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TECHNICAL PAPERS

Shell Stability Related to Pattern Formation in Plants

[+] Author and Article Information
C. R. Steele

Division of Mechanics and Computation, Department of Mechanical Engineering, Stanford University, Stanford, CA 94305

J. Appl. Mech 67(2), 237-247 (Dec 10, 1999) (11 pages) doi:10.1115/1.1305333 History: Received December 10, 1999
Copyright © 2000 by ASME
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Figures

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(a) Tip growth in the unicellular alga Acetabularia. The growth takes place in the end cap, which is nearly spherical (elliptical with a/b=1.2 for this example). The cylindrical portion remains of constant diameter, equal to about 50 μm. The load-bearing wall, composed mostly of mannan polymers, can be seen as a thin transparent layer surrounding the tip. The cytoplasm (dark granular region) and the central vacuole exert a pressure of 7–10 atmospheres on the wall and thus provide the driving force for elongation of the cell. (Photograph from Dumais 14.) (b) In Acetabularia, the elongation stage stops at regular intervals and the end cap changes from nearly hemispherical to ellipsoidal (a/b=1.9 for this example). Here only the wall is shown. When the ratio of radial to axial semi-major axes of the ellipse reaches a value near 1.5, significant circumferential compression occurs, which causes buckling of the surface. (From Dumais and Harrison, 11.) This is just as in a standard thin-walled pressure vessel. (c) Axial view of Acetabularia after buckling. It appears that the compressive circumferential stress causes a buckling pattern that initiates the development of an equally spaced array of lateral hairs. (Such a symmetric pattern is called a whorl.) Subsequently, each lateral hair elongates as in (a). (From Dumais and Harrison 11.)
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(a) Sunflower during the pattern development. The older florets are at the outer region, while the new florets are generated at the rim of the inner smooth dome. Each new floret is at the “golden section” between those of the older generation, which generates the spirals. (See Jean, 1.) At this stage, the sunflower head has a diameter of around 3–4 mm. (b) Shell model for the sunflower. The outer layer of cells (tunica) have substantially thicker walls to withstand the internal pressure, in comparison with the inner cells (corpus). The region of the floret initiation has negative Gaussian curvature of the tunica. (Geometry from Hernández 12.) (c) Stress calculated for the shell model of the sunflower. The region of floret initiation has substantial compressive stress in the circumferential direction shown by the shaded area. (From Wu 13.) (d) Effects of cuts across the sunflower capitulum. The tension regions gape open, but the region of the floret initiation is pressed together because of the compressive circumferential stress. (From Dumais 14.)
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(a) Comparison of the actual sunflower (left) and computations (right). The computation is with the use of the von Kármán equations for a plate on an elastic foundation. The plate is compressed by a uniform radial edge force. The plate is initially flat, but edge conditions on the rotation are prescribed that have the spiral pattern. For subsequent buckling, the spiral pattern propagates toward the center, as shown. Typically, the distance between calculated buckles is about twice the natural wavelength. (Calculation by S. Rennich, color enhancement of figure by J. Dumais, from the cover of the American Journal of Botany, July, 1999.) (b) Calculation of plate post-buckling, in which spiral pattern degenerates into ridges. The distance between ridges is about equal to the natural wavelength. (Calculation by S. Rennich.)
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Representation of a plant by an elastic layer, consisting of a sandwich plate representing the tunica, attached to an elastic half-space, representing the interior cells (corpus). The compressive force N in the tunica causes a buckling deformation, indicated by the dashed line, with the wavelength λ. The thickness of the tunica is t, the thickness of the walls of the tunica is tf, the thickness of the interior cell walls is tw and the cell diameter is L.
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Relation of observed peak to peak distance between primordia and the tunica thickness. The dotted line is the best fit, with a slope 14.1. The dashed lines show the natural wavelength for a plate on an elastic substrate with relative volume fractions of f=10 and 100. Generally, there seems to be little difference in the whorl and spiral patterns. (Data collected from literature by C. Schmid and J. Dumais.)
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Buckling and organ shape. (A) Wild-type stamens (st) in snapdragon. Creases traverse the formative zone, making a ring of humps. (B) The deficiens mutant of snapdragon. The formative region undulates in a plane as a ribbon, forming five cup-like primordia. (C) Out-of-plane buckling mode for a thick ring. (D) In-plane buckling of an elastic ring constrained by a wall at the outer margin. (From Green 24.)
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Thermal strain to cause in-plane buckling as a function of the harmonic index. For a given value of geometry Rg/R, the minimum gives the critical condition. For the thicker rings, the critical condition stabilizes at the harmonic n=5.
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(a) Initial hemispherical shell of radius a. The angle Φ is measured from the apex and the arc length S is measured from the equator. (b) Current configuration consisting of hemispherical shell with a cylindrical extension added.
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(a) Perturbation of hemispherical shell by the radial displacement of magnitude η. (b) Elliptical end cap with semi-major and minor axes a and b.
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Growth rate as a function of local mean stress for a simple, purely mechanical model. Growth occurs between a threshold value T and a saturation value of 1.5. For the spherical and elliptic caps, the stress is near N0, and the slope of the curve has the positive value γ. For the cylindrical region, the effective stress is higher by a factor of 1.5, and growth does not occur. The value T could represent an initial frictional resistance, while γ is the inverse of the effective viscosity.

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