Surface Elastic Waves in Cubic Crystals

Many high-angle grain boundaries in cubic crystals are thought to be either coincidence boundaries (1) or coincidence boundaries to which grain boundary dislocations have been added (1,2). Calculations of the arrangement of atoms inside coincidence boundaries suggest that the coincidence lattice will usually not be continuous across a coincidence boundary (3). There will usually be a rigid displacement of the lattice on one side of the boundary relative to that on the other. This displacement gives rise to a stacking fault in the coincidence lattice.Recently, Pond (4) and Smith (5) have measured the lattice displacement at coincidence boundaries in aluminum. We have developed (6) an alternative to the measuring technique used by them, and have used it to find two of the three components of the displacement at {112} lateral twin boundaries in gold. This paper describes our method and presents a brief account of the results we have obtained.

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Microtwinning Evidence For The Apparent Five-Fold Symmetry in T2(Al6Li3Cu)

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100125178 ◽

1987 ◽

Vol 45 ◽

pp. 24-25

Author(s):

Kenneth S. Vecchio ◽

David B. Williams

Keyword(s):

Stable Equilibrium ◽

Equilibrium Phase ◽

Convergent Beam Electron Diffraction ◽

Icosahedral Structure ◽

Cubic Crystals ◽

Icosahedral Phases ◽

Heat Treated ◽

Zero Layer ◽

Convergent Beam ◽

Zr Alloy

Since the discovery in 1984 by Shechtman et al. of crystals which display apparent five-fold symmetry, extensive effort has been given to establishing a theoretical basis for the existence of icosahedral phases (eg.2.). Several other investigations have been centered on explaining these observations based on twinning of cubic crystals (eg.3.). Recently, the existence of a stable, equilibrium phase T2Al6 Li3Cu) possessing an icosahedral structure has been reported in the Al-Li-Cu system(4-6).In the present study an Al-2.6wt.%Li-l.5wt.%Cu-0.lwt.%Zr alloy was heat treated at 300°C for 100hrs. to produce large T2 precipitates. Convergent Beam Electron Diffraction (CBED) patterns were obtained from two-fold, three-fold, and apparent five-fold axes of T2 particles. Figure 1 shows the five-fold symmetric zero layer CBED pattern obtained from T2 particles.

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Remarks on nonlinear elastic waves with null conditions

ESAIM Control Optimisation and Calculus of Variations ◽

10.1051/cocv/2020039 ◽

2020 ◽

Vol 26 ◽

pp. 121

Author(s):

Dongbing Zha ◽

Weimin Peng

Keyword(s):

Initial Data ◽

Global Existence ◽

Elastic Waves ◽

Wave Equations ◽

Alternative Proof ◽

Classical Solutions ◽

Small Initial Data ◽

Hyperelastic Materials ◽

Variational Structure ◽

Nonlinear Elastic

For the Cauchy problem of nonlinear elastic wave equations for 3D isotropic, homogeneous and hyperelastic materials with null conditions, global existence of classical solutions with small initial data was proved in R. Agemi (Invent. Math. 142 (2000) 225–250) and T. C. Sideris (Ann. Math. 151 (2000) 849–874) independently. In this paper, we will give some remarks and an alternative proof for it. First, we give the explicit variational structure of nonlinear elastic waves. Thus we can identify whether materials satisfy the null condition by checking the stored energy function directly. Furthermore, by some careful analyses on the nonlinear structure, we show that the Helmholtz projection, which is usually considered to be ill-suited for nonlinear analysis, can be in fact used to show the global existence result. We also improve the amount of Sobolev regularity of initial data, which seems optimal in the framework of classical solutions.

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