PSI - Issue 41

Victor Rizov et al. / Procedia Structural Integrity 41 (2022) 125–133 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

127 3

where i G  is the shear stress,  is the shear strain, i G and i b are material properties. The non-linear relation between the shear stress, i   , and the velocity of the shear strain of the i -th dashpot is written as i i g i       , (2) where i  and i g are material properties,   is the first derivative of the strain with respect to time. The model is under shear strain,  , that varies with time, t , according to the following law: v t    , (3) where  v is a parameter which governs the variation of the strain. By combining of (1), (2) and (3), one derives   i i b i G G v t    , (4) i i g i v      . (5) The shear stress,  , in the model is found as ) ( 1 i i G i n i          . (6) By substituting of (4) and (5) in (6), one obtains     i i g i b i i n i v G v t          1 . (7)

Fig. 2. Statically undetermined non-liner viscoelastic beam of circular cross-section with a lengthwise crack.

The non-linear viscoelastic behaviour of the beam depicted in Fig. 2 is modelled by applying (7). The beam has a circular cross-section with radius, 1 R and 2 R , in portions, 1 2 L L , and 2 4 L L , respectively. The beam is made of material that is continuously inhomogeneous along the radius of the cross-section. The distributions of i G and i  in radial direction are written as

2 R f R

Ci i i G G e 

,

(8)

2 R p R

Ci i i e   

,

(9)