Composite materials are those made of two or more constituent materials of different physical properties, whose physical characteristics differ from those of its individual components. The new material may be preferred for many reasons, for instance being stronger, lighter or less expensive when compared to traditional materials. Composite materials have been used in virtually all areas of engineering. Examples include marine structures such as boats and offshore structures, aircraft design in aerospace industry, design of wind and marine turbines, and design of sports equipment such as skis and tennis rackets. Composite materials are ubiquitous in nature. Examples include bones, horns, wood, and biological tissues such as blood vessels of major human arteries. These materials are exposed to a wide spectrum of dynamic loads. In marine structures, aerospace industry, and in many biological constructs, whether natural or human-made, the dynamic load comes from the surrounding fluid such as water, air or blood. Understanding the interaction between composite materials and the surrounding fluid is important for the understanding of normal function of the coupled fluid-structure system, as well as detection of damage and/or pathologies in their function. Prevention of catastrophic events in engineered constructs or design of medical treatments to prevent further biological tissue damage is aided by computational and experimental studies of fluid-composite structure interaction.
Very often one or two characteristic dimensions of some of the involved structures are significantly smaller than the other ones. The thin bodies in elasticity are of extreme importance in engineering (building structures) and biology (complex structure like coronary stents, biological tissues). Due to the presence of the small parameter of thickness, numerical computations dealing with 3D thin bodies are quite expensive. Hence, using lower-dimensional models for which mathematical and numerical analysis is much simpler is a natural alternative. With the advancement of analytical tools it is possible to derive simple two-dimensional (plates, shells) or one-dimensional (rods, curved rods) models starting from 3D elasticity equations.
The work plan of the project includes derivation of equations of lower dimensional models in elasticity and elasto-plasticity, analysis of fluid-structure interaction problems (well-posedness and asymptotic stability analysis), rigorous derivation of physically coupled models of plates or shells interacting with fluid. Benefits of both PDE and calculus of variation techniques are used to approach different topics. Even though the project focus is on analytical issues, its results will be of importance in applications. In particular, in multi-physics systems in engineering and biomechanics by justifying and deriving lower dimensional models and designing of stable, modular numerical schemes for their simulations.
- Simultaneous homogenization and dimensional reduction by means of Gamma - convergence.
- Simultaneous homogenization and dimensional reduction by examining Euler-Lagrange equations for stationary problems as well as evolution.
- Derivation of the lower dimensional models for high contrast materials, perforated domains, thin models of complex structures (elastic nets) in the small strain regime.
- Derivation and justification of the FSI models involving thin nonlinear structures.
- Optimal design plate models in the small strain regime.
- Simultaneous homogenization and dimensional reduction in quasistatic evolution of elasto-plasticity.
- FSI problems with nonlinear thin structures.
- Fluid structure interaction problems involving coronary stents.
- Asymptotic stability analysis for the FSI problems involving thin and composite structures.