In this paper we discuss the geometry and mechanics of discrete manifolds—so called simplicial complexes that are essentially triangulated regions—motivated by the works on discrete exterior calculus and differential geometry in computer graphics. We show that the classical ideas related to the geometry of continuous manifolds, such as metric, curvature, and affine connections take on a much simpler and more intuitive aspect when discussed in the context of such discrete triangulated manifolds. We introduce the notion of a dual mesh to describe “dual” variables (technically differential forms). We use the dual mesh to show that the geometry of the defects such as microcracks, dislocations and incoherent boundaries as well as balance laws and constitutive relations can be introduced directly, without the need for “discretization” from a continuum. This opens up the possibility of direct simulations of these bodies without the need for a continuous counterpart. We end by demonstrating how the second law of thermodynamics can be used in such a situation including discrete non-local systems.

在本文中，我们讨论了离散流形的几何学和力学原理，即所谓的单纯形络合物，本质上是三角区域，它们是由计算机图形学中的离散外部微积分和微分几何学引起的。我们表明，在这种离散的三角歧管的上下文中进行讨论时，与连续歧管的几何形状（例如公制，曲率和仿射连接）有关的经典思想具有更简单，更直观的方面。我们引入了双重网格的概念来描述“双重”变量（技术上是差分形式）。我们使用双重网格显示，可以直接引入缺陷的几何形状，例如微裂纹，位错和不连贯的边界以及平衡律和本构关系，而无需从连续体中进行“离散化”。这为直接模拟这些物体提供了可能，而无需连续的对应。最后，我们将说明如何在包括离散非局部系统在内的这种情况下使用热力学第二定律。