PSI - Issue 28

Victor Rizov et al. / Procedia Structural Integrity 28 (2020) 1226–1236 Author name / Structural Integrity Procedia 00 (2019) 000–000

1228

3

l da G dU cf   2

,

(1)

where the length of the crack front, cf l , is written as 1 2 l R cf   .

(2)

Fig. 1. Geometry and loading of inhomogeneous rod with internal longitudinal crack.

In formula (1), da is an elementary increase of the crack length. It should be mentioned that the right-hand side of (1) is doubled in view of the symmetry (Fig. 1). By substituting of (2) in (1), one obtains



G dU 1  

.

(3)

R da

The complementary strain energy stored in half of the rod is obtained as

 2 1 U U U U ,       3

(4)

 1 U ,

 2 U and

 3 U are, respectively, the complementary strain energies in the external and internal crack

where

arms, and in the un-cracked part of the rod, l l a x 2    . The complementary strain energy in the external crack arm is expressed as

R

2

  R U a u RdR  ,   2 0 1

(5)

1

where  0 u is the complementary strain energy density. In principle, the complementary strain energy density is equal to the area that supplements the area enclosed by the stress-strain curve to a rectangle