{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,27]],"date-time":"2026-09-27T14:05:39Z","timestamp":1790517939340,"version":"4.1.0"},"reference-count":62,"publisher":"MDPI AG","issue":"19","license":[{"start":{"date-parts":[[2022,9,28]],"date-time":"2022-09-28T00:00:00Z","timestamp":1664323200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Materials"],"abstract":"<jats:p>We are witnessing a tremendous transition towards a society powered by net-zero carbon emission energy, with a corresponding escalating reliance on functional materials (FM). In recent years, the application of FM in multiphysics environments has brought new challenges to the mechanics and materials research communities. The underlying mechanism in FM, which governs several fundamental characteristics, is known as martensitic phase transformation (MPT). When it comes to the application of FM in the multiphysics context, a thorough understanding of the interplay between MPT and fracture plays a crucial role in FM design and application. In the present work, a coupled problem of crack nucleation and propagation and multivariant stress-induced MPT in elastic materials is presented using a finite element method based on Khachaturyan\u2019s microelasticity theory. The problem is established based on a phase-field (PF) approach, which includes the Ginzburg\u2013Landau equations with advanced thermodynamic potential and the variational formulation of Griffith\u2019s theory. Therefore, the model consists of a coupled system of the Ginzburg\u2013Landau equations and the static elasticity equation, and it characterizes evolution of distributions of austenite and two martensitic variants as well as crack growth in terms of corresponding order parameters. The numerical results show that crack growth does not begin until MPT has grown almost completely through the microstructure. Subsequent to the initial formation of the martensite variants, the initial crack propagates in such a way that its path mainly depends on the feature of martensite variant formations, the orientation and direction upon which the martensite plates are aligned, and the stress concentration between martensite plates. In addition, crack propagation behavior and martensite variant evaluations for different lattice orientation angles are presented and discussed in-detail.<\/jats:p>","DOI":"10.3390\/ma15196744","type":"journal-article","created":{"date-parts":[[2022,9,28]],"date-time":"2022-09-28T22:53:19Z","timestamp":1664405599000},"page":"6744","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Interplay of Fracture and Martensite Transformation in Microstructures: A Coupled Problem"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4605-7489","authenticated-orcid":false,"given":"Ehsan Borzabadi","family":"Farahani","sequence":"first","affiliation":[{"name":"Department of Wind Energy, Technical University of Denmark, Frederiksborgvej 399, 4000 Roskilde, Denmark"},{"name":"Institute of Mechanics, Faculty of Mechanical Engineering, Otto von Guericke University Magdeburg, Universit\u00e4tsplatz 2, 39106 Magdeburg, Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3580-9577","authenticated-orcid":false,"given":"Behnam Sobhani","family":"Aragh","sequence":"additional","affiliation":[{"name":"School of Computing, Engineering and Digital Technologies, Teesside University, Tees Valley, Middlesbrough TS1 3BX, UK"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8997-3818","authenticated-orcid":false,"given":"Daniel","family":"Juhre","sequence":"additional","affiliation":[{"name":"Institute of Mechanics, Faculty of Mechanical Engineering, Otto von Guericke University Magdeburg, Universit\u00e4tsplatz 2, 39106 Magdeburg, Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2022,9,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Chung, D.D. (2021). Functional Materials: Electrical, Dielectric, Electromagnetic, Optical and Magnetic Applications, World Scientific.","DOI":"10.1142\/12331"},{"key":"ref_2","first-page":"99","article-title":"Observed climate variability and change","volume":"2001","author":"Folland","year":"2001","journal-title":"Clim. Chang."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Sluiter, M. (2012). First principles in modelling phase transformations in steels. Phase Transformations in Steels, Elsevier.","DOI":"10.1533\/9780857096111.3.365"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Pereloma, E., and Edmonds, D.V. (2012). Phase Transformations in Steels: Diffusionless Transformations, High Strength Steels, Modelling and Advanced Analytical Techniques, Elsevier.","DOI":"10.1533\/9780857096104"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1146\/annurev.matsci.32.112001.132041","article-title":"Phase-field models for microstructure evolution","volume":"32","author":"Chen","year":"2002","journal-title":"Annu. Rev. Mater. Res."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"104631","DOI":"10.1016\/j.ijpvp.2022.104631","article-title":"J estimation based on regression machine learning applied to circumferential surface clad pipes with V groove weld","volume":"197","author":"Armandei","year":"2022","journal-title":"Int. J. Press. Vessel. Pip."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"3805","DOI":"10.1016\/j.jmrt.2020.02.007","article-title":"Phase-field simulation of peritectic steels solidification with transformation-induced elastic effect","volume":"9","author":"Alves","year":"2020","journal-title":"J. Mater. Res. Technol."},{"key":"ref_8","first-page":"2352","article-title":"Three-dimensional finite element modelling of welding residual stresses of medium carbon steel pipes with consideration of solid-state austenite-martensite transformation and post-weld heat treatment","volume":"233","author":"Mansur","year":"2019","journal-title":"Proc. Inst. Mech. Eng."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1146\/annurev.matsci.32.101901.155803","article-title":"Phase-field simulation of solidification","volume":"32","author":"Boettinger","year":"2002","journal-title":"Annu. Rev. Mater. Res."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1016\/j.jcp.2019.03.017","article-title":"Efficient numerical scheme for a dendritic solidification phase field model with melt convection","volume":"388","author":"Chen","year":"2019","journal-title":"J. Comput. Phys."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"344","DOI":"10.1016\/j.actamat.2017.08.046","article-title":"First-principles\/Phase-field modeling of \u03b8 precipitation in Al-Cu alloys","volume":"140","author":"Kim","year":"2017","journal-title":"Acta Mater."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1196","DOI":"10.1016\/j.compscitech.2012.04.003","article-title":"Oxidation mechanism of ZrB2\/SiC ceramics based on phase-field model","volume":"72","author":"Ma","year":"2012","journal-title":"Compos. Sci. Technol."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"801","DOI":"10.3390\/met10060801","article-title":"Oxidation Induced Stresses in High-Temperature Oxidation of Steel: A Multiphase Field Study","volume":"10","author":"Toghraee","year":"2020","journal-title":"Metals"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"209","DOI":"10.1016\/j.commatsci.2017.08.043","article-title":"Phase-field study on the formation of first-neighbour topological clusters during the isotropic grain growth","volume":"140","author":"Perumal","year":"2017","journal-title":"Comput. Mater. Sci."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"109926","DOI":"10.1016\/j.commatsci.2020.109926","article-title":"Phase-field simulation of abnormal anisotropic grain growth in polycrystalline ceramic fibers","volume":"185","author":"Kundin","year":"2020","journal-title":"Comput. Mater. Sci."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"258","DOI":"10.1063\/1.1744102","article-title":"Free energy of a nonuniform system. I. Interfacial free energy","volume":"28","author":"Cahn","year":"1958","journal-title":"J. Chem. Phys."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1085","DOI":"10.1016\/0001-6160(79)90196-2","article-title":"A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening","volume":"27","author":"Allen","year":"1979","journal-title":"Acta Metall."},{"key":"ref_18","unstructured":"Landau, L.D. (1965). Collected Papers of LD Landau, Elsevier."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"592","DOI":"10.1088\/0964-1726\/9\/5\/303","article-title":"Micromechanical modeling of the martensitic transformation induced plasticity in steels","volume":"9","author":"Cherkaoui","year":"2000","journal-title":"Smart Mater. Struct."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"134206","DOI":"10.1103\/PhysRevB.66.134206","article-title":"Three-dimensional Landau theory for multivariant stress-induced martensitic phase transformations. I. Austenite martensite","volume":"66","author":"Levitas","year":"2002","journal-title":"Phys. Rev. B"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"134207","DOI":"10.1103\/PhysRevB.66.134207","article-title":"Three-dimensional Landau theory for multivariant stress-induced martensitic phase transformations. II. Multivariant phase transformations and stress space analysis","volume":"66","author":"Levitas","year":"2002","journal-title":"Phys. Rev. B"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"759","DOI":"10.1016\/S1359-6454(96)00180-2","article-title":"Three-dimensional field model and computer modeling of martensitic transformations","volume":"45","author":"Wang","year":"1997","journal-title":"Acta Mater."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1016\/j.msea.2006.04.123","article-title":"Multi-scale phase field approach to martensitic transformations","volume":"438","author":"Wang","year":"2006","journal-title":"Mater. Sci. Eng. A"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"565","DOI":"10.1016\/j.actamat.2006.08.050","article-title":"Phase field microelasticity modeling of heterogeneous nucleation and growth in martensitic alloys","volume":"55","author":"Zhang","year":"2007","journal-title":"Acta Mater."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1538","DOI":"10.1016\/j.actamat.2011.11.039","article-title":"Three-dimensional phase-field modeling of martensitic microstructure evolution in steels","volume":"60","author":"Yeddu","year":"2012","journal-title":"Acta Mater."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"221","DOI":"10.1016\/j.msea.2012.06.080","article-title":"Three dimensional elasto-plastic phase field simulation of martensitic transformation in polycrystal","volume":"556","author":"Malik","year":"2012","journal-title":"Mater. Sci. Eng. A"},{"key":"ref_27","unstructured":"Khachaturyan, A.G. (2013). Theory of Structural Transformations in Solids, Courier Corporation."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"221910","DOI":"10.1063\/1.2138358","article-title":"Elastoplastic phase field model for microstructure evolution","volume":"87","author":"Guo","year":"2005","journal-title":"Appl. Phys. Lett."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"849","DOI":"10.1007\/s00419-012-0721-6","article-title":"A phase field approach for multivariant martensitic transformations of stable and metastable phases","volume":"83","author":"Schmitt","year":"2013","journal-title":"Arch. Appl. Mech."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"5223","DOI":"10.1016\/j.actamat.2013.05.015","article-title":"Phase field modeling of the tetragonal-to-monoclinic phase transformation in zirconia","volume":"61","author":"Mamivand","year":"2013","journal-title":"Acta Mater."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"212","DOI":"10.1016\/j.commatsci.2017.11.017","article-title":"Phase field modeling for cyclic phase transition of NiTi shape memory alloy single crystal with super-elasticity","volume":"143","author":"Xie","year":"2018","journal-title":"Comput. Mater. Sci."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1177","DOI":"10.1007\/s00466-019-01699-y","article-title":"Algorithmic aspects and finite element solutions for advanced phase field approach to martensitic phase transformation under large strains","volume":"64","author":"Babaei","year":"2019","journal-title":"Comput. Mech."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1016\/j.commatsci.2019.05.036","article-title":"A non-isothermal phase field study of the shape memory effect and pseudoelasticity of polycrystalline shape memory alloys","volume":"167","author":"Sun","year":"2019","journal-title":"Comput. Mater. Sci."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1016\/j.actamat.2020.03.025","article-title":"A phase-field model for non-isothermal phase transformation and plasticity in polycrystalline yttria-stabilized tetragonal zirconia","volume":"191","author":"Zaeem","year":"2020","journal-title":"Acta Mater."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"103627","DOI":"10.1016\/j.mechmat.2020.103627","article-title":"Nanovoid induced multivariant martensitic growth under negative pressure: Effect of misfit strain and temperature on PT threshold stress and phase evolution","volume":"151","author":"Javanbakht","year":"2020","journal-title":"Mech. Mater."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"106187","DOI":"10.1016\/j.ijmecsci.2020.106187","article-title":"On the crack onset and growth in martensitic structures; a phase-field approach","volume":"194","author":"Farahani","year":"2021","journal-title":"Int. J. Mech. Sci."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1070","DOI":"10.1007\/BF01197662","article-title":"Nucleation on the crack tip and transformation toughness in crystals undergoing structural phase transitions","volume":"27","author":"Bulbich","year":"1992","journal-title":"J. Mater. Sci."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"635","DOI":"10.1016\/0020-7683(89)90030-9","article-title":"Crack-growth resistance in transformation-toughened ceramics","volume":"25","author":"Stump","year":"1989","journal-title":"Int. J. Solids Struct."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"2896","DOI":"10.1016\/j.jmps.2008.04.002","article-title":"Fracture in single crystal NiTi","volume":"56","author":"Creuziger","year":"2008","journal-title":"J. Mech. Phys. Solids"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"383","DOI":"10.1007\/s00466-014-1109-y","article-title":"A review on phase-field models of brittle fracture and a new fast hybrid formulation","volume":"55","author":"Ambati","year":"2015","journal-title":"Comput. Mech."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"342","DOI":"10.1016\/j.jmps.2008.10.012","article-title":"Laws of crack motion and phase-field models of fracture","volume":"57","author":"Hakim","year":"2009","journal-title":"J. Mech. Phys. Solids"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"118","DOI":"10.1103\/PhysRevLett.85.118","article-title":"Continuum field description of crack propagation","volume":"85","author":"Aranson","year":"2000","journal-title":"Phys. Rev. Lett."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1080\/14786431003773015","article-title":"Phase field modeling of crack propagation","volume":"91","author":"Spatschek","year":"2011","journal-title":"Philos. Mag."},{"key":"ref_44","first-page":"849231","article-title":"Computational methods for fracture in brittle and quasi-brittle solids: State-of-the-art review and future perspectives","volume":"2013","author":"Rabczuk","year":"2013","journal-title":"Int. Sch. Res. Not."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"2765","DOI":"10.1016\/j.cma.2010.04.011","article-title":"A phase field model for rate-independent crack propagation: Robust algorithmic implementation based on operator splits","volume":"199","author":"Miehe","year":"2010","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"3625","DOI":"10.1016\/j.engfracmech.2010.08.009","article-title":"A continuum phase field model for fracture","volume":"77","author":"Kuhn","year":"2010","journal-title":"Eng. Fract. Mech."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"1209","DOI":"10.1016\/j.jmps.2009.04.011","article-title":"Regularized formulation of the variational brittle fracture with unilateral contact: Numerical experiments","volume":"57","author":"Amor","year":"2009","journal-title":"J. Mech. Phys. Solids"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"100","DOI":"10.1016\/j.cma.2014.01.016","article-title":"A higher-order phase-field model for brittle fracture: Formulation and analysis within the isogeometric analysis framework","volume":"273","author":"Borden","year":"2014","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1007\/s10659-007-9107-3","article-title":"The variational approach to fracture","volume":"91","author":"Bourdin","year":"2008","journal-title":"J. Elast."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"106648","DOI":"10.1016\/j.engfracmech.2019.106648","article-title":"A phase-field fracture model based on strain gradient elasticity","volume":"220","author":"Makvandi","year":"2019","journal-title":"Eng. Fract. Mech."},{"key":"ref_51","doi-asserted-by":"crossref","unstructured":"De Lorenzis, L., and Gerasimov, T. (2020). Numerical Implementation of Phase-Field Models of Brittle Fracture. Modeling in Engineering Using Innovative Numerical Methods for Solids and Fluids, Springer.","DOI":"10.1007\/978-3-030-37518-8_3"},{"key":"ref_52","unstructured":"Lifshitz, E.M., and Pitaevskii, L.P. (2013). Statistical Physics: Theory of the Condensed State, Elsevier."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"1319","DOI":"10.1016\/S0022-5096(98)00034-9","article-title":"Revisiting brittle fracture as an energy minimization problem","volume":"46","author":"Francfort","year":"1998","journal-title":"J. Mech. Phys. Solids"},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"447","DOI":"10.1016\/j.cma.2016.01.023","article-title":"Phase field modeling and simulation of coupled fracture and twinning in single crystals and polycrystals","volume":"312","author":"Clayton","year":"2016","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"1459","DOI":"10.1007\/s00419-014-0945-8","article-title":"A combined phase field approach for martensitic transformations and damage","volume":"85","author":"Schmitt","year":"2015","journal-title":"Arch. Appl. Mech."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"22243","DOI":"10.1039\/C9NR05960A","article-title":"Phase field approach for nanoscale interactions between crack propagation and phase transformation","volume":"11","author":"Jafarzadeh","year":"2019","journal-title":"Nanoscale"},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1016\/j.engfracmech.2016.03.035","article-title":"Study of crack propagation behavior in single crystalline tetragonal zirconia with the phase field method","volume":"159","author":"Zhao","year":"2016","journal-title":"Eng. Fract. Mech."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"803","DOI":"10.1007\/s00466-022-02209-3","article-title":"Phase-field approach to evolution and interaction of twins in single crystal magnesium","volume":"70","author":"Amirian","year":"2022","journal-title":"Comput. Mech."},{"key":"ref_59","unstructured":"Mura, T. (2013). Micromechanics of Defects in Solids, Springer Science & Business Media."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"1141","DOI":"10.1007\/s00466-014-1045-x","article-title":"Phase field approximation of dynamic brittle fracture","volume":"54","author":"Kuhn","year":"2014","journal-title":"Comput. Mech."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"208","DOI":"10.1016\/j.actamat.2013.10.031","article-title":"Phase field modeling of stress-induced tetragonal-to-monoclinic transformation in zirconia and its effect on transformation toughening","volume":"64","author":"Mamivand","year":"2014","journal-title":"Acta Mater."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"521","DOI":"10.1016\/S0921-5093(01)01072-3","article-title":"Fatigue short crack behaviour in metastable austenitic stainless steels with different grain sizes","volume":"319","author":"Stolarz","year":"2001","journal-title":"Mater. Sci. Eng. A"}],"container-title":["Materials"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1996-1944\/15\/19\/6744\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:41:23Z","timestamp":1760143283000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1996-1944\/15\/19\/6744"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,9,28]]},"references-count":62,"journal-issue":{"issue":"19","published-online":{"date-parts":[[2022,10]]}},"alternative-id":["ma15196744"],"URL":"https:\/\/doi.org\/10.3390\/ma15196744","relation":{},"ISSN":["1996-1944"],"issn-type":[{"value":"1996-1944","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,9,28]]}}}