Theoretical framework and experimental procedure for modelling mesoscopic volume fraction stochastic fluctuations in fiber reinforced composites

Many engineering materials exhibit fluctuations and uncertainties on their macroscopic mechanical properties. This randomness results from random fluctuations observed at a lower scale, especially at the meso-scale where microstructural uncertainties generally occur. In the present paper, we first propose a complete theoretical stochastic framework (that is, a relevant probabilistic model as well as a non-intrusive stochastic solver) in which the volume fraction at the microscale is modelled as a random field whose statistical reduction is performed using a Karhunen–Loeve expansion. Then, an experimental procedure dedicated to the identification of the parameters involved in the probabilistic model is presented and relies on a non-destructive ultrasonic method. The combination of the experimental results with a micromechanical analysis provides realizations of the volume fraction random field. In particular, it is shown that the volume fraction can be modelled by a homogeneous random field whose spatial correlation lengths are determined and may provide conditions on the size of the meso-volumes to be considered.     

Periodic solutions of nonlinear differential equations involving the method of scalar nonlinearities

Summary The recently developed method of scalar nonlinearities is applied to establish a new type of existence proof for periodic solutions of nonlinear differential equations. It is proved that given a periodic solution of a certain linear differential equation whose coefficients are subject to some nonlinear constraint, a nonlinear differential equation, which is closely related to the linear one, has a periodic solution (of the same period) as well. While, in general, the nonlinear equation will not be explicitly resolvable, the linear equation (with constraint) will allow for explicitly given solutions.The proof is carried out by constructing a homotopy (between appropriately chosen integral operators) and is based on Leray-Schauder theory. Thus, an essential hypothesis is the a-priori boundedness of certain intermediate problems. The very definition of the homotopy, which seems to be unprecedented in the literature, bears resemblance with the introduction of Dirac's-function.The theory is applied to Duffing's equation, resulting in an abstract existence statement as well as the explicit construction of numerically tractable intermediate problems.     

Equilibrium of the reaction between dissolved sodium sulfide and biologically produced sulfur

The equilibrium of the heterogeneous reaction between dissolved sodium sulfide and biologically produced sulfur particles has been studied. Biologically produced sulfur was obtained from a bioreactor of a hydrogen sulfide removal process in which the dominating organism is Thiobacillus sp. W5. Detailed knowledge of this reaction is essential to understand its effect on the process. The results were compared with the equilibrium of the reaction of sulfide with ‘inorganic’ elemental sulfur. The equilibrium between dissolved sodium sulfide and biologically produced sulfur particles can be described by an equilibrium constant, K_x, which consists of a weighted sum of constants for polysulfide ions of different chain length, rather than a true single equilibrium constant. For biologically produced sulfur pK_x = 9.10 ± 0.08 (21 °C) and 9.17 ± 0.09 (35 °C) with an average polysulfide chain length x = 4.91 ± 0.32 (21 °C) and 4.59 ± 0.31 (35 °C). The pK_x value for biologically produced sulfur is significantly higher than for reaction of dissolved sodium sulfide with inorganic sulfur (pK_x = 8.82; 21 °C). This difference is probably caused by the negatively charged polymeric organic layer, which is present on biologically produced sulfur but absent with “inorganic” sulfur. Specific binding of polysulfide ions to the organic layer results in a higher polysulfide concentration at the reaction site compared to the bulk concentration. This results in an apparent decrease of the measured equilibrium constant, K_x.     

Zum Mechanismus massenspektrometrischer Fragmentierungsreaktionen.—I Methoxygruppenwanderungen in den Massenspektren der Methyläther von linearen und verzweigten aliphatischen Polyalkoholen

The formation of 1,1-dimethoxy-alkyl rearrangement ions in the mass spectra of methyl ethers of linear diols, 1,2,3-triols and of derivatives of pentaglycerol and pentaerythritol has been investigated by deuterium labelling and mass measurements. Methoxy group migrations do not occur, or at least only to a small amount, in the mass spectra of the diol-dimethyl ethers. The mass spectra of methyl ethers of 1,2,3-triols exhibit characteristic peaks of the rearrangement ions +CH(OCH₃)₂ and +CR(OCH₃)₂. These ions arise by a 1,3-migration of a methoxy group, probably during a one step degradation of the molecular ion to give a molecule methyl alkenyl ether and a H-atom or alkyl radical as neutral fragments. Large peaks of the rearrangement ion +CH(OCH₃)₂ are observed in the mass spectra of compounds of the following type: A radical ion, formed by loss of HY from the molecular ion, rearranges by methoxy group migration in the mass spectra of these substances. Rearrangement by migration of hydroxy and acetoxy groups are also observed, but no migration of a Cl-atom.     

Zur Thermochemie der Zinnhalogenide

The enthalpy change of the reaction at 298 K between Br₂ (l) and Sn(c) in CS₂ as solvent giving SnBr₄ (s) has been determined by calorimetry to be (?374, 2±1.4) kJ·mol^?1, [(?89.45±0.33) kcal·mol^?1]. By the same method the heat of solution of SnBr₄ (c) in CS₂ has been found to be (11.9±0.3) kJ·mol^?1, [(2.84±0.08) kcal·mol^?1]. Combining these results, a value of (?386.1±1.5) kJ·mol^?1, [(?92.3±0.4) kcal·mol^?1] is derived for the standard heat of formation of SnBr₄ (c). Substituting this figure in the thermochemical cycle hitherto used for calculating the heat of formation of SnBr₄ (c) gives ?124.3 kcal·mol^?1 for the standard heat of formation of SnCl₄ (l), which is in reasonable agreement with a recent determination of this quantity⁸.     

Synthesis of erythro-alpha-amino beta-hydroxy carboxylic acid esters by diastereoselective photocycloaddition of 5-methoxyoxazoles with aldehydes

A new photoaldol route to alpha-amino-beta-hydroxy carboxylic acid esters is initiated by the photocycloaddition of aromatic or aliphatic aldehydes to 5-methoxyoxazoles. The 4-unsubstituted 5-methyloxazole 1 gave the cycloadducts 8a-f in high yields and excellent exo-diastereoselectivities. Hydrolysis of 8a-f gives the N-acetyl alpha-amino-beta-hydroxy esters 9a-f, which could be subsequently converted into the corresponding Z-didehydro alpha-amino acids 10a-f. Quartenary alpha-amino-beta-hydroxy esters 12, 14, 16, 18, and 20, which are stable against dehydration, were obtained from the 4-alkylated 5-methoxyoxazoles 2-6, in most cases highly erythro-selective due to the high degree of stereocontrol (exo) at the photocycloaddition (to give 11, 13, 15, 17, and 19) level. The relative configurations of the N-acetyl alpha-amino-beta-hydroxy esters were determined by NMR spectroscopy and comparison with chiral pool-derived compounds as well as by X-ray structure determination of the ester 23, formed by hydrolysis of the cycloadduct 22, derived from photocycloaddition of propionaldehyde to the isoleucine-derived oxazole 21.     

Ring expansion and hydrogen scrambling in the molecular ion of 3-methylthiophene

The ratio [M ? D]/{[M-D] + [M ? H]} in the 70 eV mass spectra of six deuterated 3-methylthiophenes has been determined. From these values the mole fractions of the molecular ions that lose hydrogen atoms specifically from the various positions of the molecule were calculated, as well as the mole fraction in which the hydrogen atoms are fully scrambled before hydrogen elimination. It appears that hydrogen atoms are mainly lost from a fully scrambled [C₅H₆S]⁺· ion and from the α-position of the original molecular ion. A deuterium isotope effect of 1·60 to 1·72 was calculated for the hydrogen elimination. The reaction was also studied at low electron energies. In order to determine the degree of scrambling in the [C₅H₅S]⁺ ions, some decomposition reactions of this ion were investigated.