Dynamical properties of the growing continuum using multiple-scale method (original) (raw)
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The effect of growth in the stability of elastic materials is studied. From a stability perspective, growth and resorption have two main effects. First a change of mass modifies the geometry of the system and possibly the critical lengths involved in stability thresholds. Second, growth may depend on stress but also it may induce residual stresses in the material. These stresses change the effective loads and they may both stabilize or destabilize the material. To discuss the stability of growing elastic materials, the theory of finite elasticity is used as a general framework for the mechanical description of elastic properties and growth is taken into account through the multiplicative decomposition of the deformation gradient. The formalism of incremental deformation is adapted to include growth effects. As an application of the formalism, the stability of a growing neo-Hookean incompressible spherical shell under external pressure is analyzed. Numerical and analytical methods are combined to obtain explicit stability results and to identify the role of mechanical and geometric effects. The importance of residual stress is established by showing that under large anisotropic growth a spherical shell can become spontaneously unstable without any external loading.
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In the theory of elastic growth, a growth process is represented by the product of growth itself followed by an elastic relaxation ensuring integrity and compatibility of the body. The description of this process is local in time and only corresponds to an incremental step in the total growth process. As time evolves, these incremental growth steps are compounded and a natural question is the description of the overall cumulative growth and whether a continuous description of this process is possible. These ideas are discussed and further studied in the case of incompressible shells.
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Various differentiable models are frequently used to describe the dynamics of complex systems (see the kinetic models, fluid models, etc.). Given the complexity of all the physical phenomena involved in the dynamics of such systems, it is required to introduce the dynamic variable dependencies both on the space-time coordinates and on the scale resolutions. Therefore, in this case an adequate theoretical approach may be the use of non-linear physical models either in the form of the Scale Relativity Theory or of the Extended Scale Relativity Theory, i.e., the Scale Relativity Theory with an arbitrary constant fractal dimension. In the framework of the Extended Scale Relativity Theory, fractal velocity field is described both by topological solitons of kink type and by non-topological soliton varieties of breather type. Applications for the blood flow are proposed. The results revealed the directional flow toward the walls, which can explain the thickening effect which is one of the source of arteriosclerosis.
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Growth of living species generates stresses which ultimately design their shapes. As a consequence, complex shapes, that everybody can observe, remain difficult to predict, even when the growth biology is oversimplified. One way to tackle this question consists in limiting ourselves to quasi-planar objects like leaves in the spring. However, even in this case the diversity of shapes is really vast. Here, we focus on growing tips with the aim to compare their role in elastic growth to classical viscous fingering and dendritic growth. With the help of complex analysis, we show that a parabola under constant growth is free of stress while growing but any growth perturbation will strongly affect its final shape. Two models of finite elasticity are considered: the Neo-Hookean and the poro-elastic model with incompressibility. Résumé. La croissance biologique génère des contraintes mécaniques qui contribuent à façonner la forme des tissus, des organes et des organismes vivants. En raison de l'extrême complexité des phénomènes de croissance biologique, il est en général impossible de prédire ces formes. Dans certains cas géométriquement simples, par exemple des tissus biologiques minces en croissance quasi-planaire tels que des feuilles, les lois de la mécanique contraignent les formes possibles. Toutefois, l'espace des formes atteignables reste particulièrement vaste. Dans ce compte-rendu, nous nous intéressons au cas particulier des pointes en croissance, que nous décrivons dans le cadre de la théorie de la morpho-élasticité et de la poro-élasticité non-linéaire, et qui partage des similarités frappantes avec deux sujets d'étude classiques en physique : la croissance dendritique et la digitation visqueuse. Les outils de l'analyse complexe sont mobilisés pour montrer qu'une parabole en croissance homogène est stable et ne développe pas de contrainte mécanique. En revanche, la forme de la pointe est fortement affectée par les perturbations du champ de croissance.
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