Components Of Displacement Vector Research Articles

Detecting, mapping and monitoring of land subsidence in Jharia Coalfield, Jharkhand, India by spaceborne differential interferometric SAR, GPS and precision levelling techniques

The study aims at detection, mapping and monitoring of land subsidence in Jharia Coalfield, Jharkhand, India by spaceborne DInSAR, GPS and precision levelling techniques. Using multi-frequency C- and L-band DInSAR, both slowly and rapidly subsiding areas were identified and DInSAR-based subsidence maps were prepared. C-band DInSAR was found useful for detection of slowly subsiding areas whereas L-band DInSAR for rapidly subsiding and/or adverse land cover areas. Due to dynamic nature of mining and adverse land cover, temporal decorrelation poses a serious problem particularly in C-band DInSAR. Specially designed InSAR coherence guided adaptive filtering was found useful to highlight the deformation fringes. Collateral GPS and levelling observations were conducted in three test sites to validate DInSAR measurements and to determine the net displacement vectors. We observed an appreciable horizontal displacement component of land subsidence in all the test sites. For comparison of results, we calculated InSAR coherence weighted LOS displacement rates from the unwrapped differential interferograms of smaller spatial subsets and LOS projected ground-based displacement rates in three test sites. We found good agreement between DInSAR and ground-based measurements except for C-band observation in Dobari test site primarily due to large difference in observation periods and temporally inconsistent land subsidence. Collateral spaceborne and ground-based observations were also found useful for characterization of subsidence phenomena to determine net displacement vector and horizontal displacement component. In coal mining areas with spatially scattered and temporally irregular land subsidence phenomena, the adopted methodology can be used successfully for detection, mapping and monitoring of the subsiding areas vulnerable to future collapse. This will facilitate efficient planning and designing of surface infrastructures and other developmental structures in the mining areas and mitigation management of subsidence induced hazards.

Layered orthotropic plates. Generalized theory

In this paper a two-dimensional layer-wise generalized theory of layered orthotropic plates is presented. This theory is based on a power series expansion of the displacement vector component in each layer for the transverse coordinate. The number of terms retained in the power series is arbitrary and it is chosen depending on the problem being considered.The potentialities of the offered approach are demonstrated by examples of investigating the response of composite plates to static and dynamic loading. Also, the feasibility of the theory is shown by example of the problem of a ball impacting a composite plate.The stress–strain state of multilayer composites has been considered for distributed and space-localized loads. The results of calculations according to the generalized theory are compared with exact solutions given Pagano, as well as with data obtained from the two-dimensional theories.

Analysis of stacking faults in gallium nitride by Fourier transform of high-resolution images

We present results of studying stacking faults (SFs) in gallium nitride (GaN) with the aid of Fourier transform of high-resolution transmission electron microscopy (HRTEM) images. Using this method, it is possible both to determine the SF type and to directly measure the corresponding displacement vector. This allowed us to explain the peculiarities of the contrast of HRTEM of structures with high SF density (above 106 cm−1). It is established that the displacement vector component in the (0001) plane in these structures can significantly differ from the expected value of \((1/3)[00\bar 10]\) that is typical of single SFs.

Modeling of the stress-strain state of layered orthotropic plates on elastic foundation

The problem of the analysis of the stress-strain state of hinged layered plates on elastic foundation is considered. Bending of the layered plate on the Pasternak foundation at its mechanical loading is investigated. Original two-dimensional discrete-structural generalized theory of multi-layered plates is used to study the orthotropic plate bending. This theory is based on developing the displacement vector components of each layer in the power series along transverse coordinate. It allows to calculate displacement and stress tensor components of each layer with the required accuracy. The possibilities of the proposed approach are demonstrated on the examples of studying the response of layered plates on elastic foundation. The reliability of the results, obtained on its basis is determined through comparison with the data, presented in known scientific publications. The effect of the elastic foundation parameters on the stress-strain state of multi-layer structures is investigated. It was found that with an increase in the number of power series terms, displacement and stress demonstrate good convergence. It is shown that the elastic foundation shear stiffness significantly affects the stress-strain state of the hinged layered plates.

Finite-element harmonic experiments to model fractured induced anisotropy in poroelastic media

Fractures in a fluid-saturated poroelastic–Biot–medium can be modeled as very thin highly permeable and compliant layers within a porous background. A Biot medium containing a dense set of aligned fractures behaves as an effective transversely isotropic and viscoelastic (TIV) medium at the macroscale when the predominant wavelength is much larger than the average distance between fractures. One important mechanism in Biot media at seismic frequencies is wave-induced fluid flow generated by fast compressional waves at mesoscopic-scale heterogeneities, generating slow diffusion-type Biot waves. In this work, we present and analyze a collection of time-harmonic finite element experiments that take into account the effects of the presence of aligned fractures and interlayer fluid flow occurring at the mesoscale, allowing us to determine the complex and frequency dependent stiffnesses of the effective TIV medium at the macroscale.These numerical upscaling experiments are defined as boundary value problems on representative samples of the fractured material, with boundary conditions associated with compressibility and shear tests, which are solved using the finite element (FE) method. The FE space chosen to discretize each component of the solid displacement vector is that of globally continuous piecewise bilinear functions, while for the fluid phase the vector part of the Raviart–Thomas–Nedelec space of zero order is employed. We present results on the uniqueness of the solution of the continuous and discrete problems, and derive optimal a priori energy error estimates. First, the numerical results are validated with those of a theory valid for fluid flow perpendicular to the fracture layering and independent of the loading direction, so that the attenuation mechanism can be represented by a single relaxation function. Then, the methodology is applied to cases for which no analytical solutions are available, such as a fractured Biot medium saturated with brine and patches of CO2 and a brine saturated sample of uniform background and fractures with fractal variations in their petrophysical properties.

Model of crustal block movement in the South Yakutia geodynamic testing area based on GPS data

The results of the analysis of crustal block movement obtained from the GPS data in 2009–2012 in the South Yakutia geodynamic testing area located at the junction of two major tectonic structures—the Aldan shield of the Siberian platform and the Stanovoi fold-block area—are presented. The drift of the block in the southeastern direction corresponds to the main azimuth of the strike of the activated fault systems and is consistent with the results of the geodetic observations performed in the 1970–1980s. Based on the periodic components of the complete displacement vector, hinge-type movements along the local faults are established. It is shown that, in the annual cycles, the rotational, oscillational, and translational movements of the block have a nonlinear character, and solitary strain waves are likely to be formed in the zones of activated faults.

A Method of Function Space for Vertical Impedance Function of a Circular Rigid Foundation on a Transversely Isotropic Ground

This paper is concerned with investigation of vertical impedance function of a surface rigid circular foundation resting on a semi-infinite transversely isotropic alluvium. To this end, the equations of motion in cylindrical coordinate system, which because of axissymmetry are two coupled equations, are converted into one partial differential equation using a method of potential function. The governing partial differential equation for the potential function is solved via implementing Hankel integral transforms in radial direction. The vertical and radial components of displacement vector are determined with the use of transformed displacement-potential function relationships. The mixed boundary conditions at the surface are satisfied by specifying the traction between the rigid foundation and the underneath alluvium in a special function space introduced in this paper, where the vertical displacements are forced to satisfy the rigid boundary condition. Through exercising these restraints, the normal traction and then the vertical impedance function are obtained. The results are then compared with the existing results in the literature for the simpler case of isotropic half-space, which shows an excellent agreement. Eventually, the impedance functions are presented in terms of dimensionless frequency for different materials. The method presented here may be used to obtain the impedance function in any other direction as well as in buried footing in layered media.

Constructing generalized Cesàro formulas for finite plane strains

The problem of finding the displacement vector from a system of nonlinear differential equations which includes displacement gradient components is studied. Expressions on the right side of this system for certain parameter values have the kinematic sense of Lagrange and Euler finite strain tensors. The task is to construct generalized Cesaro formulas for finite strains. The construction of the solution consists of two stages (algebraic and differential), and the second is performed for space whose dimension is greater than or equal to two. An algorithm for the inversion of the original system is proposed, and analytical constructions for the case of two-dimensional space are performed. The problem is solved at the first (algebraic) stage, i.e., an exact analytical expression for the displacement vector components is derived through the known finite strain tensor and an unknown scalar function having the kinematic sense of rotation. Necessary conditions for the existence of this relationship are formulated.

On Reconstruction of the Three Displacement Vector Components from SAR LOS Displacements for Oil and Gas Producing Fields

SAR interferometry estimates Line-Of-Sight displacements of natural or man-made reflectors. To estimate all three components (e.g. north, east and up) of the displacement vector, it is necessary to use additional data or physical models of causative sources of the observed displacements. We propose a method to reconstruct the three components of the displacement vector at oil and gas exploration fields, based on an assumption that the registered LOS displacements are caused by a decrease or increase in seam pressure due to hydrocarbon extraction or fluid injection. We present an example of an evaluation of LOS displacements and the reconstruction of the three displacement vector components using SAR images for the Romashkino oil field near Almet’evsk city (Tatarstan Republic, Russian Federation).

Estimation of the uncertainty of elastic image registration with the demons algorithm

The accuracy of elastic image registration is limited. We propose an approach to detect voxels where registration based on the demons algorithm is likely to perform inaccurately, compared to other locations of the same image. The approach is based on the assumption that the local reproducibility of the registration can be regarded as a measure of uncertainty of the image registration. The reproducibility is determined as the standard deviation of the displacement vector components obtained from multiple registrations. These registrations differ in predefined initial deformations. The proposed approach was tested with artificially deformed lung images, where the ground truth on the deformation is known. In voxels where the result of the registration was less reproducible, the registration turned out to have larger average registration errors as compared to locations of the same image, where the registration was more reproducible. The proposed method can show a clinician in which area of the image the elastic registration with the demons algorithm cannot be expected to be accurate.

Full field displacement measurement by double symmetrical illumination with diode lasers through a pair of double exposure reflectionholograms

Compact and stable phase stepping interferometer for shape and full field displacement measurement in static and in “real time” operation mode is presented. Double symmetrical illumination of the object in two orthogonal planes with diode lasers, emitting in NIR (790nm and 830nm), through a four-exposure reflective holographic optical element (Denisiyk's volume reflection holograms of a reference plane) is applied. The proposed four channels one-beam interferometer is very compact, as the diode lasers without collimators and spatial filters are used directly for reconstruction of the reference planes from a pair of double exposure reflection holograms and for object's illumination through the same holographic optical element. Phase stepping is introduced simply by precise increments of the diode lasers current. By introduction of removable sinusoidal phase gratings and removal of the holographic optical element, the system operates as a single-shot fringe projection profilometer for shape measurement that is essential for precise estimation of displacement vector's components. The proposed system is very stable against external noise, produced by vibrations, temperature changes, air flows, as well as against the influence of object's “rigid body” motion, as the compact and low weight interferometer can be stably fixed directly onto the measured construction.

Study Underground Old Slope Activation and Influences on Environmental Safety Induced by Open Pit Mining

The long-term ground severe deformation and destruction induced by open pit and underground mining can badly affect the safety of surrounding environment. This problem is different from ground deformation and destruction induced by single mining. Based on chronological relationship between two kinds of mining and its corresponding deformation law under a practical engineering, the main reasons for long-term deformation and the deformation mechanism were revealed in this paper. Starting with the displacement mechanism, a method of horizontal movement coefficient was put forward to distinguish the movement property. Therefore, the impacts on two components of displacement vector can be distinguished, and then the basis for the proper analysis of ground deformation would be reached. At the same time, the ground deformation zone and the failure zone caused by subsequent open pit mining would be predicted, and the range of the failure zone would be gotten, providing a basis for safety decision.

On the applicability of the Biot solution to the problems of elastic propagation in saturated porous media

In this paper, the solution of the system of homogeneous Biot equations, which was derived by Biot for the displacement vectors of plane monochrome elastic waves propagating in a homogeneous infinite two-phase medium, is expanded to the case where the propagation area of the elastic waves is limited and the wavefront is a piecewise smooth curved surface. It is shown that the arbitrary system of homogeneous Biot equations for the displacement vectors of the solid and liquid phases can be reduced to three different equations pertaining to the class of Helmholtz equations. From this, irrespective of the geometry of the seismic wavefront and the boundaries of the studied two-phase medium, there is the following. (1) Each displacement vector (of the solid and liquid phase) splits into three independent vectors satisfying three different Helmholtz equations. Two of these vectors correspond to the two types of compressional waves, namely, fast waves (waves of the first kind) and slow waves (waves of the second kind). The third vector describes shear waves. (2) The similar (related to the same wave type) components of the displacement vector in the solid and liquid phases satisfy the same Helmholtz equation and are linked with each other through a corresponding scalar factor that is expressed in terms of the coefficients of the Biot equations. Taking into account the established properties of the displacement vectors in the solid and liquid phases seems to be helpful in the problems dealing with calculation of elastic fields of arbitrary sources in piecewise-homogeneous two-phase media.

The wave field of a point source that acts on the open boundary of a Biot half-plane

The initial boundary value problem of wave propagation in a half-plane filled with a fluid-saturated porous solid is considered. The Biot medium is isotropic, homogeneous, and with open pores on the boundary. Using complex analysis techniques, explicit formulas for the components of displacement vectors in elastic and fluid phases are obtained. Bibliography: 5 titles.

Elasto-Plastic Analysis of Thick Cylinders Subjected to Internal Electro-Magnetic Loading

AbstractA study of elasto-plastic deformation of circular cylindrical shells subjected to internal electromagnetic forces is presented in this paper. The five governing equations in terms of resultant forces and resultant moments with respect to basic displacement vector components u, v and w are used. Theoretical formulations, based on the first-order shear deformation theory (FSDT), take into consideration transverse shear deformation and rotary inertia. The deformation theory of plasticity is employed for constitutive equations. The cylinders are composed of an elastic-plastic material with the von Mises yield criteria and non-linear plastic behaviour. Galerkin method is employed to convert the partial differential equations (PDEs) to ordinary differential equations (ODEs). The Newmark family of methods is used to numerically time integration of system of coupled second order ODEs. In order to prove the validity of the presented method and the solving process, the results obtained with the present analysis are compared with a set of available data. Good agreement observed between the results of the two approaches. Certainly, the aim of this paper is to create a more reliable and precise mathematical model of hollow-cylinders to avoid performing several experiments.

Piezoelectric d 15 shear-response-based torsion actuation mechanism: An exact 3D Saint-Venant type solution

This paper is concerned with the detailed analysis of the behavior of a piezoceramic bi-morph torsion actuator using the d 15-effect. The bi-morph actuator is made of two oppositely polarized adjacent piezoceramic prismatic beams. The mathematical analysis is based on the Saint-Venant torsion theory; a formulation of the electromechanically coupled problem in terms of a stress function and of the electric potential is derived, which represents an exact solution of a specific three-dimensional problem; in particular, for the case when the axial stress and the axial component of the electric displacement vector are independent of the axial coordinate. The resulting boundary-value problem in the cross-section is solved using the method of finite differences. Solutions for the actuated rate of twist are presented and compared to three-dimensional electromechanically coupled finite element solutions using ABAQUS® for the case of a cantilevered bi-morph actuator. A very good agreement is found.

Notes to the proof of a weighted Korn inequality for an elastic body with peak-shaped cusps

We present the proof of the weighted anisotropic Korn inequality in a three-dimensional domain with peak-shaped cusps on the boundary. We verify the asymptotic accuracy of distribution of multipliers at the components of displacement vector and their derivatives in the corresponding weighted norm. We indicate conditions on a peak cusp under which the natural energy class is not embedded into a Sobolev or Lebesgue class. In the last case, the operator of elasticity problems possesses the continuous spectrum provoking wave processes in a finite volume (“black holes” for elastic waves). We also discuss possible generalizations of the result and open questions. Bibliography: 39 titles. Illustrations: 9 figures.

Numerical electroseismic modeling: A finite element approach

Electroseismics is a procedure that uses the conversion of electromagnetic to seismic waves in a fluid-saturated porous rock due to the electrokinetic phenomenon. This work presents a collection of continuous and discrete time finite element procedures for electroseismic modeling in poroelastic fluid-saturated media. The model involves the simultaneous solution of Biot’s equations of motion and Maxwell’s equations in a bounded domain, coupled via an electrokinetic coefficient, with appropriate initial conditions and employing absorbing boundary conditions at the artificial boundaries. The 3D case is formulated and analyzed in detail including results on the existence and uniqueness of the solution of the initial boundary value problem. Apriori error estimates for a continuous-time finite element procedure based on parallelepiped elements are derived, with Maxwell’s equations discretized in space using the lowest order mixed finite element spaces of Nédélec, while for Biot’s equations a nonconforming element for each component of the solid displacement vector and the vector part of the Raviart–Thomas–Nédélec of zero order for the fluid displacement vector are employed. A fully implicit discrete-time finite element method is also defined and its stability is demonstrated. The results are also extended to the case of tetrahedral elements. The 2D cases of compressional and vertically polarized shear waves coupled with the transverse magnetic polarization (PSVTM-mode) and horizontally polarized shear waves coupled with the transverse electric polarization (SHTE-mode) are also formulated and the corresponding finite element spaces are defined. The 1D SHTE initial boundary value problem is also formulated and approximately solved using a discrete-time finite element procedure, which was implemented to obtain the numerical examples presented.

Maxima and minima of the displacement components for the Lamb modes

This paper revisits the vanishing of the transverse component of the particle displacement vector in free surfaces of an isotropic homogeneous plate, for both symmetric and antisymmetric Lamb waves. Drawing on well-known analytical expressions from Viktorov's book [(1967) Rayleigh and Lamb Waves: Physical Theory Applications, Chap. II, pp. 67-121], two distinct frequency-thickness product expressions, in cases where this vanishing occurs, are derived: one for the symmetric modes and another for the antisymmetric modes. At these frequency-thickness products, phase and group velocities have appreciable values which are discussed herein. It appears that these velocities depend on the transverse bulk wave velocity only. This is the specific condition of the Lamé modes. Moreover, theoretical and experimental investigations of displacements in the surface of a plate in air have been carried out. The theoretical part shows that the normal and transverse displacements have, respectively, a local maximum and a local minimum in the vicinity of these frequency-thickness products. The experimental part corroborates the presence of the local maximum of the S(0) Lamb mode for various materials.

Full-field drift Hamiltonian particle orbits in 3D geometry

A Hamiltonian/Lagrangian theory to describe guiding centre orbit drift motion which is canonical in the Boozer coordinate frame has been extended to include full electromagnetic perturbed fields in anisotropic pressure 3D equilibria with nested magnetic flux surfaces. A redefinition of the guiding centre velocity to eliminate the motion due to finite equilibrium radial magnetic fields and the choice of a gauge condition that sets the radial component of the electromagnetic vector potential to zero are invoked to guarantee that the Boozer angular coordinates retain the canonical structure. The canonical momenta are identified and the guiding centre particle radial drift motion and parallel gyroradius evolution are derived. The particle coordinate position is linearly modified by wave–particle interactions. All the nonlinear wave–wave interactions appear explicitly only in the evolution of the parallel gyroradius. The radial variation of the electrostatic potential is related to the binormal component of the displacement vector for MHD-type perturbations. The electromagnetic vector potential projections can then be determined from the electrostatic potential and the radial component of the MHD displacement vector.