Moving contact problems involving a rigid punch and a functionally graded coating

Frictional contact mechanics analysis for a rigid moving punch of an arbitrary profile and a functionally graded coating/homogeneous substrate system is carried out. The rigid punch slides over the coating at a constant subsonic speed. Smooth variation of the shear modulus of the graded coating is defined by an exponential function and the variation of the Poisson's ratio is assumed negligible. Coulomb's friction law is adopted. Hence, tangential force is proportional to the normal applied force through the coefficient of friction. An analytical method is developed utilizing the singular integral equation approach. Governing partial differential equations are derived in accordance with the theory of elastodynamics. The mixed boundary value problem is reduced to a singular integral equation of the second kind, which is solved numerically by an expansion-collocation technique. Presented results illustrate the effects of punch speed, coefficient of friction, material inhomogeneity and coating thickness on contact stress distributions and stress intensity factors. Comparisons indicate that the difference between elastodynamic and elastostatic solutions tends to be quite larger especially at higher punch speeds. It is shown that use of the elastodynamic theory provides more realistic results in contact problems involving a moving punch.     

Periodically varying heat source response in a functionally graded microelongated medium

The present paper deals with the response due to periodically varying heat sources in the neighborhood of the origin of a functionally graded isotropic unbounded microelongated medium, in the context of generalized thermoelastic theory. The expressions for displacement, microelongation and temperature fields have been obtained in Laplace-Fourier transformed domain. After computing the inverse Fourier transforms by contour integration technique, the inversion of Laplace transforms has been obtained numerically. The changes of displacement, microelongation, and normal strain have been shown graphically for different types of heat source.     

Natural frequency analysis of functionally graded material beams with axially varying stochastic properties

The functionally graded material (FGM) has a potential to replace ordinary ones in engineering reality due to its superior thermal and dynamical characteristics. In this regard, the paper presents an effective approach for uncertain natural frequency analysis of composite beams with axially varying material properties. Rather than simply assuming the material model as a deterministic function, we further extend the FGM property as a random field, which is able to account for spatial variability in laboratory observations and in-field data. Due to the axially varying input uncertainty, natural frequencies of the stochastically FGM (S-FGM) beam become random variables. To this end, the Karhunen–Loève expansion is first introduced to represent the composite material random field as the summation of a finite number of random variables. Then, a generalized eigenvalue function is derived for stochastic natural frequency analysis of the composite beam. Once the mechanistic model is available, the brutal Monte-Carlo simulation (MCS) similar to the design of experiment can be used to estimate statistical characteristics of the uncertain natural frequency response. To alleviate the computational cost of the MCS method, a generalized polynomial chaos expansion model developed based on a rather small number of training samples is used to mimic the true natural frequency function. Case studies have demonstrated the effectiveness of the proposed approach for uncertain natural frequency analysis of functionally graded material beams with axially varying stochastic properties.     

Thermoelastic contact with Barber's heat exchange condition

We consider a nonlinear parabolic problem that models the evolution of a one-dimensional thermoelastic system that may come into contact with a rigid obstacle. The mathematical problem is reduced to solving a nonlocal heat equation with a nonlinear and nonlocal boundary condition. This boundary condition contains a heat-exchange coefficient that depends on the pressure when there is contact with the obstacle and on the size of the gap when there is no contact. We model the heat-exchange coefficient as both a single-valued function and as a measurable selection from a maximal monotone graph. Both of these models represent modified versions of so-called imperfect contact conditions found in the work of Barber. We show that strong solutions exist when the coefficient is taken to be a continuously differentiable function and that weak solutions exist when the coefficient is taken to be a measurable selection from a maximal monotone graph. The proofs of these results reveal an interesting interplay between the regularity of the initial condition and the behavior of the coefficient at infinity.     

Thermoelastic vibration and buckling analysis of functionally graded piezoelectric cylindrical shells

This paper presents the report of an investigation into thermoelastic vibration and buckling characteristics of the functionally graded piezoelectric cylindrical, where the functionally graded piezoelectric cylindrical shell is made from a piezoelectric material having gradient change along the thickness, such as piezoelectricity and dielectric coefficient et al. Here, utilizing Hamilton’s principle and the Maxwell equation with a quadratic variation of the electric potential along the thickness direction of the cylindrical shells and the first-order shear deformation theory, and taking into account both the direct piezoelectric effect and the converse piezoelectric effect, the thermoelastic vibration and buckling characteristics of functionally graded piezoelectric cylindrical shells composed of BaTiO₃/PZT − 4, BaTiO₃/PZT − 5A and BaTiO₃/PVDF are, respectively, calculated. The effects of material composition (volume fraction exponent), thermal loading, external voltage applied and shell geometry parameters on the free vibration characteristics are described, and the axial critical load, critical temperature and critical control voltage are obtained.     

Nonlinear vibration and instability of functionally graded nanopipes with initial imperfection conveying fluid

In this paper, the nonlinear vibration and instability of a fluid-conveying nanopipe made of functionally graded (FG) materials with consideration of the initial geometric imperfection are investigated. The material properties are assumed to vary smoothly along the radial direction according to a power-law exponent form. The fluid-conveying FG nanopipe is modeled as a Euler-Bernoulli beam, and the governing equation is derived based on the nonlocal strain gradient theory incorporating the effects of Von-Karman geometrical nonlinearity and initial imperfection. The nonlinear frequency and critical fluid velocity are achieved via He's Hamiltonian approach. After verifying the present model with comparison of several previous studies, the effect of several different system parameters including the amplitude of the nonlinear oscillator, the initial geometric imperfection, size-dependent parameters, and the power-law index on the frequency response of the fluid-conveying FG nanopipe are explored. Moreover, the critical velocity of the conveying fluid under different system parameters is also investigated and discussed in detail. The developed size-dependent nonlinear model is expected to provide a possible theoretical way to guide the application of FG nanopipe as micro/nanofluidic devices.     

Calculation of frictional heat resistance characteristics

The search for self-lubricating materials, the selection of suitable ingredients and processing techniques for such materials, and quality control require the development of methods of testing for friction and wear. The frictional heat resistance can be used to evaluate the frictional properties of self-lubricating materials and to determine their limits of applicability. It is shown with reference to AF-3am self-lubricating material that it is possible to calculate the temperature dependence of the wear rate if the physicomechanical and geometric characteristics of the contact materials are known.Mekhanika Polimerov, Vol. 3, No. 3, pp. 524–532, 1967     

Mathematical modelling of heat conduction in certain functionally graded composites

The aim of contribution is to formulate a certain extended version of the tolerance modelling technique for functionally graded composites. For the sake of simplicity, the considerations are restricted to the bidirectionally graded heat conductors. It is shown that the proposed approach enables to determine an entire class of mathematical models for which applications can be found in various specific problems. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stochastic heat conduction analysis of a functionally graded annular disc with spatially random heat transfer coefficients

The mean and variance of the temperature are analytically obtained in a functionally graded annular disc with spatially random heat transfer coefficients (HTCs) on the upper and lower surfaces. This annular disc has arbitrary variations in the HTCs (i.e., arbitrary thermal interaction with the surroundings) and gradient material composition only along the radial direction and is subjected to deterministic axisymmetrical heating at the lateral surfaces. The stochastic temperature field is analysed by considering the annular disc to be multilayered with spatially constant material properties and spatially constant but random HTCs in each layer. A type of integral transform method and a perturbation method are employed in order to obtain the analytical solutions for the statistics. The correlation coefficients of the random HTCs are expressed in the form of a linear function with respect to the radial distance as a non-homogeneous random field of discrete space. Numerical calculations are performed for functionally graded annular discs composed of stainless steel and ceramic, which comprise two types of material composition distributions. The effects of the magnitude of the means of HTCs, volume fraction distributions of the constitutive materials and correlation strengths of the HTCs on the standard deviation of the temperature are discussed.     

Viscoplasticity with frictional contact and rapid growth

We consider a problem in the inelastic deformation theory with a quasistatic deformation process of the gradient‐monotone type. We assume that the body has contact with a rigid foundation: the body moves on the foundation with friction. The frictional contact is modelled by a velocity‐dependent dissipation functional. This makes an evolution problem with two nonlinear monotone operators. We consider the gradient‐monotone inelastic constitutive function with a rapid growth at infinity. This leads us to a nonreflexive Orlicz space as an operational base. The frictional dissipation potential brings about a minimalization problem in this nonreflexive Orlicz space. Copyright © 2016 John Wiley & Sons, Ltd.     

A model of the heat conduction for functionally graded laminate with uniformly distributed micro inclusions

The aim of the contribution is to formulate an asymptotic model of heat conduction for functionally graded two component laminate reinforced by periodically spaced micro inclusions. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The solvability of a coupled thermoviscoelastic contact problem with small Coulomb friction and linearized growth of frictional heat

A coupled thermoviscoelastic frictional contact problem is investigated. The contact is modelled by the Signorini condition for the displacement velocities and the friction by the Coulomb law. The heat generated by friction is described by a non‐linear boundary condition with at most linear growth. The weak formulation of the problem consists of a variational inequality for the elasticity part and a variational equation for the heat conduction part. In order to prove the existence of a solution to this problem we first use an approximation of the Signorini condition by the penalty method. The existence of a solution for the approximate problem is shown using the fixed‐point theorem of Schauder. This theorem is applied to the composition of the solution operator for the contact problem with given temperature field and the solution operator for the heat equation problem with known displacement field. To obtain this proof, the unique solvability of both problems is necessary. Due to this reason it is necessary to introduce the penalty method. While the penalized contact problem has a unique solution, this is not clear for the original contact problem. The solvability of the original frictional contact problem is verified by an investigation of the limit for vanishing penalty parameter. Copyright © 1999 John Wiley & Sons, Ltd.     

Approximation of functionally graded plates with non-conforming finite elements

In this paper rectangular plates made of functionally graded materials (FGMs) are studied. A two-constituent material distribution through the thickness is considered, varying with a simple power rule of mixture. The equations governing the FGM plates are determined using a variational formulation arising from the Reissner–Mindlin theory. To approximate the problem a simple locking-free Discontinuous Galerkin finite element of non-conforming type is used, choosing a piecewise linear non-conforming approximation for both rotations and transversal displacement. Several numerical simulations are carried out in order to show the capability of the proposed element to capture the properties of plates of various gradings, subjected to thermo-mechanical loads.     

Analysis of two frictional viscoplastic contact problems with damage

In this work we study two quasistatic frictional contact problems arising in viscoplasticity including the mechanical damage of the material, caused by excessive stress or strain and modelled by an inclusion of parabolic type. The variational formulation is provided for both problems and the existence of a unique solution is proved for each of them. Then a fully discrete scheme is introduced using the finite element method to approximate the spatial domain and the Euler scheme to discretize the time derivatives. Error estimates are derived and, under suitable regularity assumptions, the linear convergence of the algorithm is deduced. Finally, some numerical examples are presented to show the performance of the method.     

Inverse prediction of frictional heat flux and temperature in sliding contact with a protective strip by iterative regularization method

In this study, an inverse algorithm based on the conjugate gradient method and the discrepancy principle is applied to estimate the unknown time-dependent frictional heat flux at the interface of two semi-spaces, one of them is covered by a strip of coating, during a sliding-contact process from the knowledge of temperature measurements taken within one of the semi-space. It is assumed that no prior information is available on the functional form of the unknown heat generation; hence the procedure is classified as the function estimation in inverse calculation. Results show that the relative position between the measured and the estimated quantities is of crucial importance to the accuracy of the inverse algorithm. The current methodology can be applied to the prediction of heat generation in engineering problems involving sliding-contact elements.     

Vibration analysis of functionally graded plate with a moving mass

A hybrid method is proposed to predict the dynamic behavior of functionally graded (FG) plate subjected to a moving mass. The governing equations of motion of FG plate are derived using the Kirchhoff plate theory and Lagrange equation. Improved Rayleigh–Ritz solution is used to treat the spatial partial derivatives. Penalty method is employed to deal with the constraints, and the energy terms due to boundary conditions are included in Lagrange, hence it is not necessary to particularly consider the constraints in the modeling process. And the combination of simple polynomials and trigonometric functions is selected as the admissible functions. The advantage of this improvement in Rayleigh–Ritz method is that it is not needed to find satisfied admissible functions for different boundary conditions while the convergence of the solution is improved. Meanwhile, the method can be used to handle the versatile boundary conditions. Differential quadrature method (DQM) as a step-by-step time integration scheme is employed for discretization of temporal derivatives. The validated results show that the presented method is very reliable and efficient, and its convergence and accuracy are also better compared to finite element method for solving the dynamic problems of FG plate with moving loads (force and mass). Moreover, the influences of material properties and boundary conditions on maximum dynamic deflections are investigated, as well as moving speeds and inertial effects of loads (mass and force). Although only four edge boundary conditions are addressed in the present work, the proposed procedure is applicable for any arbitrary edge boundary conditions.     

Post buckling response of functionally graded materials plate subjected to mechanical and thermal loadings with random material properties

In this paper, the second order statistics of post buckling response of functionally graded materials plate (FGM) subjected to mechanical and thermal loading with nonuniform temperature changes subjected to temperature independent (TID) and dependent (TD) material properties is examined. Material properties such as material properties of each constituent’s materials, volume fraction index are taken as independent random input variables. The basic formulation is based on higher order shear deformation theory (HSDT) with von-Karman nonlinear kinematic using modified C⁰ continuity. A direct iterative based C⁰ nonlinear finite element method (FEM) combined with mean centered first order perturbation technique (FOPT) proposed by last two authors for the composite plate is extended for Functionally Graded Materials (FGMs) plate with reasonable accuracy to compute the second order statistics (mean and coefficient of variation) of the post buckling load response of the FGM plates. The effect of random material properties with amplitude ratios, volume fraction index, plate thickness ratios, aspect ratios, boundary conditions and types of loadings subjected to TID and TD material properties are presented through numerical examples. The performance of outlined present approach is validated with the results available in literatures and independent Monte Carlo simulation (MCS).     

The capacities of certain special channels with arbitrarily varying channel probability functions

In [5] Ahlswede and Wolfowitz have obtained the capacities of a.v.ch. with binary output in a number of cases, essentially with the aid of a lemma which relates the capacity of the a.v.ch. to that of a suitable (“underlying”) d.m.c. A generalization of this lemma to a special kind of a.v.ch. with output alphabet b>2, has been given by Ahlswede (Lemma 1 of [1]) and used in [1] and [2] to prove the existence of the weak capacities of various channels under different conditions. We give a detailed proof of a weakened version of Ahlswede's lemma and show, in passing, that his lemma is incorrect. We then define certain special types of a.v.ch and, on the basis of the detailed analysis given by us earlier, we prove lemmas of a similar type for these a.v.ch. We are thus able to extend certain results given for binary output a.v.ch. in [4] and [5] to these special a.v.ch. for which b>2.     

Transient thermoelastic response in a cracked strip of functionally graded materials via generalized fractional heat conduction

This work is devoted to analyzing a thermal shock problem of an elastic strip made of functionally graded materials containing a crack parallel to the free surface based on a generalized fractional heat conduction theory. The embedded crack is assumed to be insulated. The Fourier transform and the Laplace transform are employed to solve a mixed initial-boundary value problem associated with a time-fractional partial differential equation. Temperature and thermal stresses in the Laplace transform domain are evaluated by solving a system of singular integral equations. Numerical results of the thermoelastic fields in the time domain are given by applying a numerical inversion of the Laplace transform. The temperature jump between the upper and lower crack faces and the thermal stress intensity factors at the crack tips are illustrated graphically, and phase lags of heat flux, fractional orders, and gradient index play different roles in controlling heat transfer process. A comparison of the temperature jump and thermal stress intensity factors between the non-Fourier model and the classical Fourier model is made. Numerical results show that wave-like behavior and memory effects are two significant features of the fractional Cattaneo heat conduction, which does not occur for the classical Fourier heat conduction.