Fracture Surfaces and the Regularity of Inverses for BV Deformations

Motivated by nonlinear elasticity theory, we study deformations that are approximately differentiable, orientation-preserving and one-to-one almost everywhere, and in addition have finite surface energy. This surface energy E{\mathcal{E}} was used by the authors in a previous paper, and has connections with the theory of currents. In the present paper we prove that E{\mathcal{E}} measures exactly the area of the surface created by the deformation. This is done through a proper definition of created surface, which is related to the set of discontinuity points of the inverse of the deformation. In doing so, we also obtain an SBV regularity result for the inverse.     

Synthesis of two new thioesters bearing ferrocene: Vibrational characterization and ab initio calculations. X-ray crystal structure of S-(2-methoxyphenyl)ferrocenecarbothioate

Structural and conformational properties of S-benzyl ferrocenecarbothioate (I) and S-(2-methoxyphenyl) ferrocenecarbothioate (II) are analyzed using data obtained from X-ray diffraction, vibrational data and theoretical calculations. According to chemical quantum calculations, the synperiplanar and antiperiplanar forms are found as the first and second more stable conformations, respectively, for the title compounds. The geometric parameters and normal modes of vibration were calculated using a density functional theory method (B3LYP) and the 6-31+G∗∗ basis set for all atoms except for iron. For this atom the calculations were carried out with the Lanl2dz basis set. The calculated parameters are in good agreement with the corresponding X-ray diffraction values. The combined experimental and theoretical approach allows a consistent assignment for most of the fundamental modes.     

Cavitation,Invertibility, and Convergence of Regularized Minimizers in Nonlinear Elasticity

We prove that energy minimizers for nonlinear elasticity in which cavitation is allowed only at a finite number of prescribed flaw points can be obtained, in the limit as ε→0, by introducing micro-voids of radius ε in the domain at the prescribed locations and minimizing the energy without allowing for cavitation. This extends the result by Sivaloganathan, Spector, and Tilakraj (SIAM J. Appl. Math. 66:736–757, 2006) to the case of multiple cavities, and constitutes a first step towards the numerical simulation of cavitation (in the nonradially-symmetric case).      

Energy Estimates and Cavity Interaction for a Critical‐Exponent Cavitation Model

We consider the minimization of in a perforated domain of among maps that are incompressible (det ) and invertible, and satisfy a Dirichlet boundary condition u = g on ?Ω. If the volume enclosed by g (?Ω) is greater than |Ω|, any such deformation u is forced to map the small holes B_ε( a _i) onto macroscopically visible cavities (which do not disappear as ε → 0). We restrict our attention to the critical exponent p = n, where the energy required for cavitation is of the order of and the model is suited, therefore, for an asymptotic analysis (v₁,…, v_M denote the volumes of the cavities). In the spirit of the analysis of vortices in Ginzburg‐Landau theory, we obtain estimates for the “renormalized” energy showing its dependence on the size and the shape of the cavities, on the initial distance between the cavitation points a ₁,…, a _M, and on the distance from these points to the outer boundary ?Ω. Based on those estimates we conclude, for the case of two cavities, that either the cavities prefer to be spherical in shape and well separated, or to be very close to each other and appear as a single equivalent round cavity. This is in agreement with existing numerical simulations and is reminiscent of the interaction between cavities in the mechanism of ductile fracture by void growth and coalescence. © 2012 Wiley Periodicals, Inc.     

Operator algebras and a theorem of Dieudonne

LetA be aC*-algebra with second dualA″. Let (φ _n)(n=1,...) be a sequence in the dual ofA such that limφ _n(a) exists for eacha εA. In general, this does not imply that limφ _n(x) exists for eachx εA″. But if limφ _n(p) exists whenever p is the range projection of a positive self-adjoint element of the unit ball ofA, then it is shown that limφ _n(x) does exist for eachx inA″. This is a non-commutative generalisation of a celebrated theorem of Dieudonné. A new proof of Dieudonné’s theorem, for positive measures, is given here. The proof of the main result makes use of Dieudonné’s original theorem.     

4,6‐Dimethyl‐2‐(3‐pyrid­yl)quinolin‐5‐amine

The title compound, C₁₆H₁₅N₃, shows a hindrance effect between adjacent amino and methyl groups that leads to a structural distortion, which is reflected in the non‐planarity of the quinoline entity and in the bond angles and distances. The crystal packing consists of chains along the b axis sustained by an inter­molecular hydrogen bond between the amino group and the N atom of the pyridyl ring.     

Invertibility and Weak Continuity of the Determinant for the Modelling of Cavitation and Fracture in Nonlinear Elasticity

In this paper, we present and analyze a variational model in nonlinear elasticity that allows for cavitation and fracture. The main idea in unifying the theories of cavitation and fracture is to regard both cavities and cracks as phenomena of the creation of a new surface. Accordingly, we define a functional that measures the area of the created surface. This functional has relationships with the theory of Cartesian currents. We show that the boundedness of that functional implies sequential weak continuity of the determinant of the deformation gradient, and that the weak limit of one-to-one almost everywhere deformations is also one-to-one almost everywhere. We then use these results to obtain the existence of minimizers of variational models that incorporate elastic energy and this created surface energy, taking into account orientation-preserving and non-interpenetration conditions.