Quasistatic Contact Problem Research Articles

Asymptotic analysis of the substrate effect for an arbitrary indenter

A quasistatic unilateral frictionless contact problem for a rigid axisymmetric indenter pressed into a homogeneous, linearly elastic and transversely isotropic elastic layer bonded to a homogeneous, linearly elastic and transversely isotropic half-space is considered. Using the general solution to the governing integral equation of the axisymmetric contact problem for an isotropic elastic half-space, we derive exact equations for the contact force and the contact radius, which are then approximated under the assumption that the contact radius is sufficiently small compared to the thickness of the elastic layer. An asymptotic analysis of the resulting non-linear algebraic problem corresponding to the fourth-order asymptotic model is performed in the case when the contact radius becomes vanishingly small compared with the layer thickness. A special case of the indentation problem for a blunt punch of power-law profile is studied in detail. Approximate force-displacement relations are obtained in explicit form, which is most suited for development of indentation tests.

Analysis of quasistatic frictional contact problems in nonlinear viscoelasticity with large deformations

A numerical model for the analysis of frictional contact problems in viscoelastic structures, undergoing both material and geometrical nonlinearities is presented. The nonlinear Schapery constitutive viscoelastic model is used for the stress, strain, and time relationships. The constitutive equations are expressed in an incremental form assuming a constant strain rate within each time increment. The hereditary integral is updated at the end of time increment by recursive procedure. The normal contact forces are treated as independent variables. In addition, the regularized Coulomb's friction model is adopted to model the friction throughout the contact interface. The friction resistance is represented by tangential contact stiffness. The Updated Lagrangian approach is adopted to derive the equilibrium equations for the contact systems. The Newton–Raphson iterative scheme is utilized to obtain the converged solution at the end of each time increment. The developed model is verified and compared with the available analytical methods and benchmarks. The applicability of the developed computational incremental iterative scheme is demonstrated by solving two different cases of contact problems with different geometry, loading, and boundary conditions. Numerical results show that material nonlinearity and friction have great effects on viscoelastic contact behavior.

A Generalized Model for Adhesive Contact Between a Rigid Cylinder and a Transversely Isotropic Substrate

In this paper, a solution to the quasi-static adhesive contact problem between a rigid cylinder and a transversely isotropic substrate is extended to the most general case by taking adhesion hysteresis into account. An analytical solution to the contact stress is obtained by solving the integral equations established on the basis of the Green's function for the two-dimensional transversely isotropic half-space problem. By using equilibrium conditions and Griffith's criterion, the adhesion force and resistant moment to rolling are determined as functions of contact geometries and material properties of the contacting solids. Detailed discussions on the adhesion force and resistant moment are presented for some specific cases, revealing adhesion behaviors that have not been predicted by previous models. As the most generalized solution to the discussed problem, our results would have extensive applications in predicting the adhesion behavior between solids undergoing sophisticated mechanical loadings.

Weak solvability of two quasistatic viscoelastic contact problems

We consider two mathematical models, which describe the frictional contact between a deformable body and a foundation. In both models the process is assumed to be quasistatic, the material is viscoelastic, and the friction is given by a subdifferential boundary condition. In the first model the contact is described with a univalued condition between the normal stress and the normal displacement and in the second model it is described with a subdifferential condition, which links the normal stress and the normal velocity. For each model we derive a variational formulation, which is in the form of a history-dependent hemivariational inequality for the velocity field. Then we prove the existence of a weak solution and, under additional assumptions, its uniqueness. The proof is based on a recent result on history-dependent hemivariational inequalities.

Topology optimization of quasistatic contact problems

Topology optimization of quasistatic contact problemsThis paper deals with the formulation of a necessary optimality condition for a topology optimization problem for an elastic contact problem with Tresca friction. In the paper a quasistatic contact model is considered, rather than a stationary one used in the literature. The functional approximating the normal contact stress is chosen as the shape functional. The aim of the topology optimization problem considered is to find the optimal material distribution inside a design domain occupied by the body in unilateral contact with the rigid foundation to obtain the optimally shaped domain for which the normal contact stress along the contact boundary is minimized. The volume of the body is assumed to be bounded. Using the material derivative and asymptotic expansion methods as well as the results concerning the differentiability of solutions to quasistatic variational inequalities, the topological derivative of the shape functional is calculated and a necessary optimality condition is formulated.

Evolution Mixed Variational Inclusions with Optimal Control

Evolution mixed maximal monotone variational inclusions with optimal control, in reflexive Banach spaces, are analized. Solvability analysis is performed on the basis of composition duality principles. Applications to nonlinear diffusion constrained problems, as well as to quasistatic elastoviscoplastic contact problems exemplify the theory.

A numerical solution for contact problem with finite deformation in nonlinear Schapery viscoelastic solids

The present paper presents an incremental finite element model for the solution of the two dimensional quasi-static frictionless nonlinear viscoelastic contact problems with large deformations. The nonlinear viscoelasticity is simulated using the Schapery’s single-integral viscoelastic model with stress-dependent properties. The updated Lagrangian formulation is used to model the material and geometrical nonlinearity of the problem. To eliminate the need to store the entire strain histories, the constitutive equations are transformed into an incremental recursive form. The contact constraints are enforced by using the Lagrange multiplier method where the normal contact forces are treated as independent variables. The resulting nonlinear equilibrium equations are solved by the Newton–Raphson method in an incremental-iterative procedure. The capability of the developed numerical incremental-iterative solution procedure is demonstrated by analysing two different contact problems with different nature.

Numerical Analysis of Quasistatic Contact between Rubber Roller and Rigid Roller

A quasistatic and steady state contact problems of rubber roller and rigid roller is modeled by using a stress function in the form of series. The model is derived from the paper feed unit of duplicating machine. The point interpolation meshfree method (PIM) is applied to obtain the solution of the dual series equations resulting from the boundary condition. Stress and deformation for the rubber roller are obtained based on elasticity theory. Effects of rubber thickness and normal load are observed through the numerical examples. The observation show that the contact zone is changing with the normal load F and the rubber thickness has effect on the stress distribution. In addition, the study shows that the PIM method is an efficient and promising method for simulating the problem.

A Quasistatic Contact Problem for Viscoelastic Materials with Slip‐Dependent Friction and Time Delay

A mathematical model which describes an explicit time‐dependent quasistatic frictional contact problem between a deformable body and a foundation is introduced and studied, in which the contact is bilateral, the friction is modeled with Tresca’s friction law with the friction bound depending on the total slip, and the behavior of the material is described with a viscoelastic constitutive law with time delay. The variational formulation of the mathematical model is given as a quasistatic integro‐differential variational inequality system. Based on arguments of the time‐dependent variational inequality and Banach′s fixed point theorem, an existence and uniqueness of the solution for the quasistatic integro‐differential variational inequality system is proved under some suitable conditions. Furthermore, the behavior of the solution with respect to perturbations of time‐delay term is considered and a convergence result is also given.

Study of a quasistatic contact problem in viscoelasticity

Study of a quasistatic contact problem in viscoelasticity

A class of generalized evolution variational inequalities in Banach spaces

In the present paper, a class of generalized evolution variational inequalities arising in a number of quasistatic frictional contact problems for viscoelastic materials is introduced and studied. Using Banach’s fixed point theorem, the existence and uniqueness theorem of the solution for the generalized evolution variational inequalities is proved under some suitable assumptions.

Quasistatic contact problem for cylindrical joints with minor cutting of the contours of the bodies

We solve a quasistatic contact problem of interaction of cylindrical elements with minor cutting of their contours. We study joints in which single-area or mixed (single-and-double-area) types of contact are realized. The basic regularities of changes in the maximum contact pressure caused by the rotation of a disk with certain degree of ovality in a hole with different types of cutting are established. It is well known that, in the process of manufacturing, all parts of typical cylindrical joints (plain bearings, guides, hinges, etc.) inevitably acquire a certain degree of nonroundness of the contours (ovality, trihedrality, tetrahedrality, etc.). The influence of the indicated nonroundness on the level and distribution of contact pres- sures in different joints with symmetric relative positions of contacting bodies leading to single-area or double- area contact was studied in a series of works (1-4, 8). The solution of the quasistatic problem was obtained in (5-7) for the case where a shaft with small ovality (ellipticity) rotates in an oval hole (bush), which leads to cy- clic changes in the maximum contact pressure. In the present work, we solve the quasistatic problem for joints in which the bush is characterized by the presence of cutting of complex shape.

Finite element approximation to a contact problem for a nonlinear thermoviscoelastic beam

In this paper we propose and analyze a finite element method to the solution of a quasi-static contact problem between a nonlinear beam and a rigid obstacle. Error estimates and energy decay are obtained and some numerical simulations described.

History-dependent quasi-variational inequalities arising in contact mechanics

We consider a class of quasi-variational inequalities arising in a large number of mathematical models, which describe quasi-static processes of contact between a deformable body and an obstacle, the so-called foundation. The novelty lies in the special structure of these inequalities that involve a history-dependent term as well as in the fact that the inequalities are formulated on the unbounded interval of time [0, +∞). We prove an existence and uniqueness result of the solution, then we complete it with a regularity result. The proofs are based on arguments of monotonicity and convexity, combined with a fixed point result obtained in [22]. We also describe a number of quasi-static frictional contact problems in which we model the material's behaviour with an elastic or viscoelastic constitutive law. The contact is modelled with normal compliance, with normal damped response or with the Signorini condition, as well, associated to versions of Coulomb's law of dry friction or to the frictionless condition. We prove that all these models cast in the abstract setting of history-dependent quasi-variational inequalities, with a convenient choice of spaces and operators. Then, we apply the abstract results in order to prove the unique weak solvability of each contact problem.

Numerical analysis of a quasi-static contact problem for a thermoviscoelastic beam

In this paper we revisit a quasi-static contact problem of a thermoviscoelastic beam between two rigid obstacles which was recently studied in [1]. The variational problem leads to a coupled system, composed of an elliptic variational inequality for the vertical displacement and a linear variational equation for the temperature field. Then, its numerical resolution is considered, based on the finite element method to approximate the spatial variable and the implicit Euler scheme to discretize the time derivatives. Error estimates are proved from which, under adequate regularity conditions, the linear convergence is derived. Finally, some numerical simulations are presented to show the accuracy of the algorithm and the behavior of the solution.

Frictional elastic contact with periodic loading

Quasi-static frictional contact problems for bodies of fairly general profile that can be represented as half planes can be solved using an extension of the methods of Ciavarella and Jäger. Here we consider the tangential traction distributions developed when such systems are subjected to loading that varies periodically in time. It is shown that the system reaches a steady state after the first loading cycle. In this state, part of the contact area (the permanent stick zone) experiences no further slip, whereas other points may experience periods of stick, slip and/or separation. We demonstrate that the extent of the permanent stick zone depends only on the periodic loading cycle and is independent of the initial conditions or of any initial transient loading phase. The exact traction distribution in this zone does depend on these factors, but the resultant of these tractions at any instant in the cycle does not. The tractions and slip velocities at all points outside the permanent stick zone are also independent of initial conditions, confirming an earlier conjecture that the frictional energy dissipation per cycle in such systems depends only on the periodic loading cycle. We also show that these parameters remain unchanged if the loading cycle is changed by a time-independent tangential force, provided this is not so large as to precipitate a period of gross slip (sliding).

Approximation and numerical realization of 3D quasistatic contact problems with Coulomb friction

AbstractThis paper deals with the full discretization of quasistatic 3D Signorini problems with local Coulomb friction and a coefficient of friction which may depend on the solution. After a time discretization we obtain a system of static contact problems with Coulomb friction. Each of these problems is solved by the T-FETI domain decomposition method used in auxiliary contact problems with Tresca friction. Numerical experiments show the efficiency of the proposed method.

A finite temperature bridging domain method for MD-FE coupling and application to a contact problem

A direct multiscale method coupling molecular dynamics to finite element simulations is introduced to study the contact area evolution of rough surfaces under normal loading. First, a description of the difficulties due to using the bridging domain method at finite temperatures is discussed. This approach, which works well at low temperatures, is based on a projection, in an overlap region, of the atomic degrees of freedom on the coarser continuum description. It is shown that this leads to the emergence of a strong temperature gradient in the bridging zone. This has motivated the development of a simpler approach suitable for quasi-static contact problems conducted at constant but finite temperatures. This new approach is then applied to the normal loading of rough surfaces, in which the evolution of the real contact area with load is monitored. Surprisingly, the results show little influence of the contact area on temperature. However, the plastic events, in form of atomic reshuffling at the surface and dislocation activity, do clearly depend on temperature. The results show also a strong and temperature-dependent relaxation of the initial rough surfaces. This natural mechanism which alters atomic asperities brings to question the classical atomic description of roughness.

Analysis of thermoviscoelastic frictionless contact of layered bodies

In the design of a contact system it is desirable to reduce the peak value of the contact pressure. This is in turn improving the tribological behavior of the contact system. Layering the contacting bodies by viscoelastic materials can achieve this goal. This is due to the capability of such materials to store and dissipate energy for either stress relaxation or creep condition. The response of such layering contact systems is greatly influenced by the material characteristics of the viscoelastic layers. This paper presents a computational model capable of analyzing quasistatic nonconformal viscoelastic frictionless contact problems of layered bodies with thermal loads. In addition to the capability of detecting the contact configuration, effects of the material characteristics of viscoelastic layers upon this contact configuration are also investigated.

Discontinuity of quasi-static solution in the two-node Coulomb frictional system

A single-node system introduced by Klarbring has provided insight into the non-uniqueness of solution in the quasi-static contact problem at the high coefficient of Coulomb friction. Here, we explore this issue for the two-node system under the slip displacement space in which the instantaneous condition is efficiently represented. In the paper, we identify a qualitatively different failure of the quasi-static evolution algorithm in which a more complex dynamic transition may occur. When the system evolves from the point where both-node discontinuity occurs, the transient evolution behavior involving a damping matrix is explored in order to investigate a final state of the two-node system. It is demonstrated that the final state is uniquely determined which is independent of the damping matrix.