Composite Materials Research Progress

170
⎟ ⎞
⎠
s
⎜ ⎛
⎝
=
a
E
E
Ln
f
Giangiacomo Minak and Andrea Zucchelli
PI
Ω AE,I
….
PII
….
Ω AE,II
Ω AE
PIII
Ω AE,III
….
PIV
Ω AE,IV
Displacement
Figure 2. The basic functions P I , P II , P III and P IV, used to describe the function f.
If there is more than one sub-domain ΩAE,k, k∈{I,II,III,IV}, it is possible to write the
following relation:
Ω
n I ⎛
= ⎜
UΩ
⎝ i=
1
n II
n III
n IV
⎞ ⎛ i ⎞ ⎛ i ⎞ ⎛ i ⎞
⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟
, I ⎟
U
⎜U
ΩAE,
II ⎟
U
⎜U
ΩAE,
III ⎟
U
⎜U
ΩAE
VI ⎟
(4)
⎠ ⎝ i=
1 ⎠ ⎝ i=
1 ⎠ ⎝ i=
1 ⎠
AE
i
AE
,
where nI, nII, nIII, and nIV are the number sub-domain corresponding to the trend type I,II, III
and IV respectively. Then functions PI,PII, PIII and PIV can be written as follow:
P
P
I
( x)
III
( x)
⎧ PI
⎪
= ⎨
⎪
⎩PI
,
, 1
n I
⎧ P
⎪
= ⎨
⎪
⎩
PIII
( x)
( x)
III,
1
, n III
/ x ∈ Ω
...
/ x ∈Ω
( x)
( x)
1
AE,
I
n I
AE,
I
/ x ∈Ω
...
/ x ∈ Ω
1
AE,
III
n III
AE,
III
P
P
II
( x)
IV
( x)
⎧ P
⎪
= ⎨
⎪
⎩PII
II,
1
, n II
⎧ P
⎪
= ⎨
⎪
⎩
P
( x)
IV,
1
( x)
IV,
n IV
/ x ∈ Ω
( x)
...
/ x ∈ Ω
( x)
1
AE,
II
/ x ∈ Ω
...
n II
AE,
II
/ x ∈ Ω
1
AE,
IV
n IV
AE,
IV
From the physical point of view the part of f(x) characterized by an increasing trend, type
I, represents the strain energy storing phases. The slope of PI,i (x) functions decreases during
(5)