Cohesive Traction-Separation Laws for Tearing of Ductile Metal Plates

The failure process ahead of a mode I crack advancing in a ductile thin metal plate or sheet produces plastic dissipation through a sequence of deformation steps that include necking well ahead of the crack tip and shear localization followed by a slant fracture in the necked region somewhat closer to the tip. The objective of this paper is to analyze this sequential process to characterize the traction–separation behavior and the associated effective cohesive fracture energy of the entire failure process. The emphasis is on what is often described as plane stress behavior taking place after the crack tip has advanced a distance of one or two plate thicknesses. Traction–separation laws are an essential component of finite element methods currently under development for analyzing fracture of large scale plate or shell structures. The present study resolves the sequence of failure details using the Gurson constitutive law based on the micromechanics of the ductile fracture process, including a recent extension that accounts for damage growth in shear. The fracture process in front of an advancing crack, subject to overall mode I loading, is approximated by a 2D plane strain finite element model, which allows for an intensive study of the parameters influencing local necking, shear localization and the final slant failure. The deformation history relevant to a cohesive zone for a large scale model is identified and the traction–separation relation is determined, including the dissipated energy. For ductile structural materials, the dissipation generated during necking prior to the onset of shear localization is the dominant contribution; it scales with the plate thickness and is mesh-independent in the present numerical model. The energy associated with the shear localization and fracture is secondary; it scales with the width of the shear band, and inherits the finite element mesh dependency of the Gurson model. The cohesive traction–separation laws have been characterized for various material conditions.Engineering and Applied Science

Musculoskeletal modelling of an ostrich (Struthio camelus) pelvic limb: influence of limb orientation on muscular capacity during locomotion

We developed a three-dimensional, biomechanical computer model of the 36 major pelvic limb muscle groups in an ostrich (Struthio camelus) to investigate muscle function in this, the largest of extant birds and model organism for many studies of locomotor mechanics, body size, anatomy and evolution. Combined with experimental data, we use this model to test two main hypotheses. We first query whether ostriches use limb orientations (joint angles) that optimize the moment-generating capacities of their muscles during walking or running. Next, we test whether ostriches use limb orientations at mid-stance that keep their extensor muscles near maximal, and flexor muscles near minimal, moment arms. Our two hypotheses relate to the control priorities that a large bipedal animal might evolve under biomechanical constraints to achieve more effective static weight support. We find that ostriches do not use limb orientations to optimize the moment-generating capacities or moment arms of their muscles. We infer that dynamic properties of muscles or tendons might be better candidates for locomotor optimization. Regardless, general principles explaining why species choose particular joint orientations during locomotion are lacking, raising the question of whether such general principles exist or if clades evolve different patterns (e.g., weighting of muscle force–length or force–velocity properties in selecting postures). This leaves theoretical studies of muscle moment arms estimated for extinct animals at an impasse until studies of extant taxa answer these questions. Finally, we compare our model’s results against those of two prior studies of ostrich limb muscle moment arms, finding general agreement for many muscles. Some flexor and extensor muscles exhibit self-stabilization patterns (posture-dependent switches between flexor/extensor action) that ostriches may use to coordinate their locomotion. However, some conspicuous areas of disagreement in our results illustrate some cautionary principles. Importantly, tendon-travel empirical measurements of muscle moment arms must be carefully designed to preserve 3D muscle geometry lest their accuracy suffer relative to that of anatomically realistic models. The dearth of accurate experimental measurements of 3D moment arms of muscles in birds leaves uncertainty regarding the relative accuracy of different modelling or experimental datasets such as in ostriches. Our model, however, provides a comprehensive set of 3D estimates of muscle actions in ostriches for the first time, emphasizing that avian limb mechanics are highly three-dimensional and complex, and how no muscles act purely in the sagittal plane. A comparative synthesis of experiments and models such as ours could provide powerful synthesis into how anatomy, mechanics and control interact during locomotion and how these interactions evolve. Such a framework could remove obstacles impeding the analysis of muscle function in extinct taxa

Gapless finite-TT theory of collective modes of a trapped gas

We present predictions for the frequencies of collective modes of trapped Bose-condensed $^{87}$Rb atoms at finite temperature. Our treatment includes a self-consistent treatment of the mean-field from finite-$T$ excitations and the anomolous average. This is the first gapless calculation of this type for a trapped Bose-Einstein condensed gas. The corrections quantitatively account for the downward shift in the $m=2$ excitation frequencies observed in recent experiments as the critical temperature is approached.Comment: 4 pages Latex and 2 postscript figure

Excitation spectrum in a cylindrical Bose-Einstein gas

Whole excitation spectrum is calculated within the Popov approximation of the Bogoliubov theory for a cylindrical symmetric Bose-Einstein gas trapped radially by a harmonic potential. The full dispersion relation and its temperature dependence of the zero sound mode propagating along the axial direction are evaluated in a self-consistent manner. The sound velocity is shown to depend not only on the peak density, but also on the axial area density. Recent sound velocity experiment on Na atom gas is discussed in light of the present theory.Comment: 4 pages, 5 eps figure

Coherence properties of the two-dimensional Bose-Einstein condensate

We present a detailed finite-temperature Hartree-Fock-Bogoliubov (HFB) treatment of the two-dimensional trapped Bose gas. We highlight the numerical methods required to obtain solutions to the HFB equations within the Popov approximation, the derivation of which we outline. This method has previously been applied successfully to the three-dimensional case and we focus on the unique features of the system which are due to its reduced dimensionality. These can be found in the spectrum of low-lying excitations and in the coherence properties. We calculate the Bragg response and the coherence length within the condensate in analogy with experiments performed in the quasi-one-dimensional regime [Richard et al., Phys. Rev. Lett. 91, 010405 (2003)] and compare to results calculated for the one-dimensional case. We then make predictions for the experimental observation of the quasicondensate phase via Bragg spectroscopy in the quasi-two-dimensional regime.Comment: 9 pages, 9 figure

Excitations of a Bose-condensed gas in anisotropic traps

We investigate the zero-temperature collective excitations of a Bose-condensed atomic gas in anisotropic parabolic traps. The condensate density is determined by solving the Gross-Pitaevskii (GP) equation using a spherical harmonic expansion. The GP eigenfunctions are then used to solve the Bogoliubov equations to obtain the collective excitation frequencies and mode densities. The frequencies of the various modes, classified by their parity and the axial angular momentum quantum number, m, are mapped out as a function of the axial anisotropy. Specific emphasis is placed upon the evolution of these modes from the modes in the limit of an isotropic trap.Comment: 7 pages Revtex, 9 Postscript figure

Geometric scaling in the spectrum of an electron captured by a stationary finite dipole

We examine the energy spectrum of a charged particle in the presence of a {\it non-rotating} finite electric dipole. For {\emph{any}} value of the dipole moment $p$ above a certain critical value p_{\mathrm{c}}$ an infinite series of bound states arises of which the energy eigenvalues obey an Efimov-like geometric scaling law with an accumulation point at zero energy. These properties are largely destroyed in a realistic situation when rotations are included. Nevertheless, our analysis of the idealised case is of interest because it may possibly be realised using quantum dots as artificial atoms.Comment: 5 figures; references added, outlook section reduce