The phase-field method has become popular for the numerical modeling of fluid-filled fractures, thanks to its ability to represent complex fracture geometry without algorithms. However, the algorithm-free representation of fracture geometry poses a significant challenge in calculating the crack opening (aperture) of phase-field fracture, which governs the fracture permeability and hence the overall hydromechanical behavior. Although several approaches have been devised to compute the crack opening of phase-field fracture, they require a sophisticated algorithm for post-processing the phase-field values or an additional parameter sensitive to the element size and alignment. Here, we develop a novel method for calculating the crack opening of fluid-filled phase-field fracture, which enables one to obtain the crack opening without additional algorithms or parameters. We transform the displacement-jump-based kinematics of a fracture into a continuous strain-based version, insert it into a force balance equation on the fracture, and apply the phase-field approximation. Through this procedure, we obtain a simple equation for the crack opening which can be calculated with quantities at individual material points. We verify the proposed method with analytical and numerical solutions obtained based on discrete representations of fractures, demonstrating its capability to calculate the crack opening regardless of the element size or alignment.

翻译：相场方法因其无需算法即可描述复杂裂缝几何形态的能力，在流体填充裂缝的数值模拟中日益流行。然而，裂缝几何形态的算法无关表征，为计算相场裂缝的裂缝开度（孔径）带来了重大挑战，而裂缝开度控制着裂缝渗透率，进而影响着整体的流固耦合行为。尽管已有多种方法被设计用于计算相场裂缝的开度，但它们通常需要对相场值进行复杂的后处理算法，或者需要一个对单元尺寸和排列方向敏感的额外参数。本文中，我们开发了一种计算流体填充相场裂缝开度的新方法，该方法无需额外的算法或参数即可获得裂缝开度。我们将基于位移间断的裂缝运动学关系转化为连续的基于应变的形式，将其代入裂缝上的力平衡方程，并应用相场近似。通过这一过程，我们得到了一个简单的裂缝开度方程，该方程可利用单个材料点处的量进行计算。我们通过与基于离散裂缝表征获得的解析解和数值解进行对比，验证了所提方法的有效性，证明了其能够不受单元尺寸或排列方向影响地计算裂缝开度。