Folding of an opened spherical shell (original) (raw)
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Curvature Condensation and Bifurcation in an Elastic Shell
Physical Review Letters, 2007
We study the formation and evolution of localized geometrical defects in an indented cylindrical elastic shell using a combination of experiment and numerical simulation. We find that as a symmetric localized indentation on a semicylindrical shell increases, there is a transition from a global mode of deformation to a localized one which leads to the condensation of curvature along a symmetric parabolic defect. This process introduces a soft mode in the system, converting a load-bearing structure into a hinged, kinematic mechanism. Further indentation leads to twinning wherein the parabolic defect bifurcates into two defects that move apart on either side of the line of symmetry. A qualitative theory captures the main features of the phenomena but leads to further questions about the mechanism of defect nucleation.
Buckling-induced retraction of spherical shells: A study on the shape of aperture
Scientific reports, 2015
Buckling of soft matter is ubiquitous in nature and has attracted increasing interest recently. This paper studies the retractile behaviors of a spherical shell perforated by sophisticated apertures, attributed to the buckling-induced large deformation. The buckling patterns observed in experiments were reproduced in computational modeling by imposing velocity-controlled loads and eigenmode-affine geometric imperfection. It was found that the buckling behaviors were topologically sensitive with respect to the shape of dimple (aperture). The shell with rounded-square apertures had the maximal volume retraction ratio as well as the lowest energy consumption. An effective experimental procedure was established and the simulation results were validated in this study.
Physical Review Letters, 2013
On microscopic scales, the crystallinity of flexible tethered or cross linked membranes determines their mechanical response. We show that by controlling the type, number and distribution of defects on a spherical elastic shell, it is possible to direct the morphology of these structures. Our numerical simulations show that by deflating a crystalline shell with defects, we can create elastic shell analogs of the classical Platonic solids. These morphologies arise via a sharp buckling transition from the sphere which is strongly hysteretic in loading-unloading. We construct a minimal Landau theory for the transition using quadratic and cubic invariants of the spherical harmonic modes. Our approach suggests methods to engineer shape into soft spherical shells using a frozen defect topology.
The secondary buckling transition: Wrinkling of buckled spherical shells
Eur. Phys. J. E , 2014
We theoretically explain the complete sequence of shapes of deflated spherical shells. Decreasing the volume, the shell remains spherical initially, then undergoes the classical buckling instability, where an axisymmetric dimple appears, and, finally, loses its axisymmetry by wrinkles developing in the vicinity of the dimple edge in a secondary buckling transition. We describe the first axisymmetric buckling transition by numerical integration of the complete set of shape equations and an approximate analytic model due to Pogorelov. In the buckled shape, both approaches exhibit a locally compressive hoop stress in a region where experiments and simulations show the development of polygonal wrinkles, along the dimple edge. In a simplified model based on the stability equations of shallow shells, a critical value for the compressive hoop stress is derived, for which the compressed circumferential fibres will buckle out of their circular shape in order to release the compression. By applying this wrinkling criterion to the solutions of the axisymmetric models, we can calculate the critical volume for the secondary buckling transition. Using the Pogorelov approach, we also obtain an analytical expression for the critical volume at the secondary buckling transition: The critical volume difference scales linearly with the bending stiffness, whereas the critical volume reduction at the classical axisymmetric buckling transition scales with the square root of the bending stiffness. These results are confirmed by another stability analysis in the framework of Donnel, Mushtari and Vlasov (DMV) shell theory, and by numerical simulations available in the literature.
Journal of the International Association for Shell and Spatial Structures, 2017
This paper analyses the buckling shapes of spherical shells subjected to concentrated load. A theoretical investigation, based upon geometric considerations, shows a variety of possible buckling shapes, including polygonal ones. The results show that there exists a certain difference between the geometric behavior of shells characterized by different radius-thickness ratios. An analogy between the buckling edge of the shell and a compressed planar elastic ring is also shown, which gives a better view on the point of transformation of the buckling edge from a circle to a polygon. The results of numerical and experimental analyses are also exhibited to verify the theoretical achievements.
IASS-IACM 2008 Spanning Nano to Mega, 2008
The analogy between the deployable folding pattern in a hawkmoth’s bellows (Wasserthal [10]) and the folding pattern of a cylindrical thin-walled shell, that is spontaneously buckling under torsional load, named “Kresling-pattern” (Poeppe in Stewart [9], Kresling [4,5], Hunt and Ario [2]), leads to a novel interpretation of buckling not as failure, but as a model strategy for designing and manufacturing “natural” deployable folding patterns with coupled mechanisms. Patterns of this type would operate nearly stress-free and could work reliably at both micro- and macro-level. The hawkmoth’s deployment pattern shows also some characteristics of the classic “Miura-ori” (Miura [7]), a pattern designed to deploy to 2D-arrays (maps, solar sails and other space applications…), found also in the leaves of deciduous trees (Kobayashi et al. [3]). Modified angle values of a series of fold lines would help deploy the structure in the 3-dimensional geometrical envelope of a cylinder.
Secondary polygonal instability of buckled spherical shells
EPL , 2014
When a spherical elastic capsule is deflated, it first buckles axisymmetrically and subsequently loses its axisymmetry in a secondary instability, where the dimple acquires a polygonal shape. We explain this secondary polygonal buckling in terms of wrinkles developing at the inner side of the dimple edge in response to compressive hoop stress. Analyzing the axisymmetric buckled shape, we find a compressive hoop stress with parabolic stress profile at the dimple edge. We further show that there exists a critical value for this hoop stress, where it becomes favorable for the membrane to buckle out of its axisymmetric shape, thus releasing the compression. The instability mechanism is analogous to the formation of wrinkles under compressive stress. A simplified stability analysis allows us to quantify the critical stress for secondary buckling. Applying this secondary buckling criterion to the axisymmetric shapes, we can determine the critical volume for secondary buckling. Our analytical result is in close agreement with existing numerical data.
Expansion/contraction of a spherical elastic/plastic shell revisited
Continuum Mechanics and Thermodynamics, 2014
A semi-analytic solution for the elastic/plastic distribution of stress and strain in a spherical shell subject to pressure over its inner and outer radii and subsequent unloading is presented. The Bauschinger effect is taken into account. The flow theory of plasticity is adopted in conjunction with quite an arbitrary yield criterion and its associated flow rule. The yield stress is an arbitrary function of the equivalent strain. It is shown that the boundary value problem is significantly simplified if the equivalent strain is used as an independent variable instead of the radial coordinate. In particular, numerical methods are only necessary to evaluate ordinary integrals and solve simple transcendental equations. An illustrative example is provided to demonstrate the distribution of residual stresses and strains.
Tristability of thin orthotropic shells with uniform initial curvature
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2008
Composite shells show a rich multistable behaviour of interest for the design of shape-changing (morphing) structures. Previous studies have investigated how the initial shape determines the shell stability properties. For uniform initial curvatures and orthotropic material behaviour, not more than two stable equilibria have been reported. In this paper, we prove that untwisted, uniformly curved, thin orthotropic shells can have up to three stable equilibrium configurations. Cases of tristability are first documented using a numerical stability analysis of an extensible shell model. Including mid-plane extension shows that the shells must be sufficiently curved in relation to their thickness to be multistable. Thus, an inextensible model allows us to perform an analytical stability analysis. Focusing on untwisted initial configurations, we illustrate with simple analytical results how the material parameters of the shell control the dependence of its multistable behaviour on the initial curvatures. In particular, we show that when the bending stiffness matrix approaches a degeneracy condition, the shell exhibits three stable equilibria for a wide range of initial curvatures.
Collapse of Orthotropic Spherical Shells
Physical Review Letters, 2019
We report on the buckling and subsequent collapse of orthotropic elastic spherical shells under volume and pressure control. Going far beyond what is known for isotropic shells, a rich morphological phase space with three distinct regimes emerges upon variation of shell slenderness and degree of orthotropy. Our extensive numerical simulations are in agreement with experiments using fabricated polymer shells. The shell buckling pathways and corresponding strain energy evolution are shown to depend strongly on material orthotropy. We find surprisingly robust orthotropic structures with strong similarities to stomatocytes and tricolpate pollen grains, suggesting that the shape of several of Nature's collapsed shells could be understood from the viewpoint of material orthotropy. FABRICATION OF ORTHOTROPIC SPHERICAL SHELLS FIG. S1. (a) Hemispherical cap with four layers on a sphere of radius 110 mm. (b) Complete spherical shell after curing.