This study presents the formulation, the numerical solution, and the validation of a theoretical framework based on the concept of variable-order mechanics and capable of modeling dynamic fracture in brittle and quasi-brittle solids. More specifically, the reformulation of the elastodynamic problem via variable and fractional order operators enables a unique and extremely powerful approach to model nucleation and propagation of cracks in solids under dynamic loading. The resulting dynamic fracture formulation is fully evolutionary hence enabling the analysis of complex crack patterns without requiring any a prior assumptions on the damage location and the growth path, as well as the use of any algorithm to track the evolving crack surface. The evolutionary nature of the variable-order formalism also prevents the need for additional partial differential equations to predict the damage field, hence suggesting a conspicuous reduction in the computational cost. Remarkably, the variable order formulation is naturally capable of capturing extremely detailed features characteristic of dynamic crack propagation such as crack surface roughening, single and multiple branching. The accuracy and robustness of the proposed variable-order formulation is validated by comparing the results of direct numerical simulations with experimental data of typical benchmark problems available in the literature.

中文翻译：

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变阶断裂力学及其在动态断裂中的应用

本研究提出了基于变阶力学概念并能够模拟脆性和准脆性固体中的动态断裂的理论框架的公式、数值解和验证。更具体地说，通过可变阶和分数阶算子重新表述弹性动力学问题，提供了一种独特且极其强大的方法来模拟动态载荷下固体中裂纹的成核和扩展。由此产生的动态断裂公式是完全进化的，因此能够分析复杂的裂纹模式，而无需对损坏位置和增长路径进行任何事先假设，也不需要使用任何算法来跟踪不断发展的裂纹表面。可变阶形式主义的进化性质也防止了需要额外的偏微分方程来预测损伤场，因此表明计算成本显着降低。值得注意的是，可变阶公式自然能够捕捉动态裂纹扩展的极其详细的特征，例如裂纹表面粗糙化、单分支和多分支。通过将直接数值模拟的结果与文献中可用的典型基准问题的实验数据进行比较，验证了所提出的可变阶公式的准确性和稳健性。可变阶公式自然能够捕捉动态裂纹扩展的极其详细的特征，例如裂纹表面粗糙化、单分支和多分支。通过将直接数值模拟的结果与文献中可用的典型基准问题的实验数据进行比较，验证了所提出的可变阶公式的准确性和稳健性。可变阶公式自然能够捕捉动态裂纹扩展的极其详细的特征，例如裂纹表面粗糙化、单分支和多分支。通过将直接数值模拟的结果与文献中可用的典型基准问题的实验数据进行比较，验证了所提出的可变阶公式的准确性和稳健性。