Continuum elasticity

Second, we expect that the strain energy per carbon should increase inversely proportional to the square of the nanotube radius[23]. Based on a continuum elastic model, Tibbetts[4] derived a strain energy for a thin graphitic nanotube of the general form ... 

To describe properties of solids in the nonlinear elastic strain state, a set of higher-order constitutive relations must be employed. In continuum elasticity theory, the notation typically employed differs from typical high pressure science notations. In the present section it is more appropriate to use conventional elasticity notation as far as possible. Accordingly, the following notation is employed for studies within the elastic range t = stress, t] = finite strain, with both taken positive in tension. 

Another effect of elastic deformation is that it causes a long-range interaction between steps. From the continuum elasticity theory, two steps sepa-rated by a distance have a repulsive interaction proportional to l for homo- and to In i for hetero-epitaxial cases, respectively [84]. This interaction plays an important role, for example, in step fluctuations, terrace width distribution, step bunching, and so forth [7,85-88]. 

The factors Kn are elastic constants for the nematic phase and Icb is the Boltzmann constant. Therefore a combination of molecular electronic structure, orientational order and continuum elasticity are all involved in determining the flexoelectric polarisation. Polarisation can also be produced in the presence of an average gradient in the density of quadrupoles. This is... 

Continuum Elastic Theory Membranes with Uniform Tilt Direction... 

They developed a continuum elastic-free energy model that suggests these observations can be explained as a first-order mechanical phase transition. In other recent work on steroids, Terech and co-workers reported the formation of nanotubes in single-component solutions of the elementary bile steroid derivative lithocholic acid, at alkaline pH,164 although these tubules do not show any chiral markings indicating helical aggregation. 

Miltat and Bowen showed that direct images can be synthesised from the cylinders of misorientation drawn aroimd a dislocation line using continuum elasticity theory. The image full width can be calculated from the projected width circumscribed by the contour where ( ) is equal to times the reflecting range,... 

As mentioned before and assuming the vahdity of the continuum elasticity theory at the dislocation core, F. C. Frank derived the expression for the characteristic radius of a hollow core (Frank, 1951) ... 

Because of the existence of force, under certain circumstances, the STM gap becomes unstable. This problem has been studied extensively by Pethica and Oliver (1987) and Pethica and Sutton (1988), using the classical theory of continuum elasticity. In spite of its simplicity, the theory reproduces the basic features of a large number of the observed phenomena. 

The force constants of single beam cantilevers (normal spring constant kn, torsional spring constant kv, and lateral spring constant kjJ can be calculated, assuming levers of constant thickness, based on measured cantilever dimensions, from continuum elasticity mechanics of isotropic solids [4-6]... 

The KTHNY theory is based on linear continuum elastic theory [82], so that the statistical properties of the system are determined by the reduced elastic Hamiltonian... 

Numerical calculations of dislocation pair interactions have been carried out for systems of particles with / [89] and LJ [90] potentials. For the potential, Fisher et al. [89] find that the elastic dislocation interaction potential is accurate for dislocation separations as small as 3 lattice spacings, while Joos and Duesbery [90] find that separations of 30 lattice spacings are necessary to reach the asymptotic elastic limit. The adequacy of the continuum elastic approximation in describing the short-range interactions between defects is thus still something of an open question, and may depend on the range of the interparticle potential. 

Because of its use of linear continuum elastic theory, the KTHNY theory would be expected to describe the solid phase much better than the liquid phase. Also, the KTHNY theory neglects anharmonic effects other than those due to topological defects. Nontopological anharmonic excitations may make a significant contribution to the properties of the solid and liquid phases, and may be important in determining the nature of the 2D melting transition. 

It is assumed that the limit equilibrium state is reached if cracks develop and increase on the surface of the body volume under action of external loadings. In linear fracture mechanics, Irwin s force criterion and an equivalent Griffith s energy criterion completely determine the equilibrium condition of a continuum elastic body with a crack [9],... 

Contact mechanics is both an old and a modern field. Its classical domains of application are adhesion, friction, and fracture. Clearly, the relevance of the field for technical devices is enormous. Systematic strategies to control friction and adhesion between solid surfaces have been known since the stone age [1]. In modern times, the ground for systematic studies was laid in 1881 by Hertz in his seminal paper on the contact between soHd elastic bodies [2]. Hertz considers a sphere-plate contact. Solving the equations of continuum elasticity, he finds that the vertical force, F , is proportional to where S is the indentation. The sphere-plate contact forms a nonlinear spring with a differential spring constant k = dF/dS oc The nonhnearity occurs because there is a concentration of stress at the point of contact. Such stress concentrations - and the ensuing mechanical nonhnearities - are typical of contact mechanics. 

In this study, the influence of the tip velocity and of the applied load wa.s not shown. Nevertheless, the results obtained are very similar to what has been observed on other polymer films (see below). Furthermore, the authors had studied the elTcct of the molecular weight. Polystyrene film with chains ranging from 32000 to 573 000 were investigated. The authors did not And any obvious change in the dimension and patterns of the structures. This latter result suggests that a continuum elastic (or plastic) model can be used and that the size of the polymer chain has a rather poor influence, if any. 

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