Spatial Distribution Of Displacements Research Articles

Analytical solution to quickly assess ground displacement for a pressurized or depleted deep reservoir intersected by a fault in a half space

Quick estimates of fluid-induced subsurface deformation are helpful to assess the land uplift/subsidence and to reveal precursors of induced seismicity. Here, we adopt the inclusion theory and Green's function to develop a closed-form solution in a half space for the poroelastic response of a reservoir compartmentalized by an intersecting fault that can be offset and either permeable or impermeable. Simulated results reveal that (1) fault permeability mainly impacts the spatial distribution of displacement while its effect on displacement magnitude is small; (2) ground displacement slightly increases with fault dip while slightly decreases with increasing fault offset; in contrast, reservoir geometry shows a stronger effect than fault geometry: the ground displacement is proportional to the vertical and lateral depth ratios, defined as the ratios of reservoir thickness (h) and width (w) to reservoir depth (D), respectively; (3) the maximum vertical displacement is the double of the horizontal one regardless of fault permeability, fault and reservoir geometries, and mechanical parameters. Comparing the solution in a half space with that in a full space shows that neglecting the free surface underestimates the poroelastic displacement in the overburden. The validity of full-space solutions can be assessed with the product of the lateral and vertical depth ratios, i.e., wh/D2. The full-space solutions become valid when wh/D2 decreases to an intrinsic threshold. This threshold may range from 0.01 to 0.02 for displacement, and its specific value depends on the field background and demands of projects, but can be estimated based on our solution. It is larger for stress than for displacement, and wh/D2 ≤ 0.1 is recommended as a general condition for neglecting the free-surface effects on induced stress. The analytical solution represents a useful tool for estimating ground deformation and for gaining insights of reservoir and fault geometries by analyzing surface deformation patterns.

Accurate coupled vibration analysis of a piezoelectric array element by the superposition method

In this paper, the superposition method is extended to obtain analytical solutions for the coupled vibration of a piezoelectric slender bar in a configuration corresponding to a typical ultrasonic linear array transducer element. The problem can be described mathematically by three partial differential equations with electrical and mechanical boundary conditions. To solve this, the vibrations in lateral and thickness directions are referred to as two building blocks. In each building block, the expressions of displacements and electric potential are assumed first based on their symmetry properties and then the induced dynamic responses, such as in-plane stress and electric displacements, are calculated. Finally, the vibration responses of the two building blocks are superimposed to satisfy the boundary conditions using Fourier series expansions. Electrical impedance and mode shapes, represented by the spatial distribution of displacements and electric potential, are calculated analytically and compared with the results of the finite element method. An excellent agreement is observed. The method can be applied to design and optimize piezoelectric array transducers for various applications.

Measuring the Quantitative Characteristics of Local Vortices on a Plane of a Deformed Material

A problem for determining the center, size, and spatial distribution of displacements of a vortex on the basis of a plane vector field is under consideration. A seed algorithm is proposed instead of typical contour scanning, and specific circulation is used as a criterion for determining the center of a vortex flow. In order to increase the stability of this algorithm, a parameter playing the role of a threshold is introduced. A Monte-Carlo method is used to study the influence of measurement errors on the accuracy in determining the coordinates of a hard rotation center, and the relationship between the false alarm probability and the threshold level is established. The use of the least squares method for calculating the spatial distribution of displacements is shown. This approach is demonstrated on the example of a real vector field measured in the vicinity of a main crack tip.

Spatial distribution of airway wall displacements during breathing and bronchoconstriction measured by ultrasound elastography using finite element image registration

With every breath, the airways within the lungs are strained. This periodic stretching is thought to play an important role in determining airway caliber in health and disease. Particularly, deep breaths can mitigate excessive airway narrowing in healthy subjects, but this beneficial effect is absent in asthmatics, perhaps due to an inability to stretch the airway smooth muscle (ASM) embedded within an airway wall. The heterogeneous composition throughout an airway wall likely modulates the strain felt by the ASM but the magnitude of ASM strain is difficult to measure directly. In this study, we optimized a finite element image registration method to measure the spatial distribution of displacements and strains throughout an airway wall during pressure inflation within the physiological breathing range before and after induced narrowing with acetylcholine (ACh). The method was shown to be repeatable, and displacements estimated from different image sequences of the same deformation agreed to within 5.3μm (0.77%). We found the magnitude and spatial distribution of displacements were radially and longitudinally heterogeneous. The region in the middle layer of the airway experienced the largest radial strain due to a transmural pressure (Ptm) increase simulating tidal breathing and a deep inspiration (DI), while the region containing the ASM (i.e., closest to the lumen) strained least. During induced narrowing with ACh, we observed temporal longitudinal heterogeneity of the airway wall. After constriction, the displacements and strain are much smaller than the relaxed airway and the pattern of strains changed, suggesting the airway stiffened heterogeneously.

Analytical Solution for Bending of Laminated Composites with Matrix Cracks

An analytical, closed form solution is developed for balanced (not necessarily symmetric) laminates subjected to flexural deformation. The analytical solution provides spatial distribution of displacements and curvature, from which in-plane and intralaminar strains and stresses are obtained through differentiation and constitutive equations. The deformation is shown to consist of a homogeneous deformation plus perturbations near the crack tip. A methodology is proposed to separate the perturbation from the homogeneous deformation, to eliminate ill-conditioning of the eigenvalue/eigenvector problems that occurs otherwise. It is shown that, while the homogeneous deformation provides a macroscopic measure of damage in terms of reduced flexural stiffness of the laminate, the perturbation solution provides a detailed account of the intralaminar shear induced near the crack, which is used to calculate the extent of shear lag and the maximum intralaminar shear stress. The intact portion of the laminate is modeled without lumping it into a single equivalent lamina. Furthermore, laminas can be subdivided into multiple sub-laminates to increase the accuracy of the representation of intralaminar/interlaminar shear, which is shown to improve the predicted value of maximum interlaminar shear stress, which in turn is important for the prediction of matrix-crack induced delamination.

Monitoring ground subsidence in urban environments: M-30 tunnels under Madrid City (Spain)

Big cities improvement usually requires the construction of large underground infrastructures, in order to ensure proper communication and optimize urban use. Monitoring ground subsidences is therefore one of the main challenges in changing urban environments. The "Madrid Río" project (2003-2008) is an effort to reclaim the riverfront land and improve the busy M-30 beltway that involved the construction of 7.93 km of tunnels underneath the southern center of Madrid City. This paper presents a remote-sensing approach to monitor ground subsidences induced by tunneling excavation. The Persistent Scatterers Interferometry technique (PSI) was used to estimate subsidence and displacement time series from Synthetic Aperture Radar images, acquired between August 2003 and April 2008 from ENVISAT. Remote sensed results were compared to traditional extensometric measures, fitting adequately for selected sectors. Spatial analysis of displacements allowed evaluating impacts of tunneling on surrounding buildings and facilities, highlighting critical areas. The availability of a spatial distribution of displacements in a time series allowed analyzing longitudinal, cross-sectional and temporal dynamics. The main limitations during this work were the heterogeneous spatial distribution of Persistent Scatterers, the absence of measurement points in work areas, the threshold for velocity detection and low temporal resolution of ENVISAT images. Nevertheless, these limitations of DInSAR for monitoring infrastructures are overcome by actual satellites, being a complementary technique with an exceptional added value and temporal analysis capability.

Application of PMN-32PT Piezoelectric Crystals for Novel Air-coupled Ultrasonic Transducers

Due to very high piezoelectric properties of PMN-PT crystals they may significantly improve performance of air-coupled ultrasonic transducers. For these purpose vibrations of PMN-PT rectangular plates and strips were investigated. An air-coupled ultrasonic transducer and array consisting of 8 single piezoelectric strips were designed. Operation of the transducer was simulated by the finite element method using ANSYS Mechanical APDL Product Launcher software. Spatial distributions of displacements inside piezoelectric elements and matching strip were obtained. Experimental investigations were carried out by the laser Doppler vibrometer Polytec OFV-5000 and the Bruel&Kjaer microphone 4138 with the measurement amplifier NEXUS WH 3219. It was found that performance of the ultrasonic transducer with PMN–32PT crystals was a few times better than of a PZT based ultrasonic transducer.

Multi-scale modeling of localized heating caused by ion bombardment

We have developed a multiscale modeling algorithm to incorporate both a Monte Carlo (MC) ion irradiation simulator and finite element analysis to simulate ion beam heating. The spatial distribution of displacements from the MC code was input into a 3-D FEA code to predict the temperature evolution upon ion bombardment. In order to simulate the flux effect, ions were introduced stochastically. We discuss the necessity to use both grid refinement and grid coarsening techniques to make such modeling possible, thus providing a basis to evaluate the impact on the microstructure of the substrate. The aforementioned approach was applied for the case of a 16 A cm −2 beam of 6 keV Ga + ions to simulate FIB sample sectioning and thinning in a Si substrate.

Vibration Analysis of a Reactor Driven by an Inverter Power Supply Considering Electromagnetism and Magnetostriction

To design a new model to reduce the noise of a reactor under inverter power supply, the cause of noise should be investigated. We carry out vibration analysis of a single phase of a reactor while considering electromagnetism and magnetostriction to clarify the mechanism of noise generation in the reactor. In this analysis, magnetostriction is modeled three-dimensionally with equivalent nodal forces. First, the nodal forces in the cores of reactor due to the electromagetism and magnetostriction are calculated using the flux distribution obtained from magnetic field analysis. To determine the dominant cause of noise, the spatial distributions of displacements generated by both nodal forces are then obtained by static structure analysis and they are compared in detail. Moreover, the actual movements of the cores are calculated by vibration analysis. Finally, the effect of the hardness of the cores on noise generation is calculated and measured. It is shown that the hardness of the insulator inserted in gaps between cores affects the vibration and it seems to have an optimal value. Moreover, the analyzed results are proved by measurement.

In situ deformation of cartilage in cyclically loaded tibiofemoral joints by displacement-encoded MRI

Cartilage displacement and strain patterns were documented noninvasively in intact tibiofemoral joints in situ by magnetic resonance imaging (MRI). This study determined the number of compressive loading cycles required to precondition intact joints prior to imaging, the spatial distribution of displacements and strains in cartilage using displacement-encoded MRI, and the depth-dependency of these measures across specimens. Juvenile porcine tibiofemoral joints were cyclically compressed at one and two times body weight at 0.1 Hz to achieve a quasi-steady state load-displacement response. A 7.0 T MRI scanner was used for displacement-encoded imaging with stimulated echoes and a fast spin echo acquisition (DENSE-FSE) in eight intact joints. Two-dimensional displacements and strains were determined throughout the thickness of the tibial and femoral cartilage and then normalized over the tissue thickness. Two-dimensional displacements and strains were heterogeneous through the depth of femoral and tibial cartilage under cyclic compression. Strains in the loading direction were compressive and were maximal in the middle zone of femoral and tibial cartilage, and tensile strains were observed in the direction transverse to loading. This study determined the depth-dependent displacements and strains in intact juvenile porcine tibiofemoral joints using displacement-encoded imaging. Displacement and strain distributions reflect the heterogeneous biochemistry of cartilage and the biomechanical response of the tissue to compression in the loading environment of an intact joint. This unique information about the biomechanics of cartilage has potential for comparisons of healthy and degenerated tissue and in the design of engineered replacement tissues.

Plane strain deformation in an orthotropic micropolar thermoelastic solid with a heat source

The problem of plane strain deformation in an orthotropic micropolar generalized thermoelastic half-space subjected to an arbitrary point heat source is solved. Closed-form solutions for spatial distributions of displacements, stresses, and temperature are derived by using the Fourier transform. A numerical inversion technique has been applied to obtain the solution in the physical domain. Numerical results are obtained and presented graphically along with a comparison of the ones for concentrated and distributed, as well as mechanical and thermal, sources.

Three-dimensional transient and harmonic shear-wave scattering by a soft cylinder for dynamic vascular elastography

With the objective of characterizing biological soft tissues with dynamic elastography, a three-dimensional (3D) analytical model is proposed to simulate the scattering of plane shear waves by a soft cylinder embedded in an infinite soft medium. The 3D problem of harmonic plane shear-wave scattering is first formulated and solved, and the monochromatic solution is employed to simulate transient wave scattering. Both harmonic and transient simulations are compared with experimental 3D acquisitions. The good agreements obtained between measured and calculated displacement fields allowed to conclude on the validity of the proposed 3D harmonic and transient models. The spatial distribution of displacements (diffraction lobes, displacement oscillations, wave diffraction angles, etc.) and their relative amplitudes in both inclusion and surrounding materials depended on the contrast between the viscoelastic properties of the different media. The possibility of solving an inverse problem to assess soft heterogeneous medium viscoelasticity is discussed and some future theoretical and experimental developments are proposed.

Low-energy primary knock on atom damage distributions near MeV proton beams focused to nanometre dimensions

In this preliminary study, the spatial extent of the defects introduced in Si by a normally incident 1 MeV H + nanobeam was investigated using a hybrid approach. First, the standard SRIM code was employed to calculate the angular and energy distributions of Primary Knock-on Atoms (PKA) using the Binary Collision Approximation (BCA). The long mean free path and the kinematics of scattering in a screened Coulomb potential resulted in an anisotropic PKA distribution that was mainly directed perpendicular to the primary ion trajectories. The PKA energy E dependence was close to 1/ E n with 1.5 < n < 2 such that the vast majority of PKAs had low energies (⩽40 eV) close to the threshold displacement energy. Subsequently, the low energy PKA data was used as input to a Molecular Dynamics (MD) code that was used to follow the recoil cascades. The force field resulting from Si–Si atom interactions was represented in the MD calculations by the Environmental-Dependant Interaction Potential (EDIP). Finally, the spatial distribution of Self Interstitial Atoms (SIA) and vacancies was characterised using the Pixel Mapping (PM) method. The results revealed that the low energy PKA’s resulted in localised damaged regions with Si Self-Interstitial Atoms (SIA) and associated vacancies concentrated within a few nm of the ion track. Although some clustering occurred, the majority of SIA were present as monomers. This is in agreement with previous work on low energy B + implanted Si. The contribution from the spatial distribution of displacements from high energy (>40 eV) PKAs was investigated using SRIM. It was found that within the maximum region of the SIA concentration (1–2 nm radius of the ion track) the contribution from high energy PKA’s is relatively unimportant (∼5%).

Ocean loading deformations caused by storm surges on the northwest European shelf

A model of the non‐tidal ocean loading deformations caused by storm surges on the northwest European shelf has been developed using the output from a 12 km grid storm surge model. The spatial distributions of displacements and gravity variations caused by a typical storm surge event are computed in order to show the magnitudes of the effects that can be expected in space geodetic or gravity measurements. A storm surge of just over 2 m in the southern North Sea produces vertical displacements of −20 to −30 mm and increases of gravity of 6 to 8 μgal in the coastal areas of Denmark, Germany, the Netherlands and the east coast of England. It is shown that the loading deformations due to surges affect a wide area of northwest Europe and that high resolution non‐tidal loading models are required for correcting geodetic measurements made in areas near shallow continental shelves.

The Pressurized Cylinder Problem for Nonlinear Viscoelastic Materials with a Strain Clock

A constitutive equation for nonlinear viscoelasticity is used to model the mechanical response of solid amorphous polymers such as polycarbonate. The nonlinearity arises from a reduced time, which causes stress relaxation to accelerate with increasing strain. This reduced time is referred to as a "strain clock". An important feature of the constitutive equation is that it accounts for yield under different strain and stress histories. This constitutive equation is used to study the problem of a hollow cylinder subjected to different pressure histories on its inner and outer surfaces. It is shown that if the strain clock depends only on the volumetric strain, then the governing equations admit a solution, which has a number of important consequences. First, the spatial distribution of displacements has the same form as for a linear elastic or linear viscoelastic material. The time evolution depends on the material properties. Second, if pressures are specified at the inner and outer surfaces, the resultant stress distribution is independent of material properties, and is the same as for linear elastic or linear viscoelastic response. Finally, it is found that the strain clock runs at the same rate for all radii. An experiment, based on these results, is suggested, which can be used to assess the assumption that the strain clock depends only on the volumetric strain.

THE PRESSURIZED SPHERE PROBLEM FOR NON-LINEAR VISCOELASTIC MATERIALS WITH A STRAIN CLOCK

A constitutive equation for nonlinear viscoelasticity is used to model the mechanical response of solid amorphous polymers such as polycarbonate. The nonlinearity arises from a reduced time variable which causes stress relaxation to occur faster as strain increases. This reduced time is referred to as a ‘strain clock’. This constitutive equation is used to study the problem of a hollow sphere subjected to different pressure histories at its inner and outer surfaces. It is shown that if the reduced time, or strain clock, depends only on the volumetric strain, then the governing equations admit a solution which has a number of important consequences. First the spatial distribution of displacements has the same form as for a linear elastic or linear viscoelastic material. The time evolution depends on the material properties. Second, if the pressure is specified at the inner and outer surfaces, the resultant stress distribution is independent of material properties, and is the same as for linear elastic or l...

Planar detaching faults in the southern Horn Graben, Danish North Sea

The Danish Horn Graben is located in the south-eastern North Sea and suffered fault controlled differential subsidence during the Triassic. Within the Horn Graben, several faults cut the Triassic succession but detach at or close to the Top pre-Zechstein surface. An E-W striking trend of planar detaching faults is mapped using 2-D seismic sections and the displacement on the faults is analysed with respect to timing and spatial distribution of the displacement. The spatial distribution of the displacement is analysed using the Fault Analysis Projection System (Badley Earth Sciences), whereas the timing is analysed by reconstructing the seismic sections and thus obtaining the spatial distribution of displacement at different times. The occurrence of the detaching planar faults appears to be controlled by the difference in basal friction across the pinch-out line of the mobile Zechstein sediments. A dipping detachment surface is necessary to create a driving mechanism but is of minor importance in initiating the faulting. The timing of the faulting is controlled by a general change in tectonic pattern enabling a N-S extension of the sediments covering the Top pre-Zechstein surface.

Evolution of a salt-related listric growth fault near the d-1 well, block 5605, danish north sea: displacement history and salt kinematics

The Top Chalk surface and the Tertiary sediments of the Danish North Sea are cut by only a few major faults which are often related to or controlled by the mobile Zechstein salt. A single listric growth fault (the D-1 fault) and its adjacent area have been mapped in detail in order to unravel the fault geometry and kinematics. The spatial distribution of displacements and lateral dimensions of the fault are used to examine the interaction between salt movements and the evolution of the fault. The displacement distribution on the fault is systematic, with contours of equal displacement forming concentric semi-ellipses centred on the point of maximum displacement with displacement decreasing upwards and laterally approaching the displacement configuration for the upper part of an ideal simple blind fault. Furthermore, within the volume surrounding the fault vertical displacement shows similar contour patterns for the hangingwall and footwall, respectively. The lateral displacement variation during fault growth and reconstructed displacement geometries indicate that footwall uplift was mainly a result of salt flow. Despite the complicated nature of salt dynamics the evolution of the D-1 fault resulted in a systematic fault displacement geometry within the rock volume surrounding the fault.