Shape sensitivity analysis of stress intensity factors by Generalized J-integrals

We consider an elastic plate with the non-deformed shape Ω_Σ := Ω \ Σ, where Ω is a domain bounded by a smooth closed curve Γ and Σ ⊂ Ω is a curve with the end points {γ₁, γ₂}. If the force g is given on the part Γ_N of Γ, the displacement u is fixed on Γ_D := Γ \ Γ_N and the body force f is given in Ω, then the displacement vector u(x) = (u₁(x), u₂(x)) has unbounded derivatives (stress singularities) near γ_k, k = 1, 2    u(x) = ∑²_{k, l=1} K_l(γ_k)r^1/2_kS^C_kl(θ_k) + u_R(x)     near γ_k. Here (r_k, θ_k) denote local curvilinear polar co-ordinates near γ_k, k = 1, 2, S^C_kl (θ_k) are smooth functions defined on [−π, π] and u_R(x) ∈ {H²(near γ_k)}². The constants K_l(γ_k),   l = 1, 2, which are called the stress intensity factors at γ_k (abbr. SIFs), are important parameters in fracture mechanics. We notice that the stress intensity factors K_l(γ_k) (l = 1, 2;  k = 1, 2) are functionals K_l(γ_k) = K_l(γ_k; ℒ︁, Ω, Σ) depending on the load ℒ︁, the shape of the plate Ω and the shape of the crack Σ. We say that the crack Σ is safe, if K_l(γ_k; Ω)² + K₂(γ_k; Ω)² < RẼ. By a small change of Ω the shape Σ can change to a dangerous one, i.e. we have K₁(γ_k; Ω)² + K₂(γ_k; Ω)² ⩾ RẼ. Therefore it is important to know how K_l(γ_k) depends on the shape of Ω. For this reason, we calculate the Gâteaux derivative of K_l(γ_k) under a class of domain perturbations which includes the approximation of domains by polygonal domains and the Hadamard's parametrization Γ(τ) := {x + τϕ(x)n(x);  x ∈ Γ}, where ϕ is a function on Γ and n is the outward unit normal on Γ. The calculations are quite delicate because of the occurrence of additional stress singularities at the collision points {γ₃, γ₄} = Γ _D ∩ Γ _N. The result is derived by the combination of the weight function method and the Generalized J-integral technique (abbr. GJ-integral technique). The GJ-integrals have been proposed by the first author in order to express the variation of energy (energy release rate) by extension of a crack in a 3D-elastic body. This paper begins with the weak solution of the crack problem, the weight function representation of SIF's, GJ-integral technique and finish with the shape sensitivity analysis of SIF's. GJ-integral J_ω(u; X) is the sum of the P-integral (line integral) P_ω(u, X) and the R-integral (area integral) R_ω(u, X). With the help of the GJ-integral technique we derive an R-integral expression for the shape derivative of the potential energy which is valid for all displacement fields u ∈ H¹. Using the property that the GJ-integral vanishes for all regular fields u ∈ H² we convert the R-integral expression for the shape derivative to a P-integral expression. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.     

Quantum Bitcoin Mining

This paper studies the effect of quantum computers on Bitcoin mining. The shift in computational paradigm towards quantum computation allows the entire search space of the golden nonce to be queried at once by exploiting quantum superpositions and entanglement. Using Grover’s algorithm, a solution can be extracted in time O(2256/t), where t is the target value for the nonce. This is better using a square root over the classical search algorithm that requires O(2256/t) tries. If sufficiently large quantum computers are available for the public, mining activity in the classical sense becomes obsolete, as quantum computers always win. Without considering quantum noise, the size of the quantum computer needs to be ≈104 qubits.     

Density distributions in the water shell of myoglobin

Two different approaches were used to explain the distribution of water coordinates in the crystallographically invisible part of the unit cell. Monte Carlo calculations were done starting from different initial water structures. Extended Monte Carlo calculations using equal initial structures were also used to obtain two further water structures. The differences between the Monte Carlo water structures were used to calculate the mean-square displacements of the water molecules. Monte Carlo calculations starting from different structures lead to a mean-square displacement of 0.58 Ų, whereas extended Monte Carlo moves using the same initial structure only show a mean-square displacement of 0.17 Ų. The mean-square displacement of 0.58 Ų can be used to explain the experimental data. © 1996 John Wiley & Sons, Inc.     

Optical Probes for Monitoring Intercalation and Exfoliation in Melt‐Processed Polymer Nanocomposites

Summary: Laser‐induced fluorescence spectroscopy of the optical probe Nile Blue A in polymer clay nanocomposites is described. Concentration quenching of the fluorescence dominates the probe behavior until the clay platelets are physically separated by polymer intercalation. Further separation into an exfoliated structure results in an intense increase in probe fluorescence. Preliminary results indicate the ability to discriminate between intercalated and exfoliated structures in nanocomposites formed by melt processing.

Polyamide 6 nanocomposites: Purple, 1 minute processing (left). Red, 7 minute processing (right).     

The Influence of the Preparation Method on the Physico-Chemical Properties and Catalytic Activities of Ce-Modified LDH Structures Used as Catalysts in Condensation Reactions

Mechanical activation and mechanochemical reactions are the subjects of mechanochemistry, a special branch of chemistry studied intensively since the 19th century. Herein, we comparably describe two synthesis methods used to obtain the following layered double hydroxide doped with cerium, Mg₃Al_0.75Ce_0.25(OH)₈(CO₃)_0.5·2H₂O: the mechanochemical route and the co-precipitation method, respectively. The influence of the preparation method on the physico-chemical properties as determined by multiple techniques such as XRD, SEM, EDS, XPS, DRIFT, RAMAN, DR-UV-VIS, basicity, acidity, real/bulk densities, and BET measurements was also analyzed. The obtained samples, abbreviated HTCe-PP (prepared by co-precipitation) and HTCe-MC (prepared by mechanochemical method), and their corresponding mixed oxides, Ce-PP (resulting from HTCe-PP) and Ce-MC (resulting from HTCe-MC), were used as base catalysts in the self-condensation reaction of cyclohexanone and two Claisen–Schmidt condensations, which involve the reaction between an aromatic aldehyde and a ketone, at different molar ratios to synthesize compounds with significant biologic activity from the flavonoid family, namely chalcone (1,3-diphenyl-2-propen-1-one) and flavone (2-phenyl-4H-1benzoxiran-4-one). The mechanochemical route was shown to have indisputable advantages over the co-precipitation method for both the catalytic activity of the solids and the costs.     

Homotopy after tensoring with uniformly hyperfiniteC*-algebras

The paper is devoted to the homotopy classification ofC*-algebras of continuous functions on a finite CW-complex with values in a UHF-algebra. The relevant invariants are based on (connective)K-theory.     

Branched phospha[7]triangulanes

A highly strained, thermally stable (up to 150 degrees C) branched phospha[7]triangulane was synthesized from the second-generation bicyclopropylidene and transient phosphinidene [Ph-P=W(CO)5], followed by demetalation in refluxing xylene. Bulkier transient CuCl-alkene-complexed phosphinidene gave 2-phosphabicyclo-[3.2.0]hept-1(5)-ene as an additional product. The "outer sphere" spirocyclopropanes provide a stabilizing factor for both of these novel compounds.     

Structure and stability of Ne+He(n): experiment and diffusion quantum Monte Carlo theory with "on the fly" electronic structure

New data are reported for the mass-spectrometry fragmentation patterns of helium clusters, either pure or containing a Ne or an Ar atom. The patterns for He(n)+ and Ar+He(n) show clear evidence of structure, while those of Ne+He(n) do not. To better understand the surprising result for the Ne+He(n) fragments, diffusion quantum Monte Carlo (DMC) calculations of the energies and structural properties of these ions were performed using a diatomics-in-molecule (DIM) parametrization of the potential energy. Using DIM for electronic energy evaluation allows us to sample 10(9) configurations even for a cluster as large as Ne+He14. The results of the DMC calculation are very surprising. For n > 7, the DMC random walkers rarely venture within 100 cm(-1) of the minimum potential energy. Analysis of the resulting particle density distributions shows that the zero-point energy does more than spread the wave function around the potential-energy minima, resulting in very diffuse wave functions. For some of the clusters the quantum effects nearly exclude the region of the potential minimum from the overall wave function. An important result of this effect is that the incremental bonding energy of the nth helium atom varies quite smoothly with n, for n > 5. This eliminates the expected shell structure and explains the lack of magic-number-type features in the data.