General elastic solid

These matrices define the relationships between stress and strain in a general elastic solid, whose properties vary with direction, that is an anisotropic elastic solid. In most of this book, we will be concerned with isotropic polymers all discussion of anisotropic mechanical properties will be reserved for Chapter 8. 

Another way to derive the ctHistitutive relation for a general elastic solid (eq. 1.6.4) is to start from an energy balance. We discuss the energy equation in the next chapter, but the basic idea is that for a perfectly elastic solid at equilibrium, the stress can only be a fimction of the change in die internal energy [/ of the sample away from its reference state due to a deformation... 

A large number of models that depend on rate of deformation have been developed, but they all arise logically fi om the general viscous fluid. The general viscous model can be derived by a process very similar to the derivation of the general elastic solid in Section 1.6. Here we propose that stress depends only on the rate of deformation... 

Irwin [23] developed an expression for the mode I stress intensity factor around an elliptical crack embedded in an infinite elastic solid subjected to uniform tension. The most general formulation is given by ... 

The Hertz theory of contact mechanics has been extended, as in the JKR theory, to describe the equilibrium contact of adhering elastic solids. The JKR formalism has been generalized and extended by Maugis and coworkers to describe certain dynamic elastic contacts. These theoretical developments in contact mechanics are reviewed and summarized in Section 3. Section 3.1 deals with the equilibrium theories of elastic contacts (e.g. Hertz theory, JKR theory, layered bodies, and so on), and the related developments. In Section 3.2, we review some of the work of Maugis and coworkers. 

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. 

For a given deformation or flow, the resulting stress depends on the material. However, the stress tensor does take particular general forms for experimentally used deformations (see section 2). The definitions apply to elastic solids, and viscoelastic liquids and solids. 

Whether a viscoelastic material behaves as a viscous liquid or an elastic solid depends on the relation between the time scale of the experiment and the time required for the system to respond to stress or deformation. Although the concept of a single relaxation time is generally inapplicable to real materials, a mean characteristic time can be defined as the time required for a stress to decay to 1/e of its elastic response to a step change in strain. The... 

A final remark should be made as to the validity of eq. (2.13). This equation suggests the existence of a set of independent relaxation mechanisms. A general proof for the existence of such mechanisms could be given for visco-elastic solids in terms of the thermodynamics of irreversible processes (52) at small deviation from equilibrium. For liquid systems, however, difficulties arise from the fact that in these systems displacements occur which are not related to the thermodynamic functions. 

POLYVINYL ALKYL ETHERS. These products have properties which range from sticky resins to elastic solids. They are obtained by the low-temperature cationic polymerization of alkyl vinyl ethers having the general formula ROCH=CH-. These monomers are prepared by the addition of die selected alkanol to acetylene in the presence of sodium alkoxide or mercury(ll) catalyst, As shown by the following equations, the latter yields an acetal which must be thermally decomposed to produce the alkyl vinyl ether. 

As shown in Chapter 10, molecular dynamics in polymers is characterized by localised and cooperative motions that are responsible for the existence of different relaxations (a, (3, y). These, in turn, are responsible for energy dissipation, mechanical damping, mechanical transitions and, more generally, of what is called a viscoelastic behavior - intermediary between an elastic solid and a viscous liquid (Ferry, 1961 McCrum et al., 1967). 

L. B. Freund, Crack Propagation in an Elastic Solid Subjected to General Loading IV-Stress Wave Loading, Journal of Mechanics and Physics of Solids, 21, 47-61 (1973). 

If a solid is stressed beyond its elastic limit, it will acquire a permanent deformation. The deformation can be either brittle or ductile depending on (i) the material, (ii) the hydrostatic pressure, (iii) the temperature, and (iv) the strain rate. In general, a solid is more likely to deform in a brittle manner at low hydrostatic pressures, low temperatures, and at high strain-rates. Convesely, high hydrostatic pressures and temperatures and low strain-rates favor ductile deformation. 

Consider imposing a step strain of magnitude 7 at time t = 0 (see Fig. 7.20). If the material between the plates is a perfectly elastic solid, the stress will jump up to its equilibrium value Gj given by Hooke s law [Eq. (7.98)] and stay there as long as the strain is applied. On the other hand, if the material is a Newtonian liquid, the transient stress response from the jump in strain will be a spike that instantaneously decays to zero. For viscoelastic materials, the stress after such a step strain can have some general time dependence a(t). The stress relaxation modulus G(t) is defined as the ratio of the stress remaining at time t (after a step strain was applied at time t = 0) and the magnitude of this step strain 7 ... 

As was discussed in some detail in chap. 2, the notion of an elastic solid is a powerful idealization in which the action of the entirety of microscopic degrees of freedom are subsumed into but a few material parameters known as the elastic constants. Depending upon the material symmetry, the number of independent elastic constants can vary. For example, as is well known, a cubic crystal has three independent elastic moduli. For crystals with lower symmetry, the number of elastic constants is larger. The aim of the present section is first to examine the physical origins of the elastic moduli and how they can be obtained on the basis of microscopic reasoning, and then to consider the nonlinear generalization of the ideas of linear elasticity for the consideration of nonlinear stored energy functions. 

Rheologists study the relations between stress (force per unit area) and strain (relative deformation) of a material, generally as a function of time scale or rate of strain. For elastic solids the modulus, i.e., stress over strain, is a characteristic parameter for pure liquids it is the viscosity, i.e., the ratio of stress over strain rate. An elastic solid regains its original shape after the stress is released and the mechanical energy used to deform it is regained a pure liquid retains the shape attained and the mechanical energy is dissipated into heat. 

It will be clear now that the relations governing fracture of viscoelastic materials are far more complex than those for elastic solids. The discussion above gives some qualitative relations that generally hold. Quantitative prediction of the behavior from first principles is mostly not possible. It all becomes even more complex for inhomogeneous materials. 

This analytical solution review is tractable only for very limited assumptions, such as homogeneity and linearly elastic behavior (not to mention excluding variations that are time- or temperature-dependent). The first deviation that must be examined is the elastic linearity assumption for polishing pads. Polymers, in general, show behavior that lies between that of an elastic solid and a viscous fluid. The term viscoelastic has been applied to this behavior. 

The subject matter of this book is the response that polymers exhibit when they are subjected to external forces of various kinds. Almost without exception, polymers belong to a class of substances known as viscoelastic bodies. As the name implies, these materials respond to external forces in a manner intermediate between the behavior of an elastic solid and a viscous liquid. To set the stage for what follows, it is necessary to describe in very general terms the types of force to which the viscoelastic bodies are subjected. 

R. Lakes (1995). Experimental methods for study of Cosserat elastic solids and other generalized elastic continua. In Continuum Methods for Materials with Microstructures (Ed. H. Muhlhaus), pp. 1-25. John Wiley Chichester. 

Polymers are also unique in their viscoelastic nature, a behavior that is situated between that of a pure elastic solid and that of a pure viscous liquid-like material their mechanical properties present a strong dependence on time and temperature. Given all the factors that have to be taken into account to determine the mechanical properties of polymers, their measurement would appear to be very complex. However, there is a series of general principles that determine the different mechanical properties and that give a general idea of the expected results in different mechanical tests. These principles can be organized in a systematic manner to determine the interrelation of polymer structure and the observed mechanical properties, using equations and characteristic parameters of polymeric materials. 

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