Timothy J. Healey

Biography

The grandson of immigrants, Tim Healey was the first in his extended family to attend the university. With the full support of an Evans Scholarship, he studied mathematics and civil engineering at the University of Missouri , where he was awarded the B.S. degree in 1976. He obtained an M.S. from the University of Illinois in 1978, and then worked for two years as a licensed structural engineer in Los Angeles. He returned to Illinois to pursue his doctoral studies in 1980. After receiving his Ph.D. in 1984, he spent one year as a visiting professor in the Department of Mathematics at the University of Maryland. He joined the Department of Theoretical & Applied Mechanics at Cornell in 1985. He is a member of the Editorial Boards of the SIAM Journal of Mathematical Analysis , the Journal of Elasticity , and the Zeitschrift für angewandte Mathematik und Physik (ZAMP). He has held visiting positions at the Ecole Polytechnique Federal de Lausanne, the University of Minnesota and the University of Sydney . He won a Dean's Prize for Excellence in Teaching in 1993 and a College of Engineering Teaching Award in 1996.

Research Interests

I work at the interface between the mechanics of nonlinearly elastic structures and solids, partial differential equations,nonlinear analysis and bifurcation theory. 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. My work ranges from the abstract - e.g., developing a generalized nonlinear Fredholm degree obtaining the existence of solutions “in the large” in 3-D nonlinear elasticity [5 ] – to the more concrete – e.g., modeling the helical microstructure of things like DNA in elastic rod models [11]. I am particularly interested in the role of constitutive hypotheses (modeling) within general classes of problems exhibiting instability and bifurcation phenomena. Indeed experiments on real materials often involve bifurcations from some homogeneous state to more exotic ones, e.g., the onset of super-coiling in DNA, the stress-induced formation of microstructure in shape-memory alloys, etc. We employ rigorous methods of nonlinear analysis to predict the onset, global post-critical formation and stability of patterns in the “exotic” secondary states. This interplay between constitutive hypotheses, nonlinear analysis and also computation, in comparison with the physical phenomena, enables not only the calibration of predictive models but also the potential for classifying materials according to mathematical constitutive hypotheses.  

Current Research Projects

Nonlinear Problems of Elasticity for Multiphase Solids and Shells
(NSF DMS-0406161)

In this project we are studying multi-phase equilibria in two seemingly unrelated classes of problems:

(1) Stress-induced formation of micro-structure in martenstic phase transitions of shape-memory materials:

The importance of the shape-memory effect in certain alloys and its potential application in “smart structures” and micro-actuation is well known. Our work here is inspired by experimental results – mainly those due to R. James and co-workers (U. Minnesota). A nominally homogeneous specimen is subjected to biaxial loadings - a typical microstructure is shown below. In contrast to a popular modeling approach based upon the construction of minimizing sequences, we incorporate “small” interfacial energy and employ methods of global bifurcation theory. In this way we obtain global paths of equilibria of a regularized system of elliptic pde's. In particular, we identify meta- stable equilibiria – local energy minima. The latter are crucial to the understanding of hysteresis. A typical family of computed meta-stable, symmetry-broken states – for a two-phase model in anti-plane shear – is depicted below. The analysis of more realistic models is under way. There are many open mathematical problems here in terms of existence, a-priori bounds, multiplicity of solution and singular limits (for vanishing capillarity).

Stress-induced twinned microstructure - image courtesy of R.D.James   Computed microstructure: meta-stable equilibria in anti-plane shear

(2) Formation of spatial patterns in giant unilamellar vesicles (GUV's)

GUV's are man-made lipid-bilayer membrane “spheres” – called liposomes. They mimic the behavior of cell membranes, exhibiting rich shape-transition behavior. Consequently, they have enormous potential application in the medical field – drug delivery, gene therapy, etc. Our work here is inspired by recent experiments of T. Baumgart and W. Webb (Cornell U) on GUV's. A sampling of their new imaging techniques from their experiments is given below. Each state shown corresponds to a distinct lipid composition and/or osmotic pressure. We are developing novel two-phase models for such shell structures. Although the physics and hence the models in project (1) and (2) are distinct, the same mathematical approach is employed: We begin with a nominally homogeneous (spherical) state and seek global, meta-stable, symmetry-broken states under changing pressure. There are even more open problems here than in project (1) – as far as we can tell, the analysis of geometrically exact shell problems of this genre – truly governed by pde's - is untouched!

Two-phase GUVs - image courtesy of T. Baumgart

Global Continuation and Bifurcation in 3-D Nonlinear Elasticity

We continue our previous work on a nonlinear Fredholm degree appropriate for problems in strongly-elliptic nonlinear elasticity. In particular, we are developing tools along the lines of [5], but now for incompressible materials. This is not nearly as routine as it may sound. Although the essential steps are clear – the incorporation of the nonlinear constraint causes considerable technical difficulties. The potential importance of this tool stems from the fact that detailed linearized bifurcation analyses are possible (and well known) for a large class of incompressible problems, i.e., the necessary conditions for bifurcation are readily computable (not so for the analogous compressible problems). Our new degree will enable global post-critical analyses for a large class of those problems.

Nonlinear Problems for Elastic Rods

We continue our previous work on the modeling of elastic Cosserat rods with chirality [11] (handedness) and on formulations enabling efficient computation [14]. In particular, we are currently investigating the effects of pure hemitropy (and more generally, helical symmetry) in rod models incorporating extensional strain. In [11] we demonstrate the importance of this extra degree of freedom in understanding unraveling (and raveling) in chiral rods. In contrast to the DNA-elasticity literature, where the isotropic model “reigns supreme”, we are studying the onset of supercoiling in DNA with extensional-chiral models. We demonstrate asymmetry of supercoiling in terms of the direction of imposed twisting (i.e., winding up is different than unraveling), which is observed in experiment.

Selected Publications

[1].    T.J. Healey & H. Kielhöfer, "Preservation of Nodal Structure on Global Bifurcating Solution Branches of Elliptic Equations with Symmetry", Journal of Differential Equations, vol. 106, no. 1, (1993), p. 70. ( PDF, 5.8MB)

[2].    T.J. Healey, H. Kielhöfer & C.A. Stuart, "Global branches of positive weak solutions of semilinear elliptic problems over nonsmooth domains", Royal Society of Edinburgh, 124A, (1994), p. 371. (PDF, 2.9MB)

[3].    J.C. Wohlever & T.J. Healey, "A group theoretic approach to the global bifurcation analysis of an axially compressed cylindrical shell", Comput. Meth. Appl. Mech. Engr., vol. 122, (1995), p. 315. (PDF, 20.2MB)

[4].    T.J. Healey & P. Rosakis, "Unbounded Branches of Globally Injective Solutions in the Forced Displacement Problem of Nonlinear Elasticity", J. Elasticity, vol. 49, (1997), p. 65.

[5].    T.J. Healey & H. Simpson, "Global Continuation in Nonlinear Elasticity", Arch. Rational Mech. Anal., vol. 143, (1998), p. 1.

[6].    A. Vainchtein, T.J. Healey, P. Rosakis & L. Truskinovsky, "The Role of the Spinodal Region in One-Dimensional Models of Phase Transformations", Physica D, vol. 115, (1998), p. 29.

[7].     A. Vainchtein, T.J. Healey & P. Rosakis, "Bifurcation and Metastability in a New One-Dimensional Model for Martensitic Phase Transitions", Comput. Meth. Appl. Mech. Engr., vol. 170, (1999), p. 407.

[8].    T.J. Healey & H. Kielhöfer, "Global Continuation via Higher-Gradient Regularization and Singular Limits in Forced One-Dimensional Phase Transitions", SIAM J. Math. Anal., vol. 31, (2000), p. 1307.

[9].     T.J. Healey, "Global Continuation in Displacement Problems of Nonlinear Elastostatics via the Leray-Schauder Degree", Arch. Rational Mech. Anal., vol. 152, (2000), p. 273.

[10].   G. Domokos & T.J. Healey, "Hidden Symmetry of Global Solutions in Twisted Elastic Rings", J. Nonlinear Science, vol. 11, (2001), p. 47.

[11]. Material Symmetry and Chirality in Nonlinearly Elastic Rods, Math. Mech. Solids 7 (2002) 405-420. (PDF)    

[12]. A Simple Approach to the 1:1 Resonance Bifurcation in Follower-Load Problems (with K. MacEwen), Nonlinear Dynamics 32 (2003) 143-159.

[13].  Global Bifurcation in Nonlinear Elasticity with an Application to Barrelling States of Cylindrical Columns (with E. Montes), J. Elasticity 71 (2003) 33-58.

[14].   Straightforward Computation of Spatial Equilibria of Geormetrically Exact Cosserat Rods (with P. Mehta), to appear Int., J. Bifurcation Chaos (2005). (PDF)

[15].  Multiple Helical Perversions of Finite Intrinsically Curved Rods (with Domokos), to appear  Int., J. Bifurcation Chaos (2005). (PDF)

[17].  Global Continuation in Second-Gradient Nonlinear Elasticity (with A. Mareno), submitted (2005). (PDF)

[18].  Singular Perturbation as a Selection Criterion for Young-Measure Solutions (with M. Lilli and H. Kielhöfer). (PDF)

[19].  Two-Phase Equilibria in the Anti-Plane Shear of an Elastic Solid with Interfacial Effects via Global Bifurcation (with U. Miller). (PDF)

[20].  Variational Derivation for Higher Gradient Van der Waals Fluids Equilibria and Bifurcating Phenomena (with L. Deseri). (PDF)

NSA-UPRH Regional Conference

Global Continuation Methods in Three Dimensional Elasticity - a conference supported by a grant from the National Security Agency and by the University of Puerto Rico.
I gave lectures on recent developments in degree theoretic techniques and their applications to problems in nonlinear elastostatics. Poster

Teaching

Spring 2005: T&AM 752 NONLINEAR ELASTICITY 

Last revised: 12/06