Itzhak Green

A Finite Element Study of Elasto-Plastic Hemispherical Contact Against a Rigid Flat

This work presents a finite element study of elasto-plastic hemispherical contact. The results are normalized such that they are valid for macro contacts (e.g., rolling element bearings) and micro contacts (e.g., asperity contact), although micro-scale surface characteristics such as grain boundaries are not considered. The material is modeled as elastic-perfectly plastic. The numerical results are compared to other existing models of spherical contact, including the fully plastic truncation model (often attributed to Abbott and Firestone) and the perfectly elastic case (known as the Hertz contact). This work finds that the fully plastic average contact pressure, or hardness, commonly approximated to be a constant factor of about three times the yield strength, actually varies with the deformed contact geometry, which in turn is dependent upon the material properties (e.g., yield strength). The current work expands on previous works by including these effects and explaining them theoretically. Experimental and analytical results have also been shown to compare well with the current work. The results are fit by empirical formulations for a wide range of interferences (displacements which cause normal contact between the sphere and rigid flat) and materials for use in other applications.

A Finite Element Study of the Residual Stress and Deformation in Hemispherical Contacts

This work presents a finite element model (FEM) of the residual stresses and strains that are formed after an elastoplastic hemispherical contact is unloaded. The material is modeled as elastic perfectly plastic and follows the von Mises yield criterion. The FEM produces contours for the normalized axial and radial displacements as functions of the removed interference depth and location on the surface of the hemisphere. Contour plots of the von Mises stress and other stress components are also presented to show the formation of the residual stress distribution with increasing plastic deformation. This work shows that high residual von Mises stresses appear in the material pileup near the edge of the contact area after complete unloading. Values are defined for the minimum normalized interference, that when removed, results in plastic residual stresses. This work also defines an interference at which the maximum residual stress transitions from a location below the contact region and along the axis of symmetry to one near to the surface at the edge of the contact radius (within the pileup).

Predicting the coefficient of restitution of impacting elastic-perfectly plastic spheres

The current work presents a different methodology for modeling the impact between elasto-plastic spheres. Recent finite element results modeling the static deformation of an elasto-plastic sphere are used in conjunction with equations for the variation of kinetic energy to obtain predictions for the coefficient of restitution. A model is also needed to predict the residual deformation of the sphere during rebound, or unloading, of which several are available and compared in this work. The model predicts that a significant amount of energy will be dissipated in the form of plastic deformation such that as the speed at initial impact increases, the coefficient of restitution decreases. This work also derives a new equation for the initial critical speed which causes initial plastic deformation in the sphere that is different than that shown in previously derived equations and is strongly dependant on Poisson’s Ratio. For impacts occurring above this speed, the coefficient of restitution will be less than a value of one. This work also compares the predictions between several models that make significantly different predictions. The results of the current model also compare well with some existing experimental data. Empirical fits to the results are provided for use as a tool to predict the coefficient of restitution.

A Finite Element Study of Elasto-Plastic Hemispherical Contact

This work presents a finite element study of elasto-plastic hemispherical contact. The results are normalized such that they are valid for macro contacts (e.g., rolling element bearings) and micro contacts (e.g., asperity contact). The material is modeled as elastic-perfectly plastic. The numerical results are compared to other existing models of spherical contact, including the fully plastic case (known as the Abbott and Firestone model) and the perfectly elastic case (known as the Hertz contact). At the same interference, the area of contact is shown to be larger for the elasto-plastic model than that of the elastic model. It is also shown, that at the same interference, the load carrying capacity of the elasto-plastic modeled sphere is less than that for the Hertzian solution. This work finds that the fully plastic average contact pressure, or hardness, commonly approximated to be a constant factor (about three) times the yield strength, actually varies with the deformed contact geometry, which in turn is dependant upon the material properties (e.g., yield strength). The results are fit by empirical formulations for a wide range of interferences and materials for use in other applications.Copyright

Stability Threshold and Steady-State Response of Noncontacting Coned-Face Seals

The dynamic behavior of a noncontacting coned-face seal is analyzed. Stiffness and damping properties of the fluid film and flexible support including elastomeric secondary seal are fully accounted for. Stability threshold and steady-state response in the presence of rotor axial runout and assembly misalignment are investigated. An expression is provided for the critical speed above which the seal becomes dynamically unstable. For stable operation, the relative misalignment between the mating faces is given as a function of rotor runout, assembly misalignment, design parameters, and operation conditions. An expression is provided for the critical rotor runout above which the seal will fail due to face contact. Although the analysis is based on small perturbation assumption, it is shown to be valid in many practical cases. Presented as an American Society of Lubrication Engineers paper at the ASLE/ASME Lubrication Conference in San Diego, California, October 22–24, 1984

Modeling of the rolling and sliding contact between two asperities

A semi-analytical method for the tridimensional elastic-plastic contact between two hemispherical asperities is proposed. The first part of the paper describes the algorithm used to deal with the normal contact, which can be either load-driven or displacement-driven (dd). Both formulations use the conjugate gradient method and the discrete convolution and fast Fourier transform (DC-FFT) technique. A validation of the code is made in the case of the displacement-driven formulation for an elastic-plastic body in contact with a rigid punch, simulating a nano-indentation test. Another new feature is the treatment of the contact between two elastic-plastic bodies. The model is first validated through comparison with the finite element method. The contact pressure distribution, the hydrostatic pressure and the equivalent plastic strain state below the contacting surfaces are also found to be strongly modified in comparison to the case of an elastic-plastic body in contact with a purely elastic body. The way to consider rolling and sliding motion of the contacting bodies consists of solving the elastic-plastic contact at each time step while upgrading the geometries as well as the hardening state along the moving directions. The derivations concerning the interference calculation at each step of the sliding process are then shown, and an application to the tugging between two spherical asperities in simple sliding (dd formulation) is made. The way to project the forces in the global reference is outlined, considering the macro-projection due to the angle between the plane of contact and the sliding direction, and the micro-projection due to the pile-up induced by the permanent deformation of the bodies due to their relative motion. Finally, a load ratio is introduced and results are presented in terms of forces, displacements, and energy loss in the contact.

On the Modeling of Elastic Contact between Rough Surfaces

The contact force and the real contact area between rough surfaces are important in the prediction of friction, wear, adhesion, and electrical and thermal contact resistance. Over the last four decades various mathematical models have been developed. Built on very different assumptions and underlying mathematical frameworks, model agreement or effectiveness has never been thoroughly investigated. This work uses several measured profiles of real surfaces having vastly different roughness characteristics to predict contact areas and forces from various elastic contact models and contrast them to a deterministic fast Fourier transform (FFT)-based contact model. The latter is considered “exact” because surfaces are analyzed as they are measured, accounting for all peaks and valleys without compromise. Though measurement uncertainties and resolution issues prevail, the same surfaces are kept constant (i.e., are identical) for all models considered. Nonetheless, the effect of the data resolution of measured surface profiles will be investigated as well. An exact closed-form solution is offered for the widely used Greenwood and Williamson (GW) model (Greenwood and Williamson, Proceedings of the Royal Society of London A, vol. 295, pp. 300–319), along with an alternative definition of the plasticity index that is based on a multiscale approach. The results reveal that several of the theoretical models show good quantitative and qualitative agreement among themselves, but though most models produce a nominally linear relationship between the real contact area and load, the deterministic model suggests otherwise in some cases. Regardless, all of the said models reduce the complicated surface profiles to only a few key parameters and it is therefore unrealistic to expect them to make precise predictions for all cases.

Elasto-plastic hemispherical contact models for various mechanical properties

Abstract This work uses the finite element technique to model the elastoplastic deformation of a hemisphere contacting a rigid flat for various material properties typical of aluminium, bronze, copper, titanium and malleable cast iron. Additionally, this work conducted parametric finite element method (FEM) tests on a generic material in which the elastic modulus and Poissons ratio are varied independently while the yield strength is held constant. A larger spectrum of material properties are covered in this work than in most previous studies. The results from this work are compared with two previously formulated elastoplastic models simulating the deformation of a hemisphere in contact with a rigid flat. Both of the previously formulated models use carbon steel mechanical properties to arrive at empirical formulations implied to pertain to various materials. While both models considered several carbon steels with various yield strengths, they did not test materials with various Poissons ratios or elastic moduli. The previously generated elastoplastic models give fairly good predictions when compared with the FEM results for various material properties from the current work, except that one model produces more accurate predictions overall, especially at large deformations where other models neglect important trends due to decreases in hardness with increasing deformation.

Computation of Member Stiffness in Bolted Connections

The member stiffness in a bolted connection has a direct influence upon safe design with regard to both static and fatigue loading, as well as in the prevention of separation in the connection. This work provides a simple technique for computing the member stiffness in many types of bolted connections. Finite element analyses are performed for joints having a range of materials and geometries, and the results are generalized by nondimensionalization. An exponential expression for the stiffness is determined, and the results are compared with those of some of the techniques currently used.

Crack Detection in a Rotor Dynamic System by Vibration Monitoring—Part I: Analysis

Many practical rotor dynamic systems contain shaft/rotor elements that are highly susceptible to transverse cross-sectional cracks due to fatigue. The early detection of mechanical malfunction that can be provided by an effective vibration monitoring system is essential. Two theoretical analyses, global and local asymmetry crack models, are utilized to identify characteristics of the system response that may be directly attributed to the presence of a transverse crack in a rotating shaft. A model consisting of an overhung whirling rotor is utilized to match an experimental test rig. A 2X harmonic component of the system response is shown to be the primary response characteristic resulting from the introduction of a crack. Once the unique characteristics of the system response are identified, they serve then as target observations for the monitoring system.