Stretching of an elastic strut: complex behavior in a simple problem motivated by developmental biology

Lev Truskynovski
Monday 23rd November 2015
Aula Seminari Saleri VI Piano Dip. Matematica-MOX, Politecnico di Milano
Segmentation, the repetitive division of the body axis in modular units, is a ubiquitous motif in biology. In vertebrates, segmentation is established early in embryogenesis by the formation of somites, transient structures that eventually give rise to a variety of tissues, including the spine, skeletal muscles, and the dorsal skin. Unveiling the mechanism of somite formation is one of the major challenges in developmental biology. We argue that the periodicity along the embryonic body axis anticipating somitogenesis may be controlled by mechanical rather than bio-chemical factors (as it is usually postulated). To show that regular patterning can result from a mechanical instability induced by growth-induced differential strains, we study stability of an incompressible hyper-elastic strut subjected to an uniaxial stretching in a mixed device. The necessity to account for the free surface and for the softening in the bulk makes the problem geometrically and physically nonlinear with both nonlinearities carrying their own nonconvexity. The corresponding instability modes compete, and already in a linearized setting we observe a crossover from necking (for slender struts) to wrinkling (for thick struts). The parametric dependence of the intermediate unstable modes reveals remarkable complexity reminiscent of turbulence (joint work with G. Vitale and T. Smit). contact: