Tensorial Nature Research Articles

Exploring fracture of H-BN and graphene by neural network force fields

Extreme mechanical processes such as strong lattice distortion and bond breakage during fracture often lead to catastrophic failure of materials and structures. Understanding the nucleation and growth of cracks is challenged by their multiscale characteristics spanning from atomic-level structures at the crack tip to the structural features where the load is applied. Atomistic simulations offer ‘first-principles’ tools to resolve the progressive microstructural changes at crack fronts and are widely used to explore the underlying processes of mechanical energy dissipation, crack path selection, and dynamic instabilities (e.g. kinking, branching). Empirical force fields developed based on atomic-level structural descriptors based on atomic positions and the bond orders do not yield satisfying predictions of fracture, especially for the nonlinear, anisotropic stress–strain relations and the energy densities of edges. High-fidelity force fields thus should include the tensorial nature of strain and the energetics of bond-breaking and (re)formation events during fracture, which, unfortunately, have not been taken into account in either the state-of-the-art empirical or machine-learning force fields. Based on data generated by density functional theory calculations, we report a neural network-based force field for fracture (NN-F3) constructed by using the end-to-end symmetry preserving framework of deep potential—smooth edition (DeepPot-SE). The workflow combines pre-sampling of the space of strain states and active-learning techniques to explore the transition states at critical bonding distances. The capability of NN-F3 is demonstrated by studying the rupture of hexagonal boron nitride (h-BN) and twisted bilayer graphene as model problems. The simulation results elucidate the roughening physics of fracture defined by the lattice asymmetry in h-BN, explaining recent experimental findings, and predict the interaction between cross-layer cracks in twisted graphene bilayers, which leads to a toughening effect.

Spatiotemporal energy‐density distribution of time‐reversed electromagnetic fields

AbstractTime reversal exploits the invariance of electromagnetic wave propagation in reciprocal and lossless media to localize radiating sources. Time‐reversed measurements are back‐propagated in a simulated domain and converge to the unknown source location. The focusing time (i.e. the simulation instant at which the fields converge to the source location) and the source location can be identified using field maxima, entropy, time kurtosis, and space kurtosis. This paper analyses the spatial energy‐density distribution of time‐reversed electromagnetic fields by introducing a convergence metric based on the spatial average and variance of the energy density. It is analytically proven that the proposed metric identifies the focusing time and the source location, with direct links to the source frequency content. The analytical results are verified in a free‐space numerical simulation and the proposed metric is then compared to existing ones in a simulated inhomogeneous medium. Next, this metric is applied and compared in an experimental case study to localize electromagnetic interference sources. The proposed metric outperforms existing ones to identify the focusing time and can also be used to locate the source. Finally, because of its tensorial nature, it can handle anisotropic media, opening the door to quantitative analyses of time‐reversal focusing in metamaterials.

Friction tensor for an ellipsoid enclosed in a rarefied gas

This research renders calculations of rotranslational friction and diffusion tensors in a rarefied gas, considering both specular and diffuse reflection off ellipsoidal particles. Calculations were conducted in a spherical polar coordinate system, revealing the dynamic nature of these off-diagonal tensors. The tensor elements were determined analytically, intending to compute the surface integrals. All diagrams, and pictures of two-dimensional surfaces were obtained by the numerical integration process in MATLAB. Additionally, we performed calculations without using the time parameter and time derivatives in the relevant relations. Deviations from the state of spherical symmetry were observed in the results for an ellipsoid and show that the tensors are dynamic quantities. Therefore, it is necessary to provide average values. We established that the non-zero elements of the friction tensor for an ellipsoid represent the directions of linear momentum transfer. Also, the non-zero components of the diffusion tensor signify the extent of fluctuations in relevant momentum transfers in corresponding directions.

On the tensorial nature of the hydraulic aperture of rough fractures

Current fractured porous media models may account for the roughness and geometry of the fractures to upscale hydraulic apertures that represent the local aperture distributions. Still, such approaches do not discuss the tensorial nature of the hydraulic aperture, and by employing scalar representations, these models may fail to capture the anisotropy of the fluid flow in rough fractures. By mimicking natural fractures with walls of varying roughness and contact area, we derive hydraulic aperture tensors that preserve the anisotropic flow patterns enforced by local aperture distributions. This work underscores the critical need to represent hydraulic apertures as tensors to model anisotropic fluid flow in rough fractures and proposes a method for upscaling such tensors.

HIGH-RESOLUTION SATELLITE IMAGE TRIFOCAL TENSOR SOLUTION

Abstract. This study presents an efficient algorithm for solving the affine trifocal tensor problem in high-resolution satellite imagery. Because the imaging mode of high-resolution satellites is usually a line-center projection and the field of view is small, it can be approximated as an affine camera for orientation. However, the nature of high-resolution satellite trifocal tensors has not been effectively studied or discussed. To address the shortcomings of the current trifocal tensor direct linear algorithm, this study proposes a minimally parameterized trifocal calculation method that can stably calculate the affine trifocal tensors of high-resolution satellites. Efficacy was demonstrated using ZY3-02 and Pleiades-Neo data.

A Multi-Scale Approach to Simulate the Nonlinear Optical Response of Molecular Nanomaterials.

Nonlinear optics is essential for many recent photonic technologies. Here, a novel multi-scale approach is introduced to simulate the nonlinear optical response of molecular nanomaterials combining ab initio quantum-chemical and classical Maxwell-scattering computations. In this approach, the first hyperpolarizability tensor is computed with time-dependent density-functional theory and incorporated into a multi-scattering formalism that considers the optical interaction between neighboring molecules. Such incorporation is achieved by a novel object: the Hyper-Transition(T)-matrix. With this object at hand, the nonlinear optical response from single molecules and also from entire photonic devices can be computed, including the full tensorial and dispersive nature of the optical response of the molecules, as well as the optical interaction between different molecules as, for example, in the lattice of a molecular crystal. To demonstrate the applicability of the novel approach, the generation of a second-harmonic signal from a thin film of an Urea molecular crystal is computed and compared to more traditional simulations. Furthermore, an optical cavity is designed, which enhances the second-harmonic response of the molecular film up to more than two orders of magnitude. This approach is highly versatile and accurate and can be the working horse for the future exploration of nonlinear photonic molecular materials in structured photonicenvironments.

Dirac: A command-line γ-matrix calculator

A software for simplification of Dirac matrix polynomials that arise in particle physics problems is implemented. Program summaryProgram Title: DiracCPC Library link to program files:https://doi.org/10.17632/s5w3zyhy58.1Developer's repository link:https://github.com/skutnii/diracLicensing provisions: MITProgramming language: C++20Nature of problem: Particle physics computations require simplification of Dirac matrix polynomials. Non-commutativity and tensorial nature of Dirac matrices makes the task nontrivial. Complexity of the expressions grows quickly with the number of factors making manual computations beyond a few simplest cases unfeasible and highly error-prone. Existing software solutions are either proprietary and highly expensive or have poor if any support for Dirac matrix algebra.Solution method: The problem is solved by considering a representation of Dirac matrix algebra with 5×5pseudo-matrices - matrices made of the algebra's structure constants. In this representation, multiplication of the original matrices corresponds to multiplication of pseudo-matrices and Lorentz index contraction. The problem then gets reduced to simplification of polynomials in Minkowski metric, Kronecker, and Levi-Civita symbols.Additional comments including restrictions and unusual features: The input and output have ▪-like syntax with single-expression command line input and interactive shell modes. Scripting can be used for batched input processing and advanced ▪ formatting of the results. Example scripts demonstrate this feature and provide automated correctness tests. The software supports both rational and floating-point arithmetic.

Solubility of NaCl under anisotropic stress state.

Salt solubility is generally determined under isotropic stress conditions. Yet, in the context of salt weathering of porous media, mechanical constraints on the in-pore growth of salt crystals are likely to be orientation-dependent, resulting in an anisotropic stress state on the crystal. In this paper, we determine by molecular simulation the solubility of NaCl in water when the crystal is subjected to anisotropic stress. Such anisotropy causes the chemical potential of the crystal to be orientation-dependent, and proper thermodynamic formulation requires describing the chemical potential as a tensor. The solute and crystal chemical potentials are computed from free energy calculations using Hamiltonian thermodynamic integration, and the usual condition of solubility is reformulated to account for the tensorial nature of the crystal chemical potential. We investigate in detail how the uniaxial compression of the crystal affects its solubility. The molecular simulation results led to revisiting the Correns law under anisotropic stress. Regarding the solute, the non-ideal behavior of the liquid phase is captured using Pitzer's ion interaction approach up to high concentrations of interest for in-pore crystallization and beyond the concentrations addressed in the existing literature. Regarding NaCl crystals, the validity of the generalized Gibbs-Duhem equation for a tensorial chemical potential is carefully verified, and it is found that crystallization progresses almost orthogonally to the crystal surface even under high shear stresses. Comparing uniaxial and isotropic compression highlights the major differences in solubility caused by stress anisotropy, and the revisited Correns law offers an appropriate framework to capture this phenomenon.

#3197 A NOVEL INVESTIGATION OF AVF WITH A SECOND HARMONIC GENERATION MICROSCOPY IN PATIENTS WITH END STAGE RENAL DISEASE

Abstract Background and Aims The population of patients with end-stage renal disease is rapidly growing and hemodialysis (HD) remains the most common treatment option. In clinical practice, the formation of arteriovenous fistulas (AVFs) is considered a priority in order to ensure optimal vascular access for HD patients. One of the complications in the formation of AVF is the hyperplasia of the neointimal layer of the vein, which leads to the failure of the fistula. The impact of the arrangement of collagen fibers in the veins of AVFs is little studied and opens wide opportunities for investigations. The structural tissue can be imaged with polarimetric nonlinear microscopy. Collagen fibers have a non-centrosymmetric structure, so they can be effectively imaged by second harmonic generation (SHG) microscopy [1]. The polarimetric nonlinear microscopy, exploiting the multidimentional tensor nature of light-matter interactions, is emerging as a valuable research tool in biomedical investigations and applications to digital pathology, and is only now becoming technically feasible for potential clinical translation [2,3]. This proposal represents a novel application of multimodal polarimetric nonlinear microscopy in histopathology, and biomedical imaging. The aim of this study is to determine the influence of the arrangement of collagen fibers in veins on the formation efficiency of AVFs by using the innovative method of nonlinear SHG microscopy. Method In this prospective study, we first used a novel SHG microscopy method to investigate AVFs venous intima hyperplasia. V. Cephalica segment is being obtained during arteriovenous fistula formation surgery from patients with end stage renal disease and after the removal of aneurysmatic or fibrotic AVF. Cross-sections of venous specimens are being fixed in 10% buffered formalin and stained with hematoxylin and eosin (H&E). Further, sample investigation is being performed in Laser Research Centre, Vilnius University using a home-built multiphoton laser-scanning microscope. The investigation is being performed with the permission of Lithuanian Biomedical Research Ethics Committee. Results We determined the arrangement of collagen fibers in veins before AVF formation and after the removal of aneurysmatic AVFs. It was observed that tunica intima of V. Cephalica is dominated both by endothelial cells and connective tissue made up of high-density collagen fibers. Whereas tunica adventitia consists mainly of collagen fibers. Collagen in tunica intima and adventitia are found to be highly noncentrosymmetric. Thus a strong SHG signal, resulting in high-contrast structural visualization of the V. Cephalica segment is generated. Conclusion Our initial investigations of AVF show clearly visible collagen fibers. The future work will focus on the structural changes of collagen fibers in the vein intimal hyperplasia. Further investigations need to be done to create artificial intelligence-based identification of new predictive collagen markers.

Crack-parallel stress effect on fracture energy of plastic hardening polycrystalline metal identified from gap test scaling

The gap test is a new type of fracture test developed in 2020, in which the end supports of a notched beam are installed with gaps that close only after the elasto-plastic pads next to notch introduce a desired constant crack-parallel compression σxx (also called the T-stress). The test uses the size effect method to identify how such a compression alters the material fracture energy, Gf, and the characteristic size cf of the fracture process zone (FPZ). In 2020, experiments showed that a moderate σxx doubled the Gf of a quasibrittle material (concrete) and a high σxx reduced its Gf to almost zero. A preliminary study by Nguyen et al. (2021) showed that the gap test can be extended to plastic-hardening polycrystalline metals. A generalized scaling law with an intermediate asymptote for large-scale yielding in small structures was derived, and limited tests of aluminum alloy showed its applicability. In this study, geometrically scaled gap tests of notched three-point bend fracture specimens of aluminum are conducted at three different levels of σxx. An extended structural strength scaling law that captures the transition from the micrometer-scale FPZ through millimeter-scale yielding zone (YZ) to large-scale structures which follow linear elastic fracture mechanics (LEFM) is derived and then applied to analyze the effect of σxx. Presented here are the gap tests of aluminum alloy, in which three different levels of σxx are applied to scaled notched four-point-bend beams of depths D = 12, 24, 48 and 96 mm. Using an extended size effect law for plastic-hardening metals, it is found that, at crack-parallel stress σxx≈−40% of the yield strength, the critical J-integral value gets roughly quadrupled, not only because of the well-known enlargement of the hardening YZ whose width is of millimeter scale, but also because of the increase of the FPZ width of micrometer scale. These results can be reproduced neither by line crack models, including the LEFM, cohesive crack and phase-field models, nor by peridynamic and various nonlocal models that ignore the tensorial nature of the material stress at the crack tip. The crack band models, being able to represent an FPZ of finite width and a YZ whose size evolves depending on σxx, can capture the effect of crack-parallel stresses provided that a realistic 3D tensorial damage constitutive model is used. Here, Bai–Wierzbicki’s model is shown to capture the σxx effect on the Gf and Jcr qualitatively.

Interfacial mass transfer and axial liquid dispersion in aerated column bioreactors

The upscaling of gas–liquid mass transfer and bioreaction in aerated cultures was performed within the volume averaging theory. Effective transport equations, including their associated closure problems, were derived for 1) the main substrates and biomass (non-transferable components), and 2) the gaseous substrate (transferable component). For the transferable component, two approaches were applied: 2a) the local mass equilibrium (one equation), and 2b) the two-fluid model (two equations). The nature of the dispersion tensors in the three averaged models is discussed. Numerical computations of the axial dispersion coefficient and the volumetric mass transfer coefficient were performed by solving the associated closure problems in unit cells (0.1 ≤ Re ≤ 300; 1 ≤ Sc ≤ 550; 0.1 ≤ Pe ≤ 300; Mo = 2.5 × 1011; 0.001 ≤ εG ≤ 0.05). The results of both parameters agree with trends reported in literature; some differences in orders of magnitude for DLzz are attributable to the mismatch between the experimental conditions and the occurrence of mesoscale interactions undetected in the computational cells.

Multislice computational model for birefringent scattering

Analytical modeling of optical field propagation in a scattering medium is essential for computational optics such as iterative imaging reconstructions and end-to-end photonic designs. Despite the existence of the beam propagation method framework and numerical electromagnetic simulation tools for scalar scattering processes, there has been a lack of an analytical model to include the full tensor nature of birefringent scattering. Here, we present an analytical tool with a polarization transfer function tensor to model the field propagation in a birefringent scattering medium by including a full set of field tensor elements and multiple-scattering characteristics. Birefringence-controlled field propagation experiments were successfully carried out to validate the proposed model. With the capability to solve the 3D reconstruction of the scattering potential tensor, we believe the proposed model will provide a more accurate and efficient forward analytical tool for iterative reconstruction of birefringence-related scattering processes.

A Bayesian regression analysis of in situ stress using overcoring data

Characterising the state of in situ stress at a target depth is crucial for all underground engineering projects. Consequently, on critical projects such as nuclear waste repositories extensive campaigns are implemented with the goal of estimating the in situ stress state. These campaigns often comprise both direct measurement and indirect estimation methods, but the data obtained across a project volume may exhibit significant variability. This poses significant challenges in both quantifying the variability and uncertainty of in situ stress, and determining the stress state to be used for design purposes.It is often assumed that the state of in situ stress increases linearly with depth, and thus linear regression of principal stress magnitude against depth are often found in the literature. As such methods not honouring the tensorial nature of stress are, strictly, incorrect. To show how this limitation may be overcome, here we present a Bayesian regression analysis of in situ stress with depth that uses the Cartesian stress tensor. The analysis is performed using over 100 overcoring data obtained at the SKB Forsmark site in Sweden.A comparison between the customary and Bayesian approaches is presented, which shows the superiority of the tensorial technique.

Implementation of a finite difference frequency domain mode solver incorporating subpixel smoothing.

Finite difference frequency domain (FDFD) mode solvers are straightforward to implement but can suffer from slow convergence when applied to high-contrast refractive index structures. In this work, we show how subpixel smoothing can improve the convergence properties of a full-vectorial FDFD mode solver. Based on a standard Yee grid, we formulate a generalized eigenbproblem whose solutions provide the modes of the waveguides taking into account the tensor nature of the effective dielectric constant. We investigate the convergence of the proposed FDFD mode solver in several cases including a step index fiber, a microsctuctured fiber, and a cylindrical plasmonic waveguide. The results show that tensor smoothing can significantly improve the convergence of the solver, thus allowing the use of less dense grids in the calculations. Our implementation is freely available on the web under an open-source licence.

Dispersion tensor in stratified porous media.

Dispersion in porous media is of great importance in many areas of science and engineering. While dispersion in porous media has been generally well discussed in the literature, little work has been done regarding a generalization of Taylor dispersion in stratified media. In this work, we generalized the Taylor dispersion theory and Stokes flow in porous media to derive a reduced-order model for tracer dispersion in stratified porous media. Our findings revealed that for a simple case of two-layer porous media, the hydrodynamic coupling between the two layers leads to the tensorial nature of dispersion and advection. The results showed that the obtained dispersion tensor and advection are not symmetric unless both porous layers have similar thickness, porosity, and molecular diffusion. We found that the main elements of the coefficient of the dispersion tensor remain positive while the off-diagonal elements can take negative values. On the contrary, all elements of the advection matrix may take negative values. On the basis of these observations, we report the manifestation of the dispersion barrier, uphill dispersion and advection, and osmotic dispersion during tracer transport in stratified porous media. In particular, the identified uphill advection reveals that the injected tracer in one layer could be transported countercurrent to the adjacent layer. Furthermore, we have shown that in the limiting case of Darcy flow, the Taylor dispersion is absent, and the tracer mixing between the two layers is restricted to the cross-diffusive flux between them. The results revealed that the field scale mixing may not necessarily originate from the Taylor dispersion and could be due to the modified advection terms and the cross-diffusive flux between the two layers.

Into the wild of nonlinear electromagnetism-a course on nonlinear electromagnetism, not quite from scratch, part II: tutorial.

This tutorial is aimed at introducing in a natural fashion the propagation equation system governing the scattering of an electromagnetic wave by a nonlinear medium. The purpose is first to obtain equations showing the most common nonlinear effects such as the Kerr effect or second and third harmonic generation by avoiding conventional recipes and trying to arrive at these equations with a minimum of assumptions. For this, we start from the general Maxwell's equations involving the fields E, B, D, and H, and we rigorously provide all the hypotheses needed to attain the nonlinear systems of partial differential equations involving the different complex amplitudes of the different fields associated with the different frequencies at stake. In part II, the difficult question of energy transfer between fields emitted at the various frequencies involved is discussed in detail. We then examine the tensorial nature of susceptibilities and, using Neumann's principle, show how the number of their independent components can be significantly reduced. In part III, numerical examples of scattering by nonlinear materials are given and discussed.

Into the wild of nonlinear electromagnetism-a course on nonlinear electromagnetism, not quite from scratch, part III: tutorial.

This tutorial is aimed at introducing in a natural fashion the propagation equations system governing the scattering of an electromagnetic wave by a nonlinear medium. The purpose is first to obtain the equations showing the most common nonlinear effects such as the Kerr effect or second and third harmonic generation by avoiding conventional recipes and trying to arrive at these equations with a minimum of assumptions. For this, we start from the general Maxwell's equations involving the fields E, B, D, and H, and we rigorously provide all the hypotheses needed to attain the nonlinear systems of partial differential equations involving the different complex amplitudes of the different fields associated with the different frequencies at stake. In part II, the difficult question of energy transfer between fields emitted at the various frequencies involved is discussed in detail. We then examine the tensorial nature of susceptibilities and, using Neumann's principle, show how the number of their independent components can be significantly reduced. In part III, numerical examples of scattering by nonlinear materials are given and discussed.

Into the wild of nonlinear electromagnetism-a course on nonlinear electromagnetism, not quite from scratch, part I: tutorial.

This tutorial is aimed at introducing in a natural fashion the propagation equations system governing the scattering of an electromagnetic wave by a nonlinear medium. The purpose is first to obtain the equations showing the most common nonlinear effects such as the Kerr effect or second and third harmonic generation by avoiding conventional recipes and trying to arrive at these equations with a minimum of assumptions. For this, we start from the general Maxwell's equations involving the fields E, B, D, and H and rigorously provide all the hypotheses needed to attain the nonlinear systems of PDEs involving the different complex amplitudes of the different fields associated with the different frequencies at stake. In part II, the difficult question of energy transfer between fields emitted at the various frequencies involved is discussed in detail. We then examine the tensorial nature of susceptibilities and, using Neumann's principle, show how the number of their independent components can be significantly reduced. In part III, numerical examples of scattering by nonlinear materials are given and discussed.

Stress imaging by guided wave tomography based on analytical acoustoelastic model.

A nondestructive method ( M) for stress characterization in plate-like structures is proposed. In this method, the acoustoelastic effects (AEEs) on Lamb and shear horizontal guided waves are used to reconstruct a nonuniform multiaxial stress field. The development of M starts by deriving an analytical acoustoelastic model (An-AEM) to predict AEEs induced by a triaxial stress tensor as a function of the stress components, its orientation, the wave propagation direction, and three acoustoelastic coefficients (AECs). The AECs are independent of stress but specific to each mode. The An-AEM allows one to retrieve the three components of the stress tensor and its orientation from AEEs, assuming the stress to be uniform in the plane of the plate and through its thickness. To deal with stress that is nonuniform in the plane, the An-AEM is combined with time-of-flight straight ray tomography to enable stress field reconstruction. Numerical simulation is used to illustrate how such reconstruction can be performed. It is shown that in some cases, stress components can be reconstructed with arbitrary accuracy, and in other cases, the tensorial nature of stress renders the accuracy of its reconstruction dependent on spatial variations of the stress orientation.

Critical Comparison of Phase-Field, Peridynamics, and Crack Band Model M7 in Light of Gap Test and Classical Fracture Tests

Abstract The recently conceived gap test and its simulation revealed that the fracture energy Gf (or Kc, Jcr) of concrete, plastic-hardening metals, composites, and probably most materials can change by ±100%, depending on the crack-parallel stresses σxx, σzz, and their history. Therefore, one must consider not only a finite length but also a finite width of the fracture process zone, along with its tensorial damage behavior. The data from this test, along with ten other classical tests important for fracture problems (nine on concrete, one on sandstone), are optimally fitted to evaluate the performance of the state-of-art phase-field, peridynamic, and crack band models. Thanks to its realistic boundary and crack-face conditions as well as its tensorial nature, the crack band model, combined with the microplane damage constitutive law in its latest version M7, is found to fit all data well. On the contrary, the phase-field models perform poorly. Peridynamic models (both bond based and state based) perform even worse. The recent correction in the bond-associated deformation gradient helps to improve the predictions in some experiments, but not all. This confirms the previous strictly theoretical critique (JAM 2016), which showed that peridynamics of all kinds suffers from several conceptual faults: (1) It implies a lattice microstructure; (2) its particle–skipping interactions are a fiction; (4) it ignores shear-resisted particle rotations (which are what lends the lattice discrete particle model (LDPM) its superior performance); (3) its representation of the boundaries, especially the crack and fracture process zone faces, is physically unrealistic; and (5) it cannot reproduce the transitional size effect—a quintessential characteristic of quasibrittleness. The misleading practice of “verifying” a model with only one or two simple tests matchable by many different models, or showcasing an ad hoc improvement for one type of test while ignoring misfits of others, is pointed out. In closing, the ubiquity of crack-parallel stresses in practical problems of concrete, shale, fiber composites, plastic-hardening metals, and materials on submicrometer scale is emphasized.