An equivalent growth-induced strain, highest along the edges and lessening toward the middle, occurs as a long leaf grows, leading to the elegant arc and serrated surface of the leaves in plants like the lily. This effect is widely seen, says Mahadevan, in a variety of common objects and activities.
"When knitting a scarf, as the number of stitches is increased as the knitter moves away from the center, the material forms a saddle shape. As the edge length becomes much larger ripples begin to appear. The same effect can be seen when thin potato slices are dropped into hot oil to make chips. You end up with a bulbous middle and wrinkled edges," he explains.
The researchers also dissected the leaves of the plantain lily to show that elastic strain resulting from differential growth led to the patterns seen in real leaves. From this simple experiment, the researchers then developed a mathematical model explaining the shape, using a combination of scaling concepts, stability analysis, and numerical simulations.
"While the phenomena has been studied previously, researchers did not consider the role of finite size of a leaf on the stability or the effect of boundaries. Further, our study characterizes, mathematically, the range of parameters that quantify the shape and diversity in leaf morphology," adds Mahadevan.
The resulting model has application in understanding a variety of artificial systems such as non-uniform thermal expansion, hydraulic swelling, and plasticity induced shape changes in thin laminae.
|Contact: Michael Patrick Rutter|