Issue 53

V. Rizov et alii, Frattura ed Integrità Strutturale, 53 (2020) 38-50; DOI: 10.3221/IGF-ESIS.53.04

* U M







 , (12)

where * U is the complementary strain energy in the beam. Since the upper crack arm is free of stresses, the complementary strain energy is written as

*

*

*

1 2 U U U   , (13)

* 1 U and * 2 U are, respectively, the complementary strain energies cumulated in the lower crack arm and in the un-

where

cracked beam portion, 3 a x l   . The complementary strain energy in the lower crack arm is expressed as

1 h b

a

2 2

1 2 2 b h 0      

*

*

U

01 3 1 1 u dx dy dz

, (14)

1

where * 01 u is the strain energy density, 1 y and 1 z are the centroidal axes of the cross-section of the lower crack arm (Fig. 2). The strain energy density is written as [16]

* 01

u u    , (15)

01

where  is the normal stress,  is the strain,

01 u is the strain energy density in the lower crack arm. The mechanical

behavior of the material is treated by the following stress-strain relation [21]:

m E H      , (16) where H and m are material properties which describe the material non-linearity. The strain energy density is obtained by integrating of (16)

2 m E H   

1

 

u

 . (17)

01

m

2

1

By substituting of (16) and (17) in (15), one drives

2 1 m E mH m    2

1

* 01

 

u

 . (18)

By using (4), the distribution of the modulus of elasticity in the lower crack arm is written as

g

b y



      

f

z h

  

  

1

2







1

1    1

T L E E E  

E E

E

, (19)

  

  

S

L

L

b

2 h h

where

b b y    , (20)

1

2

2

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