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.

Measurement of Elastic Lattice Distortion in PbTe/PbSnTe - Strained Layer Superlatices by Asymmetric High Angle X-ray interfernces

1985 ◽  
Vol 29 ◽  
pp. 367-374
Author(s):  
E. J. Fantner
Keyword(s):  
Elastic Strain ◽  
Lattice Mismatch ◽  
Strain Tensor ◽  
Misfit Strain ◽  
High Angle ◽  
Double Crystal ◽  
X Ray ◽  
Strained Layer ◽  
Thermal Expansion Coefficients ◽  
Quaternary Compounds

AbstractElastic strain significantly affects the electric and optical properties of PbTe/Pb1-xSnxTe - strained-layer superlattices. In the range of 10 - 350K the temperature dependence of the elastic strain present in these superlattices was measured by double-crystal x-ray diffraction. For superlattice periods smaller than 100nm High-angle x-ray interferences were observed. Using a novel method, which makes use of the High-angle interferences both for symmetrical as well as for asymmetrical reflections in a theta-twotheta scan with a narrow detector slit, the relative inclination of equivalent lattice planes due the elastic strain was measured. The components of the complete strain tensor of the constituent layers can be determined seperately even if their unstrained lattice constants are not known with sufficient accuracy as is the case in ternary and quaternary compounds. The lattice mismatch of up to 0.4% for Sn-contents smaller than 20% was found to be accommodated almost completely by elastic misfit strain. The amount of strain is shared between the constituent layers inversely to their relative thicknesses as long as the superlattice as a whole is much thicker than the buffer layer. Below room temperature an additional temperature dependent tensile strain due to differnt thermal expansion coefficients of the film and the BaF2-substrate is measured quantitatively.

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 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.

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