# Timothy J. Healey

523 Malott Hall

Ph.D. (1985) University of Illinois

### Research Area

Applied analysis and partial differential equations, mathematical continuum mechanics

I work at the interface between nonlinear analysis of pde's/calculus of variations and the mechanics of materials and elastic structures. Nonlinear (finite-deformation) 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. Although the beginnings of the subject date back to Cauchy, the current state of existence theory is generally poor; there are many open problems.

The two main goals are to establish rigorous results and to uncover new phenomena. The work invovles a symbiotic interplay between three key ingredients: careful mechanics-based modeling, mathematical analysis, and efficient computation. It 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 am 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

*Preservation of nodal structure on global bifurcating solution branches of elliptic equations with symmetry* (with H. Kielhöfer), JDE **106** (1993), 70-89

*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.

*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.

*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.

*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.

*Existence of Global Symmetry-Breaking Solutions in an Elastic Phase-Field Model for Lipid Bilayer Vesicles* (with Sanjay Dharmavaram), arXiv:1402.2314 (2014)