PSI - Issue 72

Victor Rizov / Procedia Structural Integrity 72 (2025) 113–119

114

materials is that their properties vary continuously along one or more directions in a structural member (Mahamood and Akinlabi (2017), Reichardt et al. (2020), Saiyathibrahim et al. (2016), Tokova et al. (2017)). Therefore, the distribution of the properties of these materials is treated by using continuous mathematical functions of one or more coordinates. The reliable work of these materials, especially in structural engineering applications which are under action of various types of external mechanical loadings depends in a high extend on their fracture behaviour. Analysing of fracture behaviour usually is related to determining of the SERR for given crack configurations and loads (Blake (1995), Dowling (2007), Rizov (2017), Rizov and Altenbach (2020)). This fact represents a clear indication how important is to develop solutions of the SERR for various conditions (structural geometry, kind of material, crack configuration, type of loading, etc.). The problem investigated in this paper concerns theoretical analysis of the lengthwise fracture in a non-linear elastic continuously inhomogeneous bar with considering the effect of the deflection velocity. Situations such as those studied here are of practical interest. For instance, such conditions are prevalent in engineering structures under external influences varying with time. In this paper, a bar with a lengthwise crack is under axial displacements that change with time. The bar is with two-dimensional continuous material inhomogeneity in its cross-section. The SERR is derived. The effect of the deflection velocity and two-dimensional inhomogeneity on the SERR is studied. 2. Analysis The bar shown in Fig. 1 is considered. This bar is under axial displacement, u C , that changes with time, t , at a constant velocity, v uC , i.e. u v t C uC  . (1) The constitutive law (2) that includes a term with the strain velocity,   , is applied for describing the non-linear elastic mechanical behaviour of the bar (Lukash (1978)).

  

   ,

B







1

Q







(2)

0 



2

1





where σ and ε are the stress and strain, B , Q and 0   are material properties. The bar exhibits continuous two dimensional material inhomogeneity (the material properties vary continuously along the width, b , and thickness, h , of the bar). This variation is presented by formulas (3) and (4).

b

h

y  

z

2 2

m

(3)

,

b

h

UL B B e 

b

h

y  

z

2 2

n UL Q Q e 

b

h

(4)

,

2 b y b    , h    , z 2

(5)

2 h

(6)

2

where axes, y and z , are defined in Fig. 1, m and n are parameters. In view of the symmetry, we consider only part of the bar at