Ideal elastic medium

Under very rapid mechanical actions or in observations with characteristic time t < to, the substance behaves as an ideal elastic medium. For t to the developing flow becomes stronger than the elastic deformation, and the substance can be treated as a simple Newtonian fluid. It is only if t is of the same order of magnitude as to that the elastic and viscous effects act simultaneously, and the complex nature of the deformation displays itself. 

In this section I first describe an idealized shock wave in a continuous elastic medium, and then explain how things change in real materials such as a... 

An amazing feature of shock compression is illustrated in Figs. Id-e. A driven shock front steepens up as it runs, in contrast to acoustic waves that disperse as they run [1]. Imagine a shock front that is not initially steep (Fig. Id). Think of the front as a higher pressure wave trailing a lower pressure wave. Equation (2) above shows the trailing wave moves faster. In an ideal continuous elastic medium, the shock front steepens until it becomes an abrupt discontinuity. The shock front risetime tr —> 0. 

Kj is a measure of the stress-field Intensity near the tip of an ideal crack in a linear elastic medium deformed such that the crack faces are displaced apart, normal to the crack plane (that means Mode 1). Kj is directly proportional to the applied load and depends on the ratio of the specimen dimensions. ... 

Silicon is a nearly ideal elastic material, so its intrinsic quality factor is very high. However, this quality factor is only realized if the device is operated in high vacuum, where Q may be 10,000 or more. Otherwise the damping effect of a surrounding medium reduces Q. In air, it is... 

Freezing soil is considered as porous medium of skeleton (s) of soil grains filled up with pore fluid (/) and ice (i). It is assumed that skeleton and ice are elastic solids and have equal displacements and velocities, pore fluid is an ideal solution of water (w) and dissolved salts (c) and governed by adsorption... 

Figure 2 shows the basic physical idea of the microstructure of the continuum rheologicS model we proposed earlier (2). The layers can be idealized as separated by porous slabs, which are connected by elastic springs. Liquid crystals may flow parallel to the planes in the usual Newtonian manner, as if the slabs were not there. In the direction normal to the layers, liquid crystals encounter resistance through the porous medium, proportional to the normal pressure gradient, which is known as permeation. The permeation is characterized by a body force which in turn causes elastic compression and splay of the layers. Applied strain from the compression causes dislocations to move into the sample from the side in order to relax the net force on the layers. When the compression stops and the applied stress is relaxed the permeation characteristic has no influence on stress strain field. 

Of course, in the long run, all the work transforms into heat which is dissipated in the surroundings. How do we know Well, if there were no surrounding medium to take the heat away, both a gas when compressed and a polymer when stretched would get warmer (see also below Section 7.11). Does that mean elasticity of a polymer chain depends on the environment which absorbs the heat Well, we know that the pressure of an ideal gas does not depend on the type of the environment, so maybe there is something similar for a polymer ... 

The theory of fluid flow, together with the theory of elasticity, makes up the field of continuum mechanics, which is the study of the mechanics of continuously distributed materials. Such materials may be either soKd or fluid, or may have intermediate viscoelastic properties. Since the concept of a continuous medium, or continuum, does not take into consideration the molecular structure of matter, it is inherently an idealization. However, as long as the smallest length scale in any problem under consideration is very much larger than the size of the molecules making up the medium and the mean free path within the medium, for mechanical purposes all mass may safely be assumed to be continuously distributed in space. As a result, the density of materials can be considered to be a continuous function of spatial position and time. 

The Laplace-Kelvin equation predicts that an isolated gas bubble should redissolve if its size is below a critical threshold, and conversely it should grow if its radius rj, exceeds the critical value given by Eq. (43) [50], where surface tension, Pyne is the local pressure over the bubble (it may be simply the hydrostatic pressure, but in other circumstances it may be controlled by medium elasticity [61]), and Pj, is the pressure inside the bubble, equal to the sum of the partial vapor pressures due to inert gases and volatile components if physical equilibrium and gas-phase ideality hold. So, it is possible that no bubbling occurs if the pressure is high enough or the content of volatiles is too low, but this is often not the case. 

For a complete specification of mechanical compliances, it is necessary carefully to specify the conditions under which a given measurement is carried out thus, one must state not only whether it is 5 or T that is kept fixed, but also whether , or D, and whether, electrically free when D = 0, the crystal is said to be electrically clamped. The latter condition is not readily experimentally realized for details, interested readers are referred to Nye, (Oxford, 1957) Constancy of a components demands that the crystal be mounted so as to allow the strains to take place in as unhindered a fashion as possible the material is then mechanically free. Constancy of e components requires that, ideally, the crystal be firmly attached to a medium with infinite elastic stiffness the material is then mechanically clamped. 

The gas-particle stream is considered as a flow consisting of two ideal gases 1) carrying gas 2) gas of particles. The problem of jet interaction has been solved for axially symmetric jets loaded with uniformly sized particles. The influence of particle diameter, concentration and degree of particle non-elasticity at particle-particle collisions on particle medium behavior in the milling zone has been numerically investigated. 

The main assumptions of the model are following 1) particles are spherical 2) each particle has both the directed and chaotic components of die absolute velocity 2) a chaotic particle motion is carried out according to the Boltzmann-Maxwell law (the particle medium is considered as an ideal gas having its own pressure, density etc.) 3) a chaotic particle velocity drop is caused by both a viscous particle-gas friction force and inelastic particle-particle collisions (coefficient of energy losses due to inelastic collisions has to be rather low because Bolzmann-Maxwell velocity distribution is valid only for elastic particles and can be employed only for small non-elasticity) 4) particles do not get fragmented 5) a heat exchange between gas and particles is neglected. 

---