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Equation (1.75) is a mixed governing differential equation. Note that in contrast to the
governing equations for the displacement and force formulations, Eq. (1.75) for the mixed
method contains no derivatives of the unknowns (
u
and
σ
) higher than the first and does
not involve derivatives of the material parameters.
1.7
Analysis of Stress
1.7.1 Principal Stresses
Figure (1.6) shows the state of stress at a point
P
in a body by specifying the stresses acting
on three coordinate planes passing through point
P
.Define three stress vectors on these
planes as
σ
x
=
σ
x
e
x
+
τ
xy
e
y
+
τ
xz
e
z
σ
y
=
τ
yx
e
x
+
σ
y
e
y
+
τ
yz
e
z
(1.77)
σ
z
=
τ
zx
e
x
+
τ
zy
e
y
+
σ
z
e
z
where
e
x
,
e
y
and
e
z
are the unit vectors along the
x, y,
and
z
axes. The vector
σ
x
defines
the stress on the face of the cube whose outward normal is
e
x
, and similarly for
σ
y
and
σ
z
.
We now calculate the stress vector
σ
a
on an arbitrarily oriented plane passing through the
point
P
. The orientation of this plane is specified by a unit vector
a
normal to the plane
a
=
a
x
e
x
+
a
y
e
y
+
a
z
e
z
(1.78)
where
a
x
,a
y
,
and
a
z
are the direction cosines with respect to the
x, y,
and
z
axes of the
normal to the plane. To find
σ
a
,
a tetrahedron element of volume
V
is isolated from the
body (Fig. 1.13), with three triangular sides parallel to the negative
x, y
, and
z
coordinate
y
-
e
z
a
h
-
e
x
x
-
e
y
z
FIGURE 1.13
Tetrahedron of volume
V
with sides of areas
A, a
x
A, a
y
A,
and
a
z
A
.
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