Atomistic and continuum fracture models for nanorods: equilibria anddynamics

自从20世纪90年代碳纳米管研究的第一次繁荣以来，我们已经发现了各种各样的一维纳米材料。这些包括纳米线、纳米棒、纳米柱和纳米晶须，其在电子学、光子学、传感器设计或生物医学中找到应用。用适当降维的经典杆理论，在连续弹性理论中对细杆的数学描述现在已经很好地理解了。然而，在纳米尺度上，当纯粹的连续统理论变得可疑时，人们需要求助于更基本的原子模型。如果考虑到断裂的可能性，则会出现额外的挑战。特别是，陶瓷和半导体纳米线在负载下表现出大的偏转，但也脆性或韧性断裂。它们的力学行为不同于块状材料，取决于尺寸和结构，并受表面能的影响。在前面的项目中，我们已经成功地推导出一个适当的“基尔霍夫-格里菲斯”型棒理论，这种天体的变分（伽马）收敛方法。新的能量泛函结合了广义基尔霍夫棒理论，具有原子修正项和格里菲斯型裂纹能量贡献。然而，有一些重要的问题是不能用纯粹的伽马收敛分析来回答的。特别是，这涉及（A）静态平衡的有效行为和（B）动态演变结构的有效描述。这两个问题是特别相关的薄脆性结构，无论是从理论的角度来看，并在工程中的应用。在这个项目中，我们研究了一个制度，允许有限的弯曲和扭转，同时也有有限数量的裂纹或扭结可能发展的拉杆。通过结合降维技术和一个通道从原子到连续模型，我们调查粒子系统的纳米棒在渐近制度的原子数量变得非常大，纵横比非常小。重点是（A）一般平衡组态的有效描述，我们严格地将满足力平衡条件的稳定原子组态与我们先前设计的Kirchhoff-Griffith棒理论的连续平衡联系起来。在动态区域（B）中，（高维）牛顿方程组的解将耦合到（广义）动态Kirchhoff杆理论的方程的自由边值问题的解。

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.