| This dissertation presents robust and efficient mathematical techniques for modeling and simulation of smart material systems. First, passive materials are discussed which undergo large deformations. Taking the Euler-Bernoulli cantilever beams as an example, parametric large deflection components for the case of a combined tip point loading are developed. Verified by comparison with numerical solutions, the obtained parametric deflection solutions are valid for the entire beam length, and independently and efficiently adaptable for very large loading conditions. To demonstrate the robustness of the solutions, the piecewise parametric solutions are coded as a stand-alone executable, BeamSol, which is envisioned to help engineering analyses and syntheses of beam applications. Then, smart materials, in particular, magnetostrictive materials are discussed. A reformulation of the Discrete Energy-Averaged model is given for the calculation of the 3D hysteretic magnetization and magnetostriction of iron-gallium (Galfenol) alloys, and an analytical solution procedure based on eigenvalue decomposition is developed. This procedure avoids the singularities present in the existing approximate solution by offering multiple local minimum energy directions for each easy crystallographic direction. This improved robustness is crucial for use in black-box finite-element codes. Also, analytical simplifications of the 3D model to 2D and 1D applications are presented. To find model parameters, the average of the hysteretic data is utilized. This obviates any need for anhysteretic curves, which would require additional measurements. An efficient optimization routine is developed that retains the dimensionality of the prior art, but improves the accuracy and computational efficiency. Analytical derivations of the Jacobian and Hessian matrices corresponding to this direct model are also presented. Then, a computationally efficient and robust nonlinear modeling framework is presented for fast design and optimization of nonlinear smart materials. The framework consists of a novel 3D inversion scheme for nonlinear modeling of smart material transducers, and a reduced 2D model for smart composite plate structures. Building on the Newton technique, the inversion scheme can be applied to any nonlinear smart material system with a differentiable direct model. The nonlinear 2D magnetostrictive plate model and the 3D inversion scheme are integrated with a finite-element software to analyze an aluminum plate embedded with a Galfenol strip. The resulting nonlinear finite-element framework is utilized to obtain major and minor magnetostriction curves corresponding to the tip of the Galfenol patch exposed to unbiased and biased magnetic fields. A significant advantage in computational time and numerical convergence is demonstrated via comparison with an existing approach for magnetostrictive material modeling. Finally, a globally convergent and fully coupled magnetomechanical model is developed for 3D magnetostrictive systems. The inverse model finds the unknown magnetic field and stress vectors for any specified magnetic flux density and strain vectors. This inversion procedure is iterative, and is proposed for arbitrary magnetostrictive materials. Built on continuation, the approach is globally convergent, which makes it ideal for use in finite-element frameworks. Galfenol is chosen as the magnetostrictive material, and the computational efficiency of the proposed approach is shown to compare favorably against existing models. |
