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1.38 For the following descriptions of various types of deformation, determine the strains
ij
(a) Simple dilitation
x
=
ax, y
=
y, z
=
z
(b) Pure deformation
x
=
a
1
x, y
=
a
2
y, z
=
a
3
z
(c) Simple shear
x
=
by, y
=
y, z
=
x
+
z
Hint:
Use Eqs. (1.9) to (1.15).
1.39 Suppose a thin, round, flat element lies in the
xy
plane. An external compressive
surface stress (force)
p
a
occurs on the outer radius. The outward normal
a
is along a
radius at a counterclockwise angle
α
from the
x
axis. Find the surface condition for
this element, comparable to Eq. (1.59).
Answer:
−
p
a
0
sin
2
−
α
)
cos
2
−
α
)
−
α
)
−
α
)
(
90
(
90
2 sin
(
90
cos
(
90
=
sin
2
−
(
−
α
)
(
−
α
)
(
−
α
)
(
−
α
)
−
cos
2
(
−
α
)
+
(
−
α
)
sin
90
cos
90
sin
90
cos
90
90
90
n
x
n
y
n
xy
×
Principal Stresses
1.40 For a plane inclined equally toward three principal axes, show that the normal stress
on the plane in terms of the principal stresses is
(σ
1
+
σ
2
+
σ
3
)/
3
.
1.41 At a point in a solid the stress matrix
T
has been determined as
21
−
6
−
6
T
=
−
6 7 8
−
6 8 7
Calculate the principal stresses and the extreme values of the shear stresses. Specify
a set of principal directions by a right-handed orthogonal triad of unit vectors.
1.42 From sketches of the planes on which shear stresses attain their extreme values, verify
that these planes make 45
◦
angles with the principal directions. For example, when
the normal to the plane is
1
√
2
the plane is parallel to the
a
1
axis and intersects the
a
2
a
3
axis at 45
◦
.
1.43 At a point
O
of a solid, the stress tensor referred to a right-handed coordinate system
with origin
O
and axes
x, y, z
is
a
1
=
0
a
2
=
a
3
=
320
15
0
=
15
280
0
T
0
0
360
Determine the normal and shear stresses at point
O
on a surface whose outward
normal is the bisector of the angle between the lines
Ox
and
Oz
Determine the
principal stresses, and principal axes as a right-handed coordinate system with origin
.
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