One-dimensional Continuum Research Articles

Singular decomposable continua

In this paper, we first provide an argument for the method used in [7] and [10] to blow up a point inside a subarc of a one-dimensional continuum to an arbitrary continuum. Next, we give an example of s Wilder continuum containing no strongly Wilder continua, no continuum-wise Wilder continua, no semiaposyndetic continua and no D⁎-continua. Also, we provide an example of a continuum such that each positive Whitney level of the hyperspace of the continuum is strongly Wilder, although the continuum itself does not contain any Wilder continua.

The transient dynamic wave process in unidirectional composites

The propagation of nonstationary waves in unidirectional composite material is simulated by finite-element analysis. The composite material is explicitly formulated, i.e., discrete, stiff, elastic inclusions uniformly distributed in a viscoelastic matrix are modeled. An original procedure rooted in the linear Boltzmann–Volterra equations accounts for the viscoelastic effects in the matrix behavior. The chosen approach involves finite-element simulations of the transient wave process in two indicative configurations to study the effects attributed to wave scattering by the inclusions. The transient wave propagation in the unidirectional composite material and a viscoelastic specimen without inclusions are compared. Peculiarities in the numerical results due to the finite-element algorithm employed are decoupled from the output that reveals purely mechanical phenomena. For this purpose, the transient wave propagation in a one-dimensional elastic continuum, for which the exact analytical solution is well known, is effectively obtained by finite-element analysis. Furthermore, based on the unidirectional composite a homogenized model is defined by applying the mixture rule. An additional comparison between the composite containing discrete inclusions and the analogous homogenized material provides a basis to assess the output of models obtained by implementing a self-consistent scheme in a broader context.

Applicability of two-phase modeling with compression experiments for snow compaction dynamics

Compaction is a rheological process which, in many fields, has been modeled using a 1-D two-phase continuum framework. However, only recently has it been posed as a promising method for modeling the densification of snow into glacial ice, where the conventional model is empirical or semi-empirical. Here, we explore the applicability of a standard one-dimensional two-phase continuum framework for modeling snow compaction through theoretical and laboratory methods by analyzing and simplifying theory, and then experimentally constraining the model coefficient. We find in our theory analysis that the limit of slow compaction is reached such that air evacuation during the compaction process does not impede the deformation of ice grains. Model-data comparisons are performed using data from a series of uniaxial compression experiments of snow samples under a range of compaction rates (1 × 10−6 to 3 × 10−5 m s−1) and densities (250 to 450 kg m−3) at −10° and –20 °C, which show good measures of fit (r2>0.996). By defining a linear effective pressure function, we then constrain the model parameter by tuning against the data. While our model follows proper simplification of theory, the temperature and microstructural dependence are determined exclusively by the model parameter in a rheological formulation with the strain rate, and much scatter still exists. Within the selected range of compaction rates and densities, our results indicate that a 1-D two-phase model with a continuum framework alone does not likely capture important processes involved in the compaction process.

Contextual control demands determine whether stability and flexibility trade off against each other.

Cognitive stability, the ability to focus on a current task, and cognitive flexibility, the ability to switch between different tasks, are traditionally conceptualized as opposing end-points on a one-dimensional continuum. This assumption obligates a stability-flexibility trade-off - greater stability equates to less flexibility, and vice versa. In contrast, a recent cued task-switching study suggested that stability and flexibility can be regulated independently, evoking a two-dimensional perspective where trade-offs are optional (Geddert & Egner, Journal of Experimental Psychology: General, 151, 3009-3027, 2022). This raises the question of under what circumstances trade-offs occur. We here tested the hypothesis that trade-offs are guided by cost-of-control considerations whereby stability and flexibility trade off in contexts that selectively promote stability or flexibility, but not when neither or both are promoted. This proposal was probed by analyzing whether a trial-level metric of a stability-flexibility trade-off, an interaction between task-rule congruency and task sequence, varied as a function of a broader block-level context that independently varied demands on stability or flexibility by manipulating the proportion of incongruent and switch trials, respectively. In Experiment 1, we reanalyzed data from Geddert and Egner, Journal of Experimental Psychology: General, 151, 3009-3027, (2022); Experiment 2 was a conceptual replication with a design tweak that controlled for potential confounds due to local trial history effects. The experiments produced robust evidence for independent stability and flexibility adaptation, and for a context-dependent expression of trial-level stability-flexibility trade-offs that generally conformed to the cost-of-control predictions. The current study thus documents that stability-flexibility trade-offs are not obligatory but arise in contexts where either stability or flexibility are selectively encouraged.

Computational modeling of coupled interactions of fluid membranes with embedded filaments

In this work, we present a computational formulation based on continuum mechanics to study the interaction of fluid membranes embedded with semiflexible filaments. This is motivated by systems in membrane biology, such as cytoskeletal networks and protein filaments aiding the cell fission process. We model the membrane as a fluid shell via the Helfrich–Canham energy and the filament as a one-dimensional Cosserat continuum. We assume that the filament is tethered to the surface of the membrane while being allowed to freely float on the surface. The fluidity of the membrane and the novel filament-membrane coupling gives rise to unique computational challenges, which are addressed in this work. Finally, we present validation results and apply the formulation to several problems inspired by cellular biology.

On a Generalized Wave Equation with Fractional Dissipation in Non-Local Elasticity

We analyze wave equation for spatially one-dimensional continuum with constitutive equation of non-local type. The deformation is described by a specially selected strain measure with general fractional derivative of the Riesz type. The form of constitutive equation is assumed to be in strain-driven type, often used in nano-mechanics. The resulting equations are solved in the space of tempered distributions by using the Fourier and Laplace transforms. The properties of the solution are examined and compared with the classical case.

Ramified continua as global attractors of C 1-smooth self-maps of a cylinder close to skew products

In the paper we consider -smooth self-maps of a cylinder close to -smooth skew products (and satisfying some additional conditions). We study such geometric property of the maps, as existence of -smooth invariant local lamination, and apply this geometric property to the proof of the geometric integrability of maps under consideration. Using obtained results we construct the example of the family of -smooth maps close to skew products so that each map from this family admits the global attractor, which is a one-dimensional ramified continuum with a complicated topological structure. The global attractor of every map from the family under consideration consists of arcs of two types. On the unique circle (which is the arc of first type) the map is mixing; on arcs of second type of different lengths homeomorphic to a closed interval (the family of such arcs has continuum cardinality) the map is not mixing. The topological structure of the global attractor and dynamical properties of trajectories on the attractor lead to the property of dense intermittency (in the complement to the attractor) of attraction sets of different ω-limit sets, the union of which coincides with the global attractor.

Discrete defect plasticity and implications for dissipation

Inelastic deformation of solids is almost always, if not always, associated with the evolution of discrete defects such as dislocations, vacancies, twins and shear transformation zones. The focus here is on discrete defects that can be modeled appropriately by continuum mechanics, but where the discreteness of the carriers of plastic deformation plays a significant role. The formulations are restricted to small deformation kinematics and the defects considered, dislocations and discrete shear transformation zones (STZs), are described by their linear elastic fields. In discrete defect plasticity both the stress–strain response and the partitioning between defect energy storage and defect dissipation are outcomes of an initial/boundary value problem solution. For such defects an explicit expression for the dissipation rate is presented and, because there is a length scale associated with the discrete defects, the stress–strain response and the evolution of the dissipation rate can be size dependent. Discrete dislocation plasticity modeling results are reviewed that illustrate the implications of defect dissipation evolution for friction, fracture, fatigue crack growth and thermal softening. Examples are also given of the consequences in constrained shear of three modes of the evolution of discrete defects for the size dependence of the stress–strain response and the associated dissipation. For a purely mechanical formulation the Clausius–Duhem inequality specializes to the requirement that the dissipation rate is non-negative. Requiring a non-negative dissipation rate for all points of a body and for all time imposes restrictions on kinetic relations for the evolution of discrete defects. In statistical mechanics, the Clausius–Duhem inequality can be violated for sufficiently small regions for a sufficiently short time. Explicit kinetic relations for discrete dislocation plasticity dissipation and for discrete STZ plasticity dissipation identify conditions that can lead to a negative dissipation rate. In continuum mechanics, at least in some circumstances, satisfaction of the Clausius–Duhem inequality can be regarded as a stability requirement. Nevertheless, simple one-dimensional continuum calculations illustrate that there can be a negative dissipation rate over a small region and for a short time period with overall stability maintained. Implications for discrete defect plasticity modeling are briefly discussed.

Analytic theory of coupled-cavity traveling wave tubes

Coupled-cavity traveling wave tube (CCTWT) is a high power microwave vacuum electronic device used to amplify radio frequency signals. CCTWTs have numerous applications, including radar, radio navigation, space communication, television, radio repeaters, and charged particle accelerators. Microwave-generating interactions in CCTWTs take place mostly in coupled resonant cavities positioned periodically along the electron beam axis. Operational features of a CCTWT, particularly the amplification mechanism, are similar to those of a multicavity klystron. We advance here a Lagrangian field theory of CCTWTs with the space being represented by one-dimensional continuum. The theory integrates into it the space-charge effects, including the so-called debunching (electron-to-electron repulsion). The corresponding Euler–Lagrange field equations are ordinary differential equations with coefficients varying periodically in the space. Utilizing the system periodicity, we develop instrumental features of the Floquet theory, including the monodromy matrix and its Floquet multipliers. We use them to derive closed form expressions for a number of physically significant quantities. Those include, in particular, dispersion relations and the frequency dependent gain foundational to the RF signal amplification. Serpentine (folded, corrugated) traveling wave tubes are very similar to CCTWTs, and our theory applies to them also.

Vertical Stacking Statistics of Multi-facies Object-Based Models

Equations describing facies proportions and amalgamation ratios are derived for randomly placed objects belonging to two or three foreground facies embedded in a background facies, as a function of the volume fractions and object thicknesses of independent facies models combined in a stratigraphically meaningful order. The equations are validated using one-dimensional continuum models. Evaluation of the equations reveals a simple relationship between an effective facies proportion and an effective amalgamation ratio, both measured as a function only of the facies in question and the background facies. This relationship provides a firm analytical basis for applying the compression algorithm to multi-facies object-based models. A set of two-dimensional cross-sectional models illustrates the approach, which allows models to be generated with realistic object stacking characteristics defined independently for each facies in a multi-facies object-based model.

CROSS-SCALE MODELING OF LIQUID FLOWS IN HUMAN BODIES

Multiple modeling approaches to liquid flows in human bodies are reviewed. They span from the macroscale size > 1 mm to the nanoscale size only on the 1 nm scale. They can respectively model the flows of the blood in large arteries, small arteries, arterioles, and capillaries; and the flows of water through the nanopores of the capillary wall, the cellular membrane, and the cellular connexon. They are respectively the macroscale continuum flow modeling, the mesoscale one-dimensional continuum flow modeling, the microscale dissipative particle dynamics method, the macro-nano multiscale flow modeling, and the nanoscale flow modeling. Integrating these approaches together can give comprehensive simulations of liquid flows in human bodies.

Three-Electron Dynamics of the Interparticle Coulombic Decay in Doubly Excited Clusters with One-Dimensional Continuum Confinement.

A detailed analysis of the electronic structure and decay dynamics in a symmetric system with three electrons in three linearly aligned binding sites representing quantum dots (QDs) is given. The two outer A QDs are two-level potentials and can act as (virtual) photon emitters, whereas the central B QD can be ionized from its one level into a continuum confined on the QD axis upon absorbing virtual photons in the inter-Coulombic decay (ICD) process. Two scenarios in such an ABA array are explored. One ICD process is from a singly excited resonance state, whose decay releasing one virtual photon we find superimposed with resonance energy transfer among both A QDs. Moreover, the decay-process manifold for a doubly excited (DE) resonance is explored, in which collective ICD among all three sites and excited ICD among the outer QDs engage. Rates for all processes are found to be extremely low, although ICD rates with two neighbors are predicted to double compared to ICD among two sites only. The slowing is caused by Coulomb barriers imposed from ground or excited state electrons in the A sites. Outliers occur on the one hand at short distances, where the charge transfer among QDs mixes the possible decay pathways. On the other hand, we discovered a shape resonance-enhanced DE-ICD pathway, in which an excited and localized B* shape resonance state forms, which is able to decay quickly into the final ICD continuum.

Emergence of Spinons in Layered Trimer Iridate Ba_{4}Ir_{3}O_{10}.

Spinons are well known as the elementary excitations of one-dimensional antiferromagnetic chains, but means to realize spinons in higher dimensions is the subject of intense research. Here, we use resonant x-ray scattering to study the layered trimer iridate Ba_{4}Ir_{3}O_{10}, which shows no magnetic order down to 0.2K. An emergent one-dimensional spinon continuum is observed that can be well described by XXZ spin-1/2 chains with a magnetic exchange of ∼55 meV and a small Ising-like anisotropy. With 2% isovalent Sr doping, magnetic order appears below T_{N}=130 K along with sharper excitations in (Ba_{1-x}Sr_{x})_{4}Ir_{3}O_{10}. Combining our data with exact diagonalization calculations, we find that the frustrated intratrimer interactions effectively reduce the system into decoupled spin chains, the subtle balance of which can be easily tipped by perturbations such as chemical doping. Our results put Ba_{4}Ir_{3}O_{10} between the one-dimensional chain and two-dimensional quantum spin liquid scenarios, illustrating a new way to suppress magnetic order and realize fractional spinons.

A beam model for duoskelion structures derived by asymptotic homogenization and its application to axial loading problems

Duoskelion structures have been recently introduced by Barchiesi et al. (2021) as a proof-of-concept motif for a new class of metamaterials. The properties of these periodic beam-like chiral structural elements have been investigated, up to now, by means of a discrete model formulation whose predictions are obtained by numerical methods. In this paper we select a specific scaling law for micro stiffnesses aimed at deriving, via asymptotic homogenization, an internally-constrained Cosserat one-dimensional planar continuum model as the limit of a duoskelion structure. We analyze qualitatively and quantitatively the family of equilibrium configurations of the homogenized continuum when subjected to axial loading and compare the results of the analysis with those obtained by means of the discrete model formulation.

Universal properties of boundary and interface charges in multichannel one-dimensional continuum models

We generalize our recent results for the boundary and interface charges in one-dimensional single-channel continuum [Phys. Rev. B 104, 155409 (2021)] and multichannel tight-binding [Phys. Rev. B 104, 125447 (2021)] models to the realm of the multichannel continuum systems. Using the technique of boundary Green's functions, we give a rigorous proof that the change in boundary charge upon the shift of the system towards the boundary by the distance $x_{\varphi}\in[0, L]$ is given by a perfectly linear function of $x_{\varphi}$ plus an integer-valued topological invariant $I$ -- the so called boundary invariant. For systems with weak potential amplitudes, we additionally develop Green's function-based low-energy theory, allowing one to analytically access the physics of multichannel continuum systems in the low-energy approximation.

Распространение плоских волн в упругой полуплоскости, усиленной по своей границе накладкой бесконечной длины

The dynamic contact problem of the propagation of plane elastic waves incident from infinity onto the boundary of an elastic half-plane reinforced by an infinite plate of small thickness is studied. Questions related to the dynamic mutual influence of an elastic half-plane with an stringer attached to its boundary are studied. As a physical model for the stringer, a one-dimensional elastic continuum model is taken. The solution of the problem is constructed by direct application of the Fourier transform. Analytical expressions are obtained that represent the distribution of wave components in all parts of the half-plane.

Vibration of locally cracked pre-loaded parabolic arches

We study linear dynamics of an initially parabolic arch deformed by a uniform ‘dead’ load. The arch is seen as a fully deformable one-dimensional continuum with rigid cross-sections, one of which suffers from a small local crack at its boundary. The crack is simulated by springs, the stiffnesses of which are evaluated via stress intensity factors. By two first-order perturbations we investigate a non-trivial equilibrium adjacent to the reference configuration and small vibration superposed on it. The modulation of the initial load on the natural angular frequencies and its consequences on damage detection is described and commented. It turns out that neglecting the initial load, recalling for actual ‘dead’ structural actions, can be misleading in damage identification, while its inclusion leads to better results.

Analytic theory of multicavity klystrons

Multicavity Klystron (MCK) is a high power microwave vacuum electronic device used to amplify radio frequency (RF) signals. MCKs have numerous applications, including radar, radio navigation, space communication, television, radio repeaters, and charged particle accelerators. The microwave-generating interactions in klystrons take place mostly in coupled resonant cavities positioned periodically along the electron beam axis. Importantly, there is no electromagnetic coupling between cavities. The cavities are coupled only by the flow of bunched electrons drifting from one cavity to the next. We advance here a Lagrangian field theory of MCKs with the space being represented by a one-dimensional continuum. The theory integrates into it the space-charge effects, including the so-called debunching (electron-to-electron repulsion). The corresponding Euler–Lagrange equations are ordinary differential equations with coefficients varying periodically in the space. Utilizing the system periodicity, we develop the instrumental features of the Floquet theory, including the monodromy matrix and its Floquet multipliers. We use them to derive closed form expressions for a number of physically significant quantities. Those include, in particular, the dispersion relations and the frequency dependent gain foundational to the RF signal amplification. We assume that MCKs operate in the voltage amplification mode associated with the maximal gain.

One-Dimensional Thermomechanical Model for Additive Manufacturing Using Laser-Based Powder Bed Fusion

We investigate the thermomechanical behavior of 3D printing of metals in the laser-based powder bed fusion (L-PBF) process, also known as selective laser melting (SLM). Heat transport away from the printed object is a limiting factor. We construct a one-dimensional thermoviscoelastic continuum model for the case where a thin fin is being printed at a constant velocity. We use a coordinate frame that moves with the printing laser, and apply an Eulerian perspective to the moving solid. We consider a steady state similar to those used in the analysis of production processes in the process industry, in the field of research known as axially moving materials. By a dimensional analysis, we obtain the nondimensional parameters that govern the fundamental physics of the modeled process. We then obtain a parametric analytical solution, and as an example, illustrate it using material parameters for 316L steel. The nondimensional parameterization has applications in real-time control of the L-PBF process. The novelty of the model is in the use of an approach based on the theory of axially moving materials, which yields a new perspective on modeling of the 3D printing process. Furthermore, the analytical solution is easy to implement, and allows fast exploration of the parameter space.

One-dimensional continuum finite-difference model of the dynamics of a dusty medium in aerodynamic, electric and gravitational fields

This paper presents a mathematical model of the dynamics of a gas suspension under the action of an aerodynamic field, an electric field, and a gravitational field. The continuum model is used to describe the dynamics of the disperse component. The intercomponent momentum exchange included the aerodynamic drag force, the dynamic Archimedes force, and the added mass force. The Coulomb force acting on dispersed particles was taken into account. The model assumes the solution of the equations of conservation of “average density”, momentum and energy for the dispersed phase. The electric field was described by the Poisson equation. The equations of the mathematical model were supplemented with boundary conditions. The equations of mechanics were integrated by explicit finite-difference method of McCormack.