Strain Pole Research Articles

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.

Evaluation of the Accuracy of Stop Bar Video Vehicle Detection at Signalized Intersections

Many U.S. agencies have adopted video vehicle detection technology as an alternative to inductive loops. Although many product evaluations have been performed, the majority of these evaluations concentrated on freeway applications in which speed and volume were the primary evaluation criteria. At an actuated intersection, the metrics of speed and volume do not necessarily represent how well a device will operate as a presence detector. Video detection at signalized intersections was evaluated at a test intersection in Indiana. Cameras on all approaches were located at the optimal camera position recommended by the vendors, approximately 60 ft from the strain pole. Two additional cameras were located on each mast arm at slightly less optimal positions, 36 and 48 ft from the strain pole. Traditional inductive loops were also available at the intersection and were used to provide baseline data to screen for discrepancies. Each time the detectors were not in agreement, a discrepancy was noted. A digital video recording was later viewed by a human observer to determine whether the video detector or the loop detector was in error. An analysis of the data showed that video detection was found to produce statistically significantly more false detections and missed detections than the loop detectors on most phases. A small incremental increase in performance was observed when the camera was mounted at 60 ft rather than 36 ft on two of the approaches, but this marginal improvement likely does not justify the additional expense of mast arm, pole, and pole foundation associated with this camera location.

Evaluation of the Accuracy of Stop Bar Video Vehicle Detection at Signalized Intersections

Many U.S. agencies have adopted video vehicle detection technology as an alternative to inductive loops. Although many product evaluations have been performed, the majority of these evaluations concentrated on freeway applications in which speed and volume were the primary evaluation criteria. At an actuated intersection, the metrics of speed and volume do not necessarily represent how well a device will operate as a presence detector. Video detection at signalized intersections was evaluated at a test intersection in Indiana. Cameras on all approaches were located at the optimal camera position recommended by the vendors, approximately 60 ft from the strain pole. Two additional cameras were located on each mast arm at slightly less optimal positions, 36 and 48 ft from the strain pole. Traditional inductive loops were also available at the intersection and were used to provide baseline data to screen for discrepancies. Each time the detectors were not in agreement, a discrepancy was noted. A digital video recording was later viewed by a human observer to determine whether the video detector or the loop detector was in error. An analysis of the data showed that video detection was found to produce statistically significantly more false detections and missed detections than the loop detectors on most phases. A small incremental increase in performance was observed when the camera was mounted at 60 ft rather than 36 ft on two of the approaches, but this marginal improvement likely does not justify the additional expense of mast arm, pole, and pole foundation associated with this camera location.

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.

Determination of the stress orientation distribution function using pulsed neutron sources

The stress orientation distribution function (SODF) was recently introduced as a mean-field representation to describe the grain-orientation dependence of intergranular stress. Pulsed neutron sources are ideally suited for the determination of the SODF since multiple reflections can be measured simultaneously with comparable precision. A method is developed for constructing the SODF from strain pole figures measured with a pulsed neutron source and demonstrated with cold-rolled interstitial-free steel. The experimental results are compared with those measured with a reactor-based constant-wavelength diffractometer. It is shown that access to a large number of reflections on a pulsed neutron source improves the precision of the experimental SODF and facilitatesin situstudies of the evolution of the intergranular stress during deformation and annealing.

A novel method for constructing the mean field of grain-orientation-dependent residual stress

The heterogeneity of grain-orientation-dependent residual stress (ODRS) in polycrystalline materials has received a great deal of attention in the past few years. This letter describes a novel method, called the spherical harmonics approach (SHA), for analysing ODRS. The stress orientation distribution function (SODF) is defined to describe quantitatively the mean field of the residual stress as a function of grain orientation in the Euler space. This stress field can be expanded in terms of generalized spherical harmonics, and its series coefficients can be determined directly from lattice strain maps (strain pole figures). While solving the deconvolution problem of this stress field, the stress or strain equilibrium between neighbouring grains is considered simultaneously by implementing a self-consistent model in the iterative algorithm. The application of the SHA is illustrated by the construction of the SODF for a cold-rolled and annealed austenite stainless steel, using strain pole figures obtained by neutron diffraction.

Surface spherical harmonics and intensity and strain pole figures of cubic textured materials. Erratum

The equations for diffraction strain pole figures measured on textured cubic materials exhibit an hkl dependence. This is expressed by an hkl-permutation-invariant 'surface' spherical harmonic. Four types of new harmonics are defined. These harmonics differ in essence from those for the hkl-dependent expression obtained for diffraction intensity pole figures. In the latter case associated Legendre polynomials arise whereas in the former (Jacobi type) generalizations of these polynomials occur. Equations exhibiting diffraction intensity and diffraction strain expressions are given for all cubic point groups. Structure factors arise in the expressions. The treatment is given for both anomalous and normal scattering modes. Surface spherical harmonics do not satisfy Laplace's equation. This only occurs upon conversion into 'solid spherical harmonics'. Then the harmonics associated with intensity pole figures satisfy Laplace's equation. The 'diffraction strain harmonics' do not, however. Orthonormalization is also different from the case of conventional hkl-permutation-invariant surface spherical harmonics. Stereographic projections are given for a few examples of harmonics.

Surface spherical harmonics and intensity and strain pole figures of cubic textured materials

Surface spherical harmonics and intensity and strain pole figures of cubic textured materials

Odd-order O.D.F. expansion coefficient determination. Case of diffraction strain measurements on cubic materials under macrostress loading

Diffraction intensity pole figures are often used for the determination of orientation distribution function (o.d.f.) expansion coefficients. The intensity can be seen as a convolution of the o.d.f. times unity with respect to one rotation angle (about the direction of measurement). The `normal' polycrystalline diffraction experiment only yields the even-order o.d.f. coefficients. The experiment itself imposes a centre of inversion even upon non-centrosymmetric crystals. Crystals may exhibit a centre of inversion themselves. The hkl and {\bar h}{\bar k}{\bar l}contributions to the intensity are indistinguishable then owing to the centre of inversion. As a consequence, the odd-order coefficients cannot be determined. The mean value of a general physical property determined by means of diffraction can be taken as a convolution of the o.d.f. times the single-crystal value of the physical property with respect to the rotation angle mentioned before. The dependency of the physical property on the rotation angle leads to more information being extracted from the o.d.f. in the property's mean-value pole figure. Then, all o.d.f. coefficients may be present in the mean value, i.e. the measurement. Consequently, diffraction-line-shift strain pole figures exhibit even- and odd-order o.d.f. coefficients, present or induced centres of inversion notwithstanding. If the dependency of the single-crystal strain on the rotation angle is known no model of elastic polycrystal coupling is needed. However, this does not occur in practice. The present state of the art does not allow the Kröner model to be used for textured materials. In this paper the Reuss model is used. If the (applied) macrostresses are known, the o.d.f. coefficients can be obtained from the formulae presented.

A general treatment of X‐ray (residual) macro‐stress determination in textured cubic materials: General expressions, cubic invariancy and application to X‐ray strain pole figures

AbstractAn expression in terms of general hkl and orientation distribution functions for the relationship between the elastic lattice strain measured by means of X‐ray diffraction methods and the (residual) elastic stress is given for cubic textured materials. In practice the socalled sin2 Ψ‐method is often used to determine these macro stresses. Then, for the case of textured specimens the quasi‐isotropic X‐ray elastic constants (depending on hkl only) become complicated functions of the crystallographic texture (J. Appl. Crystallogr. 16, 325 (1983)).Results are given for both orthorhombic and monoclinic specimen symmetry in terms of general (b.c.c. or f.c.c.) hkl and allow every permutation possible for a given hkl combination. From this cubic invariancy some new cubic invariant spherical surface harmonics are derived which are represented on a basis consisting of the 4‐fold symmetry axes of the cubic crystals.Special textures and special stress‐states are discussed. For a given texture of a cold‐rolled production steel sheet values of the predicted elastic strains are given in terms of “X‐ray strain pole figures”.