PSI - Issue 79
Victor Rizov et al. / Procedia Structural Integrity 79 (2026) 109–116
113
1 2 0 x l l + .
(19)
1 L , 1 L R and 1 L r are the values of , R and r in point, 1 L , of the structure, 4 L ,
In Eqs. (16), (17) and (18),
4 L R and 4 L r are the values of , R and r in point, 4 L , respectively, , R and r are parameters. The viscoelastc model in Fig. 2 is subjected to stress, , written in Eq. (20). t t e e 2 = + , (20) where and are parameters. Since (refer to Fig. 2) = , (21) Eq. (20) is transformed as t t e e 2 = + . (22) Having in mind that = R , (23) we transform Eq. (20) as t t r R e e R 2 = + . (24) The strain, r , is extracted from Eq. (24). The result is
1
1
r
t 2 )
t e e +
(
=
.
(25)
R
R
The strain, , in the viscoelastic model is found by Eq. (26). R = + .
(26)
Equation (27) is obtained by performing mathematical manipulations in Eqs. (22), (25) and (26).
1
1
r
t 2 )
2
t e e +
(
t
t
e
e
=
+
+
.
(27)
2
R
By taking into account Eq. (20), we transform Eq. (27) as
1
1
r
2
t
t
e
e
=
+
+
.
(28)
2
R
Equation (28) is used as a stress-strain-time relation for treating the non-linear viscoelastic behavior of the rotating structure in Fig. 1. The longitudinal crack in portion, 1 2 LL , of the horizontal member of the structure in Fig. 1 has length, a . The SERR, G , is derived by Eq. (29).
bda dU *
G
=
,
(29)
where * U is the complementary strain energy in the structure, b is the width of the structural member. The complementary strain energy is found by integrating of the complementary strain energy density in the volume of the structure. The complementary strain energy density in an arbitrary point of the structure is derived by Eq. (30). = u d * 0 , (30) where is expressed as a function of via Eq. (28). The rotating structure is subjected to bending around both centric axes, y and z , by the distributed inertia
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