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.

Elastic strain tensor of zirconium hydrides in Zr2.5%Nb pressure tubes by synchrotron X-ray diffraction

Zirconium alloys are used in fuel cladding and structural components of nuclear power plants. Hydrogen enters the Zr matrix during plant operation and precipitates as hydride particles that degrade the mechanical properties of the alloy, limiting service life. Knowledge of the stress state within hydride precipitates is important to understand stress-induced degradation mechanisms such as delayed hydride cracking, but no direct quantification has yet been reported in the literature. Here, measurements are reported of the average elastic strain tensor within δ zirconium hydride precipitates in Zr2.5%Nb pressure tube material from CANDU power plants. Complete intensity and strain pole figures for the hydride were obtained by synchrotron X-ray diffraction experiments on specimens with hydrogen contents ranging from ∼100 wt p.p.m. hydrogen to nearly 100% δ-hydride. Zirconium hydride precipitates by a process involving a martensitic transformation, with two hydride variants possible from a single α-Zr grain. A synthetic model of the hydride crystallographic texture allowed the interpretation of the measured strain pole figures and quantification of the elastic strain tensor for both texture components. It was found that the two variants appear in nearly equal proportion but with different stress states, differing in the sign of the shear strain components (∼±3000 µ∊). This difference is possibly associated with the shear movement of Zr atoms during the phase transformation. This suggests that hydride clusters are composed of stacks of smaller hydrides in alternating hydride variants. Stresses were estimated from a set of rather uncertain hydride elastic constants. Overall, both variants showed compressive strains along the tube axial direction (∼5000 µ∊). For low hydrogen concentrations, the hydrides' stress tensor is dominated by compressive stresses of ∼300 MPa along the axial direction, probably caused by the elongated morphology of hydride clusters along this direction, and variant-dependent shear stresses of ∼±100 MPa, probably from the shear movement of the Zr atoms involved in the phase transformation.

On continuum damage mechanics

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.

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 × 10−3depending 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.

Submicrometre-resolution polychromatic three-dimensional X-ray microscopy

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.

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

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.

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

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.

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

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.

Determinability of Complete Residual Strain Tensor from Multiple CBED Patterns

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.