We propose a gauge theoretic model for quantifying and evolving spatial defects in the form of micro-cracks in deforming solids. A non-trivial affine connection — the gauge connection, is introduced to accommodate local changes in the configuration due to spatial defects. The gauge connection enables defining the covariant derivative, a procedure known as the minimal replacement. The configuration gradients in the Lagrangian are then determined using covariant derivatives, instead of partial derivatives, thereby incorporating kinematic information pertaining to the spatial defects. Minimal replacement ensures that the invariance of the Lagrangian under the local action of the symmetry group, which in this case is the scaling of the deformed coordinates, is restored. Introduction of the gauge field Lagrangian via yet another construct — the minimal coupling, obtains the additional energy contribution pertaining to defect evolution. The resulting Euler–Lagrange (EL) equations thus describe the coupled motion of the solid and the evolving defects. The EL equation for defect evolution presently accounts for microscopic inertia. Different features of brittle damage, viz. tension–compression asymmetry, stiffness degradation and an energy functional including the contribution from defects in the form of cracks, are described within the gauge theoretic model considering the kinematic aspects of deformation and damage. The usefulness of the model and its advantages over a phase-field based damage model are assessed through peridynamics-based numerical simulations on high-speed oscillatory crack-tip instabilities as observed experimentally during dynamic crack propagation in a pre-notched polymethyl methacrylate (PMMA) plate. Simulations are also carried out for dynamic crack propagation in maraging steel subjected to impulsive loading and dynamic crack branching in a glass type material. We claim that the model offers a physically transparent and accurate tool to understand nonlinearities in the fracture process zone.

我们提出了一种规范理论模型，用于量化和演化变形固体中微裂纹形式的空间缺陷。引入了一个非平凡的仿射连接——规范连接，以适应由于空间缺陷而导致的局部配置变化。规范连接可以定义协变导数，该过程称为最小替换。然后使用协变导数而不是偏导数确定拉格朗日中的配置梯度，从而结合与空间缺陷有关的运动学信息。最小替换确保不变性在对称群的局部作用下，拉格朗日的 rgrangian 被恢复，在这种情况下是变形坐标的缩放。通过另一种构造——最小耦合引入规范场拉格朗日，获得与缺陷演化有关的额外能量贡献。由此产生的欧拉-拉格朗日 (EL) 方程描述了固体的耦合运动和不断发展的缺陷。缺陷演化的 EL 方程目前考虑了微观惯性。脆性损伤的不同特征，即。考虑到变形和损坏的运动学方面，在规范理论模型中描述了拉压不对称性、刚度退化和能量泛函，包括裂纹形式的缺陷的贡献。该模型的有用性及其相对于基于相场的损伤模型的优势通过基于近场动力学的数值模拟进行评估，该数值模拟在动态裂纹扩展过程中通过实验观察到的高速振荡裂纹尖端不稳定性在预先开槽的聚甲基丙烯酸甲酯(PMMA) 板上。还对马氏体时效钢中的动态裂纹扩展进行了模拟，该钢在玻璃类材料中受到脉冲载荷和动态裂纹分支。我们声称该模型提供了一种物理透明且准确的工具来理解断裂过程区的非线性。