{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,17]],"date-time":"2026-08-17T05:43:51Z","timestamp":1786945431546,"version":"build-2736575974"},"reference-count":46,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2023,2,17]],"date-time":"2023-02-17T00:00:00Z","timestamp":1676592000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Natural Science Foundation of Xinjiang Uygur Autonomous Region","award":["2021D01C115"],"award-info":[{"award-number":["2021D01C115"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Materials"],"abstract":"<jats:p>Hydrogen-assisted cracking is one of the most dominant failure modes in metal hydrogen-facing materials. Therefore, the hydrogen-assisted cracking mechanism has been a hot topic for a long time. To date, there is very little published research on numerical methods to describe hydrogen-assisted cracking. This paper presents a new method for the description of hydrogen embrittlement crack growth: an explicit phase-field formulation, which is based on the phase-field description of cracks, Fick\u2019s mass diffusion law, and the relationship between hydrogen content and fracture surface energy. A novel computational framework is then developed using the self-developed FEM software DYNA-WD. We numerically calculate several typical conditions in the 3-D coordinates to validate the effectiveness of the proposed computational framework. Specifically, we discuss (i) the failure of a square plate in a hydrogenous environment, (ii) the CT specimen failed with the inner hydrogen, (iii) the plate\/failed with the corrosives, and (iv) the failure of the disk test. Finally, the relationship between Mises stress, the concentration of hydrogen, the thickness of the disc, and the loading rate is investigated.<\/jats:p>","DOI":"10.3390\/ma16041708","type":"journal-article","created":{"date-parts":[[2023,2,20]],"date-time":"2023-02-20T02:02:09Z","timestamp":1676858529000},"page":"1708","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":14,"title":["Numerical Simulation for Hydrogen-Assisted Cracking: An Explicit Phase-Field Formulation"],"prefix":"10.3390","volume":"16","author":[{"given":"Di","family":"Wang","sequence":"first","affiliation":[{"name":"Department of Mechanical Engineering, Xinjiang University, Urumqi 830017, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Fangping","family":"Ma","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering, Xinjiang University, Urumqi 830017, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Hao","family":"Chen","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering, Xinjiang University, Urumqi 830017, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2023,2,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Li, Y., and Zhang, K. (2022). Analysis of Hydrogen-Assisted Brittle Fracture Using Phase-Field Damage Modelling Considering Hydrogen Enhanced Decohesion Mechanism. Metals, 12.","DOI":"10.3390\/met12061032"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"112614","DOI":"10.1016\/j.cma.2019.112614","article-title":"A phase-field regularized cohesive zone model for hydrogen assisted cracking","volume":"358","author":"Wu","year":"2020","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"742","DOI":"10.1016\/j.cma.2018.07.021","article-title":"A phase field formulation for hydrogen assisted cracking","volume":"342","author":"Golahmar","year":"2018","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"20676","DOI":"10.1016\/j.ijhydene.2022.04.260","article-title":"Hydrogen trapping and desorption affected by ferrite grain boundary types in shielded metal and flux-cored arc weldments with Ni addition","volume":"47","author":"Moshtaghi","year":"2022","journal-title":"Int. J. Hydrogen Energy"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Shimotomai, M. (2021). Heuristic Design of advanced martensitic steels that are highly resistant to hydrogen embrittlement by \u03b5-carbide. Metals, 11.","DOI":"10.3390\/met11020370"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"2794","DOI":"10.1016\/j.jmrt.2022.06.046","article-title":"Machine learning approach for prediction of hydrogen environment embrittlement in austenitic steels","volume":"19","author":"Kim","year":"2022","journal-title":"J. Mater. Res. Technol."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"74","DOI":"10.1007\/s11661-021-06491-3","article-title":"Effect of gas pressure on hydrogen environment embrittlement of carbon steel A106 in carbon monoxide mixed hydrogen gas","volume":"53","author":"Komoda","year":"2022","journal-title":"Metall. Mater. Trans. A"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Michler, T., Wackermann, K., and Schweizer, F. (2021). Review and assessment of the effect of hydrogen gas pressure on the embrittlement of steels in gaseous hydrogen environment. Metals, 11.","DOI":"10.3390\/met11040637"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"6251","DOI":"10.1007\/s10853-017-1978-5","article-title":"Understanding and mitigating hydrogen embrittlement of steels: A review of experimental, modelling and design progress from atomistic to continuum","volume":"53","author":"Barrera","year":"2018","journal-title":"J. Mater. Sci."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"734","DOI":"10.1016\/j.actamat.2018.12.014","article-title":"Enumeration of the hydrogen-enhanced localized plasticity mechanism for hydrogen embrittlement in structural materials","volume":"165","author":"Martin","year":"2019","journal-title":"Acta Mater."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"110215","DOI":"10.1016\/j.commatsci.2020.110215","article-title":"Effect of alloying elements on hydrogen enhanced decohesion in bcc iron","volume":"188","author":"Kholtobina","year":"2021","journal-title":"Comput. Mater. Sci."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"100","DOI":"10.1016\/j.commatsci.2019.05.029","article-title":"Hydrogen-enhanced decohesion mechanism of the special \u03a35 (0 1 2)[1 0 0] grain boundary in Ni with Mo and C solutes","volume":"167","author":"He","year":"2019","journal-title":"Comput. Mater. Sci."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"109497","DOI":"10.1016\/j.corsci.2021.109497","article-title":"AgCl-induced hot salt stress corrosion cracking in a titanium alloy","volume":"187","author":"Shi","year":"2021","journal-title":"Corros. Sci."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"103472","DOI":"10.1016\/j.mechmat.2020.103472","article-title":"Dislocation-density based crystal plasticity model with hydrogen-enhanced localized plasticity in polycrystalline face-centered cubic metals","volume":"148","author":"Yuan","year":"2020","journal-title":"Mech. Mater."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"035009","DOI":"10.1088\/1361-651X\/ac4a23","article-title":"Simulating intergranular hydrogen enhanced decohesion in aluminium using density functional theory","volume":"30","author":"Wilson","year":"2022","journal-title":"Model. Simul. Mater. Sci. Eng."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"105312","DOI":"10.1016\/j.engfailanal.2021.105312","article-title":"Influence of hydrogen-enhanced plasticity and decohesion mechanisms of hydrogen embrittlement on the fracture resistance of steel","volume":"123","author":"Wasim","year":"2021","journal-title":"Eng. Fail. Anal."},{"key":"ref_17","unstructured":"Djukic, M., Bakic, G., \u0160ija\u010dki-\u017derav\u010di\u0107, V., Rajicic, B., Sedmak, A., Wasim, M., and Peri\u0161i\u0107, J. (2021, January 28\u201329). The synergistic action of HELP and HEDE mechanisms of hydrogen embrittlement in steels. Proceedings of the Research Symposium on Hydrogen-Materials Interactions, Kyushu, Japan."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"107427","DOI":"10.1016\/j.engfracmech.2020.107427","article-title":"A phase field numerical manifold method for crack propagation in quasi-brittle materials","volume":"241","author":"Yang","year":"2021","journal-title":"Eng. Fract. Mech."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1273","DOI":"10.1002\/nme.2861","article-title":"Thermodynamically consistent phase-field models of fracture: Variational principles and multi-field FE implementations","volume":"83","author":"Miehe","year":"2010","journal-title":"Int. J. Numer. Methods Eng."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"486","DOI":"10.1016\/j.cma.2014.11.017","article-title":"Phase field modeling of fracture in multi-physics problems. Part II. Coupled brittle-to-ductile failure criteria and crack propagation in thermo-elastic\u2013plastic solids","volume":"294","author":"Miehe","year":"2015","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_21","first-page":"301","article-title":"On the theoretical and numerical aspects of the unified phase-field theory for damage and failure in solids and structures","volume":"53","author":"Jianying","year":"2021","journal-title":"Chin. J. Theor. Appl. Mech."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"102840","DOI":"10.1016\/j.tafmec.2020.102840","article-title":"Comparative study of phase-field damage models for hydrogen assisted cracking","volume":"111","author":"Mandal","year":"2021","journal-title":"Theor. Appl. Fract. Mech."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"504","DOI":"10.1051\/jnwpu\/20224030504","article-title":"Simulating hydrogen embrittlement fracture based on phase field method","volume":"40","author":"Chen","year":"2022","journal-title":"Xibei Gongye Daxue Xuebao\/J. Northwestern Polytech. Univ."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"107705","DOI":"10.1016\/j.engfracmech.2021.107705","article-title":"A review of phase-field models, fundamentals and their applications to composite laminates","volume":"248","author":"Bui","year":"2021","journal-title":"Eng. Fract. Mech."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1259","DOI":"10.1007\/s00466-022-02147-0","article-title":"A comparative review of peridynamics and phase-field models for engineering fracture mechanics","volume":"69","author":"Diehl","year":"2022","journal-title":"Comput. Mech."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1016\/j.ijsolstr.2020.05.003","article-title":"A statistical geometry approach to length scales in phase field modelling of fracture and strength of porous microstructures","volume":"200","author":"Carlsson","year":"2020","journal-title":"Int. J. Solids Struct."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"111380","DOI":"10.1016\/j.ijsolstr.2021.111380","article-title":"On the phase-field modeling of fully coupled chemo- mechanical deterioration and fracture in calcium leached cementitious solids","volume":"238","author":"Wu","year":"2022","journal-title":"Int. J. Solids Struct."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"16892","DOI":"10.1016\/j.ijhydene.2015.06.069","article-title":"Hydrogen embrittlement in nickel, visited by first principles modeling, cohesive zone simulation and nanomechanical testing","volume":"40","author":"Alvaro","year":"2015","journal-title":"Int. J. Hydrogen Energy"},{"key":"ref_29","unstructured":"Livermore Software Technology (2021). LS-DYNA\u00ae Keyword User\u2019s Manual Volume II Material Models, Livermore Software Technology. LS-DYNA R13."},{"key":"ref_30","unstructured":"Hallquist, J.O. (2019). Ls-Dyna Theory Manual, Livermore Software Technology."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Abdollahzadeh, M.A., Kefal, A., and Yildiz, M. (2020). A comparative and review study on shape and stress sensing of flat\/curved shell geometries using C0-continuous family of iFEM elements. Sensors, 20.","DOI":"10.3390\/s20143808"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1016\/j.tafmec.2018.08.010","article-title":"An extrinsic cohesive shell model for dynamic fracture analyses","volume":"97","author":"Wang","year":"2018","journal-title":"Theor. Appl. Fract. Mech."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1537","DOI":"10.1007\/s00466-019-01733-z","article-title":"Modeling the dynamic and quasi-static compression-shear failure of brittle materials by explicit phase field method","volume":"64","author":"Wang","year":"2019","journal-title":"Comput. Mech."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1016\/j.compstruc.2019.03.005","article-title":"An explicit phase field method for brittle dynamic fracture","volume":"217","author":"Ren","year":"2019","journal-title":"Comput. Struct."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.finel.2018.12.001","article-title":"A locally stabilized central difference method","volume":"155","author":"Soares","year":"2019","journal-title":"Finite Elem. Anal. Des."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"254","DOI":"10.1016\/j.compositesb.2018.12.023","article-title":"Consistent application of periodic boundary conditions in implicit and explicit finite element simulations of damage in composites","volume":"168","author":"Garoz","year":"2019","journal-title":"Compos. Part B Eng."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"721","DOI":"10.1007\/s00371-017-1410-9","article-title":"A novel unconditionally stable explicit integration method for finite element method","volume":"34","author":"Zheng","year":"2018","journal-title":"Vis. Comput."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1137\/19M128867X","article-title":"Rotations in Three Dimensions","volume":"63","author":"Maritz","year":"2021","journal-title":"SIAM Rev."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"108238","DOI":"10.1016\/j.engfracmech.2022.108238","article-title":"An X-FEM based framework for 3D fatigue crack growth using a B-spline crack geometry description","volume":"261","author":"Hectors","year":"2022","journal-title":"Eng. Fract. Mech."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"327","DOI":"10.1007\/s11661-003-0334-3","article-title":"Internal hydrogen embrittlement of ultrahigh-strength AERMET 100 steel","volume":"34","author":"Scully","year":"2003","journal-title":"Metall. Mater. Trans. A"},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Li, Y., Zhang, K., Lu, D., and Zeng, B. (2021). Hydrogen-Assisted Brittle Fracture Behavior of Low Alloy 30CrMo Steel Based on the Combination of Experimental and Numerical Analyses. Materials, 14.","DOI":"10.3390\/ma14133711"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"105985","DOI":"10.1016\/j.engfailanal.2021.105985","article-title":"Hydrogen-induced degradation of high-strength steel pipeline welds: A critical review","volume":"133","author":"Sun","year":"2022","journal-title":"Eng. Fail. Anal."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"313","DOI":"10.1016\/j.cma.2015.06.009","article-title":"An assessment of the phase field formulation for crack growth","volume":"294","author":"Klinsmann","year":"2015","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1115\/1.2804884","article-title":"Hydrogen embrittlement characterization by disk pressure tests: Test analysis and application to high chromium martensitic steels","volume":"118","author":"Beghini","year":"1996","journal-title":"J. Eng. Mater. Technol."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"109057","DOI":"10.1016\/j.vacuum.2019.109057","article-title":"Effect of strain rate on environmental hydrogen embrittlement susceptibility of a severely cold-rolled Al\u2013Cu alloy","volume":"172","author":"Safyari","year":"2020","journal-title":"Vacuum"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"1172","DOI":"10.1134\/S1990793122060070","article-title":"Influence of a Sealing Layer on the Physical and Mechanical Properties of the Shell of a High-Pressure Vessel for Hydrogen Storage","volume":"16","author":"Kuperman","year":"2022","journal-title":"Russ. J. Phys. Chem. B"}],"container-title":["Materials"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1996-1944\/16\/4\/1708\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T18:35:49Z","timestamp":1760121349000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1996-1944\/16\/4\/1708"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,2,17]]},"references-count":46,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2023,2]]}},"alternative-id":["ma16041708"],"URL":"https:\/\/doi.org\/10.3390\/ma16041708","relation":{},"ISSN":["1996-1944"],"issn-type":[{"value":"1996-1944","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,2,17]]}}}