Section 2.7
Solid Mechanics Part III
Kelly
258
2.7 Small Strain Theory
When the deformation is small, from 2.2.43-4,
(
)
u
I
F
u
I
U
I
F
grad
grad
Grad
+
≈
+
=
+
=
(2.7.1)
neglecting the product of
u
grad
with
U
Grad
, since these are small quantities.
Thus one
can take
u
U
grad
Grad
=
and there is no distinction to be made between the undeformed
and deformed configurations.
The deformation gradient is of the form
α
I
F
+
=
, where
α
is small.
2.7.1 Decomposition of Strain
Any second order tensor can be decomposed into its symmetric and antisymmetric part
according to 1.10.28, so that
ij
ij
i
j
j
i
i
j
j
i
j
i
x
u
x
u
x
u
x
u
x
u
Ω
+
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
−
∂
∂
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
+
∂
∂
=
∂
∂
+
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
−
∂
∂
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
+
∂
∂
=
∂
∂
ε
2
1
2
1
2
1
2
1
T
T
Ω
ε
x
u
x
u
x
u
x
u
x
u
(2.7.2)
where
ε
is the small strain tensor 2.2.48 and
Ω
, the anti-symmetric part of the
displacement gradient, is the
small rotation tensor
, so that
F
can be written as
Ω
ε
I
F
+
+
=
Small Strain Decomposition of the Deformation Gradient
(2.7.3)
It follows that (for the calculation of
e
, one can use the relation
(
)
δ
I
δ
I
−
≈
+
−
1
for small
δ
)
ε
e
E
ε
I
b
C
=
=
+
=
=
2
(2.7.4)
Rotation
Since
Ω
is antisymmetric, it can be written in terms of an axial vector
ω
,
cf
. §1.10.11, so
that for any vector
a
,
3
12
1
13
1
23
,
e
e
e
ω
a
ω
Ω
a
Ω
−
Ω
+
Ω
−
=
×
=
(2.7.5)
The relative displacement can now be written as

This ** preview** has intentionally

**sections.**

*blurred***to view the full version.**

*Sign up*