PSI - Issue 68

ScienceDirect Structural Integrity Procedia 00 (2025) 000–000 Structural Integrity Procedia 00 (2025) 000–000 Available online at www.sciencedirect.com Available online at www.sciencedirect.com ScienceDirect Available online at www.sciencedirect.com ScienceDirect

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Procedia Structural Integrity 68 (2025) 139–145

European Conference on Fracture 2024 Analysis of L-shaped functionally graded viscoelastic shafts with longitudinal crack under torsion Victor Rizov* Department of Technical Mechanics, University of Architecture, Civil Engineering and Geodesy, 1 Chr. Smirnensky blvd., 1046 – Sofia, Bulgaria Abstract The current analysis is concerned with longitudinal fracture in an L-shaped functionally graded shaft of circular cross-section. The shat is located in the vertical plane. The right-hand end of the horizontal member of the shaft is clamped. The vertical member of the shaft is supported also by a rotational spring located in the horizontal plane. The shaft is made of a material that is functionally graded along the radius of the cross-section. The material has non-linear viscoelastic behaviour. The upper end of the vertical member of the shaft is under an angle of twist that changes continuously with time. Thus, the vertical member is loaded in pure torsion, while the horizontal member of the shat is loaded in pure bending around the vertical axis. There is a longitudinal crack in the vertical member of the shaft. For this crack, the strain energy release rate is derived. The time-dependent energy balance is analyzed for verifying the solution of the strain energy release rate. The influences of various factors (L-shaped shaft geometry, parameters of loading, variation of material properties along the cross-section radius, rigidity of the spring support, etc.) are studied. © 2025 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 ECF24 organizers Keywords: L-shaped shaft; longitudinal crack; viscoelastic behavior; functionally graded structure; spring 1. Introduction Recently, functionally graded structural materials have been widely used in engineering (Chikh, 2019; El-Galy et al., 2019; Kou et al., 2012; Rizov and Altenbach, 2020; Rizov, 2020). In general, these materials represent high performance composites with continuous inhomogeneity (Saidi and Sahla, 2019; Saiyathibrahim et al., 2016; Rizov, 2017). Functionally graded materials have shown in a convincing manner their exceptional advantages over the European Conference on Fracture 2024 Analysis of L-shaped functionally graded viscoelastic shafts with longitudinal crack under torsion Victor Rizov* Department of Technical Mechanics, University of Architecture, Civil Engineering and Geodesy, 1 Chr. Smirnensky blvd., 1046 – Sofia, Bulgaria Abstract The current analysis is concerned with longitudinal fracture in an L-shaped functionally graded shaft of circular cross-section. The shat is located in the vertical plane. The right-hand end of the horizontal member of the shaft is clamped. The vertical member of the shaft is supported also by a rotational spring located in the horizontal plane. The shaft is made of a material that is functionally graded along the radius of the cross-section. The material has non-linear viscoelastic behaviour. The upper end of the vertical member of the shaft is under an angle of twist that changes continuously with time. Thus, the vertical member is loaded in pure torsion, while the horizontal member of the shat is loaded in pure bending around the vertical axis. There is a longitudinal crack in the vertical member of the shaft. For this crack, the strain energy release rate is derived. The time-dependent energy balance is analyzed for verifying the solution of the strain energy release rate. The influences of various factors (L-shaped shaft geometry, parameters of loading, variation of material properties along the cross-section radius, rigidity of the spring support, etc.) are studied. © 2025 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 ECF24 organizers Keywords: L-shaped shaft; longitudinal crack; viscoelastic behavior; functionally graded structure; spring 1. Introduction Recently, functionally graded structural materials have been widely used in engineering (Chikh, 2019; El-Galy et al., 2019; Kou et al., 2012; Rizov and Altenbach, 2020; Rizov, 2020). In general, these materials represent high performance composites with continuous inhomogeneity (Saidi and Sahla, 2019; Saiyathibrahim et al., 2016; Rizov, 2017). Functionally graded materials have shown in a convincing manner their exceptional advantages over the © 2025 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 ECF24 organizers

* Corresponding author. Tel.: +399-2-063-52-45/664; fax: +359-86-56-863. E-mail address: v_rizov_fhe@uacg.bg * Corresponding author. Tel.: +399-2-063-52-45/664; fax: +359-86-56-863. E-mail address: v_rizov_fhe@uacg.bg

2452-3216 © 2025 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 ECF24 organizers 2452-3216 © 2025 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 ECF24 organizers

2452-3216 © 2025 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 ECF24 organizers 10.1016/j.prostr.2025.06.034