State-of-the-Art Review and Prospect of Modelling the Dynamic Fracture of Rocks Under Impact Loads and Application in Blasting

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The manuscript has certain value in summarizing dynamic rock fracture modeling, but further improvement is needed in terms of transparency in literature screening, comparative depth of numerical methods, mechanism analysis, case representativeness, model validation, and engineering guidance. It is suggested that the author supplement the data, improve the analysis, and enhance the systematic discussion based on the above opinions to improve the scientific and practical value of the article. Please respond to each comment item by item in the revised draft and provide detailed modification explanations.
1. The author summarizes the advantages and disadvantages of FEM, DEM, and FDEM in Table 2 on page 24 of the manuscript, but this comparative analysis is relatively brief, only listing general descriptions such as "high computational cost" or "ability to simulate complex interactions", lacking quantitative or specific case support. The reviewer believes that the authors need to appropriately expand the content of Table 2 by adding specific cases or data to compare the performance of FEM, DEM, and FDEM in terms of computational efficiency, accuracy, and applicable scenarios. In addition, the reviewer suggests adding a comparative analysis to discuss the advantages and disadvantages of these methods in different geological conditions or engineering applications.
2. The author discussed the energy consumption and crack branching phenomena of dynamic fracture on pages 4-5 of the manuscript (approximately), but the analysis of the microscopic mechanisms of rock dynamic fracture, such as crack propagation speed and stress wave propagation, is not very sufficient. In addition, the article mentions that "energy consumption includes surface energy and fragment motion", but does not explore in detail the proportion or mechanism of energy distribution under different loading rates. Therefore, the reviewers suggest that the authors appropriately supplement the discussion on the micro mechanism of dynamic fracture in the corresponding content of the manuscript. Meanwhile, some schematic diagrams can be added appropriately.
3. The finite element method is an important method for crack propagation behavior in hydraulic fracturing processes. Therefore, the statements from lines 205 to 208 of the manuscript need to be supported by the following papers: The Crack Propagation Behaviour of CO2 Fracturing Fluid in Unconventional Low Permeability Reservoirs: Factor Analysis and Mechanism Revelation; Numerical simulation of fracture reorientation during hydraulic fracturing in perforated horizontal well in shale reservoirs.
4. Although the author reviewed the progress of dynamic fracture modeling in the manuscript, they did not provide sufficient engineering application guidance. For example, there is a lack of specific recommendations on how to select appropriate numerical methods based on rock type, loading conditions, or engineering objectives. The reviewer suggests adding a section proposing an engineering application guidance framework based on dynamic rock fracture modeling. This section can include the decision-making process for selecting numerical methods, recommendations for setting key parameters, etc.
5. In the manuscript, the author mentioned that the dynamic fracture process involves complex processes such as crack branching and wave propagation, but there is insufficient discussion on the influence of rock heterogeneity (such as fracture networks, particle structures) on fracture modes, especially in multi-scale modeling. The reviewer suggests that the author appropriately strengthen the review of the latest developments in multi-scale methods (such as micro macro DEM FEM coupling) in the manuscript. 
6. The footnote in the title of the manuscript is' state of the art review and prospect ', but the boundary between' state of the art 'and' prospect 'is extremely vague in the main body of the manuscript, and there is no systematic summary of research trends and future challenges. Therefore, the reviewer believes that the author needs to include a dedicated section on "Research Gap and Future Directions" in the manuscript, rather than just mentioning it sporadically in paragraphs.

Author Response

General comments: The manuscript has certain value in summarizing dynamic rock fracture modeling, but further improvement is needed in terms of transparency in literature screening, comparative depth of numerical methods, mechanism analysis, case representativeness, model validation, and engineering guidance. It is suggested that the author supplement the data, improve the analysis, and enhance the systematic discussion based on the above opinions to improve the scientific and practical value of the article. Please respond to each comment item by item in the revised draft and provide detailed modification explanations.

Authors’ response: Thank you very much for your constructive comments. Based on your suggestions, we have provided response to each comment item by item in the revised draft and provide detailed modification explanations. We have substantially refined the analysis and augmented the systematic discussion throughout the manuscript, including the reviewer’s invaluable suggestions to increase both the scientific rigor and practical significance of the study. Please refer to our detailed responses to each comment raised by you in the following paragraphs. In brief, the following major revisions are made:

• In Table 2, representative codes and literature of each method of FEM, DEM and FDEM are supplemented besides strengthening the advantages and disadvantages of these methods.
• A comparative analysis is added behind Table 2 to discuss the advantages and disadvantages of these methods in different geological conditions or engineering applications. Please refer to the highlighted texts in Lines Nos. 979-1002 on P. 26 of the revised manuscript for detailed information.
• Research gaps of modelling the dynamic fracture of rocks under impact loads with these computational mechanics methods are pointed out through comparing the advantages and disadvantages of FEM, DEM and FDEM presented in the extended Table 2.
• An additional section entitled “4.3. Future directions for modelling rock dynamic fracture and a systematic numerical modelling approach for rock blasting” is added to discuss the prospective developments for modelling the dynamic fracture of rock under impact loads in future.
• A new section together with a new Figure 21 is added to propose a systematic approach for modelling the dynamic fracture of rocks by blast, which include the decision-making process for selecting numerical methods and recommendations for setting key parameters following your suggestions.

 

Specific comment No. 1: The author summarizes the advantages and disadvantages of FEM, DEM, and FDEM in Table 2 on page 24 of the manuscript, but this comparative analysis is relatively brief, only listing general descriptions such as "high computational cost" or "ability to simulate complex interactions", lacking quantitative or specific case support. The reviewer believes that the authors need to appropriately expand the content of Table 2 by adding specific cases or data to compare the performance of FEM, DEM, and FDEM in terms of computational efficiency, accuracy, and applicable scenarios. In addition, the reviewer suggests adding a comparative analysis to discuss the advantages and disadvantages of these methods in different geological conditions or engineering applications.

Authors’ response: Thank you very much for your insightful suggestion. Following your suggestion, we have extended the content of Table 2 significantly, which are explained in the following:

• We have included a comparative analysis to discuss the advantages and disadvantages of these methods in different geological conditions or engineering applications as follows: “In engineering applications involving large, relatively uniform rock masses under moderate loading conditions, FEM offers an efficient approach due to its capability in simulating continuous material. FEM is prevalent because of its maturity, the advantages in handling rock inhomogeneity and nonlinearity, and the availability of many well-verified commercial codes for large- or small- scale problems. The extended, meshless or particle FEM approaches have become important subjects for further research and development of modelling rock dynamic fracture, mainly because of their flexibility for meshing and capability of fracture evolution simulations without remeshing. In cases when rocks exhibit significant discontinuities, such as jointed, multifaceted, or extensively fractured formations, DEM is a powerful numerical modelling tool simply because of its flexibility in handling a relatively large number of fractures. However, DEM requires exceptional computational effort, in terms of both computer memory and running time, for even a moderately large number of particles/blocks. The other main disadvantage of DEM is the uncertainty about the fracture system geometry and the effect of this uncertainty is difficult to quantify. In addition, DEM usually required 3D simulations to be conducted with 2D simulations only for generic studies. FDEM inherits all advantages of FEM and DEM and overcomes most of their disadvantages but is exceptionally computationally intensive. Compared with FEM, FDEM is more robust and efficient after damage and failure occurs, especially in terms of modelling fragment movement, interaction and muck-piling such as secondary and tertiary fractures resulting from the fragment interaction. Compared with DEM, FDEM is more versatile in modelling rock deformability before fracture, crack initiation and propagation during fracturing, and irregular-shaped fragment movements after fracture and fragmentation”.
• Representative codes of each method for modelling the dynamic fracture of rocks under impact loads are listed in Table 2. The representative codes of FEM include ABAQUS, ANSYS, LS-DYNA, AUTODYN, RFPA, RS and Plaxis while those of DEM are PFC, UDEC, 3DEC, DDA, Yade, NMM and LSM. The typical codes of FDEM have emerged as Y2D, ELFEN, Y-Geo, Solidity, HOSS, HFDEM, Irazu and MultiFS. The typical applications of these representative codes are reviewed in Sections 3.1, 3.3, 3.5.2 in terms of modelling the rock dynamic fracture and Section 3.2, 3.4 and 3.5.3 in terms of modelling the rock blasting with FEM, DEM and FDEM, respectively.
• Research gaps of modelling the dynamic fracture of rocks under impact loads with computational mechanics methods are pointed out through comparing the advantages and disadvantages of FEM, DEM and FDEM presented in the extended Table 2, which are
  • Research gaps in modelling the dynamic properties of rocks and discontinuities: As reviewed in Section 2, the mechanical properties of rocks and discontinuities under impact loads are related to loading rates. Unfortunately, most modellings reviewed in Section 3 adopt the static mechanical properties of rocks to study the dynamic fracture of rock by blast. Furthermore, although a great deal of research has been done on the mechanical properties of rocks under dynamic loading in the laboratory, the study of dynamic mechanical properties of discontinuities is rare and thus the dynamic mechanical properties of discontinuities are even less frequently used for the numerical simulation of rock blasting. Therefore, the employment of the dynamic mechanical parameters of rocks and discontinuities as well as their constitutive relationships considering the loading rate in numerical modelling could be a research topic for rock blast in the future.
  • Research gaps in modelling the heterogeneity of rocks under dynamic loads: Rock microstructure governs the fracture behavior of rocks under dynamic loads and therefore the phenomenological response of the rocks. To date, material heterogeneity has been introduced into some numerical models to investigate the influence of rock heterogeneity on dynamic rock failure and associated energy release as reviewed in Section 3. However, no models reviewed above are perfect as they all need to be simplified to help capture and understand the complex rock microstructure saying nothing of the heterogeneity of rock masses. Hence, the simulation of the effects of rock heterogeneity under dynamic loads in rock blast with 3D models at an engineering scale needs to be further studied. In addition, the distribution of rock heterogeneity is usually modelled with a normal or Weibull distribution [172] while the heterogeneity of rocks is generally stochastic, which may be captured with the rapidly developing 3D scanning and 3D printing technology and then implemented into numerical modellings.
  • Research gaps in modelling existing discontinuities and new fractures in 3D: All continuous and discontinuous methods have limited ability to capture the interaction between existing discontinuities and new fractures in rock masses. The hybrid methods have enhanced capabilities but usually model the interaction in 2D. However, the discontinuity consists of two rough surfaces, or two rough surfaces sandwiched with soft rocks or clays. It has both width and length, and thus it is neither accurate nor realistic to model it only as a line in 2D. Thus, further work needs to be undertaken to establish 3D models that incorporate existing discontinuities and new fractures created in rock masses.
  • Research gaps in modelling the detonation of explosives and denotation-induced gas flow through fracturing rock mass: Most studies on modelling rock blasting reviewed in Section 3 adopted the pressure-time relationships generated from empirical or theoretical equation of state (EoS) to model the detonation induced pressure applied on the boundary of borehole, which has many limitations and involves in various crude approximations of the complicated chemical reaction. Moreover, as the detonation gas expands, the rock mass around the borehole fails to form the crushed zone and cracks, and then the detonation gas flows through the crushed zone and the crack propagating out of the crushed zone, which is completely ignored or simplified in the studies. The modelling of the detonation process and the detonation-induced gas flow through fracturing rock mass is worthy of further study.
  • Research gaps in modelling the multiphysical coupling process in rock blast: The rapid deformation during impact can lead to localized temperature increases due to adiabatic heating. Elevated temperatures can cause thermal softening of the rock material, affecting its mechanical response. The interaction between thermal and mechanical effects can lead to complex material behavior under impact. In rock blasting, the detonation-induced high temperature can induce phase transitions in rock minerals around the borehole while the detonation-induced stress wave can result in complex multiphysical coupling process in the rock mass with fluids filled in discontinuities. Thus, multiphysics coupling is a critical aspect of rock blast and involves the intricate interplay between thermal (T), hydraulic (H) and mechanical (M) processes, which, however, is ignored or simplified in almost all modellings reviewed in Section 3.

 

Specific comment No. 2: The author discussed the energy consumption and crack branching phenomena of dynamic fracture on pages 4-5 of the manuscript (approximately), but the analysis of the microscopic mechanisms of rock dynamic fracture, such as crack propagation speed and stress wave propagation, is not very sufficient. In addition, the article mentions that "energy consumption includes surface energy and fragment motion", but does not explore in detail the proportion or mechanism of energy distribution under different loading rates. Therefore, the reviewers suggest that the authors appropriately supplement the discussion on the micro mechanism of dynamic fracture in the corresponding content of the manuscript. Meanwhile, some schematic diagrams can be added appropriately.

Authors’ response: Thank you very much for your comments and suggestions. Following your suggestion, the following review on stress wave propagation analysis is supplemented in Section 2: “Correspondingly, stress wave propagation serves as the cornerstone of rock dynamics and is crucial for understanding rock dynamic fracture behavior [8]: at the most fundamental level, elastic wave propagations play a vital role in understanding rock dynamics, where the study of P-waves, S-waves and surface waves provides essential insights into the engineering applications of the rock dynamics. More sophisticated models are later developed to accurately account for wave attenuation, dispersion and scatter phenomena in heterogeneous and anisotropic rocks [8]. Wave propagation analyses in rock masses with the presence of discontinuities such as fractures, joints and faults focus particularly on characterizing the transmission, reflection and mode conversion of stress wave [9]. When rock fractures under dynamic loads, non-linear wave propagation and wave-induced plasticity occur [10-11], which represents challenging frontiers in rock dynamic research and holds significant implications for practical applications [12] such as rock blast of the focus in this study”.

Following your suggestion of crack propagation speed, the following information is added in Section 2: “More significantly, impact-induced cracks can achieve remarkably high propagation velocities, approaching the rock material’s Rayleigh wave speed, which is a characteristic that fundamentally distinguishes dynamic fracture from quasi-static crack growth [31]”

Following your suggestion of exploring the propagation of energy distribution under different loading rates, the following information is added: “…understanding dynamic energy absorption and dissipation mechanisms in rocks plays a vital role in the prediction of rock dynamic failure mechanisms and the design of effective support systems…”. “…. Therefore, the quantity of energy during dynamic fracturing can be classified into multiple components such as elastic strain energy stored in the rocks, plastic dissipation energy mainly for microcrack formation, kinetic energy for fragment motion and thermal energy for friction and crack propagation, whose relative distribution depends on loading conditions and rock properties [32]. The energy-based failure criterion then predicts rocks fail when the density of total strain energy, including both elastic strain energy and damage dissipation energy, reaches a critical threshold while the critical failure threshold increases with the loading rates. Thus, the energy dissipation and release mechanisms govern the damage and failure of rocks under dynamic loads. Initially, rock absorbs energy from dynamic impact as elastic strain energy, with partial dissipation through plastic deformation and microcracking. As damage accumulates, energy dissipation capacity decreases, transitioning from absorption to release. The damage tends to concentrate in specific regions, often leading to the formation of shear bands or tensile fracture zones. With increasing strain rates, failure modes transition from tensile splitting to shear failure and eventually complete pulverization [32-34]. The fragmentation pattern often exhibits fractal characteristics, with the fractal dimension evolving as damage accumulates [32]…”

 

Specific comment No. 3: The finite element method is an important method for crack propagation behavior in hydraulic fracturing processes. Therefore, the statements from lines 205 to 208 of the manuscript need to be supported by the following papers: The Crack Propagation Behaviour of CO2 Fracturing Fluid in Unconventional Low Permeability Reservoirs: Factor Analysis and Mechanism Revelation; Numerical simulation of fracture reorientation during hydraulic fracturing in perforated horizontal well in shale reservoirs.

Authors’ response: Thank you very much for this valuable comment and suggestion regarding the important literature. The authors have gained several insights from the suggested papers and have cited them in the line No. 248 of the revised manuscript, which corresponds to the statements in the lines No. 205 to No. 208 in the original manuscript. Please refer to the highlighted texts in the line No. 248 of the revised manuscript for detailed information.

 

Specific comment No. 4: Although the author reviewed the progress of dynamic fracture modeling in the manuscript, they did not provide sufficient engineering application guidance. For example, there is a lack of specific recommendations on how to select appropriate numerical methods based on rock type, loading conditions, or engineering objectives. The reviewer suggests adding a section proposing an engineering application guidance framework based on dynamic rock fracture modeling. This section can include the decision-making process for selecting numerical methods, recommendations for setting key parameters, etc.

Authors’ response: We truly value the reviewer’s positive comments concerning the incorporation of engineering application guidance. Following your suggestion, the following changes are made to propose an engineering application guidance framework for modelling rock dynamic fracturing using various computational mechanics available nowadays:

• A small section is added behind Table 2 to comment the appropriate engineering applications of FEM, DEM and FDEM as follows: “In engineering applications involving large, relatively uniform rock masses under moderate loading conditions, FEM offers an efficient approach due to its capability in simulating continuous material. FEM is prevalent because of its maturity, the advantages in handling rock inhomogeneity and nonlinearity, and the availability of many well-verified commercial codes for large- or small- scale problems. The extended, meshless or particle FEM approaches have become important subjects for further research and development of modelling rock dynamic fracture, mainly because of their flexibility for meshing and capability of fracture evolution simulations without remeshing. In cases when rocks exhibit significant discontinuities, such as jointed, multifaceted, or extensively fractured formations, DEM is a powerful numerical modelling tool simply because of its flexibility in handling a relatively large number of fractures. However, DEM requires exceptional computational effort, in terms of both computer memory and running time, for even a moderately large number of particles/blocks. The other main disadvantage of DEM is the uncertainty about the fracture system geometry and the effect of this uncertainty is difficult to quantify. In addition, DEM usually required 3D simulations to be conducted with 2D simulations only for generic studies. FDEM inherits all advantages of FEM and DEM and overcomes most of their disadvantages but is exceptionally computationally intensive. Compared with FEM, FDEM is more robust and efficient after damage and failure occurs, especially in terms of modelling fragment movement, interaction and muck-piling such as secondary and tertiary fractures resulting from the fragment interaction. Compared with DEM, FDEM is more versatile in modelling rock deformability before fracture, crack initiation and propagation during fracturing, and irregular-shaped fragment movements after fracture and fragmentation.”
• A new section together with a new Figure 21 is added to propose a systematic approach for modelling the dynamic fracture of rocks by blast, which include the decision-making process for selecting numerical methods and recommendations for setting key parameters following your suggestions. The new section and Figure 21 are copied as follows: “Owing to the abundance of numerical methods capable of modelling rock dynamic fracturing and the complicated mechanisms of rock blasting, it is necessary to propose an approach of numerical modelling rock blasting in order to provide researchers with a systematic and reasonable numerical modelling framework. Following the pioneer work of Jing [200] and Wang et al. [201] on modelling rock engineering and rockburst, respectively, a systematic approach of modelling dynamic fracture of rock by blast is proposed, as shown in Fig. 21, in which, the selection of numerical modelling approaches, numerical programs, numerical modelling sequences, parameters, and model calibration are illustrated. The first step is the preparation which includes problem analysis and research objective definition. The second step selects numerical methods while the third step defines geometry and selected numerical software. The fourth step includes model establishment and meshing. The fifth step determines rock mass properties and selects constitutive models. The sixth step applies initial and boundary conditions as well as impact or blast loads. The seventh step includes geostatic analysis and model calibration while the last step analyzes and visualizes simulation tests.”

Figure 21 (which can be shown properly here, please refer to the revised manuscript for Fig. 21). A systematic modelling approach for the dynamic fracture of rocks by blast

Specific comment No. 5: In the manuscript, the author mentioned that the dynamic fracture process involves complex processes such as crack branching and wave propagation, but there is insufficient discussion on the influence of rock heterogeneity (such as fracture networks, particle structures) on fracture modes, especially in multi-scale modeling. The reviewer suggests that the author appropriately strengthen the review of the latest developments in multi-scale methods (such as micro macro DEM FEM coupling) in the manuscript.

Authors’ response: Thank you very much for your suggestions. Following your suggestions, a new section is added to discuss the modelling of rock heterogeneity and a review paper dedicated to the discussion on the influence of rock heterogeneity on fracture modes is cited as follows:

• Zhang, Y., Wong, L.N.Y. A review of numerical techniques approaching microstructures of crystalline rocks. Computers and Geosciences 2018, 115, 167-187

The modelling of the influence of rock heterogeneity is identified as one of the research gaps in modelling the dynamic fracture of rocks under impact loads as follows: “Rock microstructure governs the fracture behavior of rocks under dynamic loads and therefore the phenomenological response of the rocks. To date, material heterogeneity has been introduced into some numerical models to investigate the influence of rock heterogeneity on dynamic rock failure and associated energy release as reviewed in Section 3. However, no models reviewed above are perfect as they all need to be simplified to help capture and understand the complex rock microstructure saying nothing of the heterogeneity of rock masses. Hence, the simulation of the effects of rock heterogeneity under dynamic loads in rock blast with 3D models at an engineering scale needs to be further studied. In addition, the distribution of rock heterogeneity is usually modelled with a normal or Weibull distribution [172] while the heterogeneity of rocks is generally stochastic, which may be captured with the rapidly developing 3D scanning and 3D printing technology and then implemented into numerical modellings.”

Moreover, in addition to the discussion of the influence of rock heterogeneity, the influence of fracture networks is discussed separately, which is identified as another research gaps in modelling existing discontinuities and new features in 3D as follows: “All continuous and discontinuous methods have limited ability to capture the interaction between existing discontinuities and new fractures in rock masses. The hybrid methods have enhanced capabilities but usually model the interaction in 2D. However, the discontinuity consists of two rough surfaces, or two rough surfaces sandwiched with soft rocks or clays. It has both width and length, and thus it is neither accurate nor realistic to model it only as a line in 2D. Thus, further work needs to be undertaken to establish 3D models that incorporate existing discontinuities and new fractures created in rock masses.”

Furthermore, the latest development in multi-scale methods is highlighted as one of the future directions for modelling rock dynamic fracture as follows: “multi-scale modelling effectively bridges the gap between microscopic rock fracturing behavior and macroscopic engineering-scale phenomena, which can be divided into hierarchical and concurrent multi-scale modellings [8]. Hierarchical multi-scale modelling establishes connections between micro-scale simulations and continuum mechanics models of entire rock mass and thus provides a comprehensive framework for understanding rock dynamic fracture behavior across different spatial scales [198]. Concurrent multi-scale methods offer a distinctive approach by simultaneously simulating processes at different scales within a unified model framework. This simultaneous simulation capability proves particularly valuable for accurately representing localized phenomena, such as dynamic fracture propagation, where interactions across scales play a crucial role [8].”

Two recent comprehensive literature on multi-scale modelling of rock fractures are cited to further strengthen the discussion of the multi-scale methods, which are listed as follows:

• Kong, L.; Xie, H.; Li, C. Traction-based microplane model for charactering the progressive failure of rock-like material. J Mech Phys Solids 2025, 194, 105910. https://doi.org/10.1016/j.jmps.2024.105910
• He, M., Wang, L.G., Yao, W., Dang, W., Wang, Z. AI for Rock Dynamics. Publisher: Spring Nature 2025. https://doi.org/10.1007/978-981-96-5342-3

 

Specific comment No. 6: The footnote in the title of the manuscript is' state of the art review and prospect ', but the boundary between' state of the art 'and' prospect 'is extremely vague in the main body of the manuscript, and there is no systematic summary of research trends and future challenges. Therefore, the reviewer believes that the author needs to include a dedicated section on "Research Gap and Future Directions" in the manuscript, rather than just mentioning it sporadically in paragraphs.

Authors’ response: Thank you very much for your valuable comments and constructive suggestions. Following your suggestions, two new sections are added in the revised manuscript: one for highlighting the research gaps following the state-of-the-art review and other for pointing out the future directions after introducing the authors’ efforts in trying to bridge some of the identified research gaps.

The research gaps are supplemented after the state-of-the-art review as follows:

“

4.1.1. Research gaps in modelling the dynamic properties of rocks and discontinuities  

As reviewed in Section 2, the mechanical properties of rocks and discontinuities under impact loads are related to loading rates. Unfortunately, most modellings reviewed in Section 3 adopt the static mechanical properties of rocks to study the dynamic fracture of rock by blast. Furthermore, although a great deal of research has been done on the mechanical properties of rocks under dynamic loading in the laboratory, the study of dynamic mechanical properties of discontinuities is rare and thus the dynamic mechanical properties of discontinuities are even less frequently used for the numerical simulation of rock blasting. Therefore, the employment of the dynamic mechanical parameters of rocks and discontinuities as well as their constitutive relationships considering the loading rate in numerical modelling could be a research topic for rock blast in the future.

4.1.2. Research gaps in modelling the heterogeneity of rocks under dynamic loads

Rock microstructure governs the fracture behavior of rocks under dynamic loads and therefore the phenomenological response of the rocks. To date, material heterogeneity has been introduced into some numerical models to investigate the influence of rock heterogeneity on dynamic rock failure and associated energy release as reviewed in Section 3. However, no models reviewed above are perfect as they all need to be simplified to help capture and understand the complex rock microstructure saying nothing of the heterogeneity of rock masses. Hence, the simulation of the effects of rock heterogeneity under dynamic loads in rock blast with 3D models at an engineering scale needs to be further studied. In addition, the distribution of rock heterogeneity is usually modelled with a normal or Weibull distribution [172] while the heterogeneity of rocks is generally stochastic, which may be captured with the rapidly developing 3D scanning and 3D printing technology and then implemented into numerical modellings.

4.1.3. Research gaps in modelling existing discontinuities and new fractures in 3D

All continuous and discontinuous methods have limited ability to capture the interaction between existing discontinuities and new fractures in rock masses. The hybrid methods have enhanced capabilities but usually model the interaction in 2D. However, the discontinuity consists of two rough surfaces, or two rough surfaces sandwiched with soft rocks or clays. It has both width and length, and thus it is neither accurate nor realistic to model it only as a line in 2D. Thus, further work needs to be undertaken to establish 3D models that incorporate existing discontinuities and new fractures created in rock masses.

4.1.4. Research gaps in modelling the detonation of explosives and denotation-induced gas flow through fracturing rock mass   

Most studies on modelling rock blasting reviewed in Section 3 adopted the pressure-time relationships generated from empirical or theoretical equation of state (EoS) to model the detonation induced pressure applied on the boundary of borehole, which has many limitations and involves in various crude approximations of the complicated chemical reaction. Moreover, as the detonation gas expands, the rock mass around the borehole fails to form the crushed zone and cracks, and then the detonation gas flows through the crushed zone and the crack propagating out of the crushed zone, which is completely ignored or simplified in the studies. The modelling of the detonation process and the detonation-induced gas flow through fracturing rock mass is worthy of further study.

4.1.3. Research gaps in modelling the multiphysical coupling process in rock blast   

The rapid deformation during impact can lead to localized temperature increases due to adiabatic heating. Elevated temperatures can cause thermal softening of the rock material, affecting its mechanical response. The interaction between thermal and mechanical effects can lead to complex material behavior under impact. In rock blasting, the detonation-induced high temperature can induce phase transitions in rock minerals around the borehole while the detonation-induced stress wave can result in complex multiphysical coupling process in the rock mass with fluids filled in discontinuities. Thus, multiphysics coupling is a critical aspect of rock blast and involves the intricate interplay between thermal (T), hydraulic (H) and mechanical (M) processes, which, however, is ignored or simplified in almost all modellings reviewed in Section 3.

”

Moreover, an additional section entitled “4.3. Future directions for modelling rock dynamic fracture and a systematic numerical modelling approach for rock blasting” is added to discuss the prospective developments for modelling the dynamic fracture of rock under impact loads in future and propose a corresponding systematic modelling approach after the authors have tried to bridge some of research gaps identified in Section 4.1. The part of Section 4.3 about the future directions is copied from the revised manuscript as follows:

“

4.3. Future directions for modelling rock dynamic fracture and a systematic numerical modelling approach for rock blasting

                It can be seen from this review that the advances in numerical methods have revolutionized the study of rock dynamic fracture. Key future directions of modelling rock dynamic fracture may include

• Multi-scale modelling: multi-scale modelling effectively bridges the gap between microscopic rock fracturing behavior and macroscopic engineering-scale phenomena, which can be divided into hierarchical and concurrent multi-scale modellings [8]. Hierarchical multi-scale modelling establishes connections between micro-scale simulations and continuum mechanics models of entire rock mass and thus provides a comprehensive framework for understanding rock dynamic fracture behavior across different spatial scales [198]. Concurrent multi-scale methods offer a distinctive approach by simultaneously simulating processes at different scales within a unified model framework. This simultaneous simulation capability proves particularly valuable for accurately representing localized phenomena, such as dynamic fracture propagation, where interactions across scales play a crucial role [8].
• Multi-physics coupled or interaction modelling: Rock dynamic fracture and rock blasting involve complex interactions between thermal (T), mechanical (M), hydraulic (H) and chemical (C) processes. THMC coupling modelling presents significant challenges in developing efficient and accurate numerical models. These challenges become particularly pronounced when addressing problems that span extensive time scales or large spatial domains, requiring sophisticated computational approaches to capture the intricate interplay between various physical processes including the study of dynamic damage-permeability coupling [123] and more advanced dynamic interaction between rock masses and fluids in the domain of fluid-structure interaction [152-154, 196, 199].
• Hybrid modelling: FDEM including HFDEM has demonstrated that the integration of FEM with DEM enables more comprehensive modelling of rock dynamic fracturing behavior across different conditions. Thus, the development of hybrid methods represents another significant advancement in numerical modelling capabilities, combining different numerical approaches to address complex rock dynamic fracture problems. The hybrid methods leverage the strengths of multiple numerical techniques while mitigating their individual limitations.
• High-performance modelling: 3D numerical modelling of rock dynamic fracture demands extensive computational resources. Parallel computing methods have emerged as a critical solution for improving computational efficiency and revolutionized rock dynamic fracturing simulations as demonstrated by various CPU and GPU-based parallelization of FDEM with Message Passing Interface (MPI), OpenMP, OpenCL and CUDA reviewed in Section 3.5.1. Heterogeneous CPU-GPU hybrid parallelization combines multiple parallelization strategies and heterogeneous computing resources, which enable large-scale high-performance modelling in future.
• AI-enhanced modelling: The integration of machine learning techniques has emerged as a promising direction in rock dynamics simulation. This innovative approach explores the potential of artificial intelligence (AI) to enhance numerical simulations through various means, such as developing surrogate models or optimizing computational parameters, offering new possibilities for improving both the efficiency and accuracy of modelling rock dynamic fracturing behaviors [8].

…

”

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors This paper reviews current advancements in the domain of rock dynamic fracture and its use in rock blasting through computational mechanics methods. The study concentrates on the potential of modeling rock fracture using a hybrid finite-discrete element approach (HFDEM). Furthermore, the merits and demerits of the widely used finite element method (FEM), discrete element method (DEM), and the combined finite-discrete element technique (FDEM) indicate that FDEM is the most attractive approach for modeling dynamic rock fracture and blasting applications. The study efficiently demonstrates the role of computational mechanics in enhancing rock dynamic fracture. A minor revision is recommended to enhance the quality of the manuscript before to its publication in "Geosciences". The following comments require attention in the revision: 1) The comparative analysis with various computational mechanics methods should be more thoroughly addressed.   2) The original source of the author's prior study, which has been modified, should be included in Fig 16.  3) What are the constraints of the suggested mechanism?  4) Please add future directions for HFDEM, as the readers could be interested in integrating fluid and thermal effects. 5) The conclusion should begin with an introductory paragraph and subsequently explain on the primary topics.

Author Response

General comments: This paper reviews current advancements in the domain of rock dynamic fracture and its use in rock blasting through computational mechanics methods. The study concentrates on the potential of modeling rock fracture using a hybrid finite-discrete element approach (HFDEM). Furthermore, the merits and demerits of the widely used finite element method (FEM), discrete element method (DEM), and the combined finite-discrete element technique (FDEM) indicate that FDEM is the most attractive approach for modeling dynamic rock fracture and blasting applications. The study efficiently demonstrates the role of computational mechanics in enhancing rock dynamic fracture. A minor revision is recommended to enhance the quality of the manuscript before to its publication in "Geosciences". The following comments require attention in the revision:

Authors’ response: Thank you very much for your positive comments and constructive suggestions. We have further improved the manuscript according to your comments and suggestions. These revisions are briefly in the following paragraphs in response to each of your comments and suggestions.

 

Specific comment No. 1: The comparative analysis with various computational mechanics methods should be more thoroughly addressed.  

Authors’ response: The authors greatly appreciate your constructive suggestions. Following, we have further modified the comparative analysis with various computational mechanics methods, which are explained as follows:

• We have included a comparative analysis to discuss the advantages and disadvantages of these methods in different geological conditions or engineering applications as follows: “In engineering applications involving large, relatively uniform rock masses under moderate loading conditions, FEM offers an efficient approach due to its capability in simulating continuous material. FEM is prevalent because of its maturity, the advantages in handling rock inhomogeneity and nonlinearity, and the availability of many well-verified commercial codes for large- or small- scale problems. The extended, meshless or particle FEM approaches have become important subjects for further research and development of modelling rock dynamic fracture, mainly because of their flexibility for meshing and capability of fracture evolution simulations without remeshing. In cases when rocks exhibit significant discontinuities, such as jointed, multifaceted, or extensively fractured formations, DEM is a powerful numerical modelling tool simply because of its flexibility in handling a relatively large number of fractures. However, DEM requires exceptional computational effort, in terms of both computer memory and running time, for even a moderately large number of particles/blocks. The other main disadvantage of DEM is the uncertainty about the fracture system geometry and the effect of this uncertainty is difficult to quantify. In addition, DEM usually required 3D simulations to be conducted with 2D simulations only for generic studies. FDEM inherits all advantages of FEM and DEM and overcomes most of their disadvantages but is exceptionally computationally intensive. Compared with FEM, FDEM is more robust and efficient after damage and failure occurs, especially in terms of modelling fragment movement, interaction and muck-piling such as secondary and tertiary fractures resulting from the fragment interaction. Compared with DEM, FDEM is more versatile in modelling rock deformability before fracture, crack initiation and propagation during fracturing, and irregular-shaped fragment movements after fracture and fragmentation”.
• Representative codes of each method for modelling the dynamic fracture of rocks under impact loads are listed in Table 2. The representative codes of FEM include ABAQUS, ANSYS, LS-DYNA, AUTODYN, RFPA, RS and Plaxis while those of DEM are PFC, UDEC, 3DEC, DDA, Yade, NMM and LSM. The typical codes of FDEM have emerged as Y2D, ELFEN, Y-Geo, Solidity, HOSS, HFDEM, Irazu and MultiFS. The typical applications of these representative codes are reviewed in Sections 3.1, 3.3, 3.5.2 in terms of modelling the rock dynamic fracture and Section 3.2, 3.4 and 3.5.3 in terms of modelling the rock blasting with FEM, DEM and FDEM, respectively.
• Research gaps of modelling the dynamic fracture of rocks under impact loads with computational mechanics methods are pointed out through comparing the advantages and disadvantages of FEM, DEM and FDEM presented in the extended Table 2, which are
  • Research gaps in modelling the dynamic properties of rocks and discontinuities: As reviewed in Section 2, the mechanical properties of rocks and discontinuities under impact loads are related to loading rates. Unfortunately, most modellings reviewed in Section 3 adopt the static mechanical properties of rocks to study the dynamic fracture of rock by blast. Furthermore, although a great deal of research has been done on the mechanical properties of rocks under dynamic loading in the laboratory, the study of dynamic mechanical properties of discontinuities is rare and thus the dynamic mechanical properties of discontinuities are even less frequently used for the numerical simulation of rock blasting. Therefore, the employment of the dynamic mechanical parameters of rocks and discontinuities as well as their constitutive relationships considering the loading rate in numerical modelling could be a research topic for rock blast in the future.
  • Research gaps in modelling the heterogeneity of rocks under dynamic loads: Rock microstructure governs the fracture behavior of rocks under dynamic loads and therefore the phenomenological response of the rocks. To date, material heterogeneity has been introduced into some numerical models to investigate the influence of rock heterogeneity on dynamic rock failure and associated energy release as reviewed in Section 3. However, no models reviewed above are perfect as they all need to be simplified to help capture and understand the complex rock microstructure saying nothing of the heterogeneity of rock masses. Hence, the simulation of the effects of rock heterogeneity under dynamic loads in rock blast with 3D models at an engineering scale needs to be further studied. In addition, the distribution of rock heterogeneity is usually modelled with a normal or Weibull distribution [172] while the heterogeneity of rocks is generally stochastic, which may be captured with the rapidly developing 3D scanning and 3D printing technology and then implemented into numerical modellings.
  • Research gaps in modelling existing discontinuities and new fractures in 3D: All continuous and discontinuous methods have limited ability to capture the interaction between existing discontinuities and new fractures in rock masses. The hybrid methods have enhanced capabilities but usually model the interaction in 2D. However, the discontinuity consists of two rough surfaces, or two rough surfaces sandwiched with soft rocks or clays. It has both width and length, and thus it is neither accurate nor realistic to model it only as a line in 2D. Thus, further work needs to be undertaken to establish 3D models that incorporate existing discontinuities and new fractures created in rock masses.
  • Research gaps in modelling the detonation of explosives and denotation-induced gas flow through fracturing rock mass: Most studies on modelling rock blasting reviewed in Section 3 adopted the pressure-time relationships generated from empirical or theoretical equation of state (EoS) to model the detonation induced pressure applied on the boundary of borehole, which has many limitations and involves in various crude approximations of the complicated chemical reaction. Moreover, as the detonation gas expands, the rock mass around the borehole fails to form the crushed zone and cracks, and then the detonation gas flows through the crushed zone and the crack propagating out of the crushed zone, which is completely ignored or simplified in the studies. The modelling of the detonation process and the detonation-induced gas flow through fracturing rock mass is worthy of further study.
  • Research gaps in modelling the multiphysical coupling process in rock blast: The rapid deformation during impact can lead to localized temperature increases due to adiabatic heating. Elevated temperatures can cause thermal softening of the rock material, affecting its mechanical response. The interaction between thermal and mechanical effects can lead to complex material behavior under impact. In rock blasting, the detonation-induced high temperature can induce phase transitions in rock minerals around the borehole while the detonation-induced stress wave can result in complex multiphysical coupling process in the rock mass with fluids filled in discontinuities. Thus, multiphysics coupling is a critical aspect of rock blast and involves the intricate interplay between thermal (T), hydraulic (H) and mechanical (M) processes, which, however, is ignored or simplified in almost all modellings reviewed in Section 3.

 

Specific comment No. 2: The original source of the author's prior study, which has been modified, should be included in Fig 16. 

Authors’ response: Thank you very much for your meticulous checking. Following your suggestion, we have updated the caption of Fig. 16 by citing the corresponding literature. Moreover, we thoroughly checked the whole manuscript and tried our best to solve similar problems. Following the thorough check, we cited appropriate literature for Figs. 17, 19 and 20, too.

 

Specific comment No. 3: What are the constraints of the suggested mechanism? 

Authors’ response: Thank you very much for your thoughtful and constructive feedback. Following your feedback, we have further improved Table 2 to highlight the constraints of each method and the research gaps in modelling the dynamic fracture of rocks, which are detailed in response to previous review comments and thus are not repeated here. HFDEM advocated in this manuscript inherits all constraints of FDEM listed in Table 2 and the research gaps identified in Section 4.1. We have lifted some of the constraints of FDEM and bridged some of the research gaps by further developing and applying HFDEM for modelling the dynamic fracture of rocks under impact loads, which are detailed in Section 4.2. We have appreciated that HFDEM has still many constraints despite our efforts, which are pointed out as future directions in Section 4.3.

 

Specific comment No. 4: Please add future directions for HFDEM, as the readers could be interested in integrating fluid and thermal effects.

Authors’ response: Thank you very much for your constructive suggestions. Following your suggestion, an additional section entitled “4.3. Future directions for modelling rock dynamic fracture and a systematic numerical modelling approach for rock blasting” is added to discuss the prospective developments of HFDEM in particular and numerical methods in general for modelling the dynamic fracture of rocks under impact loads and propose a systematic modelling approach. The part of Section 4.3 related to the future direction is copied from the revised manuscript as follows, which include the multi-physics coupled or interaction modelling highlighted by the reviewer:

“

4.3. Future directions for modelling rock dynamic fracture and a systematic numerical modelling approach for rock blasting

                It can be seen from this review that the advances in numerical methods have revolutionized the study of rock dynamic fracture. Key future directions of modelling rock dynamic fracture may include

• Multi-scale modelling: multi-scale modelling effectively bridges the gap between microscopic rock fracturing behavior and macroscopic engineering-scale phenomena, which can be divided into hierarchical and concurrent multi-scale modellings [8]. Hierarchical multi-scale modelling establishes connections between micro-scale simulations and continuum mechanics models of entire rock mass and thus provides a comprehensive framework for understanding rock dynamic fracture behavior across different spatial scales [198]. Concurrent multi-scale methods offer a distinctive approach by simultaneously simulating processes at different scales within a unified model framework. This simultaneous simulation capability proves particularly valuable for accurately representing localized phenomena, such as dynamic fracture propagation, where interactions across scales play a crucial role [8].
• Multi-physics coupled or interaction modelling: Rock dynamic fracture and rock blasting involve complex interactions between thermal (T), mechanical (M), hydraulic (H) and chemical (C) processes. THMC coupling modelling presents significant challenges in developing efficient and accurate numerical models. These challenges become particularly pronounced when addressing problems that span extensive time scales or large spatial domains, requiring sophisticated computational approaches to capture the intricate interplay between various physical processes including the study of dynamic damage-permeability coupling [123] and more advanced dynamic interaction between rock masses and fluids in the domain of fluid-structure interaction [152-154, 196, 199].
• Hybrid modelling: FDEM including HFDEM has demonstrated that the integration of FEM with DEM enables more comprehensive modelling of rock dynamic fracturing behavior across different conditions. Thus, the development of hybrid methods represents another significant advancement in numerical modelling capabilities, combining different numerical approaches to address complex rock dynamic fracture problems. The hybrid methods leverage the strengths of multiple numerical techniques while mitigating their individual limitations.
• High-performance modelling: 3D numerical modelling of rock dynamic fracture demands extensive computational resources. Parallel computing methods have emerged as a critical solution for improving computational efficiency and revolutionized rock dynamic fracturing simulations as demonstrated by various CPU and GPU-based parallelization of FDEM with Message Passing Interface (MPI), OpenMP, OpenCL and CUDA reviewed in Section 3.5.1. Heterogeneous CPU-GPU hybrid parallelization combines multiple parallelization strategies and heterogeneous computing resources, which enable large-scale high-performance modelling in future.
• AI-enhanced modelling: The integration of machine learning techniques has emerged as a promising direction in rock dynamics simulation. This innovative approach explores the potential of artificial intelligence (AI) to enhance numerical simulations through various means, such as developing surrogate models or optimizing computational parameters, offering new possibilities for improving both the efficiency and accuracy of modelling rock dynamic fracturing behaviors [8].

”

 

Specific comment No. 5: The conclusion should begin with an introductory paragraph and subsequently explain on the primary topics.

Authors’ response: Thank you very much for your constructive suggestions. Following them, the conclusion section has been updated accordingly. It commences with an introductory paragraph that delineates the importance of the investigation, and is then succeeded by a concise elucidation of the primary subjects addressed in the manuscript. Finally, future directions are highlighted, and a systematic approach is proposed. The conclusion section of the revised manuscript is copied as follows:

“

The dynamic fracture of rocks under impact loads has many applications in civil, mining, environmental and energy engineering such rock blasting. Recent developments in computational mechanics have revolutionized rock dynamic fracturing modelling. Correspondingly, this study aims to conduct a state-of-the-art review on the recent achievements and highlight research gaps as well as future directions of modelling the dynamic fracture of rock under impact load and its application in rock blasting using computational mechanics methods.

The peculiarities of the rock fracture under dynamic loads are firstly highlighted compared with those under static loads, which are that the response of rock mechanical properties such as stress wave propagation, strength, fracture toughness, energy distribution and cracking mechanism depends on the loading rate. Then the modellings of these peculiarities of rock and its application in rock blasting using the most applied computational mechanics methods are reviewed, which focuses on reviewing those using the finite element method (FEM), discrete element method (DEM) and combined finite-discrete element method (HFDEM) as the representative continuous, discontinuous and hybrid methods, respectively. After that, the advantages and disadvantages of these computational mechanics methods in modelling the rock dynamic fracture and its application are discussed, which highlights FDEM is the most promising method for modelling rock dynamic fracture and its application in blasting as well as the research gaps in this field. Subsequently, the progress of bridging some of these research gaps by developing the hybrid finite-discrete element method (HFDEM), i.e. the authors’ version of FDEM, for modelling the rock dynamic fracture and its application in rock blasting are introduced. The key features of HFDEM for modelling rock dynamic fracture and its application in rock blasting includes modelling the effects of loading rate on rock dynamic behaviour; stress wave propagation, reflection and absorbing as well as stress wave-induced fracture; explosive-rock interaction including detonation-induced gas expansion and flow through fracturing rock; dynamic fracturing under coupled static and dynamic conditions; heterogeneous rock and rock mass with pre-existing discrete fracture network; and dynamic fracturing-induced fragment size distribution.

Furthermore, the future directions of modelling the dynamic fracture of rocks under impact loads are highlighted, which include multi-scale, multi-physics coupled/interacted, hybrid, high-performance and AI-enhanced modellings. Finally, a systemic approach is proposed for modelling the dynamic fracture of rocks by blast in light of the abundance of numerical methods available nowadays.

”

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

I have carefully reviewed the revised version, and the manuscript quality has been improved. I think it has reached a level where it can be accepted and published.