Semi-infinite Half-space Research Articles

Static Deformation of a Uniform Half-Space with Rigid Boundary Due To a Vertical Dip-Slip Line Source

The Airy stress function for a vertical dip-slip line source buried in a homogeneous, isotropic, perfectly elastic half-space with rigid boundary is obtained. This Airy stress function is used to derive closed- form analytical expressions for the stresses and displacements at an arbitrary point of the half-space caused by vertical dip-slip line source. The variation of the displacements and stress fields with distance from the fault and depth from the fault is studied numerically. Keywords - Dip-slip faulting, Half-space, Rigid boundary, Static deformation I. Introduction Strains and stresses within the Earth constitute important precursors of earthquakes. Therefore, the determination of the static deformation of an Earth model around surface faults is important for any scheme for prediction of earthquakes. Static dislocation models are used to analyze the static deformation of the medium caused by earthquake faults. Steketee (1958a, b) applied the elasticity theory of dislocations in the field of seismology. For the sake of simplicity, Steketee ignored the curvature of the Earth, its gravity, anisotropy and non-homogeneity and dealt with a semi-infinite, non-gravitating, isotropic and homogeneous medium. Homogeneity means that the medium is uniform throughout, whereas isotropy specifies that the elastic properties of the medium are independent of direction. Maruyama (1964) calculated all the sets of Green's functions required for the displacement and stress fields around faults in a half-space. Jungels and Frazier (1973) described a finite element variational method applied to plain strain analysis. This technique presents a suitable tool for the analysis of permanent displacements, tilts and strains caused by seismic events. The accuracy of technique was demonstrated by comparing the numerical results for the static field due to long dislocation in a homogeneous half-space from closed form analytical solution with those obtained from the finite element method. Sato (1971) and Sato and Yamashita (1975) derived the expressions for the static surface deformations due to two-dimensional strike slip and dip-slip faults located along the dipping boundary between the two different media. Freund and Barnett (1976) gave a two-dimensional analysis of surface deformation due to dip- slip faulting in a uniform half-space, using the theory of analytic functions of a complex variable. Singh and Garg (1986) obtained the integral expressions for the Airy stress function in an unbounded medium due to various two- dimensional seismic sources. Singh et al. (1992) followed a similar procedure to obtain closed-form analytical expression for the displacements and stresses at any point of either of two homogeneous, isotropic, perfectly elastic half-spaces in welded contact due to two-dimensional sources. Singh and Rani (1991) obtained closed-form analytical expressions for the displacements and stresses at any point of a two-phase medium consisting of a homogeneous, isotropic, perfectly elastic half-space in welded contact with a homogeneous, orthotropic, perfectly elastic half-space caused by two-dimensional seismic sources located in the isotropic half-space. Bonafede and Rivalta (1999a) obtained analytical solutions for the elementary tensile dislocation problem in a layered elastic medium composed of two welded, semi-infinite half-spaces. A plain - strain configuration was considered and different rigidities and Poisson ratios were assumed for the two half- spaces. The elementary dislocation problem refers to a dislocation surface over which a jump discontinuity with constant amplitude (Burgers vector) is prescribed for the displacement field. Similar dislocation models in homogeneous half-spaces (e.g. Okada, 1992) are often employed to model dyke injection within the crust (e.g. Bonaccorso and Davis 1993), although a constant-displacement discontinuity, in general, is not the most realistic description of dyke opening. Bonafede and Rivalta (1999b) obtained the solutions for the displacement and stress fields produced by a vertical tensile crack, opening under the effect of an assigned overpressure within it, in the proximity of the welded boundary between two media characterized by different elastic parameters. Singh et al.(2011) obtained analytical expressions for stresses at an arbitrary point of homogeneous, isotropic, perfectly elastic half-space with rigid boundary caused by a long tensile fault of finite width.

Fast model for target scattering in a homogeneous waveguide

A fast ray model for propagation in a homogenous water column tracks time-of-flight wavepackets from sources to targets and then to receivers. The model uses image sources and receivers to account for interactions with the water column boundaries, where the layer of water lies between an upper semi-infinite halfspace of air and a lower semi-infinite halfspace of a homogenous sediment. The sediment can be either an attenuating fluid with a frequency-independent loss parameter or a fluid consistent with an effective density fluid model (i.e., a fluid limit to Biot's model for a fluid-saturated poroelastic medium). The target scattering process is computed via convolution of a free-field scattering form function with the spectrum of an incident acoustic field at the target location. A simulated or measured scattered free-field pressure from a complicate target can be reduced to a scattering form function, and this form function then can be used within model via interpolation. The fast ray-based model permits the generation of sets of realistic pings suitable for synthetic aperture sonar processing for proud and partially buried target. Results from simulations are compared to measurements where the targets are an inert unexploded ordnance and aluminum cylinder. [Research supported by SERDP and ONR.]

Theoretical Approach to Surface Heat Flux Distribution Measurement from In-Depth Temperature Sensors

I NHIGH heat load environments such as combustion chambers of rocket motors or atmospheric entry, the determination of the net surface heat flux is an essential problem. Inmost cases, only in-depth temperature sensor data is available and derivation of heat flux requires the solution of what is known as the inverse heat conduction problem (IHCP). During the last 50 years, a wide array of analytical and numerical solutions for the IHCP have been developed [1,2]. However, most solutions are based on strict initial and boundary conditions. For example, some analytical solutions are only applicable to particular settings such as the semi-infinite half-space [3]. Furthermore, the knowledge of the thermophysical properties of the materials in use is a prerequisite. Finally, the temperature sensor position is very sensitive to the solution of such problems. Although promising solutions concerning the one-dimensional IHCP have been published at least for the semi-infinite half-space, there is still uncertainty in the material properties, contact resistance, and exact temperature sensor position [4]. Furthermore, given more complex geometric bodies, in which heat conduction in all three space dimensions cannot be neglected, these tools can not be applied. In this paper, an approach named the noninteger system identification (NISI) method is extended to a three-dimensional medium. The NISI method for one-dimensional problems has first been published byBattaglia et al. [5]. It has been shown that the basic idea of applying system identification as known from automatics and control as an alternative approach to solve IHCPs holds for several analytical problems, e.g., semi-infinte, sphere. It can even be applied to more complex systems since the system identification step accounts for effects as for example deviations from one-dimensional behavior or assembly issues of the thermocouple. During identification, the sensor system, i.e., the temperature sensor as mounted inside the probe body, is calibrated. That means, adhesive layers possibly in between the sensor and the probe body affecting the thermophysical properties, thermocouple position deviating from the specified position, or thermal contact resistancies of different materials are included in the calibration step and do not further influence the data analysis. More recently, the NISI method has successfully been applied to analyze the classical null-point calorimeter developed in the 1970s [6–8]. It has been shown that an asymmetry in the heat flux profile is an artificial effect arising due to radial heat fluxes to the temperature sensor. As a result of this investigation, modifications towards a miniaturized sensor aiming to measure radial heat flux profiles in other plasma flows has been developed and successfully applied [9,10]. Again this modification became possible, because the NISI method inherently takes effects as thermocouple mounting or thermal resistancies into account through the calibration step.Within the planned hypersonic flight experiment HIFiRE, a ceramic fin has been equipped with thermocouples and first investigations towards a NISI application have been conducted [11]. In the present investigation, it is tried to extend the method to a three-dimensional problem. In a simplified form a similar analysis has been published by Battaglia [12]. By simplifying the engineering problem and some further assumptions, the multidimensional approach had been considered as a series of one-dimensional problems. In the present study, we do not assume any symmetry or similarity from the experimental point of view. Using finite element analysis of a half-ellipsoid as a generic model featuring several virtually embedded temperature sensors, the method is applied in theory. In Fig. 1, the principal procedure is shown. A half-ellipsoid is the exemplary three-dimensional geometry. There are five surfaces and five temperature sensors defined. Using a commercially available finite element modelling software (Ansys), the system is first identified according to the NISI method applying a heat flux profile to each surface element and monitoring all five temperature sensors to calibrate the system. In a second step, an exemplary heatflux profile is applied to 1) all surface elements and 2) to only two surface elements. Finally, the temperature signal, which has been numerically calculated is used to determine inversely the preset heat flux profile. The following section describes the theory as an extension of the one-dimensional problem to more sensors andmore surface areas. In Sec. III the calibration based on numerical analysis is presented followed by the application to the numerical problem to solve the inverse problem.

Effect of point source and heterogeneity on the propagation of SH-Waves in a viscoelastic layer over a viscoelastic half space

The present paper is concerned with the propagation of shear waves in a homogeneous viscoelastic isotropic layer lying over a semi-infinite heterogeneous viscoelastic isotropic half-space due to point source. The inhomogeneity parameters associated to rigidity, internal friction and density are assumed to be functions of depth. The dispersion equation of shear waves has been obtained using Green’s function technique. The dimensionless angular frequency has been plotted against dimensionless wave number for different values of inhomogeneity parameters. The effects of inhomogeneity have been shown in the dispersion curves. graphical user interface (GUI) software in MATLAB has been developed to show the effect of various inhomogeneity parameters on angular frequency. The topic can be of interest for geophysical applications in propagation of shear waves on the Earth’s crust.

Seismic response and interaction of complex soil retaining systems

This paper examines the seismic response of a large and complex system comprising a Lock chamber and three neighbouring water saving basins (WSBs). The developed two-dimensional plane strain finite element model included the entire system in order to explore the interaction of the various structures. The first set of analyses was undertaken under static conditions to simulate the construction sequence and to establish the stress regime prior to the examined seismic event. Subsequently, dynamic time domain analyses were performed to examine the response of the system to seismic loading. The ground motion was applied into the finite element mesh employing a sub-structuring technique which enables economic modelling of large computation domains and accurate representation of the semi-infinite half-space. The paper explores some aspects of the numerical modelling of such complex systems, such as the impact on the predicted seismic response of the adopted constitutive model and the modelling of hydrodynamic effects. The discussion of the dynamic analyses results focuses on the main Lock structure, looking at the prevailing modes of deformation, the potential lift-off of the base of the Lock walls and the interaction of the main wall with the other retaining structures.

Reassessment of the volume of the Las Aguilas mafic-ultramafic intrusives, San Luis, Argentina, based on an alternative geophysical model

Abstract In the Sierra de San Luis, Central Argentina, a belt of small and discontinuous lenses of mafic-ultramafic rocks intrude a polydeformed basement and are thought to be the cause of a local increase of the metamorphic grade from amphibolite to granulite facies conditions. This assumption was especially based on forward modelling of a huge gravity anomaly centered over the Sierra de San Luis, which lead some workers to think that a vast volume of mafic-ultramafic rocks lay in shallow levels. Here, we propose an alternative model to explain this anomaly, in which the mafic-ultramafic intrusion is not the ultimate source. Therefore, there is no need to propose a bigger size than that observed in outcrops for the mafic-ultramafic bodies. The thermal effect of the emplacement of mafic-ultramafic sills and dikes on the host rocks was estimated applying a simple analytical solution (error function) for heating of a semi-infinite half space (the country rocks) in contact with a hotter sheet of finite thickness (the mafic-ultramafic intrusion). Results indicate that the effect of the intrusion of these hot mafic magmas is local, because beyond a few hundred meters from the contact zone temperatures never exceed 600 °C, and a few km from the intrusion they barely increase 50 °C relative to the initial temperature. These results, together with the preservation of primary igneous characteristics (such as rhythmic layering) being overprinted by metamorphic textural changes, indicate that the intrusion occurred before regional deformation. It is suggested that the thermal anomaly in the Pringles Metamorphic Complex could have been mainly caused by factors inherent to their geodynamic setting.

Structural-acoustic modeling for three-dimensional freefield and littoral environments with verification and validation

This paper describes a high-order, finite-element-based, three-dimensional time-harmonic model for large-scale exterior structural-acoustics problems. It is applicable to both freefield and littoral environments. For the freefield case, the infinite exterior is treated as a homogeneous linear acoustic medium. For littoral applications, the water or air and the sediment domains are each treated as linear homogeneous, semi-infinite half-spaces with piecewise-constant properties. Both domains admit complex-valued wave speeds to enable the inclusion of damping. The finite element formulation uses a variational statement which naturally incorporates the transmission-condition at the water or air-sediment interface. The truncation of the infinite exterior is realized using an infinite-element for the freefield case, and the perfectly-matched-layer approximation for littoral applications. Computation of the farfield quantities is done based on an integral representation which, for the littoral cases, uses efficient approximations for the appropriate Green's function. Numerical computations are presented for a series of progressively more complex problems, and are used to verify the model against analytic and other numerical solutions and validate it based on the experimental data for scattering from elastic scatterers as measured in freefield and sediment pool laboratory facilities.

Propagation of Torsional Surface Waves under the Effect of Irregularity and Initial Stress

The present paper has been framed to study the influence of irregularity, initial stress and porosity on the propagation of torsional surface waves in an initially stressed anisotropic poro-elastic layer over a semi-infinite heterogeneous half space with linearly varying rigidity and density due to irregularity at the interface. The irregularity has been taken in the half-space in the form of a parabola. It is observed that torsional sur- face waves propagate in this assumed medium. In the absence of irregularity the velocity of torsional surface wave has been obtained. Further, it has been seen that for a layer over a homogeneous half space, the velo- city of torsional surface waves coincides with that of Love waves.

Reflection and refraction of obliquely incident elastic waves upon the interface between two porous elastic half-spaces saturated by different fluid mixtures

Summary A theoretical model for the analysis of the reflection and refraction of obliquely incident elastic waves upon the interface between two semi-infinite porous elastic half-spaces saturated by different fluid mixtures is developed in the present study based on the poroelasticity theory of Lo et al. (2005) and the normal coordinates derived by Lo et al. (2010) for describing the modes of dilatory motion. The amplitude and energy ratios of the reflected and refracted waves generated from either an incident P1 wave (the first dilatational wave) or an incident SV wave (the shear wave polarized in the vertical plane) are in turn theoretically determined for the first time with respect to the angle of incidence. As a representative example, a numerical simulation is conducted for Lincoln sand permeated by an air–water mixture in the lower half-space and Columbia fine sandy loam permeated by an air–water mixture in the upper half-space. Our numerical results indicate that regardless of the type of pore fluid mixtures and porous media, the sum of the energy ratio of the reflected and refracted waves is always equal to unity, a result that indeed can not be achieved if the normal coordinates for dilatory motional modes is not taken into account as to represent the Helmholtz potential of the reflected and refracted waves. In addition, their amplitude and energy ratios are shown to be significantly affected by the angle of incidence. It is also revealed that as a SV wave is incident upon the interface, a critical angle of 31° and 33° can be found for the reflected and refracted P1 waves respectively, while the occurrence of the critical angle is not observed for the case of an incident P1 wave.

Propagation of torsional waves in an inhomogeneous layer over an inhomogeneous half-space

The present paper is concerned with the study of propagation of torsional waves in an inhomogeneous isotropic layer whose material properties vary harmonically with a space variable, lying over a semi-infinite inhomogeneous isotropic half-space. The closed form solutions for the displacement in the layer and half-space are obtained separately. The dimensionless phase velocity has been plotted against dimensionless wave number and scaled wave number for different values of inhomogeneity parameters. The effects of inhomogeneity have been shown in the dispersion curves using 2D and 3D plot.

Amplification of linear strain in a layer excited by a shear-wave earthquake pulse

Peaks of transient strains in a layer over semi-infinite half space are amplified by interference during up- and down-propagating waves, and they depend upon the impedance ratio of the layer and of the half space, as well as the wavelength. The amplification of incident motion on the ground surface is usually described in the frequency space for the response to a steady state, periodic excitation. To understand how this amplification develops for transient responses and for applications in which the linear theory leads to satisfactory answers, this paper describes the amplification in time in a layer excited by a shear-wave pulse. It is shown that the maximum amplification is equal to 2 for a “stiff” layer on a “soft” half space, and equal to 4 for a “soft” layer on a “stiff” half space, when the layer thickness corresponds to the quarter wavelength of the wave in the layer.

Effect of irregularity on the propagation of torsional surface waves in an initially stressed anisotropic poro-elastic layer

The paper studies the propagation of torsional surface waves in an initially stressed anisotropic poro-elastic layer over a semi-infinite heterogeneous half space with linearly varying rigidity and density due to irregularity at the interface. The irregularity is taken in the half space in the form of a rectangle. It is observed that torsional surface waves propagate in this assumed medium. In the absence of the irregularity, the velocity equation of the torsional surface wave is also obtained. For a layer over a homogeneous half space, the velocity of torsional surface waves coincides with that of the Love waves.

Effect of Non-Steady State Loading on Rolling-Sliding Contact Stress

The stresses, strains, and deformations produced by repeated, two-dimensional non-steady state rolling-sliding contact were analyzed using an elastic-plastic finite element model. An advanced cyclic plasticity model was used. The non-steady state rolling contact was restricted to a harmonic variation of the normal Herztian contact pressure. Repeated rolling and sliding were simulated by multiple translations of a set of varying normal and tangential surface tractions across an elastic-plastic semi-infinite half space. The non-steady state loading considered results in a wavy contact surface profile. The surface displacements and wave depth of the wavy deformation increase with increasing rolling passes, but the increases in wave depth per rolling pass (ratchetting rate) decay. The residual stresses and strains near the wave trough of the residual wavy deformation are higher than those near the wave crest. The results are in agreement with the experimental observations. The tangential force has a greater influence on the residual strains than on the residual stresses.

Profile reconstruction of a continuously-stratified layer from reflection data on acoustic waves

The paper investigates the reflection-transmission process of acoustic waves, generated by an inhomogeneous fluid layer of finite thickness, which is sandwiched between two semi-infinite homogeneous half-spaces. First a direct problem is solved by determining the reflection and transmission coefficients along with the wave solution in the layer, produced by a known incident wave. Owing to the planar stratification of the layer, the unknown acoustic pressure is looked at as a generalized plane wave. Upon the Fourier transformation, the second-order wave equation is written as a firstorder system of equations for the dependence on the depth of the pressure and the partial derivative. The corresponding Volterra integral equation gives the pressure in the layer as a series of repeated integrals of powers of the pertinent depth-dependent matrix of the system. The reflection and transmission coefficients of the layer are then determined for any incidence angle. Next an inverse problem is investigated. The derivatives of the reflection coefficient, with respect to the frequency, are shown to provide the thickness of the layer, the speed beyond the layer and the moments, of any order, of the refractive index.

Propagation of shear waves in viscoelastic medium at irregular boundaries

The aim of the paper is to study the shear wave propagation in a viscoelastic layer over a semi-infinite viscoelastic half space due to irregularity in the viscoelastic layer. It is of great interest to study the propaga-tion of shear waves in the assumed medium having a non planar boundary due to its similarity to most of the real situations. The perturbation method is applied to find the displacement field. The effect of complex wave number on dissipation factor is analysed. Finally, as an application, the result obtained has been used to get the reflected field in viscoelastic layer when the shear wave is incident on an irregular boundary in the shape of parabolic irregularity as well as triangular notch. It is observed that the amplitude of this reflected wave decreases with increasing length of the notch, and increases with increasing depth of the irregularity.

Perfect coupling of light to surface plasmons with ultra-narrow linewidths

We examine the coupling of electromagnetic waves incident normal to a thin silver film that forms an oscillatory grating embedded between two otherwise uniform, semi-infinite half spaces. Two grating structures are considered, in one of which the midpoint of the Ag film remains fixed whereas the thickness varies sinusoidally, while in the other the mid point oscillates sinusoidally whereas the film thickness remains fixed. On reducing the light wavelength from the long wavelength limit, we encounter signatures in the transmission, T, and reflection, R, coefficients associated with: (i) the short-range surface plasmon mode, (ii) the long-range surface plasmon mode, and (iii) electromagnetic diffraction tangent to the grating. The first two features can be regarded as generalized (plasmon) Wood's anomalies whereas the third is the first-order conventional (electromagnetic) Wood's anomaly. The energy density at the film surface is enhanced for wavelengths corresponding to these three anomalies, particularly for the long-range plasmon mode in thin films. When exciting the silver film with a pair of waves incident from opposite directions, we find that by adjusting the grating oscillation amplitude and fixing the relative phase of the incoming waves to be even or odd, T+R can be made to vanish for one or the other of the plasmon modes; this corresponds to perfect coupling (impedance matching in the language of electrical engineering) between the incoming light and these modes.

Indentation of bone tissue: a short review

IntroductionBone is a highly hierarchical natural composite material,which mechanical properties are investigated from thephysiological elastic behavior up to impact or fatiguedamage accumulation responsible for traumatic or stressfractures. Each hierarchical level of organization contrib-utes to the global mechanical response of a bone structure:– The mineralized collagen fibril (MCF, 200 nm)– The lamella (2–7 μm)– The bone structural unit (BSU, 60 μm)– Bone tissue, cortical shell, or trabeculae (100–3,000 μm)– Trabecular bone (TB, mm)– Organ (cm)Bone matrix quality becomes a central focus in under-standing the etiology of fractures in diseases such asosteoporosis. Macroscopic indentation was adopted in thefirst half of the 20th century to investigate hardness of boneat the tissue level (compact and trabecular) and wasextended to micro- and nano-indentation in the past 10 yearsto evaluate specific mechanical properties of bone structuralunits and single lamellae, respectively. Over this period oftime, more than 375 papers or reviews were published(Scopus, “*indentation and bone,” June 23rd 2008).Consequently, the objective of this paper is to review therecent progress of indentation techniques to evaluate themechanical properties of human bone at these intermediatelevels of organization.Materials and methodsIndentationIndentation consists in pressing a hard tip with a knownforce into a semi-infinite half-space and measuring directlyor indirectly the contact area. In the classical hardness testperformed at the macroscopic or microscopic level, thecontact area is estimated optically from the imprint createdby the tip on the material and leads to the definition ofhardness as the force divided by this area. The mostcommon hardness measuring indenters are sphere (Brinell,Rockwell), four-sided pyramid (Vickers), and asymmetricpyramid (Knoop).Recent depth-sensing technologies allow for measure-ment of the tip displacement during the indentation processof a semi-infinite half-space with micro- and even nano-precision. Indirect estimation of the contact area is obtainedby preliminary calibration of the tip shape and systemcompliance. Shallow indentation depths down to 100 nmallow for spatial resolution of the measurements in themicron range, and the sample position is typically con-trolled by high-precision motorized tables or piezoelectricscanners. Micro- and nano-indentation tips are often madeof diamond and can be found in spherical, conical, andmost commonly, three-sided pyramidal (Berkovich) shapes.The blunted extremity of the tip has a radius of approxi-mately 100 nm. Spherical tips minimize plastic deformationand damage but are difficult to manufacture, flat puncheslead to high stress concentrations, conical tips have an axialsymmetry that remains in the assumption of most theoret-

Effects of electric field gradient on the propagation of short piezoelectric interface waves

We study anti-plane or shear horizontal piezoelectric waves propagating along the interface between two semi-infinite half-spaces of polarized ceramics with consideration of the effects of electric field gradient. An equation that determines the dispersion relations is obtained. Numerical solutions to the equation show that, different from the prediction from the linear theory of piezoelectricity, short interface waves become dispersive when electric field gradient is considered.

Effect of rigid boundary on the propagation of torsional waves in a homogeneous layer over a heterogeneous half-space

In the present paper we study the effect of rigid boundary on the propagation of torsional waves in a homogeneous layer over a semi-infinite heterogeneous half-space, where the heterogeneity is both in rigidity and density. The present study demonstrates that torsional waves can propagate in the layer. The velocities of torsional waves have been calculated numerically as a functions of KH, (where K is the wave number and H is the thickness of the layer) and are presented in a number of graphs. It is also observed that, for a layer over a homogeneous half-space, the velocity of torsional waves does not coincide with that of Love waves in the presence of the rigid boundary whereas it does at the free boundary.

Size effect of indenter on determining modulus of nanowires using nanoindentation technique

This study uses a finite element modeling to simulate the behavior of copper nanowires submitted to nanoindentation and estimate their mechanical properties. The simulation results reveal that using the well-known Oliver–Pharr theory, generally applied for materials with semi-infinite half-space, yields an underestimate elastic modulus of the wire materials. Moreover, the radius of the indenter tip also influences the accuracy of the predicted elastic modulus. Such errors are mainly from the overestimate of the contact area between the indenter and the specimen. They can be corrected from the numerical modeling. The elastic modulus of the wires calculated from the corrected contact area is highly close to the bulk value.