PSI - Issue 33

4

Victor Rizov et al. / Procedia Structural Integrity 33 (2021) 428–442 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

431

where F is the axial force at the free end of the lower crack arm, U is the time-dependent strain energy cumulated in the beam, a  is a small increase of the crack. Since the upper crack arm is free of stresses, the strain energy in the beam is written as D U U U U   , (6) where D U and U U are, respectively, the strain energies in the lower crack arm and the un-cracked beam portion, a x l   3 , where 3 x is the longitudinal cdentroidal axis of the beam (Fig. 1).

Figure 2. Linear viscoelastic model with a spring and a dashpot.

The strain energy in the lower crack arm is written as

U a u dA D A D D 0 ( )   ,

(7)

D A is the lower crack arm cross-section,

D u 0 is the time-dependent strain energy density in the same crack

where

D u 0 :

arm. The following formula is used to obtain

2 1

0  D u



.

(8)

By combing of (4) and (8), one obtains

Et

2 1

 D u E e   2 0



.

(9)

The distribution of the strains in the thickness direction is treated by applying the Bernoulli’ s hypothesis for plane sections since beams of high length to thickness ratio are analyzed in the present paper. Therefore, the distribution of  that is involved in (9) along the thickness of the lower crack arm is written as   1 1 1 n D z z     , (10)

where