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

σ Y
(a) Stress element at yield
for test specimen
τ
τ
σ
σ
σY
σY
−
2 2
p2 = p3=
0
σ
Y ,
2
σ
Y
2
(b) Mohr’s circle for test
specimen at yield
p1=
Y
FIGURE 15.6 States of stress for a uniaxial tension test.
σ
σ
σ
45°
σ
Y
2
σY
2
σY
2
(c) Maximum-shear-stress
element
45°
Lüders lines on
mild steel bar
no single theory agrees with test data for all types of materials and all combinations of
loads. Several of the more common theories of failure are presented and briefly explained
in the paragraphs that follow.
Ductile materials
maximum-Shear-Stress Theory. 1 When a flat bar of a ductile material such as
mild steel is tested in uniaxial tension, yielding of the material is accompanied by lines that
appear on the surface of the bar. These lines, known as lüders lines, are caused by slipping
(on a microscopic scale) that occurs along the planes of randomly ordered grains that make
up the material. Lüders lines are oriented at 45° with respect to the longitudinal axis of the
specimen (Figure 15.5). Therefore, if one assumes that slip is the failure mechanism associated
with yielding of the material, then the stress that best characterizes this failure is
the shear stress on the slip planes. In a uniaxial tension test, the state of stress at yield can
be represented by the stress element shown in Figure 15.6a. The Mohr’s circle corresponding
to this state of stress is shown in Figure 15.6b. Mohr’s circle reveals that the maximum
shear stress in a uniaxial test specimen occurs at an orientation of 45° with respect to the
load direction (Figure 15.6c), just as the Lüders lines do.
On the basis of these observations, the maximum-shear-stress theory predicts that
failure will occur in a component that is subjected to any combination of loads when the
maximum shear stress at any point in the object reaches the failure shear stress τ f = s Y /2,
where s Y is determined by an axial tension or compression test of the same material. For
ductile materials, the shearing elastic limit, as determined from a torsion test (for pure
shear), is greater than one-half the tensile elastic limit. (An average value for τ f is about
0.57s Y .) Since the maximum-shear-stress theory is based on s Y obtained from an axial test,
the theory errs on the conservative side.
The maximum-shear-stress theory is represented graphically in Figure 15.7 for an
element subjected to biaxial principal stresses (i.e., plane stress). In the first and third quadrants,
s p1 and s p2 have the same sign; therefore, the absolute maximum shear stress acts in
an out-of-plane direction, and it has a magnitude that is equal to one-half of the numerically
larger principal stress s p1 or s p2 , as explained in Section 12.8 [see Equation (12.18)]. In the
second and fourth quadrants, where s p1 and s p2 are of opposite sign, the maximum shear
FIGURE 15.5 Lüders lines on
a ductile tension test specimen.
B
−σ
Y
σ
σ
Y
p2
−σ
Experimental data from tension test.
FIGURE 15.7 Failure diagram
for maximum-shear-stress theory
for an element subjected to plane
stress.
Y
σ
Y
A
σ
p1
1 Sometimes called Coulomb’s theory because it was originally stated by him in 1773. More frequently called
the Tresca criterion or the Tresca–Guest yield surface because of the work of French elasticity theorist
H. E. Tresca (1814–1885), which was advanced by the work of J. J. Guest in England in 1900.
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