{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,30]],"date-time":"2026-03-30T22:53:35Z","timestamp":1774911215989,"version":"3.50.1"},"reference-count":27,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2025,4,26]],"date-time":"2025-04-26T00:00:00Z","timestamp":1745625600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Sichuan Science and Technology Program","award":["2023NSFSC0907"],"award-info":[{"award-number":["2023NSFSC0907"]}]},{"name":"Sichuan Science and Technology Program","award":["2023NSFSC0980"],"award-info":[{"award-number":["2023NSFSC0980"]}]},{"name":"Equipment Integrated Research of China","award":["2023NSFSC0907"],"award-info":[{"award-number":["2023NSFSC0907"]}]},{"name":"Equipment Integrated Research of China","award":["2023NSFSC0980"],"award-info":[{"award-number":["2023NSFSC0980"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Materials"],"abstract":"<jats:p>This study develops a computational framework for evaluating the residual strength of tungsten-copper functionally graded materials following crack-arrest hole drilling. The proposed methodology features two pivotal innovations: First, a phase-field isoparametric gradient elements is established through representing the gradient effect within the finite element stiffness matrices, incorporating both Amor and Miehe elastic energy decomposition schemes to address tension-compression asymmetry in crack evolution. Second, a multi-fidelity neural network strategy is integrated with the gradient phase-field element to mitigate characteristic length dependency in residual strength predictions. Comparative analyses demonstrate that the gradient finite element achieves smoother field transitions at element interfaces compared to conventional homogeneous elements, as quantified in both stress and damage fields. The Miehe decomposition scheme outperforms the Amor model in capturing complex crack trajectories. Validation against the average strain energy criterion indicates the present approach enhances residual strength prediction accuracy by 39.07% to 44.05%, establishing a robust numerical tool for damage tolerance assessment in graded materials.<\/jats:p>","DOI":"10.3390\/ma18091973","type":"journal-article","created":{"date-parts":[[2025,4,28]],"date-time":"2025-04-28T06:23:32Z","timestamp":1745821412000},"page":"1973","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["A Method for Calculating Residual Strength of Crack Arrest Hole on Tungsten-Copper Functionally Graded Materials by Phase-Field Gradient Element Combined with Multi-Fidelity Neural Network"],"prefix":"10.3390","volume":"18","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8791-6676","authenticated-orcid":false,"given":"Bowen","family":"Liu","sequence":"first","affiliation":[{"name":"China Aerodynamics Research and Development Center, Manyang 621000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yisheng","family":"Yang","sequence":"additional","affiliation":[{"name":"China Aerodynamics Research and Development Center, Manyang 621000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2547-6921","authenticated-orcid":false,"given":"Guishan","family":"Wang","sequence":"additional","affiliation":[{"name":"China Aerodynamics Research and Development Center, Manyang 621000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yin","family":"Li","sequence":"additional","affiliation":[{"name":"China Aerodynamics Research and Development Center, Manyang 621000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,4,26]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Lei, X., Li, C., Shi, X., Xu, X., and Wei, Y. 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