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In this paper, a nonlocal energy‐informed neural network is proposed to characterize the deformation behaviors of elastic solids with cracks subjected to thermomechanical loads in the framework of ordinary state‐based peridynamics. Based on principle of virtual work, a nonlocal energy approach is developed to recast the solution to peridynamic equilibrium equation as a problem of minimizing the potential energy of the system, which automatically satisfies the traction‐free boundary conditions. Meanwhile, the energy representation of physical system can be treated as the loss function for machine learning methods. Therefore, a nonlocal energy‐informed neural network is constructed to approximate the solution of the system. A distinct advantage of the proposed neural network is that the strain energy is expressed in terms of spatial integration instead of spatial derivatives, which avoids the invalidation of automatic differentiation at the crack surfaces in original physics‐informed neural networks. To demonstrate the convergence and accuracy of the proposed neural network, a series of problems in solid and fracture mechanics are conducted, and compared with results from analytical solutions or classical numerical methods. Additionally, for elastic material containing initial cracks, displacement extrapolation method is encoded into the proposed neural network to evaluate the static stress intensity factor. [ABSTRACT FROM AUTHOR]

• Non-uniform discretization peridynamic method is proposed to investigate cracking behavior. • A novel volume correction method is proposed to eliminate unbalance force. • The force loading condition is modified to replace the displacement loading. • The relationship between the material points discretization form and cracking behavior is defined. The proposed non-uniform discretization peridynamics reduces computational costs. Cracking behavior investigation is very important in solid mechanics. In this study, a non-uniform discretization bond-based peridynamic (NUD-BBPD) method is proposed to investigate the cracking behavior, in which the new volume correction method for regenerative family material points is proposed based on Monte Carlo statistical approach. When computational domain non-uniform discretization (NUD) is employed, the proposed method eliminates the need to use the variational horizon, and naturally solves the issue of unbalanced force in NUD. Additionally, the force loading condition is revised based on the energy conservation in the proposed method. This revision eliminates the need for fictitious material points when the displacement loading condition is used in simulation, and the drawback of the conventional force loading condition, in which the force is constrained, is also solved. The proposed method is then used to analyze the static and dynamic fractures of solids. The effects of the material points arrangement on the cracking behavior are obtained via a comprehensive comparison, and the numerical results obtained by the proposed method are in good agreement with those obtained by the conventional uniform discretizational peridynamics. Moreover, NUD peridynamics with constant horizon has higher computational efficiency than traditional peridynamics. [ABSTRACT FROM AUTHOR]

In this work, the geometry-constraint-based non-ordinary state-based peridynamic (GC-NOSBPD) model is further generalized to study the mechanical responses and rate-dependent fracture behaviors of viscoelastic materials. The viscoelasticity is generated by directly incorporating the classical viscoelastic constitutive relation in the form of the Generalized Maxwell Model, which is expressed as Prony-series type expansion, into the GC-NOSBPD framework. A strain-based scalar damage variable only driven by elastic energy, whose evolution follows a rate-independent law, is proposed to describe the progressive growth of the mode-I crack. Combining the rate-independent damage variable and viscoelasticity results in a rate-dependent damage model in the hypothesis that the fracture of viscoelastic material is driven by both elastic and viscous components of the energy. The fidelity of this model is established in the absence of damage by comparison with the benchmark solution for the viscoelastic response of a 3D bar subjected to various load histories. Further, the proposed model is employed to simulate two quasi-static fracture tests under various loading rates: a three-point bend fracture test for concrete and a double-cantilever-beam delamination test for rubber interface. These numerical results agree well with experimental tests, showing that the proposed model can effectively capture rate-dependent crack propagation. [ABSTRACT FROM AUTHOR]

In this paper, a 3D non-local lattice bond model is proposed to model fracturing behaviors of materials. First, the formulations and detailed derivation for three-dimensional non-local lattice bond models are obtained by comparing the strain energy stored in a discrete lattice with the classical continuum strain energy. Then, the capabilities of three-dimensional non-local lattice bond models are verified using benchmarks. To further assess the performance of the non-local lattice bond model, fracturing behaviors in brittle solids are predicted. Compared with the previous numerical results, the proposed model demonstrates better performances, which are more consistent with the experimental observations. [ABSTRACT FROM AUTHOR]

• A nonlocal energy-informed neural network is proposed to address the crack propagation. • Minimizing the peridynamic potential energy is considered as loss function. • Transfer learning is introduced to enhance efficiency and effectiveness. Discontinuities, such as joints, faults and pores, are prevalent in rock formations, significantly impacting the fracture behavior of rocks. This paper proposes an energy-driven machine learning framework to address crack initiation and propagation in rock-like materials by considering long-range interactions. By leveraging the minimization energy principle, the displacement of physical system is determined by minimizing the peridynamic potential energy instead of directly solving the equilibrium equation, which serves as the neural network's loss function. An incremental form loss function and transfer learning technique are developed to account for the cumulative effects of all preceding loading steps. Furthermore, the critical strain energy density criterion is established from an energy perspective. A key advantage of the proposed neural network lies in evaluating peridynamic strain energy through spatial integration rather than spatial derivatives, thus bypassing the limitations of automatic differentiation encountered in original physics-informed neural networks at crack surfaces. Notably, the proposed approach inherits the ability of peridynamics to simulate spontaneous damage evolution and crack propagation without necessitating special crack treatment techniques. Finally, the accuracy and effectiveness of the proposed approach is verified via comparison with analytical solutions, experimental observations, or classical numerical methods in fracture analysis of rock-like materials subjected to various quasi-static loading conditions. [ABSTRACT FROM AUTHOR]

The purpose of this review is to discuss the development and the contribution of the experiments and numerical simulations in compression‐induced failure characteristics of flawed rock specimens. Investigation is essential to understand the fundamental failures occurring in a rock bridge, for assessing anticipated and actual performances of the structures built on or in rock masses. The review begins by representing and discussing compression‐induced crack initiation and growth in flawed rocks and explaining the significance of studying these issues in Section 1. The crack initiation and growth behaviors in flawed rocks under uniaxial, biaxial, and triaxial compressions are described in depth in Section 2 where it is expected to distinguish 2‐D crack growth from 3‐D crack growth. After that, the failure characteristics of flawed rock specimens due to temperature treatments and hydraulic pressure are systematically reviewed. Numerical studies of the compression‐induced failure characteristics of flawed rock specimens based on different numerical theories and numerical models are comprehensively evaluated in Section 3. In Section 4, the new findings obtained recently from experimental and numerical studies are drawn. Finally, some aspects of prospective research and a brief summary are presented in Sections 5 and 6. [ABSTRACT FROM AUTHOR]

In this paper, a compressive-shear fracture model of the phase-field method is proposed to simulate the compressive-shear failure behaviours of pre-cracked rock materials and investigate their fracture mechanisms. In the proposed model, a modified Hoek–Brown criterion is incorporated into the driving term in the evolution equation of the phase field to control the crack phase field. Rock-like specimens containing single flaw and double flaws were used to validate the performance of the proposed numerical model in capturing compressive-shear fractures. The numerical results were in good agreement with the experimental observations. Subsequently the effects of flaw geometries, i.e. flaw inclination angle, spacing and continuity, on the cracking behaviours and mechanical properties of rock-like specimens containing double parallel flaws were investigated. Typical shear cracks, such as coplanar or quasi-coplanar secondary cracks and oblique secondary cracks as well as coalescence types of different shear cracks, were captured. The mechanical properties were also strongly related to the flaw geometry. These performances indicated that the proposed numerical model has ability to simulate compressive-shear cracks. [ABSTRACT FROM AUTHOR]

Uniaxial compression tests with combined acousto-optical monitoring techniques are conducted on flawed granite specimens, with the aim of investigating the fracture-related acoustic emission (AE) event rate characteristics at the unstable cracking phase in flawed rocks. The interevent time (IET) function F (τ) is adopted to interpret the AE time-series from damage stress (σ cd) to ultimate failure, and photographic data are used to evaluate unstable cracking behaviours in flawed granite. The results show that a high AE event rate is always registered but intermittently interrupted by macrofracturing at the unstable cracking phase. The reversed U-shaped curve relation between the AE event rate and the loading time is documented in unstable flawed granite for the first time. The acoustic quiescence has a mechanismic linkage and quantitative correlation with stress drop, and this synchronous acousto-mechanical behaviour is a typical result of the initiation, growth and coalescence of macrocracks initiated from the flaw tips. Moreover, the reactivation and intensification of fracture process zones (FPZs) by increasing loads are the dominant mechanism triggering unstable crack growth in flawed granite. [ABSTRACT FROM AUTHOR]

In this paper, a novel geometric method combined with the piecewise linear function method is introduced into the extended finite element method (XFEM) to determine the crack tip element and crack surface element. Then, by combining with the advanced mesh technique, a novel method is proposed to improve the modelling of crack propagation in triangular 2D structure with the XFEM. The numerical tests show that the accuracy, the convergence, and the stability of the XFEM can be improved using the proposed method. Moreover, the applicability of the conventional multiple enrichment scheme is discussed. Compared with the proposed method, the conventional multiple enrichment scheme has deficiency in mixed mode I and II crack. Finally, a comparative study shows that the performance of the XFEM by using the proposed method to model the crack propagation can be greatly improved. [ABSTRACT FROM AUTHOR]

The crack initiation, growth, wrapping and coalescence of two 3D pre-existing cross-embedded flaws in PMMA specimens under uniaxial compression are investigated. The stress-strain curves of PMMA specimens with 3D cross-embedded flaws are obtained. The tested PMMA specimens exhibit dominant elastic deformation and eventual brittle failure. The experimental results show that four modes of crack initiation and five modes of crack coalescence are observed. The initiations of oblique secondary crack and anti-wing crack in 3D cracking behaviors are first reported as well as the coalescence of anti-wing cracks. Moreover, two types of crack wrapping are found. Substantial wrapping of petal cracks, which includes open and closed modes of wrapping, appears to be the major difference between 2D and 3D cracking behaviors of pre-existing flaws, which are also first reported. Petal crack wraps symmetrically from either the propagated wing cracks or the coalesced wing cracks. Besides, only limited growth of petal cracks is observed, and ultimate failure of specimens is induced by the further growth of the propagated wing crack. The fracture mechanism of the tested PMMA specimens is finally revealed. In addition, the initiation stress and the peak stress versus the geometry of two 3D pre-existing cross-embedded flaws are also investigated in detail. [ABSTRACT FROM AUTHOR]

The non-negligible fissures, cracks or pore structures, which are contained in rocks, have significant influences on their mechanical properties. In this paper, a novel method is developed to reconstruct a three-dimensional multiple phase rock model from the limited morphological information of Chongqing sandstone. Firstly, X-ray micro-CT experimental tests are applied to Chongqing sandstone to obtain the raw CT images and the mechanical parameters. Then, the two-point correlation function, lineal-path function and fractal function are combined to generate the reconstructed model. Next, the reconstructed model subjected to compressive loads is simulated by Finite Element-Smoothed Particle Hydrodynamics (FE-SPH) coupling method with a brittle model formulation. Finally, the initiation, propagation and coalescence of cracks are investigated during the numerical simulation. The topological and mechanical properties of the reconstructed model are carefully analyzed and compared with that of the reference model. The comparison of the numerical modeling results and the experimental data shows a good consistency. This indicates that the proposed method has the ability to predict macroscopic behaviors of rocks from limited experimental information. [ABSTRACT FROM AUTHOR]

• The developments and contributions of experimental studies on the FPZ in quasi-brittle materials are discussed. • Correlations of the cracking levels in quasi-brittle materials with the FPZ perspective is established. • The existence of apparent FPZs is the material basis for unstable crack growth in flawed granite. In the failure of quasi-brittle materials such as rocks, microstuctural damage always precedes macroscopic fracture and manifests itself in the formation of fracture process zones (FPZs). Investigations of the FPZ are of importance to a better understanding of quasi-brittle fracture behavior along with the size effect and failure forecast. The sensitivities of the FPZ to material microstructures, specimen geometries, loading conditions, ambient factors and monitoring techniques have been investigated extensively, as well as the effects of the FPZ on crack damage and strength/fracture toughness. The first purpose in this study is to discuss the developments and contributions of experimental studies on the FPZ in quasi-brittle materials. Besides, there is limited advance knowledge of the correlations of cracking levels with the FPZ evolution patterns due to the fact that researchers concern cracking levels seldom from the FPZ perspective. This study is also to establish a better understanding of stable and unstable crack growth in quasi-brittle materials from the FPZ perspective. Acousto-optic-mechanical data from uniaxial compression tests faithfully show that the existence of apparent FPZs is a material basis for unstable crack growth in flawed granite. [ABSTRACT FROM AUTHOR]

Propagation process of pre-existing flaws in brittle rock materials are simulated using peridynamic theory, in which it is assumed that particles in a continuum interact with each other across a finite distance. Since damage is incorporated at the level of these two particle interactions in peridynamic theory, localization and fracture of materials occur as a natural outgrowth of equations of motion and constitutive models. Numerical simulations of notched semi-circular bend (NSCB) tests and mix mode fracture in a tension–shear rock sample are performed. The effects of array of multiple pre-existing flaws on the propagation process in rock-like materials subjected to uniaxial tensile loads are investigated. The propagation process of macroflaws and microflaws in rock-like materials subjected to uniaxial tensile loads are simulated.The failure modes of rock sample containing the different array of pre-existing flaws are studied. It is found that the present numerical results obtained from peridynamic theory are in good agreement with the previous experimental and numerical results. [ABSTRACT FROM AUTHOR]

• A field-enriched finite element method is proposed to simulate cracking behaviors of brittle solids containing cracks of complex geometries. • Three different coalescence types are considered by employing reasonable strategies. • A predictor-and-corrector strategy is developed to solve the problem of competing crack growth of multiple complex cracks. • Benchmark examples containing cracks of complex geometries are performed. A field-enriched finite element method (FE-FEM) is proposed to simulate cracking behaviors of brittle solids containing cracks of complex geometries. Three different coalescence types, i.e. the coalescence of crack tip and crack tip, crack tip and crack segment, crack tip and free boundary, are considered by employing reasonable strategies. Moreover, a predictor-and-corrector algorithm is developed to solve problem of competing crack growth. Three benchmark examples containing cracks of complex geometries are performed to validate the accuracy and correctness of the proposed numerical method. Finally, the evolution of complex cracks, including a tree-shaped crack, regularly distributed intersecting cracks, and randomly distributed intersecting cracks, are simulated and investigated. All numerical results show that the proposed numerical method can effectively handle complex cracks without any special treatment around junction points. [ABSTRACT FROM AUTHOR]

The near crack line analysis method is used to investigate a center crack loaded by two pairs of point tensile forces in an infinite plate in an elastic–perfectly plastic solid, and the analytical solutions are obtained in this paper. These solutions include: the elastic–plastic stress field near the crack line, the law that the length of the plastic zone along the crack line is varied with an external loads and the bearing capacity of an infinite plate with a center crack. The results of this paper are sufficiently precise near the crack line because the assumptions of the small scale yielding theory have not been used and no other assumptions have been taken. [Copyright &y& Elsevier]

• Field-enriched finite element method to model composite lamina fracture is proposed. • Crack growth in constant stiffness and variable stiffness composite plate is studied. • Field-enriched finite element method is easier to implement. A field-enriched finite element method for crack propagation in fiber-reinforced composite lamina without remeshing is proposed. In the field-enriched finite element method, the propagation and evolution of cracks are controlled by fracture criteria, and the physical location of cracks and the influence on the whole model are realized by the field variables enriched at the nodes. Then, field-enriched finite element method is used to investigate the influence of fiber orientation on the constant stiffness and variable stiffness composite lamina under different loading conditions and configurations. The numerical results obtained by the field-enriched finite element method are in good agreement with those obtained by experimental observation and other numerical methods. Moreover, the crack propagation path is roughly along the fiber orientation direction, which means that for the fiber reinforced composite lamina, the crack propagation path is mainly dominated by the fiber orientation rather than the configuration and loading conditions of the specimen. Compared with the phase field method, the calculation efficiency of the proposed method is higher. Compared with the extended finite element method, the proposed method is easier to characterize cracks. Therefore, the field-enriched finite element method is a very promising method for simulating the fracture behavior of fiber-reinforced composite lamina. [ABSTRACT FROM AUTHOR]

This paper experimentally evaluates and quantitatively compares stress-induced damage progression and acoustic emission (AE) in granite and sandstone based on continuous acousto-optic-mechanical (AOM) observations. Experimental results reveal that the step-rise characteristics of the AE event rate preceding the entire macrofracturing process and the apparent white patching phenomena are the most significant acousto-optical evidences for process zone nucleation in granite but not in sandstone. During the unstable crack growth stage in granite, the robustly high AE event rate level has been mechanically correlated with the reactivation and intensification of fracture process zones (FPZs) by the stress build-up, and the relatively low AE event rate level has been revealed as a result of the progression of FPZs into macrocracks. It is the first time that the process zone nucleation in granite has been mechanistically related to the clustering of three types of grain-scale microcracks, i.e., grain boundary cracks, intragranular cracks and transgranular cracks, due to the grain interlocking effect. Comparatively, the clustering of microcracks is insignificant in the brittle failure of sandstone. Mechanistic correlations among the AOM characteristics in the two rock types are also investigated in detail. [ABSTRACT FROM AUTHOR]

• CSPH method is developed to tackle the cracked nonconvex bodies. • The diffraction method is embedded to modify the SPH approximation. • A small cost is required to embed the diffraction method. • Application of the dynamic fracture problems is provided. The continuous smoothed particle hydrodynamics (CSPH) method is proposed to modify the kernel approximation in the smoothed particle hydrodynamics (SPH). The diffraction criterion is embedded in the framework of standard SPH. CSPH dramatically improves the computational ability in cracked bodies. Three numerical cases of cracked bodies, including dynamic stress intensity factor (DSIF) of a central crack, tensile fracture of brittle materials and Kalthoff's experiment, are employed to verify the ability of the proposed method. It is found that the numerical results obtained by CSPH are in good agreement with the previous experimental and numerical results, implying the robustness and feasibility of the proposed method. [ABSTRACT FROM AUTHOR]

• Quasi-brittle failure of flawed granite is evaluated in real time by integrated acousto-optic-mechanics (AOM) multi-physics field characterization methods. • Procedural sample-sized localization instabilities are forecasted and evaluated using the time‐reversed Omori law. • The physics relating to the failure predictability of flawed granite is revealed. Forecasting quasibrittle failure in geomaterials is of paramount importance in the physics of fractures but rarely succeeds. To clarify the physics relating to the failure predictability in a quasibrittle regime, a combined AE (acoustic emission)-DIC (digital image correlation) technique is used in uniaxial compression experiments of flawed granite to trace its damage conditions in real time. The mechanistic correlations of the AE rate-process with the internal FPZ (fracture process zone) development are available. It is shown that the complete record of the AE activities does not meet a deterministic scaling law (e.g., typical time‐reversed Omori law), indicating a challenge in predicting ultimate failure via AE. Subsets of a power law acceleration registered much earlier than ultimate failure imply that the procedural sample-sized localization instability is predictable but still depends on the FPZ dynamic evolutions. More importantly, we find that the coexistence of FPZs and the intermittency of crack growth are the two main reasons for the difficulty of forecasting failure in flawed granite. This study sheds light on upcoming investigations in which the roles of FPZs, especially their spatiotemporal patterns, should be highlighted. [ABSTRACT FROM AUTHOR]

Rockbursts in deep tunnels subjected to high crustal stress were investigated using a general particle dynamics (GPD) code with the Holmquist–Johnson–Cook damage model (HJC). Under dynamic excavation disturbance, several numerical tests were carried out to investigate the dynamic failure patterns of surrounding rock masses with different preexisting cracks around deep tunnels. The numerical results showed that the GPD code can satisfactorily simulate the phenomenon of rockburst caused by crack evolution, including predicting the ejection velocity, rockburst location and ejected rock block volume. Moreover, the analysis of the circumferential stress in the surrounding rock of deep tunnels at the Jinping II Hydropower Station shows the efficiency of the GPD code. In addition, four typical rockburst, which occurred under different geological conditions at the Jinping II Hydropower Station, were analyzed by using the GPD code. The numerical simulations suggested that the GPD code can efficiently reproduce the 2D and 3D crack growth in the surrounding rock, which eventually leads to the occurrence of rockburst. • Rockbursts in deep tunnels subjected to high crustal stress were investigated using a general particle dynamics code. • The numerical results showed that the GPD code can satisfactorily simulate the phenomenon of rockburst. • The GPD code can efficiently reproduce the 2D and 3D crack growth in the surrounding rock. [ABSTRACT FROM AUTHOR]