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International Journal of Non-Linear Mechanics

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The growth of giant pumpkins: How extreme weight influences shapeIntroductionMethodsGiant pumpkin measurementsInstron experimentsComputational modelResultsWeight and geometry changeBreaking forceObserved plastic deformationComputed pumpkin shapesPredicting maximum pumpkin sizeConclusionsAcknowledgementsReferencesThe growth of giant pumpkins: How extreme weight influences shapeDavid L. Hua,b,, Paul Richardsa, Alexander AlexeevaaSchool of Mechanical Engineering, Georgia Institute of Technology, 801 Ferst Drive, Atlanta, GA 30332-0230, USAbSchool of Biology, Georgia Institute of Technology, 310 Ferst Drive, Atlanta, GA 30332-0230, USAarticle infoKeywords:FruitGrowthDeformationPlasticityGiantabstractGreat morphological differences exist among fruits and vegetables. In this combined experimental andtheoretical study, we predict pumpkin shape evolution and maximum size based on their materialproperties. Using time-lapse photography and measurements collected by volunteer farmers, we showthat as pumpkins grow, they m orph from spherical to pancake shapes, flattening up to 50% in height-to-width aspect ratio. By compressing whole pumpkins in material-testing machi nes, we find that theelastic response of the pumpkin is insufficient to account for the large deformations characteristic oflarge pumpkins. We hypothesize that pumpkin flattening is caused by the weight of the pumpkinretarding its normal growth processes. We test this hypothesis using a mathematical mode l thatassumes plant growth is stimulated in response to a tensile yield stress. We are able to predict pumpkinshapes consistent with those observed. The observed growth plasticity allows the fruit to redistributeinternal stresses, thereby growing to extreme sizes without breaking.& 2010 Elsevier Ltd. All rights reserved.1. IntroductionThe development of shape in plants is a century-old problem[1] that has made recent advances due to the combined inter-disciplinary efforts of plant, molecular and mathematical biology.A challenge inherent to this problem is the multiple length scalesinvolved, particularly in embryogenesis or fruit growth, where anovary of characteristic size 100mm210 mm will reproduce overtime to become a 10-cm fruit, or in the case of this investigation, aone-meter fruit. During this vast change in size, the plant genomeregulates growth through feedback with the multi-parameterchemical and physical state of the fruit [2]. The chemical andbiological changes in cells during growth [3,4] are beyond thescope of this study. Instead, we focus on the use of a simplecomputational model, which approximates elasto-plastic plantmaterial, to investigate the mechanics of extreme growth.Given the computational nature of our study, it is worthwhileto briefly review previous mathematical approaches here. Gorielyet al. [5] and Taber [6] provide comprehensive reviews ofcontinuum models used to model the growth of plant and animaltissues. A common theme among these models is the decomposi-tion of strain into components due to elasticity and growth. Inanother study, Vandiver and Goriely [7] explain how differentialgrowth in the plant can generates residual stresses, such astension, and consequently how these tensions can rigidify andstrengthen the plant. Dumais et al. [8] has shown how thebehavior of certain materials like rubber balloons can be usedas models for root tip growth. Coen et al. [9] review how thegrowth of blossoms can be computationally modeled usingformulations of elasticity and growth rules. While most previousmodels take one-dimensional or two-dimensional approaches togrowth, we take advantage of computational methods that areparticularly suited for examining three-dimensional changes.We apply a lattice spring method (LSM) [10,11], a computa-tional model, to estimate pumpkin deformation. Originallyderived for performing atomistic simulations [12], this methodcan be applied for modeling elastic solids in the continuummechanics approximation [10,11,13]. Our computational latticemodel constitutes a means for examining the influence of thevisco-plastic properties [10] on development of fruit shapes underdifferent environmental conditions. Furthermore, by removingindividual bonds, LSM allows modeling of crack formation andpropagation thorough solid materials [14,15]. Such a scenario isoften observed in giant vegetables where cracks may appear as aresult of the extremely fast growth.The study of large organisms can provide us insight into thegrowth and stress limits of tissues and can provide useful testinggrounds for hypotheses about biomechanics. The largest organismssuch as trees and dinosaurs, push the envelope of growth, metabolicand respiratory processes occurring in the whole organism and itsconstituent cells. For instance, the maximum height of redwood treesis 130 m, due to the inability of trees to syphon water at these height[16]. Water-walking insects have a maximum size of 30 cm becauseof surface tension effects [17]; the maximum size of prehistoricand extant dragonflies is 1 m and 20 cm, respectively, because ofContents lists available at ScienceDirectjournal homepage: www.elsevier.com/locate/nlmInternational Journal of Non-Linear Mec hanics0020-7462/$ - see front matter & 2010 Elsevier Ltd. All rights reserved.doi:10.1016/j.ijnonlinmec.2010.12.013Corresponding author at: School of Mechanical Engineering, Georgia Instituteof Technology, 801 Ferst Drive, Atlanta, GA 30332-0230, USA.Tel.: +1 404 894 0573.E-mail address: [email protected] (D.L. Hu).Please cite this article as: D.L. Hu, et al., The growth of giant pumpkins: How extreme weight influences shape, Int. J. Non-Linear Mech.(2011), doi:10.1016/j.ijnonlinmec.2010.12.013International Journal of Non-Linear Mechanics ] (]]]]) ]]]–]]]changes in atmospheric gas composition [18,19]. By consideringthe growth of large fruits, as will be done in this study, one canamplify the effects of gravity on their development. For all theseorganisms, mechanical forces determine their maximum size and,to some extent, their behavior. Since the length and time scales ofthese organisms exceed those of typical sizes, studying largeorganisms is inherently difficult and information on them issparse. To study giant plants we are forced to accept theuncontrolled conditions in which they are bred and grownbecause of the great personal care required by farmers to ensuregrowth at these sizes.Annual agricultural competitions starting in the 1800’s andthe


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