Frictionless Contact Research Articles

Isogeometric frictionless contact analysis with the third medium method

This paper presents an isogeometric formulation for frictionless contact between deformable bodies, based on the recently proposed concept of the third medium. This concept relies on continuum formulations not only for the contacting bodies but also for a fictitious intermediate medium in which the bodies can move and interact. Key to the formulation is a suitable definition of the constitutive behavior of the third medium. In this work, based on a number of numerical tests, the role of the material parameters of the third medium is systematically assessed. We also assess the rate of spatial convergence for higher-order discretizations, stemming from the regularization of the non-smooth contact problem inherent to the third medium approach. Finally, problems with self contact are considered and turn out to be an attractive application of the method.

Nonsmooth modal analysis of an elastic bar subject to a unilateral contact constraint

This contribution proposes a numerical procedure capable of performing nonsmooth modal analysis (mode shapes and corresponding frequencies) of the autonomous wave equation defined on a finite one-dimensional domain with one end subject to a Dirichlet condition and the other end subject to a frictionless time-independent unilateral contact condition. Nonsmooth modes of vibration are defined as one-parameter continuous families of nonsmooth periodic orbits satisfying the local equation together with the linear and nonlinear boundary conditions. The analysis is performed using the wave finite element method, which is a shock-capturing finite volume method. As opposed to the traditional finite element method with time-stepping schemes, potentially discontinuous deformation, stress and velocity wave fronts induced by the unilateral contact condition are here accurately described, which is critical for seeking periodic orbits. Additionally, the proposed strategy introduces neither numerical dispersion nor artificial dissipation of energy, as required for modal analysis. As a consequence of the mixed time–space discretization, no impact law is needed for the well-posedness of the problem in line with the continuous framework. The frequency–energy dependency nonlinear spectrum of vibration, shown in the form of backbone curves, provides valuable insight on the dynamics. In contrast to the linear system (without the unilateral contact condition) whose modes of vibration are standing harmonic waves, the nonsmooth modes of vibrations are traveling waves stemming from the unilateral contact condition. It is also shown that the vibratory resonances of the periodically driven system with light structural damping are well predicted by nonsmooth modal analysis. Furthermore, the initially unstressed and prestressed configurations exhibit stiffening and softening behaviors, respectively, as expected.

Bifurcation of elastic solids with sliding interfaces.

Lubricated sliding contact between soft solids is an interesting topic in biomechanics and for the design of small-scale engineering devices. As a model of this mechanical set-up, two elastic nonlinear solids are considered jointed through a frictionless and bilateral surface, so that continuity of the normal component of the Cauchy traction holds across the surface, but the tangential component is null. Moreover, the displacement can develop only in a way that the bodies in contact do neither detach, nor overlap. Surprisingly, this finite strain problem has not been correctly formulated until now, so this formulation is the objective of the present paper. The incremental equations are shown to be non-trivial and different from previously (and erroneously) employed conditions. In particular, an exclusion condition for bifurcation is derived to show that previous formulations based on frictionless contact or ‘spring-type’ interfacial conditions are not able to predict bifurcations in tension, while experiments—one of which, ad hoc designed, is reported—show that these bifurcations are a reality and become possible when the correct sliding interface model is used. The presented results introduce a methodology for the determination of bifurcations and instabilities occurring during lubricated sliding between soft bodies in contact.

Kinematics of a Holonomic Constraint Rod & Cube System

A loose rod of mass m1 and length l leans against one of the faces of a cube of mass m2 and side length a. The assembly is placed on a horizontal table with one end of the rod touching the table and its other end leaning against the edge of the cube. We set the rod and the center of mass of the cube on the same vertical plane, and then we release the assembly from the rest. For frictionless contacts, we calculate the separation runtime of the rod from the cube as a function of m2/m1 and a/l. This entails forming the equation describing the motion of the system. The equation of motion is analytically unsolvable nonlinear differential equation. Applying a Computer Algebra System, specifically Mathematica [1] [2], we solve the equation numerically. Utilizing the solution, in addition to evaluating the separation runtime, we quantify a list of dynamic quantities, such as the time-dependent interface forces, and, geometric quantities, such as the trajectory of the loose end of the rod. A robust Mathematica code addresses the “what if” scenarios.

Acceleration of Uzawa Method for Frictionless Contact Problems

Acceleration of Uzawa Method for Frictionless Contact Problems

Convergence Results and Optimal Control for a Class of Hemivariational Inequalities

The present paper deals with new results in the study of a class of elliptic hemivariational inequalities in reflexive Banach spaces. We start with an existence and uniqueness result. We complete it with a convergence result which describes the dependence of the solution with respect to the data. To this end, we use the notion of Mosco convergence for the set of constraints. Next, we formulate an optimal control problem for which we prove the existence of optimal pairs and state a convergence result. Finally, we exemplify the use of our results in the study of a two-dimensional boundary value problem which describes the frictionless contact of an elastic body with two reactive foundations.

FEM Design of a Cutting-Edge Support System for Micro-GT

The design of the support system (shaft, bearings, and mechanical coupling devices) of the rotor plays a key role in the development of efficient micro-gas turbines (micro-GTs) for distributed power generation. Foil air bearings are the most widespread technical solution well suited to design a reliable support system, although they cannot withstand a large number of start-stop cycles of the units. In order to overcome such limitation, we have recently proposed an innovative support system that takes advantage of spline couplings and two bearing types (e.g., air and rolling-element bearings). The devised support system employs splines as both convenient coupling systems and actuators for the load partition between the two bearing types. In the present work, the helical spline coupling is studied by means of structural FEM analyses including contact simulation in order to design the support system. Numerical results confirm previous findings in that the load transfer through the spline coupling is mainly a function of the helix angle. In addition, friction factor and structural stiffness cannot be neglected in the accurate design of the spline coupling. Such design parameters are now included in the proposed design procedure, which formerly assumed frictionless contact and rigid bodies.

The ring delamination between bodies under the action of heat sinks distributed along a circle

The frictionless contact an elastic half-space and a rigid thermo-insulated base with a local delamination between them on a ring domain under the action of heat sinks distributed uniformly along a circle and located in the half-space some distance away from its surface, is considered. The corresponding contact thermos-elasticity problem is reduced to a singular integral equation for a height of a ring gap. The solution of the singular integral equation and the internal and external radius of the ring are numerically determined using the method of collocation and the method of successive approximations. The dependence of the form of gap and normal contact stresses on the distance between the heat sinks and the surface of the half-space and the intensity of the heat sink are analyzed.

Stiffness of frictional contact of dissimilar elastic solids

The classic Sneddon relationship between the normal contact stiffness and the contact size is valid for axisymmetric, frictionless contact, in which the two contacting solids are approximated by elastic half-spaces. Deviation from this result critically affects the accuracy of the load and displacement sensing nanoindentation techniques. This paper gives a thorough numerical and analytical investigation of corrections needed to the Sneddon solution when finite Coulomb friction exists between an elastic half-space and a flat-ended rigid punch with circular or noncircular shape. Because of linearity of the Coulomb friction, the correction factor is found to be a function of the friction coefficient, Poisson's ratio, and the contact shape, but independent of the contact size. Two issues are of primary concern in the finite element simulations – adequacy of the mesh near the contact edge and the friction implementation methodology. Although the stick or slip zone sizes are quite different from the penalty or Lagrangian methods, the calculated contact stiffnesses are almost the same and may be considerably larger than those in Sneddon's solution. For circular punch contact, the numerical solutions agree remarkably well with a previous analytical solution. For non-circular punch contact, the results can be represented using the equivalence between the contact problem and bi-material fracture mechanics. The correction factor is found to be a product of that for the circular contact and a multiplicative factor that depends only on the shape of the punch but not on the friction coefficient or Poisson's ratio.

Justification and solvability of dynamical contact problems for generalized Marguerre–von Kármán shallow shells

AbstractIn this work, we consider a three–dimensional dynamical model of unilateral contact problems with Signorini conditions and Coulomb friction laws for nonlinearly elastic shallow shells with a specific class of boundary conditions of generalized Marguerre–von Kármán type. Using technics from formal asymptotic analysis, we show that the scaled three–dimensional solution still leads to a two–dimensional dynamical model with frictionless contact problems. Then, we solve the last problems, using penalization method.

A contact formulation based on a volumetric potential: Application to isogeometric simulations of atrioventricular valves

This work formulates frictionless contact between solid bodies in terms of a repulsive potential energy term and illustrates how numerical integration of the resulting forces is computationally similar to the “pinball algorithm” proposed and studied by Belytschko and collaborators in the 1990s. We thereby arrive at a numerical approach that has both the theoretical advantages of a potential-based formulation and the algorithmic simplicity, computational efficiency, and geometrical versatility of pinball contact. The singular nature of the contact potential requires a specialized nonlinear solver and an adaptive time stepping scheme to ensure reliable convergence of implicit dynamic calculations. We illustrate the effectiveness of this numerical method by simulating several benchmark problems and the structural mechanics of the right atrioventricular (tricuspid) heart valve. Atrioventricular valve closure involves contact between every combination of shell surfaces, edges of shells, and cables, but our formulation handles all contact scenarios in a unified manner. We take advantage of this versatility to demonstrate the effects of chordal rupture on tricuspid valve coaptation behavior.

A self-adaptive projection method for contact problems with the BEM

• A new and simple computational method is developed for the contact problem. • We use the variational method to analyze the convergence of proposed method. • We give the self-adaptive rule and the boundary element approximation for the method. • The performance of the method is demonstrated by the numerical results. A self-adaptive algorithm, based on the projection and boundary integral methods, is designed and analyzed for frictionless contact problems in linear elasticity. Using the equivalence between the contact problem and a variational formulation with a projection fixed point problem of infinite dimensions, we develop an iterative algorithm that formulates the contact boundary condition into a sequence of Robin boundary conditions. In order to improve the performance of the method, we propose a self-adaptive rule which updates the penalty parameter automatically. As the iteration process is given by the displacement and the stress on the boundary of the domain, the unknowns of the problem are computed explicitly by using the boundary element method. Both theoretical results and numerical experiments show that the method presented is efficient and robust.

The effect of bottom boundary condition type on the behavior of adhesive contact of spherical probe on an elastic film

This study presents an investigation on the behavior of adhesive contact between a rigid sphere and an elastic film which is either perfectly bonded (case I) or in frictionless contact (case II) with a rigid substrate. By using linear fracture mechanics, we formulate an convenient semi-analytical approach to develop relations between the applied force, penetration depth and contact radius. Finite element analysis (FEA) is used to verify the relationships. Our results reveal that the interfacial boundary conditions between the film and substrate have distinct effects on the adhesive contact behavior between the sphere and the film. The aim of the present study is to provide an instructive inspiration for controlling adhesion strength of the thin film subject to adhesive contact.

Layer-substrate system with an imperfectly bonded interface: Spring-like condition

Materials having two or more constituents of different mechanical properties are often seen in engineering practice. These constituents are connected through their interfaces where the material displacements and tractions are most likely discontinuous, subjected to one, or more, of the five general interface conditions, i.e., perfectly bonded, dislocation-like, force-like, spring-like, and frictionless contact. This paper focus on the contact problem of a rigid sphere sliding on a layered half-space with a spring-like interface between the layer and the substrate. The frequency-domain analytical solutions for the elastic field caused by surface pressure and shear traction are derived. Three stiffness coefficients are introduced to reflect the relationship between tractions and the difference in displacements at both sides of the interface. Further, a sliding contact problem of a rigid sphere and a layered half-space with the spring-like interface is numerically investigated, and the effects of stiffness coefficients on the contact behaviors are explored. Finally, the two-layer model for a coated surface with an inner layer is compared to the spring-like model. When the inner layer is thin and soft, the pressures, as well as stresses, obtained by both models are similar to each other, and the two models are equivalent.

A primal-dual active set method for solving multi-rigid-body dynamic contact problems

In this work, an active set type method is considered in order to solve a mathematical problem that describes the frictionless dynamic contact of a multi-body rigid system, the so-called nonsmooth contact dynamics (NSCD) problem. Our aim, here, is to present the local treatment of contact conditions by an active set type method dedicated to NSCD and to carry out a comparison with the various well-known methods based on the bipotential theory and the augmented Lagrangian theory. After presenting the mechanical formulation of the NSCD and the resolution of the global problem concerning the equations of motion, we focus on the local level devoted to the resolution of the contact law. Then we detail the numerical treatment of the contact conditions within the framework of the primal-dual active set strategy. Finally, numerical experiments are presented to establish the efficiency of the proposed method by considering the comparison with the other numerical methods.

A Laminated Ring on Elastic Foundation Model and a Feedback Contact Algorithm for Tire-Road Contact Analysis

ABSTRACTThis article proposes (1) a two-dimensional tire model that extends the deformable ring on elastic foundation (REF) model by treating the ring as a laminated beam and (2) a feedback compensation approach to solve the tire-road contact problem as facilitated by the laminated REF model. The internal layer of the laminated ring is formulated using the Timoshenko beam theory that can also be easily regressed to an Euler beam. The external layer of the laminated ring is modeled as a circular beam that primarily takes into account the strain energy contributed by the tire tread in the transverse or radial direction. The elastic foundations are assumed to have a predeformation in the radial direction, which can model tire inflation pressure in pneumatic tires or foundation precompression/pretension for nonpneumatic tires. The analytical solution for the static deformation response of this laminated REF model due to an arbitrary external force is detailed first. Then, a feedback p-controller algorithm that penalizes geometry errors in the contact region is outlined as a unified approach that can be used to solve frictionless contact problems between a tire and arbitrary road profiles. The performances of the proposed model and algorithm are compared against those obtained from a detailed finite element analysis. Both flat surface and cleat contact responses are shown to illustrate the utility of this laminated REF model and the contact algorithm.

Conserving algorithms for frictionless and full stick friction dynamic contact problems using the direct elimination method

SummaryIn this paper, conserving time‐stepping algorithms for frictionless and full stick friction dynamic contact problems are presented. Time integration algorithms for frictionless and full stick friction dynamic contact problems have been designed to preserve the conservation of key discrete properties satisfied at the continuum level. Energy and energy‐momentum–preserving algorithms for frictionless and full stick friction dynamic contact problems, respectively, have been designed and implemented within the framework of the direct elimination method, avoiding the drawbacks linked to the use of penalty‐based or Lagrange multipliers methods. An assessment of the performance of the resulting formulation is shown in a number of selected and representative numerical examples, under full stick friction and slip frictionless contact conditions. Conservation of key discrete properties exhibited by the time‐stepping algorithm is shown.

Short note: Method of Dimensionality Reduction for compressible viscoelastic media. I. Frictionless normal contact of a Kelvin‐Voigt solid

AbstractBased on the Method of Functional Equations by Lee and Radok we present the application of the Method of Dimensionality Reduction to frictionless axisymmetric normal contacts with a compressible Kelvin‐Voigt material. Both the contact between a rigid indenter and a viscoelastic medium and contacts between two viscoelastic bodies are studied. All mapping rules are given, strictly proven and illustrated by examples.

Analysis of the partially closed interface crack in orthotropic materials

The determination of fracture parameters such as mode I and II stress intensity factors, KI and KII, is of considerable practical importance in the analysis and failure prediction of various types of structures. For example, interfacial cracks may develop and propagate in bonded joints or lead to delaminations in composite materials. The finite element method is a powerful technique for analyzing complex structures; however, the singular behavior near crack tips is poorly approximated by conventional finite elements and convergence theorems cease to be applicable at those locations. In a previous paper, a combined analytical and numerical model based on frictionless crack face contact was used to obtain fracture parameters at interfacial cracks in composite laminates in cylindrical bending and subject to transverse loads. In this article, we further consider the method and extend the previous results by considering a variety of shear and normal loads on a cracked bimaterial plate with isotropic or orthotropic materials. Furthermore, additional validation of the method is provided by comparing the results with closed form solutions in the literature for the special case of isotropic materials in a combined tensile and shear field, and good agreement is found. Overall, we conclude that the method provides a practical way to analyze a variety of engineering structures with interfacial cracks.

Contact of two equal rigid pulleys with a belt modelled as Cosserat nonlinear elastic rod

The setting of a looped drive belt on two equal pulleys is considered. The belt is modelled as a Cosserat rod, and a geometrically nonlinear model with account for tension and transverse shear is applied. The pulleys are considered as rigid bodies, and the belt–pulley contact is assumed to be frictionless. The problem has two axes of symmetry; therefore, the boundary value problem for the system of ordinary differential equations is formulated and solved for a quarter of the belt. The considered part consists of two segments, which are the free segment without the loading and the contact segment with the full frictionless contact. The introduction of a dimensionless material coordinate at both segments leads to a ninth-order system of ordinary differential equations. The boundary value problem for this system is solved numerically by the shooting method and finite difference method. As a result, the belt shape including the rotation angle, forces, moments, and the contact pressure are determined. The contact pressure increases near the end point of the contact area; however, no concentrated contact force occurs.