An Extended Beam Theory for Smart Materials Applications

In this paper we further develop the theory of the extended Timoshenko beam model, as first introduced in Part I [5] of this work, with particular emphasis on applications of the model in formation theory [11], [12]. We begin with formal development of the equilibrium equations of static formation theory in the context of the extended Timoshenko model, giving a rigorous discussion of existence, uniqueness, and regularity of weak solutions with appropriate assumptions on the coefficients. We continue to obtain the fundamental duality relationship in the context of weak solutions and indicate its usefulness in investigations of approximate formability. Optimal formation problems and corresponding necessary conditions for optimality are discussed. We conclude with a discussion of a particular problem of joint optimization of controls and actuator densities in the context of a prismatic extended Timoshenko beam and we present the results of some computational studies.