Section 2.6
Solid Mechanics Part III
Kelly
253
2.6 Deformation Rates: Further Topics
2.6.1
Relationship between l, d, w and the rate of change of R
and U
Consider the polar decomposition
RU
F
=
.
Since
R
is orthogonal,
I
RR
=
T
, and a
differentiation of this equation leads to
T
T
R
R
R
R
Ω
R
&
&
−
=
≡
(2.6.1)
with
R
Ω
skew-symmetric (see Eqn. 1.14.2).
Using this relation, the expression
1
−
=
F
F
l
&
,
and the definitions of
d
and
w
, Eqn. 2.5.7, one finds that {
▲
Problem 1}
(
)
[
]
(
)
[
]
T
1
T
1
1
T
1
T
1
1
T
1
sym
2
1
skew
2
1
R
U
U
R
R
U
U
U
U
R
d
Ω
R
U
U
R
Ω
R
U
U
U
U
R
w
Ω
R
U
U
R
l
R
R
R
−
−
−
−
−
−
−
=
+
=
+
=
+
−
=
+
=
&
&
&
&
&
&
&
(2.6.2)
Note that
R
Ω
being skew-symmetric is consistent with
w
being skew-symmetric, and that
both
w
and
d
involve
R
, and the rate of change of
U
.
When the motion is a rigid body rotation, then
0
U
=
&
, and
T
R
R
Ω
w
R
&
=
=
(2.6.3)
2.6.2
Deformation Rate Tensors and the Principal Material and
Spatial Bases
The rate of change of the stretch tensor in terms of the principal material base vectors is
{
}
∑
=
⊗
+
⊗
+
⊗
=
3
1
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
i
i
i
i
i
i
i
i
i
i
N
N
N
N
N
N
U
&
&
&
&
&
λ
λ
λ
(2.6.4)
Consider the case when the principal material axes stay constant, as can happen in some
simple deformations.
In that case,
U
&
and
1
−
U
are coaxial (see §1.11.5):
∑
=
⊗
=
3
1
ˆ
ˆ
i
i
i
i
N
N
U
λ
&
&
and
∑
=
−
⊗
=
3
1
1
ˆ
ˆ
1
i
i
i
i
N
N
U
λ
(2.6.5)

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