Atomistic and continuum fracture models for nanorods: equilibria anddynamics

自20世纪90年代碳纳米管研究的第一次热潮以来，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、纳米柱和纳米须，它们在电子学、光子学、传感器设计或生物医学中都有应用。在连续介质弹性理论中，用充分降维的经典杆理论对细杆的数学描述现在已经很好地理解了。然而，在奈米尺度上，当纯连续体理论受到质疑时，就需要求助于更基本的原子模型。如果考虑到破裂的可能性，则会出现额外的挑战。特别是，陶瓷和半导体纳米线在载荷下表现出较大的挠曲，但也有脆性或延性断裂。它们的力学行为不同于块状材料，依赖于尺寸和结构，并受表面能的影响。在之前的一个项目中，我们已经通过变分（γ -）收敛方法成功地推导出了适用于这种超薄物体的“Kirchhoff-Griffith”型棒理论。新的能量泛函结合了具有原子校正项和griffith型裂纹能量贡献的广义Kirchhoff棒理论。然而，有一些重要的问题是不能用纯粹的伽玛收敛分析来回答的。特别地，这涉及(A)静态平衡的有效行为和(B)动态演化结构的有效描述。无论是从理论角度还是从工程应用角度来看，这两个问题都与薄脆性结构特别相关。在这个项目中，我们检查超薄杆在一个制度，允许有限的弯曲和扭转，同时也有限数量的裂缝或扭结可能发展。通过结合降维技术和从原子模型到连续体模型的过渡，我们研究了纳米棒在原子数量变得非常大而长径比非常小的渐近状态下的粒子系统。重点关注(a)一般平衡构型的有效描述，我们将满足力平衡条件的稳定原子构型严格地与我们先前设计的Kirchhoff-Griffith棒理论的连续统平衡联系起来。在动力区(B)中，（高维）牛顿方程组的解将与（广义）动态Kirchhoff杆理论方程的自由边值问题的解耦合。

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.