CAM colloquium - Friday, February 15
3:30 p.m.
655 Rhodes Hall

Speaker: Timothy Healey, Theoretical and Applied Mechanics, Cornell

 

Title: Some Problems in Second-Gradient Nonlinear Elasticity

 

Abstract: The two basic tenets of finite, nonlinear elasticity are (1) exact geometry of motion and (2) the existence of a stored energy function of an appropriate strain tensor. The latter engenders "stress" as a function of "strain". The theory has immediate applications to flexible structures (both man-made and biological), rubber-like solids and shape-memory alloys – everything from fighter jets to lingerie! In this talk we first attempt to motivate the energy formulation of the subject via the "elastic spring" of classical mechanics. We then discuss some apparent shortcomings of (2) (within the confines of models admitting a stored energy function). In classical (strongly elliptic) nonlinear elasticity, (2) and a physically realistic growth condition lead to unresolved issues concerning the existence and regularity of solutions. In "multi-well" models for phase transitions (2) leads to the lack of a length scale, enabling infinite refinement and rearrangement of phase mixtures. We present one problem from each of these two categories, in the presence of additive second-gradient regularization (in the spirit of Van der Waals and Cahn & Hilliard). We use techniques of global bifurcation theory, a-priori bounds, and computation with symmetry for a class of 2-well-potential phase-transition problems to methodically study "striped" phase mixtures. In the classical case, we obtain the existence of weak equilibrium solutions for second-gradient problems via the calculus of variations.

 

 

Refreshments at 4:30 in 657 Rhodes Hall.