Journal of Solid Mechanics
Volume:12 Issue: 2, Spring 2020

In this investigation a suitable algorithm for the detection of cracks in the pressure vessels is presented. The equations of motion for the vessel are obtained and transferred into the wavelet space in a simplified form resulted from time and position approximations. The locations of cracks are randomly distributed in different regions of the structure to cover the whole geometry of the pressure vessel. Furthermore, the pressure vessel is installed vertically with a fixed end at the bottom of each of its four leg supports. Then, the results are transferred to the wavelet space using Daubechies wavelet families. From the comparison of the displacement results associated with the intact and damaged vessels, it can be clearly seen that the crack location can be accurately detected noting the alteration in the wavelet output diagrams .The results of the crack detection show that with the proper selection of the wavelet type, the wavelet based finite element method is a suitable and nondestructive method as well as a powerful numerical tool for the detection of cracks and other discontinuities in the pressure vessels. The results of this investigation can be used in the marine and aerospace industries as well as power stations.

A generalized thermo-elastic diffusion problem in a functionally graded isotropic, unbounded, rotating elastic medium due to a periodically varying heat source in the context of fractional order theory is considered in our present work. The governing equations of the theory for a functionally graded material with GNIII model are established. Analytical solution of the problem is derived in Laplace-Fourier transform domain. Finally, numerical inversions are used to show the effect of rotation, non-homogeneity and fractional parameter on stresses, displacement, chemical potential, mass distribution, temperature, etc. and those are illustrated graphically.

The fundamental solution of the system of differential equations in bio-thermoelasticity with dual phase lag (DPL) in case of steady oscillations in terms of elementary function is constructed and basic property is established. The tissue is considered as an isotropic medium and the propagation of plane harmonic waves is studied. The Christoffel equations are obtained and modified with the thermal as well as bio thermoelastic coupling parameters. These equations explain the existence and propagation of three waves in the medium. Two of the waves are attenuating longitudinal waves and one is non-attenuating transverse wave. The thermal property has no effect on the transverse wave. The velocities and attenuating factors of longitudinal waves are computed for a numerical bioheat transfer model with phase lag. The variation with frequency, thermal parameters, blood perfusion parameter and phase lag parameter are presented graphically.   Also the reflection of plane wave from a stress free isothermal boundary of isotropic bio-thermoelastic half space in the context of DPL theory of thermoelasticity is studied. The amplitude ratios of various reflected waves are obtained and these amplitude ratios are further used to obtain the energy ratios of various reflected waves. These energy ratios are function of the angle of incidence and bio-thermoelastic properties of the medium. The expressions of energy ratios have been computed numerically for a particular model to show the effect of Poisson ratio, blood perfusion rate and phase lag parameters.

In this work, nonlocal elasticity theory is applied to analyze nonlinear free vibrations of slightly curved multi-walled carbon nanotubes resting on nonlinear Winkler and Pasternak foundations in a thermal and magnetic environment. With the aid of Galerkin decomposition method, the systems of nonlinear partial differential equations are transformed into systems of nonlinear ordinary differential equations which are solved using homotopy perturbation method. The influences of elastic foundations, magnetic field, temperature rise, interlayer forces, small scale parameter and boundary conditions on the frequency ratio are investigated. It is observed form the results that the frequency ratio for all boundary conditions decreases as the number of walls increases. Also, it is established that the frequency ratio is highest for clamped-simple supported and lowest for clamped-clamped supported. Further investigations on the controlling parameters of the phenomena reveal that the frequency ratio decreases with increase in the value of spring constant (k1) temperature and magnetic field strength. It is hoped that this work will enhance the applications of carbon nanotubes in structural, electrical, mechanical and biological applications especially in a thermal and magnetic environment.

Based on the third-order shear deformation theory (TSDT), this paper numerically investigates the natural frequencies and time response of three-layered annular plate with functionally graded materials (FGMs) sheet core and piezoelectric face sheets, under initial external electric voltage. The impressive material specifications of FGM core are assumed to vary continuously across the plate thickness utilizing a power law distribution. The equilibrium equations are obtained employing Hamilton’s method and then solved applying differential quadrature method (DQM) in conjunction with Newmark-β. Numerical studies are carried out to express the influences of the external electric voltage, aspect ratio, and material gradient on the variations of the natural frequencies and time response curves of FGM piezoelectric sandwich annular plate. It is precisely shown that these parameters have considerable effects on the free vibration and transient response.

Nowadays, with extending applications of bi-layer metallic sheets in different industrial sectors, accurate specification of each layer is very prominent to achieve desired properties. In order to predict behavior of sheets under different forming modes and determining rupture limit and necking, the concept of Forming Limit Diagram (FLD) is used. Optimization problem with objective functions and important parameters aims to find optimal thickness for each of Al3105-St14 bi-layer metallic sheet contributors. Optimized point is achieved where formability of the sheet approaches to maximum extent and its weight to minimum extent. In this paper, multi-objective Tabu search algorithm is employed to optimize the considered problem. Finally, derived Pareto front using Tabu search algorithm is presented and results are compared with the solutions obtained from genetic algorithm. Comparison revealed that Tabu search algorithm provides better results than genetic algorithm in terms of Mean Ideal Distance, Spacing, non-uniformity of Pareto front and CPU time.

Finite element formulations based generally on classical beam theories such as Euler-Bernoulli or Timoshenko. Sometimes, these two formulations could be problematic expressed in terms of restrictions of Euler-Bernoulli beam theory, in case of thicker beams due to non-consideration of transverse shear; phenomenon that is known as shear locking characterized the Timoshenko beam theory, in case of thin beams; problem of slow of convergence in regards to the element of Timoshenko beam. In responding to this problematic, a new beam finite element model is developed to study the static bending of functionally graded beams. The originality of this model lies in the use of a deformation approach with the consideration of a central node positioned in the middle of the beam. The degrees of freedom of this node are subsequently eliminated by the method of static condensation. In addition, this model is suitable for all linear structures regardless of L/h ratio. Functionally graded material beams have a smooth variation of material properties due to continuous change in micro structural details. The mechanical properties of the beam are assumed to vary continuously in the thickness direction by a simple power-law distribution in terms of the volume fractions of the constituents. A simply supported beam subjected to uniform load for different length-to-thickness ratio has been chosen in the analysis. Finite element solutions obtained with the new finite element model are presented, and the obtained results are evaluated with the existing solutions to verify the validity of the present model.

The effects of amine functionalization of carbon nanotubes (CNTs) and CNTs weight percent (wt. %), on the first bending natural frequencies and damping properties of CNT/epoxy composites are investigated in this paper. CNTs and amine functionalized CNTs (AFCNTs), with two different weight percentages, are used to manufacture the beam shaped specimens. Epoxy, CNT/epoxy (0.25 and 0.5 wt. % of CNTs) and AFCNT/epoxy (0.25 and 0.5 wt. % of AFCNTs) were fabricated. Experimental vibrational test is utilized in order to study the free vibration behavior of specimens under clamped-free boundary conditions. Natural frequencies and damping ratios are extracted from the experimental time response graphs. Results indicated that adding AFCNTs (0.5 wt. %) into the matrix material has the most effect on the natural frequency of the beam. In this case, the damping ratio has the lowest value. Moreover, scanning electron microscopy (SEM) images of the fracture surface of the specimens are prepared. The images illustrate that amine functionalization of CNTs leads to better dispersion of CNTs into the epoxy matrix. Further, it can be observed that enhancement in the value of damping ratio is more dominant than enhancement in stiffness value by dispersing AFCNTs into the epoxy resin.

In this article, the anti-plane deformation of an orthotropic sector with multiple defects is studied analytically. The solution of a Volterra-type screw dislocation problem in a sector is first obtained by means of a finite Fourier cosine transform. The closed form solution is then derived for displacement and stress fields over the sector domain. Next, the distributed dislocation method is employed to obtain integral equations of the sector with cracks and cavities under anti-plane traction. These equations are of Cauchy singular kind, which are solved numerically by generalizing a numerical method available in the literature by means of expanding the continuous integrands of integral equations with different weight functions in terms of Chebyshoff and Jacobi polynomials. A set of examples are presented to demonstrate the applicability of the proposed solution procedure. The geometric and force singularities of stress fields in the sector are also studied and compared to the earlier reports in the literature.

In the present research, a unified formulation for free vibration analysis of the bidirectional functionally graded conical and cylindrical shells and annular plates on elastic foundations is developed. To cover more individual cases and optimally tailored material properties, the material properties are assumed to vary in both the meridian/radial and transverse directions. The shell/plate is assumed to be supported by a non-uniform Winkler-type elastic foundation in addition to the edge constraints. Therefore, the considered problem contains some complexities that have not been considered together in the available researches. The proposed unified formulation is derived based on the principle of minimum total potential energy and solved using a differential transform analytical method whose center is located at the outer edge of the shell or plate; so that the resulting semi-analytical solution can be employed not only for truncated conical shells and annular plates, but also for complete conical shells and circular plates. Accuracy of results of the proposed unified formulation is verified by comparing the results with those of the three-dimensional theory of elasticity extracted from the ABAQUS finite element analysis code. A variety of the edge condition combinations are considered in the results section. A comprehensive parametric study including assessment of influences of the material properties indices, thickness to radius ratio, stiffness distribution of the elastic foundation, and various boundary conditions, is accomplished. Results reveal that influence of the meridian variations of the material properties on the natural frequencies is more remarkable than that of the transverse gradation.

In cylindrical continua, hoop stresses are induced due to the circumferential failure. This mainly happens when the cylinder is subjected to mechanical loads which vary in the circumferential directions. On the other hand, radial stress is stress in the direction of or opposite to the central axis of a cylindrical body. In the present study, the influence of curvilinear anisotropy on the radial and tangential stresses of the polar-orthotropic hollow cylinder is presented. The governing equations were derived to evaluate the radial and hoop stresses inside the material. A semi-analytical method through differential transform method (DTM) for the polar-orthotropic hollow cylinder is implemented in the solution. The findings, based on polar-orthotropy, of the effect of the radial and circumferential loads on the radial and hoop stresses of the growing cylinders, show elastic responses that assist in identifying some of the outstanding properties of the curvilinear anisotropic continuums. It is also revealed that the characteristic response of various wall thicknesses of the cylindrical segment is influenced by the fibre orientation, radial and tangential stresses. This work has shown that the curvilinear anisotropy momentously affects the radial and hoop stresses on the polar-orthotropic hollow cylinder.

In this research static deflection and free vibration of homogeneous nanobeams coated by a functionally graded (FG) layer is investigated according to the nonlocal elasticity theory. A higher order beam theory is used that does not need the shear correction factor. The equations of motion (equilibrium equations) are extracted by using Hamilton’s principle. The material properties are considered to vary in the thickness direction of FG coated layer. This nonlocal nanobeam model incorporates the length scale parameter (nonlocal parameter) that can capture the small scale effects. In the numerical results section, the effects of different parameters, especially the ratio of thickness of FG layer to the total thickness of the beam are considered and discussed. The results reveal that the frequency is maximum for a special value of material power index. Also, increasing the ratio of thickness of FG layer to the total thickness of the beam increases the static deflection and decreases the natural frequencies. These results help with the understanding such coated structures and designing them carefully. The results also show that the new nonlocal FG nanobeam model produces larger vibration and smaller deflection than homogeneous nonlocal nanobeam.

In this paper, the analytical solution of an electric and Volterra edge dislocation in a functionally graded piezoelectric (FGP) medium is obtained by means of complex Fourier transform. The system is subjected to in-plane mechanical and electrical loading. The material properties of the medium vary exponentially with coordinating parallel to the crack. In this study, the rate of the gradual change of the shear moduli and mass density is assumed to be same. At first, the Volterra edge dislocation solutions are employed to derive singular integral equations in the form of Cauchy singularity for an FGP plane containing multiple horizontal moving cracks. Then, these equations are solved numerically to obtain dislocation density functions on moving crack surfaces. Finally, the effects of the crack moving velocity, material properties, electromechanical coupling factor and cracks arrangement on the normalized mode I and mode II stress intensity factors and electric displacement intensity factor are studied.

An analysis has been presented of the effect of elastic foundation and uniform in-plane peripheral loading on the natural frequencies and mode shapes of circular plates of varying thickness exhibiting bi-directional functionally graded characteristics, on the basis of first order shear deformation theory. The material properties of the plate are varying following a power-law in both the radial and transverse directions. The numerical solutions of the coupled differential equations leading the motion of simply supported and clamped plates acquired by using Hamilton’s principle, is attained by harmonic differential quadrature method. The effect of different plate parameters namely gradient index, heterogeneity parameter, density parameter, taper parameter and thickness parameter is illustrated on the vibration characteristics for the first three modes of vibration for various values of in-plane peripheral loading parameter together with foundation parameter. Critical buckling loads in compression are calculated for both the boundary conditions by putting the frequencies to zero. The reliability of the present technique is confirmed by comparing the results with exact values and results of published work.

Given any structure, we seek to find the solution of mechanical problem as precisely and efficiently as possible. Within this condition, the BEM is robust and promising development, standing out in the analysis of cases with occurrence of large stress gradients, as in problems of fracture mechanics. Moreover, in BEM the modeling of infinite means is performed naturally, without the use of approximations. For methods involving domain integration, such as FEM, this is not possible, as models with high number of elements are usually adopted and their ends are considered flexible supports. This paper deals with the development of numerical models based on the BEM for mechanical analysis of stiffened domains. In the modeling of hardeners, immersed in a medium defined by the BEM, we tried to use both the FEM, already present in the literature, and the BEM 1D, being a new formulation. The objective was to perform the coupling of BEM with FEM and BEM 1D for elements of any degree of approximation, evaluating both isotropic and anisotropic medium. In addition, a complementary objective was to verify the effects of the adoption of different discretization and approximation degrees on the stiffeners. However, the coupling with the BEM 1D leaded to more stable results and better approximations. It was observed that the FEM results were instable for many results, mainly in the quadratic approximations. When the approximation degree rises, the methods tend to converge to equivalent results, becoming very close in fourth degree approximation. Lastly, it was observed stress concentration in the stiffeners ends. In these regions, the discretization and the approximation degree have large influence to the numerical response.

This study investigates that the sound wave propagation of multiferroic thermo elastic Nanofibers under the influence of surface effect and parametric excitation via Timoshenko form of beam equations. The equation of analytical model is obtained for Nanofiber through shear and rotation effect. The solution of the problem is reached through the coupled time harmonic equations in flexural direction. Graphs are drawn for frequency, phase velocity, piezoelectric strain, magnetic field and dynamic displacement at different vibration modes of Nanofibers. From the result obtained, it is seen that the surface effect and excitation frequency gives significant contribution to the physical variables of the Nanofiber. The frequency grows in the presence of surface effect and decay as length increases both in Euler’s and Timoshenko beam theory. Also, a comparison of numerical results is made with existing literature and good agreement is arrived. The present study is expected to be more helpful for the design of piezo-thermo-magneto-mechanical Nanofiber-based devices.