Two-dimensional Contact Problem Research Articles

A penalty-based cell vertex finite volume method for two-dimensional contact problems

A penalty-based cell vertex finite volume method for two-dimensional contact problems

On the benefits of a multiscale domain decomposition method to model-order reduction for frictional contact problems

In this paper, the efficiency of a multiscale strategy based on a domain decomposition method (DDM) for model-order reduction of time-dependent frictional contact problems is presented. The proposed strategy relies on the LArge Time INcrement (LATIN) nonlinear solver combined with model reduction based on the Proper Generalized Decomposition (PGD). The LATIN presents a robust treatment of contact conditions, sharing similarities with augmented Lagrangian approaches, and naturally leads to a mixed DDM. In addition, the global space–time formulation of the method allows PGD-based model reduction to be used during computations, creating and enriching on-the-fly reduced bases per substructure to better track sliding fronts and propagative phenomena. The introduction of a multiscale strategy in the LATIN framework is consistent with the physics of contact problems, in which phenomena with different wavelengths interact: local solutions at contact interfaces presents high gradient effects with a short wavelength compared to the characteristic length of the structure. By taking advantage of this, the coarse scale problem of the strategy enables to capture efficiently the behavior of the problem at the structural level, focusing then on capturing the local contact variations at the contact interfaces. The most important features of the approach are shown comprehensively on a simple one-dimensional frictional contact problem. Then, its robustness and effectiveness are tested on a two-dimensional multibody frictional contact problem with more complex loadings. Guidelines are also given for choosing the parameters of the strategy, in particular those driving the construction of the reduced basis.

Frictional contact analysis between two-dimensional deformable anisotropic magneto-electro-elastic bodies via a semi-analytical method

In this paper, we utilize the surface Green's function of a magneto-electro-elastic (MEE) half-plane as a general analytical kernel to develop a semi-analytical method (SAM). This SAM is designed to solve the two-dimensional contact problems of two dissimilar deformable anisotropic MEE bodies. Using the surface Green's function, the associated influence matrices in SAM can be obtained in a closed mathematical form and then computed analytically, offering significant benefits in terms of computational efficiency and accuracy. This method can be applied to general contact problems where the contact surface is frictional sliding and subjected to four different electric and magnetic conditions. By assuming the elastic and piezoelectric materials as uncoupled MEE materials, the method can also be applied to contact problems in which anisotropic elastic and piezoelectric materials are involved. To demonstrate the accuracy and applicability of the proposed SAM, four different numerical examples are provided, and their results are compared with those obtained by using other approaches, such as analytical solutions and the previous SAM in the literature. Through the numerical results, parametric studies are also performed, allowing us to explore the influences of material properties, loadings, frictional coefficient, and contact surface profile on mechanical/electrical/magnetic contact behaviors.

Peridynamic contact models for fracture analysis based on the micro-beam bond

Contact problems are ubiquitous in engineering systems. Establishing contact models to predict the crack propagation and fracture is significant. Therefore, the present study proposes two peridynamic contact models for two-dimensional (2D) and three-dimensional (3D) contact problems based on the micro-beam bond and the Hertz theory, using the relative position of particles and the contact micromoduli to calculate the contact forces. Firstly, the strain is generated on the contact surface when two bodies come into contact. Simultaneously, the contact stiffness is obtained by the Hertz theory, and the contact forces are calculated according to the relationship between the strain of contact surface and the contact stiffness. Next, based on the peridynamic theory, the contact horizon and contact micromoduli are defined for the calculation of contact forces. Finally, the contact micromoduli for 2D and 3D contact problems are derived by equating the contact forces obtained from both theories. Additionally, this study defines the correction coefficients to reduce the calculation error of the proposed models and investigates the influence of different contact horizons on the calculation of contact forces. To validate the effectiveness of the proposed contact models, three numerical examples are compared with the results from the finite element method or experiments. It is demonstrated that the proposed contact models accurately calculate the contact forces and predict the crack propagation.

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.

Dynamic indentation of viscoelastic orthotropic layer supported by a Winkler–Pasternak foundation

AbstractIn this study, the two-dimensional dynamic contact problem between a rigid flat punch and a viscoelastic orthotropic layer is investigated. The motivation of the study is to provide a better understanding of the vertical vibration of the two-parameter Winkler–Pasternak foundation, which has not yet been investigated. For the contact problem, the mixed boundary conditions on the top and bottom surfaces are transformed into linear equations using the Fourier transform technique and Helmholtz functions. Based on the Gauss–Chebyshev integration formula, the singular integral equation is obtained and solved numerically. As a result of the solutions, the effects of various parameters on the contact stresses are analyzed and examples are given. It was found that the Winkler foundation modulus does not affect the dynamic contact stress, while the Pasternak foundation modulus significantly affects the contact stress.

Power Dissipation and Wear Modeling in Wheel–Rail Contact

This paper is concerned with the modeling of power dissipation due to friction and its relation with wear estimation in wheel–rail contact. Wear is a complex multi-scale and multi-physical phenomenon appearing in rolling contact. Wear is generated by high contact stress and the work of friction forces. This phenomenon leads to the appearing of the worn material in the form of wear debris between contacting surfaces. In contact models, wear is usually described in terms of the wear depth function. This function modifies the gap between the contacting bodies as well as the shape of the surfaces of the wheel and rail in contact. In this paper, besides the wear depth function, the dissipated energy, rather than the contact stress, is taken into account to evaluate the wear impact on rail or wheel surfaces. The dissipated energy allows us to more precisely evaluate the wear debris amount as well as the depth of wear and its distribution along the contact interface. A two-dimensional rolling contact problem with frictional heat flow is considered. The elasto-plastic deformation of the rail is considered. This contact problem is governed by a coupled system of mechanical and thermal equations in terms of generalized stresses, displacement and temperature. The finite element method is used to discretize this problem. A discretized system of equations with nonpenetration and friction conditions is transformed and formulated as a nonlinear complementarity problem. The generalized Newton method is applied to numerically solve this mechanical subproblem. The Cholesky method is used to find the solution of the heat-conductive problem. The dissipated power is evaluated based on the resultant force and slip at a reference point. Numerical results including the distribution of slip velocity, power factor and wear rate are provided and discussed.

FRACTURE ANALYSIS AND SHIELDING EFFECTS IN ADVANCED MATERIALS

Due to their ability to interconnect electrical and mechanical domains, piezoelectric materials are increasingly finding applications in advanced electromechanical systems. Notably, magneto-electric composites like BaTiO3-CoFe2O4 have emerged as vital materials for the next generation of devices. Exploring this realm is a complex and significant endeavor, demanding the attention of the scientific community and further research. Over recent decades, the scope of formulating and executing computational models for continuous media has substantially broadened, encompassing a diverse array of material properties and fields in computational models. This article centers on the utilization of asymptotic analysis as a mathematical tool for constructing approximate equations and assessing the relevance of various hypotheses. It delves into the utilization of the perturbation method, pioneered by Kagadiy T. S. and others, to tackle two-dimensional contact problems in electropelasticity, particularly in the context of materials with linear anisotropy. The extensive applicability of this asymptotic approach underscores its efficacy in simplifying intricate problems by breaking them down into a sequence of boundary problem resolutions grounded in the theory of potentials. The collaborative effort of the authors in this research underscores that employing the mentioned method opens up avenues for formulating pertinent boundary problems for the fundamental equations. This, in turn, allows for the representation of the initial electropelasticity problem as a superposition of more manageable boundary problems. While mechanical and electrical components can be treated separately, they still interact via boundary conditions. To conduct a numerical analysis, the article examines scenarios where one crack surface slides parallel to the crack front against another and selects relevant materials with known characteristics. Calculations reveal that, even in the absence of mechanical load, crack surfaces undergo relative sliding due to non-zero electrical or magnetic fields. The study of the shielding effect, which mitigates crack motion by altering magnetic field distribution, holds particular significance. This effect reduces the overall stress intensity factor and impedes crack propagation. Conducting such a comprehensive analysis is pivotal for comprehending and foretelling the strength and reliability of structural components composed of piezoelectric and piezomagnetic materials. It is imperative to conduct a thorough examination of the mechanisms governing their failure.

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.

Couple stress-based flexoelectricity of frictionless contact in dielectrics

The presence of flexoelectricity in dielectric materials can affect the mechanics of contact between solid bodies, especially when viewed at the micro/nano-scale. Such an effect can also contribute significantly to contact electrification ─ a charge transfer event that takes place at the contact interface. In particular, the state of deformation and stresses in the vicinity of the contacting area can differ markedly from that anticipated from classical elastic Hertzian contact theory. In the present work, we investigate the flexoelectric effect near the contacting area of an isotropic dielectric material pressed by a frictionless rigid insulator probe, serving as a model problem to reveal the potential role of flexoelectricity in contact mechanics and electrification. The investigation is based on consistent couple stress flexoelectric theory, in which electric polarization may appear due to two-way coupling between the electric field and mean curvature vectors, even for centrosymmetric dielectric materials. To this end, we develop a new boundary element formulation to calculate the displacement, stress, electric potential, and electric displacement for two-dimensional isotropic flexoelectric contact problems. The subsequent computational mechanics investigation highlights several important characteristics of triboelectric phenomena and shows that the state of stress and polarization in contacting isotropic dielectrics is dependent on the effective length scale parameter l of couple stress elasticity and the coupling flexoelectric coefficient f.

On the Contact Problem with Deformable Stamp in the Quarter Plain

In this paper, for the first time, a two-dimensional dynamic contact problem on the action of a deformable stamp on a quarter of the plane of a multilayer medium is strictly mathematically investigated. In contrast to the case of an absolutely solid stamp, a deformable stamp introduces additional features, consisting in the possibility of the occurrence of discrete resonances predicted by academician I.I. Vorovich. The paper shows that the use of a method based on the use of block elements makes it possible to obtain an equation describing resonant frequencies. To study contact problems with a deformable stamp made of materials of complex rheology, including smart materials, it is proposed in the paper to first conduct a study for the case of a deformable stamp made of a material of simple rheology described by Helmholtz equations. Solutions of boundary value problems for stamps of complex rheology, after that, are represented by a combination of solutions of boundary value problems for stamps of simple rheology.

Optimal control of a two-dimensional contact problem with multiple unilateral constraints

In this article, we are concerned with optimal control of a frictionless contact problem with multiple unilateral constraints for a two-dimensional bar. The existence of an optimal trajectory-control pair is firstly proven under the framework of general cost functional. The Pontryagin maximum principle is then established for the investigational system equipped with many equality and inequality constraints in fixed final horizon case, owing to the Dubovitskii and Milyutin functional analytical approach. A remark concludes the article with the discussion, which address the utilization of obtained necessary optimality condition.

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.

Viscoelastic steady-state rolling contacts: A generalized boundary element formulation for conformal and non-conformal geometries

A general Boundary Element methodology is developed to study two-dimensional steady-state viscoelastic circular contact problems. The methodology relies on an ad hoc defined steady-state viscoelastic Green’s function, which accounts for the circular domain, typical, for example, of a mechanical pin joint, and can be employed to manage any real viscoelastic material, characterized by a continuum spectrum of relaxation times. The contact problem can be then easily formulated in the form of a spatial convolution product between surface stresses and displacement, which paves the way to solve contact problems of a countless number of engineering relevant systems, where multiple viscoelastic contacts occur. These include, inter alia, viscoelastic rolling element bearings, and pin joints.

The Problem of Wedge Indenter with Flat-Rounded Bottom Indenting Half-Plane Elastic Body

In this paper, the problem of a wedge indenter with flat-rounded bottom indenting an infinite half-plane elastic body is solved. It can be used to study the penetration force or the surface stress distribution of the cutter when the tool intrudes into an elastic body. First, a first-order continuity method is proposed to eliminate the stress singularity caused by the sharp corners of the wedge indenter with flat bottom. The two-dimensional elastic contact problem is transformed into a Riemann–Hilbert boundary value problem with discontinuous coefficients expressed in the form of complex variables. For the frictionless contact problem, the explicit analytical solution is obtained for contact stress on the half-plane surface. Furthermore, the expression of the stress potential function is also derived to solve the stress field in the half-plane. Verification indicates that the wedge indenter with flat-rounded bottom can degenerate to several well-studied shaped indenters, including the Hertz model and flat-rounded model. The model in this paper is also validated with its numerical counterpart built in the finite element program LS-DYNA. Finally, the sensitivity of different configuration parameters on the normal contact stress and interior stress field in numerical model was investigated.

A singularity analysis in Keer’s elastic indentation problem

A non-axisymmetric three-dimensional frictionless contact problem for an elastic half-space and a cylindrical indenter with the face in a wedge form is considered under the assumption of a circular region of contact. Explicit formulas are derived for the critical indenter displacement and contact force that are needed to ensure full contact over the entire contact region. It is shown that the contact pressure beneath the indenter maintains a logarithmic singularity, which is produced by the wedge edge and is similar to the singularity of the contact pressure in the corresponding two-dimensional contact problem for a wedge rigid indenter.

Two-dimensional frictionless large deformation contact problems using isogeometric analysis and Nitsche’s method

ABSTRACT The simulation of large deformation contact problems has been a tough subject due to the existence of multiple nonlinearities, including geometric nonlinearity and contact interface nonlinearity. In this paper, we develop a novel method to compute the large deformation of 2D frictionless contact by employing Nitsche-based isogeometric analysis. The enforcement of contact constraints as one of the main issues in contact simulation is implemented by using Nitsche’s method, and the node-to-segment scheme is applied to the contact interface discretization. We detailedly derive the discrete formulations for 2D large deformation frictionless contact where NURBS is used for geometrical modeling and the Neo-Hookean hyperelastic materials are applied to describe the deformation of the model. The collocation method with Greville points is employed to integrate the contact interface and the classical Legendre–Gauss quadrature rule is used for contact bodies’ integration. The Lagrange multiplier method and penalty method are also implemented for comparison with Nitsche’s method. Several examples are investigated, and the obtained results are compared with that from commercial software ABAQUS to verify the effectiveness and accuracy of the Nitsche-based isogeometric analysis.

The size effect in adhesive contact on a gradient nanostructured coating

The adhesive contact problem between a rigid cylindrical punch and a gradient nanostructured (GNS) coating is investigated by considering the size effect. The laminated plate model is applied to characterize the material properties of a GNS coating in plane strain couple stress elasticity. By using the Fourier integral transform and transfer matrix method, the governing integral equation(s) for the two-dimensional adhesive contact problem are obtained. Numerically calculated results are presented to analyse the effect of characteristic material length, the adhesion parameter and nonhomogeneous parameters on the mechanical response of the GNS coating for the adhesive contact problem. We explore the nanoscale contact of a GNS coating with shear modulus varying as a function of depth according to an exponential function or the power-law function. The present results provide a way to improve the contact deformation and damage to nanoelectromechanical systems by adjusting the gradient index of the GNS coating.

Contact Analysis Within the Bi-Potential Framework Using Cell-Based Smoothed Finite Element Method

This paper presents a cell-based smoothed finite element method (CS-FEM) for solving two-dimensional contact problems with the bi-potential formulation. The contact force and the relative displacements on the contact surface are coupled with each other. The Uzawa algorithm, which is a local iterative technique, is used to solve the contact force. The classic Coulomb friction rule and a unilateral contact relationship are considered. There is no need to select any user-defined parameter in the whole process. Three contact states are investigated accurately, which can be stated as sticking, separating and sliding, respectively. The CS-FEM is performed with six different kinds of smoothing domains which are constructed by dividing the background element into different regions. Only boundary integrations instead of domain integral are required in the calculation, and no coordinate mapping is needed. Three numerical examples are presented to verify the effectiveness of the method. The effect of the friction coefficient for the contact is also investigated. All the obtained numerical solutions agree well with the reference values. The results produced by the CS-FEM are more accurate than those of the traditional FEM. Moreover, the CS-FEM can provide both an upper bound and a lower bound of the strain energy solutions while using different smoothing domains.

Wear modelling in wheel–rail contact problems based on energy dissipation

ABSTRACT This work is concerned with the numerical procedure to solve two-dimensional wheel-rail contact problem between a rigid wheel and an elastic-plastic rail lying on a rigid foundation. The contact phenomenon includes Coulomb friction, frictional heat generation as well as the wear of the contacting surfaces. The wear depth function appears as an internal variable in the non-penetration condition updating the gap between the worn surfaces of the bodies. Moreover the dissipated energy due to friction is calculated to evaluate the loss of material and to determine the shape of the contacting surfaces during the wear evolution process. This contact problem is solved numerically using the finite element method as well as the operator splitting approach and semi-smooth Newton method. The distribution of contact surface stresses and temperatures as well as the evolution of the shape of the contact surfaces and the wear depth are reported and discussed.