F.F. Mahmoud

Coupling effects of nonlocal and surface energy on vibration analysis of nanobeams

This paper is proposed to study the coupled effects of surface properties and nonlocal elasticity on vibration characteristics of nanobeams by using a finite element method. Nonlocal differential elasticity of Eringen is exploited to reveal the long-range interactions of a nanoscale beam. To incorporate surface effects, Gurtin-Murdoch model is proposed to satisfy the surface balance equations of the continuum surface elasticity. Euler-Bernoulli hypothesis is used to model the bulk deformation kinematics. The surface layer and bulk of the beam are assumed elastically isotropic. Galerkin finite element technique is employed for the discretization of the nonlocal mathematical model with surface properties. An efficiently finite element model is developed to descretize the beam domain and solves the equation of motion numerically. The output results are compared favorably with those published works. The effects of nonlocal parameter and surface elastic constants on the vibration characteristics are presented. Also, the effectiveness of finite element method to handle a complex geometry is illustrated. The present model can be used for free vibration analysis of single-walled carbon nanotubes with essential, natural and nonlinear boundary conditions.

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.

Finite Element Analysis of the Deformation of Functionally Graded Plates under Thermomechanical Loads

The first-order shear deformation plate model, accounting for the exact neutral plane position, is exploited to investigate the uncoupled thermomechanical behavior of functionally graded (FG) plates. Functionally graded materials are mainly constructed to operate in high temperature environments. Also, FG plates are used in many applications (such as mechanical, electrical, and magnetic), where an amount of heat may be generated into the FG plate whenever other forms of energy (electrical, magnetic, etc.) are converted into thermal energy. Several simulations are performed to study the behavior of FG plates, subjected to thermomechanical loadings, and focus the attention on the effect of the heat source intensity. Most of the previous studies have considered the midplane neutral one, while the actual position of neutral plane for functionally graded plates is shifted and should be firstly determined. A comparative study is performed to illustrate the effect of considering the neutral plane position. The volume fraction of the two constituent materials of the FG plate is varied smoothly and continuously, as a continuous power function of the material position, along the thickness of the plate.

An incremental mathematical programming model for solving multi-phase frictional contact problems

Abstract An incremental model is developed for the simulation of elasto-static multiphase frictional contact problems. The model employs a proposed local and nonlinear friction law, and exploits the incremental convex programming method in the framework of the finite element scheme. The proposed model accommodates the two types of inequality constraints which are representing non-interpenetration and slipping conditions.

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.

On the Nonexistence of a Feasible Solution in the Context of the Differential Form of Eringen’s Constitutive Model: A Proposed Iterative Model Based on a Residual Nonlocality Formulation

The motivation of this paper is to highlight and discuss critically the details of two main aspects related to Eringen’s nonlocal constitutive model. The first aspect is to point out the inconsiste...

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.

Contour design for contact stress minimization by interior penalty method

Abstract The present study has exploited optimization techniques in the framework of the automated direct method to design optimal shapes of contact surfaces. The developed model consists of two nonlinear submodels; the first is concerned with the design problem in which the peak contact pressure is taken as an objective function to be minimized subject to a set of constraints representing the limitations on contact surface dimensions and the maximum effective stresses. The peak contact pressure is an implicit function of the design variables and state variables. Once the contact problem of two bodies of specific surfaces dimensions (design variables) is solved, the peak contact pressure can be computed directly as a qualifying function of the state variables (nodal displacements). The other submodel represents the elastostatic contact problem formulated as a nonlinear mathematical one in which the potential energy is taken as the objective function to be minimized for equilibrium state and the relevant constraints represent the noninterpenetration conditions. The proposed algorithm adopts the interior penalty method enhanced by the direct automated procedure to obtain the optimal surfaces dimensions. Two examples of different nature are presented for illustration.

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.