准连续体和有限元方法的不确定性量化J积分计算
Uncertainty-quantified $J$-integral computation for quasicontinuum and finite element methods
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中文总结 AI 辅助
研究在多尺度框架中J积分应用，提出在三维准连续体方法中严格实现和验证J积分，用柯西-博恩规则导出连续场计算，引入马尔可夫链蒙特卡罗框架量化不确定性，通过模拟展示预测能力，建立评估裂纹驱动力可靠框架。
中文摘要 AI 辅助
J积分是断裂力学中的一个基本概念，用于量化驱动裂纹扩展的能量释放率。虽然它在有限元（FE）代码中得到了广泛应用，并适用于原子计算，但其在连接原子和连续体公式的多尺度框架中的应用仍未得到探索。本文提出了一种在三维准连续体（QC3D）方法中对J积分进行严格的实现和验证，该方法在平面应变假设下计算，使用通过柯西-博恩规则从原子间势导出的连续场（应力、应变能密度）。该实现通过线性弹性断裂力学（LEFM）理论和虚拟裂纹扩展（VCE）方法在三种情况下进行了验证：（1）小应变线性弹性，在整个过程中应用规定的各向异性K场位移；（2）通过非线性柯西-博恩本构关系评估相同的场，不进行原子弛豫；（3）启用原子弛豫的相同关系，允许裂纹尖端区域达到平衡。结果显示出了极好的一致性。我们还引入了一个马尔可夫链蒙特卡罗框架，以统计量化给定网格和积分域的J积分结果的不确定性，该框架也适用于传统的有限元方法。通过对硅的三点弯曲试验的QC3D模拟展示了预测能力，其中计算出的临界能量释放率与格里菲斯准则密切吻合。这项工作建立了一个可靠的框架，用于在具有量化不确定性的多尺度断裂模拟中评估裂纹驱动力，能够在解析裂纹尖端原子机制的同时进行大规模断裂模拟。
英文摘要
The $J$-integral is a fundamental concept in fracture mechanics, quantifying the energy release rate that drives crack propagation. While extensively implemented in finite element (FE) codes and adapted for atomistic calculations, its application within multiscale frameworks bridging atomistic and continuum formulations remains unexplored. This work presents a rigorous implementation and validation of the $J$-integral within the three-dimensional quasicontinuum (QC3D) method, computed under plane strain assumptions, using continuum fields (stress, strain energy density) derived from the interatomic potential via the Cauchy-Born rule. The implementation is validated against linear elastic fracture mechanics (LEFM) theory and the virtual crack extension (VCE) method across three regimes: (1) small-strain linear elasticity, with a prescribed anisotropic $K$-field displacement applied throughout; (2) the same field evaluated through the nonlinear Cauchy-Born constitutive relation, without atomic relaxation; and (3) the same relation with atomic relaxation enabled, allowing the crack-tip region to equilibrate. Excellent agreement is shown throughout. We further introduce a Markov chain Monte Carlo framework to statistically quantify the uncertainty of $J$-integral results for a given mesh and integration domain, applicable to conventional FE methods as well. Predictive capability is demonstrated via a QC3D simulation of a three-point bending test of silicon, where the computed critical energy release rate agrees closely with the Griffith criterion. This work establishes a reliable framework for evaluating crack driving forces in multiscale fracture simulations with quantified uncertainty, enabling large-scale fracture simulations while resolving atomistic mechanisms at the crack tip.
