Anisotropic and dissipative 75
T
HEOREM
5. At every time t the velocity field, v, and the equivalent plastic factor β satisfy
the following relationships Z
t
ρ ∇v
T
ρ
· ∇w − ∇v + 4F∂
2
CC
σ C, Y[F
T
{∇v}
s
F]F
T
· {∇w}
s
−
{∇v}
s
}dx − 2 Z
p t
ρ β
h
r
F∂
C
F C, YF
T
· {∇w}
s
− {∇v}
s
d x =
13 Z
t
ρ ˙ b
· w − vdx +
Z
Ŵ
2t
˙ˆS
t
· w − vda
and −2
Z
p t
ρ h
r
δ
− β F∂
C
F C, YF
T
· {∇v}
s
d x +
Z
p t
ρ h
r
δ − ββdx ≥ 0,
14
which hold for every admissible vector field w ∈ V
ad
t , and for all δ ∈ Mt.
Proof. In the theorem of virtual power, derived from the rate quasi-static equilibrium equation 12:
Z
t
˙S
t
· ∇wdx =
Z
∂
t
˙S
t
n · wda +
Z
t
ρ ˙
b
t
· wdx, ∀w ∈ V
ad
t we substitute the rate of the nominal stress, at time t , calculated from 9, taking into account the
potentiality condition 4
2
. First of all we calculate the differential with respect to τ of the right
hand side in 4
2
, in which we replace ˙
FF
−1
= L and ˙C = 2FDF
T
, with D
= L
s
. Thus
∂ ∂τ
T
ρ
= 2LF∂
C
σ C, YF
T
+ 2F∂
C
σ C, YF
T
L
T
+ 2F∂
2
CC
σ C, Y[2FDF
T
]F
T
+ 2F∂
2
YC
σ C, Y[ ˙
Y]F
T
15 in which we introduce the modified flow rule, 7, written under the form see Remark 2
∂
2
YC
σ C, Y[ ˙
Y] = −µ∂
C
F C, Y,
16 Hence the equality 13 follows at once from 9, 15and 16.
In order to prove 14 we note that µ ≥ 0 can be express either by the inequality
˜µ − µ ˙ˆ F ≤ 0, ∀ ˜µ ≥ 0, together with µ ˆ
F = 0, 17
or under its explicit dependence on the rate of strain: µ
= β
h
r
, with
β = 2∂
6
ˆ
F 6, α · d ˆ6G, α[E
T
DE],
h
r
= 2∂
6
ˆ
F 6, α · d ˆ6G, α[{G ˜ B}
s
] − ∂
α
ˆ
F 6, α · ˜m,
where the hardening parameter h
r
0. The time derivative of ˆ F 6, α with 2 is introduced in
17. Consequently, for all x ∈
p t
we get ˜µ − µ−µh
r
+ 2∂
6
ˆ
F 6, α · d ˆ6G, α[E
T
DE] ≤ 0.
18
76 S. Cleja-T
¸ igoiu
h
r
0. We can substitute µ and ˜µ by βh
r
and δh
r
. By integrating on
p t
from 18 the inequality 14 holds, when the equality
∂
C
F · A = ∂
6
ˆ
F 6, α · d ˆ6G, α[P
−T
AP
−1
] ,
∀ A ∈ Sym is also used for A
= F
T
DF.
Let us define the convex set e
K in the appropriate functional space of the solution H
ad
, by
e
K := {w, δ | w ∈ V
ad
t , δ
: −→ R
≥0
}, and the bilinear forms, in the appropriate space H
ab
:
K [v, w]
= Z
t
ρ
∇v T
ρ
· ∇w + 4F∂
2
CC
σ C, Y[F
T
{∇v}
s
F]F
T
· {∇w}
s
d x 19
A[β, δ] =
Z
p t
ρ h
r
β δ d x
B[δ, v]
= −2 Z
p t
ρ h
r
δ F∂
C
F C, YF
T
· {∇v}
s
d x are defined
∀ v, w ∈ V
ad
t , ∀ δ, β :
t
−→ R
≥0
. As a consequence of 19 , 13 and 14 the below statement holds:
T
HEOREM
6. Find U = v, β ∈ e
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