PSI - Issue 26

Victor Rizov et al. / Procedia Structural Integrity 26 (2020) 63–74 Rizov / Structural Integrity Procedia 00 (2019) 000 – 000

66 4

The longitudinal displacements are obtained by the integrals of Maxwell-Mohr

s

l

s

a 

j p   = j

k q   =

1

,

(4)

a 

u

N dx N ij x + 

dx N +

dx

N

dx

=





+



F

x iCD

x iDH

ik x

i

j

p

r

rk

1

1

+

j

k

1

2

=

=

l

l

s

j

p

k

1

1

−

−

. In (4), p and q are, respectively, the numbers of axial forces applied on the external crack

i 1, 2, ..., =

F n

where

ij N and

arm and un-cracked part ( a x l   ) of the beam,

j x  are, respectively, the axial force induced by the unit

iCD N and

i F u and the longitudinal strain in the j -th portion of the external crack arm,

1 + p x 

loading for obtaining of

i F u and the longitudinal strain in portion,

are, respectively, the axial force induced by the unit loading for obtaining of

iDH N and

1 r x  are, respectively, the axial force induced by the unit loading for

CD , of the external crack arm,

i F u and the longitudinal strain in portion, DH , of the un-cracked part of the beam, ik N and

rk x  are,

obtaining of

i F u and the longitudinal strain in the k -th

respectively, the axial force induced by the unit loading for obtaining of

b F u , is derived as

portion of the un-cracked part of the beam (Fig. 1). The longitudinal displacement,

s

s

a  0

k q   = k

1

a 

,

(5)

u N =

dx N +

dx

N dx k x 





+

DH x

F

b x

b

r

rk

int

1

k

2

=

s

k

1

−

b N and

b F u and the

int x  are, respectively, the axial force induced by the unit loading for obtaining of

where

DH N is the axial force in portion, DH , induced by the unit loading for

longitudinal strain in the internal crack arm,

b F u , and k N is the axial force in k -portion of the un-cracked part of the beam induced by the unit

obtaining of

loading for obtaining of b F u . The strain energy in the beam is written as

UC EX IN U U U U = + + ,

(6)

where IN U , ET U and UC U are the strain energies in the internal and external crack arms, and in the un-cracked beam portion, respectively. By using polar coordinates, A R and  , the strain energy in the internal crack arm is expressed as

  = 3 0 2 0 0 R a



U

0 IN A u R dxdR d A

 .

(7)

IN

IN u 0 , is written as

where the strain energy density,

 = 

0 u IN

( ) d

   .

(8)

0

In (8), σ ( ε ) is the non-linear stress-strain relation (the constitutive law) used to describe the non-linear mechanical behaviour of the material.