Journal of Solid Mechanics
Volume:11 Issue: 2, Spring 2019

This paper discusses static and dynamic response of nanoplate resting on an orthotropic visco-Pasternak foundation based on Eringen’s nonlocal theory. Graphene sheet modeled as nanoplate which is assumed to be orthotropic and viscoelastic. By considering the Mindlin plate theory and viscoelastic Kelvin-Voigt model, equations of motion are derived using Hamilton’s principle which are then solved analytically by means of Fourier series -Laplace transform method. The parametric study is thoroughly accomplished, concentrating on the influences of size effect, elastic foundation type, structural damping, orthotropy directions and damping coefficient of the foundation, modulus ratio, length to thickness ratio and aspect ratio. Results depict that the structural and foundation damping coefficients are effective parameters on the dynamic response, particularly for large damping coefficients, where response of nanoplate is damped rapidly.

In this research, the effect of rotation on the free vibration is investigated for the size-dependent cylindrical functionally graded (FG) nanoshell by means of the modified couple stress theory (MCST). MCST is applied to make the design and the analysis of nano actuators and nano sensors more reliable. Here the equations of motion and boundary conditions are derived using minimum potential energy principle and first-order shear deformation theory (FSDT). The formulation consists of the Coriolis, centrifugal and initial hoop tension effects due to the rotation. The accuracy of the presented model is verified with literatures. The novelty of this study is the consideration of the rotation effects along with the satisfaction of various boundary conditions. Generalized differential quadrature method (GDQM) is employed to discretize the equations of motion. Then the investigation has been made into the influence of some factors such as the material length scale parameter, angular velocity, length to radius ratio, FG power index and boundary conditions on the critical speed and natural frequency of the rotating cylindrical FG nanoshell.

This paper studies axially symmetric vibrations of a liquid-filled poroelastic thin cylinder saturated with two immiscible liquids of infinite extent that is surrounded by an inviscid elastic liquid. By considering the stress free boundaries, the frequency equation is obtained. Particular case, namely, liquid-filled poroelastic cylinder saturated with single liquid is discussed.  When the wavenumber is large, the frequency equation is reduced to that of Rayleigh-type surface wave at the plane boundary of a poroelastic half-space. In this case, the asymptotic expressions of Bessel functions and modified Bessel functions are used. In both general and particular cases, the case of the propagation of Rayleigh waves in a poroelastic half-space is obtained.  The parameter values of Columbia fine sandy loam saturated with air-water mixture are used for the numerical evaluation.  In all the cases, phase velocity as a function of  wavenumber is computed and presented graphically. From the numerical results, some inferences are drawn.

The main purpose of present article is to find the fundamental solution of partial differential equations in the generalized theory of thermoelastic diffusion materials with double porosity in case of steady oscillations in terms of elementary functions.

Collocation methods are popular in providing numerical approximations to complicated governing equations owing to their simplicity in implementation. However, point collocation methods have limitations regarding accuracy and have been modified upon with the application of B-spline approximations. The present study reports the stress and deformation behavior of shear deformable functionally graded cantilever beam using B-spline collocation technique. The material grading is along the beam height and varies according to power law. Poisson’s ratio is assumed to be a constant. The equations are derived using virtual work principle in the framework of Timoshenko beams to obtain a unified formulation for such beams. A sixth order basis function is used for approximation and collocation points are generated using Greville abscissa. Deformation and stresses; bending (axial) stresses and transverse (shear) stresses, and position of neutral axis are studied for a wide range of power law index values. The results are reported along the beam cross-section and beam length.

It is well known that it is very difficult to manufacture perfect thin cylindrical shell. Initial geometrical imperfections existing in the shell structure is one of the main determining factor for load bearing capacity of thin cylindrical shell under uniform lateral pressure. As these imperfections are random, the strength of same size cylindrical shell will also random and a statistical method can be preferred to find the allowable load of these shell structures and therefore a In this work the cylindrical shell of size R/t = 228, L/R = 2 and t=1mm is taken for study. The random geometrical imperfections are modeled by linearly adding the first 10 eigen mode shapes using 2kfullfactorial design matrix of DoE. By adopting this method 1024 FE random imperfect cylindrical shell models are generated with tolerance limit of ± 1 mm. Nonlinear static FE analysis of ANSYS is used to find the buckling strength of these 1024 models. FE results of 1024 models are used to predict the reliability based on MVFOSM method.

This paper investigates the coupled axial-radial (CAR) vibration of single-walled carbon nanotubes (SWCNTs) based on doublet mechanics (DM) with a scale parameter. Two coupled forth order partial differential equations that govern the CAR vibration of SWCNTs are derived. It is the first time that DM is used to model the CAR vibration of SWCNTs. To obtain the natural frequency and dynamic response of the CAR vibration, the equations of motion are solved and the relation between natural frequencies and scale parameter is derived. It is found that there are two frequencies in the frequency spectrum and these CAR vibrational frequencies are complicated due to coupling between two vibration modes. The advantage of these analytical formulas is that they are explicitly dependent to scale parameter and chirality effect. The influence of changing some geometrical and mechanical parameters of SWCNT on its CAR frequencies has been investigated, too. It is shown that the chirality and scale parameter play significant role in the CAR vibration response of SWCNTs. The scale parameter decreases the higher band CAR frequency compared to the predictions of the classical continuum models. However, with increase in tube radius and length, the effect of the scale parameter on the natural frequencies decreases. The lower band CAR frequency is nearly independent to scale effect and tube diameter. The CAR frequencies of SWCNTs decrease as the length of the tube increases. This decreasing is higher for higher band CAR frequency. To show the accuracy and ability of this method, the results obtained herein are compared with the existing theoretical and experimental results and good agreement is observed.

The impetus of this study is to investigate the chaotic behavior of a size-dependent nano-beam with double-sided electrostatic actuation, incorporating surface energy effect and intermolecular interactions. The geometrically nonlinear beam model is based on Euler-Bernoulli beam assumption. The influence of the small-scale and the surface energy effect are modeled by implementing the consistent couple stress theory proposed by Hadjesfandiari and Dargush together with Gurtin-Murdoch elasticity theory. The governing differential equation of motion is derived using Hamilton’s principle and discretized to a set of nonlinear ODE through Galerkin’s method. Nonlinearities stemmed from different sources such as mid-plane stretching, electrostatic and interatomic forces lead to an intensive nonlinear dynamics in nano-electro-mechanical devices so that the systems exhibit rich dynamic behavior such as periodic and chaotic motions. Poincaré portrait is utilized in order to present the system dynamic response in discrete state-space. Bifurcation analysis has been performed with a change in the magnitude of AC voltage corresponding to the various values of DC voltage and excitation frequency. Then, we compare some ranges of AC voltage amplitude, in which the system response becomes stable for these cases. Fast Fourier transformation is also carried out to analyze the frequency content of the system response.

This paper presents the approach for the assessing of the operational reliability of a multi-layer thick-walled tube made of rubber with textile reinforcement. The analysis of the fatigue accumulation process is carried out within the framework of the concept of the continuum mechanics of damage. The mathematical model, which takes into account the accumulation of damages in case of a random spread of the strength characteristics of the material, as well as the process of stochastic aging for the elastomeric matrix of the composite and possible random variation of the workload has been developed. In this case, the aging process is modelled as a reduction of the endurance limit of the material. In this paper, the mean equivalent strains of the tube and their possible statistical variation in operation have been investigated on the basis of the finite element method. To solve the above problems, a submodeling method has been employed in this work. The probability of non-failure operation of the tube has been determined using the methods and models proposed. The influence of the rate of the aging process on the life-time of the tube has been estimated.

In this paper, the nonlinear dynamical behavior of an isotropic rectangular plate, simply supported on all edges under influence of a moving mass and as well as an equivalent concentrated force is studied. The governing nonlinear coupled PDEs of motion are derived by energy method using Hamilton’s principle based on the large deflection theory in conjuncture with the von-Karman strain-displacement relations. Then the Galerkin’s method is used to transform the equations of motion into the three coupled nonlinear ordinary differential equations (ODEs) and then are solved in a semi-analytical way to get the dynamical responses of the plate under the traveling load. A parametric study is conducted by changing the size of moving mass/force and its velocity. Finally, the dynamic magnification factor and normalized time histories of the plate central point are calculated for various load velocity ratios and outcome nonlinear results are compared to the results from linear solution.

This paper derives kinematic admissible bending moment – axial force (M-P) interaction relations for mild steel by considering strain hardening idealisations. Two models for strain hardening – Linear and parabolic have been considered, the parabolic model being closer to the experiments. The interaction relations can predict strains, which is not possible in a rigid, perfectly plastic idealization. The relations are obtained for all possible cases pertaining to the locations of neutral axis. One commercial rolled steel T-section has been considered for studying the characteristics of interaction curves for different models. On the basis of these interaction curves, most significant cases for the position of neutral axis which are enough for the establishment of interaction relations have been suggested. The influence of strain hardening in the interaction study has been highlighted. The strains and hence the strain rates due to bending and an axial force can be separated only for the linear-elastic case because the principle of superposition is not valid for the nonlinear case. The difference between the interaction curves for linear and parabolic hardening for the particular material is small.

In this paper, the vibrational and buckling analysis of a cylindrical sandwich panel with an elastic core under thermo-mechanical loadings is investigated. The modeled cylindrical sandwich panel as well as its equations of motions and boundary conditions is derived by Hamilton’s principle and the first-order shear deformation theory (FSDT). For the first time in the present study, various boundary conditions is considered in the cylindrical sandwich panel with an elastic core. In order to obtain the temperature distribution in the cylindrical sandwich panel in the absence of a heat-generation source, temperature distribution is obtained by solving the steady-state heat-transfer equation. The accuracy of the presented model is verified using previous studies and the results obtained by the Navier analytical method. The novelty of the present study is considering thermo-mechanical loadings as well as satisfying various boundary conditions. The generalized differential quadrature method (GDQM) is applied to discretize the equations of motion. Then, some factors such as the influence of length-to-radius ratio, circumferential wave numbers, thermal loadings, and boundary conditions are examined on the dynamic and static behavior of the cylindrical sandwich panel.

The comprehension of the anisotropy impacts on mechanical properties of the rolled steel sheets was investigated using a non-quadratic anisotropic yield function. In this study, experimental and modelling determination regarding the behaviour of an industrial rolled sheet for a ferritic stainless low-carbon steel were carried out. The parameters of the associated yield equation, derived from the three orthotropic yield functions proposed by Hill48, Yld96 and Yld2000-2d, were determined. Predictions and the evolution of normalized yield stress and normalized Lankford parameters (plastic strain ratio) obtained by the presented investigative are considered. The forecasts given by the YLD2000-2d criterion are consistent with that of the experience. In order to describe the path of strain behavior, the isotropic hardening function is described using the following four empirical standard formulae based on: Hollomon, Ludwick, Swift and Voce law. More accurately, the anisotropy coefficients of three yield functions are represented as a function of the longitudinal equivalent plastic strain.

In this article, dynamic modeling of double walled cylindrical functionally graded (FG) microshell is studied. Size effect of double walled cylindrical FG microshell are investigated using modified couple stress theory (MCST). Each layer of microshell is embedded in a viscoelastic medium. For the first time, in the present study, has been considered, FG length scale parameter in double walled cylindrical FG microshells, which this parameter changes along the thickness direction. Taking into consideration the first-order shear deformation theory (FSDT), double walled cylindrical FG microshell is modeled and its equations of motions are derived using Hamilton's principle. The novelty of this study is considering the effects of double layers and MCST, in addition to considering the various boundary conditions of double walled cylindrical FG microshell. Generalized differential quadrature method (GDQM) is used to discretize the model and to approximate the equation of motions and boundary conditions. Also, for confirmation, the result of current model is validated with the results obtained from molecular dynamics (MD) simulation. Considering length scale parameter (l=R/3) on MCST show, the results have better agreement with MD simulation. The results show that, length, thickness, FG power index, Winkler and Pasternak coefficients and shear correction factor have important role on the natural frequency of double walled cylindrical FG microshell.

The Magneto-Electro-Elastic (MEE) material exhibits pyroelectric and pyromagnetic effects under thermal environment. The effects of such pyroelectric and pyromagnetic behavior on vibration, buckling and deflection analysis of partially cracked thin MEE plate is presented and discussed in this paper. The aim of the study is to develop an analytical model for the vibration and geometrically linear thermal buckling analysis of cracked MEE plate based on the classical plate theory (CPT). The line spring model (LSM) is modified for the crack terms to accommodate the effect of electric and magnetic field rigidities, whereas the effect of thermal environment is accommodated in the form of thermal moment and in-plane forces. A classical relation for thermal buckling phenomenon of cracked MEE plate is also proposed. The governing equation for cracked MEE plate has also been solved to get central deflection which shows an important phenomenon of shift in primary resonance due to crack and temperature rise. The results evaluated for natural frequencies as affected by crack length, plate aspect ratio and critical buckling temperature are presented for first four modes of vibration. The obtained results reveal that the fundamental frequency of the cracked plate decreases with increase in temperature and crack length. Furthermore the variation of the critical buckling temperature with plate aspect ratio and crack length is also established for different modes of vibration.

The present problem deals with the propagation of Love-type surface waves in a bedded structure comprises of an inhomogeneous orthotropic layer and an elastic half-space. The upper boundary and the interface between two media are considered to be corrugated. An analytical method (separation of variables) is adapted to solve the second order PDEs, which governs the equations of motion. Equations for particle motion in the layer and half-space have been formulated and solved separately. Finally, the frequency relation has been established under suitable boundary conditions at the interface of the orthotropic layer and the elastic half-space. Obtained relation is found to be in good agreement with the classical case of Love wave propagation. Remarkable effects of heterogeneity and corrugation parameters on the phase velocity of the considered wave have been represented by the means of graphs. Moreover, the group velocity curves are also plotted to exhibit the profound effect of heterogeneity considered in the layer. Results may be useful in theoretical study of wave propagation through composite layered structure with irregular boundaries.