Civil Engineering Reference
In-Depth Information
F
3
F
2
X
3
F
1
O
X
2
X
1
Figure 1.3
A cube with faces of unit area parallel to the coordinate planes. The three
components of the forces acting on the upper and the lower faces are represented.
The variations of the components
σ
ij
are assumed to be negligible at the surface
of the cube. With the components of the positive unit normal on the upper face being
(0, 0, 1), Equations (1.19) reduce to
T
1
=
σ
31
,
T
2
=
σ
32
,
T
3
=
σ
33
(1.20)
The force
F
(F
1
,F
2
,F
3
)
acting on the upper face is equal to
T
3
. The components of
the unit normal on the lower face are (0, 0,
1). The forces on the lower and the upper
face are equal in magnitude and lie in opposite directions. The same property holds for
the two other pairs of opposite faces. The elements
σ
ij
where
i
−
=
j
correspond to normal
forces while those with
i
=
j
correspond to tangential forces.
1.5
Stress - strain relations for an isotropic elastic medium
The stress - strain relations for an isotropic elastic medium are as follows:
σ
ij
=
λθδ
ij
+
2
µe
ij
(1.21)
The quantities
λ
and
µ
are the Lame coefficients and
δ
ij
is the Kronecker delta:
δ
ij
=
if
i
=
j
1
(1.22)
δ
ij
=
0
if
i
=
j
In matrix form Equation (1.21) can be rewritten
σ
11
σ
22
σ
33
σ
13
σ
23
σ
12
C
11
C
12
C
12
000
C
12
C
11
C
12
000
C
12
C
12
C
11
000
000
C
44
00
0000
C
44
0
00000
C
44
e
11
e
22
e
33
e
13
e
23
e
12
=
(1.23)
