Piecewise Rigid Body Mechanics

We propose a new three-dimensional dynamic theory of transforming materials intended to make realistic simulations of the dynamic behavior of these materials accessible. The theory is appropriate for materials whose free energy function rises steeply from its energy wells. Essentially, the theory is the multiwell analog of ordinary rigid body mechanics with three additional features: the full stress is not treated as arbitrary (the average limiting tractions on each interface enter the theory as unknowns), a certain component of the local balance of linear momentum is used, and kinetic laws for interfacial motion are introduced based on ideas of Eshelby and Abeyaratne and Knowles. In an interesting special case of the resulting equations of motion, all material constants together with all information about the shape of the body collapse to a single dimensionless constant. We prove well-posedness up to the time of a collision between interfaces, and do a preliminary study of the problem of annihilation and nucleation of interfaces. Conservation laws and a dissipation inequality are identified. We also give generalizations of the theory to magnetic and thermodynamic piecewise rigid media. A probable application area for the theory is the assessment of the use of transforming materials at small scale as “motors” for propulsion or actuation.