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## Itzhak Green Fellow, ASME A Finite Element Study of Elasto-Plastic Hemispherical Contact Against a Rigid Flat

### BibTeX

@MISC{Jackson_itzhakgreen,

author = {Robert L Jackson and ASME Member and George W Woodruff},

title = {Itzhak Green Fellow, ASME A Finite Element Study of Elasto-Plastic Hemispherical Contact Against a Rigid Flat},

year = {}

}

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### Abstract

Introduction The modeling of elasto-plastic hemispheres in contact with a rigid surface is important in contact mechanics on both the macro and micro scales. This work presents a dimensionless model that is valid for both scales, although micro-scale surface characteristics such as grain boundaries are not considered. In the former, e.g., rolling element bearings, load may be high and the deformations excessive. In the latter, e.g., asperity contact, a model on the micro-scale is of great interest to those investigating friction and wear. In addition, the real area of contact of such asperities will affect the heat and electrical conduction between surfaces. In either scale contact is often modeled as a hemisphere against a rigid flat. Much interest is devoted in the literature to the reverse case of indentation loading where a rigid sphere penetrates an elastoplastic half-space. It is worthy to emphasize that indentation ͑other works͒ and hemispherical deformation ͑this work͒ are significantly different in the elasto-plastic and fully plastic regimes, and only the latter is the subject of this work. One of the earliest models of elastic asperity contact is that of Greenwood and Williamson ͓1͔. This ͑GW͒ model uses the solution of the frictionless contact of an elastic hemisphere and a rigid flat plane, otherwise known as the Hertz contact solution ͓2͔, to stochastically model an entire contacting surface of asperities with a postulated Gaussian height distribution. The GW model assumes that the asperities do not interfere with adjacent asperities and that the bulk material below the asperities does not deform. The Gaussian distribution is often approximated by an exponential distribution to allow for an analytical solution, although Green ͓3͔ has analytically solved the integrals using the complete Gaussian height distribution. Supplementing the GW model, many elastoplastic asperity models have been devised ͓4 -8͔. Appendix A provides a summary of these models. Many of these elasto-plastic models make use of the fully plastic Abbott and Firestone model ͓9͔, while Greenwood and Tripp derive a very similar model ͓10͔. It should be noted that Abbott and Firestone ͓9͔ intended their model to be used to describe a wear process rather then a deformation process, but literature has still traditionally attributed this fully plastic truncation model to them ͑see Appendix A for a detailed description͒. Although these previous models have proven useful, they contain clear pitfalls which may be detrimental to their validity. Additionally, the reversed case of a rigid spherical indentation of a deformable half-space has been thoroughly investigated experimentally ͓12-14͔ and numerically ͓15-19͔. Work has also been done on the contact of a rigid cylinder contacting an elastoplastic layered half-space ͓20͔. More generally, various experimental and numerical works have investigated other contacting geometries and hardness tests ͓11,21,22͔. The two works by Barber et al. ͓23͔ and Liu et al. ͓24͔ provide a more in-depth look at past and more recent findings in the field of contact mechanics. Perhaps a most important and relevant work is by Johnson ͓25͔, who experimentally measured the plastic strains between copper cylinders and spheres. Johnson's experimental results compare favorably with the findings of the current work. Despite the extensive body of works, the results, trends, and theories presented in the present work, to the authors' knowledge, have not yet been thoroughly documented. The current work uses the finite element method to model the case of an elastic-perfectly plastic sphere in frictionless contact with a rigid flat ͑see The finite element analysis presented in this work produced different results than the similar Kogut and Etsion ͑KE͒ model ͓4͔. The current work accounts for geometry and material effects which are not accounted for in the KE model. Most notable of these effects is that the predicted hardness is not a material constant as suggested by Tabor ͓11͔ and many others; rather hardness changes with the evolving contact geometry and the material properties as proven in this work. Moreover, the current work uses a mesh that is orders of magnitude finer than that in ͓4͔ which was mandated by mesh convergence. The current work models deformation surpassing / c ϭ110 ͑the limit of KE͒, and likewise models five different material strengths, S y , that showed a markedly different behavior in the transition from elasto-plastic to fully plastic deformation. The formulations derived in the current work are also continuous for the entire range of modeled interferences, whereas the KE model is discontinuous in two separate locations. There is ambiguity and a lack of a universal definition of hardness. Not only are there various hardness tests for various scales and materials ͑Brinell, Rockwell, Vickers, Knoop, Shore, etc.͒, the Metals Handbook ͓12͔ defines hardness as ''Resistance of metal to plastic deformation, usually by indentation. However, the term may also refer to stiffness or temper, or to resistance to scratching, abrasion, or cutting. It is the property of a metal, which gives it the ability to resist being permanently, deformed ͑bent, broken, or have its shape changed͒, when a load is applied. The greater the hardness of the metal, the greater resistance it has to deformation.'' Another definition is that hardness measures the resistance to dislocation movement in the material, in which case it is directly related to the yield strength ͑and thus is interchangeable and perhaps redundant͒. A common definition that has gained status in the field is that hardness equals the average indentation pressure that occurs during fully plastic yielding of the contact area. As is shown in this work, hardness of this type of definition varies with the plastic and elastic properties and the contact geometry of the surface, i.e., with the deformation itself. A hardness geometric limit will be defined and discussed in the foregoing. Critical Interference While in the elastic regime, the stresses within the hemisphere increase with P and . These stresses eventually cause the material within the hemisphere to yield. The interference at the initial point of yielding is known as the critical interference, c . The current work derives this critical interference analytically using the von Mises yield criterion ͑VM͒. The following equations, for the critical interference, contact area, and load, are all independent of the hardness, which the current work shows not to be constant with respect to S y . This is a notable improvement compared to previous elasto-plastic contact models ͓4 -6͔. The derivation is given in Appendix B, resulting in where C is derived in the Appendix to be The Poisson's ratio, , to be used in Eq. ͑2͒ is that of the material which yields first. For ϭ0.32, as is used in this work, Eq. ͑2͒ results in Cϭ1.639. The critical load, P c , is then calculated from the critical interference, c , by substituting Eq. ͑1͒ into Eq. ͑A2͒. The resulting critical contact force at initial yielding is thus Similarly, the critical contact area is calculated from Eq. ͑A1͒ and is given by These critical values predict analytically the onset of plasticity. These values are, therefore, chosen to normalize the results of all the models. The normalized parameters are Normalizing the Hertzian contact area ͓Eq. ͑A1͔͒ and force ͓Eq. ͑A2͔͒, and the AF contact area ͓Eq. ͑A5͔͒ and force ͓Eq. ͑A6͔͒, by the critical values given in Eqs. ͑3͒ and ͑4͒, results in the following simplified expressions: A AF * ϭ2* (10) Finite Element Model To improve upon the efficiency of computation, an axisymmetric 2-D model is used. The present study utilizes the commercial program ANSYS™, while ABAQUS™ produces the same results. Kogut and Etsion ͓4͔ also use ANSYS™. However, the mesh ͑see The model refines the element mesh near the region of contact to allow the hemisphere's curvature to be captured and accurately simulated during deformation. The model uses quadrilateral, four node elements to mesh the hemisphere, but the results have also been confirmed to yield identical results using a mesh of eight node elements. The resulting ANSYS™ mesh is presented in The contact force acting on the hemisphere is found from the reaction forces on the hemisphere base nodes that retain the desired interference. The radius of the contact area is determined by finding the edge of the contact, or the location of the last activated contact element. In order to validate the model, mesh convergence must be satisfied. The mesh density was iteratively increased by a factor of 2 until the contact force and contact area differed by less than 1% between iterations. The resulting mesh consists of at least 11,101 elements, since the number of meshed elements will vary with the expected region of contact. The stiffness of the contact elements was also increased by an order of magnitude in successive iterations until the contact force solution differed by no more than 1% between successive iterations. In addition to mesh convergence, the model also compares well with the Hertz elastic solution at interferences below the critical interference. The contact force of the model differs from the Hertzian solution by no more than 2%. The contact radius differs by a maximum of 8.1%, but the average error is only 4.4%. When the contact areas are calculated from the radii, the maximum error increases to 17%. The smaller error in the contact force is attributed to overall force balance ͑static equilibrium͒ enforced by the FEM packages. However, the contact radius is obtained from a discrete mesh ͑which has a finite resolution͒. Moreover, the magnitude of the contact element stiffness also has some effect upon such radii, although not on the overall force balance. Generally, though, the differences are small enough that the FEM solution practically conforms to the Hertzian solution at interferences below critical ͑and even slightly above͒. There are two ways to simulate the contact problem. The first applies a force to the rigid body and then computes the resulting displacement. The second applies a displacement and then computes the resulting contact force. In both methods, the displacement, stress, and strain in the elastic body can be determined, as well as the contact pressure. In this model the latter approach is used, where the base nodes of the hemisphere are displaced a distance or interference, , approaching the rigid flat surface. The radial displacements of the base nodes are restricted. This method is used because the resulting solution converges more rapidly than the former. The contact problem and the elasto-plastic material property make the analysis highly nonlinear and difficult to converge. An iterative scheme is used to solve for the solution, and many load steps are used to enhance solution convergence. Initially, a small interference is set of the total interference and then it is incremented after the load step converges. ANSYS™ internally controls the load stepping to obtain a converged solution by using the bisection method. This continues until a converged solution is found for the desired interference. Numerical Results and Discussion The results of the described finite element hemisphere model are presented for a variety of interferences. While the elastic modulus and Poisson's ratio are held constant at 200 GPa and 0.32, respectively, five different material yield strengths are modeled. These are designated Mat.1 through Mat.5 corresponding to their yield strengths which are 0.210, 0.5608, 0.9115, 1.2653, and 1.619 GPa. The yield strengths cover a typical range of steel materials used in engineering ͓27͔. The generated numerical data for five steel materials is given in The dimensionless contact area is normalized by the Hertz solution ͓Eq. ͑8͔͒ and plotted as functions of * in Overall though, the FEM predicted contact area generally follows the Hertz elastic solution near the critical interference and then increases past the AF model as the interference increases. Later in this work, this trend will be followed by empirical formulations fitted to the data. The FEM results also indicate a material dependence of the normalized contact area. Since the con- tact area is calculated by counting the number of elements in contact, and there are only a finite number of such elements, there is an inherent error in the data. The scatter in the data can be attributed mostly to this, and to the fact that the FEM is yet a discrete formulation. For the contact area, all the models follow the same general trend, but they differ in magnitude. The ZMC model follows the Hertz elastic solution at low and moderate interferences, but abruptly migrates to the AF model before the current model and the KE model. The KE model and the current empirical model also agree fairly well on average, except at large interferences. The KE model clearly shows a slight discontinuity at *ϭ6 and then terminates at *ϭ110. The KE model does not connect with the Hertz elastic solution at the critical interference depth. Also, the nondimensional KE model is material independent such that its contact area falls between the data of materials 1 and 5 of this work. The dimensionless contact force is normalized by the Hertz solution ͓Eq. ͑9͔͒ and plotted as a function of * in The nondimensional contact force trends of all the models are similar; however, the ZMC again crosses to the AF model prematurely. At low interferences, the KE and ZMC models predict a contact force that is greater than the elastic model. This cannot be the case, as the yield strength of the material limits the stiffness of the hemisphere. Again the KE model shows a discontinuity at *ϭ6 and then terminates at *ϭ110. Generally the KE model and the current FEM results are very similar. At about *ϭ50 the KE model crosses over the current model and continues to overestimate the contact force until *ϭ110. The KE and ZMC models also fail to capture the material dependence effects at large interferences. The average contact pressure to yield strength ratio, P/(AS y ), is calculated from the data and plotted in Empirical Formulation General empirical approximations of the FEM data are desired for use at any deflection and for any set of material properties. This will help designers in a variety of single contact problems, and it will be readily incorporated into statistical models to model rough surfaces. As mentioned previously, the FEM solution for the area of contact continues past the AF model with increasing interference. Hence, the leading coefficient in Eq. ͑10͒ is allowed to vary when equations are fitted to the FEM data. This is reasonable, since the AF model is not an exact solution ͑it is based on a truncation assumption͒. Here a power function is used in place of this leading coefficient and is fit to the numerical data. where Bϭ0.14exp͑23e y ͒ (14) In order to formulate a fit for the FEM contact force, the material-dependent trend at high interferences shown in This formulation is plotted alongside the data in This results in a formulation for H G as a function of the material properties, E, S y , and ͑not just upon S y as suggested by Tabor ͓11͔͒. To formulate an approximation of the contact force as predicted by the FEM results, the AF model for contact force must first be corrected by way of substituting Eq. ͑17͒ or ͑19͒ into Eq. ͑11͒, letting H G replace H, and by allowing the AF contact area to deviate from Eq. ͑10͒ ͓see reasoning for Eq. ͑17͔͒. This results in an equation for a corrected fully plastic model. Once again a piecewise solution is fit to the FEM data. At small interferences, the Hertz solution is assumed. The resulting piecewise equation fit to the FEM data is given as follows: For 0р*р t * and for t *р* where t *ϭ1.9. This formulation approaches asymptotically the Hertz elastic model at small interferences, and approaches and continues past the AF model at large interferences. Statistically this formulation differs from the FEM data for all five materials by an average error of 0.94% and a maximum of 3.5% when Eq. ͑19͒ is used for H G . The average pressure to yield strength ratio, P/(AS y ), can now be modeled by combining Eqs. ͑12͒-͑16͒ and Eqs. ͑20͒ and ͑21͒. Since these equations are normalized by their critical values, the resulting formulation for the average pressure is This ratio is shown in Comparison with Experimental Results Johnson ͓25͔ performed experiments on the elasto-plastic contact of copper cylinders and spheres. During one experiment, he tested the contact of a copper sphere and a comparatively rigid steel surface. These test conditions are comparable to the sphere against a rigid flat case modeled in this work. For the highest load tested, the contact has a nearly uniform pressure distribution, thus suggesting it is in the fully plastic regime. At this load, the a/R ratio is given as 0.204 and the average pressure as 2.59•S y . Interestingly, the predicted geometric hardness limit or average pressure for the same a/R using Eq. ͑17͒ is 2.61•S y . In comparison, the KE model, which assumes the AF model at this interference, predicts an average pressure of 2.8•S y . Johnson provides the contact radius and load in his results, which can also be compared with the predictions of the current formulations ͓Eqs. ͑12͒-͑21͔͒ and those of the KE model ͓Eqs. ͑A8͒ and ͑A9͔͒. Evolution of Deformation As long as the deformations are purely elastic, i.e., below the critical interferences, the entire hemisphere will abide to 3D Hooke's law. Conforming to Poisson's effect, the material volume should compress with a compressive contact pressure ͓as shown schematically in Stress Distribution and Evolution Initially, at small interferences, the sphere will deform only elastically. While in the elastic regime, the maximum von Mises stress will always occur beneath the contact surface and within the bulk material. Eventually, as the interference increases and the stresses increase, yielding will initiate at the point of maximum von Mises stress. At interferences just above the critical, the plastically deformed region is small and confined below the surface by a sizeable region of elastic material ͓see Repeated FEM analyses were performed to search for the interferences of two important cases: ͑1͒ when plastic deformation first reaches the contacting surface at the far right end point, and ͑2͒ when the contact surface first becomes entirely ͑fully͒ plastic. After plastic deformation has reached the surface, an elastic volume on the loaded tip of the sphere is still maintained ͓Fig. 10͑c͔͒ by the presence of hydrostatic stresses, which suppress yielding according to the von Mises criterion. Eventually this elastic region will turn plastic as the interference is increased. Conclusions This work presents a 2D axisymmetric finite element model of an elastic-perfectly plastic hemisphere in contact with a rigid flat surface. A comparison is also made with other existing models. The material is modeled as elastic-perfectly plastic, and yielding occurs according to the von Mises criterion. A concise form is Transactions of the ASME presented for the critical interference at which plastic deformation initiates within the hemisphere. It is derived from the Hertzian solution and the von Mises yield criterion. An a priori definition of the hardness is not needed. The resulting plots indicate that the FEM results for the contact area agree closely at small interferences with the trends of the Hertzian solution. While at large interferences the FEM predicts contact areas that surpass Abbot and Firestone's fully plastic model ͓9͔ ͑that is based upon truncation͒. The ZMC model is found to differ significantly from the FEM results, where the KE model ͑which is also based on FEM results͒ follows more closely, although still does not capture the varying hardness trend. An empirical formulation for the contact area is also fitted to the FEM data as a function of the material properties and interference. The FEM results of the contact force predict a lower load carrying capacity than the AF model for most materials and values of *. This is because the AF model assumes that the average pressure distribution is simply the hardness, which is approximated by 3•S y . It is found, however, that the fully plastic average contact pressure or hardness is not constant as is widely accepted. Rather, the limiting value of the fully plastic average pressure varies with the deformed contact geometry, which in turn is coupled to the material yield strength. This is accounted for in an empirical formulation for the limiting average pressure to yield strength ratio, H G /S y . A formulation using H G /S y is then fit to the FEM contact force data. A comparison is also made with the experimental results provided by Johnson ͓25͔. The current model compares very well, and predicts the sparse experimental results significantly better than the KE model, particularly in the fully plastic regime. The experimental results also show that the hardness trend at large deformation is a very real phenomenon that can affect practical engineering applications. This work reveals large differences between approximate analytical models and other numerical solutions. More importantly, the contact area, force, and pressure are found to be particularly dependent upon the deformed geometry in all regimes and effectively dependent upon the material properties ͑e.g., strength͒ in the elasto-plastic and plastic regimes. The fit-them-all equations that solely depend upon deformation, which are found in previous works, are imprecise when compared to current FEM results. For example, the average contact pressure to yield strength ratio in all previous work is shown to increase monotonically with deformation, and is assumed to terminate ͑or truncate͒ at the hardness. In this work it is shown that such a truncation is not warranted. Particularly, it is shown that the truncation model of Abbott and Firestone ͓9͔ cannot be justified. This work discovered significant geometrical and material nonlinearities, and that the hardness depends not just upon strength but also upon the modulus of elasticity, Poisson's ratio, and most importantly upon the deformation itself ͑i.e., hardness is not a unique or fixed material property as indicated by Tabor ͓11͔, and assumed by others after him͒. The results are based on the finest and adaptive mesh yet ͑over 11,000 four-and eight-node elements for a single hemispherical asperity in contact with a rigid flat, and 100 contact elements͒ that is necessary for finite element convergence. The results were obtained by using ANSYS™ and then independently confirmed by using ABAQUS™. In the future it would be useful to investigate the effect of material strain hardening and tangential loading ͑slid-ing͒.