Free vibration analysis of functionally graded plates resting on Winkler–Pasternak elastic foundations using a new shear deformation theory

Free vibration analysis of simply supported functionally graded plates (FGP) resting on a Winkler–Pasternak elastic foundation are examined by a new higher shear deformation theory in this paper. Present theory exactly satisfies stress boundary conditions on the top and the bottom of the plate. The material properties change continuously through the thickness of the plate, which can vary according to power law, exponentially or any other formulations in this direction. The equation of motion for FG rectangular plates resting on elastic foundation is obtained through Hamilton’s principle. The closed form solutions are obtained by using Navier technique, and then fundamental frequencies are found by solving the results of eigenvalue problems. The numerical results obtained through the present analysis for free vibration of functionally graded plates on elastic foundation are presented, and compared with the ones available in the literature.     

Free vibration and flexural response of functionally graded plates resting on Winkler–Pasternak elastic foundations using nonpolynomial higher-order shear and normal deformation theory

In the present work, the flexural and vibration response of a functionally graded plate resting on Pasternak elastic foundation is analyzed using a recently developed nonpolynomial higher-order shear and normal deformation theory by the authors. The novelty of this theory is that it contains only four unknowns and also accommodates the thickness stretching effect. Two kinds of micromechanics models, namely, the Voigt and Mori–Tanaka models, are considered. Material properties of the functionally graded plates are assumed to vary continuously in the thickness direction according to either a simple power law or an exponential law. Finite element formulation is done using C° continuous Lagrangian quadrilateral nine-noded elements with eight degrees of freedom per node. The equations of motion are derived using a variational approach. Convergence and comparison studies are carried out to establish the authenticity and reliability of the solutions. The effect of various boundary conditions, geometric conditions, micromechanics models, and foundation parameters on the flexural and vibration response of the functionally graded plate are investigated.     

Bending analysis of FG viscoelastic sandwich beams with elastic cores resting on Pasternak’s elastic foundations

The investigation of bending response of a simply supported functionally graded (FG) viscoelastic sandwich beam with elastic core resting on Pasternak’s elastic foundations is presented. The faces of the sandwich beam are made of FG viscoelastic material while the core is still elastic. Material properties are graded from the elastic interfaces through the viscoelastic faces of the beam. The elastic parameters of the faces are considered to be varying according to a power-law distribution in terms of the volume fraction of the constituent. The interaction between the beam and the foundations is included in the formulation. Numerical results for deflections and stresses obtained using the refined sinusoidal shear deformation beam theory are compared with those obtained using the simple sinusoidal shear deformation beam theory, higher- and first-order shear deformation beam theories. The effects due to material distribution, span-to-thickness ratio, foundation stiffness and time parameter on the deflection and stresses are investigated.     

Thermal buckling analysis of nanoplates based on nonlocal elasticity theory with four-unknown shear deformation theory resting on Winkler–Pasternak elastic foundation

The thermal buckling analysis of nanoplates is based on nonlocal elasticity theory with four-unknown shear deformation theory resting on Winkler–Pasternak elastic foundation. The nanoplate is assumed to be under three types of thermal loadings, namely uniform temperature rise, linear temperature rise, and nonlinear temperature rise through the thickness. The theory involves four unknown variables with small-scale effects, as against five in the case of other higher-order theories and first-order shear deformation theory. Closed-form solution for theory was also presented. Results are presented to discuss the influences of the nonlocal parameter, aspect ratio, side-to-thickness ratio, and elastic foundation parameters on the thermal buckling characteristics of analytical rectangular nanoplates.     

Mechanical buckling of two-dimensional functionally graded cylindrical shells surrounded by Winkler–Pasternak elastic foundation

In this research, buckling analysis of a two-dimensional, functionally graded, cylindrical shell that has been embedded in an outer elastic medium in the presence of combined axial and transverse loading based on third-order shear deformation shell theory is numerically investigated. Variations of the shell properties are considered to be continuous through length and thickness. Winkler–Pasternak foundation and simply supported boundary conditions have been applied. The problem has been solved using the generalized differential quadrature method. Geometrical, load, and foundation parameters beside functionally graded power indexes effects on the critical buckling load have been studied.     

Buckling Analysis of a Nanowire Lying on Winkler–Pasternak Elastic Foundation

This article studies the axial buckling of a nanowire (NW) lying on Winkler–Pasternak substrate medium with the Timoshenko beam theory. The surface effect of the NW is accounted for with the Steigmann–Ogden model. An explicit solution of the critical buckling force and its associated buckling mode are obtained analytically. The influences of the surface stress effect, the geometry of the NW, and the elastic foundation moduli on the buckling behavior are fully discussed.     

Bending of stochastic Kirchhoff plates on Winkler foundations via the Galerkin method and the Askey–Wiener scheme

In this paper, the method of Galerkin and the Askey–Wiener scheme are used to obtain approximate solutions to the stochastic displacement response of Kirchhoff plates with uncertain parameters. Theoretical and numerical results are presented. The Lax–Milgram lemma is used to express the conditions for existence and uniqueness of the solution. Uncertainties in plate and foundation stiffness are modeled by respecting these conditions, hence using Legendre polynomials indexed in uniform random variables. The space of approximate solutions is built using results of density between the space of continuous functions and Sobolev spaces. Approximate Galerkin solutions are compared with results of Monte Carlo simulation, in terms of first and second order moments and in terms of histograms of the displacement response. Numerical results for two example problems show very fast convergence to the exact solution, at excellent accuracies. The Askey–Wiener Galerkin scheme developed herein is able to reproduce the histogram of the displacement response. The scheme is shown to be a theoretically sound and efficient method for the solution of stochastic problems in engineering.     

Free damped vibrations of a viscoelastic oscillator based on Rabotnov’s model

Free damped vibrations of a linear viscoelastic oscillator based on Rabotnov’s model involving one fractional parameter and several relaxation (retardation) times are investigated. The analytical solution is obtained in the form of two terms, one of which governs the drift of the system’s equilibrium position and is defined by the quasi-static processes of creep occurring in the system, and the other term describes damped vibrations around the equilibrium position and is determined by the systems’s inertia and energy dissipation. The drift is governed by an improper integral taken along two sides of the cut of the complex plane. Damped vibrations are determined by two complex conjugate roots of the characteristic equation, which are located in the left half-plane of the complex plane. The behaviour of the characteristic equation roots as function of the system’s parameters is shown in the complex plane. Dedicated to the bright memory of Academician Yury N. Rabotnov.     

Free Vibration Analysis of Moderately Thick Rectangular Plates on Pasternak Foundation with Point Supports and Elastically Restrained Edges by Using the Rayleigh–Ritz Method

In the present paper, free vibration analysis of moderately thick rectangular plates resting on an elastic foundation of Pasternak model with point supports and elastically restrained edges based on the first-order shear deformation plate theory is presented. The Rayleigh–Ritz method is applied to derive the eigenvalue equation of the plate. The Chebyshev polynomials multiplied by a boundary function which define the displacement components are adopted in this method as the admissible functions. The accuracy of the present method is examined via lots of convergence and comparison studies with the available data in the literature, and it is demonstrated that the present method has a rapid convergence rate and high accuracy. Many numerical results are presented in tabular and graphical forms in order to investigate the effects of various parameters such as thickness–span ratio, foundation parameters, the lateral and rotational stiffness of the edge supports, and locations of the point supports on the natural frequencies.     

Effect of surface etching on interphase and elastic properties of a biocomposite reinforced using glass–silica particles

Biopolymer based composites are designed using glass–silica reinforcement. Surface etching of spherical glass–silica particles is performed using chemical and physical treatments. In particular, treatment with hydrofluoric acid proved to be efficient to achieve acceptable anchoring effect. Experimental testing of thermomoulded composites confirms that samples with chemically modified microbeads have improved mechanical properties, irrespective of phase content. A quantitative evaluation of the improvement of the starch/glass–silica interphase properties is achieved using a finite element model. Generation of typical microstructures is used to simulate phase arrangement and interphase properties. Microstructures are meshed taking into account the interphase region. Finite element results indicate that for all samples, interphase Young’s modulus is lower than those of the intrinsic materials. The thickness weighted interface modulus increases for composites where the mechanical adhesion is improved using HF chemical treatment.     

Characterization and numerical evaluation of vibration on elastic–viscoelastic sandwich structures

The objective of this paper is to study the vibration characteristic for a sandwich beam with silica/polymer blend as principal material, and pure polymer matrix as surface laminate. It is anticipated that high stiffness and structure damping of viscoelastic layer can be obtained by taking advantage of fascinating network of densely packed between silica and polymer matrix. Spherical particles of size 12–235 nm at various filler fraction (10–50 wt.%) and three different polymer matrices, polyacrylate, polyimide and polypropylene, were selected as the matrix materials. The mechanical damping and stiffness of the sandwich cantilever beam are recorded by using a Dynamic Mechanical Thermal Analyzer (DMTA). The silica’s small particle size feature and strain difference between principal and surface layers could highly enhance the energy dissipation ability of the beam structure. A numerical model is then developed and validated for the vibration of a symmetric elastic–viscoelastic sandwich beam. Experimental results show that the structure deformation for these sandwich beams with contiguous and constraining layers are in reasonable agreement with the prediction of the model. Both higher resonant vibrations are well damped in accordance with the symmetric motion of the elastic layers and relative little motion of the constraining layer.     

Effect of a heating–cooling cycle on elastic strain and Young’s modulus of high performance and ordinary concrete

In this paper, the variations of elastic strain and Young modulus of high performance concrete and ordinary concrete during a heating?Ccooling cycle is presented. For the HPC, two heating rates are applied: 1.5 and 0.1?°C/min corresponding respectively to accidental and service conditions. For ordinary concrete, the results of service conditions are given. The temperatures of 400 and 220?°C are the heating??s final temperature phase of the accidental and service conditions respectively. The present work analyses the differences between the value of the elastic strain and the Young??s modulus at the beginning of the test (at ambient temperature), the end of the heating part and the end of the cooling part of each variation. Indeed, during the heating phase, the corresponding heating rates are applied until successive constant temperature levels are achieved: 150, 200, 300 and 400?°C for the high-performance concrete under accidental conditions and 140, 190 and 220?°C for both high-performance and ordinary concrete under service conditions. Those applied temperatures are maintained for several hours to ensure the stabilisation of internal temperature and physico-chemical thermo dependent processes. Moreover, the influence of the difference in mix concretes between the two types of concretes and the heating rate influence on those variations is also presented.     

Experimental and numerical study of stress–strain state of pressurised cylindrical shells with external defects

Work presents the experimental technique with using of strain gauges for determining of strain distribution near longitudinal external defects of semielliptical shape in pressurised cylindrical shells and an appropriate procedure of numerical calculation based on a finite element method (FEM) for assessment of strain state near such defects. Numerical assessment of elastic stress distribution at bottom of external semielliptical notch in pressurised cylindrical shell is in a good agreement with the data received on the base of different analytical models. Here, the FEM results and data based on Glinka-Newport model are the most close for maximal stress. A comparison an experimental and calculation results showed on their acceptable coincidence and this fact gives the ground for using the numerical calculations instead labour-intensive and expensive experimental tests.     

Effects of polymer–cement ratio and accelerated curing on flexural behavior of hardener-free epoxy-modified mortar panels

This experimental study reports the applicability of hardener-free epoxy-modified mortar panels to permanent forms as precast concrete products. Hardener-free epoxy-modified mortars are mixed using a Bisphenoal A-type epoxy resin without any hardener with various polymer–cement ratios and steel fiber reinforcement, and subjected to different curings. Hardener-free epoxy-modified mortar panels are prepared with same polymer–cement ratios and steel fiber reinforcement on trial, and tested for flexural behavior under four-point (third-point) loading. The effects of polymer–cement ratios and curings on strength properties of hardener-free epoxy-modified mortars, and on the flexural strength, flexural stress-extreme tension fiber strain relation, flexural load–deflection relation of hardener-free epoxy-modified mortar panels were examined. The adhesion in tension (to placed concrete) of the hardener-free epoxy-modified mortar panels was also tested. As a result, the hardener-free epoxy-modified mortar panels develop a high flexural strength, large extensibility and good adhesion to the placed concrete. The epoxy-modified mortar panels are more ductile and have high load-bearing capacity than unmodified mortar panels and can be used as precast concrete permanent forms in practical applications.     

Smart control of nonlinear vibrations of doubly curved functionally graded laminated composite shells under a thermal environment using 1–3 piezoelectric composites

This paper addresses the active control of geometrically nonlinear vibrations of doubly curved functionally graded (FG) laminated composite shells integrated with a patch of active constrained layer damping (ACLD) treatment under the thermal environment. Vertically/obliquely reinforced 1-3 piezoelectric composite (PZC) and active fiber composite (AFC) are used as the materials of the constraining layer of the ACLD treatment. Each layer of the substrate FG laminated composite shell is made of fiber-reinforced composite material in which the fibers are longitudinally aligned in the plane parallel to the top or bottom surface of the layer and the layer is assumed to be graded in the thickness direction by way of varying the fiber orientation angle across its thickness according to a power law. The novelty of the present work is that, unlike the traditional laminated composite shells, the FG laminated composite shells are constructed in such a way that the continuous variation of material properties and stresses across the thickness of the shell is achieved. The Golla-Hughes-McTavish (GHM) method has been implemented to model the constrained viscoelastic layer of the ACLD treatment in time domain. Based on the first-order shear deformation theory (FSDT), a finite element (FE) model has been developed to model the open-loop and closed-loop nonlinear dynamics of the overall FG laminated composite shell under a thermal environment. Both symmetric and asymmetric FG laminated composite doubly curved shells are considered for presenting the numerical results. The analysis suggests that the ACLD patch significantly improves the damping characteristics of the doubly curved FG laminated composite shells for suppressing their geometrically nonlinear transient vibrations. It is found that the performance of the ACLD patch with its constraining layer being made of the AFC material is significantly higher than that of the ACLD patch with vertically/obliquely reinforced 1-3 PZC constraining layer. The effects of variation of piezoelectric fiber orientation in both the obliquely reinforced 1-3 PZC and the AFC constraining layers on the control authority of the ACLD patch have also been investigated.     

Effect of stamping deformation on microstructure and properties evolution of an Al–Mg–Si–Cu alloy for automotive panels

The effect of 5 % tensile deformation, which simulates the stamping process of Al–Mg–Si–Cu automotive outer panels, on the microstructural evolution during age strengthening, has been investigated. In addition, its benefit on key mechanical properties including hardness, yield strength, ductility, and corrosion resistance has been linked to the microstructural features. It was found that the aging precipitation sequence, SSSS → clusters and G.P. zones → β″ → β′ + Q′ → Q, was not influenced by the dislocations introduced through the stamping deformation prior to aging. On the other hand, stamping deformation could promote the formation of precipitates and refine the precipitates because of the enhanced heterogeneous nucleation and the accelerated precipitation kinetics, leading to superior strength of the alloy at the early stage. Meanwhile, the larger amount of Cu incorporated into nanoprecipitates leads to better intergranular corrosion resistance of the stamped alloy compared with the unstamped one. Due to the reduction in free Si amount at grain boundaries, the formation of fine subgrain structures and the increase of dislocation accumulation, the ductility of the stamped alloy was increased.     

Evolution of morphology and structure with growth of thickness of condensed films of Pd–Cu on a surface with open porosity

Cross sections obtained by the method of a focused ion beam and fractures of the PdCu/Al₂O₃ nanocomposite synthesized by magnetron sputtering (MS) of alloy-type target PdCu and condensation in vacuum on the surface of nanoporous Al₂O₃ produced by anodic oxidation of aluminum foil are investigated using the methods of high-resolution scanning electron microscopy implemented on the basis of a Helios 600i setup (FEI, United States). Regularities of the formation of the structure and morphology of crystallites of vacuum condensate of Pd–Cu solid solution with a thickness of 0.1–4 μm on a surface with open porosity are disclosed. Approaches to the formation of the gradient structures near a free surface are discussed, and the conditions of the initiation of MS mechanisms of the formation of discrete, porous, and anisotropic condensates are found. Approaches to nonselective filling of nanopores in a dielectric by metal clusters and forming of the gradient structure of the nanocomposite PdCu/Al₂O₃ are carried out.     

Modeling and calculation of slurry-chemistry effects on chemical–mechanical planarization with a grooved pad

During chemical–mechanical planarization (CMP), a rotating wafer is pressed against a rotating pad, while a slurry is dragged into the pad–wafer interface. Here, taking into account the dependence of local material removal rate (MRR) on the slurry’s chemical activity, the effects of pad groove geometry and various other process parameters on the spatial average and non-uniformity of MRR are examined. Technically, the slurry flow is calculated by following an existing approach that integrates two-dimensional fluid-film lubrication theory and contact-mechanics models. A slurry impurity transport equation is then used to calculate the impurity concentration that determines the slurry’s chemical activity and hence the local MRR. The numerical results obtained here indicate that the presence of pad grooves generally decreases the average slurry impurity concentration, and increases the average contact stress on the pad–wafer interface. However, as a grooved pad has less contact area for effective interaction with the wafer surface, the average MRR may or may not be increased, depending upon the specific setting of process parameters. Meanwhile, it appears that the retaining ring generally used to keep the wafer in place also plays an important part in reducing the MRR non-uniformity.     

Influence of the elastic stress relaxation on the microstructures and mechanical properties of metal–matrix composites

Role of the residual stresses on the mechanical properties of metal–matrix composites is studied. It is shown that the stress relaxation can be responsible for the morphologies and spatial distribution of precipitates. Direct measurements of the residual stress is also emphasized and the influence of dislocations in the accommodation process and during interface crossing is exemplified.     

Oblique penetration of a rigid projectile into a thick elastic–plastic target: theory and experiment

A model of oblique penetration of a rigid projectile into a thick elastic–plastic target has been developed (Roisman et al., Int J Impact Engng 1997; 19: 769–95) which incorporates stress-free boundary conditions at the rear surface of the target. The main objective of the present work is to validate the theoretical model by comparison with new experimental results for normal and oblique penetration of a rigid projectile into a thick plate of Al 6061-T651. Good agreement between theory and experiment is exhibited for the projectile residual velocity and the crater shape.