Mathematical working space and paradigms as an analysis tool for the teaching and learning of analysis

ZDM – Mathematics Education - Mathematical working space (MWS) is a model that is used in research in mathematics education, particularly in the field of geometry. Some MWS elements are...     

Combination of nitroxide‐mediated polymerization and SET‐LRP for the synthesis of high molar mass branched and star‐branched poly(n‐butyl acrylate) characterized by size exclusion chromatography and rheology

Branched and star‐branched polymers were successfully synthesized by the combination of two successive controlled radical polymerization methods. A series of linear and star poly(n‐butyl acrylate)‐co‐poly(2‐(2‐bromoisobutyryloxy) ethyl acrylate) statistical copolymers, P(nBA‐co‐BIEA)_x, were first synthesized by nitroxide‐mediated polymerization (NMP at T > 100 °C). The subsequent polymerization of n‐butyl acrylate by single electron transfer‐living radical polymerization (SET‐LRP at T = 25 °C), initiated from the brominated sites of the P(nBA‐co‐BIEA)_x copolymer, produced branched or star‐branched poly(n‐butyl acrylate) (PnBA). Both types of polymerizations (NMP and SET‐LRP) exhibited features of a controlled polymerization with linear evolutions of logarithmic conversion versus time and number‐average molar masses versus conversion for final M_n superior to 80,000 g mol^?1. The branched and star‐branched architectures with high molar mass and low number of branches were fully characterized by size exclusion chromatography. The Mark–Houwink Sakurada relationship and the analysis of the contraction factor (g′ = ([η]_branched/[η]_linear)_M) confirmed the elaboration of complex PnBA. The zero‐shear viscosities of the linear, star‐shaped, branched, and star‐branched polymers were compared. The modeling of the rheological properties confirmed the synthesis of the branched architectures. © 2012 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem, 2012     

Non‐Pincer‐Type Mononuclear Scandium Alkylidene Complexes: Synthesis,Bonding, and Reactivity

The first non‐pincer‐type mononuclear scandium alkylidene complexes were synthesized and structurally characterized. These complexes exhibited short Sc?C bond lengths and even one of the shortest reported to date (2.1134(18) Å). The multiple character of the Sc?C bond was highlighted by a DFT calculation. This was confirmed by experimental reactivity study where the complex underwent [2+1] cycloaddition with elemental selenium and [2+2] cycloaddition with imine. DFT calculation also revealed a strong nucleophilic behavior of the alkylidene complex that was experimentally demonstrated by the C?H bond activation of phenylacetylene.     

Upscaling Forchheimer law

We investigate the high velocity flow in heterogeneous porous media. The model is obtained by upscaling the flow at the heterogeneity scale where the Forchheimer law is assumed to be valid. We use the method of multiple scale expansions, which gives rigorously the macroscopic behaviour without any prerequisite on the form of the macroscopic equations. We show that Forchheimer law does not generally survive upscaling. The macroscopic flow law is strongly non-linear and anisotropic. A 2-point Padé approximation of the flow law in the form of a Forchheimer law is given. However, this approximation is generally poor. These results are illustrated in two particular cases: a layered composite porous media and a composite constituted by a square array of circular porous inclusions embedded in a porous matrix. We show that non-linearities are sources of anisotropy.     

Mean field modelling of the plastic behaviour of co-continuous dual-phase alloys with strong morphological anisotropy

This work addresses the plastic flow properties of a composite material in which the reinforcing phase is continuous and cannot be suitably represented by isolated ellipsoidal inclusions. The dual-phase metal under consideration is composed of a network of Inconel-601 fibres infiltrated by pure aluminium. Hence, both phases exhibit elastic–plastic behaviour and are continuous in the three dimensions of space. The fibre network presents a large morphological anisotropy that is reflected in the mechanical response of the composite. The modelling is based on Eshelby’s equivalent inclusion theory. Strain partitioning between the phases is computed incrementally based on tangent operators derived from the isotropic response of individual phases. Assessment of the model relies on extensive experimental data. Uniaxial tensile tests, involving measurement of the Lankford coefficient, have been performed at various temperatures on samples containing different volume fractions of fibres. Measurement of the phase stresses by neutron diffraction supplements the information provided by the macroscopic stress–strain curves. It is demonstrated that predictions are valid only when the micro–macro averaging scheme accounts for the co-continuous character of the constitutive phases.     

Déformation cumulée tensorielle dans le référentiel en rotation logarithmiqueCumulated tensorial strain measure in the logarithmic rotating frame.

The time integration of strain rate tensor $\stackrel{?}{\mathit{D}}$ is a central problem in large transformations even if it is often an underlying one. The cumulated tensorial strains, obtained by the time integration of strain rate tensor $\stackrel{?}{\mathit{D}}$, allow the tackling of this problem from a geometrical point of view, and independently of material behaviour considerations. The time integration here takes place in the local objective frame defined by the logarithmic spin proposed by Lehmann et al. and Xiao et al. The numerical results obtained in a closed deformation path are presented here. The advantages and drawbacks of this novel integration for the development of behaviour laws are described. To cite this article: V. Mora et al., C. R. Mecanique 332 (2004).     

An improved sample-and-hold reconstruction procedure for estimation of power spectra from LDA data

Techniques for deriving the auto or power spectrum (PSD) of turbulence from laser Doppler anemometry (LDA) measurements are reviewed briefly. The low pass filter and step noise errors associated with the sample-and-hold process are considered and a discrete version of the low pass filter for the resampled signal is derived. This is then used to develop a procedure by which the PSD estimates obtained from sample and hold measurements can be corrected. The application of the procedures is examined using simulated data and the results show that the frequency range of the analysis can be extended beyond the Nyquist frequency based on the mean sample rate. The results are shown to be comparable to those obtained using the method of Nobach et al. (1998) but the new procedures are more straightforward to implement. The technique is then used to determine the PSD of real LDA data and the results are compared with those from a hot wire anemometer.