PSI - Issue 77

Available online at www.sciencedirect.com Structural Integrity Procedia 00 (2026) 000–000 Available online at www.sciencedirect.com ScienceDirect Structural Integrity Procedia 00 (2026) 000–000 ScienceDirect

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Procedia Structural Integrity 77 (2026) 397–404

© 2026 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of ICSI organizers This paper treats the problem of longitudinal fracture in continuously inhomogeneous beam structures at increased temperature. The beams are built in the ends, i.e. they are statically indeterminate structures. Besides, the beams have non-linear elastic behaviour. Due to material inhomogeneity the material properties, including the coefficient of thermal expansion, are distributed continuously along the beam length. The integral J is applied in the analysis. Equations for resolving of statically indeterminacy at increased temperature are composed and solved. The field of stresses and strains are obtained and used for solving the integral J . The strain energy release rate is determined to check-up the integral J . In this relation, the complementary strain energy cumulated in the beams at an increased temperature is obtained and used for deriving the solution of the strain energy release rate. The combined effect of distribution of material properties along the beam length and the increased temperature on the longitudinal fracture is explored. The results obtained are reported in the form of various graphs illustrating how the value of the integral J changes when the parameters of the model of the beam with a longitudial crack are varied. Special attention is paid to exploring the effect of the ratios of the values of material properties in the end sections of the inhomogeneous beam structure on the longitudinal fracture. The exploration indicates that the value of the integral J grows when the ratio of the values of the two parameters of the non-linear constitutive law in the right and left ends of the beam structure grows. The growth of the ratio of the values of the coefficient of thermal expansion in the right and left ends of the beam leads also to increase of the J integral value . These findings clearly indicate that neglecting the influence of the increased temperature on the longitudinal fracture may have rather negative effects on the safety and reliability of the inhomogeneous beam structures under consideration in this paper. © 2026 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of ICSI organizers International Conference on Structural Integrity Longitudinal fracture study of inhomogeneous beams built in ends subjected to increased temperature Victor Rizov* Department of Technical Mechanics, University of Architecture, Civil Engineering and Geodesy, 1 Chr. Smirnensky blvd., 1046 – Sofia, Bulgaria Abstract This paper treats the problem of longitudinal fracture in continuously inhomogeneous beam structures at increased temperature. The beams are built in the ends, i.e. they are statically indeterminate structures. Besides, the beams have non-linear elastic behaviour. Due to material inhomogeneity the material properties, including the coefficient of thermal expansion, are distributed continuously along the beam length. The integral J is applied in the analysis. Equations for resolving of statically indeterminacy at increased temperature are composed and solved. The field of stresses and strains are obtained and used for solving the integral J . The strain energy release rate is determined to check-up the integral J . In this relation, the complementary strain energy cumulated in the beams at an increased temperature is obtained and used for deriving the solution of the strain energy release rate. The combined effect of distribution of material properties along the beam length and the increased temperature on the longitudinal fracture is explored. The results obtained are reported in the form of various graphs illustrating how the value of the integral J changes when the parameters of the model of the beam with a longitudial crack are varied. Special attention is paid to exploring the effect of the ratios of the values of material properties in the end sections of the inhomogeneous beam structure on the longitudinal fracture. The exploration indicates that the value of the integral J grows when the ratio of the values of the two parameters of the non-linear constitutive law in the right and left ends of the beam structure grows. The growth of the ratio of the values of the coefficient of thermal expansion in the right and left ends of the beam leads also to increase of the J integral value . These findings clearly indicate that neglecting the influence of the increased temperature on the longitudinal fracture may have rather negative effects on the safety and reliability of the inhomogeneous beam structures under consideration in this paper. © 2026 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of ICSI organizers International Conference on Structural Integrity Longitudinal fracture study of inhomogeneous beams built in ends subjected to increased temperature Victor Rizov* Department of Technical Mechanics, University of Architecture, Civil Engineering and Geodesy, 1 Chr. Smirnensky blvd., 1046 – Sofia, Bulgaria Abstract

* Corresponding author. Tel.: + (359-2) 963 52 45 / 664; fax: + (359-2) 86 56 863. E-mail address: v_rizov_fhe@uacg.bg * Corresponding author. Tel.: + (359-2) 963 52 45 / 664; fax: + (359-2) 86 56 863. E-mail address: v_rizov_fhe@uacg.bg

2452-3216 © 2026 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of ICSI organizers 2452-3216 © 2026 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of ICSI organizers

2452-3216 © 2026 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of ICSI organizers 10.1016/j.prostr.2026.01.051