Contact Problem Research Articles

A two-dimensional elastic contact problem with unilateral constraints

We consider a mathematical model which describes the equilibrium of two elastic membranes fixed on their boundary and attached to an adhesive body, say a glue. The variational formulation of the model is in a form of an elliptic quasivariational inequality for the displacement field. We prove the unique weak solvability of the model, and then we state and prove a convergence result, for which we provide the corresponding mechanical interpretation. Next, we consider two associated optimization problems for which we provide existence results. Finally, we the present numerical simulation which validates our convergence result. We end this paper with some concluding remarks and an Appendix, in which we present the preliminary material needed in this paper.

Multi-method examination of contact mechanics in orthotropic layers under gravity

Analyzing the behavior of intelligent unconventional materials under diverse contact scenarios, in comparison to conventional materials, is a critical step in the initial design of engineering systems. This paper presents the development of analytical and numerical approaches for analyzing contact mechanics in a system comprising an orthotropic layer resting on a rigid foundation. Parametric analyses include the consideration of cylindrical and flat punch profiles. Analytical formulation utilizes integral transform methods to convert planar elasticity equations into a Cauchy-type singular integral equation of the second kind. A detailed description of the solution technique for the integral equation is provided, encompassing both the analytical formulation and the required discretization for obtaining the solution. Subsequently, a finite element approach (FEA) is employed to approximate contact bodies using a collection of finite elements, while contact boundaries are approximated by utilizing a set of polygons. Alternatively, the problem is addressed using the Multilayer Perceptron approach, a form of artificial neural network frequently applied in diverse machine learning applications, including scenarios involving contact problems. Finally, the resolution to the problem is achieved by employing the multilayer perceptron (MLP) method. The study yields determinations of contact stresses under the punch, the critical separation load resulting in the detachment of the orthotropic layer from the rigid foundation, and the corresponding separation distance. This analysis considers a range of dimensionless parameters and explored the behavior of diverse orthotropic materials. Comparisons between the results of the analytical method and computations from FEA and MLP reveal the exceptional accuracy attained by all three approaches. As the pioneer study employing three distinct approaches to examine continuous contact mechanics within an orthotropic layer under the influence of gravity, this research constitutes a valuable point of reference for upcoming scholars in the field.

Numerical study on mixed lubrication performance of misaligned microgroove water-lubricated bearings considering cavitation and turbulence effects

This study presents a mixed lubrication model for misaligned microgroove water-lubricated bearings (WLBs). The model considers the effects of cavitation and turbulence to assess the mixed lubrication performance of WLBs with various microgroove morphologies. The equations are discretized using the control volume method (CVM) and solved by using the Fischer–Burmeister–Newton–Schur method for dealing with the constrained system. A comparison of the experimental data from the published literature demonstrates the model and methodology's validity. On this basis, the effects of rotational speed, load, and misalignment angle on the mixed lubrication performance of microgroove WLBs are investigated. The numerical results indicate that the left-triangular microgroove exhibits the best-mixed lubrication performance. Under elastohydrodynamic lubrication and mixed lubrication conditions, there are significant discrepancies in microgroove morphology. However, this discrepancy diminishes with increasing misalignment angles. The edge contact problem resulting from journal misalignment can be efficiently mitigated by selecting the proper microgroove morphology. This study provides useful guidance for the optimal design and mixed lubrication performance improvement of WLBs.

Experimental investigation of the effects of debris on non-conforming contact stress

Debris widely exists in contact structures, affecting the reliability of structures. In this investigation, the effects of debris on the non-conforming contact stress are investigated through two-body and three-body contact experiments. According to the relationship between the contact length and debris length, debris is divided into long debris and short debris, which show different influences on the contact behaviors. The non-conforming contact stress is reconstructed based on full-field metrology and an inverse method. The evolution of contact stress induced by debris is analyzed with various parameters and contact surface profiles. Different factors, including debris material, debris length, debris thickness, plate material, and load amplitude, are considered in experiments. Their influence rules are discussed and summarized. The plate materials and short debris materials have coupling effects on the contact stress, which is associated with the deformation characteristics of the contact surface. The results provide an experimental reference for the research of non-conforming contact problems.

A method for solving an electro-viscoelastic contact problem with damped response and friction

A method for solving an electro-viscoelastic contact problem with damped response and friction

Elastic solids under frictionless rigid contact and configurational force

A homogeneous elastic solid, bounded by a flat surface in its unstressed configuration, undergoes a finite strain when in frictionless contact against a rigid and rectilinear constraint, ending with a rounded or sharp corner, in a two-dimensional formulation. With a strong analogy to fracture mechanics, it is shown that (i.) a path-independent J–integral can be defined for frictionless contact problems, (ii.) which is equal to the energy release rate G associated with an infinitesimal growth in the size of the frictionless constraint, and thus gives the value of the configurational force component along the sliding direction. Furthermore, it is found that (iii.) such a configurational sliding force is the Newtonian force component exerted by the elastic solid on the constraint at the frictionless contact. Assuming the kinematics of an Euler–Bernoulli rod for an elastic body of rectangular shape, the results (i.)–(iii.) lead to a new interpretation from a nonlinear solid mechanics perspective of the configurational forces recently disclosed for one-dimensional structures of variable length. Finally, approximate but closed-form solutions (validated with finite element simulations) are exploited to provide further insight into the effect of configurational forces. In particular, two applications are presented which show that a transverse compression can lead to Eulerian buckling or to longitudinal dynamic motion, both realizing novel examples of soft actuation mechanisms. As an application to biology, our results may provide a mechanical explanation for the observed phenomenon of negative durotaxis, where cells migrate from stiffer to softer environments.

Aeroacoustic simulation of human phonation based on the flow-induced vocal fold vibrations including their contact

This paper deals with the numerical simulation of the flow induced sound as generated in the human phonation process including vocal folds (VF) contact. The hybrid approach is used allowing to separate the acoustic and the biomechanical description of the considered problem. The two-dimensional fluid–structure interaction (FSI) problem is modelled by a simplified linear elastic model coupled to the incompressible Navier–Stokes equations in the arbitrary Lagrangian–Eulerian form and special attention is paid to the treatment of the contact problem in the FSI context. Following approach of [Sváček and Horáček, 2021] the incompressible flow model is modified by an addition of fictitious porosity term and the penalization inlet boundary condition is used. Further, the Lighthill acoustic analogy is applied for calculation of sound sources and their propagation through a vocal tract model. All partial differential equations, describing the elastic body vibration, the fluid flow and the acoustics, are approximated by finite element method. Numerical results of one-sided vocal fold paralysis are presented.

A new class of general fractional differential quasivariational and quasivariational–hemivariational inequalities with variable constraint sets

This paper investigates a new general nonlinear system, which comprises a fractional differential equation and a history-dependent quasivariational inequality with a variable constraint set, as well as a quasivariational–hemivariational inequality with a variable constraint set. Such a general nonlinear system can be used to describe a nonlinear quasistatic thermoelastic frictional contact problem of viscoelastic materials with long memory, wear and implicit obstacles. By employing the fixed point theorem of the set-valued mapping, the solvability for such a general nonlinear system is proved under some mild conditions. Moreover, our abstract result is applied to obtain the solvability of a new nonlinear quasistatic thermoelastic frictional contact problem with long memory, wear and implicit obstacles.

Examining the contact problem of a functionally graded layer supported by an elastic half‐plane with the analytical and numerical methods

This study offers a comparative study of the analytical and numerical methods for investigating a contact problem. The contact problem comprises a functionally graded layer supported by a half‐plane and loaded with a distributed load from the top surface. First, the analytical and numerical solutions to the problem are acquired by utilizing a theory of elasticity and finite element method, respectively. The problem is transformed into a system of integral equations in which the contact stress is an unknown function. The solution of the integral equation was achieved with Gauss–Jacobi integration formulation. The finite element model of the problem is created using ANSYS software, and the two‐dimensional analysis of the problem is performed. Results were obtained from the samples for different material properties and loading conditions. The distributed load width and non‐homogeneity parameters significantly impact on contact mechanics. The results indicate that the contact area and the contact stress obtained from finite element method (FEM) are close to the analytical results. As a result, acceptable error rates were obtained. Finally, this study provides evidence of a good agreement between the two methods.

Numerical analysis of a history-dependent mixed hemivariational-variational inequality in contact problems

This paper is devoted to a study of a history-dependent mixed hemivariational-variational inequality arising in contact mechanics. The contact problem concerns the deformation of a viscoelastic body with long memory, subject to a general friction law on one part of the boundary and a frictionless Signorini condition on another part of the boundary. The solution existence and uniqueness of the history-dependent mixed hemivariational-variational inequality based on a Lagrange multiplier approach are proved. Then, a fully discrete scheme is introduced and studied. The trapezoidal rule is used to approximate the integral in the history-dependent operator. For the spatial discretization, the linear finite elements are used to discretize the displacement field, and dual basis functions are used in the approximation of the Lagrange multiplier. Optimal order error estimates are derived for the displacement and the Lagrange multiplier under appropriate solution regularity assumptions. Numerical results are presented to illustrate the theoretical prediction of the convergence orders.

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.

Nonnegative iterative reweighted method for sparse linear complementarity problem

Solution of sparse linear complementarity problem (LCP) has been widely discussed in many applications. In this paper, we consider the ℓp regularization problem with nonnegative constraint for sparse LCP, and propose algorithms based on the iterative reweighted method to approach a sparse solution of the LCP, and then show the convergence to the stationary point of ℓp regularization problem. Numerical results on simulated data exhibit an excellent performance of the proposed algorithms on approaching a sparse solution of the LCP. Finally, we apply this method to the frictional and frictionless contact problems. The numerical experiments demonstrate that the contact problems can be efficiently solved by the proposed algorithm.

Numerical simulation of ice loading characteristics of ships in a wave-current environment in a broken ice area

A numerical simulation technique with CFD-DEM coupling is used to simulate a full-scale KCS model sailing in the pancake ice region. Based on the actual polar information, the Stokes fifth-order wave model is selected. The pancake ice particle of two scales is established based on the DEM, and the corresponding broken-ice region is generated. The contact and collision problems of ice-ice and ice-wall (ship) were analyzed by solving the Hertz-Mindlin model, and the monitored data were processed and summarized. The results show that the ice load on the ship is a repetitive “loading-unloading” process, and the peak ice load curve is characterized by segmented and concentrated fluctuations. In terms of ice parameters, the pancake diameter has a great degree of influence on the crushed ice loads, and the size of the crushed ice loads increases with the diameter of the crushed ice particles.

Coupled Contact Problem of Punch Pressure Under Action of Follower Force

Coupled Contact Problem of Punch Pressure Under Action of Follower Force

On the dynamic Coulomb's frictional contact problem for thermo‐piezoelectric materials

AbstractIn this work, we study, from a variational point of view, a dynamic contact problem between a thermo‐piezoelectric body and a thermally conductive foundation. The normal compliance contact condition and Coulomb's friction law are employed to model the contact. We provide existence and uniqueness results of a weak solution to the model using adequate auxiliary problems, an abstract result on nonlinear first‐order evolution inequalities and Banach fixed point argument. Finally, the continuous dependence of the solution on the surface traction force and the surface electrical charge is studied.

A generalization of B-differentiable equations method for elastoplastic contact problems with dual mortar discretization

In this study, a novel method based on the B-differentiable equations (BDEs) is proposed to solve the three-dimensional (3D) static and dynamic elastoplastic contact problems with friction. The contact conditions are formulated as the BDEs system, which can precisely satisfy Coulomb’s law of friction. Based on the mortar segment-to-segment contact model, the contact constraints are enforced by dual Lagrange multipliers for nonconforming meshes of the contact boundary. The non-smooth and nonlinear system equations composed of equilibrium and contact equations are simultaneously solved by the B-differentiable damping Newton method (BDNM), which has global convergence property. In each iteration, firstly, some contact displacements and contact forces, called substituted contact variables, are expressed by others according to the different contact states and contact conditions. The equilibrium equation is then solved, where the stiffness matrix is updated by substituting the contact variables. At last, the substituted contact variables will be restored. In the proposed method, the contact flexibility matrix is no longer needed, which saves the time for calculating the contact flexibility matrix due to changes in the stiffness matrix resulting from the update of stresses in the iteration step. By comparing with the results obtained by finite element commercial software ANSYS, the accuracy of the proposed algorithm is verified, and the property of convergence and efficiency are demonstrated for the elastoplastic contact problems by several numerical examples.

Impact damage testing based on high-speed continuous water jet aircraft coatings

When an aircraft passes through a rainy area at high speed, the coating on the front edge of the fuselage will be continuously eroded by raindrops, causing the coating to wear, crack or even peel off. This paper uses carbon fiber T300 material as the base material, and at the different impact speeds and impact numbers, water cutting equipment was used to simulate the erosion of the coating caused by the continuous impact of water droplets. The damage morphology of samples at different damage stages was observed by digital microscope and Scanning Electron Microscope (SEM), and the damage evolution curve was established to analyze and reveal the damage behavior and damage mechanism of rain erosion. The results show that the degree of damage experienced an increasing trend with the increase of impact numbers and speed, until circular peel damage was formed; no damage occurred during the incubation period, and the curvature of the damage evolution curve increased significantly after the expansion period and eventually showed a stable expansion trend. The mechanical properties of the coating material were the main influencing factors of its rain corrosion resistance. Moreover, the axially symmetric unsteady contact problem of droplets impacting the surface of a solid deformable body was studied. And the contact area was determined based on the iterative algorithm boundary positioning method. A mathematical model and closed mathematical formula describing the unsteady interaction between a droplet and a solid deformable obstacle were proposed.

An improved tolerance analysis method for gear mechanisms considering form deviations and elastic-plastic contact behaviors

The contact quality between geometric features significantly affects the assembly accuracy and dynamic performance of mechanical products. However, accurate and efficient contact problem solving is currently a common challenge in the fields of tolerance analysis and tooth contact analysis (TCA). In this paper, a tolerance analysis method for gear mechanisms considering form deviations and elastic-plastic contact behaviors is proposed. Specifically, tolerance modeling of systematic and random deviations on tooth surfaces is represented by skin model shapes (SMS). Elastic-plastic contact behaviors including contact deformation, status, and stress distribution are modeled and calculated by a modified boundary element method (BEM). The accuracy and efficiency of this BEM algorithm are verified by comparing with the finite element method (FEM), indicating that the BEM can greatly improve the computational efficiency while maintaining high solution accuracy. A case study for a pair of spur gears demonstrates the significant effects of contact behaviors on transmission performance. This research provides an effective method to introduce mechanical contact behaviors in tolerance analysis of gear mechanisms. In addition, the contact simulation results can also be adopted in the analysis of vibration, reliability, and other aspects.

Численное исследование задачи низкоскоростного столкновения стального шара с ледяной плитой

In this paper, the dynamic deformation process during the low-speed collision was investigated. The full 3D dynamic contact problem was simulated in the Abaqus software, based on the explicit finite-element method on structured meshes. Two basic mechanical models (linear isotropic elastic and elastoplastic) are considered. The obtained results showed the possibility of the experimental data, available in the open literature for different striker velocities, reproduction. The better understanding of the ice deformation process can be useful for different applied problems, like the ice impact on the aircraft wings estimation and the ice-water contact reflected waves filtration during the seismic survey in the Arctic region.

Quadratic Discontinuous Galerkin Finite Element Methods for the Unilateral Contact Problem

Abstract In this article, we employ discontinuous Galerkin methods for the finite element approximation of the frictionless unilateral contact problem using quadratic finite elements over simplicial triangulation. We first develop a posteriori error estimates in the energy norm wherein, the reliability and efficiency of the proposed a posteriori error estimator is addressed. The suitable construction of the discrete Lagrange multiplier 𝝀 𝒉 {\boldsymbol{\lambda_{h}}} and some intermediate operators play a key role in developing a posteriori error analysis. Further, we establish an optimal a priori error estimates under the appropriate regularity assumption on the exact solution 𝒖 {\boldsymbol{u}} . Numerical results presented on uniform and adaptive meshes illustrate and confirm the theoretical findings.