Timothy J. Healey
523 Malott 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 modeling and analysis of thin elastic surfaces — in particular, wrinkling of highly stretched sheets and pattern formation of fluid-elastic vesicles.
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 71 (2013), 729-753.
Global symmetry-breaking bifurcation for the van der Waals-Cahn-Hilliard model on the sphere S2 (with H. Kielhöfer), Journal of Dynamics and Differential Equations, online (2013), 1-16.
Wrinkling behavior of highly stretched rectangular elastic films via parametric global bifurcation (with Q. Li and R.-B. Cheng), J. Nonlinear Science 23 (2013), 777-805.
Computational stability of phase-tip splitting in the presence of small interfacial energy in a simple two-phase solid (with Á. Sipos), Physica D 261 (2013), 62-69.
Existence of Global Symmetry-Breaking Solutions in an Elastic Phase-Field Model for Lipid Bilayer Vesicles (with Sanjay Dharmavaram), arXiv:1402.2314 (2014)