# Timothy J. Healey

523 Malott Hall

Ph.D. (1985) University of Illinois

### Research Area

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.

### Selected Publications

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 S*^{2} (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)