PSI - Issue 28
Zhenghao Yang et al. / Procedia Structural Integrity 28 (2020) 464–471 Author name / Structural Integrity Procedia 00 (2019) 000–000
467
4
( ) k b b ( ) w k
b
(9)
( ) k
To obtain the peridynamic equation of motion, the classical strain energy density expression given in Eqs. (4) and (5) should be converted into a peridynamic form. This can be achieved by using Taylor’s expansions and the following relationships can be obtained 2 2 ( ) ( ) ( ) 2 ( ) ( )( ) 1 k k k k i k i i i k V x A (10a)
2
( ) k i
( ) k
( ) k i w w
( ) k
k
2
2
( )( ) i k
w
1
( ) k
(10b)
V
( ) k
( ) k i
2
x
A
i
k
( )( ) i k
where is the horizon size and A is the cross-sectional area. By substituting the definitions given in Eqs. (10a,b) into Eq. (5) yields the strain energy densities of the material point k and the material point inside its horizon j as
2
( ) k i
( ) k
( ) k i w w
( ) k
2
k
2
( ) k i
( )( ) i k
1 1 2
( ) k
(11a)
W
EI
V GA
V
( ) k
( ) k i
( ) k i
2 2 A
i
i
k
k
( )( ) i k
( )( ) i k
2
( ) i j
( ) j
( ) i w w j
( ) j
2
j
2
( ) i j
( )( ) i j
1 1 2
( ) j
(11b)
W
EI
j V GA
V
( ) j
j
2 2 A
( ) i
( ) i
i
i
j
j
( )( ) i j
( )( ) i j
By using Eqs. (8a,b) and (11a,b), the peridynamic equations of motion of Timoshenko beam can be obtained from Euler-Lagrange equation given in Eq. (6) as 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( )( ) ( ) ( ) ( )( ) 1 2 2 j k j k k k b j s j k j k j k j k j j j k I c V c w w sign V b A
(12a)
( ) j w w
( ) j
w c
( ) k
( ) k
(12b)
sign
( ) j V b
( ) ( ) k k
( )( ) j k
( )
s
w k
2
j
( )( ) j k
with
2
EI
(13a)
c
b
2 2 A
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