PSI - Issue 33

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

438

11

In formulae (31) and (32), cUP E and cLW E are the values of the modulus of elasticity in compression at the upper and the lower surface of the beam, respectively. The values of the coefficient of viscosity in compression at the upper and the lower surface of the beam are denoted by cUP  and cLW  , respectively. The distributions of c E and c  along the beam thickness are controlled by the parameters, c n and c m , respectively. First, a time-dependent solution is obtained by analyzing the balance of the energy. Since the lower crack arm is loaded in tension, by using (12), the strain energy density is written as   t t E t n t D Dt u E z z e      2 1 1 2 0 1 . (34)

2 1

F u (curve 1 – for material with identical viscoelastic behaviour in

Fig. 8. The strain energy release rate in non-dimensional form plotted against

tension and compression, curve 2 – for material with different viscoelastic behaviour in tension and compression).

By using (15), the strain energy densities in the tension and compression zones of the un-cracked beam portion are expressed, respectively, as   t t E t n t U Ut u E z z e      2 2 2 2 0 2 , (35)

2 1

c c E t 

2 1







2

n c U E z z e   2 2 2

Uc u 0



.

(36)

2

In order to determine the curvatures and the neutral axes coordinates which are involved in (35) and (36), the equations for equilibrium (18) – (20) are re-written as