On the application of the Wiener–Hopf technique to problems in dynamic elasticity

Many problems in linear elastodynamics, or dynamic fracture mechanics, can be reduced to Wiener–Hopf functional equations defined in a strip in a complex transform plane. Apart from a few special cases, the inherent coupling between shear and compressional body motions gives rise to coupled systems of equations, and so the resulting Wiener–Hopf kernels are of matrix form. The key step in the solution of a Wiener–Hopf equation, which is to decompose the kernel into a product of two factors with particular analyticity properties, can be accomplished explicitly for scalar kernels. However, apart from special matrices which yield commutative factorizations, no procedure has yet been devised to factorize exactly general matrix kernels.

This paper shall demonstrate, by way of example, that the Wiener–Hopf approximant matrix (WHAM) procedure for obtaining approximate factors of matrix kernels (recently introduced by the author in [SIAM J. Appl. Math. 57 (2) (1997) 541]) is applicable to the class of matrix kernels found in elasticity, and in particular to problems in QNDE. First, as a motivating example, the kernel arising in the model of diffraction of skew incident elastic waves on a semi-infinite crack in an isotropic elastic space is studied. This was first examined in a seminal work by Achenbach and Gautesen [J. Acoust. Soc. Am. 61 (2) (1977) 413] and here three methods are offered for deriving distinct non-commutative factorizations of the kernel. Second, the WHAM method is employed to factorize the matrix kernel arising in the problem of radiation into an elastic half-space with mixed boundary conditions on its face. Third, brief mention is made of kernel factorization related to the problems of flexural wave diffraction by a crack in a thin (Mindlin) plate, and body wave scattering by an interfacial crack.     

Homogenization methods to approximate the effective response of random fibre-reinforced Composites

In this article a fibre-reinforced composite material is modelled via an approach employing a representative volume element with periodic boundary conditions. The effective elastic moduli of the material are thus derived. In particular, the method of asymptotic homogenization is used where a finite number of fibres are randomly distributed within the representative periodic cell. The study focuses on the efficacy of such an approach in representing a macroscopically random (hence transversely isotropic) material. Of particular importance is the sensitivity of the method to cell shape, and how this choice affects the resulting (configurationally averaged) elastic moduli. The averaging method is shown to yield results that lie within the Hashin–Shtrikman variational bounds for fibre-reinforced media and compares well with the multiple scattering and (classical) self-consistent approximations with a deviation from the latter in the larger volume fraction cases. Results also compare favourably with well-known experimental data from the literature.     

In situ variable temperature X-ray diffraction studies on the transformations of nano-precursors to La-Ni-O phases

In situ variable temperature XRD (VT-XRD) measurements on the transformation of nano-precursors to LaNiO phases are presented. Experimental results showed that LaNiO₃ and La₂NiO₄ phases were formed at ca. 700 °C via the reaction of La₂O₃ and NiO (from the initial nano-precursors), where a relatively low temperature of 700 °C was found for the synthesis of La₂NiO₄. The formation of La₃Ni₂O₇ at higher temperature (up to 1150 °C) appeared to proceed through a further reaction of La₂NiO₄ with unreacted NiO, whilst the formation of La₄Ni₃O₁₀ (at 1075 °C) proceeded via a further decomposition of LaNiO₃. Although phase pure La₃Ni₂O₇ and La₄Ni₃O₁₀ were not directly obtained under the processing conditions herein, the results of this study allow for a better understanding of formation pathways, particularly for the higher order La-Ni-O phases.     

Investigation of excited states of11B and7Li by means of thermal neutron capture on10B nuclei

Theγ-radiation from the¹⁰B(n,γ) reaction is studied using an unpolarized target. More accurate values for energies of transitions in¹¹B could be determined. No new levels have been found. TheQ value of this reaction: 11,454.1 (2)keV, is in agreement with earlier experiments. Also a new value for the cross section could be derived: 0.29 (4) barn, which is a factor 5 more accurate than earlier experiments. The¹⁰B(n, α)⁷ Li reaction, leading to the 478 keV state in⁷Li, is studied by means of polarized¹⁰B nuclei and polarized neutrons. The resulting anisotropy in the directional distribution of the⁷Li particles manifests itself in the Doppler broadening of the 478 keV line. Analysis of the line shape directly yields the conclusion, that the reaction proceeds for more than 96% through theJ=7/2 channel of¹¹B in case of destructive channel interference of theJ=5/2 channel. Constructive channel interference is only possible if the reaction proceeds for more than 99.5% through theJ=7/2 channel. It appeared that the outcomingα and⁷Li particles are emitted predominantly in directions perpendicular to the nuclear orientation axis.     

Evidence for Cluster Orbital Formation in CsSn2X5 Compounds (X=Cl, Br)

The formation of cluster orbitals in CsSn₂Br₅ is discussed and related more generally to tetragonal compounds of the type AB₂X₅ (A=monovalent cation; B=Sn, Pb; X=Cl, Br, I). The crystal structures of CsSn₂Cl₅ and CsSn₂Br₅ have been solved by single-crystal X-ray diffraction. These compounds are isostructural with each other and a range of AB₂X₅ structural analogues. In many AB₂X₅ compounds where B is a subvalent main group metal a tetragonal cell is observed with space group I4/mcm. The structures of CsSn₂Br₅ and CsSn₂Cl₅ are layered with polymeric sheets of [Sn₂X₅]ⁿ⁻_n separated by the Cs⁺ cations. Stereochemical considerations suggest that stabilization of this structural form, rather than the more ionic NH₄Pb₂Cl₅ or NaSn₂Cl₅ structures, is through interaction of the “nonbonding” valence electron pairs on tin with low-lying empty d-orbitals on neighboring X atoms. Electronic structure calculations based on the structural data confirm the likelihood of cluster orbital formation. Crystal data: CsSn₂Cl₅, tetragonal, I4/mcm, a=8.153(1) Å, c=14.882(4) Å, Z=4, R₁=0.0215, wR₂=0.0503 [I>2σ(I)], R₁=0.0393, wR₂=0.0536 (all data); CsSn₂Br₅, tetragonal, I4/mcm, a=8.483(6) Å, c=15.28(2) Å, Z=4, R₁=0.0607, wR₂=0.1411 [(I>2σ(I)], R₁=0.1579, wR₂=0.1677 (all data).     

K2[HCr2AsO10]: redetermination of phase II and the predicted structure of phase I

Our prediction that phase II of dipotassium hydrogen chromatoarsenate, K₂[HCr₂AsO₁₀], is ferroelectric, based on the analysis of the atomic coordinates by Averbuch‐Pouchot, Durif & Guitel [Acta Cryst. (1978), B 34 , 3725–3727], led to an independent redetermination of the structure using two separate crystals. The resulting improved accuracy allows the inference that the H atom is located in the hydrogen bonds of length 2.555 (5) Å which form between the terminal O atoms of shared AsO₃OH tetrahedra in adjacent HCr₂AsO₁₀²⁻ ions. The largest atomic displacement of 0.586 Å between phase II and the predicted paraelectric phase I is by these two O atoms. The H atoms form helices of radius ∼0.60 Å about the 3₁ or 3₂ axes. Normal probability analysis reveals systematic error in seven or more of the earlier atomic coordinates.