Isotropic material definition

In performing such experiments on isotropic materials, one is accustomed to express the elastic stiffness parameters in the experimentally more readily accessible technical parameters E (Young s modulus) and v (Poisson ratio). The relative change in length, in the direction of the tensile stress a is, by definition, given by (Al/t)i — a/E, whereas v = (Af/ )x/( A / )u. For several magnetostrictive films and substrates, E and v values are listed in table 1. Some useful relations are ... 

The microscopic image shows a juxtaposition of differently orientated areas whose sizes, varying between a few microns and several tens of microns, are associated particularly with the elementary composition of the initial carbonaceous material (4, 18, 19). The formation of a texture of this type, often called a mosaic structure, can be compared (20) to the crystallization of a supersaturated solution areas, each characterized by a definite orientation, develop from nuclei up to the total consumption of the isotropic material surrounding them. 

Refractive index — A fundamental physical property of materials through which light can travel. It is usually indicated by the symbol n, and it is defined as n = c/cQ, where c0 is the speed of light in vacuum and c corresponds to the speed at which the crests of electromagnetic radiation corresponding to a specific frequency propagate in a material [i,ii], A more rigorous definition for the refractive index of a dense and isotropic material composed of a unique kind of particles (atoms or... 

The above definition thus includes the classical fibrillar crazes (formed on surfaces, in thin films or in the bulk at lower or higher temperatures in preoriented or isotropic material, in the presence or absence of a solvant or diluant) as well as those crazes which contain so many voids and cross-ties within the heavily deformed craze matter that it becomes difficult to identify individual fibrils. The Editor is extremely grateful to all authors of this volume and to Dr. G. H. Michler for frequent discussions on the above subject and for their constructive comments which led to the above Jointly proposed definition. 

The Flux Expressions. We begin with the relations between the fluxes and gradients, which serve to define the transport properties. For viscosity the earliest definition was that of Newton (I) in 1687 however about a century and a half elapsed before the most general linear expression for the stress tensor of a Newtonian fluid was developed as a result of the researches by Navier (2), Cauchy (3), Poisson (4), de St. Venant (5), and Stokes (6). For the thermal conductivity of a pure, isotropic material, the linear relationship between heat flux and temperature gradient was proposed by Fourier (7) in 1822. For the difiiisivity in a binary mixture at constant temperature and pressure, the linear relationship between mass flux and concentration gradient was suggested by Pick (8) in 1855, by analogy with thermal conduction. Thus by the mid 1800 s the transport properties in simple systems had been defined. 

The criterion of short-term adhesion strength for the case of interphase failure under combined action of direct and tangential stresses is suggested in [382]. This criterion presumes the existence of a definite dependence between octahedral stresses, which is assumed linear in the first approximation. The hypothesis of the octahedral direct and tangential stress dependence was proposed by Nadai for continuous isotropic materials and was further developed by Balandin [384] into the strength criterion, presenting a linear function from the component of the spherical stress tensor. 

Most polycrystalline solids are considered to be isotropic, where, by definition, the material properties are independent of direction. Such materials have only two independent variables (that is elastic constants) in matrix (7.3), as opposed to the 21 elastic constants in the general anisotropic case. The two elastic constants are the Young modulus E and the Poisson ratio v. The alternative elastic constants bulk modulus B and shear modulus /< can also be used. For isotropic materials, n and B can be found from E and t by a set of equations, and on the contrary. 

The development of the physical chemistry of rubber was greatly aided by the clear definition of an "ideal" state for this material. An ideal rubber is an amorphous, isotropic solid. The liquidlike structure of rubber was discovered very soon after the technique of X-ray scattering was developed. An isotropic material is characterized by physical properties that do not depend on the orientation of the sample. The deformation of an isotropic solid can be characterized by only two unique moduli the modulus of compression, K, and the shear modulus, G. A solid is characterized by equilibrium dimensions that are functions of temperature, pressure, and the externally imposed constraints. It is convenient to define a shape vector, L, whose components are the length, width, and height of a rectangular parallelepiped. For a system with no external constraints, the shape vector can be expressed as ... 

Mechanical tests for advanced composite materials conform in many respects to the conventional test typology used for traditional isotropic materials. Despite the complication associated with the heterogeneity of composite systems, the interface between fiber and matrix, and the anisotropy at the micro- and macroscopic levels, the same characteristic property definitions generally used for conventional materials can be identified for these novel materials. In some cases additional constants are required and some differences in nomenclature are introduced especially when no isotropic counterpart exists. 

The relationships between elastic constants which must be satisfied for an isotropic material impose restrictions on the possible range of values for the Poisson s ratio of -1 < v <. In a similar manner, there are restrictions in orthotropic and transversely isotropic materials. These constraints are based on considerations of the first law of thermodynamics [15]. Moreover, these constraints imply that both the stiffness and compliance matrices must be positive-definite, i.e. each major diagonal term of both matrices must be greater than 0. 

Definition of terms relating to the non-ultimate mechanical properties of polymers. " This document defines mechanical terms of significance prior to failure of polymers. It deals in particular with bulk polymers and concentrated polymer solutions and their respective viscoelastic properties. It contains the basic definitions of experimentally observed stress, strain, deformations, viscoelastic properties, and the corresponding quantities that are commonly met in conventional mechanical characterization of polymeric materials. Definitions from ISO and ASTM publications are adapted. Only isotropic polymeric materials are considered. 

In the case of isotropic materials the stiffness matrix can be written as follows (using the definitions Cj2 = and C44 = /U) ... 

If we now consider the state of stress in an isotropic material, by definition the material has no preferred directions. In simple shear flow, we have... 

The term quasi-isotropic iaminate is used to describe laminates that have isotropic extensionai stiffnesses (the same in all directions in the plane of the laminate). As background to the definition, recall that the term isotropy is a material property whereas laminate stiffnesses are a function of both material properties and geometry. Note also that the prefix quasi means in a sense or manner. Thus, a quasi-isotropic laminate must mean a laminate that, in some sense, appears isotropic, but is not actually isotropic in all senses. In this case, a quasi-isotropic... 

Given these differences between rigid and flexible conduit, let us examine the differences between steel and RTR pipe, both of which are, of course, flexible conduits. First, steel pipe is by definition constructed from a material, steel, that for our purposes is a homogeneous isotropic substance. Therefore, steel pipe can be considered to have the same material properties in all directions that is, it is equally strong in both the hoop and longitudinal directions [Fig. 4-2(b)]. 

The purpose of this chapter is to introduce the effect of surfaces and interfaces on the thermodynamics of materials. While interface is a general term used for solid-solid, solid-liquid, liquid-liquid, solid-gas and liquid-gas boundaries, surface is the term normally used for the two latter types of phase boundary. The thermodynamic theory of interfaces between isotropic phases were first formulated by Gibbs [1], The treatment of such systems is based on the definition of an isotropic surface tension, cr, which is an excess surface stress per unit surface area. The Gibbs surface model for fluid surfaces is presented in Section 6.1 along with the derivation of the equilibrium conditions for curved interfaces, the Laplace equation. 

This definition of diffusion coefficients considers the non-isotropic diffusion behaviour in some materials. So, this stochastic modelling can easily be applied for the analysis of the oriented diffusion phenomena occurring in materials with designed properties for directional transport. 

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