PSI - Issue 41

Victor Rizov et al. / Procedia Structural Integrity 41 (2022) 94–102 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

95

2

inhomogeneous composites called functionally graded materials has attracted the attention of both practising engineers and academic circles around the globe (Chikh (2019), Ganapathi (2007), Hao et al. (2002), Hirai and Chen (1999), Kou et al. (2012), Mahamood, and Akinlabi, (2017), Marae Djouda et al. (2019)). The microstructure and the continuous variation of material properties along one or more spatial coordinates in the functionally graded structural member can be formed technologically in order to meet particular requirements in reference to strength, weight, structural stability, fracture behaviour, dynamic response, wearing quality, reaction to temperature changes, etc. In many ways, the properties of functionally graded materials are superior in comparison to the homogeneous metal materials. Therefore, the application of functionally graded materials continues to increase in various areas of engineering (Mehrali et al. (2013), Nagaral et al. (2019), Saidi and Sahla (2019), Saiyathibrahim et al. (2016)). Delamination is one of the most often failure modes of inhomogeneous (functionally graded) beams with layered structure. Prevention and assessment of delamination fracture play a very important role in the design process of inhomogeneous structural members and components. In the context of ensuring the safety, reliability and durability of layered load-bearing engineering structures, development of analyses of various delamination crack problems is of particular interest. Detailed study and careful evaluation the effects of material inhomogeneity, loading conditions, material behaviour and geometry of the structures on the delamination are of great significance (Rizov (2017), Rizov (2018), Rizov (2019), Rizov (2020), Rizov and Altenbach (2020)). This paper aims to develop a delamination analysis of inhomogeneous cantilever beam structure of rectangular cross-section which exhibits creep behaviour under external loading consisting of a torsion moment applied at the beam free end. The beam under consideration is built-up by two longitudinal vertical cracks. The delamination is located between the layers inside the beam. The analysis developed in this paper is needed since the previous studies are focussed on delamination of inhomogeneous beams of circular cross-section loaded in torsion (Rizov (2020), Rizov (2021)). Since inhomogeneous beams of rectangular cross-section subjected to pure torsion are frequently used in engineering structures, it is important to evaluate the effects of the rectangular cross-section on the delamination. In this paper, the strain energy release rate for the delamination crack is derived with considering the creep behaviour. A solution of the strain energy release rate is obtained also by analyzing the beam compliance under creep for verification. 2. Delamination analysis with considering of creep The width, thickness and length of the inhomogeneous beam in Fig. 1 are denoted by b , h and l , respectively. The beam is clamped in section, L . The beam is made of two longitudinal vertical layers of widths, 1 b and 2 b . The loading consists of a torsion moment, T , applied at the free end of the beam. There is a delamination crack of length, a 2 , between layers as shown in Fig. 1. The beam is under creep that is treated by the linear viscoelastic mechanical model in Fig. 2. The shear modulus of two springs and the coefficients of viscosity of the two dashpots are denoted by 1 G , 2 G , 1  and 2  , respectively (Fig. 2). The model is subjected to constant shear stress,  . The constitutive law of the model in Fig. 2 is written as

2 G t

    = + + − 1 2 1 1     G t G

   

−

2 

e



,

(1)

where  is the shear strain, t is time. The two layers of the beam in Fig. 1 are continuously inhomogeneous along the beam length. Therefore, the material properties are continuous functions of the abscise, x . The distribution of material properties in layer 1 along the beam length is presented as

1  l G G − l G G − 1 1  2 2  



1 

G G

x

= +

,

(2)

1

1



2





G G

x

= +

,

(3)

2

2

2

