Mixed finite element analysis of semi-coercive unilateral contact problems with given friction

A unilateral contact 2D-problem is considered provided one of two elastic bodies can shift in a given direction as a rigid body. Using Lagrange multipliers for both normal and tangential constraints on the contact interface, we introduce a saddle point problem and prove its unique solvability. We discretize the problem by a standard finite element method and prove a convergence of approximations. We propose a numerical realization on the basis of an auxiliary “ bolted” problem and the algorithm of Uzawa.     

The problem of the contact between a linear elastic body and elastic and rigid bodies (a variational approach)

The problem of the contact between a linear elastic body and a rigid body is formulated as a one-sided problem. The solution is determined from the variational inequality, equivalent to the problem of minimizing the energy functional in a set of allowable displacements. The regularity of the solution is established down to internal points of the contact boundary. A measure is constructed in the subsets of the contact boundary that enables the effect of a stamp on an elastic body to be characterized. The absolute continuity of this measure is proved at the internal point. The problem of the contact of two elastic bodies is examined in a similar formulation. The regularity of the solution is established and the nature of the effect of one body on the other is clarified.     

The Signorini problem with Coulomb friction for a hyperelastic body under finite deformations

The quasi-static three-dimensional problem of elasticity theory for a hyperelastic body under finite deformations, loading by bulk and surface forces, partial fastening and unilateral contact with a rigid punch and in the presence of time-dependent anisotropic Coulomb friction is considered. The equivalent variational formulation contains a quasi-variational inequality. After time discretization and application of the iteration method, the problem arising with “specified” friction is reduced to a non-convex miniumum functional problem, which is studied by Ball's scheme. The operator in contact stress space is determined. It is shown that a threshold level of the coefficient of friction corresponds to each level of loading, below which there is at least one fixed point of the operator. If the solution at a certain instant of time is known, the iteration process converges to the solution of the problem at the next, fairly close instant of time.     

A Holistic Simulation Approach from a Measured Load to Element Stress Using Combined Multi-body Simulation and Finite Element Modelling

The design of vehicle bodies requires the knowledge of the vehicle's structural response to external loads and disturbances. In rigid multi-body simulation the dynamic behaviour of complex systems is calculated with rigid bodies and neglect of body elasticity. On the other hand, in finite element models large degree of freedom numbers are used to represent the elastic properties of a single body. Both simulation methods can be combined, if the finite element model size is reduced to a degree of freedom number feasible to multi-body simulation. The application to practical purposes requires the use and interconnection of several different software tools. In this contribution a holistic method is presented, which starts with the measurement or synthesis of loads and excitations, continues with the integration of a reduced finite element model into a multi-body system, the dynamic response calculation of this combined model, and concludes with the result expansion to the full finite element model for calculating strain and stress values at any point of the finite element mesh. The applied software tools are Simpack, Nastran, and Matlab. An example is given with a railway vehicle simulated on measured track geometry. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Rolling Contact Problem with the Generalized Coulomb Friction

This paper deals with the numerical solution of the wheel - rail rolling contact problems. The unilateral dynamic contact problem between a rigid wheel and a viscoelastic rail lying on a rigid foundation is considered. The contact with the generalized Coulomb friction law occurs at a portion of the boundary of the contacting bodies. The Coulomb friction model where the friction coefficient is assumed to be Lipschitz continuous function of the sliding velocity is assumed. Moreover Archard's law of wear in the contact zone is assumed. This contact problem is governed by the evolutionary variational inequality of the second order. Finite difference and finite element methods are used to discretize this dynamic contact problem. Numerical examples are provided. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Investigation of a Stressed State of Shells of Revolution Contacting with Bodies under the Action of Pulsating Loads

We propose an approach to determination of a stressed state of shells of revolution contacting with rigid bodies under the action of loads periodically varying with time and consider two variants of contact interaction: bilateral and unilateral connections of a shell with a rigid body. A stressed state of the shell on a rigid support is investigated.     

The Maxwell-Contact

In multibody systems, two alternative approaches exist to compute contact forces between bodies. The unilateral constraint contact and the unilateral regularized contact do not take into account the influence of the deformation due to other contacts on a body using rigid body dynamics. In this paper, a third alternative, i.e. the Maxwell-Contact, is derived coupling the deformations of different contacts on one body quasi-statically. An academic example validates the fundamental properties of the contact model and the application in a simulation of a pushbelt CVT shows improved results. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The topology of the zero-level lines of the components of the acceleration vector in subsonic flows

The steady subsonic flow past bodies of finite dimensions, when the stream is unbounded and uniform at infinity is considered. The structure formed by the stationary points (points where both components of the acceleration vector vanishes), by the zero-level of the components of the acceleration vector emerging from them and the body past which the flow occurs is studied. It is shown that each of the above-mentioned lines must reach the surface of the body past which the flow takes place. This fact, in particular, enables one to estimate the overall number of streamlines with zero curvature emerging from the stationary points in terms of the number of zeros of the curvature of the streamlines on the body around which the flow takes place, including the branch points of a dividing streamline. With a view to refining the above mentioned number of zeros, the known solution for the neighbourhoods of the branch points of a streamline is considered and the singularity of the flow in the neighbourhoods of points of discontinuity of the curvature of the wall around which the flow occurs is investigated. In order to illustrate the above, certain properties of the flow past convex bodies are refined and a fairly broad class of so-called convex-concave bodies with zero angle of tapering of the trailing edge is constructed and considered. It is shown that, for this body, there are not more than four zeros of the curvature of the streamline and, as a consequence, there are no branch points of the isobars and isoclines in the flow field, including at infinity, an infinitely distant point is the sole stationary point and, most important of all, in the case of the flow past the given bodies the values of the circulation and the lifting force cannot vanish. The mathematical apparatus employed is based on the equations of gas dynamics constructed earlier for certain combinations of the components of the acceleration vector.     

Regularization of 2D frictional contacts for rigid body dynamics

Classic rigid body mechanics does not provide frictional forces acting in a 2D contact interface between two bodies during sticking. This is due to the statical undeterminacy related with this problem. Many technical systems, e.g. disk clutches, have such surface-to-surface contacts and it is sometimes desirable to treat them as rigid body systems despite the 2D contact. Alternatively it is possible to model the systems using elastic instead of rigid bodies, but this might lead to certain drawbacks. Here a new regularization model of such 2D contacts between rigid bodies is proposed. It is derived from a material model for elasto-plasticity in continuum mechanics. Only dry friction is taken into account. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Axisymmetric torsional fretting contact between a spherical punch and an FGPM coating

The axisymmetric torsional fretting contact between a rigid conducting spherical punch and a functionally graded piezoelectric material (FGPM) coating is studied in this paper. The exponential model is used to simulate the inhomogeneous electro-mechanical properties of the FGPMs coating. The conducting spherical punch with a constant surface electric potential is considered in the contact. A normal force and a cyclic torque are applied to the two contact bodies. The applied torque produces an outer annular slip area and an inner stick area. The torsion angle is produced within the inner stick area as a rigid body. With the aid of the Hankel integral transform technique, we can reduce the contact problem to the singular integral equations of the Cauchy type. Then the unknown electro-mechanical fields and stick/slip area can be obtained numerically. The effect of the gradient index on the surface electro-mechanical fields is discussed at loading and unloading phases. The Mises stress and principal stress at the contact surface are also discussed to predict the possible location of fretting damage and failure.     

Hybrid simulation strategy for multiple planar collisions with changing topologies and local deformation

The development of a hybrid strategy for the simulation of multiple plastic-elastic collisions is presented. The strategy attempts to bridge the gap between finite element methods (FEM), which typically require excessively long computation times for multiple impact simulations, and lumped parameter approaches that cannot provide accurate local deformation information. The proposed strategy employs a finite element routine solely to simulate the impact phase, thereby obtaining detailed local deformation information. The simulation of the flight phase between impacts, however, proceeds under rigid body dynamics, resulting in significant reduction in computation time. The transfer of control between FEM and rigid body dynamics is automatic and the points of contact need not be known a priori. The progressive object internal plastic strain, determined from FEM, is retained from one impact to the next, thereby ensuring a certain degree of continuity of the physical properties of the body. An example is presented to demonstrate the efficacy of this approach.     

The contact problem with the bulk application of intermolecular interaction forces (a refined formulation)

A refined formulation of the contact problem when there are intermolecular interaction forces between the contacting bodies is considered. Unlike the traditional formulation, it is assumed that these forces are applied to points within the body, rather than to the surface of the deformable body as a contact pressure, and that the body surface is load-free. Solutions of the contact problems for a thin elastic layer attached to an absolutely rigid substrate and for an elastic half-space are analysed. The refined and traditional formulations of the problem when there is intermolecular interaction are compared. ©2013     

Generation of surface waves through friction: FEM investigation

Sliding friction between two bodies can generate elastic vibration. This study uses a finite-element model comprising an elastic body sliding against a flat rigid surface with constant coefficient of friction. For the elastic body a structured topography is taken into account. The model shows traveling surface waves, which depend on the asperities of the sliding surface. It can be shown that the surface structure and its inertia are the cause for elastic waves in the contact region. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Implementation of the fixed point method in contact problems with Coulomb friction based on a dual splitting type technique

The paper deals with the numerical solution of the quasi-variational inequality describing the equilibrium of an elastic body in contact with a rigid foundation under Coulomb friction. After a discretization of the problem by mixed finite elements, the duality approach is exploited to reduce the problem to a sequence of quadratic programming problems with box constraints, so that efficient recently proposed algorithms may be applied. A new variant of this method is presented. It combines fixed point with block Gauss–Seidel iterations. The method may be also considered as a new implementation of fixed point iterations for a sequence of problems with given friction. Results of numerical experiments are given showing that the resulting algorithm may be much faster than the original fixed point method and its efficiency is comparable with the solution of frictionless contact problems.     

A Lagrangian formulation for steady three-dimensional Newton-Busemann flow

A new Lagrangian formulation for steady three dimensional inviscid flow over rigid bodies is developed. First, the continuity equation is eliminated by the use of two stream functions. This is followed by a transformation to new independent variables, two of which are these stream functions and the third one is a Lagrangian time distinct from the Eulerian time. This Lagrangian formulation with the use of Lagrangian time requires only three independent variables and allows the free boundary problem of flow with shock wave to be rendered a fixed boundary one thereby making it easier to solve. In the Newtonian limit the governing equations reduce to the geodesic equations of the body surface. For flow past two-dimensional bodies, bodies of revolution, and conical bodies and wings, the problems are solved to within quadrature. All known Newtonian flow solutions are found to be special cases of the present theory.     

Coupled motion of a rigid body and point vortices on a two-dimensional spherical surface

The paper is concerned with a class of problems which involves the dynamical interaction of a rigid body with point vortices on the surface of a two-dimensional sphere. The general approach to the 2D hydrodynamics is further developed. The problem of motion of a dynamically symmetric circular body interacting with a single vortex is shown to be integrable. Mass vortices on S ² are introduced and the related issues (such as equations of motion, integrability, partial solutions, etc.) are discussed. This paper is a natural progression of the author’s previous research on interaction of rigid bodies and point vortices in a plane.     

The mathematical theory of growing bodies. Finite deformations

The fundamentals of the mathematical theory of accreting bodies for finite deformations are explained using the concept of the bundle of a differentiable manifold that enables one to construct a clear classification of the accretion processes. One of the possible types of accretion, as due to the continuous addition of stressed material surfaces to a three-dimensional body, is considered. The complete system of equations of the mechanics of accreting bodies is presented. Unlike in problems for bodies of constant composition, the tensor field of the incompatible distortion, which can be found from the equilibrium condition for the boundary of growth, that is, a material surface in contact with a deformable three-dimensional body, enters into these equations. Generally speaking, a growing body does not have a stress-free configuration in three-dimensional Euclidean space. However, there is such a configuration on a certain three-dimensional manifold with a non-Euclidean affine connectedness caused by a non-zero torsion tensor that is a measure of the incompatibility of the deformation of the growing body. Mathematical models of the stress-strain state of a growing body are therefore found to be equivalent to the models of bodies with a continuous distribution of the dislocations.     

On the contact of a rigid sphere and a thin plate

We consider the problem of the contact between a rigid sphere and a thin initially flat plate. After reviewing some plate theory, we establish that a deformation where a finite piece of the plate takes the shape of the sphere is physically unrealisable, and that the contact region must be a ring. However, for both small deflections using classical linear elastic theory and large deflections using von Kármán theory, looking at some typical parameter values we find that the radius of the ring is so small that for practical purposes it should be considered as a point load.     

On equilibrium in the presence of friction I

A known equilibrium rule that refers to rigid bodies possibly subject to some external constraintsK _(e), and that is essential in many cases where friction is present, is shown to give unacceptable results in some cases that can be carried out experimentally. Then the same rule is replaced by a new one which is more complex. When the bearings that carry outK _(e) are kinematically independent in a suitable sense, the new rule is proved to simplify into the old one. The afore-mentioned cases can be treated satisfactorily by means of another simple rule. However this rule is shown to be too weak to treat other (simple) cases.