EN0175
10 / 19 / 06
Mechanical Behavior of Solids
Linear Elastic solids
ε
σ
E
=
(1D)
⇒
kl
ijkl
ij
C
ε
σ
=
or
kl
ijkl
ij
S
σ
ε
=
(3D)
where
C
is sometimes called the stiffness tensor and
S
is sometimes called the compliance
tensor. Both of them are 4
th
order elastic moduli tensors.
Symmetry of elastic moduli tensors:
jikl
ijkl
C
C
=
,
(minor symmetry)
ijlk
ijkl
C
C
=
The minor symmetries reduce the independent elastic constants from 81 to 36.
There is also a major symmetr in elastic moduli
klij
ijkl
C
C
=
,
which reduces the number of
independent elastic constants from 36 to 21.
We use the concept of energy & work to demonstrate the major symmetry of
C
and
S
:
u
F
,
Assume an increment of displacement at the bar end,
u
u
u
δ
+
→
The work done by the applied load should equal to the stored energy in the material,
(
)
ε
σδ
ε
σδ
ε
δ
σ
δ
δ
V
Al
l
A
u
F
W
=
=
=
=
ε
σδ
δ
δ
=
=
V
W
w
should be the stored elastic energy per unit volume, which is also called the
strain energy density.
( )
ε
w
w
=
,
ε
σ
∂
∂
=
w
,
( )
∫
=
ε
ε
σ
ε
0
d
w
1

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