laboratoire de physique statistique
laboratoire de physique statistique


Peristaltic patterns for swelling and shrinking of soft cylindrical gels - Ciarletta, Pasquale and Ben Amar, Martine
SOFT MATTER 81760-1763 (2012) 

Abstract : We propose a variational method for determining the surface patterns of cylindrical gels for both swelling and shrinking. Exact solutions are calculated for the initial stages of such peristaltic instabilities. The morphology and the formation mechanisms depend on a competition between bulk elastic energy and surface tension.
Mechanical Instabilities of Gels - Dervaux, Julien and Ben Amar, Martine

Abstract : Although the study of gels undoubtedly takes its roots within the field of physicochemistry, the interest in gels has flourished and they have progressively become an important object in the study of the mechanics of polymeric materials and volumetric growth, raising some fascinating problems, some of them remaining unsolved. Because gels are multiphase objects, their study represents an important step in the understanding of the mechanics of complex soft matter as well as for the process of shape generation in biological bodies. The scope of this article is to review the understanding we have of the mechanical behavior of gels, with a strong focus on the development of instabilities in swelling gels.
Pattern formation in fiber-reinforced tubular tissues: Folding and segmentation during epithelial growth - Ciarletta, P. and Ben Amar, M.

Abstract : Constrained growth processes in living materials result in a complex distribution of residual strains, which in certain geometries may induce a bifurcation in the elastic stability. In this work, we investigate the combined effects of growth and material anisotropy in the epithelial pattern formation of tubular tissues. In order to represent the structural organization of most organs, we adopt a strain energy density which accounts for the presence of a nonlinear reinforcement made of cross-ply fibers distributed inside a ground matrix. Using a canonical transformation in mixed polar coordinates, we transform the nonlinear elastic boundary value problem into a variational formulation, performing a straightforward derivation of the Euler-Lagrange equations for perturbations in circumferential and longitudinal directions. The corresponding curves of marginal stability are obtained numerically: the results demonstrate that both the three-dimensional distribution of residual strains and the mechanical properties of fiber reinforcements within the tissue are fundamental to determine the emergence of a specific instability pattern. In particular, different proportions of axial and circumferential residual strains can model the epithelial formation of mucosal folds in the esophagus and of plicae circulares in the small intestine. The theoretical predictions are compared with morphological data for embryonic intestinal tissues, suggesting that the volumetric growth of the epithelium can also drive the early stages of villi morphogenesis. (C) 2011 Elsevier Ltd. All rights reserved.
Growth instabilities and folding in tubular organs: A variational method in non-linear elasticity - Ciarletta, P. and Ben Amar, M.

Abstract : Morphoelastic theories have demonstrated that elastic instabilities can occur during the growth of soft materials, initiating the transition toward complex patterns. Within the framework of non-linear elasticity, the theory of incremental elastic deformations is classically employed for solving stability problems with finite strains. In this work, we define a variational method to study the bifurcation of growing cylinders with circular section. Accounting for a constant axial pre-stretch, we define a set of canonical transformations in mixed polar coordinates, providing a locally isochoric mapping. Introducing a generating function to derive an implicit gradient form of the mixed variables, the incompressibility constraint for the elastic deformation is solved exactly. The canonical representation allows to transform a generic boundary value problem, characterized by conservative body forces and surface traction loads, into a completely variational formulation. The proposed variational method gives a straightforward derivation of the linear stability analysis, which would otherwise require lengthy manipulations on the governing incremental equations. The definition of a generating function can also account for the presence of local singularities in the elastic solution. Bifurcation analysis is performed for few constrained growth problems of biomechanical interests, such as the mucosal folding of tubular tissues and surface instabilities in tumor growth. In a concluding section, the theoretical results are discussed for clarifying how anisotropy, residual strains and external constraints can affect the stability properties of soft tissues in growth and remodeling processes. (C) 2011 Elsevier Ltd. All rights reserved.
Papillary networks in the dermal-epidermal junction of skin: A biomechanical model - Ciarletta, Pasquale and Ben Amar, Martine

Abstract : Complex networks of finger-like protrusions characterize the dermal-epidermal junction of human skin. Although formed during the foetal development, such dermal papillae evolve in adulthood, often in response to a pathological condition. The aim of this work is to investigate the emergence of biaxial papillary networks in skin from a mechanical perspective. For this purpose, we define a biomechanical model taking into account the volumetric growth and the microstructural properties of the dermis and the epidermis. A scalar stream function is introduced to generate incompressible transformations, and used to define a variational formulation of the boundary value elastic problem. We demonstrate that incompatible growth of the layers can induce a bifurcation of the elastic stability driving the formation of dermal papillae. Such an interfacial instability is found to depend both on the geometrical constraints and on the mechanical properties of the skin components. The results provide a mechanical interpretation of skin morphogenesis, with possible applications for micropattern fabrication in soft layered materials. (C) 2011 Elsevier Ltd. All rights reserved.
Petal shapes of sympetalous flowers: the interplay between growth, geometry and elasticity - Ben Amar, Martine and Mueller, Martin Michael and Trejo, Miguel

Abstract : The growth of a thin elastic sheet imposes constraints on its geometry such as its Gaussian curvature KG. In this paper, we construct the shapes of sympetalous bell-shaped flowers with a constant Gaussian curvature. Minimizing the bending energies of both the petal and the veins, we are able to predict quantitatively the global shape of these flowers. We discuss two toy problems where the Gaussian curvature is either negative or positive. In the former case, the axisymmetric pseudosphere turns out to mimic the correct shape before edge curling; in the latter case, singularities of the mathematical surface coincide with strong veins. Using a variational minimization of the elastic energy, we find that the optimal number for the veins is either four, five or six, a number that is deceptively close to the statistics on real flowers in nature.