Timothy A. Philpot - Mechanics of materials _ an integrated learning system-John Wiley (2017)

When the third term in the brackets is expanded, this equation can be solved for u d :
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THEORIES OF FAILuRE
u
1 v
2
2 2
2 2
= + ⎡
⎣( σ
1
− 2 σ 1σ 2 + σ
2) + ( σ
2
− 2 σ 2σ 3 + σ
3) + ( σ
3
− 2 σ 3σ 1 + σ
2
1)
⎤
⎦
6E
1 v
= + (e)
⎡⎣ ( σ p1 − σ p2) 2 + ( σ p2 − σ p3) 2 + ( σ p3 − σ p1)
2 ⎤ ⎦
6E
d p p p p p p p p p p p p
The maximum-distortion-energy theory of failure assumes that inelastic action will occur
whenever the energy given by Equation (e) exceeds the limiting value obtained from a
tension test. In the tension test, only one of the principal stresses will be nonzero. If this
stress is called σ Y , then
1 v
( ud)
Y= + σ Y 2
3E
and when this value is substituted into Equation (e), the maximum-distortion-energy failure
criterion is expressed as
2
1
σY =
2
⎡⎣ ( σ p1 − σ p2 ) 2 + ( σ p2 − σ p3 ) 2 + ( σ p3 − σ p1 ) 2 ⎤ ⎦ (15.3)
or
2 2 2 2
σ = σ + σ + σ − ( σ σ + σ σ + σ σ )
Y p1
p2
p3
p1 p2 p2 p3 p3 p1
for failure by yielding. The maximum-distortion-energy failure criterion can be alternatively
stated in terms of the normal stresses and shear stress on three arbitrary orthogonal planes:
2
1
2 ( ) 2 ( ) 2
Y x y y z ( x z ) 2 6( 2 2 2
σ = ⎡⎣ σ − σ + σ − σ + σ − σ + τxy + τyz + τxz
) ⎤ ⎦ (15.4)
When a state of plane stress exists (i.e., when σ p3 = 0), Equation (15.3) becomes
Pure
shear
σ Y
σ p2
A
( σ Y , σ Y )
σ = σ − σ σ + σ 2
(15.5)
2 2
Y p1
p1 p2 p2
This last expression is the equation of an ellipse in the σ p1 − σ p2 plane with
its major axis along the line σ p1 = σ p2 , as shown in Figure 15.10. For comparison
purposes, the failure hexagon for the maximum-shear-stress yield theory is also
shown in dashed lines in Figure 15.10. While both theories predict failure at the
six vertices of the hexagon, the maximum-shear-stress theory gives the more conservative
estimate of the stresses required to produce yielding, since the hexagon
falls inside the ellipse for all other combinations of stress.
B
−σ Y
( −σ Y , −σ Y )
mises Equivalent Stress. A convenient way to employ the maximum-distortionenergy
theory is to establish an equivalent stress quantity σ M that is defined as the square
root of the right-hand side of Equation (15.3). This stress is called the Mises equivalent
stress (or the von Mises equivalent stress) 5 and is given by
2
σM =
2
⎡⎣ ( σ p1 − σ p2 ) 2 + ( σ p2 − σ p3 ) 2 + ( σ p3 − σ p1 ) 2 ⎤ 1/2
⎦ (15.6)
σ Y
σ p1
σ Y ,
σ
– Y
2 2
σY
,
σ
– Y
√3
√3
Experimental data from tension test.
FIGURE 15.10 Failure diagram for
maximum-distortion-energy theory for
an element subjected to plane stress.
If the naming convention for
principal stresses is followed
(i.e., σ p1 > σ p2 ), then all
combinations of σ p1 and σ p2 will
plot to the right of (i.e., below)
line AB shown in Figure 15.10.
5
After Richard Edler von Mises (1883–1953), Austrian mathematician and scientist who taught at the
University of Istanbul and Harvard University.