Geometry-Induced Rigidity in Nonspherical Pressurized Elastic Shells

Abstract : We present results from an experimental investigation of the indentation of nonspherical pressurized elastic shells with a positive Gauss curvature. A predictive framework is proposed that rationalizes the dependence of the local rigidity of an indented shell on the curvature in the neighborhood of the locus of indentation, the in-out pressure differential, and the material properties. In our approach, we combine classic theory for spherical shells with recent analytical developments for the pressurized case, and proceed, for the most part, by analogy, guided by our own experiments. By way of example, our results elucidate why an eggshell is significantly stiffer when compressed along its major axis, as compared to doing so along its minor axis. The prominence of geometry in this class of problems points to the relevance and applicability of our findings over a wide range of length scales. Shells are ubiquitous as both natural and engineered structures, including viral capsids [1], pollen grains, col-loidosomes [2], pharmaceutical capsules, exoskeletons, mammalian skulls [3], pressure vessels, and architectural domes. In addition to their aesthetically appealing form, shells offer outstanding structural performance. As such, they typically have the function of enclosure, containment, and protection, with an eggshell being the archetypal example [Fig. 1(a)]. The mechanics of thin elastic shells [4,5] have long been known to be rooted in the purely geometric isometric deformations of the underlying curved surface, since stretching is energetically more costly than bending [6]. However, despite a vast literature on the isometry of rigid surfaces [7,8], establishing a general direct theoretical connection between the differential geometry of surfaces and the mechanics of shells is a challenging endeavor. This is due to the difficulties in systematically quantifying the effect of a small but finite thickness on the mechanical response of a curved surface. To circumnavigate this issue, explicit boundary value problem calculations tend to be performed by deriving the local equilibrium equations of 3D continuum mechanics and then taking the limit of a small but finite thickness within the kinematics [9]. For example, closed analytical solutions have been obtained in this fashion for the indentation of spherical unpressurized [10,11] and pressurized [12] shells. For more intricate geometries and mechanical environments, numerical methods can be used such as full scale finite element simulations [13,14]. Although powerful, these computational approaches can sometimes come at the detriment of physical insight and predictive understanding of the interplay between the mechanics of the structure and the geometry of its surface. Here, we study the effect of the geometry of surfaces with a positive Gauss curvature on the linear mechanical response under the indentation of thin elastic shells, with or without an in-out pressure differential. Our goal is to quantify the geometry-induced rigidity (
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Arnaud Lazarus, H. C. B. Florijn, P. M. Reis. Geometry-Induced Rigidity in Nonspherical Pressurized Elastic Shells. Physical Review Letters, American Physical Society, 2012, 109 (14), ⟨10.1103/PhysRevLett.109.144301⟩. ⟨hal-01447380⟩

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