This study uses a recently developed phase-field approach to model fracture of arterial walls with an emphasis on aortic tissues. We start by deriving the regularized crack surface to overcome complexities inherent in sharp crack discontinuities, thereby relaxing the acute crack surface topology into a diffusive one. In fact, the regularized crack surface possesses the property of Gamma-Convergence, i.e. the sharp crack topology is restored with a vanishing length-scale parameter. Next, we deal with the continuous formulation of the variational principle for the multi-field problem manifested through the deformation map and the crack phase-field at finite strains which leads to the Euler-Lagrange equations of the coupled problem. In particular, the coupled balance equations derived render the evolution of the crack phase-field and the balance of linear momentum. As an important aspect of the continuum formulation we consider an invariant-based anisotropic constitutive model which is additively decomposed into an isotropic part for the ground matrix and an exponential anisotropic part for the two families of collagen fibers embedded in the ground matrix. In addition we propose a novel energy-based anisotropic failure criterion which regulates the evolution of the crack phase-field. The coupled problem is solved using a one-pass operator-splitting algorithm composed of a mechanical predictor step (solved for the frozen crack phase-field parameter) and a crack evolution step (solved for the frozen deformation map); a history field governed by the failure criterion is successively updated. Subsequently, a conventional Galerkin procedure leads to the weak forms of the governing differential equations for the physical problem. Accordingly, we provide the discrete residual vectors and a corresponding linearization yields the element matrices for the two sub-problems. Finally, we demonstrate the numerical performance of the crack phase-field model by simulating uniaxial extension and simple shear fracture tests performed on specimens obtained from a human aneurysmatic thoracic aorta. Model parameters are obtained by fitting the set of novel experimental data to the predicted model response; the finite element results agree favorably with the experimental findings.

中文翻译：

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一种模拟动脉壁骨折的相场方法：理论和有限元分析

这项研究使用最近开发的相场方法来模拟动脉壁的骨折，重点是主动脉组织。我们首先推导正则化裂纹表面，以克服尖锐裂纹不连续性固有的复杂性，从而将尖锐裂纹表面拓扑放松为扩散拓扑。事实上，正则化的裂纹表面具有伽玛收敛特性，即尖锐的裂纹拓扑结构以消失的长度尺度参数恢复。接下来，我们处理通过变形图和有限应变下的裂纹相场表现的多场问题的变分原理的连续公式，这导致耦合问题的欧拉-拉格朗日方程。特别是，推导出的耦合平衡方程反映了裂纹相场的演化和线性动量的平衡。作为连续体公式的一个重要方面，我们考虑了基于不变量的各向异性本构模型，该模型被加性分解为地面基质的各向同性部分和嵌入地面基质中的两个胶原纤维家族的指数各向异性部分。此外，我们提出了一种新的基于能量的各向异性破坏准则，用于调节裂纹相场的演变。耦合问题使用由机械预测器步骤（针对冻结裂纹相场参数求解）和裂纹演化步骤（针对冻结变形图求解）组成的一次性算子分裂算法解决；由失效准则支配的历史字段被连续更新。随后，传统的伽辽金过程导致了物理问题的控制微分方程的弱形式。因此，我们提供了离散残差向量，并且相应的线性化产生了两个子问题的元素矩阵。最后，我们通过模拟对从人体动脉瘤胸主动脉获得的标本进行的单轴延伸和简单剪切断裂试验来证明裂纹相场模型的数值性能。模型参数是通过将一组新的实验数据与预测的模型响应拟合而获得的；有限元结果与实验结果一致。我们提供离散的残差向量和相应的线性化产生两个子问题的元素矩阵。最后，我们通过模拟对从人体动脉瘤胸主动脉获得的标本进行的单轴延伸和简单剪切断裂试验来证明裂纹相场模型的数值性能。模型参数是通过将一组新的实验数据与预测的模型响应拟合而获得的；有限元结果与实验结果一致。我们提供离散的残差向量和相应的线性化产生两个子问题的元素矩阵。最后，我们通过模拟对从人体动脉瘤胸主动脉获得的标本进行的单轴延伸和简单剪切断裂试验来证明裂纹相场模型的数值性能。模型参数是通过将一组新的实验数据与预测的模型响应拟合而获得的；有限元结果与实验结果一致。模型参数是通过将一组新的实验数据与预测的模型响应拟合而获得的；有限元结果与实验结果一致。通过将一组新的实验数据与预测的模型响应拟合来获得模型参数；有限元结果与实验结果一致。