Simulation continuum mechanics

As the name implies, continuum mechanics is predicated on the hypothesis that one may describe the properties and behaviour of physical systems entirely in terms of continuous functions of position and time, at least for a single pure component within a single bulk phase (gas, liquid, or solid). Continuum mechanics makes no reference to the fact that real materials are composed of atoms or molecules. Strictly speaking, rationalization of transport coefficients (like viscosity, thermal conductivity, and diffusivity) in terms of molecular behaviour lies within the realm of statistical mechanics and molecular simulations. Continuum mechanics begins to lose its validity when the characteristic length and time scales in a physical system become comparable to molecular scales. Continuum mechanics thus cannot describe the initial stages of SEI film formation composed of many simultaneous, discrete, discontinuous molecular events. 

In literature, some researchers regarded that the continuum mechanic ceases to be valid to describe the lubrication behavior when clearance decreases down to such a limit. Reasons cited for the inadequacy of continuum methods applied to the lubrication confined between two solid walls in relative motion are that the problem is so complex that any theoretical approach is doomed to failure, and that the film is so thin, being inherently of molecular scale, that modeling the material as a continuum ceases to be valid. Due to the molecular orientation, the lubricant has an underlying microstructure. They turned to molecular dynamic simulation for help, from which macroscopic flow equations are drawn. This is also validated through molecular dynamic simulation by Hu et al. [6,7] and Mark et al. [8]. To date, experimental research had "got a little too far forward on its skis however, theoretical approaches have not had such rosy prospects as the experimental ones have. Theoretical modeling of the lubrication features associated with TFL is then urgently necessary. 

This section provides an alternative measurement for a material parameter the one in the ensemble averaged sense to pave the way for usage of continuum theory from a hope that useful engineering predictions can be made. More details can be found in Ref. [15]. In fact, macroscopic flow equations developed from molecular dynamics simulations agree well with the continuum mechanics prediction (for instance. Ref. [16]). 

Finally, a relatively new area in the computer simulation of confined polymers is the simulation of nonequilibrium phenomena [72,79-87]. An example is the behavior of fluids undergoing shear flow, which is studied by moving the confining surfaces parallel to each other. There have been some controversies regarding the use of thermostats and other technical issues in the simulations. If only the walls are maintained at a constant temperature and the fluid is allowed to heat up under shear [79-82], the results from these simulations can be analyzed using continuum mechanics, and excellent results can be obtained for the transport properties from molecular simulations of confined liquids. This avenue of research is interesting and could prove to be important in the future. 

Sometimes the theoretical or computational approach to description of molecular structure, properties, and reactivity cannot be based on deterministic equations that can be solved by analytical or computational methods. The properties of a molecule or assembly of molecules may be known or describable only in a statistical sense. Molecules and assemblies of molecules exist in distributions of configuration, composition, momentum, and energy. Sometimes, this statistical character is best captured and studied by computer experiments molecular dynamics, Brownian dynamics, Stokesian dynamics, and Monte Carlo methods. Interaction potentials based on quantum mechanics, classical particle mechanics, continuum mechanics, or empiricism are specified and the evolution of the system is then followed in time by simulation of motions resulting from these direct... 

Abstract In this contribution, the coupled flow of liquids and gases in capillary thermoelastic porous materials is investigated by using a continuum mechanical model based on the Theory of Porous Media. The movement of the phases is influenced by the capillarity forces, the relative permeability, the temperature and the given boundary conditions. In the examined porous body, the capillary effect is caused by the intermolecular forces of cohesion and adhesion of the constituents involved. The treatment of the capillary problem, based on thermomechanical investigations, yields the result that the capillarity force is a volume interaction force. Moreover, the friction interaction forces caused by the motion of the constituents are included in the mechanical model. The relative permeability depends on the saturation of the porous body which is considered in the mechanical model. In order to describe the thermo-elastic behaviour, the balance equation of energy for the mixture must be taken into account. The aim of this investigation is to provide with a numerical simulation of the behavior of liquid and gas phases in a thermo-elastic porous body. 

Viscoelasticity Models For characterization with viscoelasticity models, simulation models have been developed on the basis of Kelvin, Maxwell, and Voigt elements. These elements come from continuum mechanics and can be used to describe compression. 

One of the fundamental assumptions in fluid mechanical formulations of Newtonian flow past solids is the continuity of the tangential component of velocity across a boundary known as the "no-slip" boundary condition (BC) [6]. Continuum mechanics with the no-slip BC predicts a linear velocity profile. However, recent experiments which probe molecular scales [7] and MD simulations [8-10] indicate that the BC is different at the molecular level. The flow boundary condition near a surface can be determined from the velocity profile. In molecular simulations, the velocity profile is calculated in a simitar way to the calculation of the density profile. The region between the walls is divided into a sufficient number of thin slices. The time averaged density for each slice is calculated during a simulation. Similarly, the time averaged x component of the velocity for all particles in each slice is determined. The effect of wall-fluid interaction, shear rate, and wall separation on velocity profiles, and thus flow boimdary condition will be examined in the following. 

Furthermore, it has recently been found that the discrete nature of a molecule population leads to qualitatively different behavior than in the continuum case in a simple autocatalytic reaction network [29]. In a simple autocatalytic reaction system with a small number of molecules, a novel steady state is found when the number of molecules is small, which is not described by a continuum rate equation of chemical concentrations. This novel state is first found by stochastic particle simulations. The mechanism is now understood in terms of fluctuation and discreteness in molecular numbers. Indeed, some state with extinction of specific molecule species shows a qualitatively different behavior from that with very low concentration of the molecule. This difference leads to a transition to a novel state, referred to as discreteness-induced transition. This phase transition appears by decreasing the system size or flow to the system, and it is analyzed from the stochastic process, where a single-molecule switch changes the distributions of molecules drastically. 

Gayev, Ye.A. (1994) Large Spray Cooler Theoretical Simulation Based on the Continuum Mechanics Method, Proc. 9th Cooling Tower and Spraying Pond Symposium, Rhode-Saint-Genese, Belgium. 

Macroscopic phenomena are described by systems of integro-partial differential algebraic equations (IPDAEs) that are simulated by continuum methods such as finite difference, finite volume and finite element methods ([65] and references dted therein [66, 67]). The commonality of these methods is their use of a mesh or grid over the spatial dimensions [68-71]. Such methods form the basis of many common software packages such as Fluent for simulating fluid dynamics and ABAQUS for simulating solid mechanics problems. 

The obvious question is this What conditions should be imposed Without a molecular or microscopic theory for guidance, there is no deductive route to answer this question. The application of boundary conditions then occupies a position in continuum mechanics that is analogous to the derivation of constitutive equations in the sense that only a limited number of these conditions can be obtained from fundamental principles. The rest represent an educated guess based to a large extent on indirect comparisons with experimental data. In recent years, insights developed from molecular dynamics simulations of relatively simple... 

Our work on polymers using the formalism of connectivity indices has mainly focused on the physical properties. Material properties predicted from the correlations presented in this book can both be used to evaluate the inherent performance characteristics of polymers, and to specify the input parameters for continuum mechanical simulations of the performance of finished parts made from novel polymeric structures which have not yet been synthesized. 

The methods developed in this book can also provide input parameters for calculations using techniques such as mean field theory and mesoscale simulations to predict the morphologies of multiphase materials (Chapter 19), and to calculations based on composite theory to predict the thermoelastic and transport properties of such materials in terms of material properties and phase morphology (Chapter 20). Material properties calculated by the correlations presented in this book can also be used as input parameters in computationally-intensive continuum mechanical simulations (for example, by finite element analysis) for the properties of composite materials and/or of finished parts with diverse sizes, shapes and configurations. The work presented in this book therefore constitutes a "bridge" from the molecular structure and fundamental material properties to the performance of finished parts. 

Computer simulations based on lipid characterization, on continuum mechanics approaches, and on molecular theories. 

In the last chapter we discussed the relation between stress and strain (or instead rate-of-strain) in one dimension by treating the viscoelastic quantities as scalars. When the applied strain or rate-of-strain is large, the nonlinear response of the polymeric liquid involves more than one dimension. In addition, a rheological process always involves a three-dimensional deformation. In this chapter, we discuss how to express stress and strain in three-dimensional space. This is not only important in the study of polymer rheological properties in terms of continuum mechanics " but is also essential in the polymer viscoelastic theories and simulations studied in the later chapters, into which the chain dynamic models are incorporated. 

One alternative to solving the equations of change of continuum mechanics for simulating droplet collisions is the lattice-Boltzmann approach [63-67]. This technique describes the liquid dynamics on the basis of the dynamics of particle motion, which represents the liquid dynamic behavior and is governed by the lattice-Boltzmann... 

The elastoplastic multiscale analysis requires several computational modules, including (1) a microscale computation module, which consists of a set of numerical solutions for the local constitutive equation of each subphase, (2) a micromechanical computation module, which provides numerical tools to link the mechanical properties of each of the local subphases to the macroscopic responses, and (3) a macroscale computation module, in which the continuum mechanics governing equations are enforced to simulate the overall mechanical response of the material and to identify the local loading conditions over the R VE. Each of these computational modules is discussed in the following. A flowchart of the multiscale analysis is shown in Figure 5.24. 

Continuum shell models used to study the CNT properties and showed similarities between MD simulations of macroscopic shell model. Because of the neglecting the discrete nature of the CNT geometry in this method, it has shown that mechanical properties of CNTs were strongly dependent on atomic structure of the tubes and like the curvature and chirality effects, the mechanical behavior of CNTs cannot be calculated in an isotropic shell model. Different from common shell model, which is constmcted as an isotropic continuum shell with constant elastic properties for SWCNTs, the MBASM model can predict the chirality induced anisotropic effects on some mechanical behaviors of CNTs by incorporating molecular and continuum mechanics solutions. One of the other theory is shallow shell theories, this theory are not accurate for CNT analysis because of CNT is a... 

Besides the reduction in the problem size, another advantage of coarse-graining is that the forces on the beads or rods can be described by continuum mechanics instead of requiring interatomic potentials. The forces common to most simulations include the following. 

Nanoscopic particles, dispersed in a block copolymer, have dimensions that are appropriate for Brownian dynamics simulations (268). Clay composites have a range of length scales, but if the gallery spacing between the layers is not large, MD methods can be used (269) with periodicity in the directions parallel to the clay platelets. However, continuum mechanical models need to be invoked for the description of exfoliated clay systems (270). These materials have so much interfacial area that adhesion properties are very important (271). Traditional continuum bounds methods (130) usually ignore the interphases on the grounds that they comprise a very small volume fraction of the total material, and so are not expected to be very accurate for exfoliated clay systems. 

Moving from particle-based and field-based simulations to continuum mechanics is a further step of coarse-graining, after which the effea of polymer dynamics are described only in a rather unspecific manner by a set of PDEs, employing the conservation laws and phenomenological constitutive relations. Continuum mechanics relies on the fundamental notion of a mesoscopic volume element in which properties averaged over disaete particles obey deterministic relationships. Continuum-level models assume naturally that matter is a continuum that is, it can be subdivided without limit. As a result, continuum simulations can in principle handle systems of any (maaoscopic) size and dynamic processes on long timescales. 

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