Investigation of some nonlinear problems in mechanics

We investigate some nonlinear problems in mechanics, which include the dynamic problem with nonlinear interaction, the optimal design of multi-phase conductive compos- ite, and the phase transition of lattice structures governed by non-convex energy functions. The first nonlinear problem we investigate is the dynamics of a two-dimensional lattice with harmonic, weakly nonlinear, and strongly nonlinear interactions. Assuming nearest neighbor interaction, we derive the continuum approximation of the discrete system in the long wavelength regime while keeping the Hamiltonian structure of the system. For a hexagonal lattice with nontrivial shear resistance, we surprisingly find that solitary wave solutions exist in certain directions related to the underlying symmetries of the lattice. The properties of the solitary waves are also studied by numerical simulations of the original discrete system. Besides being of fundamental scientific interest, the solitary wave solutions in nonlinear hexagonal lattices are anticipated to have applications in the design of shock absorbers, acoustic lens, non-destructive structural testing devices among many others. Secondly, the optimal design of multiphase conductive composites by estimating Hashin- Shtrikman bounds (HS bounds) attainability in the constraint of volume fractions of the constituent phases. We derive the necessary condition by null lagrangian and maximum principle. We address the sufficient condition of the HS bounds attainability by proposing a new class of coated sphere comprised of three-phase and four-phase isotropic conductive materials, and generalize the coated spheres to a larger number phases. Combining the necessary and sufficient condition of HS bounds attainability, we are able to precisely characterize the G-closure and effective properties of multiphase conductive materials for a broader range. Lastly, we design and characterize a two-dimensional (2D) crystal formed by a lattice structure, which consists of repeating structure elements as called unit cell. Based on the minimization of total free energy of the unit cell, we find three stable phases coexist at the critical loading which turns out naturally result in the microstructure. The microstructure of the lattice structure may easily buckle while the macrostructure of the lattice structure is in compression, the properties of the lattice structure are also studied by numerical simulations in 2D biaxial stress system. Such lattice structures can undergo “phase transitions” mimicking the Austenite-Martensite phase transition in shape memory alloys (SMAs). More importantly, they offer directly observable material models that can shed light on the fundamental mechanisms of the first-order non-diffusive phase transitions and shape memory effects, their interactions with defects, and the physical origins of hysteresis.Ph.D.Includes bibliographical reference