# The macroscopic effect of microscopic interfaces

### Project 1. The second Stekloff eigenvalue and energy dissipation inequalities for functions with surface energy

In this project a functional with both bulk and interfacial surface energy is considered. It corresponds to the energy dissipated inside a two-phase electrical conductor in the presence of an electrical contact resistance at the two-phase interface. The effect of embedding a highly conducting particle into a matrix of lesser conductivity is investigated. We find the criterion that determines when the increase in surface energy matches or exceeds the reduction in bulk energy associated with the particle. This criterion is general and applies to any particle with Lipschitz continuous boundary. It is given in terms of the of the second Stekloff eigenvalue of the particle. This result provides the means for selecting energy-minimizing configurations.

This work has appeared in R. Lipton. SIAM J. Math. Anal. 29:3 (1998) pp. 673--680. For the existence therory of optimal designs with bulk and surface energy see, R. Lipton. On existence of energy minimizing configurations for mixtures of two imperfectly bonded conductors. Recent advances in structural modelling and optimization. Control Cybernet. 27 (1998), no. 2, 217--234.

*This research was supported by NSF grant DMS-9403866 and by the Air Force Office of Scientific Research, Air Force Material Command, USAF, under grant F49620-96-1-0055.*

### Project 2. Composites with imperfect interface

In joint work with Bogdan Vernescu we extend the Hashin Shtrikman technique to problems involving both bulk energy and interfacial energy. These bounds apply to imperfectly bonded composites with interfacial thermal contact resistance or imperfectly bonded elastic interfaces. The upper and lower bounds are used to find suspensions of neutral inclusions where the energetic effects of the imperfect interface are compensated for by the superior conductivity or elastic properties of the inclusions.

This work has appeared in R. Lipton and B. Vernescu. Mathematical Models and Methods in the Applied Sciences (M3AN), 5 (1995) pp. 1139--1173, and R. Lipton and B. Vernescu. Proceedings of the Royal Society of London A452 (1996) pp. 325--358.*This project was supported by NSF grant DMS-9403866.*

### Project 3. Reinforcement of elastic structures in the presence of imperfect bonding

The two-dimensional problem of plane strain is considered in the presence of imperfectly bonded elastic reinforcements. A geometric criterion on the shape and size of the reinforcement is found that determines when the effects of imperfect bonding overcome the benifits of the reinforcement. The criterion is given in terms of an eigenvalue problem posed on the surface of the reinforcement.

This work has appeared in R. Lipton. Reinforcement of Elastic Structures in the presence of imperfect bonding. Quarterly of Applied Mathematics Vol. 59, No. 2 (June 2001), pp. 353-364 .

*This research was supported by NSF grant DMS-9700638 and by the Air Force Office of Scientific Research, Air Force Material Command, USAF, under grant F49620-99-1-0009.*

## References

- R. Lipton. The Second Stekloff eigenvalue and energy dissipation inequalities for functionals with surface energy. SIAM Journal on Mathematical Analysis, vol. 29 (1998), no. 3, 673--680.
- R. Lipton and B. Vernescu. Variational methods, size effects and extremal microgeometries for elastic composites with imperfect interface. Mathematical Models and Methods in Applied Sciences. 5 (1995), 1139--1173.
- R. Lipton and B. Vernescu. Composites with imperfect interface. Proceedings of the Royal Society of London A 452 (1996), 329--358.
- R. Lipton. Reinforcement of Elastic Structures in the presence of imperfect bonding. Quarterly of Applied Mathematics Vol. 59, No. 2 (June 2001), pp. 353-364