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Abstract As a companion work to [1], this article presents a series of simple formulae and explicit results that illustrate and highlight why classical variational phase-field models cannot possibly predict fracture nucleation in elastic brittle materials. The focus is on “tension-dominated” problems where all principal stresses are nonnegative, that is, problems taking place entirely within the first octant in the space of principal stresses. 

We develop a model for solid objects surrounded by a fluid that accounts for the possibility of acoustic pressures giving rise to damage on the surface of the solid. The propagation of an acoustic pressure in the fluid domain is modeled by the acoustic wave equation. On the other hand, the response of the solid is described by linear elastodynamics coupled with a gradient damage model, one that is based on a cohesive-type phase-field description of fracture. The interaction between the acoustic pressure and the deformation and damage of the solid are represented by transmission conditions at the fluid-solid interface. The resulting governing equations are discretized using a finite-element/finite-difference method that pays particular attention to the spatial and temporal scales that need to be resolved. Results from model-based simulations are provided for a benchmark problem as well as for recent experiments in nanopulse lithotripsy. A parametric study is performed to illustrate how damage develops in response to the driving force (magnitude and location of the acoustic source) as a function of the fracture resistance of the solid. The results are shown to be qualitatively consistent with experimental observations for the location and size of the damage fields on the solid surface. A study of limiting cases also suggests that both the threshold for damage and the critical fracture energy are important to consider in order to capture the transition from damage initiation to complete localization. A low-cycle fatigue model is proposed that degrades the fracture resistance of the solid as a function of accumulated tensile strain energy, and it is shown to be capable of capturing damage localization in simulations of multi-pulse nano-pulse lithotripsy. 

This work focuses on the representation of model-form uncertainties in phase-field models of brittle fracture. Such uncertainties can arise from the choice of the degradation function for instance, and their consideration has been unaddressed to date. The stochastic modeling framework leverages recent developments related to the analysis of nonlinear dynamical systems and relies on the construction of a stochastic reduced-order model. In the latter, a POD-based reduced-order basis is randomized using Riemannian projection and retraction operators, as well as an information-theoretic formulation enabling proper concentration in the convex hull defined by a set of model proposals. The model thus obtained is mathematically admissible in the almost sure sense and involves a low-dimensional hyperparameter, the calibration of which is facilitated through the formulation of a quadratic programming problem. The relevance of the modeling approach is further assessed on one- and two-dimensional applications. It is shown that model uncertainties can be efficiently captured and propagated to macroscopic quantities of interest. An extension based on localized randomization is also proposed to handle the case where the forward simulation is highly sensitive to sample localization. This work constitutes a methodological development allowing phase-field predictions to be endowed with statistical measures of confidence, accounting for the variability induced by modeling choices. 

Abstract Many geo‐engineering applications, for example, enhanced geothermal systems, rely on hydraulic fracturing to enhance the permeability of natural formations and allow for sufficient fluid circulation. Over the past few decades, the phase‐field method has grown in popularity as a valid approach to modeling hydraulic fracturing because of the ease of handling complex fracture propagation geometries. However, existing phase‐field methods cannot appropriately capture nucleation of hydraulic fractures because their formulations are solely energy‐based and do not explicitly take into account the strength of the material. Thus, in this work, we propose a novel phase‐field formulation for hydraulic fracturing with the main goal of modeling fracture nucleation in porous media, for example, rocks. Built on the variational formulation of previous phase‐field methods, the proposed model incorporates the material strength envelope for hydraulic fracture nucleation through two important steps: (i) an external driving force term, included in the damage evolution equation, that accounts for the material strength; (ii) a properly designed damage function that defines the fluid pressure contribution on the crack driving force. The comparison of numerical results for two‐dimensional test cases with existing analytical solutions demonstrates that the proposed phase‐field model can accurately model both nucleation and propagation of hydraulic fractures. Additionally, we present the simulation of hydraulic fracturing in a three‐dimensional domain with various stress conditions to demonstrate the applicability of the method to realistic scenarios.