{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,18]],"date-time":"2026-08-18T04:54:21Z","timestamp":1787028861483,"version":"build-2736575974"},"reference-count":54,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2026,3,26]],"date-time":"2026-03-26T00:00:00Z","timestamp":1774483200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Project of Xizang Autonomous Region","award":["XZ202401YD0003"],"award-info":[{"award-number":["XZ202401YD0003"]}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t              https:\/\/ror.org\/01h0zpd94","doi-asserted-by":"publisher","award":["U22A20597"],"award-info":[{"award-number":["U22A20597"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t              https:\/\/ror.org\/01h0zpd94","doi-asserted-by":"publisher","award":["42407227"],"award-info":[{"award-number":["42407227"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Science and Technology Projects of Xizang Autonomous Region of China","award":["XZ202501ZY0132"],"award-info":[{"award-number":["XZ202501ZY0132"]}]},{"DOI":"10.13039\/501100012226","name":"Fundamental Research Funds for the Central Universities","doi-asserted-by":"crossref","award":["kx0020020250611"],"award-info":[{"award-number":["kx0020020250611"]}],"id":[{"id":"10.13039\/501100012226","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Materials"],"abstract":"<jats:p>Heterogeneity at the grain scale strongly influences hydraulic fracturing in crystalline rock; however, systematic studies quantifying its impacts on the evolution of injection pressure and crack propagation remain limited. To address this gap, we employ a hydro-mechanical phase-field model incorporating Voronoi-based microstructures to systematically quantify the effects of grain-scale heterogeneity on hydraulic fracturing. Two numerical experimental programs are designed to examine the effects of (i) mean grain size and (ii) mineral distribution under different axial stresses. The simulations reveal a close coupling between injection pressure and crack-length evolution, and both responses are strongly governed by grain-scale heterogeneity. When the fracture enters weak minerals, it advances rapidly and pressure drops; when it encounters on strong minerals, growth slows or arrests and pressure builds until a threshold triggers the next advance. Moreover, peak pressure statistics further indicate that mineral distribution dominates the response scatter, while axial stress plays a secondary role. Specifically, the mean peak pressures at 0 and 10 MPa are similar (about 14.31 and 14.21 MPa), whereas rearranging minerals within the same Voronoi tessellation changes peak pressure by more than 4 MPa. Higher peaks occur when strong minerals lie ahead of the initial crack tip, increasing resistance to initiation and early growth. Finally, the stress state modulates fracture trajectories: under low axial stress, fractures preferentially follow mineral boundaries, whereas higher axial stress strengthens macroscopic stress guidance and shifts the path toward a direction closer to being perpendicular to the maximum principal stress. This trend is consistent with energy minimization, since interface detouring under high axial stress incurs a larger elastic free energy penalty.<\/jats:p>","DOI":"10.3390\/ma19071322","type":"journal-article","created":{"date-parts":[[2026,3,26]],"date-time":"2026-03-26T15:10:24Z","timestamp":1774537824000},"page":"1322","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Influence of Grain-Scale Heterogeneity on Hydraulic Fracturing: A Study Based on a Hydro-Mechanical Phase-Field Model"],"prefix":"10.3390","volume":"19","author":[{"given":"Gen","family":"Zhang","sequence":"first","affiliation":[{"name":"College of Engineering, Xizang University, Lhasa 850000, China"},{"name":"Xizang Autonomous Region Plateau Major Infrastructure Intelligent Construction and Resilient Safety Technology Innovation Center, Xizang University, Lhasa 850000, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Cheng","family":"Zhao","sequence":"additional","affiliation":[{"name":"College of Engineering, Xizang University, Lhasa 850000, China"},{"name":"Xizang Autonomous Region Plateau Major Infrastructure Intelligent Construction and Resilient Safety Technology Innovation Center, Xizang University, Lhasa 850000, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0009-3272-6002","authenticated-orcid":false,"given":"Zejun","family":"Tian","sequence":"additional","affiliation":[{"name":"Department of Geotechnical Engineering, Tongji University, Shanghai 200092, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3355-7190","authenticated-orcid":false,"given":"Jinquan","family":"Xing","sequence":"additional","affiliation":[{"name":"Xizang Autonomous Region Plateau Major Infrastructure Intelligent Construction and Resilient Safety Technology Innovation Center, Xizang University, Lhasa 850000, China"},{"name":"Department of Geotechnical Engineering, Tongji University, Shanghai 200092, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jialun","family":"Niu","sequence":"additional","affiliation":[{"name":"Xizang Autonomous Region Plateau Major Infrastructure Intelligent Construction and Resilient Safety Technology Innovation Center, Xizang University, Lhasa 850000, China"},{"name":"Department of Geotechnical Engineering, Tongji University, Shanghai 200092, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zhaosen","family":"Wang","sequence":"additional","affiliation":[{"name":"College of Engineering, Xizang University, Lhasa 850000, China"},{"name":"Xizang Autonomous Region Plateau Major Infrastructure Intelligent Construction and Resilient Safety Technology Innovation Center, Xizang University, Lhasa 850000, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Wenkang","family":"Yu","sequence":"additional","affiliation":[{"name":"College of Engineering, Xizang University, Lhasa 850000, China"},{"name":"Xizang Autonomous Region Plateau Major Infrastructure Intelligent Construction and Resilient Safety Technology Innovation Center, Xizang University, Lhasa 850000, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2026,3,26]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"311","DOI":"10.1146\/annurev-fluid-010814-014736","article-title":"Mechanics of hydraulic fractures","volume":"48","author":"Detournay","year":"2016","journal-title":"Annu. 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