Elastic solid, ideal/perfect

Perfectly elastic deformation and perfectly viscous flow are idealizations that are approximately realized in some limiting conditions. In general the condensed matter has a fading structural memory, and the velocity with which a system that has been perturbed forgets the configuration that it had in the past roughly defines its solid or liquid nature. In ordinary liquids, molecular... 

Under an applied stress, which is the force per unit area, ideal liquids flow and perfectly elastic solids... 

Consider first the deformation of a perfect elastic solid. The work done on it is stored as energy of deformation and the energy is released completely when the stresses are removed and the original shape is restored. A metal spring approximates to this ideal. In contrast, when a viscous liquid flows, the work done on it by shearing stresses is dissipated as heat. When the stresses causing the flow are removed, the flow ceases and there is no tendency for the liquid to return to its original state. Viscoelastic properties lie somewhere between these two extremes. 

Fig. 2.8. Idealized elastic/perfectly plastic solid behavior results in a stress tensor in which there is a constant offset between the hydrostatic (isotropic) loading and shock compression. Such behavior is only an approximation which may not be appropriate in many cases.

The simplest model assumes ideal elastic behavior (Figure 7.12A). At a stress below the yield stress (Fy), the sample behaves perfectly elastically. In this region, a modulus of elasticity can be determined. At the yield stress, the sample flows. It continues to flow until the stress is lowered again to below the yield stress value. Therefore, both the elastic modulus and yield stress describe the behavior of a plastic material. They can be determined easily by compression testing. The continuous network of fat crystals in a fat bears the stress below the yield stress and therefore contributes solid or elastic properties to the material (Narine and Marangoni, 1999a). 

These rheological parameters have been successfully correlated to textural attributes of hardness and spreadabUity and provide information pertaining to the fat crystal network (69). The value of G is useful in assessing the solid-like stmcture of the fat crystal network. Increases in the value of G typically correspond to a stronger network and a harder fat (66). Alternatively, G" represents the fluid-like behavior of the fat system. This value can be related to the spreadability of a fat system, because increases in G" indicate more fluid-like behavior under an applied shear stress. The tan 8 is the ratio of these two values. As the value of 5 approaches 0° (stress wave in phase with stress wave), the G" value approaches zero, and therefore, the sample behaves like an ideal solid and is referred to as perfectly elastic (68). As 8 approaches 90° (stress is completely out of phase relative to the strain). 

General Point Defects, Lattice atoms can oscillate thermally about their ideal positions. This oscillation can be conceived in terms of the oscillation of an elastic body with the energy hv. Such elastic bodies are called phonons. Electrons and holes are especially important with nonmetallic semiconducting solids. A semiconductor is considered to be perfect when it has an empty semiconducting band. An isolated electron in a perfect solid will, of course, produce a defect. Holes are quantum states in a normally filled semiconducting band. They behave in an electric field like a positive charge. 

In every approach one finds a wide range of sophistication. In the continuum approach, the simplest (and most common) models are based on linear elastic fracture mechanics (LEFM), a well developed discipline that requires a linear elastic behaviour and brittle fracture, not always exhibited by fibres. Ductility and the presence of interfaces, not to mention hierarchical structures, make modelling much more involved. The same is true of the atomistic approach fracture models based on bond breaking of perfect crystals, using well established techniques of solid state physics, allow relatively simple predictions of theoretical tensile stresses, but as soon as real crystals, with defects and impurities, are considered, the problem becomes awkward. Nevertheless solutions provided by these simple models — LEFM or ideal crystals — are valuable upper or lower bounds to fibre tensile strength. 

Newton s law of viscous flow and Hooke s law for solids describe the perfect state for each. In practice however, few if any materials show this ideal behaviour and are more appropriately described as viscoelastic. That is to say, they exhibit both viscous and elastic behaviour. More importantly, the relative contribution of each with regard to a materials response will depend on the time scale of the experiment. If the experiment is relatively slow the material will appear viscous. Conversely, if the experiment is relatively fast the material will appear more elastic. 

Here y is the shear strain and G is a constant called shear modulus. A perfect elasticity occurs only in an ideal solid. 

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