Computational Mechanics of Composite Materials part 2 ppsx
Computational Mechanics of Composite Materials part 2 ppsx
... Preliminaries 15
()
[]
2
2
2
1
2
1
1
exp yyx +−=
(1.83)
1
2
2
2
1
y
y
arctgx ⋅=
π
(1.84)
with the Jacobian determinant of the form
()
)(exp
),(
),(
2
2
2
1
2
1
2
1
21
21
2
2
1
2
2
1
1
1
yy
yy
xx
y
x
y
x
y
x
y
x
+−−==
π
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
(1.85) ... specific time moment
42 Computational Mechanics of Composite Materials
[] [] []
()
[]
[] [] []
()
[]...
Computational Mechanics of Composite Materials Part 1 ppsx
... variance
[] []
22 42
)( XEXEXVar −=
(1.51)
and the expected value
[
]
[]
XEXVarXE
22
)( +=
(1. 52)
subtracted from the following equation
[]
[]
()
[] []
XEXEXVarXVarXEXVarXE
422
2
222
) (2) ( ++=+=
(1.53) ... ()
∫∫
−=−
−
tt
t
dxxdxx
0
22
exp2exp
(1.68)
it is obtained finally that
()()() ()()
2
2
0
22
2exp1exp4exp1 tdxxt
t
−−≤
⎥
⎦
⎤
⎢
⎣
⎡
−≤−−
∫
ππ
(1.69)
Then, a square...
Computational Mechanics of Composite Materials part 3 ppsx
... "bubbles"
Figure 2. 9. Expected values of probabilistically averaged Young modulus in matrix
0 .21
0 .21 2
0 .21 4
0 .21 6
0 .21 8
0 .22
0 .22 2
4.00E-03 6.00E-03 8.00E-03 1.00E- 02 1 .20 E- 02 1.40E- 02 1.60E- 02 1.80E- 02 2.00E- 02
4 ...
1.0 62
0. 023 2
18 0.985
0. 023 6
0.993
0. 023 4
0.994
0. 023 1
1.003
0. 023 0
27 0.895
0. 023 9
0.897
0. 023 5
0.908...
Computational Mechanics of Composite Materials part 7 ppsx
... lk
k
k
D
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−−=
µ
µ
()
()
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
−−+−
−+−+−+−
×
21
2
21
2
21
2
21
2
21 122 211
)()(
)()() (2) (2
llkk
kkkklkkl
µµ
µµ
(A3.16)
()
11
11
11 12
2
22
1 lk
k
k
D +
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+−=
µ
µ
()
()
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
−−+−
−+−+−+−
×
21
2
12
2
21
2
12
2
21 122 211
)()(
)()() (2) (2
llkk
kkkklkkl
µµ
µµ
(A3.17) ... +
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+−=
µ
µ
()
()
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡...
Computational Mechanics of Composite Materials part 14 ppsx
... Var[f(t)]
2 4 7.4/1.94 8.4/1.94 8.9 /2. 06
3 8 10.1 /2. 19 10.9 /2. 19 11.0 /2. 31
4 16 14.1 /2. 62 14.9 /2. 62 17.3 /2. 69
5 32 20.7/3 .25 25 .7/3.31 31.8/3.44
6 64 53.0/4.56 53.7/4.56 70.5/4.94
7 128 131.0/7.06 ... analysis of
damage at interfaces of composites. Comp. Part B: Engrg. 33 :21 5 -22 4.
28 . Ben-Amoz M, (1976) A dynamic theory for composite materials. J. Ap...
Computational Mechanics of Composite Materials part 5 ppt
... 80.95 52. 63 45 .27 38.85
E[C] 171.88 129 .97 81.43 52. 46 45 .23 38.76
α(C)
6.78e -2 4.72e -2 9 .29 e -2 6.60e -2 4.54e -2 3.45e -2
β(C)
3 .23 e-7 -2. 51e-7 5.15e-7 -1.75e-7 -2. 30e-8 -2. 50e-7
γ(C)
3 .26 3.17 ...
)(
1111
eff
C
)(
121 2
eff
C
)(
1 122
eff
C
Upper and lower bounds presented
sup(C) inf(C) sup(C) inf(C) sup(C) inf(C)
Deterministic values 100 .24 82. 24 35 .21 22...
Computational Mechanics of Composite Materials part 6 pdf
... form
138 Computational Mechanics of Composite Materials
0
)(
0
)(
0
pvpv
QqK =
(2. 2 02)
0
)(
,0
)(
,
)(
0
pv
r
pv
r
pv
qKQqK −=
(2. 203)
),(
,
)(
, )2(
)(
0 srs
pv
r
pv
bbCovqKqK −=
(2. 204)
where ... 44.84 GPa 0.0905
()
[]
121 2
inf CE
13.13 GPa 13. 12 GPa 0.09 82
()
[]
121 2
sup CE
13.88 GPa 13.88 GPa 0.0896
()
[]
1 122
inf CE
16 .27 GPa 16 .28 GPa 0.0991
()
[]...
Computational Mechanics of Composite Materials part 8 pps
...
()
{}{}()
{}{}()
∫
∫
∫∫
Ω
Ω
ΩΩ
Ω++++
Ω+++=
Ω+++=Ω=
dCCCC
dCCCC
ddG
effeffeffeff
effeffeffeff
ijij
22 22
)(
22 22
11
)(
22 11
21 12
)(
21 12
21
)(
21 21
2
1
122 1
)(
122 1
12
)(
121 2
1 122
)(
1 122
11
)(
1111
2
1
22 222 121 121 21111
2
1
2
1
εεεεεε
εεεεεε
εσεσεσεσεσ
(4.17) ... Ω, one can get
{
}
)(
22 22
)(
22 11
)(
21 12
)(
21 21
)(
122 1
)(
121 2
)(
1 122...
Computational Mechanics of Composite Materials part 9 ppt
...
h
1
h
2
0.0 025 0.0 025 20 0.0 525 <6-14>
Figure 5.8. Crack tip zone discretisation Figure 5.9. Crack tip mesh
R
Θ
Θ
T
h
1
h
2
Θ
R
Crack ti
p
1
2
2 32 Computational Mechanics of Composite Materials ... coordinates with origin at the crack tip and
24 2 Computational Mechanics of Composite Materials
()()
()
∈
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
+
−
=
π
νν
2
2
2
2
1...
Computational Mechanics of Composite Materials part 10 pps
... )
sr
rss
rrs
rs
s
rrs
rss
r
rs
rss
rrs
bbCov
SSSY
YYXX
XXXX
f
,
11
2
11
1
2
111
2
111
2
1
;
,
2
12
2
,
2
12
,
2
2
12
,
2
,
2
1
2
,
2
1
,
2
2
1
,
2
,
21
2
,
21
,
2
21
,
2
,
2
1
2
,
2
1
,
2
2
1
,
2
)2(
⎪
⎭
⎪
⎬
⎫
+
⎟
⎠
⎞
⎜
⎝
⎛
+
⎟
⎠
⎞
⎜
⎝
⎛
++
⎟
⎠
⎞
⎜
⎝
⎛
+
⎟
⎠
⎞
⎜
⎝
⎛
+−
⎟
⎠
⎞
⎜
⎝
⎛
−
⎪
⎩
⎪
⎨
⎧
⎟
⎠
⎞
⎜
⎝
⎛
−+
⎟
⎠
⎞
⎜
⎝
⎛
+
⎟
⎠
⎞
⎜
⎝
⎛
−
=
τττσ
σ...
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