Unilateral Frictional Contact Research Articles

An Improved Lyapunov Stability Test of Equilibria Under Frictional Unilateral Contact by Sums of Squares Programming

An Improved Lyapunov Stability Test of Equilibria Under Frictional Unilateral Contact by Sums of Squares Programming

A general differential quasi variational–hemivariational inequality: Well-posedness and application

In the paper we analyze a general differential quasi variational-hemivariational inequality in Banach spaces which couples the Cauchy problem for an ordinary differential equation with a quasi variational-hemivariational inequality considered with a unilateral constraint and history-dependent operators. The latter appear in all of the data: the governing nonlinear operator, the generalized directional derivative of a locally Lipschitz potential, a convex potential, and the ordinary differential equation. The results on the well-posedness and regularity of solutions are proved by exploiting a fixed point approach. The theory is illustrated by an application to a quasistatic generalized Signorini unilateral frictional contact problem for viscoplastic materials with a nonsmooth multivalued contact condition.

Unilateral frictional contact between a rigid wheel traversing on a flexible beam: An analytical investigation

This paper proposes a non-linear dynamic solver to simulate the dynamics of a rigid wheel traversing over a flexible beam involving a unilateral contact at the interface. Euler’s discretization scheme is extended incorporating Linear Complementary approach to perform explicit integration over time domain. The proposed solver is analytically designed to incorporate frictional force acting at the wheel-beam interface along with the possibility of generation of normal gap between the two bodies. Moreover, rotation of the wheel is also appended in the proposed algorithm and change in contact point between the wheel and beam is evaluated at every time step. After validation with existing literature, the solver is applied to study various dynamic responses of wheel and beam. Further, the technique is applied to moving oscillator model traversing over a flexible beam. The effect of initial wheel velocity, coefficient of friction, frequency of the oscillator on overall dynamic response of beam and wheel have been thoroughly investigated. The proposed algorithm delivers an exact solution based on the system’s fundamental mechanics, thus reducing the mesh reliance which amounts to high computational cost in finite element solver. The suggested method provides a broad future potential for solving rigid-flexible contact multibody dynamic problems.

On a frictional unilateral contact problem in nonlinear elasticity-existence and smoothing procedure.

This note is devoted to a novel frictional unilateral contact problem in finite-strain elasticity. Here, we adopt the Tresca friction model from linear elasticity. Our analysis relies on the polyconvexity approach to nonlinear elasticity due to J. Ball. We include the delicate case where the elastic body neither is fixed nor has a deformation prescribed along some part of its boundary, but rests on a rigid foundation with a free boundary and is only submitted to forces and loads acting in the interior (like gravity) and at the boundary, respectively. This leads to a loss of coercivity and necessitates an extra condition that prevents the body from escaping by the geometry of the obstacle. This new condition extends a similar condition of Ciarlet and Nečas from the frictionless case to the case of Tresca friction. In addition, as a first step towards a numerical treatment of such nonlinear problems, we present a smoothing procedure that tackles the non-smooth term from Tresca friction and provide a convergence result for the novel smoothing method. This article is part of the theme issue 'Non-smooth variational problems and applications'.

On the parametric elliptical variational-hemivariational inequality problem with applications

The goal of this paper is to introduce a new class of parametric elliptical variational-hemivariational inequality problems together with parametric inverse problems. We show the existence of solution for parametric inverse problem and well-posedness of parametric elliptical variational-hemivariationl inequality problems. An implementation in nonlinear elasticity, we address the parametric inverse problem for a parametric frictional unilateral contact problem.

Experimental Verification of Stability Theory for a Planar Rigid Body With Two Unilateral Frictional Contacts

Stability of equilibrium states in mechanical systems with multiple unilateral frictional contacts is an important practical requirement, with high relevance for robotic applications. In our previous article, we theoretically analyzed finite-time Lyapunov stability for a minimal model of planar rigid body with two frictional point contacts. Assuming inelastic impacts and Coulomb friction, conditions for stability and instability of an equilibrium configuration have been derived. In this article, we present for the first time an experimental demonstration of this stability theory, using a variable-structure rigid “biped” with frictional footpads on an inclined plane. By changing the biped's center-of-mass location, we attain different equilibrium states, which respond to small perturbations by divergence or convergence, showing remarkable agreement with the predictions of the stability theory. Using high-speed recording of video movies, good quantitative agreement between experiments and numerical simulations is obtained, and limitations of the rigid-body model and inelastic impact assumptions are also studied. The results prove the utility and practical value of our stability theory.

Rheology of soft granular materials: uniaxial compression

Soft granular materials are assemblies of highly deformable grains interacting via surface forces. The large grain deformations of these materials differ them from hard granular systems, in which, their behaviors are essentially governed by grain rearrangements. In this paper, we study the uniaxial compression of soft granular materials using a numerical approach based on the Material Point Method allowing for large grain deformations, coupled with the Contact Dynamics method for the treatment of unilateral frictional contacts between grains. Considering the neo-Hookean and elasto-plastic grains, the compaction of 2D soft granular packings is analyzed. We focus essentially on the evolution of the packing vertical stress as a function of the packing fraction and the predictive models are proposed.

Variational-hemivariational inverse problem for electro-elastic unilateral frictional contact problem

Abstract In the present paper, we study inverse problems for a class of nonlinear hemivariational inequalities. We prove the existence and uniqueness of a solution to inverse problems. Finally, we introduce an inverse problem for an electro-elastic frictional contact problem to illustrate our results.

Investigation of the effective wear of enamel accounting for microstructure and phase properties

The effective wear behavior of enamel is investigated by considering the microstructure of enamel and the wear properties of the rod and interrod phases forming it. This investigation is guided by a recently proposed micromechanics theory of wear and realized by using the finite element method. A representative surface element is determined for the unilateral frictional contact with wear between enamel and a ball assimilated to food. Accounting for the elastic properties and wear characteristics of the rod and interrod phase, using different friction coefficients and considering sliding directions, the effective (or overall) wear coefficient of enamel is quantitatively determined for different inclination angles of the rod phase. The results of the present work suggests the existence of an optimal inclination angle of the rod phase for reducing the effective wear of enamel, which can be inspiring for optimizing the fiber orientation for a better wear-resistant fiber-reinforced composites.

Variational-hemivariational inverse problems for unilateral frictional contact

ABSTRACTThe paper investigates inverse problems for a class of nonlinear elliptic variational-hemivariational inequalities. We prove results on the well posedness of a variational-hemivariational inequality and on the existence of solution to inverse problems. We illustrate our findings by an inverse problem for a frictional unilateral contact problem in nonlinear elasticity.

A comprehensive dynamic model for analysing the vibrational behaviour of a CANDU fuel string

Inside a pressure tube, under a large drag load induced by the coolant flow, CANDU fuel bundles form a coherent structure called fuel string. The presence of a large number of unilateral frictional contact (UFC) constraints makes the dynamic behaviour of the fuel string very complex and highly nonlinear. In this paper, a comprehensive dynamic model for handling a large number of UFC constraints is presented. The complex fuel string problem is formulated to be a Linear Complementarity Problem (LCP). Use of the LCP eliminates the need for iteration and satisfies all the UFC constraints in the whole fuel string simultaneously. The computational power available these days allowed us to handle all the constraints among fuel rods, between rods and pressure tube and between neighbouring bundles. Finite element method was employed in order to discretize the fuel rods and endplates. The systems of equations of motion were then discretized in the time domain using Bozzak-Newmark integration scheme. This numerical model can be used to find the dynamic response of a fuel string to any given excitation including unsteady fluid forces or acoustic pressure pulsations. In this study, the response of a 12-bundle fuel string to gravity and harmonic excitations was found; all frictional contact forces everywhere along the string are obtained and can be used to predict fretting and material loss. The numerical model presented in this study can be used to improve new fuel designs, mitigate and decrease damage, develop new safety standards and handle other engineering problems where large numbers of UFC's is a challenge, like other types of nuclear fuel rods and heat exchanger tubes.

On Weight-Prioritized Multitask Control of Humanoid Robots

We propose a formal analysis with some theoretical properties of weight-prioritized multitask inverse-dynamics-like control of humanoid robots, being a case of redundant “manipulators” with a nonactuated free-floating base and multiple unilateral frictional contacts with the environment. The controller builds on a weighted-sum scalarization of a multiobjective optimization problem under equality and inequality constraints, which appears as a straightforward solution to account for state and control input viability constraint characteristic of humanoid robots that were usually absent from early existing pseudoinverse and null-space projection-based prioritized multitask approaches. We argue that our formulation is indeed well founded and justified from a theoretical standpoint, and we propose an analysis of some stability properties of the approach. Lyapunov stability is demonstrated for the closed-loop dynamical system that we analytically derive in the unconstrained multiobjective optimization case. Stability in terms of solution existence, uniqueness, continuity, and robustness to perturbations is then formally demonstrated for the constrained quadratic program.

Hemivariational inequalities modeling electro-elastic unilateral frictional contact problem

We study a class of abstract hemivariational inequalities in a reflexive Banach space. For this class, using the theory of multivalued pseudomonotone mappings and a fixed-point argument, we provide a result on the existence and uniqueness of the solution. Next, we investigate a static frictional contact problem with unilateral constraints between a piezoelastic body and a conductive foundation. The contact, friction and electrical conductivity condition on the contact surface are described with the Clarke generalized subgradient multivalued boundary relations. We derive the variational formulation of the contact problem which is a coupled system of two hemivariational inequalities. Finally, for such system we apply our abstract result and prove its unique weak solvability.

On the coupling of local multilevel mesh refinement and ZZ methods for unilateral frictional contact problems in elastostatics

This paper addresses an efficient and robust automatic adaptive local multilevel mesh refinement strategy for unilateral frictional contact problems in elastostatics. The proposed strategy couples the Local Defect Correction multigrid method LDC (Hackbusch, 1984) with the ZZ (Zienkiewicz and Zhu, 1987) a posteriori error estimator. An extension of LDC method to frictional contact problems is introduced. An interesting feature of this extended LDC algorithm is that it still only lies on interpolations of displacement fields. Neither forces conservation nor exchange of contact status is required between the refinement levels. The ZZ a posteriori error estimator is exploited to automatically build the sub-grids of the LDC method. A criterion linked to the relative error in stress is used. The efficiency of the proposed strategy is analyzed on examples derived from nuclear engineering. Practical numerical choices are proposed and justified. The refinement process automatically varies and stops with respect to a given tolerance. Post-treatments show that the sub-grids focus around the contact areas and that the converged LDC solution always respects the prescribed tolerance.

Alternating direction method of multiplier for a unilateral contact problem in electro-elastostatics

We study an alternating direction method of multiplier (ADMM) applied to a unilateral frictional contact problem between an electro-elastic material and an electrically non conductive foundation. The frictional contact is modeled by the Tresca friction law. The resulting coupled problem is non symmetric and non coercive. By eliminating the electric potential, we obtain a symmetric and coercive problem which can be reformulated as a convex minimization problem. We then apply an alternating direction method of multiplier for the numerical approximation. To avoid explicit matrices inverse (due to the elimination of the electric potential) we use a preconditioned conjugate gradient algorithm as an inner solver. Numerical experiments are proposed to illustrate the efficiency of the proposed approach.

Model-based control of dynamic frictional contact problems using the example of hot rolling

We consider hot forming processes, in which a metal solid body is deformed by several rolls in order to obtain a desired final shape. To minimize cutting scrap and to ensure that this shape satisfies the required tolerances as precisely as possible, we formulate an optimal control problem where we use the trajectories of the rolls as control functions. The deformation of the solid body is described through the basic equations of nonlinear continuum mechanics, which are here coupled with an elasto-viscoplastic material model based on a multiplicative split of the deformation gradient. We assume that the deformations of the rolls can be neglected, thus we add unilateral frictional contact boundary conditions, resulting in an evolutionary quasi-variational inclusion. The associated control-to-observation map is non-differentiable due to changes of state between elastic and plastic material behavior, contact and separation and stick and slip motion, yet we still want to apply gradient-based methods to solve the optimal control problem and therefore have to make sure that derivatives of cost functional and constraints exist. To resolve this issue, we first regularize all non-differentiabilities and subsequently apply the direct differentiation method to obtain sensitivity information. Finally, we formulate a suitable algorithm and discuss numerical results for a real-world example to illustrate its capability.

Non-smooth model and numerical analysis of a friction driven structure for piezoelectric motors

In this contribution, typical friction driven structures are summarized and presented considering the mechanical structures and operation principles of different types of piezoelectric motors. A two degree-of-freedom dynamic model with one unilateral frictional contact is built for one of the friction driven structures. Different contact regimes and the transitions between them are identified and analyzed. Numerical simulations are conducted to find out different operation modes of the system concerning the sequence of contact regimes in one steady state period. The influences of parameters on the operation modes and corresponding steady state characteristics are also explored. Some advice are then given in terms of the design of friction driven structures and piezoelectric motors.

Analysis of dense packing of highly deformed grains

This paper concerns modeling of soft granular materials in which the grains are highly deformable. In order to simulate these materials, an approach based on an implicit formulation of the Material Point Method in the context of the finite strain theory, allowing for large deformations of grains, coupled with the Contact Dynamics method for the treatment of unilateral frictional contacts between grains, is proposed. In this context, the Mooney-Rivlin constitutive relationship is applied with two different set of elastic parameters. Considering these two material behaviors, a uniaxial compression of 2D soft granular packings is analyzed. The stress-strain relation and the evolution of the packing fraction as well as of the connectivity of the grains are discussed.

Lagrange Multipliers and mixed formulations for some inequality constrained variational inequalities and some nonsmooth unilateral problems

This paper addresses a class of inequality constrained variational inequalities and nonsmooth unilateral variational problems. We present mixed formulations arising from Lagrange multipliers. First we treat in a reflexive Banach space setting the canonical case of a variational inequality that has as essential ingredients a bilinear form and a non-differentiable sublinear, hence convex functional and linear inequality constraints defined by a convex cone. We extend the famous Brezzi splitting theorem that originally covers saddle point problems with equality constraints, only, to these nonsmooth problems and obtain independent Lagrange multipliers in the subdifferential of the convex functional and in the ordering cone of the inequality constraints. For illustration of the theory we provide and investigate an example of a scalar nonsmooth boundary value problem that models frictional unilateral contact problems in linear elastostatics. Finally we discuss how this approach to mixed formulations can be further extended to variational problems with nonlinear operators and equilibrium problems, and moreover, to hemivariational inequalities.

Dynamics of mechanical systems with two sliding contacts: new facets of Painlevé’s paradox

We investigate the dynamics of finite degree-of-freedom, planar mechanical systems with multiple sliding, unilateral frictional point contacts. A complete classification of systems with 2 sliding contacts is given. The contact-mode based approach of rigid body mechanics is combined with linear stability analysis using a compliant contact model to determine the feasibility and the stability of every possible contact mode in each class. Special forms of non-stationary contact dynamics including "impact without collision" and "reverse chattering" are also investigated. Many types of solution inconsistency and the indeterminacy are identified and new phenomena related to Painlev\'e"s non-existence and non-uniqueness paradoxes are discovered. Among others, we show that the non-existence paradox is not fully resolvable by considering impulsive contact forces. These results contribute to a growing body of evidence that rigid body mechanics cannot be developed into a complete and self-consistent theory in the presence of contacts and friction.