# William O. Williams

## Professor Emeritus

**Address:**

8130 Wean Hall

Department of Mathematical Sciences

Carnegie Mellon University

5000 Forbes Avenue

Pittsburgh, PA 15213

**P:** 412-268-2679

# Education

Ph.D., Brown University

# Research

My research generally concerns the formulation and analysis of models in mechanics and thermodynamics.

Most recently I have been involved in studies in mechanics whose central point is that of the effect of constraints. At the extreme point are systems which have no internal energy at all, and hence have internal forces which are purely reactions to constraints. A historical example is that of Tchebychev Nets, which were developed by Tchebychev, Servant and Voss in the last century as models for the deformation of cloth. In an older article listed below, I examined the stresses and concentrated forces possible in such systems. More recently, I have become interested in Tensegrity Structures, whose internal forces likewise are purely constraint-generated. These structural trusses, whose elements consist of both rigid bars and cables, were originated by Snelson and developed by Buckminster Fuller, and present interesting technical and mathematical problems. Both of these systems can be treated by relaxation from elastic or partially elastic models.

In work with Podio Guidugli and diCarlo I have considered the approach of treating shells as examples of constrained three-dimensional bodies. This point of view leads in a natural way to two-dimensional models with various degrees of microstructure.

I also have a on-going interest in mathematical models in physiology. In an older work I advanced a model for contraction of muscle fibers that can represent most observed micro-experimental responses in muscle fibers; a goal is to extend this to whole-muscle models. Currently I am investigating applications of the above-mentioned constrained systems to biological models.

# Select Publications

W. Williams, (2011) "Huxley's Model of Muscle Contraction with Compliance", J. Elasticity, to appear. Preprint

A. Micheletti and W. Williams, (2007) "An Evolutionary Procedure for Form-finding for Tensegrity Structures ", J. Mech. of Materials and Structures, 2:857--882, 2007. Preprint

W. Williams, (2003) "A Primer on the Mechanics of Tensegrity Structures ", to appear. Preprint

Oppenheim, O. J. and W. O. Williams, (2001), "Vibration of an Elastic Tensegrity Structure," Eur. J. Mech. A, 20:1023--1031. Preprint

A.diCarlo, P. Podio Guidugli and W. Williams (2000), "Shells with Thickness Distension", Int. J. Solid. Struct., 38:7405-7429.Preprint

Oppenheim, O. J. and W. O. Williams, (2000), "Geometric effects in an elastic tensegrity structure," J. Elasticity, 59:51-65. Preprint

W. Williams, (1991), "Forces in Tchebechev nets ", in Waves and Stability in Continuous Media, ed. S. Rionero, World Scientific, New Jersey, 408-418.

W. Williams. (1990), "A Purely Mechanical Model of Muscular Contraction," Math. Biosc. 100: 249-278.