# Mini-Workshop: The Mathematics of Electro-Active Smart Materials

### Ray W. Ogden

University of Glasgow, UK### Luis Dorfmann

Tufts University, Medford, United States### Giuseppe Saccomandi

Università degli Studi di Lecce, Italy

## Abstract

The mini-workshop \emph{The Mathematics of Electro-Active Smart Materials}, was attended by 15 participants (7 from Europe, 4 from USA and 1 from Israel). This workshop contained a nice blend of researchers with various backgrounds: mathematics, physics, engineering science and chemistry. The interdisciplinarity of the invited participants was essential because smart materials, such as piezoceramics, magnetostrictives and electroactive polymers, are characterized by strong couplings between their mechanical properties and applied electric, magnetic or thermal fields. Therefore skills in several areas of science are needed to understand the complicated and highly nonlinear behavior of such materials.

Science and technology have produced amazing developments in the design of electronics and machinery using smart materials. Some everyday items are already incorporating smart materials (cars, the International Space Station, eyeglasses, coffee pots, speakers, for example) and the number of applications is growing steadily. For example, electro-rheological (ER) and magneto-rheological (MR) materials are fluids that can experience a dramatic change in their viscosity. They can change from a thick fluid (similar to motor oil) to nearly solid substances within the span of a millisecond when exposed to a magnetic or electric field; the effect can be completely reversed just as quickly when the field is removed. MR fluids are being developed for use in car shock absorbers, damping of machine vibration, prosthetic limbs, exercise equipment, and surface polishing of machine parts, in particular. ER fluids have been developed mainly for use in clutches and valves, as well as in engine mounts where they are designed to reduce noise and vibration in vehicles. More recently, magneto- and electro-sensitive elastomers have been developed and commercialized. Such materials are capable of large elastic deformations and a rapid and significant change in their material properties on the application of a magnetic or electric field. The coupling in the material response is achieved and optimized by distributing within an elastomeric matrix micron size ferrous particles. Cross-linking may occur, for example, in the presence of an applied field, whereby particles form chain-like structures aligned along the applied field direction. The material is then characterized by two families of preferred direction and may result in highly controllable smart materials.

Mathematical and computational methods pervade research, development, testing, and evaluation problems encountered by researchers in the field of smart materials. Furthermore, increasing demands are being placed on research in the mathematical sciences because of their fundamental roles in the analysis and modelling issues that arise in such complex materials.

In this Mini-Workshop we have discussed recent mathematical and computational questions related to the modelling of smart materials used in structures, actuators and sensors from the point of view of material properties and control of mechanical properties.

The first talk by G. A. Maugin was on overview of existing theories capable of describing the strong coupling between mechanical properties and applied fields in electro- and magneto-active smart materials. This overview was extremely interesting also because it was punctuated with historical notes and remarks.

On the other hand M. Zrinyi gave an exhaustive review of the phenomenology of such materials and of experimental data. This was extremely useful for facilitating a clear identification of further key experiments that should be conducted in order to provide more comprehensive data for improving the modelling process.

The other talks have been devoted to

\begin{itemize} \item Recently developed specific and realistic constitutive laws for describing the response of smart materials, with particular reference to the highly nonlinear deformation, polarization and magnetization. \item Analytical formulations and numerical solution methods for a range of specific boundary-value problems for both idealized geometries and geometries that are of practical interest in the design of smart sensors and actuators. \item Illustrative numerical solutions of the coupled field equations for boundary value problems with complex geometries with particular emphasis on electric and magnetic boundary conditions at the material interfaces. \item Mathematical and numerical theories that include incremental deformations superimposed on a finite deformation as a basis for studying the effect of electro-magnetic fields on the stability of finitely deformed elastic solids, both in general terms and with a view to analysis of stability in specific boundary-value problems. \item Mathematical theories that include time dependence in the formulation of the equations in order to examine vibration and wave propagation characteristics of magneto- and electro-sensitive materials. Numerical methods for the corresponding equations of motion. \end{itemize}

## Cite this article

Ray W. Ogden, Luis Dorfmann, Giuseppe Saccomandi, Mini-Workshop: The Mathematics of Electro-Active Smart Materials. Oberwolfach Rep. 5 (2008), no. 1, pp. 473–512

DOI 10.4171/OWR/2008/10