Thermoviscoelastic Contact Research Articles

Error estimates and numerical simulations of a thermoviscoelastic contact problem with damage and long memory

This paper aims to investigate a thermal frictional contact model with damage and long memory effects. We consider a deformable body made of viscoelastic material and assume the process to be dynamic. The material is expected to adhere to the Kelvin–Voigt constitutive law, with damage and thermal effects incorporated. The variational formulation of the model results in a coupled system comprising a history-dependent hemivariational inequality governing the displacement field, a parabolic variational inequality describing the damage field and an evolution equation for the temperature field. In the analysis of this system, we initially introduce a fully discrete scheme, and then concentrate on deriving error estimates of numerical solutions. An optimal order error estimate is attained under some appropriate solution regularity assumptions. At the tail of this manuscript, numerical simulations are provided for the contact problem to validate the theoretical results.

Second order hemivariational inequality driven by evolution differential inclusion to a dynamic thermoviscoelastic contact problem

In this paper we study an abstract system which consists of a hyperbolic hemivariational inequality coupled with a differential evolution inclusion involving a history-dependent operator in Banach spaces. A hybrid iterative system corresponding to the hemivariational inequality is introduced. Combining the Rothe method, a feedback iterative technique, and a surjectivity result for pseudomonotone operators, we establish existence and a priori estimate for solutions to an approximate problem. Next, through a limiting procedure for solutions of the hybrid iterative system, we show the existence of a mild solution to the original problem. Finally, we apply the main results to a dynamic contact problem in thermoviscoelasticity.

A thermoviscoelastic contact problem with friction, damage and wear diffusion

A thermoviscoelastic contact problem with friction, damage and wear diffusion

Numerical analysis of a non-clamped dynamic thermoviscoelastic contact problem

In this work, we analyze a non-clamped dynamic viscoelastic contact problem involving thermal effect. The friction law is described by a non-monotone relation between the tangential stress and the tangential velocity. This leads to a system of second-order inclusion for displacement and a parabolic equation for temperature. We provide a fully discrete approximation of the problem and find optimal error estimates without any smallness assumption on the data. The theoretical result is illustrated by numerical simulations.

A Review on Modelling of Viscoelastic Contact Problems

Approaches to solving viscoelastic problems have received extensive attention in recent decades as viscoelastic materials have been widely applied in various fields. An overview of relevant modelling approaches is provided in the paper. The review starts with a brief introduction of some basic terminologies and theories that are commonly used to describe the contact behaviour of viscoelastic materials. By building up the complexity of contact problems, including dry contact, lubricated contact, thermoviscoelastic contact and non-linear viscoelastic contact, tentative analytical solutions are first introduced as essential milestones. Afterwards, a series of numerical models for the various types of contact problems with and without surface roughness are presented and discussed. Examples, in which computational tools were employed to assist the analysis of viscoelastic components in different fields, are given as case studies to demonstrate that a comprehensive numerical framework is currently being developed to address complex viscoelastic contact problems that are prevalent in real life.

Fully coupled thermo-viscoelastic (TVE) contact modeling of layered materials considering frictional and viscoelastic heating

This paper reports a thermo-viscoelastic (TVE) contact model of layered materials, subjected to frictional and viscoelastic heating. The mathematical formulation includes the frequency-domain solutions for TVE fields, built on velocity-frequency-temperature relationships. The numerical implementation includes the development of a contact-pressure solver, a heat-partition solver, and a temperature-dependent modulus solver. The model is verified with experimental and numerical results. Both rubber coating-elastic substrate and elastic layer-rubber substrate systems are analyzed. The results reveals that, for the former, the rubber-coating temperature rises with increasing velocity, while for the latter, the rubber-substrate temperature becomes lower with increasing elastic-layer thickness. Moreover, the modulus modification due to higher temperatures leads to higher pressures, and that ignoring the velocity effects on modulus overestimates the temperature rise.

Dynamic frictional thermoviscoelastic contact problem with normal compliance and damage

We study a dynamic problem describing the frictional contact between a thermoviscoelastic body and a foundation. The thermoviscoelastic constitutive law includes a damage effect described by the parabolic inclusion with the homogeneous Neumann boundary condition and a temperature effect described by the first order evolution equation. The contact is modeled with normal compliance condition with friction. We present a variational formulation of the problem and establish an existence and uniqueness of the weak solution. The proof is based on parabolic variational inequalities of first and second kind, first order evolutionary variational equations and fixed point arguments.

Analysis of a multidimensional thermoviscoelastic contact problem under the Green–Lindsay theory

In this paper, we investigate the existence, the stability and the numerical approximation of a multidimensional dynamic contact problem modeling the evolution of displacement and temperature in a viscoelastic body that may come into contact with a deformable foundation. The viscoelastic body is assumed to behave according to Kelvin–Voigt constitutive law with added thermal effects under the Green–Lindsay theory. We prove that the presence of viscoelastic terms in the equations provides additional regularity and then an existence and uniqueness result is obtained using the Faedo–Galerkin method. An energy decay property is also shown under the assumption of radial symmetry. Then, a numerical approximation based on the finite element method is proposed. A stability result is proved from which the decay of the discrete energy is deduced. A priori error estimates are shown from which the linear convergence is derived under suitable additional regularity conditions. Finally, some numerical experiments are described to support our results.

Nonsmooth dynamic frictional contact of a thermoviscoelastic body

This paper studies a system of two hemivariational inequalities modeling a dynamic thermoviscoelastic contact problem with general nonmonotone and multivalued subdifferential boundary conditions. Thermal effects are included in the Kelvin–Voigt thermoviscoelastic constitutive law and in the boundary conditions, and so in frictional heat generation, which takes place on the boundary and enters the condition for the temperature. The existence of a weak solution to the problem is established using a recent surjectivity result for differential inclusions associated with pseudomonotone operators.

Numerical analysis of a dynamic bilateral thermoviscoelastic contact problem with nonmonotone friction law

We study a fully dynamic thermoviscoelastic contact problem. The contact is assumed to be bilateral and frictional, where the friction law is described by a nonmonotone relation between the tangential stress and the tangential velocity. Weak formulation of the problem leads to a system of two evolutionary, possibly nonmonotone subdifferential inclusions of parabolic and hyperbolic type, respectively. We study both semidiscrete and fully discrete approximation schemes, and bound the errors of the approximate solutions. Under regularity assumptions imposed on the exact solution, optimal order error estimates are derived for the linear element solution. This theoretical result is illustrated numerically.

Mathematical and numerical approaches to a one-dimensional dynamic thermoviscoelastic contact problem

In this paper, we propose numerical schemes for solving a nonlinear system which consists of a coupled partial differential equations and two conditions, called normal compliance contact condition and Barber’s heat exchange condition. The convergence of numerical trajectories is shown by using a time discretization and passing the limit of the time step size. The uniqueness of the weak solution is proved as well. We derive the extensive form of an energy balance which will be a criterion to examine numerical stability. An example is provided to present and discuss numerical results.

A dynamic thermoviscoelastic contact problem with the second sound effect

This paper deals with a contact problem describing the mechanical and thermal evolution of a damped extensible thermoviscoelastic beam under the Cattaneo law, relating the heat flux to the gradient of the temperature. The beam is rigidly clamped at its left end whereas the right end of the beam moves vertically between reactive stops like a nonlinear spring. Existence and uniqueness of the solution is proved, as well as the exponential decay of the related energy. Then, fully discrete approximations are introduced by using the classical finite element method and the implicit Euler scheme to approximate the spatial variable and to discretize the time derivatives, respectively. An a priori error estimates result is proved, from which the linear convergence of the algorithm is deduced. The case where the two stops are rigid is also studied from the point of view of the existence and longtime behavior of the solutions. Finally, some numerical simulations are presented to demonstrate the accuracy of the approximation and the behavior of the solution.

NONLINEAR ANALYSIS OF FRICTIONAL THERMO-VISCOELASTIC CONTACT PROBLEMS USING FEM

This study presents a numerical finite element model to analyze the response of frictional thermo-viscoelastic contact systems, which experience material and geometrical nonlinearities. Thermo-rheologically complex behavior of the contacting bodies is assumed. The nonlinear viscoelastic constitutive model is expressed by an integral form of a creep function, whose elastic and time-dependent properties vary with stresses and temperatures. Adopting the assumption that the hydrostatic and deviatoric responses are uncoupled, the constitutive equation is expressed in an incremental form, with the hereditary integral updated at the end of each time increment by recursive computation. The Lagrange multiplier approach is applied to incorporate the inequality contact constraints, while friction effect along the contact interface is modeled using a local nonlinear friction law. The material and geometrical nonlinearities are modeled in the framework of the total Lagrangian formulation. The developed nonlinear viscoelastic model is verified using the available benchmarks. The applicability of the developed model is demonstrated by solving two thermo-viscoelastic frictional contact problems with different contact natures. Results show a distinct effect of the thermo-rheological behavior on viscoelastic contact status.

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.

Fractal Geometry-Based Hypergeometric Time Series Solution to the Hereditary Thermal Creep Model for the Contact of Rough Surfaces Using the Kelvin-Voigt Medium

This paper aims at constructing a continuous hereditary creep model for the thermoviscoelastic contact of a rough punch and a smooth surface of a rigid half-space. The used model considers the rough surface as a function of the applied load and temperatures. The material of the rough punch surface is assumed to behave as Kelvin-Voigt viscoelastic material. Such a model uses elastic springs and viscous dashpots in parallel. The fractal-based punch surface is modelled using a deterministic Cantor structure. An asymptotic power law, deduced using approximate iterative relations, is used to express the punch surface creep which is a time-dependent inelastic deformation. The suggested law utilized the hypergeometric time series to relate the variables of creep as a function of remote forces, body temperatures, and time. The model is valid when the approach of punch surface and half space is in the order of the size of the surface roughness. The closed-form results are obtained for selected values of the system parameters; the fractal surface roughness and various material properties. The obtained results show good agreement with published experimental results, and the methodology can be further extended to other structures such as the Kelvin-Voigt medium within electronic circuits and systems.

Theoretical analysis of thermoviscoelastic contact between friction lining and wire rope in mine friction hoists

Serious accidents of mine hoists caused by high-speed sliding between friction lining and wire rope are often seen in coal mines. In order to solve this problem, we analyzed the contact characteristics between friction lining and wire rope. Then we carried out a dynamic mechanical analysis (DMA) to explain the change in mechanical properties of the friction lining as function of temperature and load frequency and found that temperature has a stronger effect on the mechanical properties than the frequency. We used multiple regression analysis to obtain the thermoviscoelastic constitutive relations of the friction lining. As well we derived the analytic solution for the thermoviscoelastic contact radius and pressure by combining the theory of viscoelastic contact mechanics with thermoviscoelastic constitutive relations.

Numerical analysis of dynamic thermoviscoelastic contact with damage of a rod

We describe and analyse a model for a problem of thermoviscoelastic dynamic contact which allows for the general evolution of the material damage. The effects on the mechanical properties of the material due to crack expansion are described by a damage field, which measures the decrease in the load-bearing capacity of the material. The damage process is assumed to be reversible and the microcracks which develop as a result of tension or compression may grow or disappear. The geometric setting is that of a 1D rod which may contact a deformable obstacle. The contact is modelled by the normal compliance condition and the stress–strain constitutive equation is of Kelvin–Voigt type. The model consists of a coupled system of energy–elasticity equations together with a non-linear parabolic inclusion for the damage field. The existence of a local weak solution is established using penalization, a finite element algorithm for the solution is constructed and analysed and the results of numerical simulations based on this algorithm are presented. The simulations illustrate how the size of the normal compliance coefficients, the damage rate coefficients and the applied force affect the character of the evolution of the damage. In particular, cycles of bonding and debonding can occur.

Numerical analysis of a quasistatic thermoviscoelastic frictional contact problem

We study a quasistatic frictional contact problem between a viscoelastic body and a deformable obstacle. A thin lubricant layer is assumed on the contact surface and, then, a normal damped response contact condition considered. Thermal and frictional effects are also taken into account. A fully discrete scheme is proposed, using the finite element method for the spatial approximation and the Euler scheme for discretizing the time derivatives. Error estimates on the solutions are derived and the linear convergence, under suitable regularity hypotheses, is obtained. The scheme was implemented and some numerical examples are included to show the performance of the method.

Error analysis for a finite element approximation of a thermoviscoelastic contact problem

In this article, a finite element approximation, based on a variational inequality, to the solution of a one-dimensional quasi-static Signorini contact problem in linear thermoviscoelasticity is proposed. Stability and error estimates are obtained.

Numerical approximation of a thermoviscoelastic problem to the contact of two rods by a penalty method

AbstractIn this article we analyze a numerical approximation by the finite element method of a system modeling the thermoviscoelastic contact of two rods. The numerical scheme is based on an auxiliary penalized problem. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004