This study investigates a time-domain method for modeling general transient elastodynamic problems using the spectral-based finite element method (SEM) which is based upon a conforming mesh of two-dimensional quadrilaterals. Employing the Galerkin weighted residual method, detailed formulation of the SEM is derived in which various aspects involving in elastodynamic problems are discussed. The accuracy and efficiency of the method is fully demonstrated by comparing results obtained from the SEM with those reported in other studies. For this purpose, a set of wave propagation and structural dynamic problems, subjected to various load forms such as triangular load, Heaviside step load, sinusoidal impulsive load, and ramped load are modeled using the SEM. Furthermore, support motion boundary conditions are examined using the SEM. Each problem is successfully modeled using a very small number of degrees of freedom in comparison with other numerical methods. The numerical results agree very well with the analytical solutions as well the results from other numerical methods.

The aim of this paper is to present scalar potential functions for three-dimensional elastodynamic problems in transversely isotropic media defined by different cylindrical coordinate systems. To do so, the standard cylindrical coordinate system, elliptical cylindrical coordinate system and parabolic cylindrical coordinate system are considered. By virtue of Poisson’s representation, Lame solution and Chadwick-Trowbridge, the completeness of such representations is addressed. The representations given here are useful in the study of wave scattering from the circular, elliptical and parabolic edges.

SummaryIn this study, an analytical solution has been introduced to solve the explosion-induced wave propagation phenomenon in rock using elastodynamic theory.IntroductionThe explosion of explosives in the blast-hole can apply very high pressure to the surrounding rock media. The pressure can cause to propagate cracks and fractures and eventually fragment and destroy them. Rock blasting, like other dynamic phenomena, needs to propagate waves and dynamic stresses through the body. Therefore, the study of the propagation of waves is of great importance in understanding the subsequent effects of rock blasting. For this purpose, an analytical algorithm is used to obtain the appropriate Green functions to achieve the best description of the problem. To solve the blast-induced wave propagation problem.Methodology and ApproachesA two-dimensional elastodynamic Green's function has been analytically derived. Firstly, Navier's equations of motion and explosion equations have been used as governing equations. Then, using the Helmholtz theory, the Navier equations have been separated into two scleral and vector fields in terms of displacement. Explosion pressure as a body force has been added to the equations using the Poisson function. After performing the relevant calculations and solving the above equations, the corresponding Green function in terms of displacement is obtained. Moreover, a time-dependent green function is developed for the particle velocity using the basic functions and the resulting green function. After finding the appropriate Green functions, a typical problem consisting of a blast-hole in an elastic homogenous isotropic infinite rock media has been investigated and modeled. Finally, to validate, a real problem has been analytically modeled and solved using the proposed solution.Results and ConclusionsIts results have been compared with results from a finite element-discrete element numerical method and the explosion reality results. Based on the results of the comparison, there is a good agreement between the analytical, experimental, and numerical results at different distances from the blast-hole. A good match between the analytical results of this study and the experimental and numerical results of previous studies shows the validity of the proposed method and confirms the Green function obtained in this study. In general, it can be said that this method can be used to study the propagation of elastic waves in rock and rock-like materials. Also, it can be utilized from the resulting Green functions in other methods for different purposes as an input dynamic function.

In the present paper, an exact solution for transient response of an infinitely long functionally graded thick-walled cylinder subjected to dynamic. Pressures at the boundary surfaces are presented for arbitrary initial conditions. The cylinder is assumed to have a plane-strain condition and the dynamic pressures are assumed to be imposed uniformly and ax symmetrically on the boundary surfaces. Material properties of the cylinder are assumed to vary through the thickness according to a power law function. In contrast to many previous researches, the FGM cylinder is not divided into isotropic sub-cylinders. A solution approach associated with the expansion of the transient wave functions in terms of a series of the Eigen functions is employed. The dynamic radial displacement expression is divided into quasi- static and dynamic parts and for each part, an analytical solution is presented. By this method, radial displacement and stress distributions in the functionally graded thick-walled cylinders are obtained for various values of the exponent of the power law function, various radius ratios, and various dynamic loads. Finally, advantages of the proposed method are discussed.

1. Introduction In this paper, a new semi-analytical method is developed for analyzing concrete gravity dams in the frequency domain. Among different numerical methods, the finite element method (FEM), the boundary element method (BEM), and the scaled boundary finite element method (SBFEM) are more popular. BEM requires basically reduced surface discretization and may be considered as an appealing alternative to FEM for elastodynamic problems but requires fundamental solution of the governing differential equations. Although coefficient matrices of BEM are much smaller than those of FEM, they are routinely non-positive definite, nonsymmetric, and fully populated. The SBFEM combines the advantages of the FEM and the BEM. The SBFEM is a semi-analytical method for solving partial differential equations by transforming the governing partial differential equations to ordinary differential equations. In the SBFEM, similar to the BEM, the boundary of the problem’, s domain is discretized, while no fundamental solution is required. A modified form of the SBFEM with diagonal coefficient matrices has been proposed (Fakharian amd Khodakarami, 2015) for solving elastodynamic problems in the time domain. In this study, the semi-analytical approach for solving elastodynamic problems in the frequency domain has been applied, the governing equations in local coordinate system has been developed and two concrete gravity dams with rigid foundations and empty reservoir have been analyzed under the earthquake harmonic load...

After a short review of the poroelastic theory and presentation of a Finite- Element formulation, a one-dimensional elastodynamic analysis of aortic tissue has been carried out. Using a computer program ( in Visual Basic) and applying the material and geometrical parameters, elastic and poroelastic theories have been compared. Effect of certain parameters including fluid density, permeability and porosity, on tissue behavior has been studied.

The scaled boundary based methods are commonly known as semi-analytical approaches, which have very good accuracy and efficiency for solving various kinds of problems. In this paper, a new trend for improvement of the decoupled scaled boundary-finite element method (DSBFEM) in order to solve the 2D elastostatic and elastodynamic problems is provided. In this technique, only the boundaries of the problem domain are discretized by specific sub-parametric elements. Mapping functions are employed as a class of higher-order Lagrange polynomials which are set at Gauss-Lobatto-Legendre control points, so, the special shape functions, Gauss-Lobatto-Legendre numerical integration, and the integral form of the weighted residual method lead to the diagonal coefficient matrices in the governing equations. The main differences between the study conducted and the prior researches regarding decoupled scaled boundary-finite element method is that here in, geometry production procedure of the interpolation function, integration of the different is selected, and using this approach, we could reduce the complexity of the DSBFEM. Validity and accuracy of the present method are demonstrated through two benchmark elastostatic problems and three benchmark elastodynamic problems that are successfully modeled using a few numbers of DOFs. The numerical results agree very well with the analytical solutions and the results from other numerical methods.

In the present work, the elastodynamic field of the scattering of an anti-plane high frequency elastic shear wave due to an embedded nano cylindrical cavity in an infinite elastic medium is obtained by considering the effects of couple-stresses. In the theories accounting the effects of couple-stresses in their formulations, a new characteristic length of material is introduced into the formulations which enable them to capture size effect at micro and nano scale. Also, contrary to classical continuum theory which has difficulties in describing dispersion of wave at high frequencies, observed dispersive wave in experiments can be explained in the framework of these theories. In this work, the analytical expressions of elastodynamic fields around the cavity are obtained by considering equation of motion, dispersion relation and appropriate boundary conditions in the framework of two theories considering couple-stresses. Also, the dynamic stress concentration factor around the cavity within these theories is obtained, and, as a limiting case, the results of two cases of dynamic stress concentration factor in classical theory as well as static stress concentration factor in couple stress theories are recovered. In the framework of these theories, by several examples, the effects of frequency of incident wave and the ratio of couple stress characteristic length to the size of the cross section of the cavity on the displacement field, stress field and dynamic stress concentration factor around the cavity are studied, and the results are compared with the corresponding classical solutions.

Abstract: This work concerns the transient response in elastodynamic quarter plane problems with mixed boundary conditions. Two dimensional elastodynamic equations are solved analytically applying of integral transformations and Cagniard-de Hoop method and complex integration. The plots of stresses in response to step function are obtained by numerical integration.

In this paper, solution of inverse problems in a plane linear elastic bodies are investigated. In recent years, many studies have been conducted to develop effective approaches for damage detection in structural components. The efforts made over the last decades to overcome the mathematical challenges encountered in non-linear inverse problems may be categorized in two procedures: traditional methods, and qualitative methods. Although satisfactory results can be obtained using traditional approaches, they impose long reconstruction times associated with necessity of an accurate initial guess. These schemes require a priori information that may not be necessarily available. Consequently, the mentioned limitations have led to the conceptually distinct class of inverse scattering solutions, known as “, qualitative methods. ”,These methods are based on non-iterative obstacle reconstruction from far-and/or near-field measurements of the scattered field which avoids incorrect model assumption. Qualitative methods may be considered as probe/sampling methods such as linear sampling method, topological sensitivity, factorization method, and point source method, which seek to determine the geometric properties of scatterers. In this regard, the LSM and the FM introduced in the inverse scattering literature of far-field acoustics for the first time, are particularly attractive. This is due to the abilities of these methods to provide accurate reconstruction of the location and shape of the unknown scatterer from measurements of near-or far-field patterns, by monitoring the behavior of the norm of regularized solution. This norm is bounded inside the targets and unbounded elsewhere. Moreover, the most interesting feature of qualitative methods is that they do not require a priori information/assumption on the scatterer and/or the investigation domain. In addition, these methods may handle multiple scatterers as easily as single ones. Furthermore, these methods involve relatively low computational cost and can be applied to various types of defects such as non-convex and not-connected ones. For this purpose, sampling method in frequency domain is introduced for cavity/crack detection in a structural element such as plate. This goal is followed by partitioning the investigated region into an arbitrary grid of sampling points, in which a linear equation is solved. The main idea of the linear sampling method is to search for a superposition of differential displacement fields which matches with a prescribed radiating solution of the homogeneous governing equation in Ω, (D), for each sampling point. Although this method has been used in the context of inverse problems such as acoustics, and electromagnetism, there is no specific attempt to apply this method to identification of crack/cavities in a structural component. This study emphasizes the implementation of the sampling method in the frequency domain using spectral finite element method. A set of numerical simulations on two-dimensional problems is presented to highlight many effective features of the proposed qualitative identification method.