Representative volume element definition

After an introductory chapter we review in Chap. 2 the classical definition of stress, strain and modulus and summarize the commonly used solutions of the equations of elasticity. In Chap. 3 we show how these classical solutions are applied to various test methods and comment on the problems imposed by specimen size, shape and alignment and also by the methods by which loads are applied. In Chap. 4 we discuss non-homogeneous materials and die theories relating to them, pressing die analogies with composites and the value of the concept of the representative volume element (RVE). Chapter 5 is devoted to a discussion of the RVE for crystalline and non-crystalline polymers and scale effects in testing. In Chap. 6 we discuss the methods so far available for calculating the elastic properties of polymers and the relevance of scale effects in this context. 

In 1963 Hill47) defined the Representative Volume Element (RVE) in a consideration of general properties of composite materials. The definition is more exact than Sander s, which it includes. 

We need to consider the size of the representative volume element (RYE) for polymers bearing in mind the requirements of its definition. These were 1) the RVE should be entirely typical of the material on average 2) it must contain sufficient of the inclusions (phases) for the overall moduli to be independent of the surface tractions and displacements provided these are macroscopically uniform. 

It is possible to justify Eq. (4) as follows The maximum number of contacts between a macromolecule and solvent molecules is given by the first portion of the far side of Eq. (4) [(z - 2)x/i2]- The coordination about a single unit of the macromolecule is given by z, and the -2 accounts for the two directions along which the macromolecule continues. This total number of macromolecular sites is to be multiplied by the site fraction on the macromolecules which is actually occupied by the smaller molecules. The site fraction is approximated by the volume fraction of small molecules v. The total number of contacts between macromolecules and small molecules must still be multiplied by some interaction parameter, Ah>i2- Equation (4), indicates that some simplifications are possible. The first one is that the product xn 2y can be replaced by the simpler product iV2. This can easily be seen from the definitions of the volume fractions. Next, the two constants, the coordination number (z - 2) and the interaction parameter (Am/j 2) be combined and expressed in units of RT. This new interaction parameter is commonly denoted by the letter %. It represents the total interaction energy per macromolecular volume element in units of RT. Making these substitutions leads to Eqs. (5) and (6). Note that AG has a quadratic... 

Le Chatelier s principle Le ChOtelier s principle states that if a chemical system at equilibrium is stressed (disturbed), it will reestablish equilibrium by shifting of the reactions involved, limiting reactant The limiting reactant is the reactant that is used up first in a chemical reaction, line spectrum A line spectrum is a series of fine lines of colors representing wavelengths of photons that are characteristic of a particular element, liquid A liquid is a state of matter that has a definite volume but no definite shape, macromolecules Macromolecules are extremely large molecules. 

We now need to define a collection of atoms that can be used in a DFT calculation to represent a simple cubic material. Said more precisely, we need to specify a set of atoms so that when this set is repeated in every direction, it creates the full three-dimensional crystal stmcture. Although it is not really necessary for our initial example, it is useful to split this task into two parts. First, we define a volume that fills space when repeated in all directions. For the simple cubic metal, the obvious choice for this volume is a cube of side length a with a corner at (0,0,0) and edges pointing along the x, y, and z coordinates in three-dimensional space. Second, we define the position(s) of the atom(s) that are included in this volume. With the cubic volume we just chose, the volume will contain just one atom and we could locate it at (0,0,0). Together, these two choices have completely defined the crystal structure of an element with the simple cubic structure. The vectors that define the cell volume and the atom positions within the cell are collectively referred to as the supercell, and the definition of a supercell is the most basic input into a DFT calculation. 

From the definition of the moments, equation (3.1), it is apparent that physical significance can be attached to the lower moments of the PCLD. The zeroth moment represents the total molar concentration of polymer, the first moment is the total number of monomer elements/volume in the polymer, while the second moment is the weighted sum of the number of polymers present/volume. ... 

Adsorption itself is, according to Gibbs formalism, an excess quantity. The situation at the gas/solid interface is schematically represented in Fig. 1. As consequence of the adsorption potential, the average number of molecules in an element voliune near the surface is larger than in an element volume of equal size in the bulk gas. A density profile is thus estabUshed from the solid surface to the ambient gas phase. Both molecular simulation of adsorption and experimental proof reveal that this density profile vanishes over quite a short range. Therefore, the layer on the solid surface, where adsorbate molecules are concentrated, is referred to as the adsorption space or adsorbed phase. It thus assumes a definite thickness (t) for the adsorbed phase. 

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