PSI - Issue 33

Victor Rizov et al. / Procedia Structural Integrity 33 (2021) 416–427 Author name / Structural Integrity Procedia 00 (2019) 000–000

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2

structural materials such as, for instance, continuously inhomogeneous materials which possess properties that vary gradually along given directions within the material. The closest analogies of the continuously inhomogeneous structural materials are the functionally graded materials (By Xiao Kuang et al. (2019), Chikh (2019), Hirai and Chen (1999), Kou et al. (2012), Levashov et al. (2002), Mahamood, and Akinlabi, (2017), Marae Djouda et al. (2019), Mehrali et al. (2013)). In fact, the functionally graded materials are continuously inhomogeneous composites made of two or more constituent materials. A smooth variation of microstructure and mechanical properties of the functionally graded materials can be obtained by gradually changing the composition of each constituent in spatial directions during manufacturing (Nagaral et al. (2019), Saidi and Sahla (2019), Saiyathibrahim et al. (2016), Shrikantha and Gangadharan (2014), Zhou and Yan (2002), Zuiker (1994)). In this manner, a predetermined composition profile can be realized so as to satisfy non-uniform service requirements. Avoiding the mismatch of material properties and enhancing the bounding strength are among the important advantages of functionally graded materials in comparison with the laminated composites. Due to their superior properties, in the recent decades, the continuously inhomogeneous (functionally graded) materials have been widely applied in various structures and mechanisms in aeronautics, biomedicine, automotive industry, nuclear reactors and others. The integrity of continuously inhomogeneous structural members and components is closely related to their fracture behaviour. A specific problem of great practical importance is the longitudinal fracture of continuously inhomogeneous beam structures (this is due to the fact that certain kinds of inhomogeneous materials, such as functionally graded ones can be built-up layer by layer which creates conditions for appearance of longitudinal cracks between layers). The goal of the present paper is to develop a longitudinal fracture analysis of a continuously inhomogeneous viscoelastic beam with considering the recovery upon unloading. It should be mentioned that in previous papers (Rizov (2017), Ruzov (2018), Rizov (2019), Rizov and Altenbach (2020), Rizov (2020)) the recovery upon unloading and its effect on longitudinal fracture has not been analyzed. In the present paper, the time-dependent longitudinal fracture behaviour in phases of loading and unloading is studied analytically in terms of the time dependent strain energy release rate. 2. Analysis of the strain energy release rate in an inhomogeneous beam under creep An inhomogeneous beam stricture exhibiting creep behaviour is shown schematically in Fig. 1. The beam is subjected to four-point bending by two vertical forces, , applied at the end sections as shown in Fig. 1. The cross section of the beam is a rectangle of width, , and thickness, . The length of the beam is . A longitudinal crack of length, , is located symmetrically with respect the mid-span. It should be noted that . The thicknesses of the lower and upper crack arms are denoted by and , respectively. Obviously, the upper crack arm is free of stresses. The linear viscoelastic model shown in Fig. 2 is applied for analyzing the creep behaviour of the beam. The modulii of elasticity of the two springs are denoted by and . The coefficient of viscosity of the dashpot is . The strain-time relationship for the linear viscoelastic model shown in Fig. 2 at a constant applied stress, , is written as (Rabotnov (1996)) F b h ) 2( 1 l l  a 2 a l  1 h 2 h 1 E 2 E  0 

 

  

   



 



 t e 

1

1







1 0

,

(1)

E



t







where is the strain, is time. The quantities,

and , are expressed as

1 E E 





2

,

(2)

