The equations of dislocation transport at finite crystal deformation were developed, with a special emphasis on a vector density representation of dislocations. A companion thermodynamic analysis yielded a generalized expression for the driving force of dislocations that depend on Mandel (Cauchy) stress in the reference (spatial) configurations and the contribution of the dislocation core energy to the free energy of the crystal. Our formulation relied on several dislocation density tensor measures linked to the incompatibility of the plastic distortion in the crystal. While previous works develop such tensors starting from the multiplicative decomposition of the deformation gradient, we developed the tensor measures of the dislocation density and the dislocation flux from the additive decomposition of the displacement gradient and the crystal velocity fields. The two-point dislocation density measures defined by the referential curl of the plastic distortion and the spatial curl of the inverse elastic distortion and the associate dislocation currents were found to be more useful in deriving the referential and spatial forms of the transport equations for the vector density of dislocations. A few test problems showing the effect of finite deformation on the static dislocation fields are presented, with a particular attention to lattice rotation. The framework developed provides the theoretical basis for investigating crystal plasticity and dislocation patterning at the mesoscale, and it bears the potential for realistic comparison with experiments upon numerical solution.
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Shear displacement gradient in X-ray Bragg coherent diffractive imaging
Bragg coherent X-ray diffractive imaging is a cutting-edge method for recovering three-dimensional crystal structure with nanoscale resolution. Phase retrieval provides an atomic displacement parallel to the Bragg peak reciprocal lattice vector. The derivative of the displacement along the same vector provides the normal strain field, which typically serves as a proxy for any structural changes. In this communication it is found that the other component of the displacement gradient, perpendicular to the reciprocal lattice vector, provides additional information from the experimental data collected from nanocrystals with mobile dislocations. Demonstration on published experimental data show how the perpendicular component of the displacement gradient adds to existing analysis, enabling an estimate for the external stresses, pinpointing the location of surface dislocations, and predicting the dislocation motion in in situ experiments.
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- Award ID(s):
- 1944907
- NSF-PAR ID:
- 10379541
- Date Published:
- Journal Name:
- Journal of Synchrotron Radiation
- Volume:
- 29
- Issue:
- 3
- ISSN:
- 1600-5775
- Page Range / eLocation ID:
- 866 to 870
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Abstract Subwavelength imaging of elastic/acoustic waves using phononic crystals (PCs) is limited to a narrow frequency range via the two existing mechanisms that utilize either the intense Bragg scattering in the first phonon band or negative effective properties (left-handed material) in the second (or higher) phonon band. In the first phonon band, the imaging phenomenon can only exist at frequencies closer to the first Bragg band gap where the equal frequency contours (EFCs) are convex. Whereas, for the left-handed materials, the subwavelength imaging is restricted to a narrow frequency region where wave vectors in PC and background material are close to each other, which is essential for single-point image formation. In this work, we propose a PC lens for broadband subwavelength imaging of flexural waves in plates exploiting the second phonon band and the anisotropy of a PC lattice for the first time. Using a square lattice design with square-shaped EFCs, we enable the group velocity vector to always be perpendicular to the lens interface irrespective of the frequency and incidence angle; thus, resulting in a broadband imaging capability. We numerically and experimentally demonstrate subwavelength imaging using this concept over a significantly broadband frequency range.
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Diffuse scattering occurring in the Bragg diffraction pattern of a long-range-ordered structure represents local deviation from the governing regular lattice. However, interpreting the real-space structure from the diffraction pattern presents a significant challenge because of the dramatic difference in intensity between the Bragg and diffuse components of the total scattering function. In contrast to the sharp Bragg diffraction, the diffuse signal has generally been considered to be a weak expansive or continuous background signal. Herein, using 1D and 2D models, it is demonstrated that diffuse scattering in fact consists of a complex array of high-frequency features that must not be averaged into a low-frequency background signal. To evaluate the actual diffuse scattering effectively, an algorithm has been developed that uses robust statistics and traditional signal processing techniques to identify Bragg peaks as signal outliers which can be removed from the overall scattering data and then replaced by statistically valid fill values. This method, described as a `K-space algorithmic reconstruction' (KAREN), can identify Bragg reflections independent of prior knowledge of a system's unit cell. KAREN does not alter any data other than that in the immediate vicinity of the Bragg reflections, and reconstructs the diffuse component surrounding the Bragg peaks without introducing discontinuities which induce Fourier ripples or artifacts from underfilling `punched' voids. The KAREN algorithm for reconstructing diffuse scattering provides demonstrably better resolution than can be obtained from previously described punch-and-fill methods. The superior structural resolution obtained using the KAREN method is demonstrated by evaluating the complex ordered diffuse scattering observed from the neutron diffraction of a single plastic crystal of CBr 4 using pair distribution function analysis.more » « less
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Abstract Twisted 2D materials form complex moiré structures that spontaneously reduce symmetry through picoscale deformation within a mesoscale lattice. We show twisted 2D materials contain a torsional displacement field comprised of three transverse periodic lattice distortions (PLD). The torsional PLD amplitude provides a single order parameter that concisely describes the structural complexity of twisted bilayer moirés. Moreover, the structure and amplitude of a torsional periodic lattice distortion is quantifiable using rudimentary electron diffraction methods sensitive to reciprocal space. In twisted bilayer graphene, the torsional PLD begins to form at angles below 3.89° and the amplitude reaches 8 pm around the magic angle of 1. 1°. At extremely low twist angles (e.g. below 0.25°) the amplitude increases and additional PLD harmonics arise to expand Bernal stacked domains separated by well defined solitonic boundaries. The torsional distortion field in twisted bilayer graphene is analytically described and has an upper bound of 22.6 pm. Similar torsional distortions are observed in twisted WS 2 , CrI 3 , and WSe 2 /MoSe 2 .more » « less
- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1107/S1600577522002363
