Civil Engineering Reference
In-Depth Information
compressive state of stress (also known as
hydrostatic
stress) does not cause dis-
tortion and, hence, does not contribute to yielding. If the principal stresses have
been computed, total elastic strain energy is given by
1
2
U
e
=
(
1
ε
1
+
2
ε
2
+
3
ε
3
) d
V
V
2
E
1
2
+
1
3
+
2
3
)
V
1
2
1
2
2
2
3
=
+
+
−
2
(
(9.123)
To arrive at distortion energy, the average (hydrostatic) stress is defined as
av
=
1
+
2
+
3
3
(9.124)
and the corresponding strain energy is
av
2
E
3
U
hyd
=
(1
−
2
)
V
(9.125)
The distortion energy is then defined as
U
d
=
U
hyd
(9.126)
After a considerable amount of algebraic manipulation, the distortion energy in
terms of the principal stress components is found to be given by
U
e
−
(
1
/
2
1
−
2
)
2
1
−
3
)
2
2
−
3
)
2
1
+
3
E
+
(
+
(
U
d
=
V
(9.127)
2
The DET states that failure (yielding) occurs in a general state of stress when the
distortion energy per unit volume equals or exceeds the distortion energy per unit
volume occurring in a uniaxial tension test at yielding. It is relatively easy to
show (see Problem 9.20) that, at yielding in a tensile test, the distortion energy is
given by
1
+
3
E
S
y
V
U
d
=
(9.128)
and, as before, we use
S
y
to denote the tensile yield strength. Hence, Equa-
tions 9.127 and 9.128 give the failure (yielding) criterion for the DET as
(
1
/
2
1
−
2
)
2
1
−
3
)
2
2
−
3
)
2
+
(
+
(
≥
S
y
(9.129)
2
The DET as described in Equation 9.129 leads to the concept of an
equivalent
stress
(known historically as the
Von Mises stress
) defined as
(
1
/
2
1
−
2
)
2
1
−
3
)
2
2
−
3
)
2
+
(
+
(
e
=
(9.130)
2
