The singularity in weakly nonlinear fracture mechanics

Material failure by crack propagation essentially involves a concentration of large displacement-gradients near a crack's tip, even at scales where no irreversible deformation and energy dissipation occurs. This physical situation provides the motivation for a systematic gradient expansion of general nonlinear elastic constitutive laws that goes beyond the first order displacement-gradient expansion that is the basis for linear elastic fracture mechanics (LEFM). A weakly nonlinear fracture mechanics theory was recently developed by considering displacement-gradients up to second order. The theory predicts that, at scales within a dynamic lengthscale ℓ from a crack's tip, significant logr displacements and 1/r displacement-gradient contributions arise. Whereas in LEFM the 1/r singularity generates an unbalanced force and must be discarded, we show that this singularity not only exists but is also necessary in the weakly nonlinear theory. The theory generates no spurious forces and is consistent with the notion of the autonomy of the near-tip nonlinear region. The J-integral in the weakly nonlinear theory is also shown to be path-independent, taking the same value as the linear elastic J-integral. Thus, the weakly nonlinear theory retains the key tenets of fracture mechanics, while providing excellent quantitative agreement with measurements near the tip of single propagating cracks. As ℓ is consistent with lengthscales that appear in crack tip instabilities, we suggest that this theory may serve as a promising starting point for resolving open questions in fracture dynamics.     

Quantum Information Entropies on Hyperbolic Single Potential Wells

In this work, we study the quantum information entropies for two different types of hyperbolic single potential wells. We first study the behaviors of the moving particle subject to two different hyperbolic potential wells through focusing on their wave functions. The shapes of these hyperbolic potentials are similar, but we notice that their momentum entropy densities change along with the width of each potential and the magnitude of position entropy density decreases when the momentum entropy magnitude increases. On the other hand, we illustrate the behaviors of their position and momentum entropy densities. Finally, we show the variation of position and momentum entropies Sx and Sp with the change of the potential well depth u and verify that their sum still satisfies the BBM inequality relation.     

Surface reconstruction of chiral glassy oligomers under the action of volatile organic compounds (VOCs)

The response of chiral polymers to volatile organic compounds (VOCs) was studied by atomic force microscopy. It was found that under the influence of toluene vapour, the focal conic domains on the surface of chiral polymer undergo remarkable reconstruction that is at least partially reversible. This opens new ways for building VOC sensors and using surface reconstruction in creating new surface patterns for nanotechnology.     

Newton–Hensel Interpolation Lifting

The main result of this paper is a new version of Newton-Hensel lifting that relates to interpolation questions. It allows one to lift polynomials in ℤ[x] from information modulo a prime number p ≠ 2 to a power p^k for any k, and its originality is that it is a mixed version that not only lifts the coefficients of the polynomial but also its exponents. We show that this result corresponds exactly to a Newton--Hensel lifting of a system of 2t generalized equations in 2t unknowns in the ring of p-adic integers ℤ_p. Finally, we apply our results to sparse polynomial interpolation in ℤ[x].     

Comparison of Physical and Photophysical Properties of Monometallic and Bimetallic Ruthenium(II) Complexes Containing Structurally Altered Diimine Ligands

The physical and photophysical properties of a series of monometallic, [Ru(bpy)(2)(dmb)](2+), [Ru(bpy)(2)(BPY)](2+), [Ru(bpy)(Obpy)](2+) and [Ru(bpy)(2)(Obpy)](2+), and bimetallic, [{Ru(bpy)(2)}(2)(BPY)](4+) and [{Ru(bpy)(2)}(2)(Obpy)](4+), complexes are examined, where bpy is 2,2'-bipyridine, dmb is 4,4'-dimethyl-2,2'-bipyridine, BPY is 1,2-bis(4-methyl-2,2'-bipyridin-4'-yl)ethane, and Obpy is 1,2-bis(2,2'-bipyridin-6-yl)ethane. The complexes display metal-to-ligand charge transfer transitions in the 450 nm region, intraligand pi --> pi transitions at energies greater than 300 nm, a reversible oxidation of the ruthenium(II) center in the 1.25-1.40 V vs SSCE region, a series of three reductions associated with each coordinated ligand commencing at -1.3 V and ending at approximately -1.9 V, and emission from a (3)MLCT state having energy maxima between 598 and 610 nm. The Ru(III)/Ru(II) oxidation of the two bimetallic complexes is a single, two one-electron process. Relative to [Ru(bpy)(2)(BPY)](2+), the Ru(III)/Ru(II) potential for [Ru(bpy)(2)(Obpy)](2+) increases from 1.24 to 1.35 V, the room temperature emission lifetime decreases from 740 to 3 ns, and the emission quantum yield decreases from 0.078 to 0.000 23. Similarly, relative to [{Ru(bpy)(2)}(2)(BPY)](4+), the Ru(III)/Ru(II) potential for [{Ru(bpy)(2)}(2)(Obpy)](4+) increases from 1.28 to 1.32 V, the room temperature emission lifetime decreases from 770 to 3 ns, and the room temperature emission quantum yield decreases from 0.079 to 0.000 26. Emission lifetimes measured in 4:1 ethanol:methanol were temperature dependent over 90-360 K. In the fluid environment, emission lifetimes display a biexponential energy dependence ranging from 100 to 241 cm(-)(1) for the first energy of activation and 2300-4300 cm(-)(1) for the second one. The smaller energy is attributed to changes in the local matrix of the chromophores and the larger energy of activation to population of a higher energy dd state. Explanations for the variations in physical properties are based on molecular mechanics calculations which reveal that the Ru-N bond distance increases from 2.05 ? (from Ru(II) to bpy and BPY) to 2.08 ? (from Ru(II) to Obpy) and that the metal-to-metal distance increases from approximately 7.5 ? for [{Ru(bpy)(2)}(2)(Obpy)](4+) to approximately 14 ? for [{Ru(bpy)(2)}(2)(BPY)](4+).     

Complex copper(II) fluorides. XIII. Ternary system BaF2?CuF2?GaF3 and Crystal Structure of Ba3CuGa2F14

The ternary system BaF₂? CuF₂? GaF₃ is investigated by X-ray diffraction experiments. It exhibits seven quaternary phases: six of which correspond to types previously evidenced in other BaF₂? CuF₂? MF₃ systems (three “polytypic” phases obtained by adding small amounts of GaF₃ to BaCuF₄, a tetragonal non-stoichiometric phase: Ba_3+x Ga_2?2xCu_2xF₁₂ and two stoichiometric fluorides: triclinic Ba₂CuGa₂F₁₂ and monoclinic Ba₁₀Cu₁₂GaF₄₇); the seventh compound Ba₃CuGa₂F₁₄ hitherto unknown, corresponds to a new structural type. It is monoclinic (pseudo-orthorhombic) space group P2₁/n with a = 7.402(3) Å, b = 27.88(1) Å, c = 5.521(2) Å, β = 90.12(3)°, Z = 4. The structure was solved from single crystal data using 5133 independent reflections (R = 0.047, R_w = 0.051). It is built up from infinite cis-chains of GaF₆ octahedra linked by monocapped trigonal prisms CuF₇.     

The crystal structure of Pb7FeIIFe6IIIF34: A new jarlite-type compound

Pb₇Fe^IIFeF₃₄ is monoclinic: a = 16.375(2) Å, b = 11.233(2) Å, c = 7.615(1) Å, β = 102.67(1)º, Z = 2. The crystal structure was solved in the space group C2/m (nº 12), from X-ray single crystal data using 957 independent reflections (705 with F/σ(F) > 4, leading to R = 0.038). It consists in infinite helicoidal [Fe^IIFeF₃₄]_n^14n? double-chains of cornersharing octahedra running along the b-axis and separated from each other by lead ions.