Fruits and vegetables are usually composed of exocarp and sarcocarp and they take a variety of shapes when they are ripe. Buckled and wrinkled fruits and vegetables are often observed. This work aims at establishing the geometrical constraint for buckled and wrinkled shapes based on a mechanical model. The mismatch of expansion rate between the exocarp and sarcocarp can produce a compressive stress on the exocarp. We model a fruit/vegetable with exocarp and sarcocarp as a hyperelastic layer-substrate structure subjected to uniaxial compression. The derived bifurcation condition contains both geometrical and material constants. However, a careful analysis on this condition leads to the finding of a critical thickness ratio which separates the buckling and wrinkling modes, and remarkably, which is independent of the material stiffnesses. More specifically, it is found that if the thickness ratio is smaller than this critical value a fruit/vegetable should be in a buckling mode (under a sufficient stress); if a fruit/vegetable in a wrinkled shape the thickness ratio is always larger than this critical value. To verify the theoretical prediction, we consider four types of buckled fruits/vegetables and four types of wrinkled fruits/vegetables with three samples in each type. The geometrical parameters for the 24 samples are measured and it is found that indeed all the data fall into the theoretically predicted buckling or wrinkling domains. Some practical applications based on this critical thickness ratio are briefly discussed.
BACKGROUND AND MODELMany vegetables and fruits contain exocarp and sarcocarp and usually exocarp is stiffer than sarcocarp in order to protect it. There are many different morphologies for fruits and vegetables, and in particular wrinkled and globally buckled shapes are often observed, e.g. wrinkled pumpkins and buckled cucumbers (see Figs. 6 and 7). Why a fruit/vegetable takes the final shape when they ripe may be due to many different factors during the growth process. But, mechanical forces alone can play a very important role for determining the geometry of vegetables and fruits (cf. [1]). Now, it has been understood that the out layer of plant meristems often expands faster than the inner one, leading to to the whole structure under compression on the interface (see [1] and [2]). In fact, a newborn fruit or vegetable has a smooth surface, and only after the certain period of growth, the morphology features occur and remain since (cf.[3] to see details). Over the past decades or so, many authors have used purely mechanical models to study patterns in biological objects, e.g., fingerprint formation by using the Von Karman's equations ([4], [5]) and pattern formation in plants through shell instability ([6]) and shapes of sympetalous flowers through growth of a thin elastic sheet ([7]).Actually, it has been known for some time instabilities can lead to a variety of patterns in a mechanical system. For example, for a thin layer coated to a compliant substrate and a core/shell system, various...