Atomistic and continuum fracture models for nanorods: equilibria anddynamics

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Since the first boom in research on carbon nanotubes in the 1990s, we have been experiencing discoveries of a wide variety of 1d nanomaterials. These include nanowires, nanorods, nanopillars, and nanowhiskers, which find applications in electronics, photonics, sensor design or biomedicine. The mathematical description of thin rods in continuum elasticity theory in terms of adequately dimensionally reduced classical rod theories is by now well understood. At the nanoscale, however, when pure continuum theories become doubtful, one needs to resort to more fundamental atomistic models. Additional challenges arise if the possibility of fracture is taken into account. In particular, ceramic and semiconductor nanowires under loading exhibit large deflections, but also brittle or ductile fracture. Their mechanical behavior is different from that of bulk materials, size- and structure-dependent, and influenced by surface energy. In a preceding project we have succeeded in deriving an adequate "Kirchhoff-Griffith" type rod theory for such ultrathin objects by variational (Gamma-)convergence methods. The novel energy functional combines a generalized Kirchhoff rod theory featuring atomistic correction terms and Griffith-type crack energy contributions. Yet, there are important questions that cannot be answered by a pure Gamma-convergence analysis. In particular, this concerns (A) the effective behavior of static equilibria and (B) the effective description of dynamically evolving structures. Both of these questions are of particular relevance in thin brittle structures, both from a theoretical point of view and with respect to applications in engineering. In this project we examine ultrathin rods in a regime which allows for finite bending and torsion while also a limited number of cracks or kinks might develop. By combining dimension reduction techniques and a passage from atomistic to continuum models, we investigate particle systems for nanorods in the asymptotic regime as both the number of atoms becomes very large and the aspect ratio extremely small. With a focus on (A) the effective description of general equilibrium configurations, we rigorously relate stable atomistic configurations satisfying a force balance condition to continuum equilibria of our previously designed Kirchhoff-Griffith rod theory. In the dynamic regime (B), the solutions of the (high dimensional) system of Newton equations will be coupled to the solutions of a free boundary value problem for the equations of a (generalized) dynamic Kirchhoff rod theory.