Soft modes near the buckling transition of icosahedral shells

Icosahedral shells undergo a buckling transition as the ratio of Young's modulus to bending stiffness increases. Strong bending stiffness favors smooth, nearly spherical shapes, while weak bending stiffness leads to a sharply faceted icosahedral shape. Based on the phonon spectrum of a simplified mass-and-spring model of the shell, we interpret the transition from smooth to faceted as a soft-mode transition. In contrast to the case of a disclinated planar network where the transition is sharply defined, the mean curvature of the sphere smooths the transitition. We define elastic susceptibilities as the response to forces applied at vertices, edges and faces of an icosahedron. At the soft-mode transition the vertex susceptibility is the largest, but as the shell becomes more faceted the edge and face susceptibilities greatly exceed the vertex susceptibility. Limiting behaviors of the susceptibilities are analyzed and related to the ridge-scaling behavior of elastic sheets. Our results apply to virus capsids, liposomes with crystalline order and other shell-like structures with icosahedral symmetry.Comment: 28 pages, 6 figure

Programmed buckling by controlled lateral swelling in a thin elastic sheet

Recent experiments have imposed controlled swelling patterns on thin polymer films, which subsequently buckle into three-dimensional shapes. We develop a solution to the design problem suggested by such systems, namely, if and how one can generate particular three-dimensional shapes from thin elastic sheets by mere imposition of a two-dimensional pattern of locally isotropic growth. Not every shape is possible. Several types of obstruction can arise, some of which depend on the sheet thickness. We provide some examples using the axisymmetric form of the problem, which is analytically tractable.Comment: 11 pages, 9 figure

Mechanical model of the ultra-fast underwater trap of Utricularia

The underwater traps of the carnivorous plants of the Utricularia species catch their preys through the repetition of an "active slow deflation / passive fast suction" sequence. In this paper, we propose a mechanical model that describes both phases and strongly supports the hypothesis that the trap door acts as a flexible valve that buckles under the combined effects of pressure forces and the mechanical stimulation of trigger hairs, and not as a panel articulated on hinges. This model combines two different approaches, namely (i) the description of thin membranes as triangle meshes with strain and curvature energy, and (ii) the molecular dynamics approach, which consists in computing the time evolution of the position of each vertex of the mesh according to Langevin equations. The only free parameter in the expression of the elastic energy is the Young's modulus E of the membranes. The values for this parameter are unequivocally obtained by requiring that the trap model fires, like real traps, when the pressure difference between the outside and the inside of the trap reaches about 15 kPa. Among other results, our simulations show that, for a pressure difference slightly larger than the critical one, the door buckles, slides on the threshold and finally swings wide open, in excellent agreement with the sequence observed in high-speed videos.Comment: Accepted for publication in Physical Review

Strain Gradient Plasticity: Theory and Implementation

This chapter focuses on the foundation and development of various higher-order strain gradient plasticity theories, and it also provides the basic elements for their finite element implementation. To this aim,we first refer to experiments exhibiting size-effects in metals and explain them by resorting to the concept of geometrically necessary dislocations. We then bring this concept to the continuum level by introducing Nye’s dislocation density tensor and by postulating the existence of higher-order stresses associated with dislocation densities. This provides the motivation for the development of higher-order strain gradient plasticity theories. For this purpose, we adopt the generalized principle of virtual work, initially illustrated for conventional crystal plasticity and subsequently extended to both crystal and phenomenological strain gradient plasticity theories