Strain Pole Figures Research Articles

Analysis of Phase-Specific Strain Pole Figures for Duplex Steels under Elasto-Plastic Uniaxial Tension—Experiment vs. EPSC Modelling

For the duplex stainless steel X2CrNiMoN22-5-3, phase-specific strain pole figures (strain PFs) for the phases ferrite (bcc) and austenite (fcc) were analysed under uniaxial tensile loading for various loading states in purely elastic and elasto-plastic regimes. Experimentally, strain PFs were determined by means of in situ neutron diffraction strain measurements under defined uniaxial loading. These experimental results were compared with strain PFs calculated using elasto-plastic self-consistent (EPSC) modelling. The comparison was performed for two different {hkl} planes per phase. While classic load stress and load partitioning analyses for multi-phase materials are often limited to the load direction and a selected direction transverse to it, the results illustrate the added value of determining a strain PF, especially when a phase-specific texture is present. The comparison with experimental data shows how well the load partitioning behaviour can be predicted using common EPSC models, using the example of a duplex stainless steel. The EPSC model used was validated with the software ISODEC in its elastic range. Based on the results of the EPSC model, and taking into account the local phase-specific crystallographic texture, a prediction can be made as to what extent intergranular stresses and phase-specific textures could affect the results of a (residual) stress analysis by means of the diffraction method. This makes it possible to assess whether, for technical applications, meaningful residual stress results can be expected in certain component directions.

Investigation of residual stresses and modeling of tensile deformation in wire-arc additive manufactured 6061 aluminum alloy: Diffraction and elastoplastic self-consistent model

The aim of this work is to study the mechanical behavior of 6061 aluminum walls produced by Wire Arc Additive Manufacturing at the different scales of the material. The residual stresses of the parts are characterized on the surface and in the bulk thanks to X-ray and neutron diffraction techniques, respectively. Although the residual stress values determined in the close-surface are low (40 MPa for the maximum value), the mechanical state is different in depth with significant maximum residual stress levels ( ± 100 MPa). Tensile tests carried out on samples extracted from a WAAM wall in three orientations reveal that the mechanical properties of the manufactured material are slightly better compared to conventional material in T6 state. In situ X-ray diffraction experiments have also been performed under uniaxial tensile testing in order to determine strain pole figures. An elasto-plastic self-consistent model considering the microstructural characteristics and the main physical phenomena governing the material behavior, is proposed. The model agrees with the experimental data and the simulations reproduce the main features observed at the scale of the diffracting volume.

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 µɛ). 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 µɛ). 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.

Micromechanical response quantification using high-energy X-rays during phase transformations in additively manufactured 17-4 stainless steel

Recent studies of additively manufactured (AM) 17-4 stainless steel produced via laser powder bed fusion of nitrogen atomized powders have been found to contain large volume fractions of austenite (γ) compared with the fully martensitic (α') microstructure of wrought 17-4. These AM 17-4 stainless steels have metastable microstructures that transform from a mixed phase composition to predominantly martensitic during deformation. This transformation process, combined with strong preferred crystallographic orientation (texture) that arises during building, produces complex micromechanical interactions that dictate the macroscopic response. Here, high-energy X-ray diffraction performed at a synchrotron light source is used to quantify the volume fraction of austenite and martensite, texture, and the complete orientation dependence of lattice strain (strain pole figures) at various macroscopic strains levels in-situ during uniaxial tension of AM 17-4. Results from wrought 17-4 are also shown for comparison. Initial martensite volume fraction of the stress-relieved AM specimen was measured to be 0.46 and increased to 0.88 after the application of a macroscopic strain of 0.03. During the transformation process, minimal crystallographic texture evolution was observed in either the γ austenite or α' martensite. The distribution of strains in the specimen is found to be heavily influenced by both the transformation process and the initial texture. Phase transformation is found to generate tensile strains perpendicular to the applied load in untransformed γ austenite, while texture is found to produce high heterogeneity of lattice strains within lattice plane families.

Lattice Strain Pole Figures Analysis in Titanium during Uniaxial Deformation

A theoretical and experimental study was carry out to investigate deformation mechanisms in a textured titanium alloy. In situ neutron diffraction measurements were performed to analyze different {hk.l} family planes ({10.0}, {10.1}, {11.0} and {00.2}) and determine the corresponding internal strain pole figures. This method was applied to a pure titanium (a-Ti) submitted to a uniaxial tensile load up to 2 %. The experimental data was then used to validate the EPSC model in order to predict the distribution of lattice strains determined by neutron diffraction for various diffraction vector directions. This comparison reveals that the model results were in good agreement with the experimental data and the simulations reproduced the lattice strain development observed on the strain pole figures determined by neutron diffraction.

A methodology to determine the elastic moduli of crystals by matching experimental and simulated lattice strain pole figures using discrete harmonics

Determining reliable single crystal material parameters for complex polycrystalline materials is a significant challenge for the materials community. In this work, a novel methodology for determining those parameters is outlined and successfully applied to the titanium alloy, Ti-6Al-4V. Utilizing the results from a lattice strain pole figure experiment conducted at the Cornell High Energy Synchrotron Source, an iterative approach is used to optimize the single crystal elastic moduli by comparing experimental and simulated lattice strain pole figures at discrete load steps during a uniaxial tensile test. Due to the large number of unique measurements taken during the experiments, comparisons were made by using the discrete spherical harmonic modes of both the experimental and simulated lattice strain pole figures, allowing the complete pole figures to be used to determine the single crystal elastic moduli.

Determination of residual stress in a microtextured α titanium component using high-energy synchrotron X-rays

A shrink-fit sample is manufactured with a Ti-8Al-1Mo-1V alloy to introduce a multiaxial residual stress field in the disk of the sample. A set of strain and orientation pole figures are measured at various locations across the disk using synchrotron high-energy X-ray diffraction. Two approaches—the traditional [Formula: see text] method and the bi-scale optimization method—are taken to determine the stresses in the disk based on the measured strain and orientation pole figures, to explore the range of solutions that are possible for the stress field within the disk. While the stress components computed using the [Formula: see text] method and the bi-scale optimization method have similar trends, their magnitudes are significantly different. It is suspected that the local texture variation in the material is the cause of this discrepancy.

In situ lattice strains analysis in titanium during a uniaxial tensile test

In situ neutron diffraction experiments have been performed under uniaxial tensile testing in a commercially pure titanium in order to determine strain pole figures. The evolution of intergranular strains was observed in the bulk material along multiple orientations for 4 reflections: {10.0}, {10.1}, {11.0} and {00.2}. The experimental data was used to test an elasto-plastic self-consistent model. A particular focus has been devoted to the relationship between the internal strains and the deformation systems activity. The model was in agreement with the experiments, and the simulations reproduced the main features observed on the in situ measured strain pole figures.

Understanding local deformation in metallic polycrystals using high energy X-rays and finite elements

A methodology for understanding the stress and elastoplastic deformation responses within a loaded polycrystal is presented along with illustrative examples. High energy synchrotron X-rays are used to penetrate bulk metallic samples and produce diffracted intensity from each deforming crystal – revealing the evolving internal structure. A virtual representation of the microstructure is constructed using the finite element method (FEM) to simulate the evolution of the elastoplastic deformations, stress fields, and lattice orientations within the deforming crystals as the polycrystal is loaded. Simulations are compared directly to experimental diffraction data. In the case of powder experiments, lattice strain pole figures (SPFs) measured experimentally are compared to SPFs calculated by projecting X-rays through the finite element mesh. During in situ loading experiments, the stress states are found to differ from one crystal to the next and to vary from the stress being applied at the macroscale. A SPF/FEM-based methodology for quantifying residual stress fields within processed polycrystalline components is described. SPFs were measured at many points within a shrink-fit sample. Finite element discretizations of both the sample and orientation space of each diffraction volume were used to formulate an optimization for the distribution of the stress tensor within the sample. A different experiment, one in which the X-ray beam and the crystals are closer to the same size, is used to investigate the aggregate crystal by crystal. The Debye–Scherrer rings reduce to a set of spots associated with each crystal within the diffraction volume. This method is demonstrated by tracking deformation of four grains within a deforming BCC titanium aggregate loaded in situ within the elastic regime to determine the single crystal elastic moduli. Plastic deformation can also be investigated by monitoring the size and shape of individual diffraction spots. Each spot contains geometrically exact information regarding the internal structure of the crystal. Instead of reconstructing the crystal structure by inverting the diffraction data, virtual diffraction experiments are performed on the finite element mesh and the resulting simulated diffraction patterns are compared directly to the experimental results. Once the experimental/simulation methodology is validated, the approximation of the subgrain distribution of stress and lattice orientation from the finite element model can be used to construct theories for failure phenomena such as microcrack initiation. As opposed to other methods of discretizing a polycrystalline aggregate, the finite element framework enables a seamless transition to analyses associated with mechanical design.

In situmonitoring of X-ray strain pole figures of a biaxially deformed ultra-thin film on a flexible substrate

X-ray strain pole figures (SPFs) have been capturedin situduring biaxial deformation of a gold ultra-thin film (thickness = 40 nm) deposited on a polymer substrate. An area detector was used to extract one line in the reciprocal space while the strained sample was rotated azimuthally step by step to produce the SPF. Such SPFs have been obtained for a textured anisotropic ultra-thin film under controlled non-equibiaxial loading using the SOLEIL synchrotron DIFFABS tensile device. The experimental setup allows the pole figure measurements of {111} and {200} reflections to be performed simultaneously. Interestingly, those two crystallographic directions are related to the two-extreme elastic mechanical behaviour. The full directional lattice strain dependence (SPF) is obtained within 15 min and can be monitored step by step upon loading. This procedure gives an insight into ultra-thin film mechanical response under complex biaxial loading.

Quantifying Three-Dimensional Residual Stress Distributions Using Spatially-Resolved Diffraction Measurements and Finite Element Based Data Reduction

Residual stress can play a significant role in the processing and performance of an engineered metallic component. The stress state within a polycrystalline part can vary significantly between its surface and its interior. To measure three-dimensional (3D) residual stress fields, a synchrotron x-ray diffraction-based experimental technique capable of non-destructively measuring a set of lattice strain pole figures (SPFs) at various surface and internal points within a component was developed. The resulting SPFs were used as input for a recently developed bi-scale optimization scheme McNelis et al. J Mech Phys Sol 61:428–1007 449 (2013) that combines crystal-scale measurements and continuum-scale constraints to determinethe 3D residual stress field in the component. To demonstrate this methodology, the 3D residual stress distribution was evaluated for an interference-fit sample fabricated from a low solvus high refractory (LSHR) polycrystalline Ni-base superalloy.

A two-scale methodology for determining the residual stresses in polycrystalline solids using high energy X-ray diffraction data

This paper presents a new method for ascertaining residual stress fields in engineering components. Diffraction data are employed with a finite element discretization to determine the macroscopic (continuum) residual stress field over the workpiece simultaneously with the crystal-scale distribution of elastic strains at each diffraction measurement point. Stress equilibrium and traction free boundary conditions constrain the solution at the continuum scale. The thousands of lattice strain measurements made at each diffraction volume ensure that the stress solution is consistent with crystal-scale elastic distortions. Finally, integrated over all orientations within each aggregate, the crystal-scale stresses must match the continuum stress. The method was demonstrated using a shrink fit specimen with a nickel-base superalloy disk fit over a stainless steel shaft. High energy synchrotron X-ray diffraction experiments conducted at the Advanced Photon Source, Argonne National Laboratory, provided nearly 1 million lattice strain measurements. Using these data, the new methodology produced a stress field that satisfied the macroscopic constraints and matched the crystal-scale distortions at each diffraction volume as manifest by spherical harmonic expansions of the lattice strain tensor over orientation space. The residual stress distribution exhibits the general features of an axisymmetric analytical approximation, but also contains details that likely arise from the pre-existing material state and fabrication of the shrink fit assembly. Projections of the spherical harmonic expansions of the crystal-scale lattice strain tensor matched the experimental lattice strain pole figures. Issues related to diffraction volume spacing (spatial resolution) and specimen shadowing are also discussed.

Plastic incompatibility stresses and stored elastic energy in plastically deformed copper

The X-ray diffraction method and theoretical model of elastoplastic deformation were used to examine the residual stresses in polycrystalline copper. To this end, the {2 2 0} strain pole figures were determined for samples subjected to different magnitudes of tensile deformation. Using diffraction data and the self-consistent model, the tensor of plastic incompatibility stress was found for each orientation of a polycrystalline grain. Crystallographic textures, macroscopic and second-order residual stresses were considered in the analysis. As a result, the distributions of elastic stored energy and von Mises equivalent stress were presented in Euler space and correlated with the preferred orientations of grains. Moreover, using the model prediction, the variation of the critical resolved shear stress with grain orientation was determined.

Measuring and modeling distributions of stress state in deforming polycrystals

Crystal stress distributions from synchrotron X-ray diffraction experiments and crystal-based finite element simulations conducted on copper specimens loaded through yielding and the elastic–plastic transition are presented. In the experiments, the lattice strain tensor, ϵ ( R ) , and ultimately the stress tensor, σ ( R ) , for every crystal orientation, R , within the aggregate were determined at discrete load levels during a tension test by inverting measured lattice strain pole figures. The simulation conditions exactly mimicked the experiment and the underlying model employed single-crystal elasticity and restricted-slip plasticity. In the simulation, ϵ ( R ) and σ ( R ) are the average values from all elements at a particular orientation. Significant dependence in the components of σ ( R ) with orientation were found and the σ ( R ) determined from the experiment compared well with the simulation results. In addition we employed a spherical harmonic expansion of each component of stress over orientation space. The coefficients from the experiments compared well with those obtained from the simulation.

Quantitative Stress Analysis of Recrystallized OFHC Cu Subject to Deformation In Situ

Quantitative strain analysis (QSA) provides a means for assessing the orientation-dependent micromechanical stress states in bulk polycrystalline materials. When combined with quantitative texture analysis, it facilitates tracking the evolution of micromechanical states associated with selected texture components for specimens deformed in situ. To demonstrate this ability, a sheet specimen of rolled and recrystallized oxygen-free high conductivity Cu was subject to tensile deformation at APS 1-ID-C. Strain pole figures (SPFs) were measured at a series of applied loads, both below and above the onset of macroscopic yielding. From these data, a lattice strain distribution function (LSDF) was calculated for each applied load. Due to the tensorial nature of the LSDF, the full orientation-dependent stress tensor fields can be calculated unambiguously from the single-crystal elastic moduli. The orientation distribution function (ODF) is used to calculate volume-weighted average stress states over tubular volumes centered on the ⟨100⟩∥[100], ⟨311⟩∥[100], and ⟨111⟩∥[100] fibers—accounting for ≈50% of the total volume—are shown as functions of the applied load along [100]. Corresponding weighted standard deviations are calculated as well. Different multiaxial stress states are observed to develop in the crystal subpopulations despite the uniaxial nature of the applied stress. The evolution of the orientation-dependent elastic strain energy density is also examined. The effects of applying stress bound constraints on the SPF inversion are discussed, as are extensions of QSA to examine defect nucleation and propagation.

Determination of stored elastic energy in plastically deformed copper

The quantitative analysis of stresses in plastically deformed copper samples was performed using the self-consistent model and X-ray diffraction measurements of lattice strains. The {220} strain pole figures were determined for the samples subjected to different magnitude of tensile deformation. Using diffraction data and self-consistent model, the tensor of plastic incompatibility stress, was found for each orientation of a polycrystalline grain. The distribution of elastic stored energy was presented in the Euler space and correlated with preferred orientations of grains. Moreover, using the model prediction, variation of the critical resolved shear stress with grain orientation was shown.

Texture, spatial and orientation dependence of internal strains in marble: a key to understanding the bowing of marble panels?

Abstract A Carrara marble sample was measured using neutron time-of-flight diffraction on its crystallographic preferred orientation (texture), a strain scan through the sample and strain pole figures to explain the effect of deformation of marble plates. Strong compressional residual strain values of up to ɛ = -1.3 × 10 -3 measured with residual strain pole figures in a virgin bulk sample have been found. Obviously, the magnitude of residual strain depends on the sample size. Features of the observed texture pole figures and internal strain pole figures are related to each other by their sample orientations. Texture and residual strain measurements were combined with investigations of thermal expansions under dry and wet conditions in different directions to the main stress direction.

A direct method for the determination of the mean orientation-dependent elastic strains and stresses in polycrystalline materials from strain pole figures

A salient manifestation of anisotropy in the mechanical response of polycrystalline materials is the inhomogeneous partitioning of elastic strains over the aggregate. For bulk samples, the distributions of these intergranular strains are expected to have a strong functional dependence on grain orientations. It is then useful to formulate a mean lattice strain distribution function (LSDF) over the orientation space, which serves to characterize the micromechanical state of the aggregate. Orientation-dependent intergranular stresses may be recovered from the LSDFviaa constitutive assumption, such as anisotropic linear elasticity. While the LSDF may be determined directly from simulation data, its experimental determination relies on solving an inverse problem that is similar in character to the fundamental problem of texture analysis. In this paper, a versatile and robust direct method for determining an LSDF from strain pole figures is presented. The effectiveness of this method is demonstrated using synthetic strain pole figures from a model LSDF obtained from the simulated uniaxial deformation of a 1000-crystal aggregate.

Experimental measurement of lattice strain pole figures using synchrotron x rays

This article describes a system for mechanically loading test specimens in situ for the determination of lattice strain pole figures and their evolution in multiphase alloys via powder diffraction. The data from these experiments provide insight into the three-dimensional mechanical response of a polycrystalline aggregate and represent an extremely powerful material model validation tool. Relatively thin (0.5mm) iron/copper specimens were axially strained using a mechanical loading frame beyond the macroscopic yield strength of the material. The loading was halted at multiple points during the deformation to conduct a diffraction experiment using a 0.5×0.5mm2 monochromatic (50keV) x ray beam. Entire Debye rings of data were collected for multiple lattice planes ({hkl}’s) in both copper and iron using an online image plate detector. Strain pole figures were constructed by rotating the loading frame about the specimen transverse direction. Ideal powder patterns were superimposed on each image for the purpose of geometric correction. The chosen reference material was cerium (IV) oxide powder, which was spread in a thin layer on the downstream face of the specimen using petroleum jelly to prevent any mechanical coupling. Implementation of the system at the A2 experimental station at the Cornell High Energy Synchrotron Source (CHESS) is described. The diffraction moduli measured at CHESS were shown to compare favorably to in situ data from neutron-diffraction experiments conducted on the same alloys.

Measurements and predictions of strain pole figures for uniaxially compressed stainless steel

Strain pole figures representative of residual intergranular strains were determined from an −2.98% uniaxially compressed austenitic stainless steel sample. The measurements were made using neutron diffraction on the recently commissioned Spectrometer for Materials Research at Temperature and Stress (SMARTS) at Los Alamos National Laboratory, USA. The measurements were compared with predictions from an elasto-plastic self-consistent model and found to be in good agreement.