Hertz Equation Research Articles

Towards Simpler Modelling Expressions for the Mechanical Characterization of Soft Materials

Aims: The aim of this paper is to develop a new, simple equation for deep spherical indentations. Background: The Hertzian theory is the most widely applied mathematical tool when testing soft materials because it provides an elementary equation that can be used to fit force-indentation data and determine the mechanical properties of the sample (i.e., its Young’s modulus). However, the Hertz equation is only valid for parabolic or spherical indenters at low indentation depths. For large indentation depths, Sneddon’s extension of the Hertzian theory offers accurate force-indentation equations, while alternative approaches have also been developed. Despite ongoing mathematical efforts to derive new accurate equations for deep spherical indentations, the Hertz equation is still commonly used in most cases due to its simplicity in data processing. Objective: The main objective of this paper is to simplify the data processing for deep spherical indentations, primarily by providing an accurate equation that can be easily fitted to force-indentation data, similar to the Hertzian equation Methods: A simple power-law equation is derived by considering the equal work done by the indenter using the actual equation. Results: The mentioned power-law equation was tested on simulated force-indentation data created using both spherical and sphero-conical indenters. Furthermore, it was applied to experimental force-indentation data obtained from agarose gels, demonstrating remarkable accuracy. Conclusion: A new elementary power-law equation for accurately determining Young’s modulus in deep spherical indentation has been derived.

Further Validation of a Simple Mathematical Description of Wear and Contact Pressure Evolution in Sliding Contacts

The present study proposes the further validation of a simple mathematical procedure recently proposed by the authors to describe contact and wear evolution in line and point contacts. The procedure assumed that the maximum contact pressure could be determined using Hertz equations and a parabolic pressure profile. The contact half-width was obtained using the equilibrium equation and the Archard wear law. Several cases were selected from the literature, reporting experimental data or Finite Element simulations, and the results were compared to those obtained with the proposed approach. This paper confirms the reliability and potentialities of the proposed analytical procedure, which is capable of providing accurate solutions in case of frictional contacts and at the borders of the contact area, where the main discrepancies were found in the previous study.

AFM Indentation on Highly Heterogeneous Materials Using Different Indenter Geometries

Hertzian mechanics is the most frequently used theory for data processing in Atomic Force Microscopy (AFM) indentation experiments on soft biological samples, due to its simplicity and significant scientific results previously published. For instance, using the Hertz model, it has been proven that there are significant differences in the mechanical properties of normal and cancerous tissues and that cancer cells’ invasive properties are correlated with their nanomechanical properties. However, many scientists are skeptical regarding the applicability of the Hertz theory to biological materials, as they are highly heterogeneous. The main critical question to be addressed is “what do we calculate” when fitting the force-indentation data to Hertz equations. Previous studies have shown that when using cylindrical, parabolic, or conical indenters, the fitting parameter is the average Young’s modulus. In this paper, it is demonstrated that it is also valid to fit equations derived from Hertzian mechanics to force-indentation data when testing soft, heterogeneous samples for any indenter geometry. The fitting factor calculated through this approach always represents the average Young’s modulus for a specific indentation depth. Therefore, Hertzian mechanics can be extended to soft heterogeneous materials, regardless of the indenter’s shape.

Effects of Pressure Soaking on Elastic Modulus of Steamed-Cooked Glutinous Rice Kernel

The goal of this research is to assess the moduli of elasticity of steamed-cooked Thai Sanpatong 1 glutinous rice (GR) under various soaking conditions, primarily normal soaking and pressure soaking. The soaking process can influence properties of cooked rice. One of the main objectives is to determine the stiffness of cooked rice after soaking with either of the 2 methods, ASAE 368.4 was employed as main protocol for property testing and then Hooke’s and Hertz’s equations were used to estimate the moduli of elasticity. Poisson’s equation was used to estimate the Poisson’s ratios. The results show that the moduli of elasticity of the cooked normal-soaked and pressure-soaked rice determined using Hooke’s theory were 3.77 and 5.67 MPa, respectively. The moduli of elasticity of the cooked normal-soaked and pressure-soaked rice determined using Hertz’s theory were 1.58 and 2.33 MPa, respectively. Poisson’s ratio of the cooked normal-soaked and pressure-soaked rice were 0.27 and 0.28, respectively. Results revealed that the pressure treated rice has higher stiffness.
 HIGHLIGHTS
 
 Calculating modulus of elasticity of steamed-cooked Thai Sanpatong 1 Glutinous rice kernels obtained from Hertz’s theory yields get a higher modulus of elasticity than Hooke’s Theory
 Hertz's equation is suitable method to estimation of minimum and maximum values for radius of curvature of small grains
 Both of the modulus of elasticity calculated using these 2 theories theories reveal that the rice grains soaked under gauge pressure of 1 bar is stiffer than that ambient soaked
 
 GRAPHICAL ABSTRACT

Contact Stresses and Dynamic Analysis of involute Spur Gear with FEM Approach

Abstract: This paper investigates the characteristics of a gear system including contact stresses, bending stresses, and the transmission errors of gears in mesh. The objective of this paper is to compare values of contact stress and dynamic analysis obtained by theoretical hertz equation with the ANSYS result. A two stage spur gear box has been designed for material handling application by manual calculation and then performs contact stress and dynamic Simulation to ensure its reliable working. The results of the two-dimensional Finite Element Method (FEM) analysis from ANSYS and theoretical results are well comparable. Keywords: involute, spur gear, contact stress, dynamic analysis, finite element analysis

Approximation Solution for the Zener Impact Theory

Collisions can be classified as completely elastic or inelastic. Collision mechanics theory has gradually developed from elastic to inelastic collision theories. Based on the Hertz elastic collision contact theory and Zener inelastic collision theory model, we derive and explain the Hertz and Zener collision theory model equations in detail in this study and establish the Zener inelastic collision theory, which is a simple and fast calculation of the approximate solution to the nonlinear differential equations of motion. We propose an approximate formula to obtain the Zener nonlinear differential equation of motion in a simple manner. The approximate solution determines the relevant values of the collision force, material displacement, velocity, and contact time.

An investigation of the elastic cylindrical line contact equations for plane strain and stress considering friction

This work presents a finite-element model-based study of elastic cylindrical contact. The aim is to evaluate the transition between the plane stress and plane strain-based Hertz solutions when each assumption is most applicable. To accomplish this, a range of curvatures, widths, Poisson’s ratios, and friction coefficients are considered. The finite-element model results for the elastic cylindrical contact cases are compared with the Hertz contact model when assuming plane stress or plane strain. Perhaps, surprisingly, the finite-element model predictions show little dependence on Poisson’s ratio and friction coefficient. The finite-element model predictions of force as a function of deflection agree relatively well with the plane stress Hertz prediction for all cases considered. The finite-element model predictions of contact width as a function of force actually fall below all the analytical Hertz predictions. Therefore, an adapted version of the Hertz equations is provided, which shows better agreement with the cases considered in this work.

Impact Dynamics of Boulder-Enriched Debris Flow on a Rigid Barrier

AbstractBoulders entrained in debris flow induce high impact forces on a rigid barrier. In current design practice, the concentrated load from boulders is estimated using the Hertz equation with a ...

Evaluation of the Elastic Compliance of the Hardness Tester in Instrumented Indentation Tests

The elastic compliance of the instrument (hardness tester) should be taken into account when determining the mechanical properties of materials by means of instrumented indentation with registration of indentation diagrams. The values of Young’s modulus of the tested materials significantly depend on the method for evaluating and accounting for elastic compliance. Therefore, the methods should be verified during instrumented indentation of materials with known but very different Young’s moduli. Successful experience in evaluating and accounting for elastic compliance of the instrument during the instrumented indentation of materials with a diamond pyramid has already been accumulated to date. This is reflected in the relevant standards. However, this experience cannot be applied to instrumented indentation with a steel ball or hard-metal ball without additional research and experimental verification. We propose a method for evaluating the elastic compliance of the hardness tester using the instrumented diagram of ball indentation based on the Hertz equation in the case of elastic contact of the ball with the plane. It is found that additional elastic deformations of the links of instrument are directly proportional to the indentation load; this dependence is typical of each instrument and the diameter of the ball does not affect it. This dependence allows one to take into account the elastic compliance of the instrument using software when registering and processing diagrams of ball indentation. Experiments to determine Young’s modulus and hardness by instrumented ball indentation of various materials (steel, aluminum alloy, magnesium alloy, and titanium alloy) using the known and proposed methods to account for the elastic compliance of the instrument are performed. The main criterion confirming the reliability of the method is the coincidence or proximity of the values ​​of Young’s modulus of the same material determined from the diagram of ball indentation and the diagram of tension of the sample. The advantages and disadvantages of the known and proposed methods are described. Practical recommendations for use of these methods are given.

A numerical and analytical study of the stress field generated by the contact between a rail and a wheel

This article presents a study of the contact stresses between a rail and a wheel, in order to understand the mechanism related to this application. When two bodies, regardless of their forms, touch each other due to an external load, both suffer a slight deformation on the contact surface, creating a stress field in this area, and the complete comprehension of this phenomenon is fundamental to calculate critical stresses and to prevent possible failures. This article approaches this topic in two different manners: one analytical, using Hertz equations; and another numerical, using the FEM software Ansys. The results obtained from both approaches are compared, and the model can be validated. The results show that when these two cylindrical bodies are pressed against each other, a small rectangular area is created, and the highest normal stresses are found exactly where the two elements touch. It is seen that the maximum shear stress is found along the depth of the body and is the most critical for the appearance of cracks. The results from this work are in good alignment with reality and they can be used to predict possible failures and behaviors in real applications.

Contact Stress Analysis of Spur Gear for different Materials using Ansys and Hertz Equation

Contact Stress Analysis of Spur Gear for different Materials using Ansys and Hertz Equation

EVALUATION OF THE ELASTIC COMPLIANCE OF THE HARDNESS TESTER IN KINETIC INDENTATION TESTS

When determining the mechanical properties of materials in kinetic indentation tests using indentation diagrams, careful consideration of the elastic compliance of the device, i.e., the hardness tester, is required. The determined values of the Young's modulus of the tested material substantially depend on the reliability of the method of evaluation and accounting for the elastic compliance. Therefore, verification of the test techniques based on kinetic indentation should be carried out using the materials with the known, but rather different values of the Young's modulus. Successful experience has been gained to date in the evaluating and accounting for the elastic compliance of the device upon kinetic indentation of the materials by a diamond pyramid which is reflected in the relevant standards. However, there is no way of transferring this experience to the kinetic indentation by a steel or carbide ball without additional research and experimental verification. We proposes a technique for estimating the elastic compliance of a hardness tester using a kinetic ball indentation diagram based on the G. Hertz equation for the case of elastic contact of a ball with a plane. A linear correlation has been determined between the additional elastic deformations of the device and indentation load, which is characteristic of each device and independent on the ball diameter. The obtained dependence allows for correct consideration of the elastic compliance of the device using software applications in recording and processing the ball indentation diagrams. Experiments have been carried out to determine the hardness and the Young's modulus through ball instrumented indentation of different materials (steel, aluminum alloy, magnesium alloy, and titanium alloy) using the existing and developed methods of taking into account the elastic compliance of the device. The coincidence or proximity of the values of the Young's modulus of the same material determined from the ball indentation diagrams and sample tensile tests is considered the main criterion proving the accuracy of the technique. The advantages and shortcomings of the known and proposed procedures are discussed along with practical recommendations for their applications.

A Strength Analysis of Gear Train for Hydro-Mechanical Continuously Variable Transmission

The power train of hydro-mechanical continuously variable transmission(HMCVT) for the middle class forklift makes use of an hydro-static unit, hydraulic multi-wet disc brake & clutches and complex helical & planetary gears. The complex helical & planetary gears are a very important part of the transmission because of strength problems. The helical & planetary gears belong to the very important part of the HMCVT`s power train where strength problems are the main concerns including the gear bending stress, the gear compressive stress and scoring failures. The present study, calculates specifications of the complex helical & planetary gear train and analyzes the gear bending and compressive stresses of the gears. It is necessary to analyze gear bending and compressive stresses confidently for an optimal design of the complex helical & planetary gears in respect of cost and reliability. This paper not only analyzes actual gear bending and compressive stresses of complex helical & planetary gears using Lewes & Hertz equation, but also verifies the calculated specifications of the complex helical & planetary gears by evaluating the results with the data of allowable bending and compressive stress from the Stress - No. of cycles curves of gears. In addition, this paper explains actual gear scoring and evaluates the possibility of scoring failure of complex helical & planetary gear train of hydro-mechanical continuously variable transmission for the forklift.

The Strength Analysis of Differential Planetary Gears of Gearbox for Concrete Mixer Truck

The power train of mixer gearbox for concrete mixer truck includes differential planetary gears to get large reduction ratio for operating mixer a drum and simple structure. The planetary gears are very important part of a mixer gearbox where strength problems namely gear bending stress, gear compressive stress and scoring failure are the main concern. In the present study, calculating specifications of the differential planetary gears and analyzing the gear bending and compressive stresses as well as scoring factor of the differential planetary gears gearbox for an optimal design of the mixer gearbox in respect to cost and reliability are investigated. The analyses of actual gear bending and compressive stresses of the differential planetary gears using Lewes & Hertz equation and verifications of the calculated specifications of the differential planetary gears evaluate the results with the data of allowable bending and compressive stress from the Stress-No. of cycles curves of gears. In addition, we also analyze actual gear scoring factor as well as evaluate the possibility of scoring failure of the differential planetary gear.

Extended Hertz Theory of Contact Mechanics for Case-Hardened Steels With Implications for Bearing Fatigue Life

The analytical expressions currently available for Hertzian contact stresses are applicable only for homogeneous materials and not for case-hardened bearing steels, which have inhomogeneous microstructure and graded elastic properties in the subsurface region. Therefore, this article attempts to determine subsurface stress fields in ball bearings for graded materials with different ball and raceway geometries in contact. Finite element models were developed to simulate ball-on-raceway elliptical contact and ball-on-plate axisymmetric contact, to study the effects of elastic modulus variation with depth due to case hardening. Ball bearings with low, moderate, and heavy load conditions are considered. The peak contact pressure for case-hardened steel is always more than that of through-hardened steel under identical geometry and loading conditions. Using equivalent contact pressure approach, effective elastic modulus is determined for case-carburized steels, which will enable the use of Hertz equations for different gradations in elastic modulus of raceway material. Nonlinear regression tools are used to predict effective elastic modulus as a weighted sum of surface and core elastic moduli of raceway material and design parameters of ball–raceway contact area. Mesh convergence study and validation of equivalent contact pressure approach are also provided. Implications of subsurface stress variation due to case hardening on bearing fatigue life are discussed.

Tailoring the physico-chemical properties and elasticity of poly(hydroxy-functional methacrylate)-based cationically charged gel beads: Combined hydrophobicity and mechanical durability through frozen droplets

This study describes the experimental methodology to obtain hydroxy-functional methacrylate-based gel beads in millimeter-sized under production conditions by addition of the frozen pre-gel solution into the silicon oil as continuous phase using tetraethyleneglycol dimethacrylate (TEGDMA) as the crosslinking agent. Weakly basic cationic comonomer N,N-dimethylaminoethyl methacrylate (DMAEMA) was included into the polymerization procedure for the production of uniform polycationic moities exhibiting a drastic response against pH. The aim of this work is to determine the actual conditions for optimum bead production and to provide poly(hydroxypropyl methacrylate-co-N,N-dimethylaminoethyl methacrylate)-based P(HPMA-co-DMAEMA) gel beads with simple tools for the physical characterization. By the proposed experimental method, the physico-chemical properties of P(HPMA-co-DMAEMA) gel beads can be controlled by adjusting the pH, temperature of solution, ionic strength as well as the structure of the gel network. From the uniaxial compression tests, the swelling dependent stress–strain characteristics of resulting gel beads was evaluated. The obtained results were consistent with Hertzian elasticity, and were analyzed in terms of the compression modulus of the bead samples. It was found that force acting on the gel beads is the power-law function of the deformation F≈ΔD1.48 which is very close to the theoretical exponent 1.5 predicted by the Hertz equation. The water absorption capacities of P(HPMA-co-DMAEMA) gel beads with basic functional groups were repeatedly measured to obtain the optimal preparation conditions according to the variations of the bead diameters. Its maximal water absorption capacity appeared as 48 times among from 2.3mm to 11.0mm of swollen bead diameters. The swelling equilibria in various aqueous solutions were used to estimate the physico-chemical parameters asa function of gel bead size and swelling equilibrium capacity by correlating with the Flory-Rehner theory.

Predicting cashew nut cracking using hertz theory of contact stress

Abstract The cracking force is an important parameter in the design of cracking machines. A mathematical model for predicting the exact cracking force of cashew nut was presented based on the existing hertz’s theory of contact stress of two bodies under uniaxial compressive load. Hertz equation was adapted to loading of cashew nuts to cracking at three different loading orientations (X, Y, and Z) and two strain rates of 5 mm/min and 10 mm/min. The average moisture content of the dried cashew nuts was determined at 5% (db.). The physical properties of cashew nut varieties such as Length, Width, Thickness, mass, Geometric Mean diameter, Sphericity index, Surface area and Aspect ratio were 34.31 mm, 24.86 mm, 16.55 mm, 6.13 g, 24.13 mm, 0.70, 1834.65 mm 2 and 0.73 respectively. Some mechanical parameters such as cracking force, deformation and stiffness modulus at loading orientations (X, Y, and Z) (355,288 and 289 N), (1.396, 1.5002 and 1.815 mm) and (313, 195 and 160 N/mm) at 5 mm/min, and (458, 276 and 480 N), (2.28, 1.767 and 3.7197 mm) and (202, 162 and 146 N/mm) at 10 mm/min were also obtained from the force-deformation curve respectively. These physical and mechanical parameters were required in the derived model. The aspect ratio was introduced as a correcting factor to the stiffness modulus to improve the prediction of the hertz model. The highest value of mean cracking force (355 N) was determined at the X (Width) orientation at 5 mm/min while the highest value of mean cracking force (480 N) at 10 mm/min strain rate was found at the Z (Thickness) orientation. A two-sample t -test comparism of experimental cracking force values and that of the hertz model were acceptable at 5% probability level which gave good prediction of cracking force during compression. The force required for the design of suitable cashew nut shell cracking machine should be considered at the range of 400–500 N at X (width) and Z (thickness) loading orientations, therefore the magnitude of applied force with the loading orientation could determine the extent of mechanical damage and kernel recovery during cracking and separation processes.

Extending Hertz equation for an elastic contact between a layered cylindrical hollow roller and flat plate through an experimental technique

PurposeThe purpose of this paper is to extend the Hertz equation for the contact interaction between a layered cylindrical hollow roller and a flat plate through an experimental technique.Design/methodology/approachIn this work, an experimental investigation is carried out for an elastic contact between layered cylindrical hollow rollers of different hollowness, ranging from 30 to 75 per cent hollowness and a flat plate. The footprint method was used for the evaluation of the contact width corresponding to the applied load.FindingsThe contact width for the layered cylindrical hollow roller was evaluated and the Hertz equation was extended on the basis of the experimental results.Originality/valueThe value of this research work is the development of an extended Hertz equation for a cylinder-on-plate configuration for a new kind of cylindrical roller.

Large-scale successive boulder impacts on a rigid barrier shielded by gabions

Debris flows occur in multiple surges. Boulders entrained within the flow have been reported to incapacitate structures within its flow path. Single-layer cushions, such as gabions, are often installed to shield debris-resisting barriers from boulder impact. However, most relevant works only focus on single impact and the performance of gabions subjected to successive loading is still not well understood. A new large-scale pendulum facility was established to induce impact energy of up to 70 kJ on an instrumented rigid barrier shielded by 1 m thick gabions. The response of the gabions under six successive impacts was investigated. Results show that the peak boulder impact force given by the Hertz equation is at least four times the measured values. The recommended load-reduction factor (Kc) used in practice can be reduced by a factor of two. After six successive impacts at an energy level of 70 kJ, the transmitted force increases by up to 40%. Based on the Swiss guidelines, a 13% increase of gabion thickness is required when successive impacts are concerned. The results presented in this paper will be useful for practitioners designing rigid barriers.

Analytical and Numerical Tooth Contact Analysis (TCA) of Standard and Modified Involute Profile Spur Gear

Among all the common mechanical transmission elements, gears still playing the most dominant role especially in the heavy duty works offering extraordinary performance under extreme conditions and that the cause behind the extensive researches concentrating on the enhancement of its durability to do its job as well as possible. Contact stress distribution within the teeth domain is considered as one of the most effective parameters characterizing gear life, performance, efficiency, and application so that it has been well sought for formal gear profiles and paid a lot of attention for moderate tooth shapes. The aim of this work is to investigate the effect of pressure angle, speed ratio, and correction factor on the maximum contact and bending stress value and principal stresses distribution for symmetric and asymmetric spur gear. The analytical investigation adopted Hertz equations to find the contact stress value, distribution, and the contact zone width while the numerical part depends on Ansys software version 15, as a FE solver with Lagrange and penalty contact algorithm. The most fruitful points to be noticed are that the increasing of pressure angle and speed ratio trends to minimize all the induced stresses for the classical gears and the altered teeth shape with larger loaded side pressure angle than the unloaded side one behave better than the symmetric teeth concerning the stress reduction.