the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 06 Aug 2026
Mechanical interaction of geometrically linked cracks and faults
Abstract. Understanding how macroscopic fractures develop from interacting cracks is essential for describing brittle deformation in upper-crustal rocks. Although empirical failure criteria successfully describe fracturing in laboratory experiments, they do not explain how microcracks distributed within deforming rocks interact, coalesce and organize into through-going fracture networks. In this study, we examine how the geometrical distribution of pre-existing cracks controls the local stress perturbations that may influence fracture formation. We use two-dimensional Finite Element Method-based numerical simulations to quantify the elastic stress field around isolated cracks, en echelon crack arrays and randomly distributed crack systems undergoing simple shear. The results show that crack orientation, and spacing strongly control the distribution of reduced compression, effective cohesion, and localized strain around crack tips, and determine whether local stress perturbations may promote or inhibit crack linkage. Favorably arranged crack arrays produce overlapping lobes of reduced compression and local principal stress trajectories that effectively connect neighboring crack tips, defining potential linkage paths oblique to the far-field maximum principal stress. These findings suggest that shear fractures primarily develop from the overlap of tensile stress perturbation regions around pre-existing cracks rather than being controlled solely by the far-field stress or the geometry of mode I cracks formed during early stages of deformation. In highly anisotropic crack configurations, the distribution forces shear fracture evolution and fault configurations along limited paths. Conversely, rocks with isotropic crack distributions tend to preferentially activate en echelon arrays oriented approximately at 30° relative to the far-field maximum principal stress, as predicted by the Coulomb criterion. This research demonstrates that the distribution, and orientation of the pre-existing cracks exert a first-order control over the mechanical evolution of shear fractures and faults, as well as the impact of crustal and hydrostatic pressure variations on the conditions for failure and crack opening in upper-crustal rocks.
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Status: final response (author comments only)
- RC1: 'Comment on egusphere-2026-4027', Stefan Markus Schmalholz, 25 Aug 2026
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CC1: 'Comment on egusphere-2026-4027', Giacomo Medici, 18 Sep 2026
General comments
Very good research. Please, see my comments to improve the final version of the manuscript.
Specific comments
Lines 42-58. Make clear the novelty of your geo-mechanical research.
Lines 54-55. Faults as conduits and barriers for the fluid flow. Insert the most updated conceptual models on fault zone hydrogeology presented in these two review papers:
- Bense, V. F., Gleeson, T., Loveless, S. E., Bour, O., Scibek, J. (2013). Fault zone hydrogeology. Earth-Science Reviews, 127, 171-192.
- Lupi, F., Agbotui, P.Y., Medici, G. (2026). Hydraulic Conductivity in the Mesozoic Units of the Umbria–Marche Succession (Italy): Insights Towards a Sustainable Management of Carbonate Aquifers Worldwide. Sustainability, 18(18), 9297.
Line 58. Clarify the general goal of your research at the end of your introduction.
Line 58. Describe the 3 to 4 specific objectives of your research by using numbers (i, ii, and iii).
Line 226. Is equation 4 necessary?
Line 1444. The specific objectives of your research are three looking at your conclusions.
Lines 1418-1457. Make clear and evident the novelty of your research in the conclusions.
Figures and tables
Figures 1-15. Captions sometime too long. What about moving some description in the new paragraph?
Figure 2. Enlarge the figures in the lower part.
Figure 15. Too much information in one figure. I suggest to split this figure in multiple ones.
Figure 15. The orientation of the outcrop images is not clear.
Figure 15D-F. The outcrop images are difficult to see.
Figure 15D. Spatial scale unclear or undefined for the scan-area.
Citation: https://doi.org/10.5194/egusphere-2026-4027-CC1 -
RC2: 'Comment on egusphere-2026-4027', Michele Cooke, 03 Nov 2026
The study uses FEM models to explore deformation around echelon and randomly oriented slit cracks. I found the results of the randomly located and oriented cracks particularly new and interesting. The paper is well written and provides a detailed presentation of model results. The text of the paper is quite long and could be tightened up throughout to improve reader engagement. I offer some suggestions for this in my comments.
- Much of the introduction and model set up can be shortened to bring the reader more efficiently to the point of the paper. For example, all the equations on page 14 are standard equations in any undergraduate structural geology textbook. Much of the introduction is standard background knowledge for any fault mechanics study. The introduction should set up the question that this study addresses and the value of the employed approach for answering that question – assume the reader has sufficient mechanics background and doesn’t need a review of basic theory. The model setup can focus on approaches that are not common and cite other studies that also use these standard approaches.
- By using traction-free slit cracks, the cracks will interpenetrate upon compression in a manner inconsistent with the crust – this limitation appears to be the reason that the pore pressure is assumed to be high enough to prop cracks open. The model of Figure 2A (lower left) has reduce compression along the sides of the crack, suggesting that the crack sides are interpenetrating under these conditions. The same conditions also seem to arise for the cracks for figure 7 of some of the models in figure 9 and 10 (and others). This limitation of the model is first mentioned in the discussion section but should be more explicitly explained in the methods/model set up section and the implications explained with the description of results. The discussion section 5.2 presents the pore pressure and stress states needed to keep the cracks open, but is difficult to follow. A more straight forward approach may be to exclude interpenetrating results from the analysis. Alternatively, many fault mechanics modeling studies use contact mechanics (e.g. Maerten et al., 2010 1007/s10596-009-9170-x; Fattaruso et al., 2022 10.3389/feart.2022.853030) and their results might be compared with the results of this study to validate the results here.
- The normalized shear strain, Exy (introduced on line 337), only examines shear strain along the x- and y- axes. A shear strain that is invariant would serve better for this study because the faults have a wide range of orientations. For example, the second invariant of the strain tensor or the vorticity, which is 2 times curl of the displacement fields. These are both commonly used in structural geology and fault mechanics studies
- Section 4.1 about isolated cracks is set up as a test of the model against classic solutions but this comparison is not offered. How do these numerical results compare with Linear Elastic Fracture Mechanics solutions? Since nothing new is learned from this comparison, the results could be presented in a supplement.
- Similarly, other studies have investigated the stress fields around echelon cracks. How to the results here compare to those studies? Much of the results presented in the echelon cracks section does not seem new. One way to streamline the long section 4.2 (19 pages) is to highlight the results that are new and cite other studies for results that are not new (such as presence of sigmoidal stress shadows etc).
Small comments
- En echelon is a borrowed French phrase in the English language and needs to be italicized, as the authors have done. But a simple solution is available. The word echelon (without en) is an English word and can be used throughout this paper.
- Because the paper presents the results of many models, the reader may have difficulty figuring out which of the model results are new/unexpected. One way to help the reader appreciate the new insights gleaned from the results is to decorate the figures to highlight these new insights, such as potential linkage geometries.
Citation: https://doi.org/10.5194/egusphere-2026-4027-RC2
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- 1
In the manuscript of Manna and co-authors, the authors present results from two-dimensional, FEM-based linear elastic simulations to investigate how different geometries of pre-existing crack networks perturb local stress and strain fields in a rock subjected to an increment of simple shear deformation. The cracks are represented as elliptical voids. Two series of simulations are performed for different crack orientations and configurations, grouped into en échelon and random geometries. The resulting fields of quantities such as the magnitude of the minor in-plane principal stress and the first invariant of the strain tensor are presented and discussed.
The numerical results of the linear elastic simulations are sound, and the stress and strain fields around the very thin elliptical voids appear well resolved. The figures are clear and informative. Overall, the simulations provide interesting insights into how mechanical interactions among pre-existing cracks depend on their geometry, orientation, and spacing. In the Discussion, the authors further consider the applicability and implications of their results for fracture systems in natural rocks.
The presented results should be of clear interest to researchers working on brittle deformation in rocks. My main concern relates to a substantial part of the Discussion, which, in my view, discusses processes that are not explicitly represented in the numerical models, in particular the effects of pore-fluid pressure and the application of a Mohr–Coulomb failure criterion. If the authors want to discuss these effects in detail, it would strengthen the manuscript considerably to support these discussions with corresponding numerical results, for example, from a poroelastic model in the case of pore-fluid pressure, or from a model incorporating a Mohr–Coulomb criterion. Alternatively, given that the manuscript is already quite long and that the linear elastic results are very interesting, I would suggest substantially shortening these parts of the Discussion and focusing on the implications that can be drawn directly from the presented linear elastic models.
In particular, I found the current discussion of pore-fluid pressure difficult to follow in places, and I think that some of the underlying mechanical assumptions require further clarification. In the Methods section, the cracks are described as voids, which I understand to mean that the material within the cracks has no mechanical strength. Elsewhere, however, it is stated that the cracks are assumed to contain a fluid at a pressure sufficiently high to keep them open. I therefore suggest that the authors clarify precisely how the crack filling and their mechanical boundary conditions should be interpreted. Also, the authors should discuss whether cracks that are voids or cracks that are filled with a nearly incompressible fluid generate the same stress fields or not.
Related to this issue, line 935 assumes that pore-fluid pressure can be estimated using a hydrostatic pressure profile. Such a hydrostatic pressure distribution would generally require hydraulic connectivity over the vertical distance up to the surface. In the presented models, however, the cracks are hydraulically isolated. For a fluid-filled crack in mechanical equilibrium, the fluid pressure must also equal the normal traction exerted by the surrounding solid at the solid–fluid interface. I therefore think that the assumptions underlying the prescribed pore-pressure values, and their relationship to the calculated stresses, need to be explained more carefully. I also find the interpretation of Eq. (17) potentially misleading in the context of the present model. As I understand it, this type of effective-stress relation applies to a porous two-phase solid–fluid medium, where sigma_n represents an appropriate total stress of the two-phase medium. The numerical model presented here, however, does not simulate such a two-phase medium, but it represents an interface between a non-porous elastic solid and a crack represented as a void. As mentioned above, at a solid–fluid interface in mechanical equilibrium, the normal traction in the solid must balance the fluid pressure. It would therefore be helpful if the authors clearly distinguished between the total stress of a porous two-phase medium and the local stress in the non-porous elastic solid surrounding an individual fluid-filled crack.
Moreover, if the cracks were filled with a fluid, their internal pressure would generally depend on the crack’s aspect ratio and orientation relative to the far-field stress. Different cracks could therefore develop different pressure changes, including relative under- or overpressures, depending on their geometry and loading. In the present Discussion, however, the crack pressure appears, as far as I understand, to be treated independently of these effects. This further illustrates why I think that directly transferring stresses calculated for a linear elastic medium containing voids to a poroelastic medium containing fluid-filled cracks requires additional assumptions and justification. For these reasons, I suggest that the authors either provide a more rigorous mechanical treatment of pore-fluid pressure, ideally supported by simulations that explicitly account for poroelastic deformation, or substantially reduce this part of the Discussion.
Minor comments
Introduction: The Introduction is quite long and provides substantial background information on fractures and cracks. Much of this material is useful for understanding the results. The authors might consider adding a schematic figure illustrating the main structures and concepts introduced in the text, such as Mode I and Mode II fractures, wing cracks, and en échelon fractures. Such a figure could help guide the reader through the terminology used in the Introduction.
Line 39: The term “hydrostatic pressure variations” is a bit unclear because hydrostatic pressure is, by definition, controlled by fluid density, gravity, and depth. Do the authors instead mean variations in fluid pressure or indeed just the increase of hydrostatic pressure with depth?
Line 78: Please clarify that the sigma symbol represents the far-field maximum principal stress, as is explained later in the manuscript.
Line 90: This statement appears to apply specifically to non-porous rock. In porous rock, sufficiently high pore-fluid pressure may promote Mode I fracturing at greater depths.
Line 109: Could the authors extend this sentence to explain why in-plane propagation is not possible?
Line 242: “Outlook” seems somewhat unusual as the title of a paragraph in the Introduction. The paragraph appears instead to describe the aims, objectives, and/or hypotheses of the study.
Line 251: Will the numerical code be made openly available so that the simulations and results can be reproduced?
Line 306ff: Could the authors explain why they use the reduced stress rather than the deviatoric stress, i.e., the deviation from the mean stress?
Line 1021: Is this expression intended to apply to a crack represented as a void with no mechanical strength, or to a crack filled with a pore fluid? Please clarify.
Line 1045: The presented model considers only hydraulically isolated cracks.
Line 1082: Which stress is represented by this stress tensor: the stress in the solid phase or the total stress of the two-phase solid–fluid medium? This distinction should be made clear.
Best regards,
Stefan Schmalholz