Civil Engineering Reference
In-Depth Information
10.2
Frame in vacuum
Stress - strain relations
The
Z
-axis is assumed to be the axis of rotational material symmetry. Let
X
and
Y
be
two orthogonal axes in the symmetry plane. Let
u
s
be the solid displacement vector,
1
/
2
(∂u
s
i
/∂x
j
∂u
s
j
/∂x
i
)
. The stress-strain
the strain components are defined by
e
ij
=
+
relations for the frame in vacuum can be written
σ
=
(
2
G
+
A)e
xx
+
Ae
yy
+
Fe
zz
(10.1)
xx
σ
yy
=
Ae
xx
+
(
2
G
+
A)e
yy
+
Fe
zz
(10.2)
σ
zz
=
Fe
xx
+
Fe
yy
+
Ce
zz
(10.3)
2
G
e
yz
σ
yz
=
(10.4)
2
G
e
xz
σ
xz
=
ˆ
(10.5)
σ
ˆ
=
2
Ge
xy
(10.6)
xy
where
A,F,G,C
and
G
, are the rigidity coefficients (see Equation 1.23 for a compari-
son with the isotropic case where
G
G
,F
+
2
G)
. Using the engineering
notation as in the work by Cheng (1997), the Young moduli are denoted as
E
x
,E
y
,E
z
and can be rewritten
E
x
=
=
A,C
=
A
E
. The Poisson ratios are denoted as
ν
yx
,ν
zy
=
E
y
=
E,E
z
=
ν
. The coefficients
A,F,C
and
G
and
ν
zx
and satisfy the relations
ν
yx
=
ν,ν
zy
=
ν
zx
=
are related to the new coefficients by
E(E
ν
+
Eν
2
)
A
=
(10.7)
(
1
+
ν)(E
−
E
ν
−
2
Eν
2
)
EE
ν
F
=
(10.8)
E
−
E
ν
−
2
Eν
2
E
2
(
1
−
ν)
C
=
(10.9)
E
−
E
ν
−
2
Eν
2
E
2
(
1
+
G
=
(10.10)
ν)
Waves in a transversally isotropic elastic medium
As for isotropic porous media, if the wavelength is much larger than the characteristic
length of the representative elementary volume, the elastic frame in vacuum can be
replaced by a homogeneous elastic medium with the same rigidity constants. The plane
waves that propagate in a transversally isotropic elastic layer have been described by
Royer and Dieulesaint (1996). The meridian plane contains the
Z
axis and the wave
number vector. Let
X
be the direction perpendicular to
Z
in the meridian plane. With
the notations of Vashishth and Khurana (2004), the space and time dependence is
exp
jω
t
qz
x
c
−
−
