Strain Tensor Determination Research Articles

Investigation of residual strain in quartzite using neutron diffraction: Interpretation and self‐consistent modelling

The residual strain determination by neutron diffraction has been performed on a set of five natural quartzite samples collected from Betic Cordilleras (Spain) and Sivas Basin (Turkey). The starting idea was to document an earlier conclusion about the texture components related either to deformation or recrystallization phenomena. Thereafter, as a consequence of the rock inhomogeneity, the complete strain tensor has been determined for each sample and discussed by taking into consideration the present‐day knowledge in material sciences. The imposed macroscopic deformation observed at the sample locality led us to consider the form of the residual strain tensor as possibly representative of the natural finite plastic deformation suffered at the sample scale. Based on this, those tensors have been fitted to plastic tensors and introduced in texture simulations using a visco‐plastic self‐consistent (VPSC) model. The results demonstrate that the complete and meaningful strain tensor determination in rocks could be used as an experimental input in numerical modelling. This new combination of experimental and numerical approaches was validated and could help to better understand some geological phenomena.

Full elastic strain tensor determination at the phase scale in a powder metallurgy nickel-based superalloy using X-ray Laue microdiffraction

Laue microdiffraction is used to determine the full elastic strain tensor of the γ and γ′ phases in grains of a nickel-based superalloy with a coarse-grained microstructure. A `rainbow' filter and an energy dispersive point detector are employed to measure the energy of Bragg reflections. For the two techniques, an uncertainty of ±2.5 × 10−3 Å is obtained for the undetermined crystal lattice parameter. Our measurements show that the filter method provides better confidence, energy resolution, accuracy and acquisition time. The sensitivity of each method with respect to the γ–γ′ lattice mismatch is demonstrated with measurements in samples with average precipitate sizes of 200 and 2000 nm. For the 200 nm precipitate size, the lattice mismatch is less than 2 × 10−3 Å and the dilatational strains are close to ±1.5 × 10−3 depending on the considered phase. For the 2000 nm precipitate size, the lattice mismatch is close to 8 × 10−3 Å and almost no elastic strain occurs in the microstructure.

Single-shot full strain tensor determination with microbeam X-ray Laue diffraction and a two-dimensional energy-dispersive detector.

The full strain and stress tensor determination in a triaxially stressed single crystal using X-ray diffraction requires a series of lattice spacing measurements at different crystal orientations. This can be achieved using a tunable X-ray source. This article reports on a novel experimental procedure for single-shot full strain tensor determination using polychromatic synchrotron radiation with an energy range from 5 to 23 keV. Microbeam X-ray Laue diffraction patterns were collected from a copper micro-bending beam along the central axis (centroid of the cross section). Taking advantage of a two-dimensional energy-dispersive X-ray detector (pnCCD), the position and energy of the collected Laue spots were measured for multiple positions on the sample, allowing the measurement of variations in the local microstructure. At the same time, both the deviatoric and hydrostatic components of the elastic strain and stress tensors were calculated.

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.

Residual strain tensor determination within highly plastically deformed torsion samples using high energy synchrotron radiation

Abstract The residual strain tensor is measured within strongly plastically deformed torsion samples using high energy synchrotron radiation. The diameter of the solid AlMg3 samples is 5 mm. A low speed deformation with free ends to γ = 150% is carried out at room temperature. The strain mapping technique utilizes a micro-focussed incident X-ray beam, a conical slit cell, and a two dimensional detector. Extremely fast data acquisition is achieved by the simultaneous observation of six diffraction rings emanating from the same local gauge volume within the sample. The radial dependence of the complete residual strain tensor is evaluated from the deformation of the Al 422 diffraction rings.

Validity of vegard's rule for the lattice parameter and the stiffness elastic constant ratios of the AlGaAs ternary compound

A new procedure for the determination of the Poisson ratio, the elastic stiffness constant ratios and the lattice parameter of epitaxial films is presented which is based on the strain tensor determination by high-resolution X-ray diffraction by using epitaxial layers simultaneously grown on three different substrate orientations and different alloy compositions. Our method allows us to verify the validity of the Vegard's rule and/or to evidence a difference in the incorporation of group- III-elements in ternary III– V semiconductor films. We discuss the results obtained from Al x Ga 1− x As epilayers coherently grown simultaneously on (1 0 0)-, (2 1 0)- and (3 1 1)-GaAs substrate crystals. Assuming a GaAs lattice parameter d GaAs=0.565325±0.000002 nm our experiment yields the AlAs lattice parameter d AlAs=0.566139±0.000005 nm. The stiffness elastic lattice constant ratios for AlAs is c 12/ c 11=0.475±0.003, which leads to a Poisson ratio ν=0.322±0.001 and c 44/ c 11=0.517±0.003. An analogous analysis for the ternary Al x Ga 1− x As compound with x=0.5 yields a lattice parameter value and elastic constant ratios which are in agreement with the Vegard's rule. Within the accuracy of our method no deviation for the Al incorporation in the differently oriented samples has been observed.

Validity of vegard's rule for the lattice parameter and the stiffness elastic constant ratios of the AlGaAs ternary compound

A new procedure for the determination of the Poisson ratio, the elastic stiffness constant ratios and the lattice parameter of epitaxial films is presented which is based on the strain tensor determination by high-resolution X-ray diffraction by using epitaxial layers simultaneously grown on three different substrate orientations and different alloy compositions. Our method allows us to verify the validity of the Vegard's rule and/or evidence a difference in the incorporation of group-III-elements in ternary III-V semiconductor films. We discuss the results obtained from Al x Ga 1− x As epilayers coherently grown simultaneously on (1 0 0)-, (2 1 0)- and (3 1 1)-GaAs substrate crystals. Assuming a GaAs lattice parameter d GaAs=0.565325±0.000002 nm our experiment yields the AlAs lattice parameter d AlAs=0.566139±0.000005 nm. The stiffness elastic constant ratios for AlAs are c 12/ c 11=0.475±0.003, which leads to a Poisson ratio ν=0.322±0.001 and c 44/ c 11=0.517±0.003. An analogous analysis for the ternary Al x Ga 1− x As compound with x=0.5 yields a lattice parameter value and elastic constant ratios which are in agreement with the Vegard's rule. Within the accuracy of our method no deviation for the Al incorporation in the differently oriented samples has been observed.

Accurate determination of strain tensors from small shifts of reflections measured on a four-circle diffractometer

Formulas are given for the derivation of the changes in Bragg angles and the corresponding full asymmetric strain tensor from the shifts of reflections measured on a four-circle diffractometer. The effects of using either an ω- or a 2θ-scan mode are discussed. A least-squares program was written on the basis of the derived formulas and applied to data measured on the piezoelectric crystals of AgGaS2 under the influence of an external electric field. The obtained piezoelectric constants are in good agreement with values determined by conventional methods.

Limitations on the strain tensor determination by neutron diffraction using a position-sensitive detector

Diffraction techniques such as neutron diffraction allow strain components to be measured in arbitrarily chosen directions in structural components, and hence complete strain tensors can in principle be calculated from any six measured normal strain components. However, in a newly developed technique, whereby a position-sensitive detector is used for simultaneous measurements of chosen strain components at several points along a line through the specimen, there appear to be some limitations on the choice of strain components for strain tensor determination. This note verifies the nature of these limitations.

Limitations on the strain tensor determination by neutron diffraction using a position-sensitive detector

Limitations on the strain tensor determination by neutron diffraction using a position-sensitive detector