Issue 39
M. Shariati et alii, Frattura ed Integrità Strutturale, 39 (2017) 166-180; DOI: 10.3221/IGF-ESIS.39.17
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§ ©
· ¸ ¹
V
w
w
S
S
¨
³
T
l
l
r u S U ¨
r V
W
1, 2, 3, , }
(3-8)
S
dV F
l ¬¬¬¬¬¬¬¬¬¬
ns
l
l
rz
r
w
w
r
r
z
V e
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· ¸ ¹
w
w
S
S
¨
³
l
l
r w S U ¨
W
z V
1, 2, 3, , }
(3-9)
dV F
l ¬¬¬¬¬¬¬¬¬¬¬
ns
l
rz
z
w
w
r
z
V e
where ns is the number of shape functions of the element e and l
S is the component of the vector S .
^
` m m m m 2 3 4 ,
, , < < < <
S N N N N 1 2 3 4 1 2 3 4 1 , , , , , , , ¬,
(3-10)
For axisymmetric problems in cylindrical coordinates relations between the stresses and displacements can be expressed in the form below [21].
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r ¸ w w · w w ¸ w w · w w ¸ w w z r z r z · w w ¹ ¹
w w
u u w
u
O ¨
r V
P
2
r
r
u u w
u
V
¬ ¨ O
P
¬2
T
r
r
(3-11)
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w w
u u w
w
z V
¬ ¨ O
P
2
r
z
¹
§
u w z r · w w ¸ w w ©
P ¨
W
rz
¹
By substituting Eqs. (3.11) into Eqs. (3.8) and (3.9) we have
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· ¸ ¸ ¸ ¸ ¸ ¹
§ ¨ ¨ ©
· ¸ ¸ ¹
§ ¨ ¨ ©
· ¸ ¸ ¹
§ ¨ ©
· ¸ ¹
§ ¨ ©
· ¸ ¹
w
S
S
u u w w w
w w
u u w w w
u
u
l
l
O
P
O
P
2
2
w
r w w
r w w
r
r
z
r
r
r
z
r
¬
¬
¨
³
³
r u S dV r U
dV F
l
r
(3-12)
§ ¨ ¨ ©
· ¸ ¸ ¹
§ ¨
· ¸ ¹
w
S
w w
u w z r
V e
V e
l
P
w
w w ©
z
l
ns
1, 2, 3,...,
¬
¬
l S § § w ¨ ¨ ¨ w
· ¸ ¸ ¹
§ ¨ ¨ ©
· ¸ ¸ ¹
§ ¨ ©
· ¸ ¹
§ ¨ ©
· ¸ ¹
· ¸ ¹
w
w w w w
u u w w w
w w
S
¨
u w z r
w
³
³
l
r w S dV r U
P
O
P
1, 2, 3, , }
(3-13)
dV F l ¬¬¬¬¬¬¬
ns
¬
¬
¬ 2
¬
¨ ©
l
z
w
r w w
r
z
r
z
z
©
V e
V e
By substituting displacements (Eqs. (3-6) and (3-7)) into Eqs. (3-12) and (3-13), and some manipulations, equations are obtained which we can assemble them to a matrix form as below. > @ ^ ` > @ ^ ` ^ ` M K F ' ' (3-14) In this equation > @ M and > @ K are the mass and stiffness matrices, respectively. Also ^ ` ' and ^ ` F are the nodal displacements and force vectors, respectively. Generally, for the fictional element e which is enriched with both Heaviside and crack tip enrichment functions, these matrices and vectors can be written as follows:
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