Contact Problem Research Articles

A class of evolutionary history-dependent variational-hemivariational inequalities with strong and weak solutions

AbstractIn this paper we examine the well-posedness of evolutionary variational-hemivariational inequalities involving a constraint set and history-dependent operators. The strong and weak formulations of such inequalities are studied. First, the existence and uniqueness of solutions to both formulations are proved, and results on the dependence of solution on functional parameters are delivered. Next, the well-posedness is established for a general form of history-dependent variational-hemivariational inequalities with constraints by using a fixed point theorem. Finally, the results are applied to a dynamic frictional contact problem in viscoelasticity in which the contact is described by Signorini-type unilateral boundary condition with a nonmonotone Clarke’s relation.

Spur gear tooth root stress analysis by a 3D flexible multibody approach and a full-FE contact-based formulation

This paper proposes an original method to determine the gear tooth root stresses from a 3D finite element (FE) flexible multibody approach and a full-FE contact-based formulation. The contact problem is dealt with an augmented Lagrangian formulation whereas the analysis is performed by a preconditioned gradient solver (PCG). Tooth flank modifications are directly introduced within the 3D model. This one is thus able to take into account straightforwardly tooth bending and Hertzian-like deformations as well as the micro-geometry effect. Simulations are performed for several mesh periods, without making any assumptions about load distribution, tooth and gear blank flexibilities, and possible premature or delayed contacts between tooth pairs in quasi-static conditions. A precise distribution of tooth root stresses associated with instantaneous contacts conditions is then computed. For this study, a single stage spur gear with micro-geometry modifications corresponding to an arc-shaped profile crowning is modeled. Several output torques are considered. The obtained results are compared to those obtained using a 2D FE ISO-based model, where external forces are applied along the theoretical line of action.

Modelling and analysis of a viscoelastic contact problem with unilateral constraints

Modelling and analysis of a viscoelastic contact problem with unilateral constraints

Exploring the effects of finite size and indenter shape on the contact behavior of functionally graded thermoelectric materials

The performance enhancement of functionally graded thermoelectric (FGTE) devices is significantly influenced by contact studies of the FGTE materials. It is unclear how the finite thickness and the punch geometry influence the FGTE materials’ contact behaviors. This paper investigates the frictionless contact problem between three types of rigid punches (flat, triangular, and cylindrical) and the FGTE strip with finite thickness. The electric-thermo-elastic parameters of the FGTE strip vary in the thickness direction according to an exponential function. Based on the Fourier integral transform and the transformation matrix method, the problem is transformed into the numerical solution of three sets of singular integral equations. The presence of singular features on either side of the punch demands the adoption of specific collocation strategies. The distribution of the normal current density, the normal energy flux, and the normal contact stress is obtained by adjusting multiple electric-thermo-elastic parameters. The contact stresses in the case of punches with varying shapes can be effectively controlled by modulating the coefficient of thermal expansion and the strip thickness, whereas the effect of the electrical conductivity, the shear modulus, and the thermoelectric load on these stresses depends on whether they are increased or decreased.

Frictional Contacts Between Rough Grains With Fractal Morphology

AbstractSurface morphology plays a crucial role in friction between two contacting geomaterial surfaces, yet many questions remain unanswered regarding how detailed frictional responses deviate from analytical solutions for smooth surfaces due to the presence of roughness. In this study, we revisit the Cattaneo‐Mindlin problem for contacts between two fractally rough elastic or elasto‐plastic spheres generated based on ultra‐high degree (e.g., up to 2,000) spherical harmonics with the corresponding wavelength less than a thousandth of the mean grain diameter. Transverse contacts are simulated by finite element method, validated by the extended Cattaneo‐Mindlin solution to full slide regime for smooth sphere contacts. Extensive simulations are conducted to study contacts between two rough spheres with various surface geometries, micro friction coefficients, normal contact distances, relative roughness, fractal dimensions, and wavelength ranges. Out results indicate that: (a) the new analytical solution can approximately predict the macro contact response except for extremely high relative roughness and narrow wavelength range; (b) deviations induced by roughness from smooth sphere contacts can be neutralized by plasticity, high normal contact interference, and high micro friction coefficient; and (c) fractal dimension impacts frictional contacts less than relative roughness. The main cause of these phenomena can be credited to the underlying microscale contact information. Contact area and stress distributions and their evolutions provide concrete evidence of these observed behavior. This work provides a pathway for applying computational contact mechanics to many geophysical fields, such as the asperity model in earthquake science and the mechanics of granular materials.

Dynamic digital image correlation method for rolling convective contact

Digital image correlation (DIC) is an increasingly popular and effective non-contact method for measuring full-field displacements and strains of deformable bodies under load. Current DIC methods applied to bodies undergoing large displacements and rotations require a large measurement area for both the reference (i.e., undeformed) image and the deformed images. This can limit the resulting resolution of the displacement and strain fields. To address this issue, we propose a two-stage dynamic DIC method capable of measuring displacements and strains under material convection with high resolution. During the first stage, the reference image is assembled from smaller, high-resolution images of the undeformed object obtained using a spatially-fixed or moving frame. Following capture, each sub-image is rigidly translated and rotated into its appropriate place, thereby producing a full, high-resolution image of the reference body. In stage two, images of the loaded and deformed body, again obtained using a small camera frame with high resolution, are aligned with matching regions of the undeformed composite image using BRISK feature detection before performing DIC. We demonstrate the method on a contact problem whereby an elastomeric roller travels along a rigid surface. In doing so, we obtain fine-resolution measurements of the state of strain of the region of the roller sidewall in contact with the substrate, even as new material convects through the region of interest. We present these measurements as a series of images and videos capturing strain evolution as the roller transitions from static loads to a fully dynamic steady-state, documenting the effectiveness of the method.

SELF-CONSISTENCY CONDITIONS IN STATIC THREE-BODY ELASTIC TANGENTIAL CONTACT

The contact problem for an elastic third-body particle between two elastic half-spaces is considered. The contact is assumed to consist of three Hertzian contact spots. The normal and tangential contact problems are analyzed analytically considering partial slip in the contacts and the influence of third-body weight. Self-consistency conditions between global equilibrium and the contact solution are formulated to give criteria, under which circumstances static slip and stationary sliding are possible states for the third-body particle. The sliding case is solved in detail.

On the Study of Impact Plasto-Elastohydrodynamic Lubrication of Point Contact Problems Lubricated With Non-Newtonian Lubricant

Abstract In this study, the impact of a rigid ball on a substrate with lubricant in between is examined. A linear hardening model for the elastic/plastic substrate deformation is assumed. A power-law model is used to describe the lubricant rheology. Throughout the impact period, variations in the pressure distribution, the film thickness distribution, the velocity of a rigid ball, the impact load, the von Mises stress distribution, and the plastic strain distribution on the substrate are calculated. The special cases of ET = E in the present impact plasto-elastohydrodynamic lubrication (PEHL) results are in good agreement with previous impact elastohydrodynamic lubrication (EHL) results using a power-law model. The variation of central pressure over time in the PEHL model is flatter and lower compared to that in the EHL model. The significant difference shows that the plastic deformation mechanism should be considered in the simulation. The results indicate that as the flow index (n) increases, the central pressure and central film thickness increase, the pressure spike occurs earlier, and the rigid ball's rebounding velocity and maximum impact load decrease. Moreover, as the tangent modulus of the linear hardening model of the substrate increases, the rigid ball's rebounding velocity and the maximum impact load increase, and the substrate deformation and plastic strain decrease.

Semi-empirical terramechanics modelling of rough terrain represented by a height field

Dynamic simulations of various types of off-road vehicles, from planetary rovers to agricultural equipment, have long relied on well-established semi-empirical terramechanics models. While these models do have drawbacks and reliability issues that have been addressed by numerous works in the decades since the models were first introduced, semi-empirical approaches remain one of the few ways to simulate realistic wheel-soil interaction in real-time. One of their drawbacks is their assumption that the terrain is a flat plane. The models work by integrating normal and shear stresses along the wheel-terrain contact patch. The normal stress at each point along the contact patch is determined using an equation that computes soil pressure based on semi-empirical parameters, the dimensions of the wheel and the sinkage, which is determined based on the distance between the point and the plane that defines the terrain. Other works simplify the rough terrain contact problem by defining an equivalent contact plane at each time step in order to continue to be able to use semi-empirical models - modified to work with slanted planes - to compute the interaction forces. In this work, we propose a new, modified version of the semi-empirical model in which interaction forces for a wheel travelling on rough terrain can be computed without the need to use an equivalent contact plane. To highlight the advantages of our proposed approach, we compare our simulation results to the results of simulations using an existing approach for modelling a wheel travelling over rough terrain using traditional semi-empirical models.

Контактная задача для трансверсально изотропного слоя с трением

It is investigated the spatial contact problem of asymmetric interaction of two punches on a transversally isotropic layer with friction forces taken into account in an unknown contact domain. The planes of isotropy are parallel to the layer faces. The friction forces are taken into account along one coordinate axis. The lower layer face is subjected to sliding support. The layer material is characterized by five independent elastic parameters. The problem is reduced to an integral equation with respect to the contact pressure with the help of a double Fourier transformation together with the Coulomb law. The anisotropy degree is characterized be three dimensionless parameters arising in the kernel of the integral equation, two of which satisfy characteristic equation. In particular cases, the integral equation coincides with those well-known for the corresponding contact problems with friction for the isotropic layer and half-space. The B.A. Galanov method is used for numerical solutions. A system of the integral equation and integral inequality is considered. A rectangle is taken which a priori contains the unknown contact domain. By introducing special nonlinear operators, the system is reduced to only one nonlinear equation of the Hammerstein type which can be solved by the successive approximations method. The contact domain is determined by nodes at which the function required is positive. The punches are taken in the form of asymmetric elliptic paraboloids. The method allows us to investigate percolation, i.e. the process of junction of the discrete contact domains due to increasing the forces applied and settlements of the punches. Calculations are made for different friction coefficients, materials and relative thicknesses of the transversally isotropic layer.

A new contact force model for revolute joints considering elastic layer characteristics effects

The contact forces generated during the motion of revolute joints with clearances significantly impact the dynamic performance of mechanisms, making it essential to develop an accurate contact force model. Traditional contact models often neglect the influence of elastic layers or assume that the bushing thickness is equal to its radius, which limits their applicability. To address the dynamic contact problem in revolute joints with small clearances, this study employs an improved Winkler elastic foundation model and introduces an elastic layer correction coefficient to refine the static contact model. The model's validity is verified using the finite element method. Building on this, a new contact force model is proposed, incorporating the optimal damping term from multiple continuous contact models. Finally, experimental validation is conducted using the crank-slider mechanism and typical applications in the Variable Stator Vane (VSV) mechanism. The results demonstrate that the proposed contact force model effectively predicts the dynamic characteristics of mechanisms with clearances.

Nonlinear NMM analysis for large deformation and contact problems: Using full strain-rotation decomposition algorithm and augmented Lagrangian method enhanced open-closed iteration

Nonlinear analysis deals with problems that involve large displacements, strains, rotations, and contact problems. Accurate results can be obtained by choosing a reasonable method for decomposing strain and rotation to avoid errors and ensure the correct contact force. A novel full strain-rotation (S-R) decomposition and an augmented Lagrangian enhanced contact model are established within the numerical manifold method (NMM) framework. Limitations in simplified S-R NMM are overcome using our redesigned resolution procedure. A new method has been applied to analyse beams with large deflections, block columns under compression, block sliding, rock falling, and semi-ring contact with block problems. Promising results from the analysis indicate that proposed method is more accurate and effective in theory and numerically than prior approaches when resolving contact problems and large deformations.

Tribotechnical processes of the soil environment interaction with the working bodies of soil tillage and earthmoving machines reinforced with composite materials

The work presents the study results of the stress-strain state of the soil, as a continuous medium filled with abrasive particles under the action of the working bodies of soil tillage and earthmoving machines. One of the main properties of the soil, which determines the specifics of the force interaction of the working surfaces of the working bodies of soil tillage and earthmoving machines with the technological environment, is taken into account, namely the tendency of the contact layer of the treated layer to compaction. The relationship between stress in the soil and the wear of the working bodies of soil tillage and earthmoving machines has been established experimentally. A theoretical analysis is presented for the stress-strain state of the local region of the strengthened surface layer that is used in the working bodies of soil tillage and earthmoving machines, in which the filler, inclusion or strengthening phase is placed. An analysis of the contact characteristics of the stress-strain state and their changes during friction and wear was carried out based on the formulation and solution of the contact interaction problems of abrasive soil particles with the inhomogeneities of the composite coatings components based on ultra-high molecular weight polyethylene with fillers during strengthening of the working bodies of soil tillage and earthmoving machines. Computer modeling was performed to study the nature of stress distribution in the reinforced surface layer of the working bodies of soil tillage and earthmoving machines in the area of the contact zone in stationary and dynamic conditions. The contact problem is formulated, the boundary conditions and the solution in the form of components of the stress field are given. The characteristics of the filler, their content in the composite material and coating are taken into account, the relationship between the stress-strain state and wear is established.

Numerical study of a thermo-elasto-viscoplastic contact problem with adhesion using a hybrid method

Numerical study of a thermo-elasto-viscoplastic contact problem with adhesion using a hybrid method

Numerical analysis of the Thermal EHL line contact problem under intermittent motion

Abstract A mathematical model of time-varying thermal elastohydrodynamic lubrication (EHL) is developed using a sleeve chain as the object of study. The effects of thermal effect, load, speed, rest time and equivalent radius of curvature on its EHL are investigated using theoretical simulations. The results show that during intermittent motion, a portion of oil is entrapped in the contact zone at the end of the deceleration phase, after which this entrapped oil gradually moves out of the contact zone with the onset of speed, and a lower film of oil at the lubrication inlet passes through the center of the contact zone. In simple sliding intermittent motion, the temperature rise plays a crucial role in the variation of the oil film, particularly during the motion phases, and is an unavoidable factor. An increase in stop time reduces the film thickness and increases the following friction coefficient in the acceleration stage. An increase in the equivalent radius of curvature increases the film thickness and thermal rise in the contact zone as well as decreases the friction coefficient. Since the sleeve chain is one of the most commonly used mechanism in industry, high priority should be given to the lubrication design of this structure.

An ensemble method for improving robustness against the electrode contact problems in automated sleep stage scoring

In-home automated scoring systems are in high demand; however, the current systems are not widely adopted in clinical settings. Problems with electrode contact and restriction on measurable signals often result in unstable and inaccurate scoring for clinical use. To address these issues, we propose a method based on ensemble of small sleep stage scoring models with different input signal sets. By excluding models that employ problematic signals from the voting process, our method can mitigate the effects of electrode contact failure. Comparative experiments demonstrated that our method could reduce the impact of contact problems and improve scoring accuracy for epochs with problematic signals by 8.3 points, while also decreasing the deterioration in scoring accuracy from 7.9 to 0.3 points compared to typical methods. Additionally, we confirmed that assigning different input sets to small models did not diminish the advantages of the ensemble but instead increased its efficacy. The proposed model can improve overall scoring accuracy and minimize the effect of problematic signals simultaneously, making in-home sleep stage scoring systems more suitable for clinical practice.

Numerical Analysis of 3D Unilateral Quasi-static Contact: Effect of Coating Thickness and Mechanical Properties

In this study, a numerical model for unilateral quasi-static contact between a rigid sphere and an elastic coating has been developed. The contact problem was solved using a numerical procedure based on the FFT technique, considering various coatings with different thicknesses and mechanical properties, implemented through a Matlab code. The model calculates the contact surface deformation and pressure field through a double iteration process. The first iteration solves the contact problem for a given indenter penetration, while the second iteration refines this penetration by minimizing the difference between the fixed and calculated loads. To achieve this, influence coefficients are derived from the elasto-static equations using the Papkovich-Neuber potentials. The study discusses the influence of coating thickness and friction coefficient value on the tribological behavior of the coating. The results indicate that contact pressure increases (ranging from 1.9 to 2.5) as the coating becomes thicker or more rigid (0.02 mm to 0.2 mm). Additionally, the tribological behavior of the coated surface is affected by the coating's thickness, hardness, and friction coefficient value. Importantly, this model demonstrates versatility by being applicable to both smooth and rough surfaces.

Weak solvability of elliptic variational inequalities coupled with a nonlinear differential equation

In this paper we establish existence, uniqueness, and boundedness results for an elliptic variational inequality coupled with a nonlinear ordinary differential equation. Under the general framework, we present a new application modeling the antiplane shear deformation of a static frictional adhesive contact problem. The adhesion process has been extensively studied, but it is usual to assume a priori that the intensity of adhesion is bounded by introducing truncation operators. The aim of this article is to remove this restriction.The proof is based on an iterative approximation scheme showing that the problem has a unique solution. A key ingredient is finding uniform a priori bounds for each iterate. These are obtained by adapting versions of the Moser iteration to our system of equations.

Existence results for the time-incremental elastic contact problem with Coulomb friction in 2D

In this paper, the structure of the incremental quasistatic contact problem with Coulomb friction in linear elasticity (Signorini–Coulomb problem) is unraveled and sharp existence results are proved for the most general two-dimensional problem with arbitrary geometry and elasticity modulus tensor. The problem is reduced to a variational inequality involving a nonlinear operator which handles both elasticity and friction. This operator is proved to fall into the class of the so-called Leray–Lions operators, so that a result of Brézis can be invoked to solve the variational inequality. It turns out that one property in the definition of Leray–Lions operators is difficult to check and requires proving a new fine property of the linear elastic Neumann-to-Dirichlet operator. This fine property is only established in the case of the two-dimensional problem, limiting currently our existence result to that case. In the case of isotropic elasticity, either homogeneous or heterogeneous, the existence of solutions to the Signorini–Coulomb problem is proved for arbitrarily large friction coefficient. In the case of anisotropic elasticity, an example of nonexistence of a solution for large friction coefficient is exhibited and the existence of solutions is proved under an optimal condition for the friction coefficient.

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.