Variational Formulation of Softening Phenomena in Fracture Mechanics: The One‐Dimensional Case

. We show that discrete models of atoms subject to nearest‐neighbour non‐linear interactions approximate continua allowing for softening and fracture. A detailed study of local minima and stationary points is carried out. Scale effects are discussed. (Accepted April 2, 1998)     

Synthesis of Regioisomeric Functionalized Benzodifurans and Angelicins

Arenofurans have important biological and pharmacological activities. Compared to benzofurans, the reports on the synthesis of benzodifurans are rather limited. Here, we report the synthesis of a linear and an angular 3,3′‐bis(carboxymethyl)substituted benzodifuran and 4′‐carboxymethyl‐substituted angelicins from phloroglucinol, using 4‐halomethyl‐substituted dipyrones as key intermediates in the synthetic route. This strategy shows that the stability of a pyrone ring depends on the type of substituent at C(4) and the conditions used.     

A Variational Model for Plastic Slip and Its Regularization via Γ-Convergence

A variational model is presented able to interpret the onset of plastic deformations, here modeled as displacement jumps occurring along slip surfaces at constant yielding stress. The corresponding strain energy functional, leading to a free-discontinuity problem set in the space of SBV functions, is then approximated by a sequence of regularized elliptic functionals following the seminal work by Ambrosio and Tortorelli (Commun. Pure Appl. Math. 43, 999–1036, 1990) within the framework of Γ-convergence. Comparisons between the results obtainable with the free-discontinuity model and its regularized approximation, in terms of stability of the pure elastic phase, irreversibility of plastic slip and response under unloading, are presented, in general, for the 2-D case of antiplane shear and exemplified, in particular, for the 1-D case.     

The Variational Approach to Fracture Mechanics. A Practical Application to the French Panthéon in Paris

Recently, Francfort and Marigo (J. Mech. Phys. Solids 46, 1319–1342, 1998) have proposed a novel approach to fracture mechanics based upon the global minimization of a Griffith-like functional, composed of a bulk and a surface energy term. Later on the same authors, together with Bourdin, introduced (in J. Mech. Phys. Solids 48, 797–826, 2000) a variational approximation (in the sense of Γ-convergence) of such functional, essentially for computational purposes. Here, we utilize this new variational approach to show how it might be altered to incorporate the idea of less brittle, “deviatoric-type fracture” and apply to materials such as confined stone. To do so, we modify the original formulation of Francfort and Marigo, in particular its approximation of Bourdin, Francfort and Marigo, to only allow for discontinuities in the deviatoric part of the strain. We apply such modified model to gain insight on the deterioration and cracking in the ashlar masonry work of the French Panthéon, which are so repetitious and particular to be a distinguishable symptom of ongoing damage. Numerical experiments are performed and the results compared to those obtained using the original Francfort-Marigo model and to actual crack patterns from the Panthéon. The modified formulation allows one to reproduce fracture paths surprisingly similar to that observed in situ, to sort out the possible causes of damage, and to confirm, with a quantitative analysis, the main structural deficiencies in the French monument. This practical example enhances the importance of this promising new theory based in the mathematical sciences.     

Effects of MgCl2 Crystallographic Structure on Active Centre Formation in Immobilized Single‐Centre and Ziegler–Natta Catalysts for Ethylene Polymerization

The ability of a MgCl₂ support to activate a transition metal catalyst has been found to depend both on the crystallographic structure of the support and on the nature of the catalyst. A high degree of crystallographic disorder can be very effective for the immobilization and activation of titanium and vanadium complexes, but is not necessarily effective for zirconocene activation. A highly disordered support prepared by the reaction of MgBu₂ with HCl gave high activity with TiCl₄ but low activity with (n‐PrCp)₂ZrCl₂. High polymerization activities with the zirconocene were only obtained with supports of type MgCl₂/AlR_n(OEt)_3−n prepared from the reaction of AlR₃ with MgCl₂ · 1.1EtOH. These supports are characterized by additional peaks in the X‐ray diffraction pattern, indicating the presence of a crystalline structure which is absent in the other supports and contains highly Lewis acidic sites able to generate the active metallocenium species.

     

Computer-Assisted Methods for the Study of Stationary Solutions in Dissipative Systems, Applied to the Kuramoto–Sivashinski Equation

We develop some computer-assisted techniques for the analysis of stationary solutions of dissipative partial differential equations, of their stability, and of their bifurcation diagrams. As a case study, these methods are applied to the Kuramoto–Sivashinski equation. This equation has been investigated extensively, and its bifurcation diagram is well known from a numerical point of view. Here, we rigorously describe the full graph of solutions branching off the trivial branch, complete with all secondary bifurcations, for parameter values between 0 and 80. We also determine the dimension of the unstable manifold for the flow at some stationary solution in each branch.     

The Aminotriazole Antagonist Cmpd‐1 Stabilises a Distinct Inactive State of the Adenosine 2A Receptor

The widely expressed G‐protein coupled receptors (GPCRs) are versatile signal transducer proteins that are attractive drug targets but structurally challenging to study. GPCRs undergo a number of conformational rearrangements when transitioning from the inactive to the active state but have so far been believed to adopt a fairly conserved inactive conformation. Using ¹⁹F NMR spectroscopy and advanced molecular dynamics simulations we describe a novel inactive state of the adenosine 2A receptor which is stabilised by the aminotriazole antagonist Cmpd‐1. We demonstrate that the ligand stabilises a unique conformation of helix V and present data on the putative binding mode of the compound involving contacts to the transmembrane bundle as well as the extracellular loop 2.     

Bifurcation Instability in Linear Elasticity with the Constraint of Local Injectivity

There are problems in linear elasticity theory whose corresponding deformations, usually associated with singular stress fields, are open to question because they are not one-to-one and predict self-intersection. Recently, a theory has been advanced to handle such situations, which consists in minimizing the quadratic energy functional of linear elasticity subject to the constraint of local injectivity. In particular, the Jacobian of the deformation gradient is required to be not less than an arbitrarily small positive quantity, and, thus, the local orientation is preserved. Here, this theory is applied to the classical Lekhnitskii problem of an elastic aelotropic circular disk which is loaded on its boundary by a uniform radial pressure. Without the injectivity constraint, this classical linear problem has a unique solution. This example, with the injectivity constraint, already has been considered in previous works, but radial symmetry was assumed in order to reduce the problem from 2D to 1D. Here, making use of an interior penalty formulation, a numerical scheme is implemented that solves a full 2D problem. Remarkably, it is shown that there are values of the material moduli for which the minimal potential energy solution is no longer symmetric, producing a strong azimuthal shear and nominally a 180° rotation of an internal central core of the disk. Although the elastic strain energy is quadratic and convex, the strongly nonlinear character of the constraint allows for bifurcation instabilities. We gratefully acknowledge the partial support of the Minnesota Supercomputing Institute and the Italian “Ministero per l’Università e la Ricerca Scientifica” under the program PRIN 2005 “Affidabilità di elementi in vetro strutturale: indagini teoriche e sperimentali sulla risposta termo-meccanica del materiale e di strutture trasparenti di tipo innovativo”. R.F. gratefully acknowledges the Department of Civil and Environmental Engineering at the Politecnico di Bari, Italy, for their kind hospitality and support during his visit of 2006. We appreciate the helpful comments and suggestions of Paolo Podio-Guidugli on an earlier draft of this work.     

Time-discretization and global solution for a doubly nonlinear Volterra equation

This note addresses the analysis of an abstract doubly nonlinear Volterra equation with a nonsmooth kernel and possibly unbounded and degenerate operators. By exploiting a suitable implicit time-discretization technique, we obtain the existence of a global strong solution. As a by-product, the discrete scheme is proved to be conditionally stable and convergent.     

Characterization of phospho-proteins in bovine and buffalo caseins using atmospheric pressure ionization mass spectrometry

The specific bovine and buffalo caseins have been characterized with the aim of using atmospheric pressure ionization mass spectrometry as an alternative to electrophoretic and immunological methods in the detection of the fraudulent addition of bovine milk to buffalo milk used in the production of speciality cheeses.