Progress in the understanding and modelling of components that could drive the overall fragility of a nuclear power plant

International audienc

Scaling of the buckling transition of ridges in thin sheets

When a thin elastic sheet crumples, the elastic energy condenses into a network of folding lines and point vertices. These folds and vertices have elastic energy densities much greater than the surrounding areas, and most of the work required to crumple the sheet is consumed in breaking the folding lines or ``ridges''. To understand crumpling it is then necessary to understand the strength of ridges. In this work, we consider the buckling of a single ridge under the action of inward forcing applied at its ends. We demonstrate a simple scaling relation for the response of the ridge to the force prior to buckling. We also show that the buckling instability depends only on the ratio of strain along the ridge to curvature across it. Numerically, we find for a wide range of boundary conditions that ridges buckle when our forcing has increased their elastic energy by 20% over their resting state value. We also observe a correlation between neighbor interactions and the location of initial buckling. Analytic arguments and numerical simulations are employed to prove these results. Implications for the strength of ridges as structural elements are discussed.Comment: 42 pages, latex, doctoral dissertation, to be submitted to Phys Rev

Observation of wave turbulence in vibrating plates

The nonlinear interaction of waves in a driven medium may lead to wave turbulence, a state such that energy is transferred from large to small lengthscales. Here, wave turbulence is observed in experiments on a vibrating plate. The frequency power spectra of the normal velocity of the plate may be rescaled on a single curve, with power-law behaviors that are incompatible with the weak turbulence theory of D{\"u}ring et al. [Phys. Rev. Lett. 97, 025503 (2006)]. Alternative scenarios are suggested to account for this discrepancy -- in particular the occurrence of wave breaking at high frequencies. Finally, the statistics of velocity increments do not display an intermittent behavior

Multiple-length-scale elastic instability mimics parametric resonance of nonlinear oscillators

Spatially confined rigid membranes reorganize their morphology in response to the imposed constraints. A crumpled elastic sheet presents a complex pattern of random folds focusing the deformation energy while compressing a membrane resting on a soft foundation creates a regular pattern of sinusoidal wrinkles with a broad distribution of energy. Here, we study the energy distribution for highly confined membranes and show the emergence of a new morphological instability triggered by a period-doubling bifurcation. A periodic self-organized focalization of the deformation energy is observed provided an up-down symmetry breaking, induced by the intrinsic nonlinearity of the elasticity equations, occurs. The physical model, exhibiting an analogy with parametric resonance in nonlinear oscillator, is a new theoretical toolkit to understand the morphology of various confined systems, such as coated materials or living tissues, e.g., wrinkled skin, internal structure of lungs, internal elastica of an artery, brain convolutions or formation of fingerprints. Moreover, it opens the way to new kind of microfabrication design of multiperiodic or chaotic (aperiodic) surface topography via self-organization.Comment: Submitted for publicatio

Network development in biological gels: role in lymphatic vessel development

In this paper, we present a model that explains the prepatterning of lymphatic vessel morphology in collagen gels. This model is derived using the theory of two phase rubber material due to Flory and coworkers and it consists of two coupled fourth order partial differential equations describing the evolution of the collagen volume fraction, and the evolution of the proton concentration in a collagen implant; as described in experiments of Boardman and Swartz (Circ. Res. 92, 801–808, 2003). Using linear stability analysis, we find that above a critical level of proton concentration, spatial patterns form due to small perturbations in the initially uniform steady state. Using a long wavelength reduction, we can reduce the two coupled partial differential equations to one fourth order equation that is very similar to the Cahn–Hilliard equation; however, it has more complex nonlinearities and degeneracies. We present the results of numerical simulations and discuss the biological implications of our model

Noise and Robustness in Phyllotaxis

A striking feature of vascular plants is the regular arrangement of lateral organs on the stem, known as phyllotaxis. The most common phyllotactic patterns can be described using spirals, numbers from the Fibonacci sequence and the golden angle. This rich mathematical structure, along with the experimental reproduction of phyllotactic spirals in physical systems, has led to a view of phyllotaxis focusing on regularity. However all organisms are affected by natural stochastic variability, raising questions about the effect of this variability on phyllotaxis and the achievement of such regular patterns. Here we address these questions theoretically using a dynamical system of interacting sources of inhibitory field. Previous work has shown that phyllotaxis can emerge deterministically from the self-organization of such sources and that inhibition is primarily mediated by the depletion of the plant hormone auxin through polarized transport. We incorporated stochasticity in the model and found three main classes of defects in spiral phyllotaxis – the reversal of the handedness of spirals, the concomitant initiation of organs and the occurrence of distichous angles – and we investigated whether a secondary inhibitory field filters out defects. Our results are consistent with available experimental data and yield a prediction of the main source of stochasticity during organogenesis. Our model can be related to cellular parameters and thus provides a framework for the analysis of phyllotactic mutants at both cellular and tissular levels. We propose that secondary fields associated with organogenesis, such as other biochemical signals or mechanical forces, are important for the robustness of phyllotaxis. More generally, our work sheds light on how a target pattern can be achieved within a noisy background

A finite strain fibre-reinforced viscoelasto-viscoplastic model of plant cell wall growth

A finite strain fibre-reinforced viscoelasto-viscoplastic model implemented in a finite element (FE) analysis is presented to study the expansive growth of plant cell walls. Three components of the deformation of growing cell wall, i.e. elasticity, viscoelasticity and viscoplasticity-like growth, are modelled within a consistent framework aiming to present an integrative growth model. The two aspects of growth—turgor-driven creep and new material deposition—and the interplay between them are considered by presenting a yield function, flow rule and hardening law. A fibre-reinforcement formulation is used to account for the role of cellulose microfibrils in the anisotropic growth. Mechanisms in in vivo growth are taken into account to represent the corresponding biologycontrolled behaviour of a cell wall. A viscoelastic formulation is proposed to capture the viscoelastic response in the cell wall. The proposed constitutive model provides a unique framework for modelling both the in vivo growth of cell wall dominated by viscoplasticity-like behaviour and in vitro deformation dominated by elastic or viscoelastic responses. A numerical scheme is devised, and FE case studies are reported and compared with experimental data