A fourth-order phase-field model of crack initiation and propagation under thermomechanical loading solved with strong-form meshless method

Ali, Izaz (author), Vuga, Gašper (author), Mavrič, Boštjan (author), Šarler, Božidar (author)

This study aims to develop a combination of the fourth-order phase-field method (PFM) and a strong-form meshless solution procedure for crack initiation and propagation under thermomechanical loading in two-dimensional brittle elastic material. The formulation involves thermal, mechanical and fourth-order phase-field equations. A spectral-split method decomposes the elastic strain tensor for physically correct crack propagation under mixed-mode thermomechanical loading conditions. The three coupled equations are solved in a staggered form. The domain discretization is based on regular or scattered nodes, overlapping subdomain division of the domain, and interpolation of the involved subdomain fields using third-order polyharmonic splines combined with second-order polynomial augmentation. The three numerical benchmarks include the pure thermal case, single-edge crack propagation under combined thermomechanical loading and quenching. The suitability and robustness of the proposed meshless approach are demonstrated on both regular and scattered node arrangements, and the corresponding second-order h-convergence of the displacement fields was shown for the subdomains with 13 nodes. The scattered node arrangement more effectively captures crack propagation under complex loading conditions. The originality of this study is twofold. It represents a pioneering approach using the fourth-order PFM to the thermomechanical brittle elastic system and the first solution to the related physical situation by a meshless method. The proposed method possesses a strong performance in terms of accuracy. The required dense node arrangement around the crack influences its computational efficiency. Further study will include an adaptive node arrangement scheme to enhance computational efficiency, enabling the extension to three dimensions.

• Language: English
• Keywords: thermomechanical cracks, brittle elastic material, fourth-order phase-field, strong-form meshless method, radial basis function interpolation
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FS - Faculty of Mechanical Engineering
• Publication status: Published
• Publication version: Version of Record
• Year: 2025
• Number of pages: 16 str.
• Numbering: Vol. 139, pt. B, art. 105078
• PID: 20.500.12556/RUL-177898
• UDC: 621.7.019.1
• ISSN on article: 1872-7638
• DOI: 10.1016/j.tafmec.2025.105078
• COBISS.SI-ID: 242471171
• Publication date in RUL: 12.01.2026

Record is a part of a journal

• Title: Theoretical and applied fracture mechanics
• Publisher: Elsevier
• ISSN: 1872-7638
• COBISS.SI-ID: 175278083

Licences

• License: CC BY-NC-ND 4.0, Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International
• Link: http://creativecommons.org/licenses/by-nc-nd/4.0/
• Description: The most restrictive Creative Commons license. This only allows people to download and share the work for no commercial gain and for no other purposes.