PSI - Issue 77

Victor Rizov et al. / Procedia Structural Integrity 77 (2026) 397–404 Author name / Structural Integrity Procedia 00 (2026) 000–000

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Keywords: Inhomogeneous beam; Longitudinal fracture; Temperature; Non-linear elastic behaviour; Load-bearing structure

1. Introduction The rapid advance of the science of continuously inhomogeneous structural materials is closely connected to the needs of the modern technique (Dowling (2007)). In this relation, completely new materials with predefined properties have been developed recently (El-Galy et al. (2019), Kou et al. (2012), Rizov and Altenbach (2020)). The necessary technologies for producing of such materials have been developed and sophisticated in the recent decades which is an important factor for increasing their use in various spheres of up-to-date engineering. One example of widely used continuously inhomogeneous structural materials is the functionally graded materials (Hirai and Chen (1999), Saiyathibrahim et al. (2016), Rizov (2017), Yan et al. (2020)). In principle, the functionally graded materials are obtained by composing of constituent materials of different properties (Bohidar et al. (2014), Mahamood and Akinlabi (2017), Reichardt et al. (2020)). The composition of the constituent materials changes smoothly in the solid. Besides, this change can be tailored in order to obtain the desired distribution of the material properties. In this way, new materials representing continuously inhomogeneous composites of excellent properties superior to these of constituent materials can be obtained. It is natural that various aspects of the behaviour (fracture, strength, etc.) of the continuously inhomogeneous materials have to be studied in detail to improve their performance under different loading conditions and influences. Currently, the requirements towards the fracture behaviour are very high in view of the fact that namely the fracture is one of the main failure modes of these materials. The structural integrity depends in a high extent on the fracture behaviour. Studying fracture behaviour is an important contribution for improving the integrity of engineering structures and facilities made by continuously inhomogeneous materials. The main goal of this paper is to study the problem of longitudinal fracture in statically indeterminate continuously inhomogeneous beam structures subjected to increased temperature. In particular, a beam structure built in both ends is considered. There is a longitudinal crack in the beam under consideration. The beam has a non linear elastic behaviour and exhibits continuous material inhomogeneity in longitudinal direction. The material properties including the coefficient of thermal expansion change continuously in length direction of the beam. The integral J is used for exploring the longitudinal fracture under increased temperature. The strain energy release rate in the beam under increased temperature is obtained for checking-up the integral J . The effects of temperature and the distribution of material properties along the beam length on the longitudinal fracture are explored through the integral J . 2. Theoretical analysis The beam structure shown in Fig. 1 hosts a longitudinal crack of length, a .

Fig. 1. Beam structure with a longitudinal crack

The crack is located in portion, 2 3 DD , of the beam. Both ends of the beam are built in. Thus, the beam is statically indeterminate. The beam is subjected to increased temperature. The temperature at the upper and lower surfaces of