Measurement Of Strain Tensor Research Articles

Full strain tensor measurements with X-ray diffraction and strain field mapping: a simulation study.

Strain tensor measurements are important for understanding elastic and plastic deformation, but full bulk strain tensor measurement techniques are still lacking, in particular for dynamic loading. Here, such a methodology is reported, combining imaging-based strain field mapping and simultaneous X-ray diffraction for four typical loading modes: one-dimensional strain/stress compression/tension. Strain field mapping resolves two in-plane principal strains, and X-ray diffraction analysis yields volumetric strain, and thus the out-of-plane principal strain. This methodology is validated against direct molecular dynamics simulations on nanocrystalline tantalum. This methodology can be implemented with simultaneous X-ray diffraction and digital image correlation in synchrotron radiation or free-electron laser experiments.

Plural spectral frequency divisions for ultrasonic tissue displacement vector measurement.

For a human tissue displacement vector or strain tensor measurement, the plural spectral frequency division method (PSFDM) is presented. In this report, the effectiveness of the processing is demonstrated by performing the measurements on an agar phantom interrogated by wideband ultrasonic (US) single beam scanning and compounding of steered plane wave transmissions.

Full elastic strain and stress tensor measurements from individual dislocation cells in copper through-Si vias.

Nondestructive measurements of the full elastic strain and stress tensors from individual dislocation cells distributed along the full extent of a 50 µm-long polycrystalline copper via in Si is reported. Determining all of the components of these tensors from sub-micrometre regions within deformed metals presents considerable challenges. The primary issues are ensuring that different diffraction peaks originate from the same sample volume and that accurate determination is made of the peak positions from plastically deformed samples. For these measurements, three widely separated reflections were examined from selected, individual grains along the via. The lattice spacings and peak positions were measured for multiple dislocation cell interiors within each grain and the cell-interior peaks were sorted out using the measured included angles. A comprehensive uncertainty analysis using a Monte Carlo uncertainty algorithm provided uncertainties for the elastic strain tensor and stress tensor components.

Strain tensor determination of compressed individual silica sand particles using high-energy synchrotron diffraction

The three-dimensional X-ray diffraction (3DXRD) nondestructive technique was used to measure lattice strains within individual sand particles subjected to compressive loading. Three experiments were conducted on similar single columns of silica sand particles with particle sizes between 0.595 and 0.841 mm. In each experiment, three sand particles were placed inside an acrylic mold with an inner diameter of 1 mm. Multiple in situ 3DXRD scans were acquired for each sand column as compressive load was increased. The volume-averaged lattice strain tensor was calculated for each sand particle. In addition, particle orientation and volumetric strain were calculated for individual sand particles. The axial normal strain $$\upvarepsilon _\mathrm{zz}$$ exhibited a linear response in the range of 0 to $$10^{-3}$$ when the applied compressive axial load (F) increased from 0 to $$\sim $$ 30 N when one particle in the sand column fractured. Stress tensor of individual particles was calculated from the acquired lattice strain measurements and elastic constants of silica sand that were reported in the literature. To the best of our knowledge, there have been no reported experimental measurements of the lattice strain tensor measurements within individual silica sand particles. The quantitative measurements reported in this paper at the particle level are very valuable for developing, validating or calibrating micromechanics-based finite element and discrete element models to predict the constitutive behavior of granular materials. 3DXRD represents an exciting new non-destructive technique to directly measure constitutive behavior at the scale of individual particles.

Measurement of the 3D arterial wall strain tensor using intravascular B-mode ultrasound images: a feasibility study

Intravascular ultrasound (IVUS) elastography is a promising tool for studying atherosclerotic plaque composition and assessing plaque vulnerability. Current IVUS elastography techniques can measure the 1D or 2D strain of the vessel wall using various motion tracking algorithms. Since biological soft tissue tends to deform non-uniformly in 3D, measurement of the complete 3D strain tensor is desirable for more rigorous analysis of arterial wall mechanics. In this paper, we extend our previously developed method of 2D arterial wall strain measurement based on non-rigid image registration into 3D strain measurement. The new technique registers two image volumes acquired from the same vessel segment under different levels of luminal pressure and longitudinal stress. The 3D displacement field obtained from the image registration is used to calculate the local 3D strain tensor. From the 3D strain tensor, radial, circumferential and longitudinal strain distributions can be obtained and displayed. This strain tensor measurement method is validated and evaluated using IVUS images of healthy porcine carotid arteries subjected to a luminal pressure increase and longitudinal stretch. The ability of the algorithm to overcome systematic noise was tested, as well as the consistency of the results under different longitudinal frame resolutions.

Elastic strain tensor measurement using electron backscatter diffraction in the SEM

The established electron backscatter diffraction (EBSD) technique for obtaining crystallographic information in the SEM has been adapted to permit elastic strain measurement. Basically, the displacement of crystallographic features in an EBSD pattern, such as zone axes, which result from strain in a crystal, is determined by comparing those same features as they appear in a pattern from an unstrained region of the crystal. The comparison is made by cross-correlation of selected regions in the two patterns. Tests show that the sensitivity to displacement measurement is 1 part in 10 000, which translates to a strain sensitivity of 2 parts in 10 000. Eight components of the strain tensor are determined directly and the ninth is calculated using the fact that the free surface of the sample is traction-free. Examples discussed are taken from studies of a lenticular fracture in germanium, the strain distribution surrounding a carbide precipitate in a nickel base alloy and grain boundary studies in another nickel base alloy.

Relative shear modulus reconstruction for visualization with no geometrical artifact

We previously reported several useful shear modulus reconstruction methods obtained from motion or static equations and reference shear moduli that yield an absolute reconstruction of shear modulus, density or inertia using the finite difference method (FDM) or finite element method (FEM) with strain tensor measurement (i.e., methods referred to as Methods A to C and F). Methods A to C are particularly useful for three-dimensional (3D) shear modulus reconstruction in an incompressible case or in the case of an inhomogeneous Poisson’s ratio by using the mean normal stress as an unknown, whereas Method F using a typical Poisson’s ratio is particularly useful when the reconstruction dimensionality is low or when the Poisson’s ratio is homogeneous. For instance, when using FDM, although Method B permits the absolute reconstruction of a mean normal stress by using a reference mean normal stress, Methods A and C yield biased mean normal stress reconstructions using no reference value and a quasi-reference value (e.g., zero), respectively. In this report, we describe our newly developed Methods D and E obtained from Methods A to C and F, which yield a unique relative shear modulus reconstruction with no geometrical artifact. In Method D, no reference shear modulus is used with Methods A to C and F, whereas in Method E, an arbitrary finite value (e.g., unity) is used with Methods A to C and F as a quasi-reference shear modulus at one or more arbitrary points (i.e., a quasi-reference point) or in an arbitrary region (i.e., a quasi-reference region) in the region of interest. That is, both Method D and Method E permit the visualization of the shear modulus distribution without using an absolute reference value (e.g., no reference material or typical value), which is particularly useful when dealing with deeply located tissues such as the liver and heart tissues. Methods D and E use iterative methods such as the conjugate gradient method to solve algebraic equations. As confirmed from simulated phantom deformation data, each version of Method E results in a higher convergence speed and reconstruction accuracy than the corresponding Method D. Although several versions of Method D using FEM are practically useful, Method E obtainable should be used for such relative reconstructions rather than the corresponding version of Method D, in combination with Methods A to C and F.

Shear modulus reconstruction by strain tensor measurement using lateral modulation

Two-dimensional (2D) shear modulus reconstructions are performed on an agar phantom using our previously developed ultrasound lateral Gaussian envelope cosine modulation method (LGECMM) together with multidimensional displacement vector measurement methods, i.e., the multidimensional cross-spectrum phase gradient method (MCSPGM), the multidimensional autocorrelation method (MAM) and the multidimensional Doppler method (MDM). The accuracies of the obtained 2D reconstructions are compared with each other in addition to those of 1D reconstructions obtained from the respective ratios of lateral and axial strains.

Effective lateral modulations with applications to shear modulus reconstruction using displacement vector measurement

High accuracy in measuring target motions can be realized by combined use of our previously developed lateral Gaussian envelope cosine modulation method (LGECMM) and displacement vector measurement methods that enable simultaneous axial and lateral displacement measurements, such as the multidimensional autocorrelation method (MAM). In this paper, LGECMM is improved by using parabolic functions and Hanning windows instead of Gaussian functions in the apodization function, i.e., parabolic apodization and Hanning apodization. The new modulations enable decreases in effective aperture length (i.e., channels) and yield more accurate displacement vector measurements than LGECMM due to increased echo signal-to-noise ratio and lateral spatial resolution. That is, on the basis of a priori knowledge about ultrasound propagation using the focusing scheme and shape of the apodization function, we stopped using Fraunhofer approximation. As practical applications of the modulations, for an agar phantom that is deformed in a lateral direction, stable and accurate 2-D shear modulus reconstructions are performed using our previously developed direct inversion approach together with 2-D strain tensor measurements using MAM.

Regularization for ultrasonic measurements of tissue displacement vector and strain tensor

In this report, we present the regularization results of displacement vector measurement by the multidimensional cross-spectrum phase gradient method (MCSPGM), multidimensional autocorrelation method (MAM), and multidimensional Doppler method (MDM) without lateral modulation. In addition to the results of spatially variant regularization in displacement vector measurement using displacement variances, the results of displacement component-dependent regularization are presented. That is, according to the measurement accuracy of displacement components, the regularization is properly applied to the respective displacement components. For instance, only the regularization on the lateral displacement is effective. That is, for the lateral nonmodulation case, the accuracies and stabilities of lateral/elevational displacement measurements significantly increase. In conjunction, the convergence speed of phase matching also increases. The demonstrated measurements of the displacement vector distributions in experiments using inhomogeneous shear modulus agar phantoms confirm that displacement-componentdependent regularization enables better strain tensor measurement and shear modulus reconstruction from the viewpoints of accuracy and stability. The contrast-to-noise ratio (CNR) is useful to set regularization parameters properly for the displacement regularization, whereas the CNR is useless for shear modulus regularization in the sense that the accuracy of reconstruction value cannot be evaluated.

P1-57 Improvement of accuracy and stability for measurement of tissue displacement vector and strain tensor by regularization

P1-57 Improvement of accuracy and stability for measurement of tissue displacement vector and strain tensor by regularization

X-ray Microbeam Diffraction Measurements in Polycrystalline Aluminum and Copper Thin Films

ABSTRACTThermally induced residual strains in polycrystalline Cu and Al films on single crystal Si and glass substrates, respectively, have been examined on a grain-by-grain basis by x-ray microbeam diffraction. The crystallographic orientation and the deviatoric strain tensor, εij*, are determined for each grain by white beam Laue diffraction. From grain orientation mapping and strain tensor measurements, information is obtained about the distributions of strains for similarly oriented grains, about strain variations within single grains, and about grain-to-grain correlations of strains. This type of information may be useful in developing and testing theories for intergrain effects in strain evolution in polycrystals.

Proposal for a multi-detector system for residual stress measurements

Abstract The scheme of a multi-detector system dedicated to full strain tensor measurements by neutron diffraction is suggested. The proposed system can considerably shorten measurement time in comparison with the conventional one-detector system and simplify the measurement procedure. A “ring” of detectors disposed along the circumference of a diffraction cone which can simultaneously scan different parts of the diffraction cone is suggested. The detectors constituting the “ring” measure strain in different directions, with the number of directions equal to the number of detectors. The instrument has maximum advantage for the measurement of the full strain tensor when the gauge volume is small and fine collimation (∼0.2°) is used to provide spatial resolution in the horizontal and vertical direction. The experiments carried out using the one-dimensional position sensitive detector showed the feasibility of the proposed multi-detector system.

How to choose sample orientations in full strain/stress tensor determination with a time-of-flight goniometer

Abstract A computer program helping to plan strain tensor measurements is presented. It searches for those goniometer positions, which yield the most accurate result for a given number of measurements, a given specimen and the goniometer used. The concept of the program is very open so that it can easily be adapted for any TOF goniometer, no matter how many rotation axes it has, no matter also if it is built with one or two detectors and how these are situated. It can also take account of texture of the specimen or other influences on the intensity of the diffracted beam. The program is available from the authors or at www.oeaw.ac.at/esi/ as source code so that it can be compiled and run in any operating system and can also be changed for the special need of the user.