Configuration Tensor Research Articles

Revisiting the governing equations of a magnetic suspension of polar particles: From microhydrodynamics analysis to rheological response

In this review, we describe a formulation for the stress tensor of a monodisperse magnetic suspension of polarized neutrally buoyant spheroidal particles suspended in a non-magnetic liquid. A magnetic suspension affords a rare example of a material for which the stress tensor is non-symmetric. The present formulation is based on a microhydrodynamics description of a spherical particle suspended in a Newtonian fluid subjected to magnetic forces and torques. The magnetic suspension is considered statistically homogeneous and treated as being a homogeneous equivalent fluid. Under this condition, a volume average over all particles in the carrier fluid is used in order to obtain the magnetization equation evolution and the constitutive equation for the stress tensor of the magnetic suspension, in particular the magnetic stress contribution. The average effects on the homogeneous continuum fluid due to particle pressure, particle dipole, and the applied magnetic field on each particle are computed by our constitutive equation. In this approach, the particles are not considered force or torque free since their permanent magnetization allows them to experience the effects of an applied magnetic field. The calculated stress tension can be used for modeling common flows of symmetric or non-symmetric magnetic fluids flowing in arbitrary geometries and in rheological applications for determination of important properties such as the rotational viscosity of non-symmetric magnetic fluids. The final expression of the constitutive equation for the stress tensor based on a particle scale approach presents some difference as compared with current constitutive models proposed in the current literature. Our constitutive equation considers the effect of a magnetic particle pressure, the average particle stresslet contribution in terms of an effective viscosity, the average particle rotlet in terms of a rotational viscosity, and a configurational tensor associated with dipole–dipole interactions. In addition, we discuss the situation in which the dipole moment of the particle is not frozen on it which leads to the necessity of an internal balance of angular momentum in a fluid element to close the governing equations of the model. An extension of the model for emulsions of polar deformable droplets is also proposed.

Fracture Criteria and Crack Initiation Mechanism of Material Based on Configurational Stress

This study aims to build a new bridge between configurational stress/force and material fracture. The migrating control volume and thermodynamics are used to develop the Eshelby relation, and the relationship between the conservative integral in fracture mechanics and configurational stress/force for elastic or elastic-plastic materials is further clarified. Additionally, the configurational stresses, including circumferential configurational stress at the crack tip taking T-stress into consideration, are determined, and the [Formula: see text] integral vector is then calculated further. The results indicate that [Formula: see text] integral is path-independent while [Formula: see text] is path-dependent when T-stress is considered. We preliminarily present the relationship between the configurational stress and crack initiation and the zero circumferential configurational stress fracture criterion (ZCCS) is proposed based on the local properties of the crack-tip configurational stress tensor and fracture mechanics. To estimate the fracture loads, we also develop two fracture criteria based on the critical area of crack-tip plastic zone determined by the Mises configurational stress (MCSPA) and the principal configurational stress difference (PCSDPA), respectively. It is found that the initiation angle assessed by the ZCCS fracture criterion is in good agreement with that by both the MTS fracture criterion and experimental observations, as well as T stresses could affect the initiation angles for mixed-mode cracks under tension-shear loads. Furthermore, the fracture loads evaluated by the MCSPA and PCSDPA fracture criteria are consistent with that by both the MTS fracture criterion and experimental results. Finally, the initiation angles determined based on the characteristics of crack-tip plastic zone by configurational stress coincide with that by MTS and ZCCS fracture criteria.

Rheology of edible soft glassy materials

In this paper we investigate the rheology of three different classes of edible yield stress fluids: a) cookie doughs, b) vegetable/fruit purees, and c) protein-rich doughs as used in meat analogs and 3D food printing. Strains sweeps of different formulations within one class can be mapped to a (partial) master curve, with at least the same strain softening index, and a similar crossover point. For classes showing only a partial master curve, the formulation differs in the response of the loss modulus in the linear viscoelastic (LVE) regime, and the transition regime between LVE and strain softening regimes, where several formulations also show a weak strain overshoot. Amongst different classes, different behaviour in strain softening has been observed. Meat analogs and fruit purees show a ratio of 2 between strain softening indices of elastic and loss modulus, while for cookie doughs and protein-rich inks, the ratio is near unity. A constitutive model using a configurational tensor with shear rate-dependent viscosity and strain-dependent elastic modulus can reproduce all features observed for the different classes and for the various formulations within a class. We conclude that our constitutive model is a good basis to design edible yield stress fluids for fiber-enrichment, starch and/or sugar replacement.

Automated Urban Planning for Reimagining City Configuration via Adversarial Learning: Quantification, Generation, and Evaluation

Urban planning refers to the efforts of designing land-use configurations given a region. However, to obtain effective urban plans, urban experts have to spend much time and effort analyzing sophisticated planning constraints based on domain knowledge and personal experiences. To alleviate the heavy burden of them and produce consistent urban plans, we want to ask that can AI accelerate the urban planning process, so that human planners only adjust generated configurations for specific needs? The recent advance of deep generative models provides a possible answer, which inspires us to automate urban planning from an adversarial learning perspective. However, three major challenges arise: (1) how to define a quantitative land-use configuration? (2) how to automate configuration planning? (3) how to evaluate the quality of a generated configuration? In this article, we systematically address the three challenges. Specifically, (1) We define a land-use configuration as a longitude-latitude-channel tensor. (2) We formulate the automated urban planning problem into a task of deep generative learning. The objective is to generate a configuration tensor given the surrounding contexts of a target region. In particular, we first construct spatial graphs using geographic and human mobility data crawled from websites to learn graph representations. We then combine each target area and its surrounding context representations as a tuple, and categorize all tuples into positive (well-planned areas) and negative samples (poorly-planned areas). Next, we develop an adversarial learning framework, in which a generator takes the surrounding context representations as input to generate a land-use configuration, and a discriminator learns to distinguish between positive and negative samples. (3) We provide quantitative evaluation metrics and conduct extensive experiments to demonstrate the effectiveness of our framework.

Elasto-plastic evolution of single crystals driven by dislocation flow

This work introduces a model for large-strain, geometrically nonlinear elasto-plastic dynamics in single crystals. The key feature of our model is that the plastic dynamics are entirely driven by the movement of dislocations, that is, 1-dimensional topological defects in the crystal lattice. It is well known that glide motion of dislocations is the dominant microscopic mechanism for plastic deformation in many crystalline materials, most notably in metals. We propose a novel geometric language, built on the concepts of space-time “slip trajectories” and the “crystal scaffold” to describe the movement of (discrete) dislocations and to couple this movement to plastic flow. The energetics and dissipation relationships in our model are derived from first principles drawing on the theories of crystal modeling, elasticity, and thermodynamics. The resulting force balances involve a new configurational stress tensor describing the forces acting against slip. In order to place our model into context, we further show that it recovers several laws that were known in special cases before, most notably the equation for the Peach–Koehler force (linearized configurational force) and the fact that the combination of all dislocations yields the curl of the plastic distortion field. Finally, we also include a brief discussion on how a number of other effects, such as hardening, softening, dislocation climb, and coarse-graining, could be incorporated into our model.

The Oldroyd-B fluid in elastic instabilities, turbulence and particle suspensions

The Oldroyd-B fluid has become the starting point for almost all complex flow calculations and analysis involving the behavior of dilute polymer solutions. The reasons for this are clear: based on the kinetic theory of high molecular weight “phantom” chains near equilibrium, the resulting model, in its simplest form, produces a single mode description of the solution constitutive equation in terms of a single symmetric dyad order parameter. The physical connection to the individual chains is therefore unambiguous through the end-to-end configuration tensor. The flow of the fluid resulting from the mathematical solution of the constitutive equation coupled to Cauchy’s equation of motion and continuity has been examined for decades and its strengths and weaknesses in predicting the features of non-linear viscometric and non-viscometric polymer solution flows are well known. Moreover, “simple” extensions (e.g. the FENE-P model) can ameliorate the most serious of these shortcomings. For example, the multi-mode Oldroyd-B model can provide quantitative predictions of the linear viscoelasticity of a wide range of polymer solutions. Remarkably, even with all its shortcomings, the Oldroyd-B fluid has been invaluable in predicting at least qualitatively, features as complex as purely elastic instabilities, turbulent drag modification, and even the effective rheology of particles suspended in polymer solutions. While remaining mindful of the model’s shortcomings, any student of these diverse fields may find the Oldroyd-B fluid an important starting point.

The derivation of the FENE-P model within the context of a thermodynamic perspective for bodies with evolving natural configurations

In this paper, we provide a method for developing the FENE-P constitutive relation and generalizations of such constitutive relations from a totally different perspective than that used to develop such constitutive relations. The thermodynamic framework within which the constitutive relations are generated is a generalization of the structure within which rate type fluid models to describe the viscoelastic response of fluids were developed (JNNFM, 88, 207–227 (2000)). It is interesting to note that widely different approaches such as those based on the notion of configuration tensors and the study carried out here are able to lead to the same classes of models, providing one basis for some conviction concerning the appropriateness of the model. Also, the fact that the general thermodynamic structure used here has also been used successfully to establish constitutive relations to describe the response of bodies in a wide range of areas in inelasticity, twinning, phase transitions, chemically reacting mixtures, shape memory alloys and polymers, etc., gives additional credibility to the construction of constitutive relations such as those obtained in this study.

Direct Eulerian Formulation of Anisotropic Hyperelasticity

Abstract Many soft materials and biological tissues comprise isotropic matrix reinforced by fibers in the characteristic directions. Hyperelastic constitutive equations for such materials are usually formulated in terms of a Lagrangian strain tensor referred to the initial configuration and Lagrangian structure tensors defining characteristic directions of anisotropy. Such equations are “pushed forward” to the current configuration. Obtained in this way, Eulerian constitutive equations are often favorable from both theoretical and computational standpoints. In the present note, we show that the described two-step procedure is not necessary, and anisotropic hyperelasticity can be introduced directly in terms of an Eulerian strain tensor and Eulerian structure tensors referring to the current configuration. The newly developed constitutive equation is applied to the particular case of the transverse isotropy for the sake of illustration.

О 6-мерных AH-подмногообразиях класса W1⊕W2⊕W4 алгебры Кэли

Six-dimensional submanifolds of Cayley algebra equipped with an almost Hermitian structure of class W1 W2 W4 defined by means of three-fold vector cross products are considered. As it is known, the class W1 W2 W4 contains all Kählerian, nearly Kählerian, almost Kählerian, locally conformal Kählerian, quasi-Kählerian and Vaisman — Gray manifolds. The Cartan structural equations of the W1 W2 W4 -structure on such six-dimensional submanifolds of the octave algebra are obtained. A criterion in terms of the configuration tensor for an arbitrary almost Hermitian structure on a six-dimensional submanifold of Cayley algebra to belong to the W1 W2 W4 -class is established. It is proved that if a six-dimensional W1 W2 W4 -submanifold of Cayley algebra satisfies the quasi-Sasakian hypersurfaces axiom (i.e. a hypersurface with a quasi-Sasakian structure passes through every point of such submanifold), then it is an almost Kählerian manifold. It is also proved that a six-dimensional W1 W2 W4 -submanifold of Cayley algebra satisfies the eta-quasi-umbilical quasi-Sasakian hypersurfaces axiom, then it is a Kählerian manifold.

О 6-мерных подмногообразиях Вайсмана — Грея алгебры октав

The W1 W4 class of almost Hermitian manifolds (in accordance with the Gray — Hervella classification) is usually named as the class of Vaisman — Gray manifolds. This class contains all Kählerian, nearly Kählerian and locally conformal Kählerian manifolds. As it is known, Vaisman — Gray manifolds are invariant under the conformal transformations of the metric. A criterion in the terms of the configuration tensor for an arbitrary six-dimensional submanifold of Cayley algebra to belong to the Vaisman — Gray class of almost Hermitian manifolds is established. The Cartan structural equations of the almost contact metric structures induced on oriented hypersurfaces of six-dimensional Vaisman — Gray submanifolds of the octave algebra are obtained. It is proved that totally geodesic hypersurfaces of six-dimensional Vaisman — Gray submanifolds of Cayley algebra admit nearly cosymplectic structures (or Endo structures). This result is a generalization of the previously proved fact that totally geodesic hypersurfaces of nearly Kählerian manifolds also admit nearly cosymplectic structures.

Development of a predictor for fatigue crack nucleation of dielectric viscoelastomers under electromechanical loads

Capable of yielding large deformation under electrical stimuli, dielectric elastomers (DEs) have extensive applications. In most applications, DEs undergo cyclic or time-varying loads over a long period. As a result, durability and fatigue analysis of DEs is a critical issue needs to be addressed. However, there are few models in the literature for predicting the fatigue damage of materials subjected to electromechanical coupled field. Moreover, the existing models fail to capture the effect of the intrinsic viscoelasticity of elastomers. The current work aims to bridge this gap. A configurational stress tensor for DEs is formulated for the first time and adopted as a predictor for their fatigue crack nucleation, which is based on the theoretical framework of configurational mechanics. Borrowing the ideas from the coupled field theory for dielectrics and the theory of finite-deformation viscoelasticity, the developed predictor can adopt most of the strain energy density functions for rubber-like materials and thermodynamics evolution equations for viscoelastic solids. With the developed fatigue predictor, the effects of different factors that may contribute to the fatigue damage of DEs are examined. It is found that the electrical part and the viscous part of the configurational stress do not lead to fatigue damage of DEs. Furthermore, simulation results show that the strain-softening behavior of elastomers exert a significant effect on the fatigue life of DEs. The modeling framework in this work is anticipated to be a useful platform for further study on the fatigue life of DEs under different loading conditions, and even applicable to other soft materials when subjected to coupled loads.

Low-lying dipole strength in Cr52

The low-lying dipole strength in $^{52}\mathrm{Cr}$ was measured in nuclear resonance fluorescence experiments using a quasimonochromatic, linearly polarized photon beam. The parities of the excited dipole states were determined by the intensity asymmetry of resonantly scattered $\ensuremath{\gamma}$ rays with respect to the polarization plane of the incident photon beam. We observed 62 resonances at excitation energies between 7.5 and 12.1 MeV. The observed $M1$ and $E1$ strengths were compared via random-phase-approximation calculations using the Skyrme interaction. The effects of two-particle--two-hole configuration mixing and tensor force on dipole strength distributions were investigated.

Geometry of 6-Dimensional Hermitian Manifolds of the Octave Algebra

This paper is a survey of recent results in the geometry of 6-dimensional Hermitian manifolds of Cayley algebras. We describe Gray–Hervella classes of almost Hermitian structures on 6-dimensional oriented submanifolds of Cayley algebras and propose a characterization of these in terms of the Kirichenko tensors and the configuration tensor.

On some fundamental properties of the L-integral in plane elasticity

The explicit definition of the L-integral is reviewed by the curl operation of the Lagrangian energy density moment in plane elasticity. The physical interpretation of the configurational stress tensor associated with the integrand of the L-integral is explored, and it is identified as the change of potential energy due to the rotation of one infinitesimal material element with respect to the fixed point. Further, the path-independence of the L-integralisanalyzedbyanexplicitform.Itisdemonstratedthatthe L-integralshowsthepath-independent properties for an isotropic elasticity or the isotropic plane in a transversely isotropic material while the path- dependence is found for the anisotropic elasticity. Finally, it is explicitly concluded that the L-integral will be independent of the coordinate system attributing to the conservation laws of J1 and J2 integrals.

Piezoelectric, elastic, structural and dielectric properties of the Si1−xGexO2 solid solution: a theoretical study

We apply first principles quantum mechanical techniques to the study of the solid solution Si1−xGexO2 of α-quartz where silicon atoms are progressively substituted with germanium atoms, to different extents, as a function of the substitutional fraction x. For the first time, the whole range of the substitution (x = 0.0, 0.1, 0., 0.5, 0., 0.8, 1.0), including pure end-members α-SiO2 and α-GeO2, is explored. An elongated supercell (doubled along the c crystallographic axis) is built with respect to the unit cell of pure α-quartz and a set of 13 symmetry-independent configurations is considered. Their structural, energetic, dielectric, elastic and piezoelectric properties are computed and analyzed. All the calculations are performed using the CRYSTAL14 program with a Gaussian-type function basis set with pseudopotentials, and the hybrid functional PBE0; all geometries are fully optimized at this level of theory. In particular, for each configuration, fourth-rank elastic and compliance tensors and third-rank direct and converse piezoelectric tensors are computed. It has already been shown that the structural distortion of the solid solution increases, almost linearly, as the substitutional fraction x increases. The piezoelectric properties of the Si1−xGexO2 solid solution are found to increase with x, with a similar quasi-linear behavior. The electromechanical coupling coefficients are enhanced as well and the linear trend recently predicted by Ranieri et al (2011 Inorg. Chem. 50 4632) can be confirmed from first principles calculations. These doped crystals do represent good candidates for technological applications requiring high piezoelectric coupling and high thermal stability.

Injury of the spinothalamic tract in a patient with mild traumatic brain injury: Diffusion tensor tractography study

To report on a patient found to have injury of the spinothalamic tract on diffusion tensor tractography following traumatic brain injury. A 29-year-old male patient with head trauma resulting from a pedestrian car accident presented with pain in multiple areas (both subscapular areas, posterior head and neck, both upper trapezius areas, and the right arm and leg). His pain had not improved with various types of conservative management. Evaluations (conventional brain magnetic resonance imaging, electromyography, and whole spine magnetic resonance imaging), performed 2 years after the head trauma, did not reveal any specific abnormality. Fibromyalgia and myofascial pain syndrome were ruled out by physical examination. Injuries of the spinothalamic tracts in both hemispheres were observed on diffusion tensor imaging in terms of the configuration (thinning) and diffusion tensor tractography parameters (decreased fractional anisotropy or tract volume). Some of the pain in the trunk and extremities in this patient could be ascribed to central pain caused by injury of both spinothalamic tracts. We conclude that diffusion tensor tractography provides a useful means of detecting injury of the spinothalamic tract in patients with traumatic brain injury.

Effect of fibre curvature on the rheology of particulate suspensions

Rheological properties of non-Brownian straight and curved fibre suspensions in simple shear flow have been investigated by particle-level simulation. An inertialess fibre particle in Newtonian fluid is modelled as a collection of spheres connected with Hookean-type constraint force. The fibre motions are governed by hydrodynamic interaction based on Rotne–Prager correction to velocity disturbance and short-ranged repulsive force. An isolated curved fibre in simple shear flow shows a decrease in the period of rotation with increasing the curvature. Initial stress overshoot is observed as fibres undergo transition from isotropic orientation. Shear viscosity and first normal stress of straight fibre suspensions exhibit scaling with respect to concentration in a manner consistent with comparable literature results. The overall viscosity enhancement with increasing the curvature has been observed over the studied range of aspect ratio and concentration. This is accompanied by the decrease of flow-aligned component of configuration tensor. Straight and curved fibre simulation exhibits an average positive first normal stress difference but the data fluctuation is too large to distinguish curvature effect.

Use of tensorial description in tissue remodeling: examples of F-actin distributions in pulmonary arteries in hypoxic hypertension.

A molecular configuration tensor Pij was introduced to analyze the distribution of fibrous proteins in vascular cells for studying cells and tissues biomechanics. We have used this technique to study the biomechanics of vascular remodeling in response to the changes of blood pressure and flow. In this paper, the remodeling of the geometrical arrangement of F-actin fibers in the smooth muscle cells in rat's pulmonary arteries in hypoxic hypertension was studied. The rats were exposed to a hypoxia condition of 10% for 0, 2, 12, and 24 hr at sea level. Remodeling of blood vessels were studied at the in vivo state under normal perfusion, no-load state when small rings from blood vessels were excised, and zero-stress state after the rings were cut open radially to release the residual stress. Tissue remodeling in response to changes in blood pressure is reflected in the zero-stress state. The tensor components were determined by analyzing the configuration of phalloidin stained F-actin fibers in the media layer of pulmonary arteries. The values of P31, P32, P33 in the in-vivo state, the no-load state, and the zero-stress state are obtained. This study demonstrated the distributions of fibrous molecules in tissue remodeling can be described quantitatively using the molecular configuration tensor.

Tensorial Description of the geometrical arrangement of Endothelin‐1 in response to the changes of blood pressure in rat's aorta

For better studying the vascular remodeling in hypertension, the geometrical arrangement of the cellular molecules, which are the foundation of the physical behavior of the cells and tissues, has to be described quantitatively. Endothelin‐1 (ET‐1) is a vasoconstrictor and mitogenic agent, and has been found in the vascular endothelium, the vascular smooth muscle cells, and the fibroblasts. This study aimed to investigate quantitatively the changes of the geometrical distribution of ET‐1 in blood vessel wall in response to the changes of blood pressure. By clamping one cross section of the rat's abdominal aorta, the blood pressure in the aorta was raised in the portion above the clamp, and lowered in the portion below the clamp. The effects of the raising or the lowering of the blood pressure on these two portions of the aorta simultaneously on the tissue remodeling in the abdominal aortas were studied after aortic banding for 0, 0.5, 4, or 8 days. Aortic cross sections in circumferential and longitudinal directions and En Face were prepared and immunostained with ET‐1 antibody for examination by a fluorescent microscope. Digital images were taken for analysis. The geometrical distributions of ET‐1 in the vessel wall was described by a Molecular configuration tensor, Pij, where the subscripts i, j refer to the Cartesian coordinate axes x1, x2, x3. All components in the molecular configuration tensor of ET‐1 are presented.

Effect of lattice topology on the adsorption of benzyl alcohol on kaolinite surfaces: Quantum chemical calculations of geometry optimization, binding energy, and NMR chemical shielding

We investigate the effect of the lattice topology (ideal hexagonal rings vs. ditrigonal rings) and cluster size of model clusters (three-ring vs. seven-ring clusters) on the nature of benzyl alcohol adsorption on kaolinite surfaces using quantum chemical calculations with an emphasis on the equilibrium configuration, binding energy, and NMR chemical shielding tensors. The optimized structure of benzyl alcohol adsorbed on the tetrahedral layer of kaolinite varies according to the type and size of model cluster. While the calculated binding energy varies with the level of theory and the basis sets used for the calculations, the binding energies between benzyl alcohol and seven-ring clusters are smaller than those between benzyl alcohol and three-ring clusters partly due to the edge hydrogen for the latter. The results also indicate a stronger binding energy between benzyl alcohol and octahedral surfaces than between benzyl alcohol and tetrahedral layers. Although the calculated binding energies for seven-member rings with varying lattice topologies are rather similar, the detailed optimized structures are distinct, demonstrating the effect of lattice topology on the nature of adsorption. The optimized structures and binding energies indicate that an intermediate degree of hydrogen bonding is dominant for the three-member silicate rings and that the interaction between the benzene ring and basal O atoms in the seven-member rings is characterized by a weak hydrogen bond and dispersion force. The calculated 17O isotropic chemical shieldings of some basal O atoms decrease up to ~4–5 ppm after the adsorption (with an estimated uncertainty of ~2 ppm). Since the high-resolution 17O 3QMAS NMR spectroscopy of layer silicates yielded a resolution of 1–2 ppm for the basal oxygen sites in the layer silicates in previous work, the NMR technique may be useful in exploring the nature of adsorption between organic molecules and silicate surfaces, whereas further computational studies on the effect of the basis sets, the surface coverage, and the types of diverse organic molecules with larger model clusters for surfaces remain to be explored.