Buckling model

Figure 4.26 Compressive failure models in high-petformance fibers (a) micro-buckling model and (b) misorientation model (after Kozey and Kumai 1994).

The possibility of diamond surface reconstruction without hydrogen termination has been discussed in Ref [461] based on an analogy with Si and Ge. The surface reconstruction models are Hanemann s (111) 2x1 buckling model [462], Tandy s (111) 2x1 model, Pandy s jr-bonded model [463], and Chadi s rr-bonded (111) 2 X 1 model [464]. It was suggested that the buckling and 7t-bonded structures are in equilibrium at temperatures above 1000 °C. In Ref [465], a reconstruction from an H-terminated (111) 1 x 1 to an H-free (111) 2 x 1 structures after heating at 1000 °C is described. 

Figure 2. Buckling model constant k versus EDAS/TEOS for xerogels ( ) and aerogels ( ),...

Experimentaly, the Si 2p core-level spectroscopies have been interpreted within the buckling model, the presence of two different charges on the two silicon atoms of the dimer leading to two core levels [21-24]. However, other explanations have been proposed within symmetric models. Artacho [9] and Redondo [12] have found an excellent agreement with experiment when the spin-orbit coupling is introduced. 

Because of the complexity of the 7x7 geometry, most theoretical research has been directed to the 2x1 reconstruction. Until recently, the commonly accepted model for this surface was a buckling model in which alternate rows of surface atoms are raised and lowered. 

E(k), for the surface states which is in disagreement with theory constrained to a buckled 2x1 reconstruction. Recently, total energy and force calculations also show that the buckling model is unfavorable. 

Buckles et al. suggested tentative structural assignments for 53a and 53b and their respective benzamido acids on the basis of ultraviolet spectral data and by comparison of physical properties with those of model compounds. They pointed out that it is not possible to establish structural relationships from configurations of the diastereomeric 2-benzamido-3-methoxy-3-phenylpropionic acids (54), each of which, on treatment with acetic anhydride, give mixtures of the azlactones. Similar observations have been made by others. ... 

This paper describes application of mathematical modeling to three specific problems warpage of layered composite panels, stress relaxation during a post-forming cooling, and buckling of a plastic column. Information provided here is focused on identification of basic physical mechanisms and their incorporation into the models. Mathematical details and systematic analysis of these models can be found in references to the paper. 

The numerical model consisted of two alternating procedures During the first one, the creep under applied dead load of a 2-dimensional bar, with an initial small deviation from the straight shape, was simulated the second procedure was the solution to the eigenvalue buckling problem for a bar with a shape developed due to the creep, this approach allowed a prediction of the buckling time with the assumption of the initial imperfection accepted as an unavoidable handicap. 

Buckled surface models for the two other copper sulfur phases were proposed that rationalised the STM, LEED and SEXAFS data (Figure 10.5). Note that the local adsorption site of the sulfur adatoms in both the p(3 x 2) and p(5 x 2) structures is the four-fold hollow in agreement with the SEXAFS18 measurements but looking at the structure from above with the STM, the periodicity of the structure suggests a variety of different adsorption sites. 

Figure 5.47 Electrostatic model for buckling of bilayer ribbon of length l and width w based on attraction between lines of charge on opposite edges.168 Left Side view. Right Top view of curvature of ribbon. Reprinted with permission from Ref. 139. Copyright 2001 by the American Chemical Society.

Micromechanics theories for closed cell foams are less well advanced for than those for open cell foams. The elastic moduli of the closed-cell Kelvin foam were obtained by Finite Element Analysis (FEA) by Kraynik and co-workers (a. 14), and the high strain compressive response predicted by Mills and Zhu (a. 15). The Young s moduli predicted by the Kraynik model, which assumes the cell faces remain flat, lie above the experimental data (Figure 7), while those predicted by the Mills and Zhu model, which assumes that inplane compressive stresses will buckle faces, lie beneath the data. The experimental data is closer to the Mills and Zhu model at low densities, but closer to the Kraynik theory at high foam densities. 

The faces in low density LDPE foams are partly buckled or wrinkled, as a result of processing (a.l7). This affects both the bulk modulus and the Young s modulus. The foam bulk modulus Kp is predicted, using the Kelvin closed cell foam model, to be ... 

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