Continuum Mechanics Point Of View Research Articles

Souriau’s general covariant formulation of Relativistic Hyperelasticity revisited

We present and modernize Souriau’s 1958 geometric framework for Relativistic continuous media, and enlighten the necessary and the ad hoc modeling choices made since. We focus as much as possible on the Continuum Mechanics point of view and describe the general covariant formulation of Hyperelasticity in (Variational) General Relativity. At this level of generality, different relativistic strain and stress tensors are formulated and discussed. Then, the choice of an observer, through the introduction of a spacetime structure allows us to make deeper insights into the foundations of Continuum Mechanics. We extend Souriau’s calculations, initially performed in the flat Minkowski spacetime, to the Schwarzschild spacetime accounting for gravitation. Finally, we recover the Classical Galilean Hyperelasticity with gravity, as the Newton–Cartan infinite light speed limit of this formulation.

Lubrication\u2013Contact Interface Conditions and Novel Mixed/Boundary Lubrication Modeling Methodology

Under severe conditions, solid contacts take place even when parts are lubricated. Precise mathematical conditions are needed to describe the interior interface between fluid lubrication and solid-contact zones. In order to distinguish the conditions for this interface from conventional lubrication boundary conditions, they are named lubrication–contact interface conditions (LCICs). In this work, mathematical LCICs are derived with local flow continuity from the continuum mechanics point of view and pressure inequality across the interface. Numerical implementations are developed and tested with problems having simple geometries and configurations, and they are integrated into a new mixed/boundary elastohydrodynamic lubrication solver that uses a new method to determine solid-contact pressures. This solver is capable of capturing film thickness and pressure behaviors involving solid contacts.

Statistical foundation of EMMS-based two-fluid models for heterogeneous gas-solid flow

In response to the general existence of mesoscale structures in gas–solid fluidized beds, two versions of EMMS-based two-Fluid Model (EFM) (Hong et al., 2012; Wang et al., 2012a) have been proposed simultaneously from the viewpoint of continuum mechanics via different options of interpenetrating continua. A statistical foundation is however not available yet. To this end, an attempt was made to lay a unified statistical foundation of EFMs: four Boltzmann equations were used to describe respectively the kinetics of particles and gas molecules in dilute and dense phases, the governing equations of EFMs and their corresponding constitutive relations were derived theoretically. It was shown that (i) all governing equations and constitutive laws are structure-dependent; (ii) the solid and gas stresses include the kinetic stress, the phase-internal and interphase collisional stresses and the pseudo-Reynolds stress; and (iii) the interphase mass, momentum and energy transfer between the dilute phase and the dense phase can be quantified by assuming that collisions between the particles or gases from the dilute phase and dense phase have a certain probability to result in the mass transfer and vice versa.

Checkerboard free topology optimization for compliance minimization applying the finite-volume theory

The purpose of this research is to demonstrate that the numerical issue associated with the checkerboard patterns can be entirely controlled by applying the finite-volume theory. Usually, in the gradient-based topology optimization algorithms, it is common to occur some problems associated with numerical instabilities, such as checkerboard pattern, mesh dependence, and local minima. The occurrence of checkerboard subdomains is directly related to the assumptions of the finite-element method, as the satisfaction of equilibrium equations and continuity conditions through the element nodes. Differently, the finite-volume theory satisfies the equilibrium equations at the subvolume level, and the continuity conditions are established through the subvolumes adjacent interfaces, as expected from the Continuum Mechanics point of view. Thus, a topology optimization approach based on the standard (or zeroth order) finite-volume theory for linear elasticity is proposed, resulting in a numerically efficient computational modeling, able to obtain checkerboard free topologies in the absence of filtering techniques. A sensitivity filtering technique is employed to solve issues associated with mesh dependence and length scale in the finite-volume approach, providing optimized topologies with desired manufacturing features.

Checkerboard-free topology optimization for compliance minimization of continuum elastic structures based on the generalized finite-volume theory

Topology optimization is a well-suited method to establish the best material distribution inside an analysis domain. It is common to observe some numerical instabilities in its gradient-based version, such as the checkerboard pattern, mesh dependence, and local minima. This research demonstrates the generalized finite-volume theory's checkerboard-free property by performing topology optimization algorithms without filtering techniques. The formation of checkerboard regions is associated with the finite element method's displacement field assumptions, where the equilibrium and continuity conditions are satisfied through the element nodes. On the other hand, the generalized finite-volume theory satisfies the continuity conditions between common faces of adjacent subvolumes, which is more likely from the continuum mechanics point of view. Also, the topology optimization algorithms based on the generalized finite-volume theory are performed using a mesh independent filter that regularizes the subvolume sensitivities, providing optimum topologies that avoid the mesh dependence and length scale issues.

Strongly anisotropic surface elasticity and antiplane surface waves.

Within the new model of surface elasticity, the propagation of anti-plane surface waves is discussed. For the proposed model, the surface strain energy depends on surface stretching and on changing of curvature along a preferred direction. From the continuum mechanics point of view, the model describes finite deformations of an elastic solid with an elastic membrane attached on its boundary reinforced by a family of aligned elastic long flexible beams. Physically, the model was motivated by deformations of surface coatings consisting of aligned bar-like elements as in the case of hyperbolic metasurfaces. Using the least action variational principle, we derive the dynamic boundary conditions. The linearized boundary-value problem is also presented. In order to demonstrate the peculiarities of the problem, the dispersion relations for surface anti-plane waves are analysed. We have shown that the bending stiffness changes essentially the dispersion relation and conditions of anti-plane surface wave propagation. This article is part of the theme issue 'Modelling of dynamic phenomena and localization in structured media (part 2)'.

Si Nanopowder Based Anode Material for the Lithium Ion Battery Cell

We consider a silicon nanopowder based anode for a lithium ion battery cell. We present the design of the battery cell ready for in situ Raman and X-ray experiments and a technical procedure for the cell manufacturing. From the continuum mechanics point of view, this type of anode can be represented by a spherical nanoparticle surrounded by viscoelastic matrix. During the charging process this nanoparticle undergoes a chemical reaction. Based on the chemical affinity concept we describe how the mechanical properties of the matrix material influence the kinetics of the charging process. We study spherically symmetry problems numerically for different sets of matrix material parameters and show their influence on the reaction front kinetics.

The directional motion of nano-objects induced by an inhomogeneous strain field

Recent molecular dynamics simulations have shown that an inhomogeneous strain field in a substrate could drive nano-objects to move on its surface. A theoretical model is established in this paper in order to find the mechanical mechanism from the continuum mechanics point of view, but with the help of atomic interaction potential. The substrate is represented by graphene, and the above nano-object is represented by a graphene flake or a carbon nanotube. Three typically inhomogeneous strain fields are considered, including linear, power and exponential. It was found that a driving force and a couple moment can be induced on the flat flake and the circular nanotube, respectively. Furthermore, both the driving force and the couple moment increase with the increase in strain gradient in the substrate. The size effect of nano-objects and boundary effect on the movement are also investigated in order to explain some interesting results found in the existing numerical simulations. The present theoretical study cannot only reveal the source of the driving mechanism for the movement induced by an inhomogeneous strain field but also give insights for the design of nano-transport devices.

Generalized Boltzmann kinetic theory for EMMS-based two-fluid model

It has long been recognized that the solid particles in circulating fluidized bed risers are distributed heterogeneously in the form of clusters. In response to this fundamental phenomenon, an EMMS-based two-fluid model has been developed recently from the viewpoint of continuum mechanics, however, its microscopic foundation remains unknown. In this study, the statistical mechanics foundation of EMMS-based two-fluid model was presented using generalized Boltzmann kinetic theory. With respect to the gas phase, a new method was developed by considering the fluctuations at different scales simultaneously, with which we can for the first time derive the correct governing equations of gas phase via kinetic theory, in the sense that both the molecular stress and the Reynolds (or pseudo-Reynolds) stress can be obtained simultaneously, whereas all previous kinetic theory analyses failed to predict the appearance of Reynolds (or pseudo-Reynolds) stress in the momentum conservation equation of gas phase due to the assumption of uniform structure, although it is physically always existent no matter how small the Reynolds number is. In case of particle phase, the generalized Boltzmann equation considering the spatio-temporal variation of the volume, density and velocity of clusters was firstly derived, a set of macroscopic transport equations was then derived in different phase spaces. It was shown that the governing equations of dense phase in the EMMS-based two-fluid model derived from continuum mechanics viewpoint corresponds to the macroscopic transport equations at (r,t) space. Therefore, present study launches a solid microscopic foundation of EMMS-based two-fluid model. Finally, CFD simulations have been carried out to validate EMMS-based two-fluid model and to study the effect of gas phase pseudo-turbulence.

On the equations of motion for curved slender beams using tubular coordinates

The equations of motion for a beam are derived from the three-dimensional continuum mechanical point of view. The kinematics involve bending, torsion and shearing as well as cross-sectional displacements. The transplacement of the beam from an arbitrary curved reference placement is considered. For slender beams so called tubular coordinates may be used as global coordinates in the reference placement. An explicit geometrical characterization of slender beams is given. The equations of motion for the slender beam are derived and some consequences of the power theorem are presented. Using the principle of virtual power the classical local beam equations are obtained along with equations representing the local balance of bi-momentum. Constitutive assumptions are introduced where the dependence of stress on the natural deformation measures for the beam is obtained by assuming that the beam consists of a St Venant-Kirchhoff elastic material. Simplified stress-strain relations may be obtained using so called torsion free coordinates.

A new look on wetting models: continuum analysis

In this study, we considered the wetting phenomenon on a general substrate from a new viewpoint of continuum mechanics. The analyses first show how the Wenzel and the Cassie models deviate the practical results in some special substrates, and then elucidate the mechanism of the triple contact line (TCL) moving. Based upon variational theory of the total free functional dealing with the movable boundary condition, we show that the macroscopic contact angle (MCA) expression is the corresponding transversality condition. It manifests that the MCA depends only on the chemical and geometric property at the TCL, and is not affected by the gravity of the droplet and the contact area beneath the liquid. Our continuum model also shows the exploration of the pinning effect on a sharp wedge or the interface between two different phases. This investigation will help designing super-hydrophobic materials for novel micro-fluidic devices.

On the equations of motion in constrained multibody dynamics

The equations of motion for a constrained multibody system are derived from a continuum mechanical point of view. This will allow for the presence of rigid, as well as deformable, parts in the multibody system together with kinematical constraints. The approach leads to the classical Lagrange–d’Alembert equations of motion under constraint conditions. The generalized forces appearing in the equations of motion are given as expressions involving contact, internal and body forces in the sense of continuum mechanics. A detailed analysis of physical constraint conditions and their implication for the equations of motion is presented. A precise distinction is made between constraints on the motion on the one hand and resistance to the motion on the other. Transformation properties – covariance and invariance under changes of configuration coordinates – are elucidated. The elimination and calculation of the so-called Lagrangian multipliers is discussed and some useful reformulations of the equations of motion are presented. Finally a Power theorem for the constrained multibody system is proved.

On local indentation and impact compliance of isotropic auxetic materials from the continuum mechanics viewpoint

Implications of the negative Poisson ratio with regard to the quasi-static and dynamic indentation compliances for isotropic homogeneous materials are considered. It has been shown that analogously to some solutions for three-dimensional stress or displacement fields concentrations, the indentation, vibrational, and impact compliances are significantly affected when Poisson’s ratio of the material assumes negative values. It was observed that the friction at the contact interface between the indenter and the surface of an elastic medium strengthens the negative Poisson ratio effect. In the Hertzian contact problem, it was found out that the location of the maximum shear stress approaches the contact interface as Poisson’s ratio is made sufficiently negative. In view of this fact, the local indentation of a coated elastic half-space, which is reinforced with an elastic plate, has been considered. It was shown that the reinforcement preserves the negative Poisson ratio effect, and the perfect bonding of the plate to the surface of an auxetic half-space significantly strengthens it. This theoretical work be considered as a first step towards an indentation and impact analysis of real auxetic materials.

Modeling microbending of thin films through discrete dislocation dynamics, continuum dislocation theory, and gradient plasticity

Abstract

On the equations of motion in multibody dynamics

The equations of motion for a multibody system are derived from a continuum mechanical point of view. This will allow for the presence of rigid, as well as deformable, parts in the multibody system. The approach, using the principle of virtual power, leads to the classical Lagrange equations of motion. The generalized forces appearing in the equations are given as expressions involving contact, internal and body forces in the sense of continuum mechanics. A detailed analysis of the interaction between parts and their implication for the equations of motion is presented. Transformation properties, covariance and invariance under changes of configuration coordinates, are elucidated and a power theorem for the multibody system is proved. The equivalence between the standard balance equations for momentum and moment of momentum and the principle of virtual power is demonstrated.

Numerical simulation of coupled heat and mass transfer in wood dried at high temperature

The mutual effect between heat and mass transfer is investigated for wood dried at high temperature. A numerical model of coupled heat and mass transfer under the effect of the pressure gradient is presented. Based on the macroscopic viewpoint of continuum mechanics, the mathematical model with three independent variables (temperature, moisture content and gas pressure) is constructed. Mass transfer in the pores involves a diffusional flow driven by the gradient of moisture content, convectional flow of gaseous mixture governed by the gradient of gas pressure, the Soret effect and phase change of water. Energy gain or loss due to phase change of water is taken as the heat source. Numerical methods, the finite element method and the finite difference method are used to discretize the spatial and time dimension, respectively. A direct iteration method to solve the nonlinear problem without direct evaluation of the tangential matrix is introduced. The local convergence condition based on the contraction–mapping principle is discussed. The mathematical model is applied to a 3-D wood board dried at high temperature with the Neumann boundary conditions for both temperature and moisture content, and the Dirichlet boundary conditions for gas pressure.

Mechanical properties of DNA biofilms adsorbed on microcantilevers in label-free biodetections

Biomolecule adsorption is a fundamental process in the design of biosensors. Mechanical/electrical/thermal properties of biofilms have great influences on biodetection signals. The double-stranded DNA (dsDNA) biofilm adhered on microcantilever is treated as a bending beam with a macroscopic elastic modulus in the viewpoint of continuum mechanics. Accounting for hydration force, electrostatic repulsion and conformational entropy, moment-angle diagrams of dsDNA biofilm in pure bending state are depicted with the help of the energy conservation law and a mesoscopic liquid crystal theory presented by Strey et al. An analytical model is provided to predict macroscopic elastic modulus of dsDNA biofilm as a function of nanoscopic properties of dsDNA, packing density, buffer salt concentration and etc. The parameters for microcantilever-DNA system are obtained by curve fitting with Stachowiak’s experimental data based on a modified Stoney’s formula. Elastic modulus grows exponentially with the enhancement of packaging density, but diminishes with the increase of buffer salt concentration, and its order is about 1 ∼ 10 MPa. Conformational entropy is one of predominant factors considered in near-surface system whether in high or low salt consternation.

Transformation Plasticity and the Effect on Quenching Process Simulation

Mechanism of transformation plasticity (TP) is discussed from continuum mechanics viewpoint, and derivation of TP law from the unified thermo-mechanical and transformation plasticity constitutive equation. Result of identified TP coefficient for a chromium steel (JIS SCr420) by use of multi-functional testing machine is introduced as one of the material data together with other data to the simulation of a quenching process by use of newly developed code COSMAP. The simulated distribution of temperature, phases and stress/distortion are compared with the experimentally measured values to verify the accuracy.

The mechanics of spall fracture in rock and concrete

Spalling is a wave-induced dynamic fracture phenomenon. The waves can be either one: elastic, elasto-plastic, or shock waves. From a continuum mechanics point of view, fracture mechanics and wave propagation form the main ingredients in the formation of spalls. Recently, however, micro-structural effects have become important in the initial stages of spall formation in a variety of engineering materials ranging from metals to rock and concrete. From a structural geology point of view, the rock mass cannot be modelled as a continuum. In this case, a discontinuum approach has to be taken where the individual features of the rock mass such as joints and faults need to be taken into account. From an application point of view, spallation is important where rapid loading by explosives, impact, or energy deposition, occurs. The range of applications stretches from blasting in mining engineering to damage prevention to structures under explosive excitation. This contribution offers a multi-faceted and multi-disci...

Viscoelastic (Non‐Fickian) Diffusion

AbstractThis review paper deals with mass transport in macromolecular materials. The study of mass transport through polymeric composite materials for example has applications in a variety of areas such as coatings, packaging and gas separations, to name just a few. Here we discuss several models for diffusion as well as several experimental techniques. In particular, we discuss models for case II diffusion, from a continuum mechanics point of view as well as via a mesoscopic theory. Variables such as temperature, molecular structure and mechanical deformation, affecting mass transport are also discussed.