This series of 5 seminars will provide an introduction to the analysis of nonlinear problems in elasticity. Based on examples from recent research works and/or from daily life (such as the inflation of a cylindrical ballon, the shape of human hair, or the hexagonal patterns produced by the elastic Rayleigh-Taylor instability), we will derive the main methods applicable to nonlinear elasticity problems: the calculation of critical loads by a linear bifurcation analysis, the selection of buckling patterns using Koiter's method, the analysis of localized buckling using amplitude equations, etc. We will provide examples of both geometrical instabilities (as Euler's buckling) and material instabilities (as in the striction of bars and other localization phenomena).
5. Material instabilities - Tuesday, May 15, 2018 at 12:00PM 115 Gates-Thomas
The inflation of a cylindrical party balloon leads to localization, with two almost uniform cylinders of different diameters connected by a sharp interface: as the inflation proceeds, the interface propagates and the large cylinder progressively invades the entire length of the balloon. The coexistence of the two 'phases' is classically explained by the analysis of homogeneous cylindrical solutions, whereby the pressure appears to be an up-down-up function of the volume. Maxwell's construction accounts for the pressure plateau at which propagation takes place. We complement this classical picture by a description of the transition region, which (i) brings in an analogy with van der Waals' theory of the liquid-vapor phase transition and (ii) accounts for the finite length of the balloon.