Isotropic elastic continua

As shown in Figure 12.13, we have performed the pressure dependence study of the Ge-Ge modes of the NCs for one pressure cycle and there is no hysteresis observed. This confirms that the pressure-induced strain in the NCs is reversible. With this elastic behavior of our Ge/Si02 NCs-matrix system, we assume both the NCs and the matrix as isotropic elastic continua [46]. We modeled the NC as a sphere of radius ri in a spherical Si02 matrix of radius t2, where r2 2> ri. The system is subjected to a hydrostatic pressure P. Using spherical co-ordinates with the origin at the center of the NC sphere, our system has a spherical symmetry where the displacement vector u is everywhere radial and can be written d u = ar + b/r. The components of the strain tensor are Urr = a — and uee = =... 

In our simulation, we used a finite element package (ABAQUS/Standard) and an axisymmetric model stracture as shown in Figure 12.16, where the Ge NCs sphere is located along the rotational z-axis, and the Si substrate and Si02 matrix cylindrical layers are separated by a sharp interface. In the calculation of the elastic field distribution in the nanosystem, an external hydrostatic pressure of 10 kbar was assumed. For simplicity, the Ge NCs, Si02 matrix, and Si substrate were considered as isotropic elastic continua, with elastic constants of Ge (Pi = 1316.6 kbar, cti = 0.20748, Ki = 750.2 kbar) [32], Si02 (P2 = 730 kbar, [Pg.292]

III Simple continuum, such as isotropic elastic continuum. Including layering and empirical models Conventional laboratory and in-situ tests... 

The atoms in a Debye solid are treated as a system of weakly coupled harmonic oscillators. Normal modes with wavelengths that are large compared to the atomic spacing do not depend on the discrete nature of the crystal lattice, and consequently these normal modes can be obtained by treating the crystal as an isotropic elastic continuum. In the Debye treatment of a solid all of the normal modes are treated as elastic waves. The partition function for a Debye solid cannot be obtained In closed form, but the thermodynamic functions for a Debsre solid have been tabulated as a function of 9p/T- For the pair of Isotopic metals Li(s)... 

The qualitative basis of Eq. (8) is the assumption that ev i layer and chain structures behave like an isotropic elastic continuum for long wavelength vibrations. For shorter wavelength, vibrations in different layers or chains are assumed to be completely decoupled. The Debye expression derives from Eq, (8) if we assume ... 

When a themally expansible homogeneous isotropic elastic continuum is deformed iso thermally the mechanical work done on the body is partly stored as internal energy partly converted to heat. In the range of strain up to about 10 percent (which is typical of natural rubber) the internal energy storage exceeds the work done so that heat must be added to the sample. Beyond the strain of 10 percent heat is increasingly liberated. An equivalent statement conveys the information that if the sample be stretched adiabatically there is an initial temperature drop then subsequent rise. 

Next, let us compile some quantitative relations which concern the stress field and the energy of dislocations. Using elastic continuum theory and disregarding the dislocation core, the elastic energy, diS, of a screw dislocation per unit length for isotropic crystals is found to be... 

Note that for the asymptotic equations of Eqns. (2) and (3) to be valid, r

characteristic length, and is normally the crack length or the remaining ligament, whichever is the smaller, of a fracture specimen. Also, the above asymptotic equations are not valid for an orthotropic elastic continuum, such as a ceramic fiber/ceramic matrix composite. While the static crack tip state for an orthotropic elastic continuum has been derived, to the author s knowledge, no dynamic counterpart is available to date. Nevertheless, the above crack tip state should be applicable to particulate/whisker-filled ceramic matrix composites which macroscopically behave like an isotropic homogeneous continuum. 

We begin by examining what continuum mechanics might tell us about the structure and energetics of point defects. In this context, the point defect is seen as an elastic disturbance in the otherwise unperturbed elastic continuum. The properties of this disturbance can be rather easily evaluated by treating the medium within the setting of isotropic linear elasticity. Once we have determined the fields of the point defect we may in turn evaluate its energy and thereby the thermodynamic likelihood of its existence. 

The basic assumptions of fracture mechanics are (1) that the material behaves as a linear elastic isotropic continuum and (2) the crack tip inelastic zone size is small with respect to all other dimensions. Here we will consider the limitations of using the term K = YOpos Ttato describe the mechanical driving force for crack extension of small cracks at values of stress that are high with respect to the elastic limit. 

Let us examine the instability oi strained thin films. In Fig. 3, thin films of30 ML are coherently bonded to the hard substrates. The film phase has a misfit strain, e = 0.01, relative to the substrate phase, and the periodic length is equal to 200 a. The three interface energies are identical to each other = yiv = y = Y Both phases are elastically isotropic, but the shear modulus of the substrate is twice that of the film (p = 2p). On the left-hand side, an infinite-torque condition is imposed to the substrate-vapor and film-substrate interfaces, whereas torque terms are equal to zero on the right. In the absence of the coherency strain, these films are stable as their thickness is well over 16 ML. With a coherency strain, surface undulations induced by thermal fluctuations become growing waves. By the time of 2M, six waves are definitely seen to have established, and these numbers are in agreement with the continuum linear elasticity prediction [16]. 

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]... 

We now consider a more quantitative model of the vibrational density of states which makes a remarkable linkage between continuum and discrete lattice descriptions. In particular, we undertake the Debye model in which the vibrational density of states is built in terms of an isotropic linear elastic reckoning of the phonon dispersions. Recall from above that in order to effect an accurate calculation of the true phonon dispersion relation, one must consider the dynamical matrix. Our approach here, on the other hand, is to produce a model representation of the phonon dispersions which is valid for long wavelengths and breaks down at... 

Continuum Models of Point Defects with Tetragonal Symmetry One of the approaches we used to examine the structure and energetics of point defects from the continuum perspective was to represent the elastic consequences of the point defect by a collection of force dipoles. The treatment given in the chapter assumed that the point defect was isotropic. In this problem, derive an equation like that given in eqn (7.38). Assume that the forces at (a/2)ei and (a/2)e2 have strength /o while those at (a/2)e3 have strength /i. Evaluate the displacement fields explicitly and note how they differ from the isotropic case done earlier. 

The significance of the results given above may be seen in a number of different contexts. Indeed, the solution for the perfect screw dislocation in an isotropic linear elastic medium provides a window on much of what we will wish to know even in the more generic setting of an arbitrary curved dislocation. Further, this simple result also serves as the basis for our realization that all is not well with the continuum solution. One insight that emerges immediately from our consideration of the screw dislocation is the nature of the energies associated with dislocations. 

The Compliant Joint Model (CJM) was chosen for the mechanical calculations for this analysis. The CJM uses an equivalent continuum approach to model the behavior of jointed media. An equivalent continuum approach captures the average response of a jointed rock mass by distributing the response of the individual joints throughout the rock mass. The CJM in JAS3D can model up to four joint sets of arbitrary orientation, with the fractures in each set assumed to be parallel and evenly spaced. The intact rock between joints is treated as an isotropic linear-elastic material. More detailed descriptions of the CJM model can be found in Chen (1991). 

Constrained sintering can be well described macroscopically by using continuum mechanics. If a powder compact is elastically isotropic and linearly viscous, the constitutive equations are written as... 

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 JKR theory, similar to the Hertz theory, is a continuum theory in which two elastic semi-infinite bodies are in a non-conforming contact. Recently, the contact of layered solids has been addressed within the framework of the JKR theory. In a fundamental study, Sridhar et al. [32] analyzed the adhesion of elastic layers used in the SFA and compared it with the JKR analysis for a homogeneous isotropic half-space. As mentioned previously and depicted in Fig. 5, in SFA thin films of mica or polymeric materials ( i, /ji) are put on an adhesive layer Ej, I12) coated onto quartz cylinders ( 3, /i3). Sridhar et al. followed two separate approaches. In the first approach, based on finite element analysis, it is assumed that the thickness of the layers and their individual elastic constants are known in advance, a case which is rare. The adhesion characteristics, including the pull-off force are shown to depend not only on the adhesion energy, but also on the ratios of elastic moduli and the layers thickness. In the second approach, a procedure is proposed for calibrating the apparatus in situ to find the effective modulus e as a function of contact radius a. In this approach, it is necessary to measure the load, contact area... 

The phenomenological theory of rubberlike elasticity is based on continuum mechanics. It provides a mathematical structure from which, in principle, the deformation produced within a vulcanized elastomer by applied surface and bulk forces can be calculated. In the theory, the material is idealized by the assumption that it is perfectly elastic, isotropic in the undeformed state and incompressible. The most general form of the strain energy function (which vanishes at... 

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