PSI - Issue 54

Victor Rizov et al. / Procedia Structural Integrity 54 (2024) 468–474 Victor Rizov/ Structural Integrity Procedia 00 (2019) 000 – 000

469

2

1. Introduction Functionally graded structural materials with continuously varying properties along coordinates are widely used in different areas of the present-day engineering mainly because these materials can satisfy a variety of functionalities (Gururaja Udupa et al. (2014), Hirai and Chen (1999), Kieback et al. (2003), Reichardt et al. (2020), Saidi, and Sahla (2019), Yan et al. (2020)). In fact, the functionally graded materials are advanced functional composites with graded distribution of properties in a selected direction (Ganapathi (2007), Kou et al. (2012), Kyung-Su Na and Ji-Hwan Kim (2004), Najafizadeh and Eslami, (2002), Saiyathibrahim et al. (2016), Shrikantha and Gangadharan (2014)). Many problems are associated with design of modern continuously inhomogeneous functionally graded structural systems. One of these problems is the fracture behaviour. Knowledge of fracture mechanics is of paramount importance for practicing engineers since it can aid significantly in structural design and application of these advanced materials. Functionally graded materials can be set-up by layers (this is a commonly used technologies for producing of these materials) which makes them rather vulnerable of appearance of multiple longitudinal cracks between layers (Mahamood and Akinlabi (2017)). Therefore, in many cases the collapse and failure of various functionally graded engineering structures and facilities is due to longitudinal fracture. The load carrying capacity and reliability of structures in such cases is largely influenced by longitudinal fracture behaviour of functionally graded materials. Design of safe and reliable functionally graded structures demands knowledge of the basic mechanisms of longitudinal fracture. A full understanding of longitudinal fracture behaviour requires a thorough analysis of various fracture problems. The effect of different factors and parameters on longitudinal fracture has to be carefully studied and assessed. The aim of this work is to perform a longitudinal fracture analysis by using a non-linear rheological model structured by springs, dashpots and a frictional slider. The beam is with an arbitrary number of longitudinal vertical cracks and has a visoplastic time-dependent behaviour in contrast to previous publications which are devoted to longitudinal fracture analyses of continuously inhomogeneous viscoelastic beam structures with one crack only (Rizov (2020), Rizov (2020), Rizov (2022)). Solutions of the time-dependent strain energy release rate (SERR) for each crack in the beam are found here. These solutions take into account viscoplastic behaviour of the material and the progressive variation of properties in the functionally graded beam structure. The energy balance is examined for control of the solutions derived. Ascendency of inhomogeneity and time over the SERR are investigated by using the solutions. 2. Theoretical model A functionally graded cantilever beam structure with an arbitrary number of longitudinal vertical cracks is depicted in Fig. 1. The length of the i -th crack is denoted by i a . The beam width, thickness and length are b , h and l . The cross-section is rectangle. The i -crack arm width is i b . The bending moment, M , varies with time, t , as M v t M  where M v is the speed (Fig. 1). The beam viscoplastic behaviour is analyzed by making use of the rheological model depicted in Fig. 2. The model is structured by two units, ( ) I and ( ) II . The frictional slider with yield strength, S  , simulates the plastic strains. Stress, vt   , is applied on the model ( v is the speed of the stress). At S    , only unit ( ) I deforms and the stress-strain-time constitutive relationship is written as   t t e v t          3 2 2 1 1 1     , (1)

  

 

  

 

1 2  

1 2  

  

2 3 1

2



1

1

where

     2 1 1  E

   ,

1 2  

1 

(2)