Significant Lattice Rotation Research Articles

Finite-strain scale-free phase-field approach to multivariant martensitic phase transformations with stress-dependent effective thresholds

A scale-free phase-field model for martensitic phase transformations (PTs) at finite strains is developed as an essential generalization of small-strain models in Levitas et al. (2004) and Idesman et al. (2005). The theory includes finite elastic and transformational strains and rotations as well as anisotropic and different elastic properties of phases. The gradient energy term is excluded, and the model is applicable for any scale greater than 100 nm. The model tracks finite-width interfaces between austenite and the mixture of martensitic variants only; volume fractions of martensitic variants are the internal variables rather than order parameters. The concept of the effective threshold for the driving force is introduced, which can be either positive (e.g., due to interface friction) or negative (e.g., due to defects and stress concentrators promoting PTs). To reproduce PT conditions obtained from experiments or atomistic simulations under general stress tensor, the effective threshold depends on the stress tensor components. Material parameters are calibrated for martensitic PT between single crystalline cubic Si I and tetragonal Si II phases, which has large transformation strains (ɛt1=0.1753;ɛt2=0.1753;ɛt3=−0.447). Finite element algorithms and numerical procedures are implemented in the code deal.II. Multiple 3D problems are solved to study the effect of mesh size, holding time during quasi-static loading, and strain rate on the multivariant microstructure evolution in Si I → Si II PT under uniaxial and hydrostatic loadings. The solution exhibits significant lattice rotations both in Si I and Si II, reproducing the appearance of diffuse grain boundaries in Si I and Si II and transforming them in polycrystals, which corresponds to known experiments. While finer mesh can produce a more detailed microstructure, the solution becomes practically mesh-independent after the mesh size is 80 times smaller than the sample size. When approaching the stationary solution, rough mesh leads to convergence to the correct microstructure faster than the fine mesh, because it neglects fine details in the microstructure. In some regions, reverse PT occurs at continuous compression, despite large transformation hysteresis. For most stationary interfaces, local thermodynamic equilibrium conditions (thermodynamic driving force for the interface motion equal to the effective threshold) are satisfied.

Residual stress and localized lattice rotation under fatigue loading

A significant change in residual stress around a fatigue crack has been observed in Al alloys. For Al 7075, the residual stress increases by 200% closer to the crack. However, in commercially pure Al (Al 1100), the residual stress decreases by 80% closer to the crack. The residual stresses are mostly the stored elastic energy as a result of deformation around the fatigue crack. To understand these results, we estimate the dislocation density as a function of distance from the crack using the full width at half maxima of 111 reflection for both alloys. The dislocation density increases by 25–30 % for Al 7075. However, it actually decreases by 10% for Al 1100. Such change in dislocation density cannot explain the observed change in residual stress closer to the crack. Additionally, the deformation leads to a significant lattice rotation, as observed by the x-ray diffraction and the electron backscatter diffraction measurements. We argue that dislocations are needed to produce the lattice rotation and the estimated lattice rotation is ≈ 11°, and conclude that the deformation associated with the lattice rotation would be a major contributing factor to the residual stress. This provides a new insight on the role of lattice rotation on the residual stress under fatigue loading.

Fatigue mediated lattice rotation in Al alloys

A significant lattice rotation in the plastic zone of a fatigue crack at room temperature in highly oriented Al 1100 and Al 7075 alloys has been observed by employing X-ray diffraction (XRD) and transmission electron microscopy (TEM). The intensity of 111 peak gradually increases as compared to the intensity of 200 peak as we approach the crack, suggesting lattice rotation. The texture was seen to change gradually from nearly cubic, away from the crack, to a more random orientation close to the crack. We ascribe the lattice rotation to the increase in dislocation density in the plastic zone. The estimated average angle of rotation is approximately 11°, which is sufficient to disturb the cube texture close to the crack. In addition, the observed plastic zone, as estimated by the lattice rotation, is order of magnitude higher as compared to the theoretically predicted plastic zone. This is of great importance, as it leads to a new insight of the underlying physics of the deformation process in the plastic zone under cyclic loading.

Lattice strains at cracks in single crystal titanium: Elastic distortion and GND contributions

There is evidence from diffraction experiments that significant peak broadening is measured local to crack tips and this has been attributed to the development of geometrically necessary dislocations (GNDs) which are retained upon unloading. This is reasonable due to the stress singularity found locally at the crack which is expected to activate slip on favourably oriented slip systems, potentially resulting in plastic strain gradients and geometrically necessary dislocation development. Hence, a systematic study is presented here to ascertain the contributions of both elastic distortional strain and GND density to lattice deformation local to the crack at loaded and subsequently unloaded states. The results show that whilst elastic strains dominate lattice distortion in comparison to GNDs at the loaded state i.e. at the peak load applied, these strains are largely recovered upon unloading and the contribution from GND development subsequently dominates the broadening seen. Two initial crystallographic configurations were considered. In the example in which the crystal c-axis was oriented parallel to the loading direction, the <c+a> pyramidal systems contributed most to slip with <a> basal slip system contributing to a lesser extent. However, in the example where the c-axis was oriented perpendicular to the loading direction, the <c+a> pyramidal and <a> prismatic systems were the more significant contributors to slip and the <a> basal contributing to a lesser extent. However, basal, prismatic and c+a pyramidal slip systems were found to be active in both examples and this was attributed to significant lattice rotation driven by locally high stresses which enabled otherwise badly oriented slip systems to become favourable for slip. Finally, increases in GND density were seen upon unloading for c-axis orientation parallel with loading. This was attributed to the influence of <c+a> pyramidal slip on reverse plasticity leading to diffuse GND density distributions and significant resulting lattice strains compared to that from <a> prism slip loading

Transverse Creep of Nickel-Base Superalloy Bicrystals

A liquid metal cooled (LMC) solidification technique was used to produce nickel-base GTD 444 and Rene N4 superalloy bicrystals with varying degrees of misorientation. Creep experiments with loading normal to the bicrystal boundaries were conducted at 1255 K (982 °C). Despite the similar overall compositions of these two alloys, the GTD 444 alloy with higher levels of carbon and boron displayed significantly higher tolerance to high-angle boundaries. Creep ductilities at rupture of greater than 5 pct were observed in GTD444 for boundaries misoriented by greater than 20 deg. Electron backscatter diffraction analysis showed significant lattice rotation adjacent to the bicrystal boundary in GTD444. In contrast, the Rene N4 bicrystal accumulated damage along the grain boundary early in creep, failing at less than 2 pct strain.

EFFECTS OF THE FORMATION METHOD OF EARLY GAP BUFFER MONOLAYERS ON THE STRAIN STATE OF GAAS FILMS ON VICINAL SI(001) SUBSTRATES

A significant dependence of the strain state of GaAs film lattice grown by molecular−beam epitaxy (MBE) on the nucleation method of early GaP buffer layers (50 nm) on the vicinal substrate Si(001) 4° around the axis was discovered. GaP growth started layer−by−layer with a gallium or a phosphorus sublayer. If GaP nucleated with a gallium sublayer, the GaAs film has a significant lattice rotation around the axis. If the buffer starts forming with a phosphorus layer the GaAs film evidently rotates around the axis. The film relaxation degree ex- ceeds 100%, and the film is in a laterally strained state. Analysis was carried out using the triclinic distortion model. A reciprocal space scattering map was obtained using X−ray diffraction in a three−axis low resolution setup. The map clearly shows that the GaAs film lattice is rotated.

Texture effects on development of shear bands in rolled AZ31 alloy

We investigate the effects of texture on shear banding in highly textured Mg–3Al–1Zn alloy. The plates with different textures were rolled to 10% reduction in thickness at various temperatures, and the microstructure, texture and misorientation distributions within the regions undergoing shear banding were analyzed with optical microscopy and X-ray diffraction and electron backscattering diffraction techniques. It is found that texture prior to rolling significantly affects the various features of shear bands such as the formation frequency of shear bands, the inclined angle of shear bands to the rolling plane, and orientations of grains in shear bands. We infer that shear bands in AZ31 rolled to a relatively low strain occur as a result of geometric (texture) softening. Shear banding is most pronounced in the plate with conventional strong basal texture, whereas the plate oriented favorably for the prism 〈a〉 slip shows no sign of shear banding. The inclined angle of shear bands in the plate with basal texture is around 30°, while those in the plates oriented for basal <a> slip or tension twinning is around 55°. The regions of shear bands receive more strain than the region outside the bands, which results in significant lattice rotation of the grains in the bands, leading to off-basal texture.

Measurement of residual elastic strain and lattice rotations with high resolution electron backscatter diffraction

A set of dynamically simulated electron backscatter patterns (EBSPs) for α-Ti crystals progressively rotated by 1° steps were analysed using cross-correlation to determine image shifts from which strains and rotations were calculated. At larger rotations the cross-correlation fails in certain regions of the EBSP where large shifts are generated. These incorrect shifts prevent standard least square error procedures from obtaining a valid solution for the strain and rotation, where the applied rotation exceeds ∼ 8°. Using a robust iterative fitting routine reliable strains and rotations can be obtained for applied rotations of up to and including ∼ 11° even though some image shifts are measured incorrectly. Finally, high resolution electron backscatter diffraction has been used to analyse the residual elastic strain, lattice rotations and density of stored geometrically necessary dislocations in a sample of copper deformed to 10% total strain. The robust iterative fitting analysis allows reliable analysis of a larger proportion of the map, the difference being most obviously beneficial in regions where significant lattice rotations have been generated.

An investigation of dynamic recrystallisation on Cu–Sn bronze using 3D EBSD

A focused ion beam and electron backscatter diffraction were used to investigate the three-dimensional microstructure of small dynamically recrystallised grains in a deformed Cu–Sn alloy and to determine their orientation relationships. It was found that many of the grains, which form as thin sheets at the prior boundaries, are twin related. It is concluded that dynamical recrystallisation usually starts in regions where plastic deformation has resulted in significant lattice rotations beside the grain boundaries. Once such a new grain has formed, dynamic recrystallisation is propagated by successive twinning from the new grain, along the prior grain boundary, resulting in families of several hundred twin related dynamically recrystallised grains.

On lattice plane rotation and crystallographic structure of the expanded austenite in plasma nitrided AISI 316L steel

Crystallographic structure and lattice rotation of the ‘expanded’ austenite produced by means of low temperature plasma nitriding is investigated. The microstructure of the nitrogen enriched layer has been investigated by means of X-ray diffraction (XRD) while the lattice rotations in the nitrided zone were assessed by electron backscattered diffraction (EBSD). The nitrogen depth profiles have been determined by means of glow discharge optical emission spectroscopy and nuclear reaction analysis. XRD shows the presence of the ‘expanded’ austenite or γ N phase in all the nitrided samples characterized by average larger lattice spacing in relation to non-nitrided steel matrix. EBSD investigation demonstrates that in addition to the lattice expansion nitrogen incorporation into the stainless steel matrix induces significant lattice rotations. The amount and direction of these crystallographic plane rotations are function of the initial orientation. An unusual evolution of the 220 γ N line as a function of the nitriding duration is observed together with an anomalously high intensity ratio of the 111 and 200 γ N and matrix lines. The XRD results are interpreted on the basis of the lattice rotations of diffracting planes, nitrogen concentration gradient, nitrogen diffusion anisotropy and residual stress. It shows that these rotations are a pertaining feature for the understanding of the γ N microstructure.

Disclinations in Plastically Deformed Metallic Materials

Cold-working of metallic materials up to large strains is usually characterised by simultaneous substructure evolution on different length scales and accompanied by the formation of significant lattice rotations. A promising tool for the description of such microstructure development is the concept of partial disclinations. Transmission electron microscopy (TEM) studies illustrate clearly, that defects of this kind are frequently existent in cold-worked metals and have to be accepted as an important defect entity in the substructure evolution at larger strains. Moreover, it is shown that substructure modelling on the base of a coupled dislocation-disclination dynamics leads to satisfying correspondence of calculated substructure characteristics with experimental results obtained by TEM, X-ray diffractometry, and EBSD (electron backscattering diffraction), and to a satisfying prediction of the macroscopic deformation behaviour, i.e., especially the transition from stage III to stage IV of crystal plasticity.