268
25 Elastic Displacements and Waves
β
T
= ω
T
+ ε
T
= ω
T
+ ε ,
with
ω
T
=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎝
0
1
2
∂s x
∂ y
−
∂s y
∂x
1
2
∂s x
∂z
−
∂s z
∂x
1
2
∂s y
∂x
−
∂s x
∂ y
0
1
2
∂s y
∂z
−
∂s z
∂ y
1
2
∂s z
∂x
−
∂s x
∂z
1
2
∂s z
∂ y
−
∂s y
∂z
0
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎠
,
ε =
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎝
∂s x
∂x
1
2
∂s x
∂ y
+
∂s y
∂x
1
2
∂s x
∂z
+
∂s z
∂x
1
2
∂s y
∂x
+
∂s x
∂ y
∂s y
∂ y
1
2
∂s y
∂z
+
∂s z
∂ y
1
2
∂s z
∂x
+
∂s x
∂z
1
2
∂s z
∂ y
+
∂s y
∂z
∂s z
∂z
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎠
.
⎫
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
(331)
We now write
s = ω
T
· x + ε · x .
(332)
If one constructs a vector − → ω from the three independent components of ω according
to
− → ω =
1
2
∂s z
∂ y
−
∂s y
∂z
,
∂s x
∂z
−
∂s z
∂x
,
∂s y
∂x
−
∂s x
∂ y
= (ω yz , ω zx , ω xy ) := (ω x , ω y , ω z )
(333)
then one can calculate for the first term on the right-hand side of (332) that
ω
T
· x = − → ω × x =
ω y z − ω z y , ω z x − ω x z , ω x y − ω y x
.
(334)
The vector product on the right-hand side of (334) describes nothing else than a
rotation of the whole volume element V in the direction defined by − → ω . This part
of the shift of V must be ignored and this leads us to the tensor ε of elastic strain
according to (331),
ε =
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎝
∂s x
∂x
1
2
∂s x
∂ y
+
∂s y
∂x
1
2
∂s x
∂z
+
∂s z
∂x
1
2
∂s y
∂x
+
∂s x
∂ y
∂s y
∂ y
1
2
∂s y
∂z
+
∂s z
∂ y
1
2
∂s z
∂x
+
∂s x
∂z
1
2
∂s z
∂ y
+
∂s y
∂z
∂s z
∂z
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎠
.
(335)
We recognise from (335):