80 S. Cleja-T
¸ igoiu
The rate evolution equation for plastic deformation expressed by Mandel’s nine- dimen- sional flow rule, i.e. there is a particular representation of the modified flow rule given in 8,
˙PP
−1
= µ∂
6
F 6, α, κ is associated to the orthotropic yield function, generated by 23, which describe the proportional
and kinematic hardening given by F 6, α, κ ≡ f 6, κ − 1 ≡
ˆ
Mn
1
⊗ n
1
, n
2
⊗ n
2
, κ6 · 6 − 1 = 0,
6 = 6 − α.
24 Here we put into evidence the possible dependence on κ of the yield function through the fourth
order tensor M.
We provide the constitutive relations for the plastic strain rate, D
p
, as well as for the plastic
spin W
p
, defined by
D
p
= 12L
p
+ L
p T
, W
p
= 12L
p
− L
p T
, where
L
p
= ˙PP
−1
For orthotropic material the plastic strain rate is given by
D
p
= µ ˆN
p
6, α, κ, n
1
⊗ n
1
, n
2
⊗ n
2
with
ˆN
p
= 2C
1
6
s
· II + 2C
2
6
s
+ C
4
[6
s
· In
1
⊗ n
1
+ 25
+ 6
s
· n
1
⊗ n
1
I] + C
5
[6
s
· In
2
⊗ n
2
+ 6
s
· n
2
⊗ n
2
I] +
+ 2C
6
{6
s
n
1
⊗ n
1
}
s
+ 2C
7
{6
s
n
2
⊗ n
2
}
s
+ + C
8
{n
1
⊗ n
1
6
a
}
s
+ C
9
{n
2
⊗ n
2
6
a
}
s
+ + 2C
10
6
s
· n
1
⊗ n
1
n
1
⊗ n
1
+ + 2C
11
6
s
· n
2
⊗ n
2
n
2
⊗ n
2
+ + 2C
12
[6 · n
1
⊗ n
1
n
2
⊗ n
2
+ 6 · n
2
⊗ n
2
n
1
⊗ n
1
] and the plastic spin is expressed under the form
W
p
= µ ˆ
p
6, α, κ, n
1
⊗ n
1
, n
2
⊗ n
2
with ˆ
p
= −2C
3
6
a
+ C
8
{n
1
⊗ n
1
6
s
}
a
+ C
9
{n
2
⊗ n
2
6
s
}
a
− −2C
13
{6
a
n
1
⊗ n
1
}
a
− 2C
14
{6
a
n
2
⊗ n
2
}
a
26
R
EMARK
4. W
p
involves the terms generated by the symmetric part of 6, while D
p
con- tains terms generated by the skew- symmetric part of 6, with two coupling coefficients C
8
, C
9
. R
EMARK
5. In the case of 6 ∈ Sym, i.e. for small elastic strains and α ∈ Sym, directly
from 26 we derive the following expression for orthotropic plastic spin
W
p
= µ
p
= µ {C
8
{n
1
⊗ n
1
6
s
}
a
+ C
9
{n
2
⊗ n
2
6
s
}
a
} 27
But in this case, the yield condition 23 does not depend on the parameters which enter the expression 27 of the plastic spin.
Anisotropic and dissipative 81
P
ROPOSITION
3. From the orthotropic Mandel’s flow rule 26 the flow rule characterizing the g
4
− transversely isotropic material is derived when C
5
= C
7
= C
9
= C
11
= C
12
= 0, i.e. dependent on six material constants. The plastic spin is given by 25, in which C
9
= C
14
= 0, i.e. dependent on three constant only.
Evolution equation for internal variable can be described, see [6], by some new generaliza- tion to finite deformation of Armstrong- Frederick hardening rule.
From the orthotrop representation g
4
− transversely isotropic case only can be obtained. Thus for plastically incompressible material, i.e.
˜ρ = ρ ,
the representatio from [21] can be obtained by taking into account small deformation theory. The fibre-inextensible case given in
[22] can be also derived from our general representation, when the appropriate yield constant is much grater then the others.
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