Timothy J. Healey
523 Malott Hall
Ph.D. (1985) University of Illinois
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
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 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.
Existence of Global Symmetry-Breaking Solutions in an Elastic Phase-Field Model for Lipid Bilayer Vesicles (with Sanjay Dharmavaram), arXiv:1402.2314 (2014)