PSI - Issue 83

Victor Rizov et al. / Procedia Structural Integrity 83 (2026) 85–94

89

gr m on the upper surface of the beam to

where m is the mass per unit area of the beam. The change of m from

dl m on the lower surface follows the law defined in Eq. (16).



 h m m dl 

2 h

  

  

gr

m m

z

 



,

(16)

1

gr

where

z h    .

2 h

(17)

1

2

Here,  is a parameter. The inertia terms, Rn q and Rs q , are shown in Fig. 1. Equations (11), (13), (14) and (15) indicate that the inertia terms do not change with time. Under these inertia terms the beam undergoes non-linear creep treated by the creep equation (3). The longitudinal crack in portion, 1 2 LL , of the beam has length, a . Our main goal is to derive the SERR, G , for the crack under non-linear creep. For achieving of this goal, we apply the differential relationship (18) between the SERR, the complementary strain energy, * U , in the moving beam and the area, A , of the longitudinal crack.

*

dU

G



.

(18)

dA

The complementary strain energy and, therefore, SERR change with time because of the creep in the beam. The complementary strain energy, * 1 U , in the lower arm of the crack is

U u dV lw V lw   ( ) * 0 * 1 .

(19)

The notations in Eq. (19) are: *

0 lw u - specific complementary strain energy, lw V - volume of the lower arm of the

crack. The specific complementary strain energy is given by     d u lw * 0 , where  is related with  in Eq. (3). The change of  along the thickness of the lower arm is   n up z z 2 2     ,

(20)

(21)

where