Hertzian Stress Research Articles

Effect of elastic gradients on the fracture resistance of tri-layer restorative systems

ObjectivesTo assess the impact of elastic gradients formed among restorative material, cement, and substrate on the fracture resistance of tri-layer restorative systems. MethodsFour CAD/CAM materials were utilized, two glass-ceramics (IPS e.max CAD, Vita Suprinity) and two resin-ceramic hybrids (Vita Enamic, Lava Ultimate). Their fracture resistance was examined by biaxial flexure (n = 8) and Hertzian indentation (n = 10) tests. Statistical analysis was conducted using ANOVA and Tukey tests (p = 5 %). Finite element analysis (FEA) was employed to simulate the Hertzian indentation test and elucidate the stress-fields formed on the intaglio surface below the loading area. ResultsThe biaxial flexural strength (MPa) of glass-ceramics exceeded the hybrid materials (e.max 417a, Suprinity 230b, Enamic 138c, and Lava Ultimate 183bc). Conversely, the load-bearing capacity (N) of the materials bonded to dentin analog demonstrated the opposite trend, with the hybrid materials achieving superior results (e.max 830 C, Suprinity 660D, Enamic 1822B, and Lava Ultimate 2593 A). The stress-fields observed by FEA were coherent with the experimental results for Hertzian flexural stresses (MPa): e.max 501 A, Suprinity 342 C, Enamic 406B, whereas no tensile stress was observed at the intaglio surface of Lava Ultimate. SignificanceDetailed analysis of the fracture resistance of the tri-layer restorative systems showed that the elastic gradients play a more significant role than the flexural strength of the restorative materials. The coherence of the elastic moduli between the restorative material and supporting structures results in reduced tensile stress concentration at the intaglio surface beneath the loading area and enhances the ability to withstand load.

Study of the effect of macrogeometry of rubbing steel bodies at their point contact on the tribological characteristics of lubricants under boundary lubrication conditions

The effect of the shape of contacting nonconformal elements of friction units that form an initial point contact on the tribological characteristics of lubricants during friction in the boundary lubrication mode. Point contact of rubbing bodies is realized on four-ball friction machines when testing tribological characteristics of lubricants in accordance with Russian GOST 9490–75 and GOST 23.221–84 Standards.However, these tests are carried out with the friction of standard bearing balls and does not allow taking into account the impact of materials of rubbing bodies on the results of the experiment. Therefore, two alternative designs of mandrels were developed, which allow implementation of two friction schemes: «rotating ball clamped in the spindle of the machine – three roller face planes» and «rotating ball — three cylindrical rollers». The load on the friction unit for both friction schemes, as well as in parallel on the mandrel under testing, in which three balls are installed, is the same (108 N), the spindle rotation speed is equal to 1 rpm, duration of testing is 60 min. The diameter of balls used in these experiments was 12.7 mm, and diameters of rollers were 5 and 8 mm, respectively. The material of samples was ShKh-15 tool steel or its 100Cr6 analogue. The designs of mandrels and the dimensions of samples were chosen in such a way that in all cases the contact areas of the experimental samples with the wearing (upper) ball would lie on a circle with a diameter of 7.32 mm. The tests were carried out under dry friction, and in boundary lubricated conditions in a PAO-4 polyalphaolefin oil medium, as well as in the same oil medium with the additives of oleic acid and DF-11 (a solution of zinc dialkyldithiophosphate in a low-viscosity oil). It is shown that the composition of a lubricating medium has a decisive effect on the test results (i.e., on the values of the friction coefficients and on the values of the areas of wear scars), while the geometry of the rubbing bodies, as well as the maximum Hertz stresses in the tribological contact, had a little effect on the antifriction and anti-wear characteristics of oils.

Fully ceramic versus steel gears: Potential, feasibility and challenges

The most common gear strength calculations performed in accordance with ISO/TC 60/SC 2 include surface durability and root bending strength. Typical geared power transmissions are characterized by high Hertzian contact stresses (surface durability) on the flanks – frequently in the order of 103 MPa – versus almost half an order magnitude lower first principal stresses on the tooth root (tooth bending strength). Therefore, the use of high strength special steel alloys is almost mandatory, especially in high-end applications, mainly to withstand the high contact stress values. Ceramic materials are typically characterized by high compressive strength versus very low flexural strength that at first glance makes them inefficient for gear transmissions since they are unlikely to withstand the tensile root stresses. Nevertheless, ceramic materials possess a number of advantages that makes them potential candidates for geared transmissions, such as their exceptional thermal properties, their low density (weight), their excellent wear characteristics and limited lubrication demands. The present work investigates the main challenges of using high-end ceramic gears and attempts to provide an insight on defining the design envelope of such gears to counter the main disadvantages. The increase of the normal pressure angle beyond the 25° limit set by ISO 6336 is primarily investigated, since in minimizes the tensile tooth root stresses, while also increasing the curvature of the contact surfaces without greatly increasing Hertzian stresses. Comparative assessment of the performance of fully ceramic gear designs versus their conventional steel counterparts is performed through quasi-static FEA simulations to set the basic design requirements for such gears and to promote further investigation of the feasibility of their usage in high power density applications.

A life prediction model for rolling contact fatigue based on the improved moment of load path method

AbstractAn improved moment of load path model for rolling contact fatigue life prediction is proposed in this paper. First, the complex multiaxial stress variation of the subsurface materials under rolling contact conditions was analyzed by the finite element method. Next, the equivalent stress parameter was presented to characterize the complex stress state based on the improved moment of load path method. The key concept of the proposed method is building a new coordinate system for analyzing the load path of the complex multiaxial stress state. The efficiency of the proposed model was validated with different kinds of fatigue test results in the existing literature. The model results show a good agreement with the test data. Finally, the model was used to obtain the relationship of the Hertz stress versus fatigue life under rolling contact conditions according to the fatigue life data.

Zone of influence in Hertzian contacts and stress analysis in a finite-width body

Zone of influence has been defined as a region beyond which stresses due to loads applied on an elastic half-space are below a given fraction of the peak stress. These zones are significant since the half-space assumption is valid if the width of contacting body is beyond these zones of influence, while it may need correction if the width is less than the zone width. Further, the half-space solution for stresses needs revision at certain depths since they do not integrate up to the corresponding applied loads within the finite body width, thus violating equilibrium. This paper presents analytical solutions for zones of influence for different stresses and peak percentages and the revised stress distribution at various depths of the contacting body, for Hertzian contacts under: (a) normal loading only; and (b) full-sliding condition. They are found to match with solutions from finite element analysis.

Tribological parameters analysis of cam and follower pair used in IC engine using ANSYS

The cam and follower pair play an essential role in the valve train mechanism in IC engines. An IC engine has a specific form of contact condition called cam and follower contact. The tribological parameters related to this contact are vital for engine performance. In this paper, the cam and follower pair are analyzed, and improvements to the traditional material used for making the cam and follower are suggested. A 3D geometry of the disc type cam and spherical follower used in a typical IC engine is designed using Solidworks software, and Finite Element Analysis (FEA) of the cam and follower assembly is done using the CAE software ANSYS Mechanical. The materials taken for this analysis are structural steel and grey cast iron. We have done analyses of two of the most important tribology parameters: contact pressure and Hertzian contact stress. The maximum value of contact pressure and Hertzian contact stress is calculated for both types of material, and the results are presented and compared in tabular form and using graphical plots. The results showed that the contact pressure, Hertzian contact stress increase linearly with each other for both type of materials. Also, the maximum values of Hertzian contact stress and contact pressure were lower for grey cast iron material. So, it is a preferable substitute to structural steel for making the cam and follower pair of a motorcycle IC engine. A good agreement of results showed that our model provided a reliable prediction of the tribological characteristics of the cam follower contact.

Microstructure and rolling contact fatigue behavior of martensitic nodular cast irons with different graphite sizes and densities

AbstractMartensitic nodular cast irons (MNCIs) with different graphite sizes and number densities are produced through continuous casting and cyclic heat treatments to refine their microstructures and to obtain a hardness of 60 HRC, and their rolling contact fatigue (RCF) properties, fractography, and subsurface microstructures have been investigated in the present study. RCF tests show that the RCF life is mainly determined by the applied Hertzian stress. Based on the calculation of characteristic parameter size and the analysis of stress concentration factor, it is found that the lower nodularity rate of graphite brings about the larger stress concentration at matrix‐graphite interfaces. This stress concentration induces a debonding of the matrix and graphite, cracks, deformation, and carbon redistribution in a local volume, which results in the formation of a white etching area. The present study indicates that MNCIs are a potential material for lubricant‐free bearings after chemical homogenization, microstructure refinement, and hardness enhancement.

Estimation of contact stresses in EN31 rolling contact bearings for screw compressor using Gauss quadrature & statistical analysis

ABSTRACT The aim of this work is to estimate Hertzian contact stresses in EN31 alloy steel and to obtain the line of best fit using statistical analysis. In refrigerant screw compressors, EN31 is used as a material of construction for cylindrical roller bearings. The Hertzian contact with friction between the outer surface of the inner race of the rolling contact bearing & the outer surface of the cylindrical roller is simulated using pin-on-disk tribometer. Measurement of friction coefficient was carried out by sliding different cylinders of hardened EN31 material against the hardened EN31 disk of the pin-on-disk tribometer. A total 16 tests, each of 1 minute time duration, are performed for the estimation of friction coefficient with varying loads from 20 N to 100 N and speeds in the range of 50 RPM to 200 RPM. The magnitude of Von Mises stress induced in the inner race of the bearing is estimated using Gauss quadrature method with the help of a self-developed MS Excel program. Statistical analysis showed that the stress magnitude increases with the increase in the load & decreases with an increase in speed.

Contact stress analysis of external non-variable multi-modulus spur gear pairs

The conditions of correct gear engagement of a pair of involute gears and the principle of geometric relationship of involute tooth profile are derived to determine the pitch point of externally meshing non-variable multi-modulus spur gear pairs into the double tooth engagement region. The Hertzian contact stress model is used as the basis to derive the contact stress calculation equations for pitch point and other characteristic contact points, and the variation of the contact stress distribution on the contact line with the module ratio is analyzed. The results demonstrate that when the number of teeth of the gear pairs is determined, the larger the value of the module difference, the easier it is for the pitch point of the gear pairs to be in the double tooth meshing zone. Except for the double tooth meshing out point, the contact stress decreases with the increase of module ratio, and the maximum point of contact stress is at the upper boundary point of the single tooth meshing zone.

Re‐recognition of deep spalling damage mechanism of rolling contact fatigue under heavy load

AbstractThis paper presents a detailed analysis of the nucleation and propagation of surface‐initiated crack in rolling contact fatigue (RCF) of bearing steel under heavy load. The cracks initiated from the junction of machining marks and trailing edge of dents formed by the plastic deformation. Crack propagation and spalling were mainly composed of three stages: (1) cracks initiating from surface defects propagate downward to form small spalling pits; (2) due to the surface stress and the maximum Hertz stress form a stress bridge, the cracks continue to expand towards the Hertz stress peak forming big spall; (3) the trailing edge of the spalling pit is used as the stress raiser to initiate next subsurface cracks and spalling expansion in successive. Different from previous understanding on the formation of subsurface‐initiated spalling, the results indicate that under heavy load the spalling at depth of the maximum shear stress zone in the subsurface can initiate from surface cracks.

Multi-objective optimisation with finite element analysis of profile shifted altered tooth sum spur gear

ABSTRACT Traditional gear design approaches provide design parameters that meet only one design requirement at a time. However, a major problem with this kind of approach is the deficit for high consistency and reliability. The objective of the article is to present a multi-objective design optimisation procedure for addendum modified spur gear with altered tooth sum. The design procedure involves implementation of Particle Swarm Optimisation (PSO) and a hybrid PSO- Teaching-learning based optimisation (PSO-TLBO) algorithm for simultaneous optimisation of the weight and contact or hertzian stress along line of action of the spur gear drive. The design optimisation was validated through stress analysis of the optimised spur gear set, using Finite element analysis (FEA) with various mesh types and grain size. Afterwards, manufacturing of the optimised non-standard gear set with metallic (EN24) and a composite (Hylum) gear material using standard tooling. This indicates that with the proposed design optimisation approach there is a possibility of meeting the requirement for efficient, compact, and reliable gear set with reduced computational cost. This study suggests that it is possible to determine a set of optimised design variables that can meet more than one design criteria at a time.

Optimal Design and Dynamic Analysis of a Spring-Actuated Cam-Linkage Mechanism in a Vacuum Circuit Breaker

This study attempted to establish an optimal design and perform dynamic analysis for a spring-actuated cam-linkage composite mechanism in a rated 12 kV, 25 kA vacuum circuit breaker (VCB). The optimal design of the VCB mechanism involves two steps: the first step involves the optimal design of the stiffness of closing springs and the cam profile, based on three-order polynomial motion curve of the follower; the second step involves the optimal design of a four-bar linkage. To minimize the surplus work in the VCB operating mechanism, the overall difference between the general output force and general resistant force of the VCB operating mechanism during the closing operation was adopted as the objective function to be minimized. Furthermore, the differential evolution algorithm with the golden ratio (DE-gr), an efficient and accuracy metaheuristic algorithm, was employed as the optimization method. The dynamic analysis of the optimal operating mechanism was conducted using a MATLAB-coded program and verified by the multibody dynamic software MSC-ADAMS. The dynamic analysis revealed that the optimal design of the VCB mechanism can considerably decrease the average closing velocity of the movable contact and the Hertz stress between the contact surface of the cam and the roller follower.

Prediction of rolling contact fatigue crack propagation in bearing steels using experimental crack growth data and linear elastic fracture mechanics

Rolling contact fatigue (RCF) is a major life limiting factor for machine elements that employ non-conformal, rolling sliding, lubricated contacts such as rolling bearings and gears. This paper explores the application of linear elastic fracture mechanics (LEFM) principles, as commonly used in structural fatigue, for prediction of RCF crack propagation. A triple-disc contact fatigue machine is used to generate RCF cracks of varying lengths in AISI 52100 bearing steel roller specimens. Crack propagation rates across the surface are measured using optical inspection of test specimens and the final crack geometry is established through specimen sectioning. A numerical finite element model of surface breaking RCF cracks based on LEFM methods is devised to predict the evolution of stress intensity factors (SIFs) during over-rolling of the contact over the experimentally observed crack geometries. The model employs a suitable fracture mechanics mesh to resolve stresses at the crack tip and accounts for Hertzian contact stresses, contact friction and crack face friction. Potential effects of lubricant pressurisation within the crack are not modelled. The predicted SIFs are then related to the experimentally measured crack propagation rates to establish the applicability of the LEFM principles to RCF crack propagation. Results show that LEFM can be used to predict the growth of surface braking RCF cracks. For cracks longer than about 100 μm, a Paris law relationship with the stress intensity exponent of about 4 is derived. Mode II was seen to be the dominant mode of propagation for surface braking RCF cracks. Mode I SIFs are much smaller but can exhibit significant values when the contact is located just ahead of the crack mouth. Decreasing the crack face friction significantly increases mode II stress intensity suggesting that this is one important mechanism by which lubricant entry into the surface braking crack can accelerate its propagation. The findings can help in improving the reliability of mechanical systems by supporting the development of new tools for prediction of remaining useful life of machine components such as bearings and gears.

Geometric design, modelling and finite element analysis of spur gear pairs having normal teeth based on the modification of the pressure angle

Abstract The aim of the thesis is the geometric design, CAD modelling and TCA of spur gear pairs having normal teeth based on the modification of the pressure angle. The first task is the geometric design and CAD modelling of the gear pairs (5 pairs are designed) where only the pressure angle is modified beside the constancy of the other initial parameters. The second task is to analyse the comparison possibility and the accuracy similarity of the 2D and the 3D models by the Hertz (equivalent) stress analysis. Finally, I give analysis of the maximum equivalent stress, normal stress and contact pressure for each pair while three teeth are rolling down on each other.

Failure analysis of overdrive gear of passenger car gearbox fabricated from powder metallurgy

In this paper, investigations into the premature failure of gearbox overdrive gears were done in three passenger car models. The premature failure occurred when a replaced gear model was used instead of the original gear. A series of experiments were carried out on gears made by two manufacturers, including failed gear after an average operation of 100,000 km and gear with partial failure but usable with an average operation of 200,000 km. Numerical and experimental results showed that sharp hardness slope (reduction of 300 HV in 0.15 mm), high porosity, and Hertz stress distribution of about 90% of the allowable contact stress σHa are the factors affecting premature failure. In addition, replacement gear samples with low density compared to original gear have low yield and ultimate strength due to high porosity and interconnected structure of pores. Pores with sharp corners act as the initiation points of the cracks, resulting in final fracture.

The effects of hardness on the spalling property of railway wheel steel

Spalling is a typical rolling contact fatigue (RCF) failure of locomotive wheels. The effects of hardness on the spalling properties of three wheel steels with different hardness levels were investigated. The RCF tests were conducted using two cylindrical contact specimens under water lubrication at a slip ratio of 1%. As a result, the fatigue strength and wear property increased with increasing hardness. The outer surface and cross section of the failure specimens were observed using optical and scanning electron microscopes. Cracks and obvious plastic deformation regions were observed under the contact surface. Both the depth of the main crack and crack branching increase rapidly with Hertzian stress. A model based on the Hirakawa failure model was proposed to describe the effects of hardness on the spalling behavior of wheel steels. With increasing hardness, the critical condition of the spalling increases, which agrees with the experimental results.

Contact pressure analysis of pin-loaded lug with clearance

In engineering design, none of the existing models for calculating the contact pressure in pin-loaded lug is fully compatible with the actual working conditions. In order to obtain a more accurate formula for calculating the maximum contact pressure in pin-loaded lug, after verifying the reliability of the finite element model with PRESCALE pressure measurement film, single-parameter and multi-parameter analysis of two types of factors affecting the contact pressure distribution in the compressive surface of the pin-loaded lug were conducted in the fully elastic state. The influence of the relevant design parameters of the pin-loaded lug on its maximum contact pressure was obtained, and a formula for calculating the maximum contact pressure of the pin-loaded lug based on the Hertz stress distribution model was introduced. It is verified by a large number of finite element samples that the predicted values of the formula are in good agreement with the finite element solutions in the range of common scales for engineering design, and have high accuracy, which can be used for engineering design reference.

Simulation of the Fatigue Crack Initiation in SAE 52100 Martensitic Hardened Bearing Steel during Rolling Contact

An investigation on the White Etching Crack (WEC) phenomenon as a severe damage mode in bearing applications led to the observation that in a latent pre-damage state period, visible alterations appear on the surface of the raceway. A detailed inspection of the microstructure underneath the alterations reveals the existence of plenty of nano-sized pores in a depth range of 80 µm to 200 µm. The depth of the maximum Hertzian stress is calculated to be at 127 µm subsurface. The present study investigates the effect of these nanopores on the fatigue crack initiation in SAE 52100 martensitic hardened bearing steel. In this sense, two micro-models by means of the Finite Element Method (FEM) are developed for both a sample with and a sample without pores. The number of cycles required for the crack initiation for both samples is calculated, using the physical-based Tanaka–Mura model. It is shown that pores reduce the number of cycles in bearing application to come to an earlier transition from microstructural short cracks (MSC) to long crack (LC) propagation significantly.

Influence of operating speed on the thermo-mechanical behaviour of cylindrical roller bearing

The high load-bearing ability of cylindrical roller bearing made its usage in rotating machinery popular. However, the performance of this bearing is impeded by the contact stress, stiffness and the temperature rise experienced operation. In this study, the thermomechanical behaviour of martensitic stainless steel (X20) fabricated roller bearing with polyamide pin type cage subjected to different rotating/operating speeds was simulated using finite element analysis software, Abaqus. The outcome of the analyses shows that the temperature, heat flux, contact or Hertzian stress and frictional energy developed in the bearing during operation increases with a corresponding increase in speed. It was further observed that a slight increase in the operating speed of the bearing leads to a significant rise in the temperature and frictional energy developed in the bearing. Also, the maximum Hertzian/contact stress was observed to developed on the outer ring of the roller bearing assembly (at the point of contact between the outer ring and the balls) in all the operational speeds considered. Thus, making this outer ring more susceptible to failure during operation as compared to the other components of the bearing.

Numerical simulation of the effect of angular misalignment on the dynamic behaviour of bearing

Bearing has been used extensively in numerous applications and their unplanned failure has a consequential effect on the smooth operation of the machinery. A slight misalignment in bearing has a detrimental effect on the smooth running of most machines. Hence, the paper leverages finite element technique to simulate the consequential effect of different degrees (0.1°, 0.2°, 0.3°, 0.4°, 0.5°) of misalignment on the dynamic behaviour of a cylindrical roller bearing subjected to typical operating conditions of an airflow root blower. The results of the study show that during operation, the temperature and Hertzian stress developed increased with an increase in the degree of misalignment and operating/rotating speed, and the maximum Hertzian stress was developed on the outer ring of the bearing in all the degrees of misalignments and operational speeds considered. Thus, making the outer ring of the bearing component, the most prone to failure during operation in the presence of misalignment.