Elastic behavior atomic structure

The initial part of the curve, OA in Fig. 1, is the characteristic linear-elastic behavior of the material, i.e., the extension that occurs is fully reversible and the relationship between the force and the extension is linear. At an atomic level the bonds between the atoms of the crystal structure are just flexing. The extension in this region is however very small and can only be measured using special extensometers. This linearity ceases at point A and the material starts to behave irreversibly, i.e., permanent or plastic deformation occurs. This phenomenon is known as yielding. In this region the atoms take up new position relative to each other by the mechanism of dislocation activation. 

Continuum shell models used to study the CNT properties and showed similarities between MD simulations of macroscopic shell model. Because of the neglecting the discrete nature of the CNT geometry in this method, it has shown that mechanical properties of CNTs were strongly dependent on atomic structure of the tubes and like the curvature and chirality effects, the mechanical behavior of CNTs cannot be calculated in an isotropic shell model. Different from common shell model, which is constmcted as an isotropic continuum shell with constant elastic properties for SWCNTs, the MBASM model can predict the chirality induced anisotropic effects on some mechanical behaviors of CNTs by incorporating molecular and continuum mechanics solutions. One of the other theory is shallow shell theories, this theory are not accurate for CNT analysis because of CNT is a... 

Chrysotile NTs were synthesized and characterized by Piperno and co-workers (2007) using atomic force microscopy and transmission electron Microscopy (TEM). The results have shown that chrysotile NTs exhibit elastic behavior at small deformation. The chrysotile Young s modulus evaluated by (Piperno et al, 2007) are 159 + 125 GPa. The stoichiometric chrysotile fibers demonstrate a hollow structure with quite uniform outer diameter around 35 nm and inner diameter about 7-8 nm. The NTs are open ended with several hundred nanometers in length. 

As briefly mentioned in Sect. 2.2, the bond valence parameter b represents the compliance of a bond to external forces. Approximating by a universal value therefore eliminates the crystal-chemical information on elastic behavior from bond valence parameters (or more precisely reduces the information from an approximation that takes into account structure type and atomic properties to a crude estimate solely based on the coordination type). Whether such information is relevant for a given application purpose and available for specific cation-anion pair may depend on individual circumstances. Here it will be assumed that retaining this information available is desired, and thus it is necessary to elaborate suitable procedures to systematically determine the respective b values. 

We recall that our wave equation represents a long wave approximation to the behavior of a structured media (atomic lattice, periodically layered composite, bar of finite thickness), and does not contain information about the processes at small scales which are effectively homogenized out. When the model at the microlevel is nonlinear, one expects essential interaction between different scales which in turn complicates any universal homogenization procedure. In this case, the macro model is often formulated on the basis of some phenomenological constitutive hypotheses nonlinear elasticity with nonconvex energy is a theory of this type. 

The basic law of viscosity was formulated before an understanding or acceptance of the atomic and molecular structure of matter although just like Hooke s law for the elastic properties of solids the basic equation can be derived from a simple model, where a flnid is assumed to consist of hypothetical spherical molecules. Also like Hooke s law, this theory predicts linear behavior at low rates of strain and deviations at high strain rates. But we digress. The concept of viscosity was first introduced by Newton, who considered what we now call laminar flow and the frictional forces exerted between layers within a fluid. If we have a fluid placed between a stationary wall and a moving wall and we assume there is no slip at the walls (believe it or not, a very good assumption), then the velocity profile illustrated in Figure... 

Second, the Q-dependence of the measured elastic incoherent structure factor (EISF) appears to be in excellent agreement with the predictions of the model of localized atomic motion over a hexagon (Eq. (26.13)) with the distance between the nearest-neighbor sites equal to the experimental value. As an example of these results, Eig. 26.5 shows the behavior of the EISE for TaV2Hj j as a function of Q at several temperatures. The solid curves represent the fits of the six-site model to the data. In these fits the distance between the nearest-neighbor sites has been fixed to its value resulting from the structure, = 0.99 A, so that the... 

The second approach enforces a finite-pressure rear boundary condition, i.e., a supported shock wave, by sending the AB target at a velocity -Up towards a few layers of a rigid A2 solid. Since the approaching AB molecules see the piston A2 molecules as soon as they are within the potential cutoff distance, this provides a smoother version of the sudden momentum mirror which specularly reflects atoms reaching the z = 0 boundary [7]. (Using the perfect momentum mirror can result in peculiar behavior in 2D at low velocities, it collapses the herringbone lattice into an amorphous or even melted structure, rather than the expected elastic response.)... 

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