Hertzian Contact Problem Research Articles

Axisymmetric Hertzian contact problem accounting for surface tension and strain gradient elasticity

In this paper, we investigate the axisymmetric Hertzian contact problem at micro-/nanoscale. The deformation of material bulk is described by a simplified theory of strain gradient elasticity, and the influence of surface tension is integrated based on the surface elasticity theory. Using the Mindlin's potential function method and double Fourier integral transform, the normal surface displacement induced by a concentrated force is derived in a closed form. Following this, the contact between a rigid sphere and an elastic half-space is formulated in terms of singular integral equation, which is numerically solved by applying the Gauss-Chebyshev method. The results indicate that the distribution of contact pressure is distinctly different from that in classical elasticity theory. The indented substrate tends to perform stiffer due to the effects of surface tension and strain gradient elasticity. When the contact radius is comparable with the material length parameter, the indentation force (depth) can be ten (three) times of that given by classical Hertz theory.

A comprehensive isogeometric analysis of frictional Hertz contact problem

The development of computational methods has advanced the solving of complex contact problems. Despite this, reproducing the analytical solution of a classical Hertz contact problem using computational methods remains challenging. Finite element method (FEM) has gained popularity, with many studies solving frictionless Hertz contact problems. However, the more general Hertz contact problem with finite friction lacks an analytical solution and is more complex. FEM was first used for this in the 1970s. Recently, isogeometric analysis (IGA) has shown enhanced efficiency than FEM. However, comprehensive studies on the frictional Hertz contact problem using IGA are lacking. The present study aims to fill this gap by examining a benchmark example and exploring contact simulation parameters using the developed in-house MATLAB® code.

Solving forward and inverse problems of contact mechanics using physics-informed neural networks

This paper explores the ability of physics-informed neural networks (PINNs) to solve forward and inverse problems of contact mechanics for small deformation elasticity. We deploy PINNs in a mixed-variable formulation enhanced by output transformation to enforce Dirichlet and Neumann boundary conditions as hard constraints. Inequality constraints of contact problems, namely Karush–Kuhn–Tucker (KKT) type conditions, are enforced as soft constraints by incorporating them into the loss function during network training. To formulate the loss function contribution of KKT constraints, existing approaches applied to elastoplasticity problems are investigated and we explore a nonlinear complementarity problem (NCP) function, namely Fischer–Burmeister, which possesses advantageous characteristics in terms of optimization. Based on the Hertzian contact problem, we show that PINNs can serve as pure partial differential equation (PDE) solver, as data-enhanced forward model, as inverse solver for parameter identification, and as fast-to-evaluate surrogate model. Furthermore, we demonstrate the importance of choosing proper hyperparameters, e.g. loss weights, and a combination of Adam and L-BFGS-B optimizers aiming for better results in terms of accuracy and training time.

Semi-Analytical Results for the Linear Nonaging Viscoelastic Contact Behaviors

This paper investigates the importance of creep and relaxation functions in modeling the Hertzian contact problem between linear nonaging viscoelastic materials and viscoelastic rough surfaces. We provide novel analytical and semi-analytical solutions to these functions for various materials, including Generalized Maxwell, Generalized Kelvin, and Burgers’ rheological materials. Closed-form and semi-analytical solutions are developed for several specific cases with high practical applications, such as the contact between incompressible solids, a rigid material and an incompressible material, and two solids with no volumetric viscous strain. They are extensions of existing solutions given in the literature. A simple computation procedure based on the discretized integral technique is proposed for the general case of contact between linear nonaging viscoelastic materials. This proposed method is easy to implement in practical applications such as the interpretation of viscoelastic indentation tests for material characterization and the implementation of viscoelastic contact in finite/discrete element simulations.

Stable model reduction for linear variational inequalities with parameter-dependent constraints

We consider model reduction for linear variational inequalities with parameter-dependent constraints. We study the stability of the reduced problem in the context of a dualized formulation of the constraints using Lagrange multipliers. Our main result is an algorithm that guarantees inf-sup stability of the reduced problem. The algorithm is computationally effective since it can be performed in the offline phase even for parameter-dependent constraints. Moreover, we also propose a modification of the Cone Projected Greedy algorithm so as to avoid ill-conditioning issues when manipulating the reduced dual basis. Our results are illustrated numerically on the frictionless Hertz contact problem between two half-disks with parameter-dependent radius and on the membrane obstacle problem with parameter-dependent obstacle geometry.

An investigation of Hertzian contact in soft materials using photoelastic tomography

Hertzian contact of a rigid sphere and a highly deformable soft solid is investigated using integrated photoelasticity. The experiments are performed by pressing a styrene sphere of 15 mm diameter against a 44 × 44 × 47 mm3 cuboid made of 5% wt. gelatin, inside a circular polariscope, and with a range of forces. The emerging light rays are processed by considering that the retardation of each ray carries the cumulative effect of traversing the contact-induced axisymmetric stress field. Then, assuming Hertz's theory is valid, the retardation is analytically calculated for each ray and compared to the experimental one. Furthermore, a finite element model of the process introduces the effect of finite displacements and strains. Beyond the qualitative comparison of the retardation fields, the experimental, theoretical, and numerical results are quantitatively compared in terms of the maximum equivalent stress, surface displacement, and contact radius dimensions. A favorable agreement is found at lower force levels, where the assumptions of Hertz theory hold, whereas deviations are observed at higher force levels. A major discovery of this work is that, at the maximum equivalent stress location, all three components of principal stress can be determined experimentally and show satisfactory agreement with theoretical and numerical ones in our measurement range. This provides valuable insight into Hertzian contact problems since the maximum equivalent stress controls the initiation of plastic deformation or failure. The measured displacement and contact radii also reasonably agree with the theoretical and numerical ones. Finally, the limitations that arise due to the linearization of this problem are explored.

Integrated photoelasticity in a soft material: phase retardation, azimuthal angle, and stress-optic coefficient

Integrated photoelasticity is investigated for a soft material subjected to a three-dimensional stress state with large deformation. Our measurement target is designed based on the axisymmetric stress field (Hertzian contact problem). In the experiment, a solid sphere is pressed against a gelatin gel (Young’s modulus is about 4.2 kPa) with varying applied forces from zero to the maximum force that deforms the gel up to approximately 4.5 mm. For soft materials, two-dimensional photoelasticity, as conventionally practiced, is difficult to be used. This is because two-dimensional slices of highly deformable materials, such as the gelatin used in this study, do not allow stable experiments. This requires a three-dimensional body with sufficient thickness for stability, where integrated photoelasticity is demanded. The stressed gel’s photoelastic parameters (phase retardation and azimuthal angle) are measured using a polarization camera. The measured phase retardation and azimuthal angle are compared with the analytical prediction based on Hertzian contact problem. Remarkably, experimental and analytical results of the photoelastic parameters show a reasonable agreement not only in the phase retardation but also in the azimuthal angle that is related to the direction of secondary principal stresses. Never before validated in previous studies, which is crucial for reconstructing three-dimensional stress fields in soft materials. In addition, the stress-optic coefficient of the gelatin gel used is measured to be 3.12×10−8 1/Pa. Such findings proved that integrated photoelasticity benefits measure the three-dimensional stress field in soft materials, which is essential in biomedical engineering and cell printing applications.

Numerical calculation and finite element analysis for anisotropic elastic properties of carbon fibers: dependence of integration subinterval and mesh size on indentation-derived elastic modulus

The elastic modulus measured by indentation of carbon fibers with various anisotropic elasticity is calculated by two numerical approaches, the Vlassak–Nix model and finite element analysis, to reveal the acceptable calculation condition for highly anisotropic materials. Five commercially available carbon fibers that varied in anisotropy index in the range of 0.5–7.9 are used (either polyacrylonitrile- or pitch-based). The numerical error in the calculated modulus increases with the decrease in fiber angle and with the increase in the anisotropy index under the same mesh condition, indicating finer mesh is required for a highly anisotropic material. The acceptable mesh size linearly increases with anisotropic index. The Vlassak–Nix model overestimates the elastic modulus at a small tilt angle if few integration subintervals are used. Conversely, finite element analysis of the Hertz contact problem with coarse mesh underestimates the modulus at a small tilt angle, and a maximum modulus is observed when the fiber is tilted a few degrees against the indentation axis. These findings are expected to assist the future determination of ideal calculation conditions for materials with large anisotropic elasticity including fibers and composites.

Influence of Wear Profile Geometry on Critical Plane Fatigue Crack Initiation Criteria in Plane and Axisymmetric Elastic Fretting Contacts

The subsurface elastic stress fields in plane and axisymmetric contacts with friction under oscillating tangential loading are calculated via a very robust, high-precision method, which operates with appropriate superpositions of analytic solutions for the respective Hertzian contact problems. Based on the stress fields, two critical plane fatigue crack initiation criteria—the Smith-Watson-Topper (SWT) parameter and the Findley parameter—are evaluated for three types of contact profile geometries: (unworn) parabolic contact, the partial slip limiting wear profile of an initially parabolic contact, and truncated parabolic contact. Appropriate scaling laws are introduced to formulate a general solution in terms of non-dimensional variables. The crack initiation criteria are determined in the full subsurface loading plane. It is found that the truncated profile—which may originate from sliding wear—has practically the same local distribution of crack initiation criteria as the unworn profile, despite the (weak) stress singularity at the edge of the flat face. The partial slip limiting wear profile, on the other hand, exhibits a strong edge stress singularity at the boundary of the permanent stick zone, the crack initiation criteria are drastically increased (and theoretically infinite). Also, while for the unworn and truncated profiles high values of the crack initiation criteria are extremely localized around “hotspots” at the surface, for the partial slip limiting wear profile they reach much deeper into the subsurface material. This offers a new explanation for the dominance of fatigue failure in the partial slip regime of fretting. The differences between plane and axisymmetric cases are generally small. The SWT parameter is generally more localized than the Findley parameter.

Stresses in multi-layer coatings in Hertzian contact with a moving circular punch

This article presents analytical solution procedures for moving Hertzian contact problems involving multi-layer and functionally graded coatings. A frictional circular punch is assumed to slide down the surface of the coating with a specified velocity. Formulation is based on wave equations of plane elasticity, and reduced to a singular integral equation by Galilean and Fourier transformations. Developed procedures are verified through comparisons to the findings available in the literature. Numerical results illustrate influences of factors such as dimensionless punch speed, coefficient of kinetic friction, property variation profile, and contact length-to-thickness ratio upon the contact stresses, and the required contact force.

Evaluating the performance of enriched finite elements for Hertzian contact problem

Solution of contact problems in the field of mechanics is often very challenging, as it is hard to predict the contact forces and stresses due to non-constant boundary conditions. It is observed that when standard finite elements of linear order are used to represent the contact boundary, it is difficult to capture the contact forces within the contact region accurately. To improve this situation, different techniques usually are adopted. One such technique is using highly refined mesh at the contact zone. But it increases the computational time. Thus, it is more efficient to adopt the enrichment strategies for the contact elements lying within the contact region. Such elements are also referred to as the enriched contact finite elements. To the best of the authors' knowledge, the effects of contact enrichment with various interpolation orders on the solution accuracy of contact problems have not been exploited completely and hence is the scope for the present work. The current contribution evaluates the performance of enriched finite elements in solving the Hertz contact problem. The enrichment strategy is based on the introduction of extra degrees of freedom for the contact elements in the slave surface. The contact surface is locally enriched to represent the surface more accurately than the bulk. Thus, the improved computational contact formulation helps to correctly approximate the contact forces while keeping the total degrees of freedom low. In this work, the penalty method enforces contact constraints, and material behaviour follows the Neo-Hookean hyperelastic material model. A benchmark problem on the Hertzian contact is considered to demonstrate the FE enrichment technique. The numerical results exhibit the better performance of the FE enrichment over the mesh refinement for the solution accuracy. This work also shows that the higher-order finite elements after enrichment outperform the linear order finite elements in the contact region.

Чисельне моделювання НДС конструкційних матеріалів фільтроелементів в умовах пошкоджень від кавітаційної корозії

Розроблено імітаційні моделі та методику для розрахунку силової взаємодії рухомих частинок забруднювача з нерухомою поверхнею фільтроелементу, що деформується. Застосовано чисельні алгоритми, в основані яких лежить розв’язок контактної задачі Герца. Представлено результати чисельних експериментів по визначенню еквівалентних за Мізесом напружень для фрагментів імітаційних моделей фільтроелементів виготовлених з матеріалів Д16АТ, АМг61гк, 12Х18Н10Т при гідросилових навантаженнях.
 Simulation models and methods for calculating the force interaction of moving contaminant particles with the fixed surface of the deformable filter element have been developed. Numerical algorithms based on the solution of the Hertz contact problem are used. The results of numerical experiments to determine the Mises equivalent stresses for fragments of simulation models of filter elements made of materials D16AT, AMg61gk, 12Х18Н10T at hydropower loads are presented.

The hybrid action tool for operations of cleaning of turbine units cavities

The paper deals with the hybrid action tool for cleaning of cavities and elements of turbine units whose operation is based on a combination of action of a water-ice flow with mechanical shock influence of the small concentrated masses mounted on elastic suspensions. Receiving energy from the flow, the masses, performing self-oscillating motion, come into contact with the treated surface having a layer of strong contamination, and create a multipoint shock-cyclic loading of the surface, which results in active development of initial defects of the contamination film, due to which the following action of the water-ice stream produces better and more productive cleaning. It is shown that the generated local stresses, determined on the basis of Hertz contact problems, reach 15-20 MPa, do not have a significant effect on the surface of the base, which is a thin curves shell, do not change the state of its surface in the plane of adhesion, but result in defects in surface film in the form of cracks and delaminations . In this case the film is eliminated by water-ice flow more dynamically. The use of water-ice jet, formed by the original tubeless device, enables the formation of a wide flow (with the top angle of p/12… p/6) and the work of its ice particles is spent on grinding the surface film. The jet stream cleans the surface and removes the products of destruction beyond the impact.The use of a hybrid tool has increased the productivity of treatment with the wetting angles of the jet flow, different from p/2, by more than 30%, while the consumption of cryogenic liquid (liquid nitrogen) is reduced by 20-25%.The process modeling is performed, the conditions of the destruction of the adhesive bond of the contaminant film with the surface are estimated, the conditions of its rational execution are determined.

Adaptive radial basis function–generated finite differences method for contact problems

SummaryThis paper proposes an original adaptive refinement framework using radial basis function–generated finite differences method. Node distributions are generated with a Poisson disc sampling–based algorithm from a given continuous density function, which is altered during the refinement process based on the error indicator. All elements of the proposed adaptive strategy rely only on meshless concepts, which leads to great flexibility and generality of the solution procedure. The proposed framework is tested on four gradually more complex contact problems. First, a disc under pressure is considered and the computed stress field is compared to the closed‐form solution of the problem to assess the behaviour of the algorithm and the influence of free parameters. Second, a Hertzian contact problem is studied to analyse the proposed algorithm with an ad hoc error indicator and to test both refinement and derefinement. A contact problem, typical for fretting fatigue, with no known closed‐form solution is considered and solved next. It is demonstrated that the proposed methodology produces results comparable with finite element method without the need for manual refinement or any human intervention. In the last case, generality of the proposed approach is demonstrated by solving a three‐dimensional Boussinesq's problem.

The Hertz contact problem in couple-stress elasticity

The classical three-dimensional Hertz contact problem is re-examined in the present work within the context of couple-stress elasticity. This theory introduces characteristic material length scales that emerge from the underlying microstructure and has proved to be very effective yet rather simple for modeling complex microstructured materials. An exact solution of the axisymmetric contact problem is obtained through Hankel integral transforms and singular integral equations. The goal is to examine to what extend this gradient theory captures the experimentally observed indentation response and the related size effects in materials with microstructure. Furthermore, the present work extends the classical Hertz contact solution in the framework of a generalized continuum theory providing thus a useful theoretical background for the interpretation of spherical indentation tests of microstructured materials. The results obtained in the present work can be used in order to model nano/micro indentation experiments of several materials, such as polymers, ceramics, composites, cellular materials, foams, masonry, and bone tissues, of which their macroscopic response is influenced by the microstructural characteristic lengths.

Fretting in orthodontics and its test methods

Purpose: A review of the literature regarding the occurrence of fretting wear in orthodontics and its test methods have been presented. Design/methodology/approach: The influence of micro-movements occurring in the oral cavity on the occurrence of fretting wear in a fixed orthodontic appliance is discussed. The fretting test methods were analysed, taking into account the shape of the samples and the amplitude of the movements, calculated according to the Hertz contact problem. Fretting-corrosion tests on the wear of materials are included. Findings: Fretting occurs between the bracket and the orthodontic archwire in a fixed appliance. The test of the amount of wear material mainly uses samples created for the needs of the device. The test of ready-made components of a fixed appliance usually relate to the value of the coefficient of friction. The use of coatings increases the coefficient with the simultaneous reduction of the amount of wear material. Fretting-corrosion occurring in the oral environment has a negative impact on the wear of materials. The value of the total area of abraded material after fretting is unknown. Research limitations/implications: Particles of fretting wear get through to the organism from the elements of a fixed orthodontic appliance and they may cause hypersensitivity of surrounding tissues. The amount of wear material and its total area should be tested. Originality/value: Presented are the methods of fretting wear testing in a fixed orthodontic appliance, taking into account the cognitive gap in the knowledge of the amount of abraded material from ready-made components and the value of their total area.

Relations between Parameters of Fracture Processes on Different Scale Levels

The processes of ultrasonically-assisted drilling (UAD) and the dynamic tests on split Hopkinson pressure bar (SHPB), fracture in which is implemented at various structural-scale levels, are considered. The simulation of UAD based on the Hertz contact problem and the structural−time criterion is presented. The problem of using the value of the fracture incubation time and its linear size obtained from the tests on SHPB in the simulation is considered. A principle of equal power is used for converting the strength parameters into another structural−scale level. The theoretical curve obtained in the simulation is compared with the results of experiments on conventional drilling (CD) and UAD.

Dynamic response analysis of the vertical elastic impact of the spherical rock on the metal plate

In the process of top coal caving mining, the metal sheet of the tail beam of the hydraulic support will vibrate when coal and rock particles start to impact the tail beam. In the impact between the particles and the metal plate of finite dimension, there are many complicated problems such as short duration of impact, rapid enhancement of transient stress, local large deformation of particles and macroscopic deflection of the metal plate are involved. However, the analysis of the contact force between the spherical particles and other objects is still simplified to the ideal Hertz contact problem. In order to study the real contact states of coal and rock particles impact on the metal plate, we simplified the working conditions from the coal and rock particles impact the tail beam in top coal caving mining to the system that spherical particles have similar properties to coal or rocks impact vertically on the metal plate. Based on the Flores contact theory and the energy absorbed by the bending deformation of the metal plate, the contact model of spherical rock at the compression stage was established. At the same time, we researched the initial yield stress and initial yield impact velocity of spherical rock that obeys the Drucker–Prager (D–P) criterion and proposed the approximate theoretical calculation method of maximum contact compression of spherical rocks. In the range of elastic impact velocity, the influence of impact velocity, restitution coefficient, material parameters and structure sizes on the dynamic response of the system were also analyzed. Finally, the correctness of the theoretical model has been verified by the virtual prototype simulation combined by ADAMS and Hypermesh. The research will provide the basic theory reference for the recognition technology of a single particle of coal or rock that based on the impact vibration of the tail beam.

Surface effect in axisymmetric Hertzian contact problems

Surface effect in three different axisymmetric Hertzian contact models is investigated in this paper with a recently developed elastic theory for nanostructured materials, including a Boussinesq problem, contact problem between a rigid flat-ended cylindrical indenter and an elastic half space as well as contact problem between a rigid spherical indenter and an elastic half space. With the help of the Love's strain function method and Hankel integral transformation, closed-form solutions of the stress and displacement fields at the surface of an elastic half space subjected to a concentrated force are achieved, based on which the interface tractions and displacements in the three different axisymmetric contact problems can be further obtained. It is found that surface effect in these contact problems can be characterized only by an intrinsic length, i.e., the ratio of the bulk surface energy density to the bulk shear modulus of the indented material. When the contact radius is comparable with the intrinsic length, surface effect is much obvious, leading to a serious deviation between the two solutions predicted respectively by the theoretical model developed for nanomaterials and the classical contact model. A more interesting phenomenon is about surface effect on the indentation hardness, which is found to increase with the reduction of the indenter radius when the external load is fixed, or to increase with the decrease of the external load when the indenter radius keeps unchanged. All the results in this paper should be helpful not only for deep understanding of the surface effect on nano-contact behaviors but also for further revealing the nature of surface effect of nano-indentation hardness.

A new criterion for loading time determination with minimum dynamic effects in quasi-static contact modeling

PurposeThe explicit finite element method (FEM) is one of the most popular approaches in quasi-static contact analysis which involves highly nonlinear friction and large deformation. Usually, a high loading rate is expected to improve computation efficiency in FEM. However, a higher loading rate often results in significant dynamic effects in the simulations. This study aims to propose a new criterion to achieve a good balance between a high loading rate and minimal dynamic effects.Design/methodology/approachThe proposed criterion is based on the fluctuation of total strain energy as well as the smoothness of its first derivative to determine the proper loading time with an acceptable level of dynamic effect.FindingsAsperities’ sliding contact and Hertz contact problems have been solved with the proposed criterion to verify its validity. The simulations show that the computation efficiency with the proposed criterion can be improved by up to 80 per cent compared to the regular energy ratio criterion.Originality/valueThis criterion will provide a valuable tool in determining the proper loading time to improve the computation efficiency for quasi-static analysis of asperities’ contacts.