Timothy J. Healey
523 Malott Hall / 224 Kimball Hall
Ph.D. (1985) University of Illinois
Nonlinear elasticity, nonlinear analysis, partial differential equations
I work at the interface between nonlinear analysis of pde’s — mostly elliptic systems — and the mechanics of elastic structures and solid continua. Nonlinear (finite) elasticity is the central model of continuum solid mechanics. It has a vast range of applications, including flexible engineering structures, biological structures — both macroscopic and molecular, and materials like elastomers and shape-memory alloys — everything from fighter jets to lingerie! Although the beginnings of the subject date back to Cauchy, the current state of existence theory is quite poor; properly formulated problems of the subject are often out of the range of present day mathematics. In other words there are many open problems.
My work ranges from the abstract — e.g., developing a generalized nonlinear Fredholm degree to obtain solutions “in the large” in 3-D nonlinear elasticity — to the more concrete — e.g., modeling the helical microstructure of DNA in elastic rod models. Most recently I have become quite interested in the analysis of elastic surfaces in models inspired by “biological membranes.”
Global continuation in nonlinear elasticity (with H. Simpson), Arch. Rat. Mech. Anal. 143 (1998), 1–28.
Global continuation in displacement problems of nonlinear elastostatics via the Leray-Schauder degree, Arch. Rat. Mech. Anal. 152 (2000), 273–282.
Global continuation via higher-gradient regularization and singular limits in forced one-dimensional phase transitions (with H. Kielhöfer), SIAM J. Math. Anal. 31 (2000), 1307.
Material symmetry and chirality in nonlinearly elastic rods, Math. Mech. Solids 7 (2002), 405–420.
Global bifurcation in nonlinear elasticity with an application to barrelling states of cylindrical columns (with E. Montes), J. Elasticity 71 (2003), 33–58.
Bifurcation with a two-dimensional kernel (with S. Krömer and H. Kielhöfer), J. Diff. Eq. 220 (2006), 234–258.
Two-phase equilibria in the anti-plane shear of an elastic solid with interfacial effects via global bifurcation (with U. Miller), Proceeding of the Royal Society A 463 (2007), 1117–1134.
Injective weak solutions in second-gradient nonlinear elasticity (with S. Krömer), ESAIM: COCV 15 (2009), 863–871.
A generalized computational approach to the stability of equilibria of nonlinearly elastic rods in the presence of constraints (with Ajeet Kumar), Comp. Meths. Appl. Mech. Engr. 199 (2010), 1805–1815.
A rigorous derivation of hemitropy in nonlinearly elastic rods, Discrete and Continuous Dynamical Systems B 16 (2011), 265–282.
Bifurcation of hemitropic elastic rods under axial thrust (with C. Papadopoulos), Quarterly of Applied Mathematics (to appear).
Global symmetry-breaking bifurcation for the van der Waals-Cahn-Hilliard model on the sphere S2 (with H. Kielhöfer), preprint.
Wrinkling behavior of highly stretched rectangular elastic films via parametric global bifurcation (with Q. Li and R.-B. Cheng), J. Nonlinear Science (to appear).