Elastic Contact Problems Research Articles

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.

Stochastic process model for interfacial gap of purely normal elastic rough surface contact

In purely normal elastic rough surface contact problems, Persson’s theory of contact shows that the evolution of the probability density function (PDF) of contact pressure with the magnification is governed by a diffusion equation. However, there is no partial differential equation describing the evolution of the PDF of the interfacial gap. In this study, we derive a convection–diffusion equation in terms of the PDF of the interfacial gap based on stochastic process theory, as well as the initial and boundary conditions. A finite difference method is developed to numerically solve the partial differential equation. The predicted PDF of the interfacial gap agrees well with that by Green’s Function Molecular Dynamics (GFMD) and other variants of Persson’s theory of contact at high load ranges. At low load ranges, the obvious deviation between the present work and GFMD is attributed to the overestimated mean interfacial gap and oversimplified magnification-dependent diffusion coefficient used in the present model. As one of its direct application, we show that the present work can effectively solve the adhesive contact problem under the DMT limit. The current study provides an alternative methodology for determining the PDF of the interfacial gap and a unified framework for solving the complementary problem of random contact pressure and random interfacial gap based on stochastic process theory.

Explicit frictional stick–slip dynamics of elastic contact problem incorporating the LuGre model

Dynamical contact problem with friction is of continuous concern and have been investigated over the years. The effective and accurate description of the development of the friction force is the key to this problem. However, the stick–slip phenomenon, as typical dynamical behaviour, has rarely been considered when developing contact algorithms for frictional contact problems of elastic bodies. The conventionally used Coulomb’s model, with differentiating the static and kinetic coefficient of friction, can lead to inaccurate results due to errors generated when switch between the stick or slip states. In this work, a three-dimensional mortar-based frictional contact algorithm is proposed in an explicit scheme. The proposed algorithm incorporates the LuGre model to account for dynamical frictional behaviours such as stick–slip. The effectiveness and accuracy of the proposed algorithm are validated through several simplified two-dimensional cases. The sticking time ratio and the evolution of the global coefficient of friction with time of a three-dimensional contact problem with a randomly rough contact interface are investigated as an application.

NUMERICAL SIMULATIONS OF ELASTIC-PLASTIC CONTACT CONSIDERING FRICTION

The solution of the elastic-plastic contact is of major importance in contact mechanics because contact damage and wear are responsible for the failure of many machine elements subjected to contact load. In many situations including gears and bearings, the load is transmitted through a small contact region, i.e. a concentrated contact, and the resulting high gradients of stress often exceed the plastic yield limit of the softer material. Plastic flow is initiated at a certain depth, resulting in modification of contact geometry and of the stress state that can lead to crack inception and propagation. Understanding the contact behaviour in the presence of plastic strains is fundamental for predicting the contact resistance. Considering the mathematical model complexity for the contact between bodies of arbitrary boundary, as well as the dissipative nature of plasticity, an analytical solution for the elastic plastic contact problem is hard to accomplish. Alongside finite element analysis, semi-analytical methods are the only existing simulation tools for the elastic-plastic contact problems. This paper extends a previously reported elastic-plastic contact solver developed by the same author, by considering friction. In the previous solver iteration, residual displacements were calculated only in the normal contact direction, thus assuming that the shear displacements are small enough to be neglected, in other words, the contact was assumed frictionless. By considering friction, a slip-stick contact process is introduced, adding a new iterative loop in which the contact pressure and shear stresses are iterated until convergence is reached. The new computer simulation tool is expected to give more realistic predictions for the elastic-plastic contact between materials with dissimilar elastic properties, in which case the shear tractions disturb the stress and displacements fields calculated for the frictionless contact case. The results are expected to contribute to a better understanding of the contact processes in the presence of both plastic strains and friction.

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.

A FETI B-differentiable equation method for elastic frictional contact problem with nonconforming mesh

A FETI B-differentiable equation method for elastic frictional contact problem with nonconforming mesh

Approximate contact force model for viscoelastic materials based on generalized Maxwell model

This paper proposes a novel approximate model for estimating the normal contact force between a rigid sphere and a viscoelastic half-space based on the generalized Maxwell model. The relaxation function derived from the generalized Maxwell model is applied to the solution to the elastic contact problem based on the elastic–viscoelastic correspondence to obtain an integral expression of pressure for viscoelastic contact. Using the approximate method on the basis of Taylor series, the integral expression is translated into an explicit function of displacement, velocity, time and material properties. A normal contact force model is established by integrating the pressure over contact area. A relaxation test of Nafion® XL is conducted to capture the viscoelastic parameters, and these parameters are employed to simulate the contact behaviors using the finite element method (FEM) to determine the scale factor in the contact force model. In order to verify the validity of the approximate model, the outcomes obtained from the model are compared with those generated by the FEM. A parametric study is conducted to analyze the effects of variations in parameters on normal contact behaviors. The results indicate that the established model is able to predict the contact responses of the viscoelastic materials with complex mechanical characteristics.

Analysis of adhesive contact of heterogeneous elastic materials

This paper reports a three-dimensional numerical model to solve adhesive elastic contact problem of heterogeneous materials with an engineered rough surface. This model integrates the Maugis-Dugdale (MD) adhesive contact model, the Boussinesq-Cerruti integral equations, the equivalent inclusion method (EIM), and the discrete convolution and fast Fourier transform (DC-FFT) algorithm, aiming at accurate determination of contact results for heterogeneous materials. An adhesive displacement-driven conjugate gradient method (CGM) was also developed to support rapid solutions of contact pressure and contact area. Four cases were analyzed to verify this model with satisfaction. This model was used to analyze the effects of inhomogeneity (location, size, and material properties) and adhesion on micro-adhesive-elastic contact, and the results lead to a curve-fitting equation for quick evaluation of the effects of inhomogeneity properties and adhesion characteristics on normal traction. It was also utilized to study the influence of the roughness level of a type of rough surface on the surface adhesion behavior.

A generalized penalty method for a new class of differential inequality system

The primary objective of this paper is to study a nonlinear system involving a parabolic variational inequality, a history-dependent hemivariational inequality and a differential equation constraints in a Banach space. First, we derive a unique solvability theorem for such problem under some mild hypotheses. Second, we construct a penalized problem for such nonlinear system, and show the existence and uniqueness of its solution to obtain an approximating sequence for the nonlinear system. Moreover, we prove the strong convergence of the sequence of approximate solution to the solution of the original system when the penalty parameter converges to zero. Finally, these results are applied to a quasistatic elastic frictional contact problem with heat equation with memory, and damage.

FFT-Based Numerical Method for Nonlinear Elastic Contact

In theoretical research pertaining to sealing, a contact model must be used to obtain the leakage channel. However, for elastoplastic contact, current numerical methods require a long calculation time. Hyperelastic contact is typically simplified to a linear elastic contact problem, which must be improved in terms of calculation accuracy. Based on the fast Fourier transform, a numerical method suitable for elastoplastic and hyperelastic frictionless contact that can be used for solving two-dimensional and three-dimensional (3D) contact problems is proposed herein. The nonlinear elastic contact problem is converted into a linear elastic contact problem considering residual deformation (or the equivalent residual deformation). Results from numerical simulations for elastic, elastoplastic, and hyperelastic contact between a hemisphere and a rigid plane are compared with those obtained using the finite element method to verify the accuracy of the numerical method. Compared with the existing elastoplastic contact numerical methods, the proposed method achieves a higher calculation efficiency while ensuring a certain calculation accuracy (i.e., the pressure error does not exceed 15%, whereas the calculation time does not exceed 10 min in a 64 × 64 grid). For hyperelastic contact, the proposed method reduces the dependence of the approximation result on the load, as in a linear elastic approximation. Finally, using the sealing application as an example, the contact and leakage rates between complicated 3D rough surfaces are calculated. Despite a certain error, the simplified numerical method yields a better approximation result than the linear elastic contact approximation. Additionally, the result can be used as fast solutions in engineering applications.

A convergence criterion for elliptic variational inequalities

We consider an elliptic variational inequality with unilateral constraints in a Hilbert space X which, under appropriate assumptions on the data, has a unique solution u. We formulate a convergence criterion to the solution u, i.e. we provide necessary and sufficient conditions on a sequence { u n } ⊂ X which guarantee the convergence u n → u in the space X. Then we illustrate the use of this criterion to recover well-known convergence results and well-posedness results in the sense of Tykhonov and Levitin–Polyak. We also provide two applications of our results, in the study of a heat transfer problem and an elastic frictionless contact problem, respectively.

Deterministic Model of Elastic Rough Contact Taking into Account the Mutual Influence of Asperities

The analytical solution of the elastic contact problem for the bodies with rough surfaces is presented using a deterministic description of the surface roughness. Particularly, a model of the normal contact of a periodic system of axisymmetric punches of different heights with an elastic half-space is developed. The solution is obtained taking into account the real contact pressure distribution only under nearby ones, and considering the action of remote ones as the nominal pressure (the localization method). The effect of the shape of the punches and their spatial positions on the dependencies of the real contact area and the contact pressure distribution at the contact spots on the applied load are studied. The additional displacement function, which characterizes the effect of surface roughness on the relative approach of contacting bodies at the macrolevel, is also analysed.

A coupled SBFETI-BDEs method for solving 2D static and dynamic contact problem with friction

In this study, we propose a coupled approach called the SBFETI-BDEs method for solving the 2D elastic frictional contact problem. The contact system is decomposed into multiple subdomains, similar to the finite-element tearing and interconnecting (FETI) algorithm. Each subdomain can be regarded as a scaled boundary finite element (SBFE), where only the boundaries are discretized. To avoid computing the global contact flexibility matrix using the force method, we consider the generalized inverse of the stiffness matrix for each subdomain as a subcontact-flexibility matrix. We uniformly formulate the contact equations, displacement continuity equations, and the equilibrium equations of force for the floating subdomains, as a B-differentiable equations (BDEs) system, which we solve using the B-differentiable damped Newton-Raphson method that possesses a global convergence property. The proposed method offers increased computational efficiency without compromising accuracy when compared to the original BDEs algorithm without domain decomposition. We present the dynamic scheme of the SBFETI-BDEs method within the framework of the HHT time-integration scheme, where the equilibrium equations of force for the floating subdomains disappear due to their nonsingular equivalent stiffness matrices. Numerical examples are provided to demonstrate the convergence, accuracy, efficiency, and capability in solving engineering problems of the proposed method.

"THE ELASTIC CONTACT PROBLEM INVOLVING A BILATERALLY LOADED THIN STRIP "

"For the solution of the elastic contact problem, it is generally assumed that the contacting solids can be assimilated to elastic half-spaces. This assumption is reasonable when the contact area is small compared to the dimensions of the contacting solids, and when the contact stresses are restricted to a small vicinity of the initial point of contact, without reaching the boundaries of the solids. These conditions are not met when a thin strip is bilaterally and symmetrically compressed between two punches, as the dimensions of the contact area might be of the same order of magnitude as the strip thickness. Moreover, the stresses induced in the strip will reach the boundary without a significant decay in intensity. Consequently, in this contact scenario, the classical solutions for a point force acting on the boundary of the half-space cannot be directly applied. The starting point for the problem solution is a modified Boussinesq-type solution for the thin elastic layer, expressing the displacement and stress fields induced in an elastic strip by two opposed normal forces perpendicular to the layer boundaries. To this end, supplementary displacements are added to the half-space solution to satisfy both geometric and loading symmetry, as well as the linear elasticity equations. Superposition principle is then applied, resulting in a contact model for an elastic strip compressed between two identical indenters with aligned axes. An algorithm for the contact of solids that can be assimilated to elastic half-spaces is modified and applied to the contact involving a thin strip. To this end, the required influence coefficients for displacements are derived. A calculation example involving a thin strip compressed between two spheres is presented, and a comparison with the half-space solution is performed. "

Response of elastic/viscoelastic layers on an elastic half-space in rolling contacts: Towards a new modeling approach for elastohydrodynamic lubrication

The 3D quasi-static contact problem of a rigid sphere rolling on an elastic half-space covered by a thin viscoelastic/elastic layer is studied as representative of soft layered contacts in engineering, physics, and biomedical applications, as well as for its potential merit to model Elasto-Hydrodynamically Lubricated (EHL) contacts in specific EHL film behavior suggested in earlier research. The viscoelastic layer behavior is modeled with a standard linear solid (SLS) model with a single relaxation time. Two approaches are used, the foundation (Winkler) reduced model assuming unidirectional stress–strain behavior normal to the surface only and the Papkovich–Neuber potential model which, based on the complete Navier–Cauchy equations, accounts for non-local support and bending effects. The models are validated against literature and compared. Whereas coated/layered contact problem studies mostly consider relatively thick coatings, and results focus on the pressure distribution and the contact area size, in this paper we consider layers with a thickness much smaller than the contact radius, i.e. of the order of a thickness of a conventional EHL film. The details of the layer deformation, pressure profiles and subsurface stresses are presented and interpreted in terms of the underlying physics, in particular the thin layer limit common also in thin fluid layers. It is shown that two dimensionless parameters, the ratio of the elastic modulus between the layer and the substrate and the ratio of the layer thickness to the corresponding Hertz contact width, dominate the systematic response for elastic layered contact problems. And four dimensionless parameters with two extras, the Deborah number based on the Hertz contact width and the ratio of the two elastic limits of the SLS viscoelastic material, are needed and analyzed for viscoelastic layered contacts. The results presented provide a good framework for the understanding and interpretation of the effects of viscoelastic layers on the deformation and pressure distribution of contact problems. Finally, the capability of actually modeling EHL film behavior via a viscoelastic layer suggested by van Emden et al. (2017) is discussed.

A coupled FETI-BDNM for solving 3D elastic frictional contact problem

Based on the B-differentiable Newton method (BDNM) and finite element tearing and interconnecting (FETI) algorithm, a coupled approach named FETI-BDNM is developed to solve the 3D elastic contact problem with friction. In this method, the contact bodies are decomposed into a number of subdomains. The interfaces between the two adjacent substructures will be skillfully considered as the special contact surfaces that are always in the stick state. By resorting to the generalized inverse of stiffness matrices of the subdomains, the relationships between the relative displacements and forces on contact/interface, i.e., contact flexibility matrices will be computed efficiently. The contact equations, displacement continuity equations and equilibrium equations for each subdomain are expressed in the form of B-differentiable equations (BDEs) uniformly, which can be solved by the damped Newton-Raphson method with 1D Armijo type linear search. The proposed method makes it possible to solve large-scale contact problems without compromising the solution accuracy compared to the original BDNM, and the good convergence of BDEs and parallel scalability of the FETI algorithm will be preserved. Numerical examples validate the accuracy, efficiency, robustness and effects of the number of subdomains on the computational time of the proposed method.

A custom arc‐length Finite Element solver for large deformation adhesive contacts using a k‐d tree accelerated volumetric interaction scheme

AbstractThis paper introduces a Finite Element framework for large deformation adhesive contact between elastic bodies based on volume‐to‐volume interaction. A key component of this framework is a novel, customized arc‐length algorithm designed to follow the solution path through multiple snap‐through and snap‐back instabilities. Another new component is a k‐d tree based algorithm which significantly accelerates the computationally expensive volume‐to‐volume interaction calculations. This optimized Finite Element framework is then used to solve two‐body and three‐body elastic adhesive contact problems with realistic material and interaction parameters, using a 6‐12 Lennard‐Jones interaction potential. The improved efficiency of the k‐d tree based scheme makes it indispensable for interaction calculations. This paper also shows the importance of implementing adhesion‐specific customizations in the arc‐length solver.

Shape sensitivity analysis of an elastic contact problem: Convergence of the Nitsche based finite element approximation

In a recent work, we introduced a finite element approximation for the shape optimization of an elastic structure in sliding contact with a rigid foundation where the contact condition (Signorini’s condition) is approximated by Nitsche’s method and the shape gradient is obtained via the adjoint state method. The motivation of this work is to propose an a priori convergence analysis of the numerical approximation of the variables of the shape gradient (displacement and adjoint state) and to show some numerical results in agreement with the theoretical ones. The main difficulty comes from the non-differentiability of the contact condition in the classical sense which requires the notion of conical differentiability.

Rapid Methods for the Resolution of Contact Problems in Static Linear Elasticity

In this paper, the two‐dimensional Signorini static contact problem in linear elasticity is presented. We present the weak formulation of the frictional contact problems, and the boundary integral operators are used to propose a boundary variational formulation whose resolution by the generalized Newton method is presented. Moreover, a particular formulation by the fixed point method associated with the augmented Lagrangian is proposed for efficient analysis of contact problems with Coulomb friction, and powerful algorithms are constructed. The discretization is carried over by using the Galerkin method. The resulting linear system is solved by using a preconditioned conjugate gradient (CG) iterative solver.

A Self-Adaptive Alternating Direction Multiplier Method for Frictionless Elastic Contact Problems

A Self-Adaptive Alternating Direction Multiplier Method for Frictionless Elastic Contact Problems