An approximation technique is considered for computing transmission and reflection coefficients for propagation of an elastic pulse through a planar slab of finite width. The propagation of elastic pulse through a planar slab is derived from first principles using straightforward time-dependent method. The paper ends with calculations of enhancement factor for the elastic plane wave and it is shown that it depends on the velocity ratio of the wave in two different media but not the incident wave form. The result, valid for quite arbitrary incident pulses and quite arbitrary slab inhomogeneities, agrees with that obtained by time-independent methods, but uses more elementary methods.

In this paper, a Mindlin rectangular nanoplate model is developed for the bending and vibration analysis of a graphene nanoplate based on a modified couple stress theory. In order to consider the small scale effects, the modified couple stress theory, with one length scale parameter, is used. In modified couple stress theory, strain energy density is a function of strain tensor, curvature tensor, stress tensor and symmetric part of couple stress tensor. After obtaining the strain and kinetic energy, external work and substituting them in the Hamilton’ s principle, the main and auxiliary EQUATIONS of the nanoplate are obtained. Then, by manipulating the boundary conditions the governing EQUATIONS are solved using Navier approach for bending and vibration of the nanoplate. The bending rates and dimensionless bending values under uniform surface traction and sinusoidal load and different mode frequencies are all obtained for various plate's dimensional ratios and material length scale to thickness ratios. The effect of material length scale, length, width and thickness of the nanoplate on the bending and vibration ratios are investigated and the results are presented and discussed in details.

In this paper, the influence of the elastic foundation on the free vibration and buckling of thin-walled piezoelectric-based functionally graded materials (FGM) cylindrical shells under combined loadings is investigated. The EQUATIONS of motion are obtained by using the principle of Hamilton and Maxwell's EQUATIONS and the NAVIER'S type solution used to solve these EQUATIONS. Material properties are changed according to power law in the direction of thickness. In this study, the effects of Pasternak elastic foundation coefficients and also the effects of material distribution, geometrical ratios and loading conditions on the natural frequencies are studied. It is observed that by increasing Pasternak elastic medium coefficients, the natural frequencies of functionally graded piezoelectric materials (FGPM) cylindrical shell always increases. The mode shapes of FGPM cylindrical shell has been shown in this research and the results show that the distribution of the radial displacements is more significant than circumferential and longitudinal displacements.

In this research, applications of the Boundary Element Method to viscous flow are investigated. The BEM formulation allows a boundary-only solution for linear stokes flow. For higher speed flows in which the non-linear convective effects cannot be ignored, a volume integral must be retained. The proposed formulation is based on analogy between NAVIER'S EQUATIONS in elastostatics and Navier-Stokes EQUATIONS expressed by using a penalty function. By using penalty function formulation, pressure term is eliminated and Navier-Stokes EQUATIONS are converted to Navier EQUATIONS in elastostatics. In many previous works, potential fundamental solution was used for solving viscous fluid flow with BEM but in this research elastostatics fundamental solution is used. Finally, some two-dimensional examples are provided to validate the presented approach. It is found that the boundary element method gives accurate solutions and it is more economical than other methods since the application of the method requires discretization only on the boundaries of the domain and thus it reduces the spatial dimensions of the problem by one.

Differential EQUATIONS are used to represent different scientific problems are handled efficiently by integral transformations, where integral transforms represent an easy and effective tool for solving many problems in the mentioned fields. This work utilizes the integral transform of the Complex SEE integral transformation to provide an efficient solution method for the difference and differential-difference EQUATIONS by benefiting from the properties of this complex transform to solve some problems related to difference and differential-difference EQUATIONS. The 3D, contour and 2D surfaces, as well as the related density plot surfaces of some acquired data, are used to draw the physical aspect of the obtained findings. The proposed approach offers an efficient and rapid solution for addressing the inherent complexity of differential-difference problems with initial conditions.

This study attempts to shed light on the analysis of the static behavior of simply supported FG type property gradient material beams according to an original refined 2D shear deformation theory. Young's modulus is considered to vary gradually and continuously according to a power-law distribution in terms of volume fractions of the constituent materials. The equilibrium EQUATIONS are obtained by applying the principle of virtual work. The governing equilibrium EQUATIONS obtained are thus solved by using the analytical model developed here and NAVIER'S solution technique for the case of a simply supported sandwich beam. Moreover, Using the numerical results of the non-dimensional stresses and displacements are calculated and compared with those obtained by other theories. Two studies are presented, comparative and parametric, the objective of which is the first to show the accuracy and efficiency of the theory used and the second to analyze the mechanical behavior of the different types of beams under the effect of different parameters. Namely boundary conditions, the material index , the thickness ratio and the type of beam.