Elastic solid description

Actually, some fluids and solids have both elastic (solid) properties and viscous (fluid) properties. These are said to be viscoelastic and are most notably materials composed of high polymers. The complete description of the rheological properties of these materials may involve a function relating the stress and strain as well as derivatives or integrals of these with respect to time. Because the elastic properties of these materials (both fluids and solids) impart memory to the material (as described previously), which results in a tendency to recover to a preferred state upon the removal of the force (stress), they are often termed memory materials and exhibit time-dependent properties. 

Polymers are viscoelastic materials meaning they can act as liquids, the visco portion, and as solids, the elastic portion. Descriptions of the viscoelastic properties of materials generally falls within the area called rheology. Determination of the viscoelastic behavior of materials generally occurs through stress-strain and related measurements. Whether a material behaves as a viscous or elastic material depends on temperature, the particular polymer and its prior treatment, polymer structure, and the particular measurement or conditions applied to the material. The particular property demonstrated by a material under given conditions allows polymers to act as solid or viscous liquids, as plastics, elastomers, or fibers, etc. This chapter deals with the viscoelastic properties of polymers. 

If a sample shows elastic, solid-like deformation below a certain shear stress ay and starts flowing above this value, ay is called a yield stress value. This phenomenon can occur even in solutions with quite low viscosity. A practical indication for the existence of a yield stress value is the trapping of bubbles in the liquid Small air bubbles that are shaken into the sample do not rise for a long time whereas they climb up to the surface sooner or later in a liquid without yield stress even if their viscosity is much higher. A simple model for the description of a liquid with a yield stress is called Bingham s solid ... 

Electromigration is the displacement of atoms in a conductor due to an electric current. A metal, such as aluminum, consists of positively charged ions, and Z conduction electrons per ion (Z = 3 for aluminum). To simplify the discussions, we will consider an idealized homogenous conductor, rather than the more realistic polycrystalline description. Our mixture is then made of two components, an electron gas and an ionic body, which we will describe as a linearly elastic solid. 

In this paper the negative Celsius temperatures were assumed, when the material can be regarded as elastic solid with sufficient accuracy. In present paper were used classical notions of Griffiths and Orowan. These notions, suitable for description of the cracks behavior in homogeneous materials, were modified for description of crack development in heterogeneous materials. In heterogeneous materials, as AC, the embryonic voids with the characteristic sizes a 20-50 nm with a concentration c. 

Filled polymer rheology is basically concerned with the description of the deformation of filled polymer systems under the influence of applied stresses. Softened or molten filled polymers are viscoelastic materials in the sense that their response to deformation lies in varying extent between that of viscous liquids and elastic solids. In purely viscous liquids, the mechanical energy is dissipated into the systems in the form of heat and cannot be recovered by releasing the stresses. Ideal solids, on the other hand, deform elastically such fliat the deformation is reversible and the energy of deformation is fully recoverable when the stresses are released. 

It is necessary here to point out that the rheology of all materials, especially the rheology of surfactant solutions, can depend strongly on the time-scale on which the material is studied. This can be demonstrated quite easily. For example, Newtonian liquid water behaves as an elastic solid if the stress is applied for nanoseconds or picoseconds on the other hand, granite, which can be regarded as an Euclidean solid to a first approximation, can flow like a liquid if a strong stress acts over a period of 10 -10 years. Therefore, it must be stated at this point that the description of the rheological effects of surfactant solutions is valid for observation times which are necessary for the measurements, i.e. several seconds up to several days, or weeks in some cases. 

Since according to the indicated above reasons two order parameters are required, as a minimum, for solid-phase polymers elastic constants description, then variable percolation threshold should be introduced in the Eq. (3. 1), that is,p should be replaced on A. Besides, it has been shown earlier, that for polymers structure Vp 1 (see Table 1.1) [10] and therefore, T[=d - 1 can be assumed in the Eq. (3.2) as the first approximation. Then the Eq. (3.1) assumes the following form [6, 7] ... 

Between the extremes of viscous fluids and elastic solids are materials that seem to exhibit both traits. These are called viscoelastic materials or memory fluids, and their dual nature becomes most evident when we subject them to time-dependent (unsteady) tests. The three major types of unsteady tests are the so-called relaxation, creep and dynamic tests. In the previous sections, we gave definitions and descriptions for stress, strain and deformation rates. These quantities are now used in defining the various unsteady tests. Thus, in a relaxation test the sample is subjected to a sudden, constant, strain. The stress shoots up in response and then gradually decays ( relaxes ). In the creep test, a sudden stress is applied and held constant. Now the strain picks up quickly and then, while continuing to increase, slows down on its rate of increase. We say the material creeps under the constant stress. In dynamic tests, one confining wall is made to move periodically with respect to another. One monitors both the strain and the stress as a function of time. 

Melt rheology is concerned with the description of the deformatitm of the material under the influence of stresses. Deformation and flow naturally exist when the thermoplastics are melted and then reformed into solid products of various shapes. All polymer melts are viscoelastic materials that is, their response to external load lies in varying extent between that of a viscous liquid and an elastic solid. In an ideal viscous liquid, the energy of deformation is dissipated in the form of heat and cannot be recovered just by releasing the external forces whereas, in an ideal elastic solid, the deformation is fiilly recovered when the stresses are released. 

The fluid mechanics origins of shock-compression science are reflected in the early literature, which builds upon fluid mechanics concepts and is more concerned with basic issues of wave propagation than solid state materials properties. Indeed, mechanical wave measurements, upon which much of shock-compression science is built, give no direct information on defects. This fluids bias has led to a situation in which there appears to be no published terse description of shock-compressed solids comparable to Kormer s for the perfect lattice. Davison and Graham described the situation as an elastic fluid approximation. A description of shock-compressed solids in terms of the benign shock paradigm might perhaps be stated as ... 

To develop a terse, broad description of mechanical, physical, and chemical processes in solids, this book is divided into five parts. Part I contains one chapter with introductory material. Part II summarizes aspects of mechanical responses of shock-compressed solids and contains one chapter on materials descriptions and one on experimental procedures. Part III describes certain physical properties of shock-compressed solids with one chapter on such effects under elastic compression and one chapter on effects under elastic-plastic conditions. Part IV describes work on chemical processes in shock-compressed solids and contains three chapters. Finally, Part V summarizes and brings together a description of shock-compressed solids. The information contained in Part II is available in much better detail in other reliable sources. The information in Parts III and IV is perhaps presented best in this book. 

It is instructive to describe elastic-plastic responses in terms of idealized behaviors. Generally, elastic-deformation models describe the solid as either linearly or nonlinearly elastic. The plastic deformation material models describe rate-independent behaviors in terms of either ideal plasticity, strainhardening plasticity, strain-softening plasticity, or as stress-history dependent, e.g. the Bauschinger effect [64J01, 91S01]. Rate-dependent descriptions are more physically realistic and are the basis for viscoplastic models. The degree of flexibility afforded elastic-plastic model development has typically led to descriptions of materials response that contain more adjustable parameters than can be independently verified. 

In this book those ferroelectric solids that respond to shock compression in a purely piezoelectric mode such as lithium niobate and PVDF are considered piezoelectrics. As was the case for piezoelectrics, the pioneering work in this area was carried out by Neilson [57A01]. Unlike piezoelectrics, our knowledge of the response of ferroelectric solids to shock compression is in sharp contrast to that of piezoelectric solids. The electrical properties of several piezoelectric crystals are known in quantitative detail within the elastic range and semiquantitatively in the high stress range. The electrical responses of ferroelectrics are poorly characterized under shock compression and it is difficult to determine properties as such. It is not certain that the relative contributions of dominant physical phenomena have been correctly identified, and detailed, quantitative materials descriptions are not available. 

Among the newer probes now being developed, spectroscopic observations of crystals in the elastic-plastic regime hold promise for limited development of atomic level physical descriptions of local defects [91S02]. It is yet to be determined how generally this probe can be applied to solids. The electrochemical probe appears to have considerable potential to describe shock-compressed matter from a radically different perspective. 

Before dealing with reinforcement of elastomers we have to introduce the basic molecular features of mbber elasticity. Then, we introduce—step-by-step—additional components into the model which consider the influence of reinforcing disordered solid fillers like carbon black or silica within a rabbery matrix. At this point, we will pay special attention to the incorporation of several additional kinds of complex interactions which then come into play polymer-filler and filler-filler interactions. We demonstrate how a model of reinforced elastomers in its present state allows a thorough description of the large-strain materials behavior of reinforced mbbers in several fields of technical applications. In this way we present a thoroughgoing line from molecular mechanisms to industrial applications of reinforced elastomers. 

The solid state polymerisation of diacetylenes (2) with U.V. radiation, heating or shear force is most indicative of the predominant influence of electron-lattice coupling. The details of the chemical changes that occur during th polymerisation process are crucial (2,40) but the overall description only needs part of this chemical information. The kinetics and thermodynamics of the polymerisation process using an elastic strain approach have been worked out in (41). 

An alternative approach involves integrating out the elastic degrees of freedom located above the top layer in the simulation.76 The elimination of the degrees of freedom can be done within the context of Kubo theory, or more precisely the Zwanzig formalism, which leads to effective (potentially time-dependent) interactions between the atoms in the top layer.77-80 These effective interactions include those mediated by the degrees of freedom that have been integrated out. For periodic solids, a description in reciprocal space decouples different wave vectors q at least as far as the static properties are concerned. This description in turn implies that the computational effort also remains in the order of L2 InL, provided that use is made of the fast Fourier transform for the transformation between real and reciprocal space. The description is exact for purely harmonic solids, so that one can mimic the static contact mechanics between a purely elastic lattice and a substrate with one single layer only.81... 

Fig. 4.34 Comparison between the descriptions of the elastic intensity at 260 K (filled triangle) and 280 K filled circle) in terms of the EISF corresponding to a methyl-group rotation (solid lines) and to a 2-site jump (dashed lines), (Reprinted with permission from [195]. Copyright 1998 American Chemical Society)...

Solids undergoing martensitic phase transformations are currently a subject of intense interest in mechanics. In spite of recent progress in understanding the absolute stability of elastic phases under applied loads, the presence of metastable configurations remains a major puzzle. In this overview we presented the simplest possible discussion of nucleation and growth phenomena in the framework of the dynamical theory of elastic rods. We argue that the resolution of an apparent nonuniqueness at the continuum level requires "dehomogenization" of the main system of equations and the detailed description of the processes at micro scale. 

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