Anti-plane Case Research Articles

Wave Propagation through a Piezoelectric Semiconductor Infinite Space

The propagation characteristics of anti-plane waves, i.e., the quasi-shear horizontal (QSH) waves and the carrier (CA) waves through a piezoelectric semiconductor (Piezo-SEMI) infinite space are discussed in this paper. First, the elastic, piezoelectric, dielectric, carrier mobility and diffusion constants in the considered propagation coordinate system are obtained by Bonde transformation from those in the crystal axis coordinate system. For anti-plane case, the secular equation with the mechanical, electric and the carrier concentration fields of the anti-plane waves through a Piezo-SEMI infinite space is derived. We obtain all possible wave modes, i.e., QSH waves and CA waves, that propagate in the Piezo-SEMI material by solving the secular equation. Then, the wave speeds and attenuation of QSH waves and CA waves are obtained. The steady-state carrier concentration and biasing electric field have significant effects on dispersion and attenuation of the QSH waves and the CA waves, and the sensitive areas to the wave velocity and attenuation are obtained.

Quasistatic crack growth in elasto-plastic materials with hardening: The antiplane case

Abstract We study a variational model for crack growth in elasto-plastic materials with hardening in the antiplane case. The main result is the existence of a solution to the initial value problem with prescribed time-dependent boundary conditions.

Surface‐Wave Retrieval from Generalized Diffuse Fields in 2D Synthetic Models of Alluvial Valleys

Abstract Among other methods, passive imaging technique is widely applied to obtain surface‐wave velocities. This technique implies that the average cross correlations between diffuse wavefields recorded at two observers is proportional to the imaginary part of the Green’s function. For this purpose, most applications rely on both seismic ambient noise and the coda of earthquakes. Instead, we use a generalized diffuse field (GDF), defined as the waves produced by a multiplicity of distant seismic sources. These wavefields undergo multiple scatterings along their way and at the local surface geology. In this communication, we use GDF to extract the locally generated surface waves in a 2D alluvial valley model for both inplane and antiplane cases from the retrieved Green’s function. For the inplane case, an equipartitioned cocktail of plane P , SV , and Rayleigh waves is used, whereas for the antiplane case, the incidence is a set of plane SH waves. In addition to isotropic illumination, we explore the partial illumination from one side of the valley. In both cases, we obtain dispersion curves for the Rayleigh and Love waves’ group velocities from the retrieved Green’s functions and found good agreement with the exact result for the fundamental modes of both Love and Rayleigh waves in an infinite horizontal layer. This theoretical validation is a proof of concept within an ongoing project whose goal is to improve the characterization of Mexico City subsoil throughout tomography maps of surface‐wave velocities using a collection of historical strong earthquakes recorded by the Mexico City Accelerometric Network.

Fracture models for elasto-plastic materials as limits of gradient damage models coupled with plasticity: the antiplane case

We study the asymptotic behavior of a variational model for damaged elasto-plastic materials in the case of antiplane shear. The energy functionals we consider depend on a small parameter $$\varepsilon $$ , which forces damage concentration on regions of codimension one. We determine the $$\Gamma $$ -limit as $$\varepsilon $$ tends to zero and show that it contains an energy term involving the crack opening.

The indirect boundary element method to simulate elastic wave propagation in a 2-D piecewise homogeneous domain

SUMMARY A method is presented to simulate the propagation of elastic waves in a 2-D piecewise homogeneous domain composed of several regions of arbitrary shapes. The regions can be isolated within an elastic background or also embedded among themselves. The method is based on the indirect boundary element method (IBEM) in which the diffracted field is obtained as the contributions of distributed sources along the regions boundaries. This is somehow reminiscent of Huygens’s Principle for wave fronts. The fields emitted by the sources are computed in frequency domain using analytical expressions of Green’s functions for displacements and tractions. This approach is validated, in the antiplane case, by comparing against a semianalytical solution for a basin composed of concentric semi-circular layers and embedded in a half-space. Finally, an example of in-plane wave propagation is given for a complex configuration involving five interlocked regions.

A new interfacial imperfection coupling model (IICM) and its effect on the facture behavior of a layered multiferroic composite: Anti-plane case

Abstract The across-interface diffusion and permeation of the ingredients in the manufacturing process inevitably induce magneto-electro-mechanical couplings in the interfacial regions of multiferroic composites that consist of alternate ferromagnetic (FM) and ferroelectric (FE) layers. Therefore, it is impossible for the mechanical, the magnetic and the electric interfacial imperfections to occur and grow independently. Up till now, the imperfection couplings have not been investigated. By extending the classical spring model, this article presents a new one to formulate these three kinds of imperfections and their inter-couplings in the anti-plane deformation case. Six coefficients are proposed to construct the model, three representing the imperfections and the other three characterizing the inter-couplings. On this basis, fracture analysis is further performed on a bi-layered multiferroic composite with two groups of cracks parallel to the imperfect interfacial region and each in a layer. Numerical results of the stress intensity factor (SIF) are obtained from the Cauchy singular integral equations derived by the methods of Fourier integral transform and Green's function. The effects of the imperfections and their inter-couplings on the SIFs are surveyed, yielding three main conclusions: (a) the mechanical imperfection can enhance the SIFs independently; (b) the magnetic or electric imperfection can reduce the SIFs only if it is coupled with the mechanical one; (c) the magneto/electro-mechanical imperfection coupling can help to lower down the SIFs, but the magneto-electric imperfection coupling always has no effect on the SIFs. These conclusions can serves as theoretical references for the optimal design against intra-layer fracture of the multiferroic composites.

A viscosity-driven crack evolution

We present a model of crack growth in brittle materials which couples dissipative effects on the crack tip and viscous effects. We consider the 2-dimensional antiplane case with pre-assigned crack path, and firstly prove an existence result for a rate-dependent evolution problem by means of time-discretization. The next goal is to describe the rate-independent evolution as limit of the rate-dependent ones when the dissipative and viscous effects vanish. The rate-independent evolution satisfies a Griffith's criterion for the crack growth, but, in general, it does not fulfil a global minimality condition; its fracture set may exhibit jump discontinuities with respect to time. Under suitable regularity assumptions, the quasi-static crack growth is described by solving a finite-dimensional problem.

About identification of defects in an elastic-plastic medium from boundary measurements in the antiplane case

This article is devoted to a new method to find defects in some elastic-plastic structure in the antiplane case from overdetermined data on a subpart of its boundary. This iterative method merges the method of quasi-reversibility and a simple level set technique. The foundations of the method are justified from a theoretical point of view and its efficiency is shown with the help of numerical experiments.

Propagation of elastic waves in layered composites with microdefect concentration zones and their simulation with spring boundary conditions

The possibility is studied of applying spring boundary conditions to describe propagation of elastic waves in layered composites with nonperfect contact of components or in the presence of groups of microdefects at the interface. Stiffnesses in spring boundary conditions are determined by crack density, the average size of microdefects, and the elastic properties of the materials surrounding them. In deriving the values of the effective stiffness parameters, the Baik-Thompson and Bostrom-Wickham approaches are applied, as well as the integral approach. The components of the stiffness matrices are derived from consideration of an incident, at a random angle to the interface, plane wave in the antiplane case, and at a normal angle in the plane case. The efficiency of this model and the possibility of using its results in the three-dimensional case are discussed.

Theoretical and numerical stress analysis at edges of interacting faults: application to quasi-static fault propagation modelling

SUMMARY We study theoretically and numerically the process of fault interactions within a quasi-static faulting model at long, tectonic timescale. The model handles birth, growth and sliding of multiple straight but non-coplanar interacting faults, regardless of any coseismic dynamic slip events. The study is restricted to the 2-D elastic antiplane case, an idealization of the normal faulting process. The model handles a general slip-dependent friction law for faults, to take into account a possible long-term fault weakening process. At fault tips, finite stress and progressive weakening lead to fault tip cohesive zones and ensure stability. We introduce a new numerical method based on a series development of slip profiles using a Chebyshev basis, which provides an accurate computation of large stress gradients at fault tips. Here simulations are limited to two parallel faults, which is enough to investigate many important features such as slip partitioning between faults, variable fault tip velocities and the state of stress in fault relay zones, responsible for the fault linking process. We study both a quasi-static problem and the associated spectral problem and show the link between them. We compare our quasi-static simulations with experimental results concerning the acceleration/deceleration of fault tips submitted to stress interaction and concerning the geometrical parameters that favour the linking of two normal faults. We find that the linking (coalescence) process should most likely occur during the deceleration phase of the faults tips subject to stress shadowing in the fault relay. Furthermore, for large ratios of fault lengths to separation, the linking process should begin for fault overlaps comparable to the values observed by Soliva & Benedicto in small natural fault relays (typically 2.9 times the fault separation).

Relaxation of an optimal design problem in fracture mechanic: the anti-plane case

In the framework of the linear fracture theory, a commonly-used tool to describe the smooth evolution of a crack embedded in a bounded domain Ω is the so-called energy release rate defined as the variation of the mechanical energy with respect to the crack dimension. Precisely, the well-known Griffith's criterion postulates the evolution of the crack if this rate reaches a critical value. In this work, in the anti-plane scalar case, we consider the shape design problem which consists in optimizing the distribution of two materials with different conductivities in Ω in order to reduce this rate. Since this kind of problem is usually ill-posed, we first derive a relaxation by using the classical non-convex variational method. The computation of the quasi-convex envelope of the cost is performed by using div-curl Young measures, leads to an explicit relaxed formulation of the original problem, and exhibits fine microstructure in the form of first order laminates. Finally, numerical simulations suggest that the optimal distribution permits to reduce significantly the value of the energy release rate.

Constrains on stresses in isotropic homogeneous infinite half-spaces being in welded contact: 2D anti-plane and in-plane cases

Formulas are derived for two-dimensional problems relating stresses across a plane boundary that divides infinite homogeneous half-spaces being in welded contact. The calculations are made for both anti-plane and in-plane stress cases. The results obtained for the former case that involve only two stress components are useful in the analysis of fracture of strike-slip type. For the in-plane case, the relations that link stresses in one half-space with the corresponding homogeneous stresses in the other half-space are presented for arbitrarily oriented shear and normal stresses and for the center of compression (dilatation). The above relations provide a compete set of expressions that, among other things, make it possible to analyze stresses involved in faulting of deep-slip type in an inhomogeneous medium. The quantitative preliminary evaluations based on the results obtained demonstrate the great role of low rigidity media in fracture processes of all kinds within the Earth’s crust.

On the Theory of Direct and Inverse Problems in the Elastic Rectangle: Antiplane Case

In this article, we study the antiplane deformation of the boundary surface of a rectangular domain in the presence of a void and a shear force on the outer boundary surface. For a formulated inverse problem, we develop some analytical results and use them to solve the problem numerically for various elliptic geometrical configurations. The analytical method allows us to give an efficient representation for Green's function in the rectangular domain. Then we derive the same Green's function by an alternative method based on Fourier series expansions. Finally, for a number of configurations, we demonstrate the comparison between real and reconstructed defects.

An efficient boundary integral equation method applicable to the analysis of non-planar fault dynamics

Abstract We develop a novel and efficient boundary integral equation method based on the spatio-temporal formulation for the two-dimensional dynamic and quasistatic analyses of an earthquake fault in a single scheme. A major advantage of this method is its applicability to the analysis of non-planar faults with the same degree of accuracy as to that of planar faults. Calculation time and memory requirement are reduced through the employment of asymptotic representations of the integration kernels appearing in the convolution integral. Asymptotic kernels are factorized into terms dependent on space or time alone, resulting in efficient numerical computations. In addition, the dependence on time is found to vanish in the asymptotic kernels far behind the S-wave front, which also contributes to the time-saving efficiency of the calculations. We show that, in a dynamic analysis, if a 3% error is allowed for the slip rate, computation time and memory requirement are reduced by 25% and 45%, respectively, in an in-plane fault case, and by 60% and 75%, respectively, in an anti-plane fault case. This method can be employed as a powerful numerical tool in simulating an entire earthquake cycle consisting of both quasi-static and dynamic processes with a more realistic non-planar geometry of faults.

Inverse problem on crack reconstruction in the elastic half-space: anti-plane case

The present article is concerned with the reconstruction of the position and the size of a linear crack located in the elastic half-space parallel to its boundary surface, in the case of anti-plane (shear) deformation. It is assumed that the input data is given by the results obtained from measurements of the boundary surface settlement. A respective direct problem is first formulated as a certain hyper-singular integral equation, and then it is stated that the studied inverse problem can be reduced to an optimization problem for strongly nonlinear functional. Three different methods of optimization are constructed, and the results predicted numerically by these methods are compared on some examples.

Microcrack initiation at the tip of a finite rigid conducting line in piezoelectric media

In this paper is proposed a dislocation emission mechanism for microcrack initiation at the tip of a finite rigid conducting line in a piezoelectric solid. When a finite rigid conducting line is embedded in a piezoelectric matrix, because of the highly concentrated stress and electric displacement fields at its tips, dislocations of one sign are driven away from the tip, while the stationary dislocations of the opposite sign are left behind. As a result, a micro Zener‐Stroh crack is initiated at each tip for the in-plane case, and two microcracks at each tip for the antiplane case. We obtain analytical solutions of both in-plane and anti-plane extension forces for microcracks initiated at the tip of a finite rigid conducting line. By obtaining the stress and electric displacement fields at the tip under nonzero net Burgers vectors, we observe two critical crack lengths. We find that the inplane and anti-plane critical extension forces for a finite rigid conducting line are related to those for a conventional crack in the same piezoelectric materials.

Elastodynamic guided wave scattering in infinite plates

AbstractThe scattering of elastic guided waves by defects in two‐dimensional infinite plates is analysed in the plane and antiplane cases, corresponding, respectively, to Lamb and SH modes. A hybrid boundary element–finite element technique is used, where the defect neighbourhood is discretized with quadratic boundary elements and the radiation condition in the plate is satisfied through a normal mode expansion. A semi‐analytical finite element technique is applied in the infinite plate to calculate its dispersion curves and normal modes. This hybrid technique, which showed excellent performance in the solution of Lamb wave reflection at the edge of semi‐infinite plates, is extended in this paper to a wider range of problems, such as Lamb mode scattering by delaminations and surface defects and SH mode interaction with step discontinuities. Copyright © 2003 John Wiley & Sons, Ltd.

Interaction of harmonic waves with a periodic array of interface cracks in a multi-layered medium: anti-plane case

The interaction of anti-plane elastic waves with a periodic array of interface cracks in a multi-layered medium is analyzed in this paper. The number of the layers is arbitrary and the cracks may be distributed on any one of the interfaces. Transfer matrix and Fourier series techniques are used to formulate the mixed boundary-value problem in terms of a Hilbert singular integral equation. Numerical solutions are presented for some typical cases: (i) two bonded half-spaces, (ii) a layer bonded to a half-space, (iii) two half-spaces bonded through a layer, (iv) two layers bonded to a half-space. The dependence of the dynamic stress intensity factors (DSIFs) on the frequency of the incident wave, and the influences of geometric configuration, material combination and incident angle are discussed in detail. The results show that the DSIF–frequency curves involve sharp dips and peaks in many cases, i.e. the DSIF drops to a very low value or rises to a very high value in a surprisingly narrow region of frequency. We analyze this phenomenon in detail and find that the rapid change of the DSIF in a quite narrow frequency region is caused by particular modes of Love waves propagating in the elastic layers.

Transient response of a crack in piezoelectric strip subjected to the mechanical and electrical impacts: mode-I problem

Mode-I transient responses of a piezoelectric ceramic strip containing a center-situated crack under the in-plane mechanical and electrical impacts are investigated based on integral transforms and the singular integral equation techniques. Numerical results reveal the great dependence of crack extension on both the loading combination and the ratio of crack length to strip width. It can be expected that the crack extension may be retarded by adjusting the loading combination parameter for a certain ratio of crack length to strip width. In addition, numerical results also indicate that the intrinsic mechanical–electrical coupling effects play more important roles in the dynamic fracture analysis of the in-plane problem than in the anti-plane case.

Transfer Matrix Method of Wave Propagation in a Layered Medium With Multiple Interface Cracks: Antiplane Case

The paper develops a universal method for SH-wave propagation in a multilayered medium with an arbitrary number of interface cracks. The method makes use of the transfer matrix and Fourier integral transform techniques to cast the mixed boundary value problem to a set of Cauchy singular integral equations of the first type which can be solved numerically. The paper calculates the dynamic stress intensity factors for some simple but typical examples.