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

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COMbINEd LOAdS
If the naming convention for
principal stresses is followed
(i.e., s p1 > s p2 ), then all
combinations of s p1 and s p2 will
plot to the right of (i.e., below)
line AB shown in Figure 15.7.
The load P must be applied
slowly so that there is no kinetic
energy associated with the
application of the load. All work
done by P is stored as potential
energy in the strained bar.
stress is equal to one-half of the arithmetic sum of the two principal stresses (i.e., simply
the radius of the in-plane Mohr’s circle).
Therefore, the maximum-shear-stress theory applied to a plane stress state with
in-plane principal stresses s p1 and s p2 predicts that yielding failure will occur under the
following conditions:
• If s p1 and s p2 have the same sign, then failure will occur if s p1 ≥ sY or s p2
≥ sY.
• If s p1 is positive and s p2 is negative, then failure will occur if s p1 − s p2 ≥ s Y .
maximum-Distortion-Energy Theory. 2 The maximum-distortion-energy theory
is founded on the concept of strain energy. The total strain energy per unit volume can be
determined for a specimen subjected to any combination of loads. Further, the total strain
energy can be broken down into two categories: strain energy that is associated with a change
in volume of the specimen and strain energy that is associated with a change in shape, or
distortion, of the specimen. This theory predicts that failure will occur when the strain energy
causing distortion reaches the same intensity as the strain energy at failure found in axial tension
or compression tests of the same material. Supporting evidence comes from experiments
which reveal that homogeneous materials can withstand very high hydrostatic stresses (i.e.,
equal-intensity normal stresses in three orthogonal directions) without yielding. Based on
this observation, the maximum-distortion-energy theory assumes that only the strain energy
which produces a change of shape is responsible for the failure of the material. The strain
energy of distortion is most readily computed by determining the total strain energy of the
stressed material and subtracting the strain energy associated with the change in volume.
The concept of strain energy is illustrated in Figure 15.8. A bar of uniform cross section
subjected to a slowly applied axial load P is shown in Figure 15.8a. A load–deformation
diagram for the bar is shown in Figure 15.8b. The work done in elongating the bar by an
amount δ 2 is
W
∫
2
= δ Pdδ
(a)
0
P
σ
L
P 2
P
σ
σ
2
δ
P 1
σ
1
P
δ
1 2
d
δ
δ
δ
ε
ε
1 2
d
(a) (b) (c)
FIGURE 15.8 Concept of strain energy.
ε
ε
2
Frequently called the Huber–von Mises–Hencky theory or the von Mises yield criterion because it was
proposed by M. T. Huber of Poland in 1904 and, independently, by R. von Mises of Germany in 1913. The
theory was further developed by H. Hencky and von Mises in Germany and the United States.