Issue 47

V. Rizov, Frattura ed Integrità Strutturale, (2047) 468-481; DOI: 10.3221/IGF-ESIS.47.37

where da is an elementary increase of the delamination crack length.

Figure 2 : Cross-section of the left-hand crack arm in the beam mid-span.

B B , the complementary strain energy cumulated in the beam

Since the delamination crack is located in beam portion, 2 4

B B and 4 5

B B , does not depend on the delamination crack length (Fig. 1). Thus, it is enough to calculate the

portions, 1 2

B B , only. Since the two segments of the right-hand crack arm

complementary strain energy cumulated in beam portion, 3 4 are free of stresses, the complementary strain energy,

* U , is written as

* * L R U U U   *

(4)

* L U and * R U are the complementary strain energies cumulated in the left-hand crack arm and the un-cracked beam

where

portion, 1 2 l l a x l l      . The complementary strain energy cumulated in the left-hand crack arm is expressed as 3 1 2 2

h

ya

i n 

2

1 1 i 

L

1       1 0 i i 

*

* 0 1 1 1 u dx dy dz L

U

(5)

L

i

h y

2

n is the number of layers in the left-hand crack arm, 1 i y and 1 1 i y 

where L

are the coordinates, respectively, of the left

hand and right-hand lateral surfaces of the i -th layer, * 0 i L u

is complementary strain energy density in the same layer, the axes,

1 x , 1 z , are shown in Fig. 2. The Ramberg-Osgood stress-strain relation which is used to model the material non-linearity is written as y and 1

1

i i E H           i i

m

i



(6)

 is the distribution

where  is the distribution of the lengthwise strains in the cross-section of the left-hand crack arm, i

E is the modulus of elasticity in the same layer, i

H and i

m are

of the normal stresses in the cross-section of the i -th layer, i

471