A finite-element analysis of the indentation of an elastic-work hardening layered half-space by an elastic sphere

An elastic–plastic contact problem in elastic-work hardening layered half-space indented by an elastic sphere was solved numerically using the finite element method. The case of a surface layer stiffer than the substrate is considered, and general solutions for the subsurface stresses and deformation fields are presented for a relatively thin elastic layer. Differences between the elastic and elastic–plastic solutions for the contact pressure distribution have been investigated for various layer thicknesses. Crack initiation and decohesion of the layer was also discussed with reference to the growth of the plastic zone.     

Subsurface crack mechanisms under indentation loading

A linear elastic fracture mechanics analysis of plane-strain indentation of a homogeneous half-space with a subsurface horizontal crack was performed using the finite element method. Stress intensity factor results obtained for an infinite plate with a central crack subjected to far-field tension and a half-space with a frictionless subsurface horizontal crack under a moving surface point load are shown to be in good agreement with corresponding analytical results. Crack mechanism maps illustrating the occurrence of separation, forward and backward slip, stick, and separation at the crack interface are presented for different indentation load positions and crack face friction coefficients. Results for the stresses in the vicinity of the crack tips and the mode I and mode II stress intensity factors are given for different indentation positions, crack face friction coefficients, and both concentrated and distributed surface normal tractions. Although indentation produces a predominantly shear and compressive stress field, mode I loading conditions are shown to occur for certain indentation positions. However, the magnitude of the mode I stress intensity factor is significantly smaller than that of mode II, suggesting that in-plane shear mode crack growth is most likely to occur in the absence of microstructural defects. The significance of crack face friction and sharpness of the indenter on the subsurface shear mode crack propagation rate is interpreted in terms of the mode II stress intensity factor range and material behavior.     

A new algorithm for computing the indentation of a rigid body of arbitrary shape on a viscoelastic half-space

In this paper the contact problem between a rigid indenter of arbitrary shape and a viscoelastic half-space is considered. Under the action of a normal force the penetration of the indenter and the distribution of contact pressure change. We wish to find the relations which link the pressure distribution, the resultant force on the indenter and the penetration on the assumption that the surfaces are frictionless. For indenters of arbitrary shape the problem may be solved numerically by using the Matrix Inversion Method (MIM), extended to viscoelastic case. In this method the boundary conditions are satisfied exactly at specified “matching points” (the mid-points of the boundary elements). It can be validated by comparing the numerical results to the analytic solutions in cases of a spherical asperity (loading and unloading) and a conical asperity (loading only). Finally, the method was implemented for a finite cylindrical shape with its curved face indenting the surface of the half-space. This last example shows the efficiency of the method in case of a prescribed penetration as well as a given normal load history.     

Thermoelastic contact of a rotating sphere and a half-space

An approximate solution is given for the thermoelastic contact problem in which a rotating sphere is loaded against a half-space. Frictional heating causes a redistribution of contact pressure away from the axis of rotation. The contact radius falls with increasing rotational speed, whilst for a given speed there is a limiting value which the radius cannot exceed, no matter how high the load.     

Stress analysis of an elastic half-space perforated by twin circular holes and under gravitational loading

The plane elastic problem which simulates gravitational loading, with an accompanying horizontal pressure, of an identical pair of circular tunnels at a common depth below a stress-free horizontal surface is treated. The analysis employs a sequence of Airy stress functions to construct an analytical solution which is then evaluated for a full range of geometries and Poisson's ratios, and used to gauge the effect of the coefficient of horizontal pressure, k₀. Results for hoop stress concentrations at the tunnel surfaces quantify the significant interactions that can occur between the tunnels and the free surface and also define the marked dependence on k₀.     

On partial contact of a thin-walled circular cylinder and a rigid half-space

A long circular cylinder with radius R and thickness t transmits a longitudinal eccentrically applied force to a rigid half-space. We study the influence of the eccentricity of the load and the ratio on the stress state of the cylinder using the theory of thin shells in bending. The elastic displacements are presented in the form of a Fourier expansion, the coefficients of which are found by the variational formulation of the problem combined with the penalty approach for a numerical implementation of contact conditions. Results are applied to certain problems of buckling of a circular bar and compared with previous approximate solutions.     

Vibrations of a deformable ball rolling on a lubricated rigid half-space

The problem of vertical vibrations of a deformable ball rolling on a lubricated rigid half-space has been stated and numerically solved. It is assumed that the force, which causes the ball’s elastic deformation in the zone of lubricated concentrated contact, is applied to the ball. The problem’s mathematical model is reduced to the system of nonstationary integrodifferential equations and inequalities with initial and boundary conditions. The computational algorithm of the problem’s solution was based on Newton’s method and application of an implicit scheme to approximate nonstationary equations. Vibrational solutions, which describe the behavior of the system consisting of the ball and the lubrication layer both under excitation of the initial conditions and under the excitation force, have been obtained. It is shown that as opposed to the traditional elastohydrodynamic lubrication theory approach, in which contacting bodies are considered as massless ones, taking the ball’s mass and corresponding inertia into account makes it possible to extend the category of obtained solutions and investigate vibrational processes in the contact zone. The results may be used to analyze processes occurring in lubricated concentrated contacts of gears and rolling bearings.     

Prediction of the film thickness for the normal approach of a rigid sphere towards a thin soft layer

A semi-analytical solution to the problem of the squeeze film between a rigid sphere and a thin soft layer attached firmly to rigid foundation is presented. It is assumed that the shapes of the solids are the same as the elastically deformed shapes under the unlubricated conditions (the Grubin’s approximation). Formulae for the variation of the film thickness with time are presented in the full range of Poisson’s ratio using two different pressure boundary conditions. It was found that only the magnitude of the film thickness predicted for an incompressible layer (ν=0.5), is influenced by the type of the pressure condition considered in the analysis.     

Plate bending wave propagation behavior under metal sphere impact loading

A loose part can cause component damages and material wear on nuclear power plants; thus, a mass estimation of the loose part is crucial to safety management. A bending wave propagation of a structure under the loose part impact loading is precisely simulated to accurately estimate the mass of a loose part. Lamb’s general solution for an arbitrary impact force function and Hertz impact theory have been used to identify the characteristics of the bending wave that is impacted by a metallic loose part in reactor pressure boundary components. However, these approaches cannot provide accurate information on the acceleration response that is required to identify the impact source. In this study, the bending wave propagation behavior of plate structures under a simulated loose part (Metal sphere) impact loading was modeled using a Finite element analysis (FEA) technique. The characteristics (e.g., Maximum acceleration amplitude, primary frequency, and bending wave velocity) of the impact response signal from a metal sphere were analyzed with the FEA results and were verified with experimental results. In addition, the correlation between plate thickness and characteristic length was presented. Results from this study can be utilized to estimate the location and mass of a loose part for condition monitoring and diagnostics in nuclear power plants.     

Boundary element numerical analysis of elastic indentation of a sphere into a bi-layer material

The initial part of load/penetration plot in depth sensing spherical indentation (nano-indentation) is analysed. The results of a numerical study using a specifically developed simulation tool based on the boundary element method are presented.They reveal that the usual linear relationship between the indentation depth and the square of the contact radius for homogeneous materials is also valid in the case of a bi-layer material.It is also shown that the elastic response of the bi-layer material is specific of the coating alone only if the indentation depth, h, is less than 2% of the coating thickness, t.For higher indentation depths (10%>h/t>1%), the macro-elastic response of the composite is seen to saturate with an overall elastic response still containing a contribution of roughly 10% of the coating modulus.The hypothesis of indentation limit h/t<10% of bi-layer material insuring coating-specific response for penetration hardness (Bückle's rule) is roughly two orders of magnitude high when applied by default to the corresponding coating-specific elastic response.     

Parameters of fracture mechanics under combined loading by normal separation and In-plane shear

The methods of separable functions and finite elements have been used to generate design formulas of the J-integral for a tensing plate made of material reinforced by a deformation with a centrally inclined through crack. The crack angle allows for a variation of the loading conditions from normal separation up to the combined loading by normal separation and in-plane shear. The correcting coefficients η _pl and η _pl ^COD that enter into the formulas of the J integral depend on the relation between normal separation and in-plane shear parameters and on the deformation reinforcement factor. At the same time, the normalized correcting coefficients η _pl (θ)/η _pl (θ = 0) are practically independent of the sample geometry.     

On the oblique impact of a rigid sphere against a rigid-plastic solid

Previous workers have sought to explain the variation in crater volume observed when hardened steel spheres impact at different angles against mild steel plates by numerically modelling a rigid sphere striking an ideal rigid-plastic half-space. The present note shows that the discrepancy between experiment and their throretical prediction from this model can be significantly reduced by incorporating a more accurate calculation of the area of contact into the analysis.     

Transient torsional vibrations of a rigid mass connected to an elastic half-space by an elastic circular rod

Applying Laplace transformations with respect to time and numerical inverse Laplace transformations, the transient torsional vibrations of a rigid mass connected to an elastic half-space by an elastic rod are analyzed under two specific incident pulse loads. The time histories of the rigid mass rotation are presented. Application of the numerical inverse Laplace transformations enables us to obtain the time histories of the rigid mass rotation accurately and fairly quickly by utilizing a digital computer. This method is applicable to the transient problem of a rigid mass connected to a half-space by an elastic medium or a finite elastic medium on a half-space, e.g. machines such as a press or a pile driver footed on a foundation, if the shape of the incident pulse is not very complicated and the maximum time T_max is not too long. When the slenderness ratio of the rod is small, the damping effect of the half-space on the rigid mass rotation due to waves radiating to infinity is small, regardless if the shear modulus ratio of the half-space and the rod is large or not. The amplitude of the rigid mass roation increases and its period lengthens as the shear modulus ratio increases.     

Theoretical aspects of technological mechanics. 10. Hardness and axisymmetric indentation of a half-space. Part 2

Examples are presented to demonstrate the high precision of the formulas derived for the insertion of an axisymmetric punch in a blank with large transverse dimensions.     

Theoretical aspects of technological mechanics. 10. Hardness and axisymmetric indentation of a half-space. Part 1

Reliable determination of the strength of materials on the basis of hardness measurements is shown to be impossible. The stress, kinematic, and strain states of a half-space during the insertion of an axisymmetric body are determined.     

精密小转矩加载机构的研制

研制了一种压电驱动的柔性铰链微位移放大机构,以满足航天微型扭杆刚度测试装置中实现精密小转矩加载的要求,时这种机构建立了精确的有限元模型,运用有限元方法对这种的机构的强度和放大比进行了校核和分析。该机构具有350μm的行程,位移分辨率大于0．25μm,最大输出力不低于4N,满足了测试装置的要求。     

Operation of a tight joint under a torque

A tight joint subjected to a torque is considered, in the case of pliability of the joint components and tangential pliability of the contact layer. On that basis, the torque distribution over the joint length may be determined more precisely, and fretting wear may be predicted.     

Upper and lower bounds to indentation of rigid ball into semi-infinite solid

A closed-form solution for upper and lower bounds (‘dual bounds’) to a slow indentation of rigid ball (under increasing normal load) into rigid plastic solids are presented. The solutions enlarge the theoretical background to the familiar Brinell hardness tests used in assessing strength of materials and add few new features.The new features are mainly: (i) constructing a lower-bound solution to ball indentation, (ii) a closed-form estimation of the dual bounds with any prescribed frictional shear along the indenter/solid contact, (iii) providing dual bounds to two different situations of the material response: (a) the removed material is falling off from the bulk (akin to indentation into brittle-like materials) and (b) the removed material is piled up along the indenter face (akin to indentation into ductile-like materials).The dual bound solutions of the indenting mean ball pressure for both material cases seem to capture properly well the material constrain to local plastic indentation and the effect of the interfacial friction. Having both bounds simultaneously, despite their unavoidable deviation, provide much more informative results than a single bound.Comparisons between the suggested bounding analysis and experiments (taken from Tabor [Tabor D. Hardness of Metals. Oxford: Clarendon Press; 1951], Johnson [Johnson KL. Contact mechanics. Cambridge: Cambridge University Press; 1985 (new edition, 2001)] and others), numerical FEM solutions (Mesarovic and Fleck [Mesarovic SDj, Fleck N. Spherical indentation of elastic-plastic solids. Proceedings of Royal Society of London 1999; A455: 2707–28] and others) and some self-made measurements are all favorable.     

Isoviscous flow past a rigid sphere partially immersed in a thin oil film

Numerical solutions to steady state circular contact elastohydrodynamic lubrication problems normally employ Reynolds equation, forcing the assumption that the flow through the conjunction is two‐dimensional and single‐phase. In this paper, the full Navier–Stokes equations are used on a simpler rigid isoviscous sliding point contact. The aim of the study was to compute the lubricant flow patterns through and around the conjunction of a point contact when the oil inlet is a free surface that has a similar wavy shape to the wake. This model is like some seen experimentally in a ball and plate machine or industrially in a radial ball bearing with insufficient lubricant supply. There is first a discussion on the alteration with time of the oil wake geometry after it has left the conjunction. Then, the computed results mapped the areas of varying concentration between oil and air. They showed an accumulation of oil and vapour forming a reservoir at the inlet to the conjunction. Within it was a vortex attached to the stationary surface, just downstream of which, the main pressure distribution commences. The computed flow patterns were compared with experimental pictures wherever possible. Copyright © 2007 John Wiley & Sons, Ltd.     

Stresses in hard coating due to a rigid spherical indenter on a layered elastic half-space

The axisymmetrical contact problem of elasticity connected with an indentation of a rigid spherical indenter in an elastic semi-space covered by an elastic layer is considered. Stress tensor components in interior points of the non-homogeneous half-space by numerical calculation of some integrals was obtained. Detailed analysis of the maximal tensile stress distributions and Huber-von Mises reduced stress distributions produced by contact pressure is presented. The dependence between these stresses and the ratio between the layer thickness and contact area width is explored. The obtained results for stresses are compared with results obtained for half-space loaded by the Hertz pressure.