Gaussian element,elasticity

If Gc chains crystallize (partially) there will remain G-Gc completely amorphous chains and G + 3Gc/2 total elastic elements (amorphous chains and subchains). This is true only for the model described with 1/2 Gc chains folding once and 1/2 Gc chains not folding at all. If each elastic element is Gaussian in its behavior, the elastic free energy Fg can be written as... 

The theory of relaxation spectra in polarized luminescence for various dynamic models of a flexible polymer chain has been developed by several groups of workers. Wahl has proposed a theory for the model of Gaussian subchains. The authors and coworkers used dynamic chain models consisting of rigid or deformable elements with continuous visco-elastic mechanism of mobility and rotational-isomeric lattice chain... 

Moreover, the equation can only be accurate for small strains, since considerable change in the end-to-end distance of the cords would distort the Gaussian distribution of statistical chain elements. This happens more readily for a smaller value of It also implies that at increasing strain, the chemical bonds in the primary chain become increasingly distorted. Consequently, the increase in elastic free energy is due not merely to a decrease in conformational entropy but also to an increase in bond enthalpy. If the value of is quite small, even a small strain will cause an increase in enthalpy. (In a crystalline solid, only the increase in bond enthalpy contributes to the elastic modulus.)... 

Under this assumption, the bending energy Eg can be represented in terms of the membrane s curvature. For this reason. Eg is also referred to as the curvature elastic energy. The curvature of smooth surfaces is characterized by two functions that depend on the local canonical curvatures, h t) and h lr), in a surface element dS centered at r. These functions are the mean curvature, H = ( 1 + hi) , and the Gaussian curvature, K = hih2- In general, H and K change with the point r. 

In contrast to the scarce knowledge on hardness and elastic properties of intermetallic clathrates, there is an enormous amount of data on Einstein and Debye temperatures available in the literature. The exceptional vibrational features of these cage compounds were extensively smdied in recent years and will be discussed in the frame of this work. Needless to say that DPT calculations of electron and phonon density of states, of thermoelectric properties and of elastic properties have greatly supported clathrate research. In this context attention should be drawn to a recent work of Karttunen et al. [29], who employed the Perdew-Burke-Emzerhof hybrid density functional with localized atomic basis sets composed of Gaussian-type functions, to calculate the elastic properties of 14 different types of clathrate frameworks (for elemental structures of C, Si, Ge, Sn) predicting bulk and Young s moduli comparing them with their diamond-like, dense so called ot-phases. 

If of the G network chains crystallize, leaving G-G amorphous chains, there are G+Gc amorphous elastic elements (amorphous chains plus amorphous sub-chains) that comprise the seml-crystalllne network. It Is here assumed that a chain enters a crystallite only once or not at all. By and large, multiple entry Into crystalline regions Is probably rare except In very lightly crossllnked networks. Let each elastic element (chain or subchain) behave In a Gaussian fashion so that we may write for the elastic free energy F of a specified network... 

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