Elastic properties continuum theory modelling

There are two basic theories to describe the LC state, the continuum theory mainly proposed by Oseen, Zocher, and Frank and the swarm theory supported by researchers such as Bose, Bom, Omstein, Maier, and Saupe [10,11]. The continuum theory models the liquid crystal as an anisotropic elastic medium with properties varying as a function of position. The swarm theory emphasizes molecular interactions and interprets the LC state as the result of a statistically driven thermodynamic equilibrium. In the recent work of de Gennes, Leslie and Ericksen, LC theories integrate aspects of both the continuum theory and the swarm theory [11]. 

Material Parameters. The key means whereby material specificity enters continuum theories is via phenomenological material parameters. For example, in describing the elastic properties of solids, linear elastic models of material response posit a linear relation between stress and strain. The coefficient of proportionality is the elastic modulus tensor. Similarly, in the context of dissipative processes such as mass and thermal transport, there are coefficients that relate fluxes to their associated driving forces. From the standpoint of the sets of units to be used to describe the various material parameters that characterize solids, our aim is to make use of one of two sets of units, either the traditional MKS units or those in which the e V is the unit of energy and the angstrom is the imit of length. 

Researchers have performed experiments on CNT-polymer bulk composites at the macroscale and observed the enhancements in mechanical properties (like elastic modulus and tensile strength) and tried to correlate the experimental results and phenomena with continuum theories like micro-mechanics of composites or Kelly Tyson shearing model [105,115-120]. 

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... 

The function f contains all the constitutive (material-dependent) properties and describes how the internal forces depend on the deformation. A standard assumption is that for a given material, f = 0 for all u when jr -rj > S for some > 0 called the horizon. This characteristic distance represents a length scale for the nonlocality of force. Nonlocal models allow to incorporate directly the description of atomistic effects and long-range interactions into a continuum theory. In contrast, classical elasticity theory does not have a length scale - the horizon consists of a point because of the assumption of a contact force. 

To account for the nonideal nature of real soUds and liquids, the theory of Unear viscoelasticity provides a generaUzation of the two classical approaches to the mechanics of the continuum-that is, the theory of elasticity and the theory of hydromechanics of viscous Uquids. Simulation of the ideal boundary properties elastic and viscous requires mechanical models that contain a combination of the ideal element spring to describe the elastic behavior as expressed by Hooke s law, and the ideal element dash pot (damper) to simulate the viscosity of an ideal Newton Uquid, as expressed by the law of internal friction of a liquid. The former foUows the equation F = D -x (where F = force, x = extension, and D = directional force or spring constant). As D is time-invariant, the spring element stores mechanical energy without losses. The force F then corresponds to the stress a, while the extension x corresponds to the strain e to yield a = E - e. 

The elastic properties of liquid crystal phases can be modelled using continuum theory. As its name su ests, this involves treating the medium as a continuum at the level of the director, neglecting the structure at the molecular scale. 

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