PSI - Issue 28
Wei Lu et al. / Procedia Structural Integrity 28 (2020) 1559–1571 Author name / Structural Integrity Procedia 00 (2019) 000–000
1564 6
2
2 2 r
2
H
0 0
0
(13)
2 l l r l
3 l l dl
2
cos
q x x
dV
ldld
which only depends on r . By considering Eq. (13), the factors associated with ij in Eq. (12) can be calculated as
l
2
2
1
3
q
4 1
2 2 r
0 0
0
0
(14)
3 l l dl
2
cos 4
r ldld
d
2
2
8
l
l
2
2
1
3
q
4 2
2 2 r
0 0
0
0
(15)
3 l l dl
2
sin 4
r ldld
d
2
2
8
l
l
2
2
1
8 q
2 2
2 2 r
0 0
0
0
(16)
3 l l dl
2
sin 2 cos 2
r ldld
d
1
2
2
2
l
Substituting Eqs. (14)-(16) in Eq. (12), the second term of the PD strain energy density expression can be rewritten as
2
q
q
e e
* 2
(17)
ij ij
2
2 8
8
, 1,2
i j
Thus, the PD strain energy density expression can be rewritten as
2
2 16 q
2 2
u r
r u r
q
8
* 2 * r
(18)
W
ij ij
PD
, 1,2
i j
Equating the PD strain energy density with the strain energy density from classical continuum mechanics, the undetermined parameters can be calculated as
8 q , 3 ,
,
(19)
After obtaining the PD strain energy density expression, we can calculate the scalar force state vector from the derivative of the peridynamic strain energy density PD W . However, PD W here is not only related to bond elongation, but also depends on the radial displacement r u . Therefore, the increment of the PD energy is divided into two parts. One part is influenced by the bond elongation, e PD W e . The Frechet derivative of PD strain energy density,
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