Tangential Surface Tractions Research Articles

Elastic Stress Field beneath a Sticking Circular Contact under Tangential Load

Based on a potential theoretical approach, the subsurface stress field is calculated for an elastic half-space which is subject to normal and uniaxial tangential surface tractions that—in the case of elastic decoupling—correspond to rigid normal and tangential translations of a circular surface domain. The stress fields are obtained explicitly and in closed form as the imaginary parts of compact complex-valued expressions. The stress state in the surface and on the central axis are considered in detail. As, within specific approximations that have been discussed at length in the literature, any tangential contact problem with friction can be understood as a certain incremental series of such rigid translations, the solutions presented here can serve as the basis of very fast superposition algorithms for the analysis of subsurface stress fields in general tangential contact problems with friction. This idea is demonstrated by means of the frictional tangential contact between an elastic half-space and a rigid cylindrical flat punch with rounded corners.

Numerical solution of smooth and rough contact problems

A general procedure for the numerical solution of three-dimensional normal and tangential contact problems for non-conforming bodies of arbitrary shape is presented. The proposed procedure is based on the interpolation of known analytical solutions relating surface displacements of an elastic half-space subjected to pyramidal distributions of normal and tangential surface tractions. These formulas are used to interpolate unknown pressure distributions on the contact zone of two elastic bodies in contact. The proposed interpolation significantly reduces the computational burden of the numerical procedure required to interpolate the actual pressure distribution and to determine the initially unknown contact region. It amounts to expressing the contact pressure as a function of the elastic parameters of the two bodies, the distance between their surfaces and the relative displacement between two far-points pertaining to the bodies in contact. The procedure has been validated by comparison with classical contact problems and the results show excellent agreement with existing analytical and numerical solutions.

Finite deformations of full sine-wave St.-Venant beam due to tangential and normal distributed loads using nonlinear TSNDT

We analyze deformations of a beam initially in the form of a full sine wave and loaded by distributed tangential and normal surface tractions with the goal of finding how the beam curvature and the consideration of all nonlinearities affect the maximum values of stresses and deflections. The curvature of the beam in the left half is opposite of that in the right half of the beam, and the radius of curvature varies from point to point. The problem has been analyzed by using the third order shear and normal deformable beam theory (TSNDT) that accounts for all geometric nonlinearities. The beam material is St. Venant–Kirchhoff for which the second Piola–Kirchhoff stress tensor is a linear function of the Green–Lagrange strain tensor. It is found that for quasistatic deformation with non-dimensional pressure, \( \bar{q}_{0} = \frac{{q_{0} {\mathbb{L}}^{4} }}{{100E_{L} H^{4} }} = 2. 6 \), the axial stress at the point \( \left( {\frac{{3{\mathcal{L}}}}{4},\frac{H}{2}} \right) \) from the nonlinear theory equals nearly 4 times that from the linear theory. The lateral deflection at the point \( \left( {\frac{{3{\mathcal{L}}}}{4},0} \right) \) from the nonlinear theory is about 1.5 times that from the linear theory. Here q 0 is the uniform pressure applied on the bottom surface of the beam, E L Young’s modulus in the longitudinal direction for infinitesimal deformations, H the beam height, \( {\mathbb{L}}^{ } \) the horizontal distance between the two end faces, and \( {\mathcal{L}} \) the arc length. Significant features of the work include using the TSNDT, accounting for all geometric nonlinearities, using a materially objective constitutive relation, considering curvature varying from positive to negative, and applying both tangential and normal surface tractions.

Effect of Non-Steady State Loading on Rolling-Sliding Contact Stress

The stresses, strains, and deformations produced by repeated, two-dimensional non-steady state rolling-sliding contact were analyzed using an elastic-plastic finite element model. An advanced cyclic plasticity model was used. The non-steady state rolling contact was restricted to a harmonic variation of the normal Herztian contact pressure. Repeated rolling and sliding were simulated by multiple translations of a set of varying normal and tangential surface tractions across an elastic-plastic semi-infinite half space. The non-steady state loading considered results in a wavy contact surface profile. The surface displacements and wave depth of the wavy deformation increase with increasing rolling passes, but the increases in wave depth per rolling pass (ratchetting rate) decay. The residual stresses and strains near the wave trough of the residual wavy deformation are higher than those near the wave crest. The results are in agreement with the experimental observations. The tangential force has a greater influence on the residual strains than on the residual stresses.

Analytical solutions for functionally graded incompressible eccentric and non-axisymmetrically loaded circular cylinders

We study analytically plane strain static deformations of functionally graded eccentric and non-axisymmetrically loaded circular cylinders comprised of isotropic and incompressible linear elastic materials. Normal and tangential surface tractions on the inner and the outer surfaces of a cylinder may vary in the circumferential direction . The shear modulus is taken to vary either as an exponential function or as a power law function of the radius only. The radial and the circumferential displacements , and the hydrostatic pressure are expanded in Fourier series in the angular coordinate, and expressions for their coefficients are derived from equations expressing the balance of mass (or the continuity equation) and the balance of linear momentum . Boundary conditions are satisfied in the sense of Fourier series. For the exponential variation of the shear modulus, the method of Frobenius series is used to solve 4th-order ordinary differential equations for coefficients of the Fourier series. It is shown that the series solutions for displacements and the hydrostatic pressure converge rapidly. Results for eccentric cylinders and non-axisymmetrically loaded circular cylinders are computed and exhibited graphically. Effects on stress distributions of the eccentricity in the cylinders and of the gradation in the shear modulus are illuminated. It is found that in a thin cylinder subjected to cosinusoidally varying pressure on the inner surface, segments of the cylinder between two adjacent cusps in the pressure deform due to bending rather than stretching.

A numerical model for the point contact of dissimilar materials considering tangential tractions

This paper presents a three-dimensional numerical model for the simulation of the contacts of elastically dissimilar materials. This model is based on the Boussinesq–Cerruti integral equations relating normal and tangential surface tractions to surface displacements and employs the static friction law to identify the onset of local micro-slip. An iterative conjugate gradient method (CGM) is utilized to determine the unknown contact and stick area efficiently and accurately with the assistance of the discrete convolution and fast Fourier transform (DC-FFT) algorithm. The model leads to the solutions of surface real contact and stick areas, pressure, tangential tractions, and the subsurface stress field. It is verified through the comparison of the numerical results with analytical solutions. Simulations are performed for a ball and half-space contact with the normal loading alone first, and then the results for the contact with increasing tangential loading are also presented. The effects of shear tractions on the contact area, the stick zone, pressure and the intensity of von-Mises stress are discussed.

Maxwell stress distributions in chiral sculptured thin films due to obliquely incident plane waves

When a circularly polarized plane wave is obliquely incident on a chiral sculptured thin film (STF), the Maxwell stress dyadic exhibits a decreasing periodic variation across the thickness of the film. Normal and tangential surface tractions exist on the two faces of the chiral STF, as well as a net normal pressure across the film. These stresses are affected by the incidence angle of light, and are maximized when (i) the incident plane wave and the chiral STF are co-handed, (ii) the wavelength falls within a regime called the Bragg regime, (iii) the ratio of film thickness to the structural period of the chiral STF reaches a saturation value, and (iv) the deviation from normal incidence is small.

A semi-analytical approach to three-dimensional normal contact problems with friction

We present a semi-analytical approach for three-dimensional elastostatic normal contact problems with friction. The numerical approach to iteration on contact area and stick zone size is supported by an underlying analytical solution relating normal and tangential surface tractions to surface displacements in three coordinate directions. The governing equations are fully coupled. The analytical surface displacement solutions for a basic loading element have been derived elsewhere (Li and Berger 2001), and the total surface displacements are constructed as a superposition of deflections due to overlapping pyramid load segments. This approach requires no interpolation scheme for the field variables, which distinguishes it from other numerical techniques such as the FEM, BEM, and meshless methods. A background grid is defined only on the contact surfaces, and iteration approaches are used to determine a convergent configuration for contact domain and stick zone size. The approach is exercised on several normal contact problems, with and without friction, and the results compare favorably to existing analytical and numerical solutions.

THERMOELASTIC EVOLUTION OF HOT SPOTS IN THERMALLY NONLINEAR CARBON GRAPHITE SLIDERS

Abstract Presented are formulations and calculations of elastic stress and displacement fields within an evolving thermal mound present on the interface between a carbon graphite brush sliding against a fast moving rotor. Thermal mound temperatures, due to frictional and electrical power dissipation about the discrete brush to rotor contact spots, evolve from cold to hot and are based on the thermally nonlinear temperature fields calculated by Yune and Bryant. Presented are thermoelastic surface displacements and thermoelastic stresses that correspond to the aforementioned temperature fields. Also presented are values of contact length, normal approach, and contact pressures that exist when the thermoelastically induced surface curvatures are pressed flat by prescribed normal and tangential surface tractions.

Three‐dimensional infinite boundary elements for contact problems

AbstractA three‐dimensional infinite boundary element method for the modelling of half space contact problems is presented. The infinite surface of the half space is discretized into a finite number of elements with the elements extending to infinity mapped onto finite elements. A special mapping scheme that handles the singular integrals over those infinite boundary elements is introduced. The infinite boundary element scheme treats the effect of infinity correctly, leading to excellent accuracy which is verified for a half space subjected to uniform pressure in a circular region and a square region on the surface. The utility of the program is then illustrated through the analysis of the stress concentration at the surface of an oblate spheroidal cavity, serving as a model for a naturally occurring void, embedded in the half space subjected to normal and tangential surface tractions.

An Analysis of Deformation of Steel Coated With Ceramics in Rolling-Sliding Contact

The stresses, strains, and deformations produced by repeated, two-dimensional rolling-sliding contacts are analyzed both in the coated layer of ceramics SiC and in the substrate of two types of steels using a finite element method. Rolling and sliding are simulated by translating an appropriate set of normal and tangential surface tractions across the surface of an elastic-plastic model. A peak-pressure to shear-strength ratio of p0/k = 5 and normal to tangential force ratios of T/P = 0.2 are examined. The effect of thickness of the coated layer on the mechanical behavior of the substrate is discussed.

Elasto-Plastic Finite Element Analysis of Repeated, Two-Dimensional Rolling-Sliding Contacts

The stresses, strains, and deformations produced by repeated, two-dimensional rolling-sliding contact are analyzed using a modified finite element model developed by Bhargava et al. [1]. Rolling and sliding are simulated by translating an appropriate set of normal and tangential surface tractions across an elastic-perfectly plastic half space. The study examines a peak-pressure-to-shear strength ratio of po/k = 4.5 and normal to tangential force ratios of T/N = 0.20 and T/N = 0.17. The calculations describe the residual stresses, displacements and the continuing cyclic radial, shear and equivalent strains generated at various depths in the rim. The results are compared with previous calculations by Johnson and Jefferis [2] of rolling-sliding contact and with pure rolling. The present work predicts much higher deformations than previously calculated.

The Role of Hard Layers in Lubricated and Dry Sliding

Lubricated and dry experiments on titanium and steel surfaces with and without TiN sputtered coatings of various thicknesses have been conducted. The significance of the layer thickness, interfacial “friction”, magnitudes of normal and tangential surface tractions, and the mechanical properties of the layer and of the substrate (e.g., elastic modulus and hardness) are critically examined. The conditions under which the deformation mode at the solid-solid contacts is elastic or plastic are addressed in light of the experimental evidence and a finite element analysis. It is shown that surfaces with very low friction, especially for unlubricated sliding, and practically zero wear rates can be obtained in both lubricated and dry sliding by coating the surfaces with sufficiently thick TiN layers. Removal of the protective TiN layer resulted in plowing, severe damage, and delamination.

A Direct Tensile Loading Apparatus Combined with a Cubical Test Cell for Testing Rocks and Concrete

Abstract A novel tensile loading apparatus is described, which when used in conjunction with an existing fluid-cushion cubical test cell, allows the simultaneous application of a pure principal tensile stress and transverse principal compressive stresses to concrete and rock specimens with a minimum of boundary constraints. A tensile stress is applied along one axis of a cubical specimen through brush platens glued to the specimen surface while compressive stresses are applied along the two remaining axes through fluid cushion systems. In this manner, three totally independent principal stresses can be applied with virtually no tangential surface tractions so that the actual stress state in the specimen is identical to the applied stresses. Stress-strain response curves and strength relations from biaxial tension-compression tests on Indiana limestone and steel-fiber reinforced concrete are presented as examples of the capabilities of the new device.

Elastodynamic Stress-Intensity Factors for a Crack Near a Free Surface

Elastodynamic Mode I and Mode II stress-intensity factors are presented for a subsurface crack in an elastic half space. The plane of the crack is normal to the surface of the half space. The half space is subjected to normal and tangential time-harmonic surface tractions. Numerical results show the variation of KI and KII at both crack tips, with the dimensionless frequency and the ratio a/b, where a and b are the distances to the surface from the near and the far crack tips, respectively. The results are compared with corresponding results for a crack in an unbounded solid.

Some mixed boundary value problems for the semi-infinite elastic solid subjected to tangential surface tractions

A method is developed for representing the tangential displacements and tractions at the surface of the semi-infinite solid in terms of potential functions. In this form, a mathematical analogy is revealed between corresponding mixed boundary value problems involving tangential and normal surface displacements respectively. This analogy enables a general solution to be obtained to the problem in which the surface tangential displacements are specified axisymmetric functions inside the circle a⩾ r⩾0 and the tangential surface traction is zero outside this circle. The method can also be used for certain non-axisymmetric problems, but it fails if the indentation analogue has a stress singularity at the boundary of the stressed area.

The microslip due to tangential surface tractions applied to rolling cylinders

The microslip due to tangential surface tractions applied to rolling cylinders

Role of wear debris in the wear characteristics of a rolling element subjected to tangential surface tractions : S. N. Giolmas and J. Halling, Proc. Inst. Mech. Engrs., 179, Part 3J (1964–65) 9 pp., 10 figs., 12 refs.

Role of wear debris in the wear characteristics of a rolling element subjected to tangential surface tractions : S. N. Giolmas and J. Halling, Proc. Inst. Mech. Engrs., 179, Part 3J (1964–65) 9 pp., 10 figs., 12 refs.

Second Paper: Analysis of Spin/Roll Conditions and the Frictional-Energy Dissipation in Angular-Contact Thrust Ball Bearings

In current analyses of the contact conditions in angular-contact ball bearings it is assumed that sliding occurs in all regions where tangential surface tractions are operative. Other work in similar rolling-contact situations has, however, demonstrated that some of this slip requirement may be accommodated by the elastic surface deformations. This type of analysis leads to areas of sliding and sticking coexisting within the contact areas. These concepts are here applied to angular-contact thrust ball bearings and lead to some interesting deviations from the results obtained using the complete slip analysis. In particular, these concepts of microslip enable the determination of the spin/roll ratio at each race contact. Detailed calculations for conical ball thrust bearings illustrate the main discrepancy of the full slip analysis in over-estimating the frictional-energy losses, such discrepancies being most marked with decreasing contact angle, increasing pitch/ball radii, and increasing coefficient of sliding friction.

The effect of surface finish on the creep and wear of a rolling ball subjected to normal and tangential surface tractions

Experimental results have been obtained for the creep of a relatively smooth steel ball rolling along planes of various surface finish. The ball was subjected to combined normal and tangential surface tractions over a wide range of loading. It is found that the measured creep is only marginally dependent on the degree of surface roughness. The coefficient of sliding friction derived from the rolling tests, and also as measured by independent sliding tests, shows no significant variation with surface texture. These results are considered in relation to other workers'evidence relating to the effects of surface finish and it is concluded that theoretical predictions of these phenomena based on the assumption of ideal smooth surfaces may generally be used for real engineering surfaces. It is suggested that this assumption may be invalid at small normal loads. The wear of a rolling ball subjected to these loading conditions was also studied by using radioactive tracer techniques. These results again show virtually no dependence on the surface texture except in such cases as when the surface is modified by the removal of the hard amorphous surface layer created by the machining process.