Thermal buckling and post-buckling analysis of functionally graded beams based on a general higher-order shear deformation theory

In the present work, attention is focused on the prediction of thermal buckling and post-buckling behaviors of functionally graded materials (FGM) beams based on Euler–Bernoulli, Timoshenko and various higher-order shear deformation beam theories. Two ends of the beam are assumed to be clamped and in-plane boundary conditions are immovable. The beam is subjected to uniform temperature rise and temperature dependency of the constituents is also taken into account. The governing equations are developed relative to neutral plane and mid-plane of the beam. A two-step perturbation method is employed to determine the critical buckling loads and post-buckling equilibrium paths. New results of thermal buckling and post-buckling analysis of the beams are presented and discussed in details, the numerical analysis shows that, for the case of uniform temperature rise loading, the post-buckling equilibrium path for FGM beam with two clamped ends is also of the bifurcation type for any arbitrary value of the power law index and any various displacement fields.     

Analysis of wave propagation in functionally graded piezoelectric composite plates reinforced with graphene platelets

This paper presents a semi-analytical approach to investigate wave propagation characteristics in functionally graded graphene reinforced piezoelectric composite plates. Three patterns of graphene platelets (GPLs) describe the layer-wise variation of material properties in the thickness direction. Based on the Reissner-Mindlin plate theory and the isogeometric analysis, elastodynamic wave equation for the piezoelectric composite plate is derived by Hamilton’s principle and parameterized with the non-uniform rational B-splines (NURBS). The equation is transformed into a second-order polynomial eigenvalue problem with regard to wave dispersion. Then, the semi-analytical approach is validated by comparing with the existing results and the convergence on computing dispersion behaviors is also demonstrated. The effects of various distributions, volume fraction, size parameters and piezoelectricity of GPLs as well as different geometry parameters of the composite plate on dispersion characteristics are discussed in detail. The results show great potential of graphene reinforcements in design of smart composite structures and application for structural health monitoring.     

Natural frequencies of a functionally graded cracked beam using the differential quadrature method

The present work is concerned with the free vibration analysis of an elastically supported cracked beam. The beam is made of a functionally graded material and rested on a Winkler–Pasternak foundation. The line spring model is employed to formulate the problem. The method of differential quadrature is applied to solve it. The obtained results agreed with the previous similar ones. Further, a parametric study is introduced to investigate the effects of the geometric and elastic characteristics of the problem on the values of natural frequencies and mode shape functions.     

One-dimensional analysis for magneto-thermo-mechanical response in a functionally graded annular variable-thickness rotating disk

In this paper, the magneto-thermo-mechanical response of a functionally graded magneto-elastic material (FGMM) annular variable-thickness rotating disk is investigated. The material properties namely material stiffness, heat conduction coefficient, thermal expansion coefficient, mass density and magnetic permeability are assumed to vary continuously along the radial direction according to a power law. The thickness profile of the disk placed in a uniform magnetic field and subjected to the thermal load is assumed to be hyperbolic in nature. The effects of the magnetic field, grading index and geometric nonlinearity on the mechanical and thermal stresses of the disk are investigated. For a specific value of the grading index the maximum radial stress due to magneto-mechanical load in a mounted FGMM disk with hyperbolic convergent profile is found away from the center. This result is different from other thickness profile disks where the radial stresses are always at the center. It is observed that unlike radial stress in a mounted FGM disk subjected to mechanical load only where it is always tensile, the radial stress due to magneto-thermal load in a mounted FGMM disk can be both tensile and compressive type. It is seen that a decrease in the value of grading index invokes shifting of the location of the maximum temperature in FGMM disk with hyperbolic convergent profile towards the outer surface of the disk.     

Rayleigh type wave dispersion in an incompressible functionally graded orthotropic half-space loaded by a thin fluid-saturated aeolotropic porous layer

This paper is concerned with the Rayleigh wave dispersion in an incompressible functionally graded orthotropic half-space loaded by a thin fluid-saturated aeolotropic porous layer under initial stress. Both the layer and half-space have subjected to the incompressible in nature. The particle motion of the Rayleigh type wave is elliptically polarized in the plane, which described by the normal to the surface and the focal point along with wave generation. The dispersion of waves refers typically to frequency dispersion, which means different wavelengths travel at a different velocity of phase. To deal with the analytical solution of displacement components of Rayleigh type waves in a layer over a half-space, we have taken the assistance of different methods like exponential, characteristic polynomial and undetermined coefficients. The dispersion relation has been derived based upon suitable boundary conditions. The finite difference scheme has been introduced to calculate the phase velocity and group velocity of the Rayleigh type waves. We also have derived the stability condition of the finite difference scheme (FDS) for the phase and group velocities. If a wave equation has to travel in the time domain, it is necessary to achieve both accuracy and stability requirements. In such cases, FDS is preferred because of its power, accuracy, reliability, rapidity, and flexibility. The effect of various parameters involved in the model like non-homogeneity, porosity, and internal pre-stress on the propagation of Rayleigh type waves have been studied in detail. Graphical representations for the effects of various parameters on the dispersion equation have been represented. Numerical results demonstrated the accuracy and versatility of the group and phase velocity depending on the stability ratio of the FDS.     

Numerical analysis of a water wave model with a nonlocal viscous dispersive term using the diffusive approach

In this paper, we numerically study the water wave model with a nonlocal viscous term where is the Riemann‐Liouville half‐order derivative in time. We propose and compare different numerical schemes based on the diffusive realizations of fractional operators.     

Persistence of traveling wave solutions in a bio-reactor model with nonlocal delays

By considering the random migration of individuals and the period of consuming the captured nutrient, we first introduce the nonlocal delays into a bio-reactor model. The persistence of nontrivial traveling wave solutions then is proved by combining the geometric singular perturbation theory with the center manifold theorem. From the viewpoint of biology, our results indicate that the nonlocality induced by small average delays is harmless to the growth of the species.     

Wave propagation and transient response of a functionally graded material plate under a point impact load in thermal environments

In this paper, the wave propagation and transient response of an infinite functionally graded plate under a point impact load in thermal environments are studied. The thermal effects and temperature-dependent material properties are taken into account. The temperature field considered is assumed to be a uniform distribution over the plate surface and varies in the thickness direction only. Material properties are assumed to be temperature-dependent, and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents. Considering the effects of transverse shear deformation and rotary inertia, the governing equations of the wave propagation in the functionally graded plate are derived from Hamilton’s principle. The analytic dispersion relation of the functionally graded plate is obtained by means of integral transforms and a complete discussion of dispersion for the functionally graded plate is given. Using the dispersion relation and integral transforms, exact integral solutions of the functionally graded plate under a point impact load in thermal environments are obtained. The influences of the volume fraction distributions and temperature field on the wave propagation and transient response of functionally graded plates are discussed in detail. The results carried out can be used in the ultrasonic inspection techniques and provide a theoretical basis for engineering applications.     

On the longitudinal wave propagation in a viscoelastic cylinder

Résumé L'auteur étudie le calcul des déplacements d'un matériau visco-élastique linéaire dans un cylindre d'étendue finie qui sont produits par des oscillations longitudinales. Il montre qu'avec une approximation pour le cas des basses fréquences, on arrive au même résultat que celui obtenu de façon rigoureuse par un autre auteur. De plus, il montre que cette méthode peut être utilisée dans la region de hautes fréquences.     

Elastic solutions of a functionally graded cantilever beam with different modulus in tension and compression under bending loads

This paper considers a functionally graded cantilever beam with different modulus in tension and compression. The beam is subjected to bending loads, including pure bending, shear force at the free end and uniform pressure on the upper lateral, respectively. Its modulus values in tension and compression both change with the thickness coordinate as arbitrary functions, which could bring the beam a broader range of applications in engineering. The problem is treated as a plane stress case and described by Airy stress function. By using semi-inverse method, the elastic solutions for the beam are obtained, which can be easily degenerated into the ones for homogeneous beams. An example is finally presented to show the effect of nonhomogeneous materials with different modulus on the elastic field in a cantilever beam.     

On wave propagation in a random conducting magneto-non-simple thermo-viscoelastic medium

The paper examines the problem of wave propagation in a random conducting magneto-non-simple thermo-viscoelastic medium. The medium has been assumed to be weakly conducting and weakly thermal. The thermomechanical coupling parameter and the conductivity are random functions, proportional to ε, with non-zero mean values, ε measuring the smallness of the scale of random fluctuation of inhomogeneities of the medium. The smooth perturbation technique enunciated by Keller (1964) has been employed to analyze the appropriate dispersion equation in non-simple thermoelastic medium. The longitudinal and transverse waves were discussed by using a particular form of thermomechanical coupling parameter representing the corresponding auto-correlation function. The effect of magnetic conductivity has been investigated. The phenomena of attenuation of waves and change of phase speed were discussed numerically in details.     

On a wave map equation arising in general relativity

We consider a class of space‐times for which the essential part of Einstein's equations can be written as a wave map equation. The domain is not the standard one, but the target is hyperbolic space. One ends up with a 1 + 1 nonlinear wave equation, where the space variable belongs to the circle and the time variable belongs to the positive real numbers. The main objective of this paper is to analyze the asymptotics of solutions to these equations as t → ∞. For each point in time, the solution defines a loop in hyperbolic space, and the first result is that the length of this loop tends to 0 as t^?1/2 as t → ∞. In other words, the solution in some sense becomes spatially homogeneous. However, the asymptotic behavior need not be similar to that of spatially homogeneous solutions to the equations. The orbits of such solutions are either a point or a geodesic in the hyperbolic plane. In the nonhomogeneous case, one gets the following asymptotic behavior in the upper half‐plane (after applying an isometry of hyperbolic space if necessary):

• 1 The solution converges to a point.
• 2 The solution converges to the origin on the boundary along a straight line (which need not be perpendicular to the boundary).
• 3 The solution goes to infinity along a curve y = const.
• 4 The solution oscillates around a circle inside the upper half‐plane.

Thus, even though the solutions become spatially homogeneous in the sense that the spatial variations die out, the asymptotic behavior may be radically different from anything observed for spatially homogeneous solutions of the equations. This analysis can then be applied to draw conclusions concerning the associated class of space‐times. For instance, one obtains the leading‐order behavior of the functions appearing in the metric, and one can conclude future causal geodesic completeness. © 2004 Wiley Periodicals, Inc.     

Surface wave propagation in a liquid-saturated porous solid layer lying over a heterogeneous elastic solid half-space

Dispersion equation is derived for the propagation of Rayleigh type surface waves in a liquid saturated porous solid layer lying over an inhomogeneous elastic solid half-space. Effect of heterogeneity on the phase velocity is studied by taking different numerical values of heterogeneity factor for particular models. Dispersion curves have been drawn showing the effect of heterogeneity on the phase velocity.     

Some remarks on the Biot system of equations describing wave propagation in a porous medium

The ill-posedness of the Cauchy problem for the Biot system of equations in the case of energy dissipation is shown (the Cauchy problem does not have a unique solution).     

On whispering gallery wave propagation in a medium with a vertical separating boundary

The solution of the problem of transmission of a whispering gallery wave through the separating boundary is constructed in the parabolic equation approximation. Formulas for the transformation coefficients of the normal modes are constructed in the case of the Dirichlet and Neumann problems. Computational results are presented.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova, Vol. 179, pp. 147–151, 1989.     

Modelling and simulation of wave transformation in porous structures using VOF based two-phase flow model

This paper represents the results of wave transformation in porous structures and hydraulic performance of a vertical porous seawall. The study was carried out using a VOF based two-phase numerical hydrodynamic model. The model was developed by coupling an ordinary porous flow model based on extended Navier–Stokes equations for porous media, and a two-phase flow model. A unique solution domain was established with proper treatment of the interface boundary between water, air and the structure. The VOF method with an improved fluid advection algorithm was used to trace the interface between water and air. The resistance to flow caused by the presence of structural material was modeled in terms of drag and inertia forces. The parameters that govern resistance to flow in a porous media were calibrated for a typical structural setup and then the computational efficacy of the model was evaluated for several wave and structural conditions other than the calibrated setup. A set of comparisons of wave properties in and around the structure showed that the model reproduced reasonably good agreement between computed results and measured data. The model was then applied to investigate wave transformation in a vertical porous structure. The role of porosity and width of a structure in reducing wave reflection and increasing energy dissipation was investigated. It is confirmed that there exists an optimum value of structure width and porosity that can maximize hydraulic performances of a porous seawall.     

Possibility of Love wave propagation in a porous layer under the effect of linearly varying directional rigidities

The present paper investigates the Love wave propagation in an anisotropic porous layer under the effect of rigid boundary. Effect of initial stresses on the propagation of Love waves in a fluid saturated, anisotropic, porous layer having linear variation in directional rigidities lying in contact over a pre-stressed, inhomogeneous elastic half-space has also been considered. The dispersion equation of phase velocity has been derived and the influence of medium characteristic such as porosity, rigid boundary, initial stress, anisotropy and inhomogeneity over it has been discussed. The velocities of Love waves have been calculated numerically as a function of KH (where K is the wave number and H is the thickness of the layer) and are presented in a number of graphs.     

Study of non-linear dynamic behavior of open cracked functionally graded Timoshenko beam under forced excitation using harmonic balance method in conjunction with an iterative technique

The nonlinear modeling and subsequent dynamic analysis of cracked Timoshenko beams with functionally graded material (FGM) properties is investigated for the first time using harmonic balance method followed by an iterative technique. Crack is assumed to be open throughout. During modeling, nonlinear strain–displacement relation is considered. Rotational spring model is adopted in order to model the open cracks. Energy formulations are established using Timoshenko beam theory. Nonlinear governing differential equations of motion are derived using Lagrange's equation. In order to incorporate the influence of higher order harmonics, harmonic balance method is employed. This reduces the governing differential equations into nonlinear set of algebraic equations. These equations are solved using two different iterative techniques. Methodology is computationally easier and efficient as well. This is observed that although assumption of simple harmonic motion (SHM) simplifies the problem, it yields to erroneous results at higher amplitude of motion. However, accuracy of the solution is improved considerably when the contribution of higher order harmonic terms are considered in the analysis. Results are compared with the available results, which confirm the validity of the methodology. Subsequent to that a parametric study on influence of forcing term, material indices and crack parameters on large amplitude vibration of Timoshenko beams is performed for two different boundary conditions.