Elastodynamic Fundamental Solution Research Articles

Dynamic BEM analysis of elastic anisotropic nano‐sheet with surface imperfections and cavities

AbstractThe paper considers elastic anisotropic nano‐sheet with nanotopography and containing embedded nanocavities under time‐harmonic in‐plane wave motion. The mechanical model is based on the classical elastodynamic theory for the anisotropic bulk solid and the nonclassical boundary conditions along its boundary derived by Gurtin and Murdoch. A localized constitutive equation for the half‐plane boundary is considered in the frame of surface elasticity theory. The computational tool is an efficient direct displacement boundary element method (BEM) based on the frequency‐dependent elastodynamic fundamental solution for elastic generally anisotropic material. To show the versatility of the proposed methodology, wave propagation in an elastic anisotropic heterogeneous nano‐sheet with nanorelief presented by hill‐canyon nanotopography is studied. The simulations illustrate the dependence of the wave field on the material anisotropy, on the surface elasticity properties, on the nanorelief peculiarities, on the nanocavities existence and their dynamic interaction, and on the incident wave characteristics. The obtained results reveal the potential of the developed mechanical model based on the boundary integral equations in the frame of surface elasticity theory to produce highly accurate results by using strongly reduced discretization mesh in comparison with the domain‐based methods.

Scattering Analysis and Detection of Layered Plate Debonding Using Guided SH Waves with Boundary Element Method

This paper investigates the scattering behavior of guided shear horizontal (SH) waves in a two-dimensional, isotropic, and linear elastic layered plate with partially debonded interface by analyzing the reflection and transmission coefficients of scattered waves. The partial wave technique is established to form the displacement and stress of guided wave functions, and the boundary element method (BEM) is utilized to handle the numerical calculation with elastodynamic fundamental solutions in the frequency domain. After applying proper boundary conditions including continuity condition on the interface with traction-free debonding, the scattering coefficients can be obtained in terms of boundary element solutions. Two different materials (steel and aluminum) with various debonding lengths and locations in a 1 mm double-layered plate are considered. With several modes of the incident wave over a frequency range up to 4.5 MHz, the variations of scattering coefficients and scattering phenomena are numerically investigated as several parameters such as mode of the incident wave, materials, locations, and length of debonding are changed. The numerical results also suggest the potential of the suitable wave mode for the debonding detection, which can be useful for non-destructive inspection.

3D hybrid model of foundation‐soil‐foundation dynamic interaction

Abstract3D dynamic interaction of two adjacent elastic foundations embedded in a finite layered soil region rested in a homogeneous elastic isotropic half‐space with a transient dynamic source is studied. The hybrid computational model, the corresponding numerical scheme and the accompanied software are developed, verified and inserted in detail parametric study. The hybrid approach is based on (a) finite element method (FEM) describing the scattered wave field in a finite layered soil profile with two foundations; (b) boundary element method (BEM) considering waves radiating from a dynamic transient source in elastic semi‐infinite range. The aim is to propose an efficient hybrid methodology for evaluation of the dynamic response of a foundation‐soil‐foundation system, taking into account (a) the whole wave path from the dynamic source, through the layered soil region, till the underground structures; (b) the damage state of the geological material. The BEM model is based on the 3D elastodynamic fundamental solution in Fourier domain and it is applied in order to obtain frequency‐dependent stiffness matrix and load vector of the dynamically active semi‐infinite geological zone. Once the BEM model is formulated, it is inserted as a macro‐finite element in the FEM software package ABAQUS. The frequency‐dependent FEM model describing the wave field in finite layered soil profile with two elastic foundations is realized by ABAQUS. Solutions in time domain are obtained through application of the inverse fast Fourier transform. As a final result an efficient hybrid model comprising all in one: dynamic (seismic or other type) source‐homogeneous elastic half‐space‐finite heterogeneous by layers and foundations soil profile, is presented. Parametric study illustrating the sensitivity of the dynamic field to key factors such as the type and properties of the load, the soil layering, the material properties of the far‐ and near‐field soil regions, the damaged state of the geological material and the foundation‐soil‐foundation interaction is shown and discussed.

Wave Scattering by Cracks at Macro- and Nano-Scale in Anisotropic Plane by Boundary Integral Equation Method

AbstractElastic wave scattering by cracks at macro- and nano-scale in anisotropic plane under conditions of plane strain is studied in this work. Furthermore, time-harmonic loads due to incident plane longitudinal P- or shear SV- wave are assumed to hold. In a subsequent step, the elastodynamic fundamental solution for general anisotropic continua derived in closed-form via the Radon transform is implemented in a numerical scheme based on the traction boundary integral equation method (BIEM). The surface elasticity effect in the case of nano-crack is taken into consideration via non-classical boundary condition along the crack surface proposed by Gurtin and Murdoch [1]. The numerical results obtained herein reveal substantial differences between anisotropic materials containing a macro- and a nano-crack in terms of their dynamic stress response, where the latter case demonstrates clearly the strong influence of the size-effects. Finally, these types of examples serve to illustrate the present approach and to show its potential for evaluating the stress concentration fields (SCF) inside cracked nanocomposites. The obtained results concern the reliability and safety of the advancing nanomaterials.

A low frequency elastodynamic fast multipole boundary element method in three dimensions

This paper presents a fast multipole boundary element method (FMBEM) for the 3-D elastodynamic boundary integral equation in the `low frequency' regime. New compact recursion relations for the second-order Cartesian partial derivatives of the spherical basis functions are derived for the expansion of the elastodynamic fundamental solutions. Numerical solution is achieved via a novel combination of a nested outer---inner generalized minimum residual (GMRES) solver and a sparse approximate inverse preconditioner. Additionally translation stencils are newly applied to the elastodynamic FMBEM and an implementation of the 8, 4 and 2-box stencils is presented, which is shown to reduce the number of translations per octree level by up to $$60\,\%$$60%. This combination of strategies converges 2---2.5 times faster than the standard GMRES solution of the FMBEM. Numerical examples demonstrate the algorithmic and memory complexities of the model, which are shown to be in good agreement with the theoretical predictions.

Seismic response of laterally inhomogeneous geological region by boundary integral equations

The proposed study deals with synthesis of seismograms by BIEM (boundary integral equation method) taking into account all three base components-seismic source, wave path and local region of interest. Consider a laterally inhomogeneous geological profile situated in a half-plane with non-parallel layers. Seismic load is time-harmonic or transient in time. It is presented by incident SH wave or wave radiating from an embedded line seismic source. Two types of lateral inhomogeneities with arbitrary shape and located in the inhomogeneous half-plane are considered: (i) free-surface relief as a canyon or a hill; (ii) alluvial basin with properties different from those of the layered half-plane. The computational tool is BIEM based on the frequency-dependent elastodynamic fundamental solutions. A relation between displacements and tractions along the free surface and arbitrary interface of the soil stratum is derived, which is applicable for arbitrary geometry of the interfaces between soil layers. Validation and convergence study is presented. All simulations reveal the sensitivity of the synthetic seismic signals on the type and characteristics of the seismic time-harmonic or transient load, on the wave path inhomogeneity and on the specific geotechnical properties of the local geological region.

Comparative study of time and frequency domain BEM approaches in frictional contact problem for antiplane crack under harmonic loading

Abstract Two different boundary element methods (BEM) for crack analysis in two dimensional (2-D) antiplane, homogeneous, isotropic and linear elastic solids by considering frictional contact of the crack edges are presented. Hypersingular boundary integral equations (BIE) in time-domain (TD) and frequency domain (FD), with corresponding elastodynamic fundamental solutions are applied for this purpose. For evaluation of the hypersingular integrals involved in BIEs a special regularization process that converts the hypersingular integrals to regular integrals is applied. Simple regular formulas for their calculation are presented. For the problems solution while considering frictional contact of the crack edges a special iterative algorithm of Udzava's type is elaborated and used. Numerical results for crack opening, frictional contact forces and dynamic stress intensity factors (SIFs) are presented and discussed for a finite III-mode crack in an infinite domain subjected to a harmonic crack-face loading and considering crack edges frictional contact interaction using the TD and FD approaches.

Three-dimensional dynamic Green’s functions in transversely isotropic tri-materials

An analytical derivation of the elastodynamic fundamental solutions for a transversely isotropic tri-material full-space is presented by means of a complete representation using two displacement potentials. The complete set of three-dimensional point-load, patch-load, and ring-load Green’s functions for stresses and displacements are given, for the first time, in the complex-plane line-integral representations. The formulation includes a complete set of transformed stress-potential and displacement-potential relations in the framework of Fourier expansions and Hankel integral transforms, that is useful in a variety of elastodynamic as well as elastostatic problems. For the numerical computation of the integrals, a robust and effective methodology is laid out. Selected numerical results for point-load and patch-load Green’s functions are presented to portray the dependence of the response on layering, the frequency of excitation, and type of loading.

A wideband fast multipole accelerated boundary integral equation method for time‐harmonic elastodynamics in two dimensions

SUMMARYThis article presents a wideband fast multipole method (FMM) to accelerate the boundary integral equation method for two‐dimensional elastodynamics in frequency domain. The present wideband FMM is established by coupling the low‐frequency FMM and the high‐frequency FMM that are formulated on the ingenious decomposition of the elastodynamic fundamental solution developed by Nishimura's group. For each of the two FMMs, we estimated the approximation parameters, that is, the expansion order for the low‐frequency FMM and the quadrature order for the high‐frequency FMM according to the requested accuracy, considering the coexistence of the derivatives of the Helmholtz kernels for the longitudinal and transcendental waves in the Burton–Muller type boundary integral equation of interest. In the numerical tests, the error resulting from the fast multipole approximation was monotonically decreased as the requested accuracy level was raised. Also, the computational complexity of the present fast boundary integral equation method agreed with the theory, that is, Nlog N, where N is the number of boundary elements in a series of scattering problems. The present fast boundary integral equation method is promising for simulations of the elastic systems with subwavelength structures. As an example, the wave propagation along a waveguide fabricated in a finite‐size phononic crystal was demonstrated. Copyright © 2012 John Wiley & Sons, Ltd.

Transient elastodynamic analysis of nonhomogeneous anisotropic plane bodies

A boundary-only BEM procedure is employed to solve the transient dynamic analysis of nonhomogeneous anisotropic plane elastic bodies. The response of such bodies is governed by two coupled linear, second-order hyperbolic PDEs with spatially dependent coefficients. The lack of a reliable 2D time-domain elastodynamic fundamental solution is overcome using the principle of the Analog Equation, a method by which the equations of motion of the problem are substituted by two coupled quasi-static Poisson-type equations having as nonhomogeneous terms the components of a fictitious time-dependent load distribution in the specified domain. The standard BEM is employed for the solution of the substitute equations. To avoid the appearance of the domain integral in the integral representation of the solution, the fictitious load distribution is approximated by multiquadrics with unknown time-dependent expansion coefficients, which are calculated at discrete timepoints by collocating the equations of motion at a predefined set of domain interpolation nodes. The obtained numerical results by the proposed method demonstrate its stability and accuracy over other numerical methods.

Analytical 3D transient elastodynamic fundamental solution of unsaturated soils

AbstractUnsaturated soils are considered as porous continua, composed of porous skeleton with its pores filled by water and air. The governing partial differential equations (PDE) are derived based on the mechanics for isothermal and infinitesimal evolution of unsaturated porous media in terms of skeleton displacement vector, liquid, and gas scalar pressures. Meanwhile, isotropic linear elastic behavior and liquid retention curve are presented in terms of net stress and capillary pressure as constitutive relations. Later, an explicit 3D Laplace transform domain fundamental solution is obtained for governing PDE and then closed‐form analytical transient 3D fundamental solution is presented by means of analytical inverse Laplace transform technique. Finally, a numerical example is presented to validate the assumptions used to derive the analytical solution by comparing them with the numerically inverted ones. The transient fundamental solutions represent important features of the elastic wave propagation theory in the unsaturated soils. Copyright © 2010 John Wiley & Sons, Ltd.

Transient dynamic crack analysis in piecewise homogeneous, anisotropic and linear elastic composites by a spatial symmetric Galerkin-BEM

Transient elastodynamic analysis of two-dimensional, piecewise homogeneous, anisotropic and linear elastic solids containing interior and interface cracks is presented in this paper. To solve the initial boundary value problem, a spatial symmetric time-domain boundary element method is developed. Stationary cracks subjected to impact loading conditions are considered. Elastodynamic fundamental solutions for homogenous, anisotropic and linear elastic solids are implemented. The piecewise homogeneous, anisotropic and linear elastic solids are modeled by the multi-domain technique. The spatial discretization is performed by a symmetric Galerkin-method, while a collocation method is utilized for the temporal discretization . An explicit time-stepping scheme is obtained for computing the unknown boundary data. Numerical examples are presented and discussed to show the effects of the interface cracks, the material anisotropy , the material combination and the dynamic loading on the dynamic stress intensity factors .

Cracks in Magnetoelectroelastic Solids under Impact Loading

In this paper, transient dynamic crack analysis in two-dimensional, linear magnetoelectroelastic solids is presented. For this purpose, a time-domain boundary element method (BEM) is developed and the elastodynamic fundamental solutions for linear magnetoelectroelastic and anisotropic materials are derived. The spatial discretization of the boundary integral equations is performed by a Galerkin-method while a collocation method is implemented for the temporal discretization of the arising convolution integrals. An explicit time-stepping scheme is developed to compute the discrete boundary data and the generalized crack-opening-displacements. To show the effects of the coupled fields and the different dynamic loading conditions on the dynamic intensity factors, numerical examples will be presented and discussed.

Dynamische Rissanalyse in geschichteten anisotropen Verbundwerkstoffen mit einer Zeitbereichs-Randelementmethode

AbstractIn this paper we present a novel time-domain boundary element method (BEM) for dynamic crack analysis in two-dimensional (2-D) linear elastic and layered anisotropic materials. Stationary cracks subjected to impact loading are considered. The homogeneous layers are considered by the multi-domain BEM formulation. On the external boundary and the interfaces, the classical displacement boundary integral equations are used, while the hypersingular traction boundary integral equations are applied on the crack-faces inside the homogeneous layers. The time-domain elastodynamic fundamental solutions for anisotropic solids are used in the present method. The spatial discretization is performed by a Galerkin-method while a collocation method is adopted for the temporal discretization. An explicit time-stepping scheme is developed to compute the discrete boundary data and the crack-opening-displacements.

A time-harmonic fundamental solution for a class of inhomogeneous transversely isotropic media

We present here some numerical results for a two-dimensional fundamental solution in a class of inhomogeneous transversely isotropic media. The inhomogeneity is assumed to be the same not only for the stiffnesses, but also for the density. The derivation, which is based on a previous work by Rangelov et al. [T.V. Rangelov et al., Elastodynamic fundamental solutions for certain families of 2d inhomogeneous anisotropic domains: basic derivations, Eur. J. Mech. A Solids 24 (2005) 820–836], is accomplished in terms of the Radon transform and numerical integration procedures. The time-harmonic fundamental solution reveals its non-wave nature for sources with lower frequencies than the critical one. We identify a subcase of the fundamental solution which is amenable to numerical evaluation, requiring however additional constraints with respect to the elasticity constants. The general fundamental solution reveals a more complicated structure, with additional effects in comparison to the above mentioned subcase, as e.g. the loss of symmetry of the Green’s Tensor.

A hypersingular time‐domain BEM for 2D dynamic crack analysis in anisotropic solids

AbstractA hypersingular time‐domain boundary element method (BEM) for transient elastodynamic crack analysis in two‐dimensional (2D), homogeneous, anisotropic, and linear elastic solids is presented in this paper. Stationary cracks in both infinite and finite anisotropic solids under impact loading are investigated. On the external boundary of the cracked solid the classical displacement boundary integral equations (BIEs) are used, while the hypersingular traction BIEs are applied to the crack‐faces. The temporal discretization is performed by a collocation method, while a Galerkin method is implemented for the spatial discretization. Both temporal and spatial integrations are carried out analytically. Special analytical techniques are developed to directly compute strongly singular and hypersingular integrals. Only the line integrals over an unit circle arising in the elastodynamic fundamental solutions need to be computed numerically by standard Gaussian quadrature. An explicit time‐stepping scheme is obtained to compute the unknown boundary data including the crack‐opening‐displacements (CODs). Special crack‐tip elements are adopted to ensure a direct and an accurate computation of the elastodynamic stress intensity factors from the CODs. Several numerical examples are given to show the accuracy and the efficiency of the present hypersingular time‐domain BEM. Copyright © 2008 John Wiley & Sons, Ltd.

Transient dynamic analysis of interface cracks in layered anisotropic solids under impact loading

Transient elastodynamic crack analysis in two-dimensional (2D), layered, anisotropic and linear elastic solids is presented in this paper. A time-domain boundary element method (BEM) in conjunction with a multi-domain technique is developed for this purpose. Time-domain elastodynamic fundamental solutions for homogenous, anisotropic and linear elastic solids are applied in the present time-domain BEM. The spatial discretization of the boundary integral equations is performed by a Galerkin-method, while a collocation method is adopted for the temporal discretization of the arising convolution integrals. An explicit time-stepping scheme is developed to compute the unknown boundary data and the crack-opening-displacements (CODs). To show the effects of the crack configuration, the material anisotropy, the layer combination and the dynamic loading on the dynamic stress intensity factors and the scattered elastic wave fields, several numerical examples are presented and discussed.

A multi-level fast multipole BEM for 3-D elastodynamics in the frequency domain

To reduce computational complexity and memory requirement for 3-D elastodynamics using the boundary element method (BEM), a multi-level fast multipole BEM (FM-BEM) is proposed. The diagonal form for the expansion of the elastodynamic fundamental solution is used, with a truncation parameter adjusted to the subdivision level, a feature necessary for achieving optimal computational efficiency. Both the single-level and multi-level forms of the elastodynamic FM-BEM are considered, with emphasis on the latter. Crucial implementation issues, including the truncation of the multipole expansion, the optimal number of levels, the direct and inverse extrapolation steps are examined in detail with the backing of numerical experiments. A complexity analysis for both the single-level and multi-level versions is conducted. The correctness and computational performances of the proposed elastodynamic FMM are demonstrated on numerical examples, featuring up to O ( 10 6 ) DOFs run on a single-processor PC and including the diffraction of an incident P plane wave by a semi-spherical or semi-ellipsoidal canyon, representative of topographic site effects.

Fast multipole method applied to 3-D frequency domain elastodynamics

This article is concerned with the formulation and implementation of a fast multipole-accelerated BEM for 3-D elastodynamics in the frequency domain, based on the so-called diagonal form for the expansion of the elastodynamic fundamental solution, a multilevel strategy. As usual with the FM-BEM, the linear system of BEM equations is solved by GMRES, and the matrix is never explicitly formed. The truncation parameter in the multipole expansion is adjusted to the level, a feature known from recently published studies for the Maxwell equations. A preconditioning strategy based on the concept of sparse approximate inverse (SPAI) is presented and implemented. The proposed formulation is assessed on numerical examples involving O ( 10 5 ) BEM unknowns, which show in particular that, as expected, the proposed FM-BEM is much faster than the BEM traditional, and that the GMRES iteration count is significantly reduced when the SPAI preconditioner is used.

BEM Solutions of Frequency Domain Gradient Elastodynamic 3-D Porblems

A boundary element methodology is presented for the frequency domain elastodynamic analysis of three-dimensional solids characterized by a linear elastic material behavior coupled with microstructural effects taken into account with the aid of the simple gradient elastic theory of Aifantis. A variational statement is established to determine all possible classical and non-classical (due to gradient terms) boundary conditions of the general boundary value problem. The gradient frequency domain elastodynamic fundamental solution is explicitly derived and used to construct the boundary integral representation of the solution with the aid of a reciprocal integral identity. In addition to a boundary integral representation for the displacement, a boundary integral representation for its normal derivative is also necessary for the complete formulation of a well posed problem. All the kernels in the integral equations are explicitly provided. Surface quadratic quadrilateral boundary elements are employed and the discretization is restricted only to the boundary. The solution procedure is described in detail. A numerical example serves to illustrate the method and demonstrate its accuracy. The present version of the method does not provide explicit expressions for the computation of interior stresses.