Rigid Body Translation Research Articles

Identities for the fundamental solution of thin plate bending problems and the nonuniqueness of the hypersingular BIE solution for multi-connected domains

Four integral identities for the fundamental solution of thin plate bending problems are presented in this paper. These identities can be derived by imposing rigid-body translation and rotation solutions to the two direct boundary integral equations (BIEs) for plate bending problems, or by integrating directly the governing equation for the fundamental solution. These integral identities can be used to develop weakly-singular and nonsingular forms of the BIEs for plate bending problems. They can also be employed to show the nonuniqueness of the solution of the hypersingular BIE for plates on multi-connected (or multiply-connected) domains. This nonuniqueness is shown for the first time in this paper. It is shown that the solution of the singular (deflection) BIE is unique, while the hypersingular (rotation) BIE can admit an arbitrary rigid-body translation term in the deflection solution, on the edge of a hole. However, since both the singular and hypersingular BIEs are required in solving a plate bending problem using the boundary element method (BEM), the BEM solution is always unique on edges of holes in plates on multi-connected domains. Numerical examples of plates with holes are presented to show the correctness and effectiveness of the BEM for multi-connected domain problems.

Assessment of Gradient-based Digital Speckle Correlation Measurement Errors

The optical method Digital Speckle Correlation Measurement (DSCM) has been extensively applied due its capability to measure the entire displacement field over a body surface. A formula of displacement measurement errors by the gradient-based DSCM method was derived. The errors were found to explicitly relate to the image grayscale errors consisting of sub-pixel interpolation algorithm errors, image noise, and subset deformation mismatch at each point of the subset. A power-law dependence of the standard deviation of displacement measurement errors on the subset size was established when the subset deformation was rigid body translation and random image noise was dominant and it was confirmed by both the numerical and experimental results. In a gradient-based algorithm the basic assumption is rigid body translation of the interrogated subsets, however, this is in contradiction to the real circumstances where strains exist. Numerical and experimental results also indicated that, subset shape function mismatch was dominant when the order of the assumed subset shape function was lower than that of the actual subset deformation field and the power-law dependence clearly broke down. The power-law relationship further leads to a simple criterion for choosing a suitable subset size, image quality, sub-pixel algorithm, and subset shape function for DSCM.

EVALUATING STRESS INTENSITY FACTOR OF POLYCARBONATE USING DIGITAL IMAGE CORRELATION

The application of the digital image correlation (DIC) method to determine the stress intensity factor (SIF) for cracks in Polycarbonate is proposed in this paper. Standard compact tension specimen was tested under tensile loading and the full-field displacement fields of the test sample were recorded using DIC. According to linear elastic fracture mechanics, the SIF associated with unavoidable rigid-body displacement translation and rotation were calculated simultaneously from the experimental data by fitting the theoretical displacement field using the method of least-squares. Selection of displacement and convergence values is discussed. The effectiveness and accuracy of the proposed technique is verified by comparison with theoretical results, which reveals that the DIC is a practical and effective tool for full-field deformation and SIF measurement.

Speckle Cross-Correlation Method in Measuring Fine Surface Displacements

Industrial applications need regular testing for the lifetime, movement, strength, and performance of manufacturing machines during production process. Since speckle photography is a simple economic technique, it is used in investigating object response under mechanical and thermal effects depending on the movement of the speckle patterns with respect to the deformation strength and direction. In the present work, the cross-correlation technique is used to analyze the speckle patterns by iterative method to define both values and directions of rigid body translation and expansion. In order to check the accuracy of the cross-correlation technique, the results are compared with the displacement values given by analyzing the Young's interference fringes resulted from the Fourier transformation of the speckle patterns. This noncontact technique is found to be accurate and informative depending on the stability and sensitivity of the optical system. This method of measurement is an effective tool in studying the hard cases of objects and machines under various effects.

On the influence of a translating inner core in models of outer core convection

It has recently been proposed that the hemispheric seismic structure of the inner core can be explained by a self-sustained rigid-body translation of the inner core material, resulting in melting of the solid at the leading face and a compensating crystallisation at the trailing face. This process induces a hemispherical variation in the release of light elements and latent heat at the inner-core boundary, the two main sources of thermochemical buoyancy thought to drive convection in the outer core. However, the effect of a translating inner core on outer core convection is presently unknown. In this paper we model convection in the outer core with a nonmagnetic Boussinesq fluid in a rotating spherical shell driven by purely thermal buoyancy, incorporating the effect of a translating inner core by a time-independent spherical harmonic degree and order 1 (Y11) pattern of heat-flux imposed at the inner boundary. The analysis considers Rayleigh numbers up to 10 times the critical value for onset of nonmagnetic convection, a parameter regime where the effects of the inhomogeneous boundary condition are expected to be most pronounced, and focuses on varying q∗, the amplitude of the imposed boundary anomalies. The presence of inner boundary anomalies significantly affects the behaviour of the model system. Increasing q∗ leads to flow patterns dominated by azimuthal jets that span large regions of the shell where radial motion is significantly inhibited. Vigorous convection becomes increasingly confined to isolated regions as q∗ increases; these regions do not drift and always occur in the hemisphere subjected to a higher than average boundary heat-flux. Effects of the inner boundary anomalies are visible at the outer boundary in all inhomogeneous models considered. At low q∗ the expression of inner boundary effects at the core surface is a difference in the flow speed between the two hemispheres. As q∗ increases the spiralling azimuthal jets driven from the inner boundary are clearly visible at the outer boundary. Finally, our results suggest that, when the system is heated from below, a Y11 heat-flux pattern imposed on the inner boundary has a greater overall influence on the spatio-temporal behaviour of the flow than the same pattern imposed at the outer boundary.

On the Treatment of Statical Indeterminacy in the Elementary Statics of Particles and Rigid Bodies

The inability to determine reactions beyond a maximum of six for a three-dimensional rigid body (or three for a planar rigid body) is termed statical indeterminacy and represents a flaw in the elementary statics of particles and rigid bodies because the indeterminate reactions cannot be obtained by methods developed from within the subject. As noted in virtually every modern textbook on statics, statical indeterminacy may be resolved by relaxing the assumption of structural rigidity. This theory forms the content of the mechanics of deformable solids, which considers the stretching, twisting and bending of elastic bars. The indeterminacy may also be resolved by maintaining structural rigidity but relaxing the assumption of support rigidity. Because deformable supports may be modeled by lumped linear and torsional springs (which are traditionally part of the statics of particles and rigid bodies), analysis of statically indeterminate reactions in such systems utilizes no new concepts except that of rigid body translation and rotation. The purpose of this paper is to demonstrate, by means of example, that this class of statically indeterminate problem is amenable to simple solution and therefore should be treated in standard presentations of the subject.

Analytical prediction of stepped feature generation in multi-pass single point incremental forming

Single point incremental forming (SPIF) is a new sheet metal forming process characterized by higher formability, product independent tooling and greater process flexibility. The inability of conventional single pass SPIF to form vertical walls without failure is overcome by forming multiple intermediate shapes before forming the final component, i.e., multi-pass single point incremental forming (MSPIF). A major issue with MSPIF is significant geometric inaccuracy of the formed component, due to the generation of stepped features on the base. This work proposes analytical formulations that are shown to accurately and quantitatively predict the stepped feature formation in MSPIF. Additionally, a relationship is derived among the material constants used in these analytical equations, the yield stress and thickness of the blank material, such that the computational effort required for the calibration of these constants can be minimized. Finally, the physical effects of yield stress and sheet thickness on the rigid body translation are further discussed.

Digital Image Correlation Using Stochastic Parallel-Gradient-Descent Algorithm

This article proposes a digital image correlation (DIC) method based on the stochastic parallel gradient descent (SPGD) algorithm. Stochastic parallel perturbations are imposed on deformation parameters to make the correlation coefficients converge to a global extremum; thus, this allows the final measured values of the deformation parameters to be obtained and the DIC measurement to be made. Both simulated and real data processing, including rigid body and strain deformation, show that the proposed method can achieve nearly the same accuracy as the Newton–Raphson (NR) method in most cases and higher accuracy in some cases, such as the simulated experiments of rigid body translation with and without noise. It also has a good noise-robustness. Furthermore, a series of experiments have been designed to evaluate the convergence characteristics of the proposed method, and it has been proved able to process large displacement and have a stable convergence process, good robustness, and a high convergence speed when bilinear interpolation is adopted.

SU-D-217A-05: Auto-Registration of Cardiac PET/CT Images with a 3D Weighted Gradient Correlation Algorithm.

To design a novel 3D automatic registration algorithm for cardiac PET/CT registration using gradient information and to evaluate the performance of the algorithm with clinical PET/CT datasets. The 512×512×47 CT images are at first resized to 128×128×47 to match the matrix size of PET. In order to maximize the gradient information at boundaries in CT, a conventional fuzzy c-means clustering algorithm (number of cluster = 7) is implemented to suppress signals from tissues that do not contribute much useful information for the registration purpose (fat, lung, and bones). The 3D Image gradient map, consisting of all three orthogonal components, is derived from the PET images and the post- clustering CT images. The mis-registration is modeled as 3D rigid body translation in this study, though it can be extended to include rotations as well. The details of the gradient-based objective function are described in the support document. Optimal registration is determined by searching for the maximum of the objective function over a range of potential translation positions (e.g. 8.2×8.2×2.8cm). This process is repeated at a higher matrix size (512×512×47) to refine the result of registration. We applied this auto registration technique on 55 patient data sets of cardiac PET/CT images. The CT images were average-CT images, and the PET images were without attenuation correction. 54 out of the 55 cases produced satisfactory registration. The proposed weighted gradient correlation algorithm is a viable solution for auto registration of cardiac PET/CT images. More work is needed to further improve the robustness of thealgorithm.

Fabrication technique of micro/nano-scale speckle patterns with focused ion beam

The fabrication technique of micro/nano-scale speckle patterns with focused ion beam (FIB) system is studied for digital image correlation (DIC) measurement under a scanning electron microscope (SEM). The speckle patterns are fabricated by directly etching the counterpart of the specimen to the black part of a template. Mean intensity gradient is used to evaluate the quality of these SEM images of speckle patterns fabricated based on different templates to select an optimum template. The pattern size depending on the displacement measurement sensitivity is adjusted by altering the magnification of FIB according to the relation curve of the etching size versus magnification. The influencing factors including etching time and ion beam current are discussed. Rigid body translation tests and rotation tests are carried out under SEM to verify the reliability of the fabricated speckle patterns. The calculated values are in good agreement with the imposed ones.

Measuring displacement fields by cross-correlation and a differential technique: experimental validation

A digital image correlation (DIC) algorithm for displacement measurements combining cross-correlation and a differential technique was validated through a set of experimental tests. These tests consisted of in-plane rigid-body translation and rotation tests, a tensile mechanical test, and a mode I fracture test. The fracture mechanical test, in particular, was intended to assess the accuracy of the method when dealing with discontinuous displacement fields, for which subset-based image correlation methods usually give unreliable results. The proposed algorithm was systematically compared with the Aramis® DIC-2D commercial code by processing the same set of images. When processing images from rigid-body and tensile tests (associated with continuous displacement fields), the two methods provided equivalent results. When processing images from the fracture mechanical test, however, the proposed method obtained a better qualitative description of the discontinuous displacements. Moreover, the proposed method gave a more reliable estimation of both crack length and crack opening displacement of the fractured specimen.©

Measurement of mixed-mode stress intensity factors using digital image correlation method

Applications of the digital image correlation method (DIC) for the determination of the mixed-mode stress intensity factors (SIF) is investigated in this paper. Experiments were performed on an edge fatigue cracked aluminum specimen using a special loading device, which is an appropriate apparatus for experimental mixed-mode fracture analysis. The full-field displacements around the crack-tip region of the test sample were calculated using DIC. And then the SIF associated with unavoidable rigid-body displacement motion were calculated simultaneously from the experimental data. The effect of the rigid body motion on the measured displacements was then eliminated using the computed rigid body translation and rotation. A coarse-fine searching method was developed to determine the crack-tip location. For validation, the SIF thus determined is compared with theoretical results, confirming the effectiveness and accuracy of the proposed technique. Therefore it reveals that the DIC is a practical and effective tool for full-field deformation and SIF measurement.

Single-shot profilometry of rough surfaces using hyperspectral interferometry

The combination of white light interferometry with hyperspectral imaging ("hyperspectral interferometry") is a recently proposed technique for single-shot measurement of 3D surface profiles. We consider for the first time its application to speckled wavefronts from optically rough surfaces. The intensity versus wavenumber signal at each pixel provides unambiguous range information despite the speckle-induced random phase shifts. Experimental results with samples undergoing controlled rigid body translation demonstrate a measurement repeatability of 460 nm for a bandwidth of approximately 30 nm. Potential applications include roughness measurement and coordinate measurement machine probes where rapid data acquisition in noncooperative environments is essential.

A linear soft tissue artefact model for human movement analysis: Proof of concept using in vivo data

We investigated the accuracy of a linear soft tissue artefact (STA) model in human movement analysis. Simultaneously recorded bone-mounted pin and skin marker data for the thigh and shank during walking, cutting and hopping were used to measure and model the motion of the skin marker clusters within anatomical reference frames (ARFs). This linear model allows skin marker movements relative to the underlying bone contrary to a rigid-body assumption. The linear model parameters were computed through a principal component analysis, which revealed that 95% of the variance of the STA motion for the thigh was contained in the first four principal components for all three tasks and all subjects. For the shank, 95% of the variance was contained in the first four principal components during walking and cutting and first five during hopping. For the thigh, the maximum residual artefact was reduced from 27.0mm to 5.1mm (walking), 22.7mm to 3.0mm (cutting) and 16.2mm to 3.5mm (hopping) compared to a rigid-body assumption. Similar reductions were observed for the shank: 24.2mm to 1.9mm (walking), 20.3mm to 1.9mm (cutting) and 14.7mm to 1.8mm (hopping). A geometric analysis of the first four principal components revealed that, within the ARFs, marker cluster STA is governed by rigid-body translations and rotations rather than deformations. The challenge remains, however, in finding the linear model parameters without bone pin data, but this investigation shows that relatively few parameters in a linear model are required to model the vast majority of the STA movements.

Bridge Deflection Measurement Using Digital Image Correlation with Camera Movement Correction

When displacement measurement by digital image correlation is performed in outside for the inspection of real structures, the position and the direction of a camera are often changed slightly because of wind, oscillations and the lack of stability of ground. In order to realize the bridge deflection measurement by digital image correlation, a method for correcting the effect of camera movement is proposed in this study. The relationship between images before and after the camera movement is described by an equation of perspective transformation. The unknown coefficients of the equation are determined from undeformed regions of the images. Then, the effect of the camera movement is eliminated by using the perspective transformation. The effectiveness is validated by applying the proposed method to the rigid body rotation and translation measurement of a planar specimen, the deflection measurement of a wide-flange beam, and the bridge deflection measurement. Results show that the effect of the camera movement can be corrected by the proposed method. It is emphasized that noncontact displacement measurement is possible by simple and easy procedure with digital image correlation for the structural evaluation of infrastructures. [doi:10.2320/matertrans.I-M2011843]

Initial guess by improved population-based intelligent algorithms for large inter-frame deformation measurement using digital image correlation

Digital image correlation (DIC) has received a widespread research and application in experimental mechanics. In DIC, the performance of subpixel registration algorithm (e.g., Newton–Raphson method, quasi-Newton method) relies heavily on the initial guess of deformation. In the case of small inter-frame deformation, the initial guess could be found by simple search scheme, the coarse-fine search for instance. While for large inter-frame deformation, it is difficult for simple search scheme to robustly estimate displacement parameters and deformation parameters simultaneously with low computational cost. In this paper, we proposed three improving strategies, i.e. Q-stage evolutionary strategy (T), parameter control strategy (C) and space expanding strategy (E), and then combined them into three population-based intelligent algorithms (PIAs), i.e. genetic algorithm (GA), differential evolution (DE) and particle swarm optimization (PSO), and finally derived eighteen different algorithms to calculate the initial guess for qN. The eighteen algorithms were compared in three sets of experiments including large rigid body translation, finite uniaxial strain and large rigid body rotation, and the results showed the effectiveness of proposed improving strategies. Among all compared algorithms, DE-TCE is the best which is robust, convenient and efficient for large inter-frame deformation measurement.

Thickness-shear vibration of an elastic plate carrying an array of rigid microbeams with consideration of couple stresses

We study thickness-shear vibration of an elastic plate carrying an array of rigid microbeams with their bottoms attached to the top surface of the plate. The beams undergo rigid-body translation and rotation when the plate is in thickness-shear motion. The plate is modeled by the couple-stress theory of elasticity to properly take into account both the shear forces and the bending moments at the bottoms of the beams. A equation that determines the resonant frequencies of the structure is derived. Analytical and numerical examinations of a few special cases of the general frequency equation reveal a few basic effects of the microbeams and the couple stress. The results are useful in measuring geometric/physical properties of the microbeams using plate thickness-shear mode acoustic wave sensors.

Shape and displacement measurement of discontinuous surfaces by combining fringe projection and digital image correlation

We describe how a shape-measurement system (SMS) based on fringe projection can be combined with a two-dimensional digital image correlation (DIC) technique to accurately measure both surface profile and displacement fields at the same time. Whereas the measurement of all three displacement components by traditional DIC requires the use of at least two cameras, the approach presented here provides the full three-dimensional (3-D) displacement field from a single-camera, single-projector SMS with no additional hardware requirements. Furthermore, the single-pixel spatial resolution of the fringe projection technique can be exploited to prevent the correlation peak-splitting phenomenon that occurs when a DIC subimage straddles a global geometrical discontinuity. Thus, unlike traditional 3-D DIC techniques, the proposed method can measure displacement fields on discontinuous surfaces as easily as on smooth ones. Details of the algorithm are given together with experimental results of a rigid-body translation test. Measurements made during a routine fatigue test on a part of a wing panel are also presented.

Assessment of the Deformation of Low Density Polymeric Auxetic Foams by X-Ray Tomography and Digital Volume Correlation

This paper deals with the use of X-ray tomography and Digital Volume Correlation in order to study the mechanical behaviour of a low density polymeric auxetic foam. First, the metrological performances of the procedure are assessed using rigid body translation. Then, tensile test results are analyzed for two load steps and comments on the deformation process of the foam are given with a specific view to Poisson’s effect.

The Stress Intensity Factor of Opening Mode Determined by Digital Image Correlation

Applications of the digital image correlation method (DIC) for the determination of the opening mode stress intensity factor (SIF) is investigated using an edge cracked aluminum plate in this paper. Standard compact tension test specimen was tested under tensile loading and the full-field displacement fields of the test sample were recorded using DIC. The SIF associated with unavoidable rigid-body displacement translation were calculated simultaneously from the experimental data by fitting the theoretical displacement field using the method of least-squares. Selection of displacement and convergence values is discussed. For validation, the SIF thus determined is compared with theoretical results, confirming the effectiveness and accuracy of the proposed technique. Therefore it reveals that the DIC is a practical and effective tool for full-field deformation and SIF measurement.