Elasticity model

In this section we find the derivative of the energy functional in the three-dimensional linear elasticity model. The derivative characterizes the behaviour of the energy functional provided that the crack length is changed. The crack is modelled by a part of the two-dimensional plane removed from a three-dimensional domain. In particular, we derive the Griffith formula. 

Alternatively, if detachment is associated with a brittle failure, then one must first determine if the fracture followed an elastic loading where an elastic model such as the JKR theory is appropriate or if it follows a plastic or elastic-plastic loading. In this latter case, the force needed to detach the particle from the substrate depends on the specific properties of the materials and the details of the deformations [63]. 

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 ... 

The preceding restrictions on engineering constants for orthotropic materials are used to examine experimental data to see if they are physically consistent within the framework of the mathematical elasticity model. For boron-epoxy composite materials, Dickerson and DiMartino [2-3] measured Poisson s ratios as high as 1.97 for the negative of the strain in the 2-direction over the strain in the 1-direction due to loading in the 1-direction (v 2)- The reported values of the Young s moduli for the two directions are E = 11.86 x 10 psi (81.77 GPa) and E2 = 1.33x10 psi (9.17 GPa). Thus,... 

Linear elastic models for rafting are reviewed in J. C. Chang, S. M. Allen. J... 

Whereas the quasi-chemical theory has been eminently successful in describing the broad outlines, and even some of the details, of the order-disorder phenomenon in metallic solid solutions, several of its assumptions have been shown to be invalid. The manner of its failure, as well as the failure of the average-potential model to describe metallic solutions, indicates that metal atom interactions change radically in going from the pure state to the solution state. It is clear that little further progress may be expected in the formulation of statistical models for metallic solutions until the electronic interactions between solute and solvent species are better understood. In the area of solvent-solute interactions, the elastic model is unfruitful. Better understanding also is needed of the vibrational characteristics of metallic solutions, with respect to the changes in harmonic force constants and those in the anharmonicity of the vibrations. 

Eckart, criteria, 264, 298 procedure, 267 Effective charge, 274, 276 Effective Hamiltonian, 226 Elastic model, excess entropy calculation from, 141 of a solid solution, 140 Electric correlation, 248 Electric field gradient, 188, 189 Electron (s), 200... 

Van Orman JA, Grove TL, Shimizu N (2001) Rare earth element diffusion in diopside influence of temperature, pressure and ionic radius, and an elastic model for diffusion in silicates. Contrib Mineral Petrol 141 687-703... 

Monte Carlo computer simulations were also carried out on filled networks [50,61-63] in an attempt to obtain a better molecular interpretation of how such dispersed fillers reinforce elastomeric materials. The approach taken enabled estimation of the effect of the excluded volume of the filler particles on the network chains and on the elastic properties of the networks. In the first step, distribution functions for the end-to-end vectors of the chains were obtained by applying Monte Carlo methods to rotational isomeric state representations of the chains [64], Conformations of chains that overlapped with any filler particle during the simulation were rejected. The resulting perturbed distributions were then used in the three-chain elasticity model [16] to obtain the desired stress-strain isotherms in elongation. 

The earliest approach to explain tubule formation was developed by de Gen-nes.168 He pointed out that, in a bilayer membrane of chiral molecules in the Lp/ phase, symmetry allows the material to have a net electric dipole moment in the bilayer plane, like a chiral smectic-C liquid crystal.169 In other words, the material is ferroelectric, with a spontaneous electrostatic polarization P per unit area in the bilayer plane, perpendicular to the axis of molecular tilt. (Note that this argument depends on the chirality of the molecules, but it does not depend on the chiral elastic properties of the membrane. For that reason, we discuss it in this section, rather than with the chiral elastic models in the following sections.)... 

H. Delingette, S. Cotin and N. Ayache, Efficient linear elastic models of soft tissues for realtime surgery simulation, Stud. Health Technol. Inform., 1999, 62, 100-101. 

Benham, C.J. (1977) Elastic model of supercoiling. Proc. Natl. Acad. Sci. USA 74, 2397-2401. Olson, W.K. and Zhurkin, V.B. (2000) Modeling DNA deformations. Curr. Opin. Struct. Biol. 10, 286-297. 

Pagano, N.J. and Tandon. G.P. (1990). Thermo-elastic model for multidirectional coated fiber composites Traction formulation. Composites Sci. Technol. 38, 247 269. 

Table 8.2 Input Parameters (Experimental) for Elastic Model...

Figure 8.15 Comparison of dimensionless curvature (q) predicted by elastic model and experimental data...

Sekimoto and Kawasaki showed that an elastic model with uniaxial symmetry can be linearly unstable if its lower surface is clamped and its upper surface is deformable [17,88]. The present author started with the free energy functional presented in Sect. 3 and found that it can be lowered in uniaxial gels by periodic folding of the surface [20,89]. Subsequent numerical simulations showed that the surface tends to touch and fold as corrugations grow [90]. 

Ciferri, A., and K. J. Smith Phase changes in fibrous macromolecular systems and associated elasticity. Model phase diagrams. J. Polymer Sci. Pt. A-2, 731 (1964). 

These results indicate that in the present linear elastic model, the limiting velocity for the screw dislocation will be the speed of sound as propagated by a shear wave. Even though the linear model will break down as the speed of sound is approached, it is customary to consider c as the limiting velocity and to take the relativistic behavior as a useful indication of the behavior of the dislocation as v — c. It is noted that according to Eq. 11.20, relativistic effects become important only when v approaches c rather closely. 

Manning, G.S. (1985) Packaged DNA An elastic model. Cell Biophys., 7, 57-89. 

Figure 3 presents the time evolution of tangential stresses at the surface and the core of the cylinder according to the viscoelastic and elastic models. The stress reverse can be explained as follows when the body dries, the drier surface attempts to shrink but is restrained by the wet core. The surface is stressed in tension and the core in compression and inelastic strain occurs. Later, under a surface with reduced shrinkage, the core dries and attempts to shrink causing the stress state to reverse [4],... 

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