Tensorial Model Research Articles

Quantized plastic deformation

In engineering crystal plasticity inelastic mechanisms correspond to tensorial zero-energy valleys in the space of macroscopic strains. The flat nature of such valleys is in contradiction with the fact that plastic slips, mimicking lattice-invariant shears, are inherently discrete. A reconciliation has recently been achieved in the mesoscopic tensorial model (MTM) of crystal plasticity, which introduces periodically modulated energy valleys while also capturing in a geometrically exact way the crystallographically-specific aspects of plastic slips. In this paper, we extend the MTM framework, which in its original form had the appearance of a discretized nonlinear elasticity theory, by explicitly introducing the concept of plastic deformation. The ensuing model contains a novel matrix-valued spin variable, representing the quantized plastic distortion, whose rate-independent evolution can be described by a discrete (quasi-)automaton. The proposed reformulation of the MTM leads to a considerable computational speedup associated with the use of a robust and efficient hybrid Gauss-Newton–Cauchy energy minimization algorithm. To illustrate the effectiveness of the new approach, we present a detailed case-study focusing on the aspects of crystal plasticity that are beyond reach for the classical continuum theory. Thus, we provide compelling evidence that the re-formulated MTM is fully adequate to deal with the intermittency of plastic response under quasi-static loading. In particular, our numerical experiments show that the statistics of dislocational avalanches, associated with plastic yield in 2D square crystals, exhibits a power-law tail with a critical exponent matching the value predicted by general theoretical considerations and also independently observed in discrete-dislocation-dynamics (DDD) simulations.

Rheology of brain tissue and hydrogels: A novel hyperelastic and viscoelastic model for forensic applications

A new strain-energy function W, which possesses the strain energy expressible as a rational function of the principal invariants of the Cauchy tensor C, is proposed. It generates a hyperelastic constitutive equation with characteristics of brain tissues: a much stronger resistance to compression than to stretching and strongly nonlinear response in simple shear, including non-zero first and second normal differences. This model exponent α resembles the Ogden model in uniaxial stretching/compression and reveals plausible predictions for brain tissue with even values of α < 0 with sufficiently high magnitude (say, at α = −20). However, the dependence of the strain-energy function W on the principal invariants of C links it to hyperelastic hydrogel models (the Special and General Blatz–Ko models, neo-Hookean materials, incompressible Mooney–Rivlin and the Yeoh models). For α = −8, the present model reveals a compression/stretching behavior close to the tensorial Special Blatz–Ko model used for description of hydrogels. Furthermore, the present hyperelastic model is used as a kernel of the corresponding tensorial viscoelastic model with exponential fading memory. It belongs to the class of the integral Bernstein–Kearsley–Zapas (BKZ) models. In a number of important cases (the uniaxial stretching/compression, simple shear), it can be transformed into a differential viscoelastic model and predict viscoelastic liquid-like behavior under sustained deformations. The stress relaxation following an imposed strain reduces to the hyperelastic model with the elastic parameters exponentially fading in time. These tensorial hyperelastic and viscoelastic constitutive equations aim applications in modeling of blast-induced traumatic brain injuries and bullet penetration and spatter of brain tissue in forensic context.

Inelastic rotations and pseudoturbulent plastic avalanches in crystals.

Plastic deformations in crystals produce microstructures with randomly oriented patches of unstressed lattice forming complex textures. We use a mesoscopic Landau-type tensorial model of crystal plasticity to show that in such textures rotations can originate from crystallographically exact microslips which self organize in the form of laminates of a pseudotwin type. The formation of such laminates can be viewed as an effective internal "wrinkling" of the crystal lattice. While such "wrinkling" disguises itself as an elastically neutral rotation, behind it is inherently dissipative, dislocation-mediated process. Our numerical experiments reveal pseudoturbulent effective rotations with power-law distributed spatial correlations which suggests that the process of dislocational self-organization is inherently unstable and points toward the necessity of a probabilistic description of crystal plasticity.

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.

AI-assisted de novo design of selective κ-opioid receptor antagonists for the treatment of opioid addiction.

Daunting estimates of overdose fatalities, costs of healthcare, lost productivity, and criminal justice involvement for the misuse of prescription opioids have converted opioid use disorders into a major national crisis in the United States. Among likely efficacious pharmacological interventions for the treatment of opioid dependence are those that can attenuate brain reward deficits experienced during periods of abstinence. Pharmacological blockade of κ-opioid receptors (KOR) has been shown to abolish brain reward deficits detected in rodents during withdrawal from opioids, as well as to reduce the escalation of opioid use in rats with extended access to opioids. Although KOR antagonists represent promising candidates for the treatment of opioid addiction, very few potent selective KOR antagonists are known to date and most of them exhibit significant safety concerns. Here, we used a deep learning framework for the de novo design of ligands with predefined molecular properties to predict new chemotypes with KOR antagonistic activity. Specifically, we pre-trained a deep generative tensorial model to learn a mapping of the chemical space from a ZINC subset of purchasable drug-like molecules, as well as sets of effective (IC50 < 1 µM) or less effective (IC50 ≥ 1 µM) known KOR inhibitors from the ChEMBL database, while simultaneously encoding the relationship between the molecules and their properties. Subsequently, we biased the resulting generative model towards the development of putative KOR antagonists with a reinforcement learning algorithm that rewarded similarity to the antagonist binding mode revealed by the JDTic-bound KOR crystal structure, as well as novelty, synthetic accessibility, absence of unstable and reactive moieties, drug-like solubility, and blood-brain barrier permeability. The generated molecules were prioritized for chemical synthesis and functional evaluation based on their predicted optimal interactions with the receptor.

Ion-beam radiation-induced Eshelby transformations: The mean and variance in hydrostatic and shear residual stresses

Ion beam plays a pivotal role in ion implantations and the fabrication of nanostructures. However, there lacks a quantitative model to describe the residual stresses associated with the ion-beam radiation. Radiation-induced residual stress/transformation strain have been mostly recognized in the hydrostatic sub strain space. Here, we use molecular dynamics (MD) simulations to show that the response of a material to irradiation is generally anisotropic that depends on the ion-beam direction, and should be described using tensorial quantities. We demonstrate that accelerator-based ion beam irradiation, combined with the intrinsic lattice anisotropy and externally induced anisotropy (such as anisotropic mechanical loadings), causes radiation-actuated shear transformation strains in addition to hydrostatic expansion. We map out these complex correlations for several materials. Radiation-induced defects are shown to be responsible for residual shear stresses in the manner of Eshelby inclusion transformation. We propose such tensorial response model should be considered for accurate nanoscale fabrication using ion-beam irradiation.

Homogeneous nucleation of dislocations as a pattern formation phenomenon

Dislocation nucleation in homogeneous crystals begins as a symmetry-breaking elastic instability. In the absence of explicit nucleation centers, such instability develops simultaneously all over the crystal and due to the dominance of long range elastic interactions it advances into the nonlinear stage as a collective phenomenon through pattern formation. In this paper we use a novel mesoscopic tensorial model (MTM) of crystal plasticity to study the delicate role of crystallographic symmetry in the development of the dislocation nucleation patterns in defect free crystals. The model is formulated in 2D and we systematically compare the unfolding of the pattern formation phenomenon in lattices with square and triangular symmetry. To avoid the dominance of the conventional plastic mechanisms, we consider the loading paths represented by pure shears applied on the boundary of the otherwise unloaded body. These peculiar loading protocols can be qualified as exploiting the ‘softest’ and the ‘hardest’ directions and we show that the associated dislocation patterns are strikingly different.

Effects of surface rigidity and metallicity on dielectric properties and ion interactions at aqueous hydrophobic interfaces.

Using classical molecular dynamics simulations, we investigate the dielectric properties at interfaces of water with graphene, graphite, hexane, and water vapor. For graphite, we compare metallic and nonmetallic versions. At the vapor-liquid water and hexane-water interfaces, the laterally averaged dielectric profiles are significantly broadened due to interfacial roughness and only slightly anisotropic. In contrast, at the rigid graphene surface, the dielectric profiles are strongly anisotropic and the perpendicular dielectric profile exhibits pronounced oscillations and signchanges. The interfacial dielectric excess, characterized by the shift of the dielectric dividing surface with respect to the Gibbs dividing surface, is positive for all surfaces, showing that water has an enhanced dielectric response at hydrophobic surfaces. The dielectric dividing surface positions vary significantly among the different surfaces, which points to pronounced surface-specific dielectric behavior. The interfacial repulsion of a chloride ion is shown to be dominated by electrostatic interactions for the soft fluid-fluid interfaces and by non-electrostatic Lennard-Jones interactions for the rigid graphene-water interface. A linear tensorial dielectric model for the ion-interface interaction with sharp dielectric interfaces located on the dielectric dividing surface positions works well for graphene but fails for vapor and hexane, because these interfaces are smeared out. The repulsion of chloride from the metallic and nonmetallic graphite versions differs very little, which reflects the almost identical interfacial water structure and can be understood based on linear continuum dielectric theory. Interface flexibility shows up mostly in the nonlinear Coulomb part of the ion-interface interaction, which changes significantly close to the interfaces and signals the breakdown of linear dielectric continuum theory.

Derivation of a constitutive model for the rheology of jammed soft suspensions from particle dynamics

Considering the rheology of two-dimensional soft suspensions above the jamming density, we derive a tensorial constitutive model from the microscopic particle dynamics. Starting from the equation governing the N-particle distribution, we derive an evolution equation for the stress tensor. This evolution equation is not closed, as it involves the pair and three-particle correlation functions. To close this equation, we first employ the standard Kirkwood closure relation to express the three-particle correlation function in terms of the pair correlation function. Then we use a simple and physically motivated parametrization of the pair correlation function to obtain a closed evolution equation for the stress tensor. The latter is naturally expressed as separate evolution equations for the pressure and for the deviatoric part of the stress tensor. These evolution equations provide us with a non-linear tensorial constitutive model describing the rheological response of a jammed soft suspension to an arbitrary uniform deformation. One of the advantages of this microscopically-rooted description is that the coefficients appearing in the constitutive model are known in terms of packing fraction and microscopic parameters.

Dynamic tensorial eddy viscosity model: Effects of compressibility and of complex geometry

A previous paper by Cimarelli et al. [“General formalism for a reduced description and modelling of momentum and energy transfer in turbulence,” J. Fluid Mech. 866, 865–896 (2019)] has shown that every decomposition of turbulent stresses is naturally approximated by a general form of tensorial eddy viscosity based on velocity increments. The generality of the formalism is such that it can also be used to give a reduced description of subgrid scalar fluxes. In the same work, this peculiar property of turbulent stresses and fluxes has been dynamically exploited to produce tensorial eddy viscosity models based on the second-order inertial properties of the grid element. The basic idea is that the anisotropic structure of the computational element directly impacts, although implicitly, the large resolved and small unresolved scale decomposition. In the present work, this new class of turbulence models is extended to compressible turbulence. A posteriori analysis of flow solutions in a compressible turbulent channel shows very promising results. The quality of the modeling approach is further assessed by addressing complex flow geometries, where the use of unstructured grids is demanded as in real world problems. Also in this case, a posteriori analysis of flow solutions in a periodic hill turbulent flow shows very good behavior. Overall, the generality of the formalism is found to allow for an accurate description of subgrid quantities in compressible conditions and in complex flows, independent of the discretization technique. Hence, we believe that the present class of turbulence closures is very promising for the applications typical of industry and geophysics.

Emergent cosmology from quantum gravity in the Lorentzian Barrett-Crane tensorial group field theory model

We study the cosmological sector of the Lorentzian Barrett-Crane (BC) model coupled to a free massless scalar field in its Group Field Theory (GFT) formulation, corresponding to the mean-field hydrodynamics obtained from coherent condensate states. The relational evolution of the condensate with respect to the scalar field yields effective dynamics of homogeneous and isotropic cosmologies, similar to those previously obtained in SU(2)-based EPRL-like models. Also in this manifestly Lorentzian setting, in which only continuous SL(2,ℂ)-representations are used, we obtain generalized Friedmann equations that generically exhibit a quantum bounce, and can reproduce all of the features of the cosmological dynamics of EPRL-like models. This lends support to the expectation that the EPRL-like and BC models may lie in the same continuum universality class, and that the quantum gravity mechanism producing effective bouncing scenarios may not depend directly on the discretization of geometric observables.

Rheological response of compressible SMCs under various deformation kinematics: Experimental aspects and simple modelling approach

Flow conditions during compression moulding of Sheet Moulding Compounds (SMCs) govern the microstructure and the resulting mechanical properties of composite parts. Low-density and high-fibre content SMCs were subjected to simple compression, œdometric compression and through-thickness shear loadings, using dedicated rheometers. Simple compression and œdometric compression experiments revealed the compaction behaviour of both types of SMCs. During simple compression experiments, the compaction of low-density SMCs was accompanied by in-plane elongational flow phenomena, whereas the high-fibre content SMCs exhibited compaction below a characteristic axial strain and elongation and shear above. Shear tests showed that shear stress did not vary over a wide range of shear strain while being accompanied by axial compression stress. Both SMCs exhibited a shear-thinning behaviour with compression viscosities that depended on the fibre volume fraction. Finally, a transversely isotropic tensorial model for SMCs seen as compressible materials was proposed. Its predictions are consistent with the experiments.

Tensorial conditional gradient method for solving multidimensional ill-posed problems

We consider solving a class of tensorial ill-conditioned problems. This problem is treated as a convex constrained minimization problem. This kind of ill-conditioned problems appears in several applications as color images and video restoration. A new tensor degradation model to recover color image and video from blur and noise is given. Tikhonov regularization approach is used to reduce the effect of noise in the computed solution. This approach leads to a convex tensor minimization problem, that can be solved basing on the conditional gradient method. We adapted the Generalized Cross Validation method to the tensorial model for appropriate choice of the regularization parameter. Numerical tests for image and video restoration are given to illustrate the effectiveness of the proposed approach compared with the classical method.

Shear Jamming and Fragility of Suspensions in a Continuum Model with Elastic Constraints.

Under an applied traction, highly concentrated suspensions of solid particles in fluids can turn from a state in which they flow to a state in which they counteract the traction as an elastic solid: a shear-jammed state. Remarkably, the suspension can turn back to the flowing state simply by inverting the traction. A tensorial model is presented and tested in paradigmatic cases. We show that, to reproduce the phenomenology of shear jamming in generic geometries, it is necessary to link this effect to the elastic response supported by the suspension microstructure rather than to a divergence of the viscosity.

Direct simulations and subgrid modeling of turbulent channel flows asymmetrically heated from both walls

Thermal large-eddy simulations (T-LES) and a direct numerical simulation are carried out in a bi-periodical channel with hot and cold wall temperatures of, respectively, 900 and 1300 K. The mean fluid temperature is lowered below the cold wall temperature thanks to a heat source, resulting in a both walls heating of the fluid. The hot and cold wall friction Reynolds numbers are, respectively, 640 and 1000. These conditions are representative of the working conditions of gas-pressurized solar receiver of solar power tower. The low Mach number Navier–Stokes equations are solved. The coupling between the dynamic and the temperature effects is considered. In the T-LES, both the momentum convection and the density–velocity correlation subgrid terms are modeled. Functional models, structural models, and mixed models are considered. A tensorial version of the anisotropic minimum-dissipation (AMD) model is also investigated. The Quick and the second-order-centered schemes are tested for the discretization of the mass convection term. First, an overview of the results of 17 T-LES on first- and second-order statistics is proposed. It permits selecting 6 of these simulations for a detailed analysis consisting in the investigation of profiles of mean quantities and turbulent correlations. Particular attention is given to the wall heat fluxes because they are a critical point for the design and the optimization of solar receivers. Overall, the first-order statistics are better predicted than the second-order's. The tensorial AMD model takes advantage of the classical AMD model properties and better reproduces the anisotropy of the flow thanks to its formulation. The tensorial AMD model produces the most reliable and efficient results among the considered models.

Investigation of thermal large-eddy simulation approaches in a highly turbulent channel flow submitted to strong asymmetric heating

This study deals with thermal large-eddy simulation (T-LES) of anisothermal turbulent channel flow in the working conditions of solar receivers used in concentrated solar power towers. The flow is characterized by high-temperature levels and strong heat fluxes. The hot and cold friction Reynolds numbers of the simulations are, respectively, 630 and 970. The Navier–Stokes equations are solved under the low-Mach number approximation and the thermal dilatation is taken into account. The momentum convection and the density–velocity correlation subgrid terms are modeled. Functional, structural, and mixed subgrid-scale models are investigated. A tensorial version of the classical anisotropic minimum-dissipation (AMD) model is studied and produces good results. A Quick scheme and a second-order-centered scheme are tested for the discretization of the mass convection term. First, a global assessment of 22 large-eddy simulations is proposed, then six are selected for a careful analysis including profiles of mean quantities and fluctuation values as well as a comparison of instantaneous fields. Probability density functions of wall heat fluxes are plotted. The results point out that T-LESs performed with the Quick scheme tend to underestimate the wall heat flux whereas the second-order-centered scheme significantly improves its estimation. T-LESs tend to overestimate the peaks of velocity correlations. When regarding the dimensionless profiles of fluctuations, the tensorial AMD model provides better results than the other assessed models. For the heat flux estimation, the best agreement is found with the AMD model combined with the second-order-centered scheme.

Tensorial permeability microstructure model considering crystallographic texture and grain size for evaluation of magnetic anisotropy in polycrystalline steels

ABSTRACT A finite element microstructure model with permeability tensors that considers crystallographic texture and grain size based on magnetic domain theory has been developed for the evaluation of magnetic anisotropy in polycrystalline steels. The model has proved capable of capturing the crystallographic texture, the grain size and the vector induction effects on the effective permeability behaviours for typical textures in steels. The predicted magnetic properties as a function of the magnetic field direction enables a quantitative characterisation of the magnetic anisotropy. The predicted effective permeability maps can serve as a visual indication of the crystallographic texture from magnetic values. These features have been experimentally validated against a commercial grain oriented electrical steel featuring strong texture and magnetic anisotropy.

Comparison Between Linear and Tensor Models of EEG Signals Representation

Electroencephalography (EEG) is an important toolfor the study of the human brain because it provides potentiallyuseful signals for understanding the spatial and temporal dynam-ics of neural information processing. These signals are commonlyrepresented by vector or matrix mathematical structures, whichmay counteract their natural behaviour for a multidimensionalrepresentation. Thus, in this case, the information from an EEGsignal should be represented using tensors. This study presentsan analysis of how these different mathematical structures canbe explored to obtain functional brain information. Two matrixmodels and one tensor model were investigated and assessed usingbrain maps and classification results. Our results show at leastthree different and complementary ways for the representationof cognitive brain maps and, as far as our exploratory analysis isconcerned, the tensorial model stands out in terms of the highestlevel of compression and precision in comparison to the othermodels.

Normal stress differences in the consolidation of strong colloidal gels

Macroscopic models for the uniaxial consolidation of strong colloidal gels typically characterise the compressive strength of the particulate network in terms of the compressive yield stress Py(ϕ) or uniaxial elastic modulus K ′ (ϕ). Almost all industrial applications involve multi-dimensional (MD) configurations with arbitrary tensorial stress states, and it is unclear how to generalise these 1D constitutive models to MD consolidation. Several studies have attempted to extend these 1D constitutive models to MD by assuming either isotropic consolidation or zero Poisson’s ratio, but the validity of these assumptions is currently unknown. Lacking is a validated tensorial rheology for the consolidation of strong colloidal gels that is capable of predicting the consolidation of these materials. One step toward the development of such tensorial rheology is the consideration of normal stress differences (NSDs) during uniaxial consolidation. Thus, a tensorial constitutive model for the consolidation of colloidal gels cannot be developed without accounting for these NSDs. We address this problem by performing discrete element model (DEM) simulations of the uniaxial consolidation of a two-dimensional (2D) strong colloidal gel and investigate evolution of the tensorial stress state during consolidation. We show that during consolidation, the Poisson ratio increases from zero near the gel point to almost unity near close-packing and uncover the particle-scale mechanisms that underpin these observations. These results provide the first steps toward a complete tensorial rheology of colloidal gels that is capable of resolving the evolution of these complex materials under superposed differential compression and shear. Graphical abstract

A simple scalar directional hardening model for the Bauschinger effect compared with a tensorial model

A simple scalar directional hardening model for the Bauschinger effect compared with a tensorial model