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

Although stress is applied only in the x direction, normal strains are produced in the y and
z directions because of the Poisson effect:
x
ey
ν σ ez
ν σ x
=− =−
E
E
Note that these strains in the transverse direction are negative (i.e., they denote contraction).
If the element elongates in the x direction, then it contracts in the transverse directions,
and vice versa.
Similarly, the normal stress σ y produces strains not only in the y direction, but also in
the transverse directions (Figure 13.12b):
σ y
y
y
e y = ex
= − ν σ ez
= −ν σ
E E E
Likewise, the normal stress σ z produces the strains (Figure 13.12c)
σ z
z
z
e z = ex
= − ν σ ey
= −ν σ
E E E
If all three normal stresses σ x , σ y , and σ z act on the element at the same time, the total deformation
of the element can be determined by summing the deformations resulting from each
normal stress. This procedure is based on the principle of superposition, which states that
the effects of separate loadings can be added algebraically if two conditions are satisfied:
1. Each effect is linearly related to the load that produced it.
2. The effect of the first load does not significantly change the effect of the second load.
The first condition is satisfied if the stresses do not exceed the proportional limit for the
material. The second condition is satisfied if the deformations are small, in which case the
small changes in the areas of the faces of the element do not produce significant changes in
the stresses.
Using the superposition principle, we can state the relationship between normal strains
and normal stresses as follows:
z
σ z
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gENERALIzEd HOOkE’S LAw
FOR ISOTROPIC MATERIALS
τ yz
y
τ zy τ xy
τ zx
σ x
FIGURE 13.11
σ y
τ xz
τ yx
x
ex 1
= [ σx E
− νσ ( y + σ z )]
1
ey = [ σ y
E
− νσ ( x + σ z )]
1
ez = [ σz E
− νσ ( x + σ y )]
(13.16)
y
y
σ y
y
σ x
x
x
x
z
z
z
σ z
FIGURE 13.12a
FIGURE 13.12b
FIGURE 13.12c
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