Discrete structures in continuum descriptions of defective crystals

I discuss various mathematical constructions that combine together to provide a natural setting for discrete and continuum geometric models of defective crystals. In particular I provide a quite general list of `plastic strain variables', which quantifies inelastic behaviour, and exhibit rigorous connections between discrete and continuous mathematical structures associated with crystalline materials that have a correspond-ingly general constitutive specification

Stiffest Elastic Networks

The rigidity of a network of elastic beams crucially depends on the specific details of its structure. We show both numerically and theoretically that there is a class of isotropic networks which are stiffer than any other isotropic network with same density. The elastic moduli of these \textit{stiffest elastic networks} are explicitly given. They constitute upper-bounds which compete or improve the well-known Hashin-Shtrikman bounds. We provide a convenient set of criteria (necessary and sufficient conditions) to identify these networks, and show that their displacement field under uniform loading conditions is affine down to the microscopic scale. Finally, examples of such networks with periodic arrangement are presented, in both two and three dimensions

Rotational elasticity and couplings to linear elasticity

It is the aim of the paper to present a new point of view on rotational elasticity in a nonlinear setting using orthogonal matrices. The proposed model, in the linear approximation, can be compared to the well known equilibrium equations of static linear elasticity. An appropriate kinetic energy is identified and we present a dynamical model of rotational elasticity. The propagation of elastic waves in such a medium is studied and we find two classes of waves, transversal rotational waves and longitudinal rotational waves, both of which are solutions of the nonlinear partial differential equations. For certain parameter choices, the transversal wave velocity can be greater than the longitudinal wave velocity. Moreover, parameter ranges are identified where the model describes an auxetic material. However, in all cases the potential energy functional is positive definite. Finally, we couple the rotational waves to linear elastic waves to study the behaviour of the coupled system. We find wave like solutions to the coupled equations and can visualise our results with the help of suitable figures.Comment: 19 pages, 2 figures, heavily revised and largely extended versio

Yielding and irreversible deformation below the microscale: Surface effects and non-mean-field plastic avalanches

Nanoindentation techniques recently developed to measure the mechanical response of crystals under external loading conditions reveal new phenomena upon decreasing sample size below the microscale. At small length scales, material resistance to irreversible deformation depends on sample morphology. Here we study the mechanisms of yield and plastic flow in inherently small crystals under uniaxial compression. Discrete structural rearrangements emerge as series of abrupt discontinuities in stress-strain curves. We obtain the theoretical dependence of the yield stress on system size and geometry and elucidate the statistical properties of plastic deformation at such scales. Our results show that the absence of dislocation storage leads to crucial effects on the statistics of plastic events, ultimately affecting the universal scaling behavior observed at larger scales.Comment: Supporting Videos available at http://dx.plos.org/10.1371/journal.pone.002041

Sea level: measuring the bounding surfaces of the ocean

The practical need to understand sea level along the coasts, such as for safe navigation given the spatially variable tides, has resulted in tide gauge observations having the distinction of being some of the longest instrumental ocean records. Archives of these records, along with geological constraints, have allowed us to identify the century-scale rise in global sea level. Additional data sources, particularly satellite altimetry missions, have helped us to better identify the rates and causes of sea level rise and the mechanisms leading to spatial variability in the observed rates. Analysis of all of the data reveals the need for long-term and stable observation systems to assess accurately the regional changes as well as to improve our ability to estimate future changes in sea level. While information from many scientific disciplines is needed to understand sea level change, this paper focuses on contributions from geodesy and the role of the ocean’s bounding surfaces: the sea surface and the Earth’s crust

Skin friction blistering: computer model

BACKGROUND/PURPOSE: Friction blisters, a common injury in sports and military operations, can adversely effect or even halt performance. Given its frequency and hazardous nature, recent research efforts appear limited. Blistering can be treated as a delamination phenomenon; similar issues in materials science have been extensively investigated in theory and experiment. An obstacle in studying blistering is the difficulty of conducting experiment on humans and animals. Computer modeling thus becomes a preferred tool. METHOD: This paper used a dynamic non-linear finite-element model with a blister-characterized structure and contact algorithm for outer materials and blister roof to investigate the effects on deformation and stress of an existing blister by changing the friction coefficient and elastic modulus of the material in contact with the blister. RESULTS: Through the dynamics mode and harmonic frequency approach, we demonstrated that the loading frequency leads to dramatic changes of displacement and stress in spite of otherwise similar loading. Our simulations show that an increased friction coefficient does not necessarily result in an increase in either the stress on the hot spot or blister deformation; local maximum friction stress and Von Mises stress exist for some friction coefficients over the wide range examined here. In addition, the stiffness of contact material on blistering is also investigated, and no significant effects on deformation and Von Mises stress are found, again at the range used. The model and method provided here may be useful for evaluating loading environments and contact materials in reducing blistering incidents. CONCLUSION: The coupling finite-element model can predict the effects of friction coefficient and contacting materials' stiffness on blister deformation and hot spot stress