 9 a and 9 b, respectively. Similarly, the r U values are obtained after the end of the impact and are shown in Figs. 9 c and 9 d, for normal and oblique impacts, respectively. Comparing normal and oblique impact cases, there is negligible difference in max values but a considerable difference exists for r. The residual depth is larger for the oblique impact case at 0.19 m compared with the normal impact where it is 0.1 m. This is because once gross slip occurs during oblique impact, the higher stress, which originally exists beneath the surface, will move closer to the surface resulting in increased plasticity near the surface. The case analyzed in Fig. 9 could be considered extreme and r differences between normal and oblique impacts are lower for lower V y values. Thus, the CM-based normal impact model presented earlier can be reasonably extended to also model oblique impact cases. 7 CM Impact Model Validation Using FEA Parametric comparison was performed to determine the validity of the CM impact model by comparing with the FEA results for both normal and oblique impacts. Figure 10 shows the comparison of the penetration depths obtained from the elastic-plastic CM impact and FEA models with a cylinder of R=10 m. A comparison of max and r for a cylinder with L/R=10, R=10 m, and V y =0. 1 m/s is shown in Figs. 10 a and 10 b, respectively. There is good agreement for max between the CM impact model and FEA normal and oblique impact models, as shown in Fig. 10 a. In Fig. 10 b, r values obtained using Lim and Stronge residual depth model 13, the improved residual depth model along with the FEA results is shown. Lim and Stronge residual depth model overestimates r, whereas the improved residual depth model results are closer to the FEA impact results. Similar comparison is made with a longer cylinder L/ R=100, R=10 m, and V y =1 5 m/s. For most micromechanical systems like magnetic storage disks, V y range will be lower, with 1 m/s being the limit. However, for long cylinders like in Fig. 10 c, there will be no plastic deformation observed at this range. Hence, in order to test the validity of the CM model, a higher V y range was employed in this work when L/R=100. The agreement for max between the CM model and FEA is good, as seen in Fig. 10 c. In Fig. 10 d, r values from the improved elastic-plastic contact model, for most cases, are closer to both normal and oblique impact FEA compared with the Lim and Stronge model. However, in certain cases at higher impact velocities, the improved elastic-plastic contact model seems to overestimate r. The maximum and average percentage difference between the depths max and r estimated by the CM impact model and FEA are shown in Table 1. Compared with Lim and Stronge s estimation of the residual depth, a considerable improvement in accuracy is obtained by using the improved residual depth model in the CM impact model. Since the magnitude of r values is small, a small difference will lead to a high percentage difference. The normal coefficient of restitution e y values obtained from the impact model Eqs for the same cases as discussed in Fig. 10 are compared with the FEA normal impact results in Fig. 11. In Fig. 11 a, the e y from the impact of the cylinder with R=10 m and L/R=10 is shown, whereas Fig. 11 b plots the same for a cylinder with R=10 m and L/R=100. In both figures, we can observe that at low V y, e y 1, which implies that most of the impact is elastic and thus, there is no plastic deformation energy loss. As V y increases, e y is decreasing, implying increased plastic deformation energy loss. We can also see that in Fig. 11 b, the decrease in e y values is steeper than in Fig. 11 a. This is because, though the cylinder is larger than in Fig. 11 a, the V y values are also much higher 1 5 m/s. A fairly good comparison is obtained between the CM impact model and FEA results, which validates the elastic-plastic CM-based cylinder impact model. The maximum and average percentage difference between e y estimated by the CM impact model and FEA are shown in Table. The improved residual depth contact model and improved unloading contact model used in the CM impact model will result in a more accurate estimation of e y compared with when using Lim and Stronge s residual depth and unloading contact models. Fig. 11 Normal impact FEA versus impact model coefficient of restitution comparison e y for R=10 m: a L/R=10 and b L/ R=100 Journal of Tribology OCTOBER 010, Vol. 13 /

10 Table Maximum and average percentage difference of e y estimated by impact model compared with the FEA model results Maximum % difference Lim and Stronge e y, Eq. 9 Proposed model e y, Eq. 33 Lim and Stronge e y, Eq. 9 Average % difference Proposed model e y, Eq. 33 L/ R= L/ R= Conclusions A contact mechanics-based impact model of a rigid cylinder with finite length and radius of curvature impacting a homogeneous elastic-plastic finite length deformable body was developed. The model accounts for the finite length of the cylinder contacting the body, as long as the cylinder length is considerably higher than its radius of curvature, ensuring plane strain conditions are satisfied during the contact. The model can be used to estimate the resulting residual depth after the impact. For this purpose, an improved method of estimating the residual depth was proposed. Also, on the elastic unloading and the estimate of the residual depth, an improved normal impact coefficient of restitution model was presented. The main focus of the model was to study impacts in micromechanical systems. In this work, the impact model was used to study a case of a cylindrical feature on the slider of a magnetic storage hard disk drive, impacting the disk to predict various critical impact contact parameters. The impact model was validated using a plane strain finite element model by comparing the maximum contact depth, residual depth, and coefficient of restitution from the finite element model results. Acknowledgment This work was partially supported by the National Center for Supercomputing Applications NCSA under Grant No. TG- MSS05005N and utilized the Dell Intel 64 Cluster Abe computing resources. The authors would also like to acknowledge help from Mr. Melih Eriten, University of Illinois at Urbana- Champaign regarding symbolic computations and derivations. The motivation of this work was through a sponsored research program from Seagate Technology LLC through Grant No. SRA Nomenclature a half-contact width a el elastically unloaded half-contact width after plastic deformation a max maximum half-contact width a r residual half-contact width a Y half-contact width at initial yield C dimensionless parameter see text d thickness of the deformable body e x tangential or horizontal coefficient of restitution e y normal coefficient of restitution E elastic modulus E r = 1 /E 1 effective elastic modulus i current time-step L length of the cylinder m mass of cylinder m m/l n L/R p local contact pressure p m mean contact pressure p my =1.5Y mean contact pressure at initial yield P contact force P P/L P i contact force at time-step i P max maximum contact force P Y contact force at initial yield R cylinder radius R radius of the crater formed on the body during elastic-plastic contact V x horizontal or lateral impact velocity V y normal impact velocity V yi normal impact velocity during impact phase V yr normal impact velocity during restitution or rebound phase W el work done during elastic phase of contact W ep work done during elastic-plastic phase of contact W fp work done during fully plastic phase of contact W tot total work done during contact W reb work done during the restitution or rebound phase unloading Y yield strength of substrate material penetration depth velocity of cylinder acceleration of cylinder i penetration depth at current time step i pl penetration depth during fully plastic contact r residual depth max maximum depth n r Y penetration depth at the initial yield d/r surface shear stress friction coefficient Poisson s ratio of homogeneous deformable body References 1 Gilardi, G., and Sharf, I., 00, Literature Survey of Contact Dynamics Modeling, Mech. Mach. Theory, 37, pp Johnson, K. L., 1985, Contact Mechanics, Cambridge University Press, New York. 3 Yu, N., Polycarpou, A. A., and Hanchi, J. V., 008, Elastic Contact Mechanics-Based Contact and Flash Temperature Analysis of Impact-Induced Head Disk Interface Damage, Microsyst. Technol., 14, pp Streator, J. L., 003, Dynamic Contact of Rigid Sphere With an Elastic Half- Space: A Numerical Simulation, ASME J. Tribol., 15, pp Johnson, K. L., 1970, The Correlation of Indentation Experiments, J. Mech. Phys. Solids, 18, pp Kogut, I., and Etsion, I., 00, Elastic-Plastic Contact Analysis of a Sphere and a Rigid Flat, ASME J. Appl. Mech., 69, pp Jackson, R. L., and Green, I., 005, A Finite Element Study of Elasto-Plastic Hemispherical Contact Against a Rigid Flat, ASME J. Tribol., 17, pp Ye, N., and Komvopoulos, K., 001, Three-Dimensional Contact Analysis of Elastic-Plastic Layered Media With Fractal Surface Topographies, Microsyst. Technol., 13, pp Kogut, L., and Komvopoulos, K., 004, Analysis of the Spherical Indentation Cycle for Elastic-Perfectly Plastic Solids, J. Mater. Res., 19 1, pp / Vol. 13, OCTOBER 010 Transactions of the ASME

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