Ph.D., Northwestern
University
E-mail address: weo@northwestern.edu
Professor of Applied Mathematics and Mathematics
Charles Deering McCormick Professor of Teaching Excellence (1994-97).
Editorial Board - SIAM Journal on Applied Mathematics (1998 - )
Board of Directors - Society of Engineering Science (1997-2000)
Recent graduate students (year finished/first professional position):
L.R. Ritter
(2003/ Post Doc, Texas A&M U.)
C.M. Kirk (1999/Assistant
Professor, Montclair State U.)
R.P. Flemming (1998/Assistant
Professor, Harvey Mudd College)
Research
Interests:
Blow-up in a reactive-diffusive medium due to a
moving, localized energy source. Shear localization effects in high
strength materials.
When a localized energy source such as a laser is passed over the surface of a material that has combustible properties, there is the possibility of a thermal blow-up. Such problems can be modeled mathematically by a nonlinear parabolic equation. By applying analytical methods to the model problem, it is possible to determine the roles played by the size, strength and motion of the energy source in determining whether or not a blow-up occurs. A basic result shows that moving the source at a sufficiently high speed can prevent a blow-up.
Shear localization effects occur when high-strength metals are subject to very large strain rates. Such conditions arise in various machining processes as well as in situations of impact or fragmentation. The localized plastic flow of the metal in very narrow zones is called shear banding (see figure 1). Understanding the formation of a shear band is important since it represents a possible site of material failure. We are taking advantage of the extreme thinness of the shear band to model it mathematically as a boundary layer. This leads to a considerable simplification of the system of partial differential equations which govern the mechanical behavior of highly stressed materials. The calculations made from this theoretical model confirm the experimentally observed collapse of the stress in the shear band as the metal undergoes thermal softening (see figure 2).
Recent publications:
“Blow-up solutions of the two-dimensional heat equation due to a localized moving source”. Analysis and Applications, 3 (2005) (with C.M. Kirk).
“A Saint-Venant principle for shear localization”. Z. angew. Math. Phys. 54 (2003). (with C.O. Horgan).
“Initiation of Free-Radical Polymerization Waves”. SIAM J. Appl. Math. 64 (2003) (with L.R. Ritter and V.A. Volpert).
"Numerical Solution of Shear Localization in Johnson-Cook materials”. Mech. of Materials 35 (2003) (with J. A. DiLellio).
Last modified: May 3, 2005