Ahmed G. El-Shafei

An Incremental Adaptive Procedure for Viscoelastic Contact Problems

Contact pressure distribution throughout the contact interface has a vital role on the tribological aspects of the contact systems. Generally, contact of deformable bodies is a nonlinear problem. Viscoelastic materials have a time-dependent response, since both viscous and elastic characteristics depend on time. Such types of materials have the capability of storing and dissipating energy. When at least one of the contacting bodies is made of a viscoelastic material, contact problems become more difficult, and a nonlinear time-dependent contact problem is obtained. The objective of this paper is to develop an incremental adaptive computational model capable of handling quasistatic viscoelastic frictionless contact problems. The Wiechert model, as an effective model capable of describing both creep and relaxation phenomena, is adopted to simulate the linear behavior of viscoelastic materials. The resulting constitutive integral equations are linearized and, therefore, complications that arise during the direct integration of these equations, specially with contact problems, are avoided. In addition, the incremental convex programming method is adopted and modified to accommodate the contact problem of viscoelastic bodies. The Lagrange multiplier method is adopted to enforce the contact constraints. Two different contact problems are presented to demonstrate the efficient applicability of the proposed model.

A Numerical Solution for Quasistatic Viscoelastic Frictional Contact Problems

The tribological aspects of contact are greatly affected by the friction throughout the contact interface. Generally, contact of deformable bodies is a nonlinear problem. Introduction of the friction with its irreversible character makes the contact problem more difficult. Furthermore, when one or more of the contacting bodies is made of a viscoelastic material, the problem becomes more complicated. A nonlinear time-dependent contact problem is addressed. The objective of the present work is to develop a computational procedure capable of handling quasistatic viscoelastic frictional contact problems. The contact problem as a convex programming model is solved by using an adaptive incremental procedure. The contact constraints are incorporated into the model by using the Lagrange multiplier method. In addition, a local-nonlinear nonclassical friction model is adopted to model the friction at the contact interface. This eliminates the difficulties that arise with the application of the classical Coulomb’s law. On the other hand, the Wiechert model, as an effective model capable of describing both creep and relaxation phenomena, is adopted to simulate the linear behavior of viscoelastic materials. The resulting constitutive integral equations are linearized; therefore, complications that arise during the integration of these equations, especially with contact problems, are avoided. Two examples are presented to demonstrate the applicability of the proposed method.

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.

Numerical Analysis of Frictionless Nano-Contact Problems Based on Surface Elasticity

A nonlinear finite element model is developed to predict the response of frictionless contact of isotropic, elastic nano-bodies incorporating the effect of surface energy. The boundary value problem is formulated based on the classical theory of linear elasticity of the bulk material, while the complete Gurtin–Murdoch constitutive relation is adopted to accommodate the effect of surface energy. The Lagrange multiplier approach is employed to enforce inequality contact constraints without any need of an appropriate value for the penalty parameter, where the contact forces are treated as independent variables. The proposed finite element model accounts for both advancing and receding contact problems. The output results are compared favorably with those published analytical solutions. The influence of surface energy and its size-dependency on the behavior of conformal and nonconformal conformal/nonconformal nano-contact systems is demonstrated by carrying out by analyzing three different contact problems.

Modeling of Quasistatic Thermoviscoelastic Frictional Contact Problems

Frictional contacts of thermoviscoelastic bodies are complicated nonlinear temperature-and time-dependent problems. The introduction of friction with its irreversible character makes the problem more difficult. Additionally, the consideration of temperature, as an independent variable, destroys the convolution integral form of the viscoelasticity constitutive relations. This paper presents a computational model capable of predicting the nonlinear quasistatic response of uncoupled thermoviscoelastic frictional contact problems. The contact problem, as a variational inequality constrained model, is handled by using the Lagrange multiplier method to incorporate the inequality contact constraints. A local nonlinear friction law is adapted to model friction at the contact interface. This, in turn, eliminates difficulties that arise with the application of the classical friction laws. The temperature-dependency of viscoelasticity is modeled by applying the time-temperature superposirion principle. The constitutive equations are transformed to be a function of the reduced time as the only independent variable, maintaining the convolution integral form. Two different illustrative examples are presented to demonstrate the applicability of the proposed model to analyze both nonconformal and conformal thermoviscoelastic frictional contact problems.

Analysis of Steady-State Rolling Contact Problems in Nonlinear Viscoelastic Materials

A comprehensive numerical model is developed using Lagrangian finite element (FE) formulation for investigating the steady-state viscoelastic (VE) rolling contact response. Schaperys nonlinear viscoelastic (NVE) model is adopted to simulate the VE behavior. The model accounts for large displacements and rotations. A spatially dependent incremental form of the VE constitutive equations is derived. The dependence on the history of the strain rate is expressed in terms of the spatial variation of the strain. The Lagrange multiplier approach is employed. The classical Coulombs friction law is used. The developed model is verified and its applicability is demonstrated.

NONLINEAR ANALYSIS OF VISCOELASTICALLY LAYERED ROLLS IN STEADY STATE ROLLING CONTACT

The present paper analyzes the steady state rolling contact (SSRC) response of nonlinear viscoelastically layered rigid roll indented by a rigid cylindrical indenter. Both material and geometrical nonlinearities are accounted for in the framework of the updated Lagrangian finite element formulation. The Schaperys viscoelastic creep model is adopted to model the viscoelastic behavior. To accommodate the steady state rolling condition, the constitutive equations are recast into a spatially dependent incremental form. Throughout the contact interface, the Lagrange multiplier method is used to enforce the contact constraints, while the classical Coulombs law is adopted to simulate friction. The resulting nonlinear equilibrium equations are solved by the Newton–Raphson method. The developed model is applied to analyze a viscoelastically layered rigid roll in steady state rolling and intended by a rigid cylindrical indenter. Results showed the distinct effects of angular velocity, retardation time, indenter radius, and viscoelastic layer thickness on the SSRC configuration.

Effect of the Material Parameters on Layered Viscoelastic Frictional Contact Systems

In the design process, one of the main targets is to reduce the peak values of the contact stresses. This can be attained by layering the contacting bodies by layers of different material characteristics. Viscoelastic materials are characterized by either a stress relaxation or a creep deformation; therefore, the contacting bodies can be layered with such materials to attain this target. This paper discusses effects of the material characteristics of viscoelastic layers upon the unbounded contact configuration. Three material parameters are considered: the layer/contact solids stiffness ratio, the delayed/instantaneous elasticity ratio, and the material relaxation time. The results are obtained by using a two-dimensional time-dependent nonlinear computational model, developed by the authors, capable of analyzing quasistatic viscoelastic frictional contact problems.

Effect of Material Characteristics on Viscoelastic Frictionless Contact Configuration

The tribological status of contact systems is affected by the contact configuration; contact stress distribution throughout the contact interface. Viscoelastic materials have the capability of storing and dissipating energy. When the contacting bodies are made of viscoelastic material, viscous and elastic properties of the material have a vital effect upon the contact pressure distribution and the extent of the contact interface. This paper illustrates the effect of viscoelastic material parameters on the contact configuration. Two material parameters are considered; the ratio of the delayed elasticity to the instantaneous elasticity and the material relaxation time. The results are obtained by using a time-dependent nonlinear computational model capable of analyzing quasistatic viscoelastic frictionless contact problems. This computational model adopts the Wiechert model to simulate the linear behavior of viscoelastic materials and the modified incremental convex programming method to accommodate the contact problem of viscoelastic bodies.Copyright

Analysis of steady-state frictional rolling contact problems in Schapery–nonlinear viscoelasticity

In the context of an updated Lagrangian formulation, a computational model is developed for analyzing the steady-state frictional rolling contact problems in nonlinear viscoelastic solids. Schaperys nonlinear viscoelastic model is adopted to simulate the viscoelastic behavior. In addition to the material nonlinearity, the model accounts for geometrical nonlinearities, large displacements, and rotations with small strains. To satisfy the steady-state rolling contact condition, a spatially dependent incremental form of the viscoelastic constitutive equations is derived. Consequently, the dependence on the past history of the strain rate in the stress–strain law is expressed in terms of the spatial variation of the strain. The contact conditions are exactly satisfied by employing the Lagrange multiplier approach to enforce the contact constraints. The classical Coulombs friction law is used to simulate friction. The developed model is verified and compared and good agreement is obtained. The applicability of the developed model is demonstrated by analyzing the steady-state rolling contact response of viscoelastically walled-wheel over rigid foundation. Moreover, the obtained results show remarkable effects of the rotational velocity and the viscoelastic material parameters on the mechanical response of steady-state frictional rolling contact.