The prediction of crack initiation and propagation in ductile failure processes are challenging tasks for the design and fabrication of metallic materials and structures on a large scale. Numerical aspects of ductile failure dictate a sub-optimal calibration of plasticity- and fracture-related parameters for a large number of material properties. These parameters enter the system of partial differential equations as a forward model. Thus, an accurate estimation of the material parameters enables the precise determination of the material response in different stages, particularly for the post-yielding regime, where crack initiation and propagation take place. In this work, we develop a Bayesian inversion framework for ductile fracture to provide accurate knowledge regarding the effective mechanical parameters. To this end, synthetic and experimental observations are used to estimate the posterior density of the unknowns. To model the ductile failure behavior of solid materials, we rely on the phase-field approach to fracture, for which we present a unified formulation that allows recovering different models on a variational basis. In the variational framework, incremental minimization principles for a class of gradient-type dissipative materials are used to derive the governing equations. The overall formulation is revisited and extended to the case of anisotropic ductile fracture. Three different models are subsequently recovered by certain choices of parameters and constitutive functions, which are later assessed through Bayesian inversion techniques. A step-wise Bayesian inversion method is proposed to determine the posterior density of the material unknowns for a ductile phase-field fracture process. To estimate the posterior density function of ductile material parameters, three common Markov chain Monte Carlo (MCMC) techniques are employed: (i) the Metropolis–Hastings algorithm, (ii) delayed-rejection adaptive Metropolis, and (iii) ensemble Kalman filter combined with MCMC. To examine the computational efficiency of the MCMC methods, we employ the \(\hat{R}{-}convergence\) tool. The resulting framework is algorithmically described in detail and substantiated with numerical examples.

延性破坏过程中裂纹萌生和扩展的预测对于大规模金属材料和结构的设计和制造来说是一项具有挑战性的任务。延性破坏的数值方面要求对大量材料属性的塑性和断裂相关参数进行次优校准。这些参数作为正向模型进入偏微分方程系统。因此，对材料参数的准确估计能够精确确定不同阶段的材料响应，特别是对于发生裂纹萌生和扩展的后屈服状态。在这项工作中，我们开发了一个用于韧性断裂的贝叶斯反演框架，以提供关于有效断裂的准确知识。机械参数。为此，使用合成和实验观察来估计未知数的后验密度。为了模拟固体材料的延性破坏行为，我们依赖于断裂的相场方法，为此我们提出了一个统一的公式，允许在变分基础上恢复不同的模型。在变分框架中，使用一类梯度型耗散材料的增量最小化原则来推导控制方程。重新审视整个公式并将其扩展到各向异性韧性断裂的情况。随后通过参数和本构函数的某些选择来恢复三个不同的模型，然后通过贝叶斯反演技术对其进行评估。一个逐步提出了贝叶斯反演方法来确定韧性相场断裂过程的材料未知数的后验密度。为了估计韧性材料参数的后验密度函数，采用了三种常见的马尔可夫链蒙特卡罗 (MCMC) 技术：(i) Metropolis-Hastings 算法，(ii) 延迟拒绝自适应 Metropolis，以及 (iii) 组合卡尔曼滤波器与 MCMC。为了检查 MCMC 方法的计算效率，我们使用\(\hat{R}{-}convergence\)工具。由此产生的框架在算法上进行了详细的描述，并用数值例子加以证实。