Isotropic media, elastic properties

One prominent example of rods with a soft interaction is Gay-Berne particles. Recently, elastic properties were calculated [89,90]. Using the classical Car-Parrinello scheme, the interactions between charged rods have been considered [91]. Concerning phase transitions, the sohd-fluid equihbria for hard dumbbells that interact additionally with a quadrupolar force was considered [92], as was the nematic-isotropic transition in a fluid of dipolar hard spherocylinders [93]. The influence of an additional attraction on the phase behavior of hard spherocylinders was considered by Bolhuis et al. [94]. The gelation transition typical for clays was found in a system of infinitely thin disks carrying point quadrupoles [95,96]. In confined hquid-crystalline films tilted molecular layers form near each wall [97]. Chakrabarti has found simulation evidence of critical behavior of the isotropic-nematic phase transition in a porous medium [98]. 

If the diffusion medium is isotropic in terms of diffusion, meaning that diffusion coefficient does not depend on direction in the medium, it is called diffusion in an isotropic medium. Otherwise, it is referred to as diffusion in an anisotropic medium. Isotropic diffusion medium includes gas, liquid (such as aqueous solution and silicate melts), glass, and crystalline phases with isometric symmetry (such as spinel and garnet). Anisotropic diffusion medium includes crystalline phases with lower than isometric symmetry. That is, most minerals are diffu-sionally anisotropic. An isotropic medium in terms of diffusion may not be an isotropic medium in terms of other properties. For example, cubic crystals are not isotropic in terms of elastic properties. The diffusion equations that have been presented so far (Equations 3-7 to 3-10) are all for isotropic diffusion medium. 

We begin by examining what continuum mechanics might tell us about the structure and energetics of point defects. In this context, the point defect is seen as an elastic disturbance in the otherwise unperturbed elastic continuum. The properties of this disturbance can be rather easily evaluated by treating the medium within the setting of isotropic linear elasticity. Once we have determined the fields of the point defect we may in turn evaluate its energy and thereby the thermodynamic likelihood of its existence. 

For cubic symmetry materials, three independent elastic properties that are orientation dependent are required to describe the mechanical behavior of the material. This anisotropy effect increases significantly the number of the nonzero elements in the FE stiffness matrix leading to alteration in the calculated stress components and the wave speed. In order to test these anisotropy effects, we plot the wave profiles of three different orientations and compare it with the isotropic behavior with a loading axis in the [001] directions as shown in Fig 8. We observed that under the same loading condition, the peak stress of [111] and [Oil] orientations are slightly higher than those of the [001] which is lower that that of isotropic material. Furthermore, wave speed varies moderately with orientation with the fastest moving wave in the [ 111 ] followed by [011 ], isotropic medium and [001 ] respectively. 

The self-consistent method is based on a classical solution of Eshelby (1957) that spawned a remarkably large number of different apphcations. It states that for an ellipsoidal isotropic elastic inclusion in an infinite elastic medium of different but uniform isotropic elastic properties the state of stress (or strain) inside the inclusion is uniform when the distant body is subjected to a uniform stress (or strain). 

At the same time, when a physical property is represented by a tensor of a given order, it can be characterized by a particular number of components in a given space. For example, the three spatial components and time form a 4D field vector, which is a first-order tensor. Another noteworthy example is that of the fonrth-order elasticity tensor, which in an isotropic medium is degenerated into two scalar quantities the Young s modulus and the Poisson ratio. 

For pure tensile and shear-tensUe sources, the ISO/CLVD ratio depends on the elastic properties of the medium surrounding the source. In isotropic media, this ratio is (Vavrycuk 2001, 2011)... 

The two-shell model of Kerner [65] conforms to the conditions of the second group of models. The dilatation of a spherical inclusion surrounded by a homogeneous medium is derived subject to the condition that displacements and tractions at the surface of the inclusion are continuous. The homogeneous medium is supposed to have the elastic properties of the composite as a whole. The model interrelates shear (Gj) and compressive (Kj) moduli (or Poisson s ratios p ) of an arbitrary number of isotropic elements with the macroscopic moduli Gc and Kc. 

Collective motions are elastic deformations in liquid crystalline samples, but can also exist in their isotropic phases with a finite coherence length, giving rise to pretransitional phenomena [6.2]. These motions are perceived as hydrodynamic phenomena and are influenced by molecular properties such as elastic constants and viscosities of the liquid crystalline medium. At best, director fluctuations can only provide indirect information on the anisotropic intermolecular interactions. On the contrary, motion on a molecular level must reflect the shape of the instantaneous potential of mean torque on each molecule. Both both molecular rotation and translation are expected to be sensitive to the nature of anisotropic interactions, which determine the formation of various liquid crystalline structures. 

---