On morphoelastic rods (original) (raw)
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Tip growth in morpho-elasticity
Comptes Rendus. Mécanique
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.
Biomechanics and Modeling in Mechanobiology, 2006
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.
Towards a unified theory for morphomechanics
Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2009
Mechanical forces are closely involved in the construction of an embryo. Experiments have suggested that mechanical feedback plays a role in regulating these forces, but the nature of this feedback is poorly understood. Here, we propose a general principle for the mechanics of morphogenesis, as governed by a pair of evolution equations based on feedback from tissue stress. In one equation, the rate of growth (or contraction) depends on the difference between the current tissue stress and a target (homeostatic) stress. In the other equation, the target stress changes at a rate that depends on the same stress difference. The parameters in these morphomechanical laws are assumed to depend on stress rate. Computational models are used to illustrate how these equations can capture a relatively wide range of behaviours observed in developing embryos, as well as show the limitations of this theory. Specific applications include growth of pressure vessels (e.g. the heart, arteries and brain...
Growth and instability in elastic tissues
Journal of the Mechanics and Physics of Solids, 2005
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.
On a new model for inhomogeneous volume growth of elastic bodies
Journal of the Mechanical Behavior of Biomedical Materials, 2014
In general, growth characterises the process by which a material increases in size by the addition of mass. In dependence on the prevailing boundary conditions growth occurs in different, often complex ways. However, in this paper we aim to develop a model for biological systems growing in an inhomogeneous manner thereby generating residual stresses even when growth rates and material properties are homogeneous. Consequently, a descriptive example could be a body featuring homogeneous, isotropic material characteristics that grows against a barrier. At the moment when it contacts the barrier inhomogeneous growth takes place. If thereupon the barrier is removed, some types of bodies keep the new shape mainly fixed. As a key idea of the proposed phenomenological approach, we effort the theory of finite plasticity applied to the isochoric part of the Kirchhoff stress tensor as well as an additional condition allowing for plastic changes in the new grown material, only. This allows us to describe elastic bodies with a fluid-like growth characteristic. Prominent examples are tumours where the characteristic macro mechanical growth behaviour can be explained based on cellular arguments. Finally, the proposed framework is embedded into the finite element context which allows us to close this study with representative numerical examples.
Journal of the Mechanics and Physics of Solids, 2009
The shape of plants and other living organisms is a crucial element of their biological functioning. Morphogenesis is the result of complex growth processes involving biological, chemical and physical factors at different temporal and spatial scales. This study aims at describing stresses and strains induced by the production and reorganization of the material. The mechanical properties of soft tissues are modeled within the framework of continuum mechanics in finite elasticity. The kinematical description is based on the multiplicative decomposition of the deformation gradient tensor into an elastic and a growth term. Using this formalism, the authors have studied the growth of thin hyperelastic samples. Under appropriate assumptions, the dimensionality of the problem can be reduced, and the behavior of the plate is described by a two-dimensional surface. The results of this theory demonstrate that the corresponding equilibrium equations are of the Fö ppl-von Ká rmá n type where growth acts as a source of mean and Gaussian curvatures. Finally, the cockling of paper and the rippling of a grass blade are considered as two examples of growth-induced pattern formation.
Thermomechanics of volumetric growth in uniform bodies
International Journal of Plasticity, 2000
A theory of material growth (mass creation and resorption) is presented in which growth is viewed as a local rearrangement of material inhomogeneities described by means of ®rst-and second-order uniformity``transplants''. An essential role is played by the balance of canonical (material) momentum where the¯ux is none other than the so-called Eshelby material stress tensor. The corresponding irreversible thermodynamics is expanded. If the constitutive theory of basically elastic materials is only ®rst-order in gradients, diusion of mass growth cannot be accommodated, and volumetric growth then is essentially governed by the inhomogeneity velocity``gradient'' (®rst-order transplant theory) while the driving force of irreversible growth is the Eshelby stress or, more precisely, the``Mandel'' stress, although the possible in¯uence of``elastic'' strain and its time rate is not ruled out. The application of various invariance requirements leads to a rather simple and reasonable evolution law for the transplant. In the second-order theory which allows for growth diusion, a second-order inhomogeneity tensor needs to be introduced but a special theory can be devised where the time evolution of the second-order transplant can be entirely dictated by that of the ®rst-order one, thus avoiding insuperable complications. Dierential geometric aspects are developed where needed. #
Material evolution in plasticity and growth
Solid Mechanics and Its Applications, 2002
The theories of plasticity and growth, as well as other theories of anelastic behaviour, are shown to share the same formal structure. The unifying concept is that of material evolution, rooted in the theory of material inhomogeneities. Particular attention is given to models based on first and second-grade elastic prototypes. A detailed treatment of the formal restrictions to be imposed on any possible evolution law is followed by the formulation of balance equations in a volumetrically growing body. A short final section is devoted to the discussion of thermodynamic inequalities and their consequences.