Continuum Dynamics on Manifolds: Application to Elasticity of Residually-Stressed Bodies

This paper is concerned with the dynamics of continua on differentiable manifolds. We present a covariant derivation of the equations of motion, viewing motion as a curve in an infinite-dimensional Banach manifold of embeddings of a body manifold in a space manifold. Our main application is the motion of residually-stressed elastic bodies, where the residual stresses result from a geometric incompatibility between body and space manifolds. We then study a particular example of elastic vibrations of a two-dimensional curved annulus embedded in a sphere.     

Functional Concepts in Continuum Mechanics

This paper is an attempt to give a concise presentation of the main concepts of continuum mechanics and to show their articulation. Functional definitions have been favoured.The first section is devoted to a review of continuum mechanics.The second section deals with the mechanics of materials.Constitutive equations of the material are given first by equations of state and then by complementary equations written in order to fulfil the fundamental inequality concerning the production of entropy and the physical properties of the material (viscosity, plasticity, damage etc.).Section 3 gives the Lagrangian and the Hamiltonian formulations for a moving body. Section 4 is devoted to the motion of surfaces through which discontinuities appear, to show briefly two examples of application of the previous concepts. One can easily define the source of intrinsic inhomogeneity, of heat, of irreversible entropy on a surface of phase transition and also for a shockwave.SommarioIn questo lavoro si cerca di dare una presentazione concisa dei concetti fondamentali della meccanica dei continui e di mostrare le loro articolazioni,con particolare rilievo alle definizioni funzionali. La prima sezione è dedicata ad un esame generale introduttivo. La seconda alla meccanica dei materiali, le cui equazioni costitutive sono date come equazioni di stato e come equazioni complementari scritte al fine di soddisfare le diseguaglianze concernenti la produzionedi entropia e le proprietà fisiche dei materiali (qualiviscosità, plasticità, guasti...).La terza sezione dà le formulazioni Lagrangiana ed Hamiltoniana per il moto di un corpo.La quarta sezione è dedicata al moto di superfici di discontinuità per mostrare brevemente due esempi di applicazione dei concetti precedenti.Viene quindi definita la sorgente di disomogeneità intrinseca, di calore, di entropia irreversibile su di una superficie, di transizione di fase ed anche di un' onda d'urto.     

Balanced Powers in Continuum Mechanics

An approach to weak balance laws in Continuum Mechanics is presented, involving densities with only divergence measure, which relies on the balance of power. An equivalence theorem between Cauchy powers and Cauchy fluxes is proved. As an application of this method, the construction of the stress tensor when the body is an orientable differential manifold is achieved under very general assumptions.     

Constitutive Equations in Continuum Mechanics

One of the main research areas in continuum mechanics is constitutive equations. There is no generally accepted viewpoint on or method for this research. Several concepts have appeared, but further studies are still necessary. The paper elucidates some problems and viewpoints of such research     

Continuum modelling of polyelectrolytes in solution

A phenomenological model for solutions of polyelectrolytes accounting for electromechanical interactions is proposed within the framework of continuum thermodynamics. The modelling involves the conformation of macromolecules in the form of a tensorial internal variable and the electric polarization of the solution. The paper aims to demonstrate the possible competition between flow and electricity effects on rigorous phenomenological grounds. Three kinds of dissipative mechanisms are accounted for: viscosity of the solution, electric relaxation and the relaxation of the conformation of macromolecules. Anisotropic effects induced by the applied electric field and flow-induced polarization through changes in conformation are exhibited. These effects are small. By way of application, the problem of the orientation and conformation taken by macromolecules of polyelectrolytes under the combined influence of a simple shear flow and a longitudinal or orthogonal electric bias field is treated in detail and illustrated graphically. Some optical properties are mentioned.     

On Molecular Modelling and Continuum Concepts

Continuum concepts and field values are related to local (scale-dependent) spacetime atomistic averages. Spatial averaging is effected by both weighting function and cellular localisation procedures, and resulting forms of linear momentum balance are compared. The former yields a local balance directly, with several candidate interaction stress fields. The latter results in a global balance involving a traction field expressible in terms of an interaction stress tensor field. In both approaches the Cauchy stress incorporates distinct interaction and thermokinetic contributions. Inter alia are addressed physically-distinguished choices of weighting function; the scale-dependence of the boundary of a body, its motion and material points thereof; physical interpretations of various candidate interaction stress tensors; temporal averaging and material systems whose content changes with time; and the possible relevance of the latter to investigating a molecular context for configurational forces.     

Surface effects in non-uniform nanobeams: Continuum vs. atomistic modeling

Nanobeams are expected to be one of the key structural elements in nanotechnology. Contrary to macroscopic structures, surface effects can strongly influence the stress and deformation properties of nano-devices. In addition, at such small scales, material non-uniformity becomes significant and must be considered.In this work, a continuum model for nanobeams, including both surface effects and material heterogeneity is developed. The model treats the surfaces as separate material layers with finite thickness. The continuum solution is compared with atomistic simulations, from which the effective bulk and surface properties are computed independently. A special case of self-deflection due to surface non-uniformities, which is important for design of nanosensors, is studied. A comparison between continuum and atomistic solutions reveals differences, which originate from local transition effects in the neighborhood of strong non-uniformities. This discrepancy is addressed by defining an effective length, found by correlating the beam deflections from both methods.     

Balance Laws and Weak Boundary Conditions in Continuum Mechanics

A weak formulation of the stress boundary conditions in Continuum Mechanics is proposed. This condition has the form of a balance law, allows also singular measure data and is consistent with the regular case. An application to the Flamant solution in linear elasticity is shown.     

Mesoscopic Simulation of Rarefied Flow in Narrow Channels and Porous Media

Rarefied gas flow in channels and computer-aided reconstructions of porous media is simulated using the direct simulation Monte Carlo (DSMC) method and a modified lattice-Boltzmann (LB) method that can account for rarefaction effects. An increase of two orders of magnitude was noted for the gas permeability as the Knudsen number increased from 0.1 to 10. It was found that incorporation of a Bosanquet expression for the viscosity in the dusty gas flux equations leads to the recovery of the well known Klinkenberg expression for the gas permeability, revealing an explicit relation of the, thus far empirical, permeability correction factor to the fluid and structure properties. The expression for the effective gas viscosity in the transition flow regime is also incorporated in the LB method, which is then validated against the DSMC method by comparing predictions for the velocity profiles in straight channels over the entire transition flow regime. The new LB method offers the additional advantages of simplicity in the code implementation and great savings in computational time and memory compared to the DSMC method. It is shown that the rough adjustment of a single parameter suffices to make the LB method suitable for the reliable prediction of the gas permeability in porous media over the entire transition flow regime.     

Stability of crystalline solids—I: Continuum and atomic lattice considerations

Many crystalline materials exhibit solid-to-solid martensitic phase transformations in response to certain changes in temperature or applied load. These martensitic transformations result from a change in the stability of the material's crystal structure. It is, therefore, desirable to have a detailed understanding of the possible modes through which a crystal structure may become unstable. The current work establishes the connections between three crystalline stability criteria: phonon-stability, homogenized-continuum-stability, and the presently introduced Cauchy-Born-stability criterion. Stability with respect to phonon perturbations, which probe all bounded perturbations of a uniformly deformed specimen under “hard-device” loading (i.e., all around displacement type boundary conditions) is hereby called “constrained material stability”. A more general “material stability” criterion, motivated by considering “soft” loading devices, is also introduced. This criterion considers, in addition to all bounded perturbations, all “quasi-uniform” perturbations (i.e., uniform deformations and internal atomic shifts) of a uniformly deformed specimen, and it is recommend as the relevant crystal stability criterion.     

From Discrete Visco-Elasticity to Continuum Rate-Independent Plasticity: Rigorous Results

We show that continuum models for ideal plasticity can be obtained as a rigorous mathematical limit starting from a discrete microscopic model describing a visco-elastic crystal lattice with quenched disorder. The constitutive structure changes as a result of two concurrent limiting procedures: the vanishing-viscosity limit and the discrete-to-continuum limit. In the course of these limits a non-convex elastic problem transforms into a convex elastic problem while the quadratic rate-dependent dissipation of visco-elastic lattice transforms into a singular rate-independent dissipation of an ideally plastic solid. In order to emphasize our ideas we employ in our proofs the simplest prototypical system mimicking the phenomenology of transformational plasticity in shape-memory alloys. The approach, however, is sufficiently general that it can be used for similar reductions in the cases of more general plasticity and damage models.     

碳纤维束-圆辊细观接触行为研究

为探索碳纤维束-圆辊的接触机制,本文中采用自制摩擦模拟试验装置,结合薄膜压痕法分析了纤维束接触部位的三维(3D)表面形貌,研究了预加张力和接触角度对碳纤维束-圆辊接触面积的影响规律. 结果表明:碳纤维束-圆辊接触面内的纤维束取向度是影响接触面积的关键因素,当接触角度从60°增加到170°时,纤维束取向度逐渐减小,碳纤维束-圆辊接触面积也逐渐减小,且接触角度为60°时的接触面积是170°时的22倍;当预加张力从0.19 N增加到1.47 N时,纤维束取向度逐渐增大,碳纤维束-圆辊接触面积逐渐增大,且预加张力为1.47 N时的接触面积是0.19 N时的2倍. 在Hertz接触理论的基础上,考虑纤维束取向度建立碳纤维束-圆辊接触面积的理论预测模型,通过试验数据验证了该模型能够准确预测碳纤维束-圆辊的接触面积.      

Continuum Mechanics Modeling and Simulation of Carbon Nanotubes

The understanding of the mechanics of atomistic systems greatly benefits from continuum mechanics. One appealing approach aims at deductively constructing continuum theories starting from models of the interatomic interactions. This viewpoint has become extremely popular with the quasicontinuum method. The application of these ideas to carbon nanotubes presents a peculiarity with respect to usual crystalline materials: their structure relies on a two-dimensional curved lattice. This renders the cornerstone of crystal elasticity, the Cauchy–Born rule, insufficient to describe the effect of curvature. We discuss the application of a theory which corrects this deficiency to the mechanics of carbon nanotubes (CNTs). We review recent developments of this theory, which include the study of the convergence characteristics of the proposed continuum models to the parent atomistic models, as well as large scale simulations based on this theory. The latter have unveiled the complex nonlinear elastic response of thick multiwalled carbon nanotubes (MWCNTs), with an anomalous elastic regime following an almost absent harmonic range.     

Continuum mechanical aspects of phase transformations in solids

Summary Some aspects of classical thermodynamics of phase transformations are discussed. Then typical solid state transformations as the displacive and diffusional transformation in metals are explained. A general formulation of the Gibbs free energy is presented including all energy terms required. Based on the classical nucleation, the triggering-off and the dissipation condition, various transformation conditions are formulated taking into account the elasto-plastic deformation of both phases. Transformation conditions presented in the literature over the last 40 years are reviewed and compared to the transformation conditions derived here. The transformation conditions for a spherical region growing under a certain volume change in an elasto-plastic matrix are studied as an example. The relevant analytical expressions are presented and discussed.

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Kontinuumsmechanische Aspekte von Phasenübergängen in Festkörpern

Übersicht Einige Aspekte der klassischen Thermodynamik von Phasenumwandlungen werden behandelt. In der Folge werden typische Festkörperumwandlungen wie die displazive und die diffusive Umwandlung in Metallen erörtert. In allgemeiner Formulierung wird die Gibbs Energie unter Berücksichtigung aller erforderlichen Energieterme hergeleitet Basierend auf der klassischen Keimentwicklungsbedingung, einer Formulierung über das Wachstum von Keimen und einem auf der Dissipationsleistung beruhenden Konzept werden verschiedene Umwandlungsbedingungen hergeleitet. Dabei wird in beiden Phasen elasto-plastisches Materialverhalten vorausgesetzt. Die hier formulierten Umwandlungsbedingungen werden mit einigen der in den letzten 40 Jahren publizierten Beziehungen verglichen und bewertet. In einem Beispiel werden diese Bedingungen für eine in einer elasto-plastischen Matrix wachsende Kugel bei einer umwandlungsbedingten Volumsdehnung angewandt. Analytische Ausdrücke werden präsentiert und näher erläutert.