Full local elastic strain tensor from Laue microdiffraction: simultaneous Laue pattern and spot energy measurement

2011 ◽  
Vol 44 (4) ◽  
pp. 688-696 ◽  
Author(s):  
Odile Robach ◽  
Jean-Sébastien Micha ◽  
Olivier Ulrich ◽  
Patrice Gergaud
Keyword(s):  
Strain Tensor ◽  
Polycrystalline Sample ◽  
Laue Pattern ◽  
Test Cases ◽  
Hydrostatic Strain ◽  
Measurement Strategies ◽  
Elastic Strain Tensor ◽  
Point Detector ◽  
Beam Technique ◽  
Laue Microdiffraction

In sample-scanning Laue microdiffraction, the local crystal orientation and local deviatoric strain tensor are obtained by illuminating the polycrystalline sample with a broadband `white' (5–30 keV) X-ray microbeam and analyzing the spot positions in the resulting local Laue pattern. Mapping local hydrostatic strain is usually slower, owing to the need to alternate between white and tunable-energy monochromatic microbeams. A technique has been developed to measure hydrostatic strain while keeping the white beam. The energy of one of the Laue spots of the grain of interest is measured using an energy-dispersive point detector, while simultaneously recording the Laue pattern on the two-dimensional detector. The experimental spot energy,Eexp, is therefore measured at the same time asEtheor, the theoretical spot energy for zero hydrostatic strain, which is derived from the analysis of the Laue pattern. The performance of the technique was compared with that of the monochromatic beam technique in two test cases: a Ge single crystal and a micrometre-sized UO2grain in a polycrystal. Accuracies on the hydrostatic strain Δa/aof ±0.4 × 10−4and ±1.3 × 10−4were obtained for Ge and UO2, respectively. Measurement strategies to limit the remaining uncertainties onEtheorare discussed.

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

2017 ◽  
Vol 50 (6) ◽  
pp. 1754-1765 ◽  
Author(s):  
Gader Altinkurt ◽  
Mathieu Fèvre ◽  
Odile Robach ◽  
Jean-Sébastien Micha ◽  
Guillaume Geandier ◽  
...  
Keyword(s):  
Elastic Strain ◽  
Lattice Mismatch ◽  
Strain Tensor ◽  
Coarse Grained ◽  
Precipitate Size ◽  
Crystal Lattice Parameter ◽  
Acquisition Time ◽  
Nickel Based Superalloy ◽  
Elastic Strain Tensor ◽  
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 × 10−3depending 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.

scholarly journals On continuum damage mechanics

2019 ◽  
Vol 52 (3) ◽  
pp. 125-147
Author(s):  
Kari Juhani Santaoja
Keyword(s):  
Stress Tensor ◽  
Effective Stress ◽  
Elastic Strain ◽  
Damage Mechanics ◽  
Volume Fraction ◽  
Strain Tensor ◽  
Continuum Damage Mechanics ◽  
Continuum Damage ◽  
The Matrix ◽  
Elastic Strain Tensor

A material containing spherical microvoids with a Hookean matrix response was shown to take the appearance usually applied in continuum damage mechanics. However, the commonly used variable damage D was replaced with the void volume fraction f , which has a clear physical meaning, and the elastic strain tensor \Bold {ε}^e with the damage-elastic strain tensor \Bold {ε}^{de}. The postulate of strain equivalence with the effective stress concept was reformulated and applied to a case where the response of the matrix obeys Hooke’s law. In contrast to many other studies, in the derived relation between the effective stress tensor \Bold {\Tilde{σ}} and the stress tensor \Bold {σ}, the tensor \Bold {\Tilde{σ}} is symmetric. A uniaxial bar model was introduce for clarifying the derived results. Other candidates for damage were demonstrated by studying the effect of carbide coarsening on creep rate.

Submicrometre-resolution polychromatic three-dimensional X-ray microscopy

2012 ◽  
Vol 46 (1) ◽  
pp. 153-164 ◽  
Author(s):  
B. C. Larson ◽  
L. E. Levine
Keyword(s):  
Spatial Resolution ◽  
Elastic Strain ◽  
Strain Tensor ◽  
Three Dimensional ◽  
Direct Determination ◽  
Phase Identification ◽  
Sn Whisker ◽  
X Ray ◽  
Elastic Strain Tensor

The ability to study the structure, microstructure and evolution of materials with increasing spatial resolution is fundamental to achieving a full understanding of the underlying science of materials. Polychromatic three-dimensional X-ray microscopy (3DXM) is a recently developed nondestructive diffraction technique that enables crystallographic phase identification, determination of local crystal orientations, grain morphologies, grain interface types and orientations, and in favorable cases direct determination of the deviatoric elastic strain tensor with submicrometre spatial resolution in all three dimensions. With the added capability of an energy-scanning incident beam monochromator, the determination of absolute lattice parameters is enabled, allowing specification of the complete elastic strain tensor with three-dimensional spatial resolution. The methods associated with 3DXM are described and key applications of 3DXM are discussed, including studies of deformation in single-crystal and polycrystalline metals and semiconductors, indentation deformation, thermal grain growth in polycrystalline aluminium, the metal–insulator transition in nanoplatelet VO2, interface strengths in metal–matrix composites, high-pressure science, Sn whisker growth, and electromigration processes. Finally, the outlook for future developments associated with this technique is described.

Microdiffraction Experiments and Modeling for Analyzing Multiscale Dislocation Ensembles in Materials

2003 ◽  
Vol 779 ◽  
Author(s):  
G.E. Ice ◽  
R.I. Barabash ◽  
J. Pang
Keyword(s):  
Intensity Distribution ◽  
Elastic Field ◽  
Polycrystalline Sample ◽  
Dislocation Slip ◽  
Laue Pattern ◽  
Model Parameters ◽  
Dislocation Densities ◽  
Good Agreement ◽  
Dislocation Walls

AbstractThe intensity distribution of Laue diffraction is analyzed as a function of local misorientation. We show how unpaired dislocations alter the white beam Laue patterns for isolated dislocations, for dislocation walls, and for a combination of both. We consider the effect of different statistically and geometrically necessary dislocation densities on the intensity distribution along and perpendicular to the Laue streak. A 3D x-ray crystal microscope is used to analyze the complicated plastic-elastic field in a grain of a Ni polycrystalline sample during in-situ uniaxial pulling. A change of dislocation activity with depth is demonstrated. The dislocation slip systems and their densities are determined at various depths. The model parameters are used to simulate the whole Laue pattern including details about the contours for specific Laue spots; good agreement is found between simulated and experimental contours.

Determinability of Complete Residual Strain Tensor from Multiple CBED Patterns

2006 ◽  
Vol 524-525 ◽  
pp. 115-120 ◽  
Author(s):  
Adam Morawiec
Keyword(s):  
Strain Tensor ◽  
Convergent Beam Electron Diffraction ◽  
Reasonable Accuracy ◽  
Multiple Diffraction ◽  
Numerical Tests ◽  
Diffraction Patterns ◽  
Convergent Beam ◽  
The Impact ◽  
Elastic Strain Tensor

The ambiguity in determination of complete elastic strain tensor by convergent beam electron diffraction can be overcome by simultaneous use of multiple diffraction patterns. Numerical tests of strain determining procedure based on multiple patterns have been carried out. Patterns were simulated using both kinematic and dynamic approaches, and then they were used as input in the tested procedure. The tests indicate that, in practice, at least three patterns are needed in order to determine a complete strain tensor with reasonable accuracy. The strain resolution of two parts per ten thousand was achieved with five diffraction patterns. Moreover, the impact of errors in voltage and camera length is considered. It is shown that within the kinematic description, the deviations from the correct voltage are equivalent to errors in the isotropic part of strain.

scholarly journals Stress Analysis by Kossel Microdiffraction on a Nickel-Based Single Crystal Superalloy during an In Situ Tensile Test – Comparison with Classical X-Ray Diffraction

2011 ◽  
Vol 681 ◽  
pp. 1-6 ◽  
Author(s):  
Denis Bouscaud ◽  
Raphaël Pesci ◽  
Sophie Berveiller ◽  
Etienne Patoor
Keyword(s):  
Crystallographic Orientation ◽  
Elastic Strain ◽  
Strain Tensor ◽  
Ccd Camera ◽  
X Ray Diffraction ◽  
Charge Coupled Device ◽  
X Ray ◽  
Set Up ◽  
Elastic Strain Tensor

A Kossel microdiffraction experimental set up is under development inside a Scanning Electron Microscope (SEM) in order to determine the crystallographic orientation as well as the inter- and intragranular strains and stresses. An area of about one cubic micrometer can be analysed using the microscope probe, which enables to study different kinds of elements such as a grain boundary, a crack, a microelectronic component, etc. The diffraction pattern is recorded by a high resolution Charge-Coupled Device (CCD) camera. The crystallographic orientation, the lattice parameters and the elastic strain tensor of the probed area are deduced from the pattern indexation using a homemade software. The purpose of this paper is to report some results achieved up to now to estimate the reliability of the Kossel microdiffraction technique.

Strain

2020 ◽  
pp. 1-8
Author(s):  
Adrian P. Sutton
Keyword(s):  
Elastic Strain ◽  
Displacement Vector ◽  
Strain Tensor ◽  
Deformation Tensor ◽  
Continuum Approximation ◽  
Linear Elastic ◽  
Strain Ellipsoid ◽  
Definition Of ◽  
Elastic Strain Tensor ◽  
Homogeneous Strain

A discussion of the continuum approximation is followed by the definition of deformation as a transformation involving changes in separation between points within a continuum. This leads to the mathematical definition of the deformation tensor. The introduction of the displacement vector and its gradient leads to the definition of the strain tensor. The linear elastic strain tensor involves an approximation in which gradients of the displacement vector are assumed to be small. The deformation tensor can be written as the sum of syymetric and antisymmetric parts, the former being the strain tensor. Normal and shear strains are distinguished. Problems set 1 introduces the strain ellipsoid, the invariance of the trace of the strain tensor, proof that the strain tensor satisfies the transformation law of second rank tensors and a general expression for the change in separation of points within a continuum subjected to a homogeneous strain.

REGULARITY RESULTS FOR THREE-DIMENSIONAL ISOTROPIC AND KINEMATIC HARDENING INCLUDING BOUNDARY DIFFERENTIABILITY

2009 ◽  
Vol 19 (12) ◽  
pp. 2231-2262 ◽  
Author(s):  
JENS FREHSE ◽  
DOMINIQUE LÖBACH
Keyword(s):  
Elastic Strain ◽  
Dirichlet Boundary ◽  
Kinematic Hardening ◽  
Strain Tensor ◽  
Three Dimensional ◽  
Yield Function ◽  
Isotropic Hardening ◽  
Von Mises ◽  
Regularity Results ◽  
Elastic Strain Tensor

For a flat Dirichlet boundary we prove that the first normal derivatives of the stresses and internal parameters are in L∞(0, T; L1+δ) and in L∞(0, T; H½-δ) up to the boundary. This deals with solutions of elastic–plastic flow problems with isotropic or kinematic hardening with von Mises yield function. We show that the elastic strain tensor ε(u) of three-dimensional plasticity with isotropic hardening is contained in the space [Formula: see text] and in L∞(0,T;H4-δ) up to the flat Dirichlet boundary. We obtain related results concerning traces of ε(u). In the case of kinematic hardening we present a simple proof of the [Formula: see text] inclusion of the elastic strain tensor.

Sign in / Sign up

Export Citation Format

Share Document