PSI - Issue 72

Victor Rizov / Procedia Structural Integrity 72 (2025) 128–134

132

.

(24)

J

st dn J J

  min

Hereafter, the strain energy release rate (SERR) under harmonic load is deduced from the energy balance for check-up of J . Equation (25) gives the SERR, G st , induced by static load, P .   G GP st 2  (25)

The energy balance results in

b a b P    1 

a U



 

,

(26)

G P





where b is the beam width, U is the strain energy, a is the crack length. Equation (27) can be used to estimate U .

l

      l

      ,

a



z i

z i

 i n

 i n

 1 1

 1 1

2

2

1

1

  

  

 U b 2

i 0 1 1 u dz dx

ai u dz dx 0 1 1

(27)



1

1

i

i





l

0

z i 1

z i 1

a



2

where specific energies are determined by Eqs. (28) and (29).

2 1

,

(28)

i u

0 

  i

2 1

,

(29) (30) (31) (32)

ai u

0 

ai a  

 i i E  ,



,

i a E 

ai  

1 1,2,..., n i  ,

where σ i and ε are in part, H 3 H 4 , of the structural component, while σ i and ε are in part, H 4 H 5 . The SERR induced by load, F q , is   q Fq G GF 2  ,

(33)

where G ( F q ) is estimated by Eqs. (26) and (27) after replacement of P with F q . The SERR induced by harmonic load is

(34)

dn Fq dn G k G  .

Equations (35) and (36) can be applied for estimation of maximum and minimum of the total SERR, respectively.

(35)

st dn G G G   max ,

(36)

st dn G G G   min .

The extremals of the total SERR induced by harmonic load confirms the maximum and minimum of the total J .