Showing papers on "Displacement field published in 1991"

Full-field representation of discretely sampled surface deformation for displacement and strain analysis

Abstract: A detailed evaluation of the feasibility of determining displacements and displacement gradients from measured surface displacement fields is presented. An improved methodology for both the estimation and elimination of noise is proposed. The methodology is used to analyze the gradients for three tests: (1) uniform rotation, (2) uniform strain, and (3) crack-tip displacement fields. Results of the study indicate that the proposed methodology can be used to extract the underlying two-dimensional displacements and their corresponding gradients from the noisy data with reasonable accuracy. Specifically, it is shown that (a) the digital correlation method for acquiring displacement fields has an error in strain of approximately 150 μ strain at each point, (b) the average strain in a region of uniform strain has much less error, typically on the order of 20 μ strain, (c) the displacement ‘nolse’ present in digital correlation is very small, approximately 0.01 pixels, (d) the proposed methodology for reducing noise in the data is essential to the accurate evaluation of displacement gradients and (e) the successful evaluation of displacement and displacement gradients for all three cases indicates that the proposed methodology can be used both to quantify the displacement fields and to reasonably estimate the overall gradient trends.

A coherent gradient sensor for crack tip deformation measurements: analysis and experimental results

Abstract: A first order diffraction analysis of an optical interferometer, Coherent Gradient Sensor (CGS), for measuring surface gradients is presented. Its applicability in the field of fracture mechanics is demonstrated by quantitatively measuring the gradients of out-of-plane displacements around a crack tip in a three point bent fracture specimen under static loading. This method has potential for the study of deformation fields near a quasi-statically or dynamically growing crack.

Update: Application of the Finite Element Method to Linear Elastic Fracture Mechanics

Abstract: Use of the finite element method to treat two and three-dimensional linear elastic fracture mechanics problems is becoming common place. In general, the behavior of the displacement field in ordinary elements is at most quadratic or cubic, so that the stress field is at most linear or quadratic. On the other hand, the stresses in the neighborhood of a crack tip in a linear elastic material have been shown to be square root singular. Hence, the problem begins by properly modeling the stresses in the region adjacent to the crack tip with finite elements. To this end, quarter-point, singular, isoparametric elements may be employed; these will be discussed in detail. After that difficulty has been overcome, the stress intensity factor must be extracted from either the stress or displacement field or by an energy based method. Three methods are described here: displacement extrapolation, the stiffness derivative and the area and volume J-integrals. Special attention will be given to the virtual crack extension which is employed by the latter two methods. A methodology for calculating stress intensity factors in two and three-dimensional bodies will be recommended.

Finite element with inner softening band

Abstract: A new technique for modeling localized deformations within a softening band is described, where softening is attributed to a displacement discontinuity within an element. Concepts such as fracture strain are not included in the formulation of the model, and consequently nonlocal parameters such as internal length measures are not needed. It is shown that the shape functions within the element provide the necessary information normally given with internal length. In this manner, objectivity with regard to element configuration seems to be automatically satisfied, which is demonstrated by numerical studies in which Rankine failure criterion is employed. It is also noted that the displacement field rather than the strain field is additively decomposed into elastic and inelastic parts. This additivity is valid independently of the magnitude of displacement continuity in the softening band, which implies that the technique can be extended in a straight‐forward fashion to finite displacements.

Motion displacement estimation using an affine model for image matching

Abstract: A model is developed for estimating the displacement field in spatio-temporal image sequences that allows for affine shape deformations of corresponding spatial regions and for affine transformations of the image intensity range. This model includes the block matching method as a special case. The model parameters are found by using a least-squares algorithm. We demonstrate experimentally that the affine matching algorithm performs better in estimating displacements than other standard approaches, especially for long-range motion with possible changes in scene illumination. The algorithm is successfully applied to various classes of moving imagery, including the tracking of cloud motion.

傾斜機能材料平板におけるラム波伝播とその衝撃応答 : 第一報,解析手法

Abstract: The numerical methods which have been proposed by the present authors for Lamb waves in an anisotropic laminated plate and its transient responses are expanded for the wave propagation analysis of functionally gradient material (FGM) plates. The material properties of the plate change gradually in the thickness direction, and are anisotropic in the plane of the plate. The plate is divided into N plate elements. In the element, we assume that the material properties change linearly in the thickness direction, and that the displacement field is expressed by second-order polynomials. The principle of virtual work is used to develop approximate dynamic equilibrium equations. The dispersion relation and the mode shape of the Lamb waves are obtained by using the free boundary conditions. The energy velocities of the Lamb waves are formulated with the aid of the Rayleigh quotient. The method of Fourier transforms in conjunction with the modal analysis is used to determine the response of displacements. The formulation of the theory is described in this paper.

Dynamic response of buildings due to trains in underground tunnels

Abstract: The interaction of a tunnel-soil-building system due to trains is investigated by a substructure technnique. The soil medium is assumed to be a viscoelastic halfspace. The method of wave function expansion is used to construct the displacement fields in terms of potentials. The total soil-structure interaction problem is decomposed into a foundation radiation problem and a tunnel radiation problem. The impedance matrices for the corresponding substructure problems are obtained using a collocation technique. The steady state response of buildings for a given tunnel-foundation geometry is determined using the impedance matrix. Hence, the response of the building to train loading at different speeds is evaluated and compared with allowable vibration limits.

Strongly nonlinear composite dielectrics: A perturbation method for finding the potential field and bulk effective properties.

Abstract: A class of strongly nonlinear composite dielectrics is studied. We develop a general method to reduce the scalar-potential-field problem to the solution of a set of linear Poisson-type equations in rescaled coordinates. The method is applicable for a large variety of nonlinear materials. For a power-law relation between the displacement and the electric fields, it is used to solve explicitly for the value of the bulk effective dielectric constant ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{e}}$ to second order in the fluctuations of its local value. A simlar procedure for the vector potential, whose curl is the displacement field, yields a quantity analogous to the inverse dielectric constant in linear dielectrics. The bulk effective dielectric constant is given by a set of linear integral expressions in the rescaled coordinates and exact bounds for it are derived.

Numerical simulation of electromagnetic acoustic transducer in the time domain

Abstract: A novel multistage numerical modeling approach to simulate acoustic‐wave generation and propagation of an electromagnetic acoustic transducer (EMAT) is presented The model assumes a uniform static magnetic field in a conducting half‐space A meander coil winding situated above the half‐space is driven by a transient high‐frequency current pulse The numerically computed eddy currents in the conducting specimen are combined with the static magnetic field to yield spatially and temporally diffusive Lorentz forces which in turn are coupled into the elastic‐wave equation This hyperbolic elastodynamic system is solved for the displacement field vector which gives rise to propagating elastic waves Numerical transduction results are discussed for a generic two‐wire transient EMAT configuration and for an isotropic half‐space with electric and acoustic material parameters equivalent to those of aluminum

Three-dimensional surface singularity of an interface crack

Abstract: The three-dimensional singularity field near the terminal point of an interface crack at the free surface of an elastic bimaterial is investigated. The Finite Element Iterative Method (FEIM) is used for evaluating the asymptotic field. A spherical coordinate system r, θ, φ is used and the singular displacement field is assumed to be of a product form rλg(θ, φ), where λ and g(.) are in general complex. To validate the model, the method is first applied to the three dimensional surface crack in a homogeneous elastic material. The results for this case show excellent agreement with previously published analytical and numerical results. For an extreme effect of bimaterial property mismatch, on the surface crack singularity, an elastic material bonded to a rigid substrate is investigated (E1/E2=∞). The results show that the complex power singularity depends strongly on Poisson's ratio ν. The real part of the stress singularity is greater than 0.5 of the plane strain case and the imaginary part becomes almost zero at ν ≥ 0·25 instead of at ν=0.5. The second term in the expansion of the asymptotic field was shown to have a singularity of 0.5.

Homogenization, Plasticity and Yield Design

Abstract: We consider an epi-convergence problem arising from the theory of yield design. The functional under consideration has a linear growth with respect to the deformation tensor of the displacement field, and the problem is naturally posed in a space of displacement fields with bounded deformation. The problem includes a linear constraint which can be closed or not closed, depending on the type of boundary conditions considered. In the case where the constraint is not closed (applied forces on a part of the boundary) a relaxation term appears. Physically the strength of the loaded boundary turns out to be smaller than the natural guess deduced from the well known Average Variational Principle.

Propagation of shear horizontal surface acoustic waves parallel to the grooves of a random grating

Abstract: An elastic medium that occupies the region x3≳ζ(x1), where the surface‐profile function ζ(x1) is assumed to be a stationary, stochastic process, is studied. The displacement field u(x;t) in this region is assumed to satisfy stress‐free boundary conditions on the surface x3=ζ(x1). First the boundary conditions satisfied by the mean displacement field 〈u(x;t)〉 on the plane x3=0 are obtained, where the angle brackets 〈 〉 denote an average over the ensemble of realizations of ζ(x1). The results are then used to obtain the dispersion relation for surface acoustic waves of shear horizontal polarization propagating in the x2 direction, i.e., parallel to the grooves of the random grating defined by the surface‐profile function ζ(x1). It is found that the surface acoustic waves of this polarization, which cannot exist on a planar, stress‐free surface of an elastic medium, are trapped by the random roughness, but are bound to the surface for only a finite range of values of the wave number k2 characterizing their p...

Elastic wave scattering by anisotropic obstacles: Application to fractured volumes

Abstract: The Born approximation is frequently applied to isotropic media to determine wave fields scattered from heterogeneous regions. We extend this theory to the general case of an anisotropic obstacle embedded within an anisotropic matrix and show that a perturbation to any of the 21 independent elastic constants acts as a secondary moment tensor source which radiates energy as it is encountered by the incident wave. We consider the case of an anisotropic obstacle in an isotropic background medium in more detail, since the well-known Green's tensor for isotropic, homogeneous media and the Born approximation allow an expression of the radiation patterns, P and S wave, of Rayleigh scattering due to a perturbation to any elastic constants. The wave fields scattered from an anisotropic obstacle differ significantly from the isotropic case in dependence on both the direction of observation and the direction of the incident wave. In order to provide a concrete example of these results, we examine the case of a small fractured volume. If the fractures within the volume are randomly oriented, the resulting material is isotropic, but if the cracks are aligned, the material is anisotropic with five independent elastic constants. In the latter case, the value of C44, analogous to the isotropic rigidity, is unchanged from the background value (δC44 = 0) when the cracks are aligned perpendicular to the x; axis. We present the P, SV and SH radiation patterns for the two scatterers and discuss the implications for observation of fractured volumes. The most important result is that the the zero perturbation δC44 in the aligned crack case causes the scattered displacement field to vanish completely for incident shear waves polarized parallel to the crack plane. Determination of this direction for observations of a fractured zone would allow important insights into the nature of the fracturing.

Source location and valley shape effects on the P-SV near displacement field using a boundary integral equation-discrete wavenumber representation method

Abstract: SUMMARY We use a numerical simulation to investigate the response of sediment-filled valleys to P–SV seismic excitation. The calculation is carried out using the discrete wavenumber–boundary integral equation method. We study in particular the case of explosion sources located within a basin or in its vicinity. We show that large amplification or defocusing effects are related to the source location and to the shape of the basin. In the case of complex basins made up of several sedimentary layers, waves propagating along the various interfaces play an important role in shaping the seismic response of the basin.

Epicardial Deformation From Coronary Cinéangiograms

Abstract: A quantitative method is developed for estimating epicardial deformation in the intact heart using the motions of the coronary arteries. The method makes full use of a deformable structural model of the time-varying epicardial surface to guide the analysis of coronary cineangiograms. A finite element model was adopted, consisting of nodal geometric parameters interpolated by bicubic Hermite basis functions in the spatial domain and sinusoidal (Fourier) basis functions in the temporal domain. The parameters of an initial static surface were fitted to the three-dimensional (3-D) locations of the coronary arteries at diastasis. The parameters of the time-varying displacement field were then fitted to the tracked displacements of the bifurcation points of the coronary arteries. The locations of the vessel centerlines were then tracked from frame to frame throughout the cycle, using the fit to the bifurcations as a reference state. Owing to the nonuniform distribution of the superficial arteries over the epicardium, weighted spline smoothness constraints were added to the error function in order to regularize the least-squares solution. The time-varying surface model provides a complete description of epicardial deformation over the entire region spanned by the ensemble basis functions. The Green’s strain tensor, referred to in-plane material coordinates, was used to calculate principal (major and minor) surface strains at any point.

Limits on static shape control for space structures

Abstract: This paper deals with correction of shape distortion due to zero-mean normally distributed errors in structural sizes. The concept of ideal actuators - actuators that can produce any desired displacement field - is introduced. Using this concept, a bound on the possible improvement in the expected value of the root-mean-square shape error is obtained. The shape correction associated with the ideal actuators is also characterized. An actuator effectiveness index is developed by comparing the displacement field generated by the actuator to the ideal. The results are specialized to a simple form for truss structures composed of nominally identical members. The bound and effectiveness index are tested on a 55-m radiometer antenna truss structure. It is found that previously obtained results for optimum actuators had a performance close to the bound obtained here. Also, it is found that large numbers of actuators are needed for large reductions in shape errors. Furthermore, the actuators associated with the optimum design are shown to have high effectiveness indices. Since only a small fraction of truss elements tend to have high effectiveness indices, the use of the effectiveness index can greatly reduce the number of truss members that need to be considered actuator sites.

p-version hierarchical three dimensional curved shell element for elastostatics

Abstract: This paper presents a hierarchical three dimensional curved shell finite element formulation based on the p-approximation concept. The element displacement approximation can be of arbitrary and different polynomial orders in the plane of the shell (ξ, η) and the transverse direction (ξ). The curved shell element approximation functions and the corresponding nodal variables are derived by first constructing the approximation functions of orders pξ, pη and pξ and the corresponding nodal variable operators for each of the three directions ξ, η and ξ and then taking their products (sometimes also known as tensor product). This procedure gives the approximation functions and the corresponding nodal variables corresponding to the polynomial orders pξ, pη and pξ. Both the element displacement functions and the nodal variables are hierarchical; therefore, the resulting element matrices and the equivalent nodal load vectors are hierarchical also, i.e. the element properties corresponding to the polynomial orders pξ, pη and pξ are a subset of those corresponding to the orders (pξ + 1), (pη +1) and (pξ +1). The formulation guarantees C° continuity or smoothness of the displacement field across the interelement boundaries. The geometry of the element is described by the co-ordinates of the nodes on its middle surface (ξ = 0) and the nodal vectors describing its bottom (ξ = −1) and top (ξ = +1) surfaces. The element properties are derived using the principle of virtual work and the hierarchical element approximation. The formulation is equally effective for very thin as well as very thick plates and curved shells. In fact, in many three dimensional applications the element can be used to replace the hierarchical three dimensional solid element without loss of accuracy but significant gain in modelling convenience. Numerical examples are presented to demonstrate the accuracy, efficiency and overall superiority of the present formulation. The results obtained from the present formulation are compared with those available in the literature as well as analytical solutions.

Phase-stepped fractional moiré

Abstract: Two computer-aided methods have been developed for analysis of in-plane and out-of-plane surface displacements of structures under load. Both methods are whole-field techniques which combine phase-stepped geometric moire with video and computer technologies. With these methods, a displacement field of interest is determined by computer-processing phase stepped, geometric moire image data with fringe ordering done automatically within the software. The theory of the techniques is described and results of accuracy tests and application problems are given. It is shown that very good agreement is obtained between theory and experiment for in-plane strain determinations. For out-of-plane displacement determinations errors are only a few percent over the entire field of view. The application problems discussed are: (1) the measurement of composite column buckling, and (2) the determination of the shape of a slightly distorted, thin aluminum plate.

Sensitivity analysis and optimum design of elastic-plastic structural systems

Abstract: The paper deals with elastic-plastic optimization of flexural structural systems, subjected to kinematic restrictions A finite holonomic piecewise linear elastic-hardening constitutive law is adopted Sensitivity analysis for the displacement field is also performed, and a suitable finite element formulation, allowing for the spreading of plasticity, is also given Finally, some meaningful numerical applications, together with their physical interpretation, are presented

Elastic stress analysis of three-dimensional solids with small holes by BEM

Abstract: An advanced level development of the boundary element method is presented for the elastic stress analysis of a three-dimensional solid containing a large number of small diameter, tubular shaped holes. The formulation has been developed such that these holes can be modeled by a system of curvilinear line elements, resulting in substantial savings in both data preparation and computing costs. This is accomplished by assuming a variation in the displacement field along the circumference defined in terms of trigonometric functions together with a linear or quadratic variation of displacements along the longitudinal direction.

Performance Characteristics of Externally Pressurized Orifice Compensated Flexible Journal Bearing

Abstract: This paper presents a theoretical study of the elastohydrostatic performance of an orifice-compensated multirecess journal bearing. The finite element method has been used to solve Reynolds equation governing the lubricant flow in the clearance space of a hydrostatic journal bearing and three dimensional elasticity equations governing the displacement field in the bearing shell. Results have been presented for a wide range of values of deformation coefficient ([Cbar]d). An example is also given to enhance understanding of the effects of bearing shell flexibility on performance characteristics.