A “quasi-elastic” affine formulation for the homogenised behaviour of nonlinear viscoelastic polycrystals and composites

The derivation of the overall behaviour of nonlinear viscoelastic (or rate-dependent elastoplastic) heterogeneous materials requires a linearisation of the constitutive equations around uniform per phase stress (or strain) histories. The resulting Linear Comparison Material (LCM) has to be linear thermoviscoelastic to fully retain the viscoelastic nature of phase interactions. Instead of the exact treatment of this LCM (i.e., correspondence principle and inverse Laplace transforms) as proposed by the “classical” affine formulation, an approximate treatment is proposed here. First considering Maxwellian behaviour, comparisons for a single phase as well as for two-phase materials (with “parallel” and disordered morphologies) show that the “direct inversion method” of Laplace transforms, initially proposed by Schapery (1962), has to be adapted to fit correctly exact responses to creep loading while a more general method is proposed for other loading paths. When applied to nonlinear viscoelastic heterogeneous materials, this approximate inversion method gives rise to a new formulation which is consistent with the classical affine one for the steady-state regimes. In the transient regime, it leads to a significantly more efficient numerical resolution, the LCM associated to the step by step procedure being no more thermoviscoelastic but thermoelastic. Various comparisons for nonlinear viscoelastic polycrystals responses to creep as well as relaxation loadings show that this “quasi-elastic” formulation yields results very close to classical affine ones, even for high contrasts.     

trans-[Pt(BCat')Me(PCy3)2]: an experimental case study of reductive elimination processes in Pt-Boryls through associative mechanisms

A stable trans‐(alkyl)(boryl) platinum complex trans‐[Pt(BCat′)Me(PCy₃)₂] (Cat′=Cat‐4‐tBu; Cy=cyclohexyl=C₆H₁₁) was synthesised by salt metathesis reaction of trans‐[Pt(BCat′)Br(PCy₃)₂] with LiMe and was fully characterised. Investigation of the reactivity of the title compound showed complete reductive elimination of Cat′BMe at 80 °C within four weeks. This process may be accelerated by the addition of a variety of alkynes, thereby leading to the formation of the corresponding η²‐alkyne platinum complexes, of which [Pt(η²‐MeCCMe)(PCy₃)₂] was characterised by X‐ray crystallography. Conversion of the trans‐configured title compound to a cis derivative remained unsuccessful due to an instantaneous reductive elimination process during the reaction with chelating phosphines. Treatment of trans‐[Pt(BCat′)Me(PCy₃)₂] with Cat₂B₂ led to the formation of CatBMe and Cat′BMe. In the course of further investigations into this reaction, indications for two indistinguishable reaction mechanisms were found: 1) associative formation of a six‐coordinate platinum centre prior to reductive elimination and 2) σ‐bond metathesis of B? B and C? Pt bonds. Mechanism 1 provides a straightforward explanation for the formation of both methylboranes. Scrambling of diboranes(4) Cat₂B₂ and Cat′₂B₂ in the presence of [Pt(PCy₃)₂], fully reductive elimination of CatBMe or Cat′BMe from trans‐[Pt(BCat′)Me(PCy₃)₂] in the presence of sub‐stoichiometric amounts of Cat₂B₂, and evidence for the reversibility of the oxidative addition of Cat₂B₂ to [Pt(PCy₃)₂] all support mechanism 2, which consists of sequential equilibria reactions. Furthermore, the solid‐state molecular structure of cis‐[Pt(BCat)₂(PCy₃)₂] and cis‐[Pt(BCat′)₂(PCy₃)₂] were investigated. The remarkably short B? B separations in both bis(boryl) complexes suggest that the two boryl ligands in each case are more loosely bound to the Pt^II centre than in related bis(boryl) species.     

Optimization of the collocation inversion method for the linear viscoelastic homogenization

The effective behaviour of linear viscoelastic heterogeneous material can be derived from the correspondence principle and the inversion of the obtained symbolic homogenized behavior. Various numerical methods were proposed to carry out this inversion. The collocation method, widely used, within this framework rests on a discretization of the characteristic spectrum in a sum of discrete lines for which it is necessary to determine the intensities and the positions by the minimization of the difference between the exact temporal function and its approximation. The classical method is based on a priori choice of the lines positions and on the optimization of their intensities. It is shown here that the combined optimization of the positions and the (positive) intensities lead to a minimization problem under constraints. In the simple case of an incompressible isotropic two-phase material, the assessment of the effective relaxation function with a continuum spectra or made up of discrete lines proves that the proposed method improves the predictions of the classical approach.     

The rationality of the Hilbert-Kunz multiplicity in graded dimension two

We show that the Hilbert-Kunz multiplicity is a rational number for an R₊−primary homogeneous ideal I=(f₁, . . . , f_n) in a two-dimensional graded domain R of finite type over an algebraically closed field of positive characteristic. More specific, we give a formula for the Hilbert-Kunz multiplicity in terms of certain rational numbers coming from the strong Harder-Narasimhan filtration of the syzygy bundle Syz(f₁, . . . , f_n) on the projective curve Y=ProjR.     

Atomic-level studies of superconducting and nonsuperconducting YBa2Cu3O7−x

Atomic-resolution images from superconducting and nonsuperconducting samples of YBa₂Cu₃O_7–x have been obtained by field ion microscopy. Both phases of the material exhibit a similar layered structure in the field ion images. Transmission electron micrographs of superconducting field ion tips show vividly the twinned structure of the orthorhombic phase. Time-of-flight (atom-probe) mass spectra from the two materials differ primarily in an enhanced molecular oxygen signal from the superconducting samples indicating that oxygen in the O(4) sites (CuO chains) is field desorbed preferentially as molecular oxygen ions.     

Evaluation of an ultrasonic nebulizer-membrane separation interface (USN-MEMSEP) with ICP-AES for the determination of trace elements by solvent extraction

The introduction of volatile organic solvents and metal organic complexes into an inductively coupled plasma (ICP) is problematic due to overloading and pyrolysis effects. These include carbon built up in the torch and spectral interferences. As a consequence, solvent extraction as a method for preconcentrating trace metals for the determination by ICP has been limited. In this report a commercial ultrasonic nebulizer-membrane separation interface (USN-MEMSEP) for the direct introduction and separation of organic solvents using ICP atomic emission spectrometry (AES) and a sequential spectrometer has been evaluated for solvent extraction of chelated trace metals. The ability of the MEMSEP to separate volatile organic flows from metal aerosols has been demonstrated by determining the recoveries of several transition metals in an oil-based methyl-isobutyl ketone (MIBK) standard relative to an aqueous solution. However, low recoveries of several metal chelates have been found evidently due to the volatilization of the organic metal species at the boiling point of MIBK (160° C). Moreover, the multielement capability and limits of detection have been limited due to sequential atomic emission detection. Advantages of the technique include enhanced limits of detection (LODs) and reduced plasma and spectral interferences.     

Lower bounds for three-dimensional nonoverlapping domain decomposition algorithms

Summary. Lower bounds for the condition numbers of the preconditioned systems are obtained for the wire basket preconditioner and the Neumann-Neumann preconditioner in three dimensions. They show that the known upper bounds are sharp. Received January 28, 2001 / Revised version received September 3, 2001 / Published online January 30, 2002 This work was supported in part by the National Science Foundation under Grant Nos. DMS-9600133 and DMS-0074246     

Effective conductivities of composite materials composed of cubic arrangements of spherical particles embedded in an isotropic matrix

Results are presented for the effective conductivities of simple, face-centered, and body-centered cubic arrays of isotropic spherical particles embedded in an isotropic medium possessing a different conductivity. With the use of a multipole expansion technique, numerical and conceptual errors in the prior work of others are corrected. Agreement with existing experimental data is found to be excellent.

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Zusammenfassung Für die Leitfähigkeit einer einfachen körper-zentrierten kubischen Anordnung von isotropen kugelfömigen Teilchen, die in einem isotropen Medium mit einer verschiedenen Lefähigkeit eingebettet sind, werden Resultate angegeben. Mit Hilfe einer Multipole-Entwicklung werden gewisse numerische und konzeptuelle Fehler früherer Arbeiten korrigiert. Sehr gute Uebereinstimmung mit den vorhandenen experimentellen Resultaten wurde festgestellt.

     

Measuring the motions in the human middle ear by Laser Doppler Vibrometry

For the detection of tiny motions which are caused by the tympanic membrane under normal hearing conditions, the touch-free method of Laser Doppler Vibrometry was used. Spectra containing information about the motions of the middle ear bones were recorded within 1 min when the umbo was chosen as the detection point and acoustic stimulation was performed via white noise excitation. It was observed that these spectra correlate to middle ear diseases, which had been artificially induced by manipulations in the chain of the middle ear bones in human temporal bones. The dosimetry of the applied laser radiation was found to be harmless to the tympanic membrane, which opened the way for successful in vivo measurements.