A Griffith-Euler-Bernoulli theory for thin brittle beams derived from nonlinear models in variational fracture mechanics

We study a planar thin brittle beam subject to elastic deformations and cracks described in terms of a nonlinear Griffith energy functional acting on $SBV$ deformations of the beam. In particular we consider the case in which elastic bulk contributions due to finite bending of the beam are comparable to the surface energy which is necessary to completely break the beam into several large pieces. In the limit of vanishing aspect ratio we rigorously derive an effective Griffith-Euler-Bernoulli functional which acts on piecewise $W^{2,2}$ regular curves representing the midline of the beam. The elastic part of this functional is the classical Euler-Bernoulli functional for thin beams in the bending dominated regime in terms of the curve's curvature. In addition there also emerges a fracture term proportional to the number of discontinuities of the curve and its first derivative

On the infinite particle limit in Lagrangian dynamics and convergence of optimal transportation meshfree methods

We consider Lagrangian systems in the limit of infinitely many particles. It is shown that the corresponding discrete action functionals Gamma-converge to a continuum action functional acting on probability measures of particle trajectories. Also the convergence of stationary points of the action is established. Minimizers of the limiting functional and, more generally, limiting distributions of stationary points are investigated and shown to be concentrated on orbits of the Euler-Lagrange flow. We also consider time discretized systems. These results in particular provide a convergence analysis for optimal transportation meshfree methods for the approximation of particle flows by finite discrete Lagrangian dynamics

Ground states of the 2D sticky disc model: fine properties and N3/4N^{3/4} law for the deviation from the asymptotic Wulff shape

We investigate ground state configurations for a general finite number $N$ of particles of the Heitmann-Radin sticky disc pair potential model in two dimensions. Exact energy minimizers are shown to exhibit large microscopic fluctuations about the asymptotic Wulff shape which is a regular hexagon: There are arbitrarily large $N$ with ground state configurations deviating from the nearest regular hexagon by a number of $\sim N^{3/4}$ particles. We also prove that for any $N$ and any ground state configuration this deviation is bounded above by $\sim N^{3/4}$. As a consequence we obtain an exact scaling law for the fluctuations about the asymptotic Wulff shape. In particular, our results give a sharp rate of convergence to the limiting Wulff shape

Closure and commutability results for Gamma-limits and the geometric linearization and homogenization of multi-well energy functionals

Under a suitable notion of equivalence of integral densities we prove a $\Gamma$-closure theorem for integral functionals: The limit of a sequence of $\Gamma$-convergent families of such functionals is again a $\Gamma$-convergent family. Its $\Gamma$-limit is the limit of the $\Gamma$-limits of the original problems. This result not only provides a common basic principle for a number of linearization and homogenization results in elasticity theory. It also allows for new applications as we exemplify by proving that geometric linearization and homogenization of multi-well energy functionals commute

Biosynthetic pathway of mitochondrial ATPase subunit 9 in Neurospora crassa

Subunit 9 of mitochondrial ATPase (Su9) is synthesized in reticulocyte lysates programmed with Neurospora poly A-RNA, and in a Neurospora cell free system as a precursor with a higher apparent molecular weight than the mature protein (Mr 16,400 vs. 10,500). The RNA which directs the synthesis of Su9 precursor is associated with free polysomes. The precursor occurs as a high molecular weight aggregate in the postribosomal supernatant of reticulocyte lysates. Transfer in vitro of the precursor into isolated mitochondria is demonstrated. This process includes the correct proteolytic cleavage of the precursor to the mature form. After transfer, the protein acquires the following properties of the assembled subunit: it is resistant to added protease, it is soluble in chloroform/methanol, and it can be immunoprecipitated with antibodies to F1-ATPase. The precursor to Su9 is also detected in intact cells after pulse labeling. Processing in vivo takes place posttranslationally. It is inhibited by the uncoupler carbonylcyanide m- chlorophenylhydrazone (CCCP). A hypothetical mechanism is discussed for the intracellular transfer of Su9. It entails synthesis on free polysomes, release of the precursor into the cytosol, recognition by a receptor on the mitochondrial surface, and transfer into the inner mitochondrial membrane, which is accompanied by proteolytic cleavage and which depends on an electrical potential across the inner mitochondrial membrane

On a discrete-to-continuum convergence result for a two dimensional brittle material in the small displacement regime

We consider a two-dimensional atomic mass spring system and show that in the small displacement regime the corresponding discrete energies can be related to a continuum Griffith energy functional in the sense of Gamma-convergence. We also analyze the continuum problem for a rectangular bar under tensile boundary conditions and find that depending on the boundary loading the minimizers are either homogeneous elastic deformations or configurations that are completely cracked generically along a crystallographic line. As applications we discuss cleavage properties of strained crystals and an effective continuum fracture energy for magnets

An analysis of crystal cleavage in the passage from atomistic models to continuum theory

We study the behavior of atomistic models in general dimensions under uniaxial tension and investigate the system for critical fracture loads. We rigorously prove that in the discrete-to-continuum limit the minimal energy satisfies a particular cleavage law with quadratic response to small boundary displacements followed by a sharp constant cut-off beyond some critical value. Moreover, we show that the minimal energy is attained by homogeneous elastic configurations in the subcritical case and that beyond critical loading cleavage along specific crystallographic hyperplanes is energetically favorable. In particular, our results apply to mass spring models with full nearest and next-to-nearest pair interactions and provide the limiting minimal energy and minimal configurations.Comment: The final publication is available at springerlink.co

Transport of the precursor to neurospora ATPase subunit 9 into yeast mitochondria

Isolated yeast mitochondria were able to take up Neurospora ATPase subunit 9 in vitro although the homologous yeast protein is synthesized within the mitochondria and inserted into the membrane from the matrix side (Tzagoloff, A., and Meagher, P. (1972) J. Biol. Chem. 247, 594- 603). The transfer of the protein was dependent on an energized mitochondrial inner membrane. It was accompanied by proteolytic processing of the precursor to the mature protein with the correct NH2 terminus as determined by Edman degradation of the transferred protein. The possibility is discussed that there are common features in the uptake machinery neither specific for one species nor specific for individual precursor proteins in the same species

Existence and Convergence of Solutions of the Boundary Value Problem in Atomistic and Continuum Nonlinear Elasticity Theory

We show existence of solutions for the equations of static atomistic nonlinear elasticity theory on a bounded domain with prescribed boundary values. We also show their convergence to the solutions of continuum nonlinear elasticity theory, with energy density given by the Cauchy-Born rule, as the interatomic distances tend to zero. These results hold for small data close to a stable lattice for general finite range interaction potentials. We also discuss the notion of stability in detail.Comment: new version with only minor change