Classical Elastodynamics Research Articles

Classical Elastodynamics as a Linear Symmetric Hyperbolic System in Terms of $({\mathbf{u}}_{\mathbf{x}}, {\mathbf{u}}_{t})$

Classical Elastodynamics as a Linear Symmetric Hyperbolic System in Terms of $({\mathbf{u}}_{\mathbf{x}}, {\mathbf{u}}_{t})$

Semi-analytical solution for the Lamb’s problem in second gradient elastodynamics

In the present paper, we consider the Lamb-type problem of the dynamic response of isotropic half-space subjected to the time-harmonic vertical point load within the simplified strain gradient elasticity theory (SGET). The classical elastodynamics solution of this problem predicts singular displacements and stresses at the point of the load application. In the present study, we use the technique of Lamé potentials and Hankel transform to solve the high-order motion equations of SGET. An explicit solution accounting for traction and double traction boundary conditions is obtained in the transformed domain. The inverse transformation is performed numerically with accurate capturing of the solution behavior around the Rayleigh pole, which position is defined by the modified scale-dependent Rayleigh equation. Based on obtained results, it is shown that SGET solution is bounded at the epicenter of the load, and that it takes into account the spatial dispersion effects for the propagated waves. These effects are important for the analysis of the high-frequency processes in the structured materials. The intensity of non-classical effects in the solution is controlled by the length scale parameters of gradient model. Based on numerical calculations, we derive the universal non-dimensional relations between the amplitudes of the applied load and maximum displacements on the surface, which can be used to identify the length scale parameters of SGET based on the atomistic calculations or experimental data.

Seismic surface wave attenuation by resonant metasurfaces on stratified soil

AbstractThis study aims to theoretically and numerically investigate the dispersion relations of Rayleigh waves propagating through vertical oscillators periodically distributed on stratified media. The classical elastodynamics theory and an effective medium approximation method are adopted to describe the dynamic behavior of metasurfaces and hybridization between the local oscillators and the foundational surface wave modes. The Abo‐zena algorithm and delta‐matrix method are combined to simplify the Eigen equation to overcome the accuracy problem in solving the closed‐form dispersion laws and improve the computational efficiency. Subsequently, plane‐strain finite element (FE) models with three configurations are developed to confirm the analytical predictions and obtain further insight into the resonator‐Rayleigh wave coupling mechanism. The numerical results are in good agreement with the analytical solutions, revealing that only the foundational mode is strongly coupled with the vertical resonators at resonance, while the surface wave band gap reported in homogeneous media is crossed by the remaining higher‐order surface modes. The attenuation performance and mechanical behavior of a finite‐length metasurface are investigated, and it is demonstrated that the output surface ground motion can be significantly reduced in a narrow frequency band near resonance. Moreover, a graded resonant metasurface with decreasing frequency is simulated to assess the feasibility of broadband attenuation. In summary, the aforementioned analytical framework and numerical simulation results show that the vertical oscillators placed atop a stratified soil system can be designed as resonant metasurfaces for shielding seismic surface waves to protect multiple large infrastructures or special structures from earthquake hazards.

Mathematical models of supersonic and intersonic crack propagation in linear elastodynamics

This paper presents mathematical models of supersonic and intersonic crack propagation exhibiting Mach type of shock wave patterns that closely resemble the growing body of experimental and computational evidence reported in recent years. The models are developed in the form of weak discontinuous solutions of the equations of motion for isotropic linear elasticity in two dimensions. Instead of the classical second order elastodynamics equations in terms of the displacement field, equivalent first order equations in terms of the evolution of velocity and displacement gradient fields are used together with their associated jump conditions across solution discontinuities. The paper postulates supersonic and intersonic steady-state crack propagation solutions consisting of regions of constant deformation and velocity separated by pressure and shear shock waves converging at the crack tip and obtains the necessary requirements for their existence. It shows that such mathematical solutions exist for significant ranges of material properties both in plane stress and plane strain. Both mode I and mode II fracture configurations are considered. In line with the linear elasticity theory used, the solutions obtained satisfy exact energy conservation, which implies that strain energy in the unfractured material is converted in its entirety into kinetic energy as the crack propagates. This neglects dissipation phenomena both in the material and in the creation of the new crack surface. This leads to the conclusion that fast crack propagation beyond the classical limit of the Rayleigh wave speed is a phenomenon dominated by the transfer of strain energy into kinetic energy rather than by the transfer into surface energy, which is the basis of Griffiths theory.

Analytical solution to scattering of SH waves by a circular lined tunnel embedded in a semi–circular alluvial valley in an elastic half–space

In this study, the dynamic interaction between a surficial alluvial valley and an underground tunnel embedded in it under plane shear horizontal (SH) waves is investigated. Within the framework of classical elastodynamics, an analytical solution to the wave propagation and scattering around a circular lined tunnel embedded in a semi–circular alluvial valley in an elastic half space is proposed. Using the wavefunction expansion and the mirror image methods, the wave fields with unknown coefficients in three regions (i.e., the tunnel lining, alluvial valley, and rutted half space) are constructed separately in different polar coordinates to satisfy the traction–free boundary conditions along the horizontal ground surface as well as the inner surface of the tunnel lining. Using a series of transformations of the Graf’s addition formula, the coefficients of the wave fields are obtained by matching the three regions to satisfy the stress and displacement continuity conditions at the interfaces. Both the surface displacement amplitude of the alluvial valley and the dynamic stress concentration factor of the tunnel are calculated to determine the dynamic interaction effects between the surficial valley and underground tunnel. A mutual enhancement of the dynamic responses in the alluvial valley and embedded tunnel is found owing to the wave reflection, scattering, interference, and amplification.

A generalized fractional-order elastodynamic theory for non-local attenuating media.

This study presents a generalized elastodynamic theory, based on fractional-order operators, capable of modelling the propagation of elastic waves in non-local attenuating solids and across complex non-local interfaces. Classical elastodynamics cannot capture hybrid field transport processes that are characterized by simultaneous propagation and diffusion. The proposed continuum mechanics formulation, which combines fractional operators in both time and space, offers unparalleled capabilities to predict the most diverse combinations of multiscale, non-local, dissipative and attenuating elastic energy transport mechanisms. Despite the many features of this theory and the broad range of applications, this work focuses on the behaviour and modelling capabilities of the space-fractional term and on its effect on the elastodynamics of solids. We also derive a generalized fractional-order version of Snell's Law of refraction and of the corresponding Fresnel's coefficients. This formulation allows predicting the behaviour of fully coupled elastic waves interacting with non-local interfaces. The theoretical results are validated via direct numerical simulations.

Dynamic response of buried pipelines in randomly structured soil

Buried pipelines in soil undergoing ground vibrations respond by simple kinematic interaction and essentially follow the motion of the surrounding ground. More specifically, the strains that develop in the soil are imparted to the outer surface of the pipeline, which is the celebrated Newmark assumption dating from the 1960's that appears in design codes for pipelines. Of course, this assumption is invalid if the pipeline has bends and other geometric discontinuities. Furthermore, there remains the basic question of modeling the surrounding soil, which is a complex, two-phase medium with an inhomogeneous and anisotropic composition. In this work, we examine the dynamic response of a continuous pipeline by employing the waveguide model from classical elastodynamics. This implies that the pipeline is a continuously supported, beam-type structural element with distributed mass undergoing both axial and flexural vibrations. In here, we retain the influence of the axial vibrations on the flexural vibrations and work with a coupled system of two partial differential equations. The end boundaries of the pipeline are assumed to be fixed at a large distance from its center. We first examine the eigenvalue problem and then focus on the transient vibrations of the pipeline to support motion. In order to account for the complex composition of the ground, the soil impedance is converted into a random variable. Assuming uniform, Gaussian and log-normal distributions, we use Monte Carlo simulations to generate sample values for the soil impedance and compute the statistics (mean, variance and skewness) for the eigenvalue problem. Once the statistics of the eigenproperties of the buried pipeline example are recovered, the validity of the assumption that the soil is a deterministic medium is discussed.

Soil and topographic effects on ground motion of a surficially inhomogeneous semi-cylindrical canyon under oblique incident SH waves

To elucidate the ground motion amplification due to combined soil and topographic effects, an analytical formulation in the framework of classical elastodynamics is derived for the scattering of oblique incident plane SH waves by a semi-cylindrical canyon covered by a local inhomogeneous soil layer with a radially-varying modulus. Allowing the shear modulus of the finite soil layer covering the circular surface of the canyon as a power of the radial distance, the governing equation of motion for the anti-plane shear problem is derived and solved analytically by the method of wave function expansion. The ground motions for both of the homogeneous and inhomogeneous canyons under oblique incident waves can be computed efficiently and a comprehensive set of numerical examples are presented as illustrations. The degree of inhomogeneity of the soil layer and its thickness are found to affect the magnitude and the pattern of ground motion amplification of the cylindrical canyon surface depending on the frequency content, the irregular topography and obliquity of the wave incidence.

Elastodynamics and Elastostatics by a Unified Method of Potentials for x 3-Convex Domains

A new general solution in terms of two scalar potential functions for classical elastodynamics of x3-convex domains is presented. Through the establishment and usage of a set of basic mathematical lemmas, a demonstration of its connection to Kovalevshi–Iacovache–Somigliana elastodynamic solution, and thus its completeness, is realized with the aid of the theory of repeated wave equations and Boggio’s theorem. With the time dependence of the potentials suppressed, the new decomposition can, unlike Lame’s, degenerate to a complete solution for elastostatic problems.

Resolution of linear viscoelastic equations in the frequency domain using real Helmholtz boundary integral equations

Boundary integral equations are well suitable for the analysis of seismic waves propagation in unbounded domains. Formulations in elastodynamics are well developed. In contrast, for the dynamic analysis of viscoelastic media, there are very seldom formulations by boundary integral equations. In this Note, we propose a new and simple formulation of time harmonic viscoelasticity with the Zener model, which reduces to classical elastodynamics if a compatibility condition is satisfied by boundary conditions. Intermediate variables which satisfy the classical elastodynamic equations are introduced. It makes it possible to utilize existing numerical tools of time harmonic elastodynamics. To cite this article: S. Chaillat, H.D. Bui, C. R. Mecanique 335 (2007).

On the completeness of a method of potentials in elastodynamics

In this paper, the theoretical foundation of a compact scalar potential method in three-dimensional classical elastodynamics is substantiated. Beginning with a derivation of two basic lemmas on the decomposition and integration of wave solutions and vector fields which are apt to be of interest to general mechanics and analysis, the treatment proceeds to a proof of the completeness of the proposed representation as well as its extension to non-zero body forces.

Stability analysis of linear multistep methods for classical elastodynamics

We propose an analysis of the stability of linear multistep methods when they are employed for the numerical integration in time of the equations resulting from the finite element spatial discretization of the problem of linear elastodynamics. First, the notion of stability is defined for the global equations of the space–time discrete problem. Second, a spectral analysis is employed to identify simpler modal criteria that are necessary and sufficient for stability. These new criteria are stronger than the classical spectral stability criterion which, furthermore, is shown not to be sufficient for global energy stability. The results of the analysis are applied to the study of several widely employed linear multistep integrators.

Surface displacement characteristics of line crack trespassing the Rayleigh wave speed barrier as influenced by velocity and applied stress dependent local zone of restrain

Surface displacement characteristics of line crack trespassing the Rayleigh wave speed barrier as influenced by velocity and applied stress dependent local zone of restrain

STOCHASTIC PERTURBATION APPROACH TO ENGINEERING STRUCTURE VIBRATIONS BY THE FINITE DIFFERENCE METHOD

The main idea of the paper is to introduce the second order perturbation second probabilistic moment analysis in the context of the finite difference method (FDM) modelling of vibrations. The approach can be successfully applied in all those engineering analyses where FDM modelling of engineering structures vibrations is still useful and, at the same time, some structural parameters are random variables or fields. The general advantage of the stochastic finite difference method (SFDM) proposed is the relatively easy extension of the existing deterministic results of the classical elastodynamics on the random or stochastic case. However, similarly to stochastic boundary or finite element methods, the approach proposed has its limitations on the second order random uncertainties measures of input random variables.

Correction for Rahman, Point singularities and the singularity method in classical elastodynamics

Correction for ‘Point singularities and the singularity method in classical elastodynamics’ by M. Rahman (Proc. R. Soc. Lond. A 456, 2021–2042. (doi: 10.1098/rspa.2000.0600)). The correct form of the expression on p. 2025, line 12 of §3 is presented here.

Correction to: Point singularities and the singularity method in classical elastodynamics

Correction for ‘Point singularities and the singularity method in classical elastodynamics’ by M. Rahman (Proc. R. Soc. Lond. A 456 , 2021–2042. (doi: 10.1098/rspa.2000.0600)). The correct form of the expression on p. 2025, line 12 of §3 is presented here.

Point singularities and the singularity method in classical elastodynamics

The object of the article is to elucidate the application of the singularity method in classical elastodynamics. To this end, solutions for higher–order point singularities, e.g. time–harmonic concentrated moment, centre of dilatation, centre of rotation and force tensor, are derived from the influence tensor by a method usually employed in the theory of electrostatics for the study of potential of a system of charged particles. Solutions of some elastodynamic problems concerning spherical cavities and rigid spherical inclusions are then derived by combining these point singularities.

Degeneracies in the theory of plane harmonic wave propagation in anisotropic heat–conducting elastic media

This paper explores the unusual hierarchy of degeneracies in the linear theory of thermoelasticity. In classical elastic wave theory all degeneracies take the form of acoustic axes, that is directions in which two or all three plane bulk waves have equal speeds. In dynamical thermoelasticity four plane harmonic waves can travel in an arbitrary direction, and there are two types of degeneracy. The first type arises when two or more waves have equal slownesses, normally complex, and the second type when the coefficient matrix of the governing system of differential equations has a repeated zero eigenvalue. Each type of degeneracy is of two possible kinds, so the number of cases in which at least one degeneracy occurs is eight. It is shown that only three of these possibilities can actually exist and in only one of them are both types of degeneracy present. The effects of thermomechanical interaction on the modes of wave propagation are then minimal. An analysis of the degeneracies, their interconnexion and their influence on the nature of thermoelastic waves occupies the first part of the paper. In the second part the relationship of classical elastodynamics to linear thermoelasticity is studied, in respect of degeneracy, by considering small thermoelastic perturbations of an acoustic axis. The underlying degeneracy is either removed by the perturbation or divided into one or two pairs of thermoelastic degeneracies. The directions in which the new degeneracies appear are determined and the properties of the associated degenerate waves discussed in detail.

Time domain fundamental solution to Biot's complete equations of dynamic poroelasticity. Part I: Two-dimensional solution

This paper develops transient fundamental solution for Biot's full dynamic two-dimensional equations of poroelasticity. An explicit, well-posed Laplace transform domain fundamental solution is obtained for the governing differential equations which are established in terms of solid displacements and fluid pressure. In some limiting cases, the solutions are shown to reduce to those of classical elastodynamics and steady state poroelasticity, thus ensuring the validity of our result. The closed-form transient fundamental solutions both for the limiting case (early time approximation) and for the general case are derived from corresponding ones in the Laplace transform domain by means of Laplace transform techniques. They represent the very first satisfactory fundamental solutions in the real-time domain for full dynamic poroelasticity. Some characteristics of the solutions are investigated. Selected numerical results are presented to demonstrate the features of the waves, and the accuracy of the solution is established by comparing them with Laplace transform domain solutions. To complete the ingredients for developing the BEM, the Betti's reciprocal theorem is extended to the full dynamic poroelasticity, based on which a time domain boundary integral equation is obtained.

A complete solution in elastodynamics

In the context of classical elastodynamics, complete representations for the displacement vector and the stress tensor are obtained in terms of a single vector obeying a wave equation. The connection between these representations and other known representations is exhibited.