Issue 54
T. I. J. Brito et alii, Frattura ed Integrità Strutturale, 54 (2020) 1-20; DOI: 10.3221/IGF-ESIS.54.01
being [ F 0 ] b the elastic flexibility matrix, expressed as follows [38]:
2
2
2
b U M M M N b U
U
b
2 M
i
j
i
i
i
2
2
2
U
U
U
0 F
b
b
b
2
b
M M
M N
M
i
j
j
i
j
2
2
2
b U M N M N N b U U
b
2
i
i
j
i
i
2
4 sin 3 2 cos
b
3sin cos
b
R U
2
sin cos
b
b
1 2
1 2
b
b
b
b
b
b
b
2
2
2
b
b
M
EI
R AE
sin
sin
i
b
b
b
sin cos
b
2 U M M b
R
2sin
sin cos
b
b
1 2
1 2
b
b
b
2
2
b
b
EI
R AE
sin
sin
i
j
b
2
2
5sin 5 cos
b
b
b
3 2 cos
b
R U
cos
2
1 2
b
b
b
b
b
b
2
M N
b
EI
sin
i
i
b
sin sin cos
b
b
b
cos
1 2
b
b
2
b
AE
sin
b b R U
sin cos
b
2
in cos
b
b
s
1 2
1 2
b
b j
b
2
2
2
b
b
M
EI
R AE
sin
sin
b
2
sin R U
b
b
b
sin cos
b
2
cos
1 2
b
b
b
b
2
M N
b
sin sin sin cos EI
j
i
b
b
b
b
cos
1 2
b
b
2
b
AE
sin
(16)
3 R U
2
b
b
b
b
b
2 3sin cos
b
3sin
cos
cos
2
b
b
b
b
2
2
b b
N
sin sin sin cos EI
i
b R
b
b
b
c
os
b
b
2
b
AE
sin
Deformation equivalence hypothesis For the sake of simplicity, consider initially a uniaxial problem where a damaged straight bar is axially loaded. By definition, the damage variable ω represents the micro-cracks density [11]. Then, such bar resists to the axial force by means of the effective stress i.e. the Cauchy stress σ divided by (1 – ω ). The following expression, named strain equivalence hypothesis [11], is given by substituting the effective stress in the Hooke’s law: 1 p E (17) where E is the Young’s modulus, ε is the total strain and ε p is the plastic strain. The previous relation can be rewritten as follows:
p
p e d
p
(18)
E E
E
1
1
6
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