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

y
y
y
τ xy
τ yz
τ zx
x
x
x
z
FIGURE 13.13a
z
FIGURE 13.13b
z
FIGURE 13.13c
The deformations produced in an element by the shear stresses τ xy , τ yz , and τ xz are
respectively shown in Figures 13.13a–c. There is no Poisson effect associated with shear
strain; therefore, the relationship between shear strain and shear stress can be stated as
1 1 1
γ xy = τxy γ yz = τyz γ zx = τzx
(13.17)
G G G
where G is the shear modulus, which is related to the elastic modulus E and Poisson’s
ratio ν by
G =
E
2(1 + ν)
(13.18)
Equations (13.16) and (13.17) are known as the generalized Hooke’s law for isotropic
materials. Notice that the shear stresses do not affect the expressions for normal strain
and that the normal stresses do not affect the expressions for shear strain; therefore, the
normal and shear relationships are independent of each other. Furthermore, the shear strain
expressions in Equation (13.17) are independent of each other, unlike the normal strain
expressions in Equation (13.16), where all three normal stresses appear. For example, the
shear strain γ xy is affected solely by the shear stress τ xy .
In addition, Equations (13.16) and (13.17) can be solved for the stresses in terms of
the strains as follows:
E
σ x =
[(1 − v) εx + ν( εy + εz)]
(1 + ν)(1−
2 ν) E
σ y =
[(1 − νε ) y + νε ( x + ε z)]
(1 + ν)(1−
2 ν) E
σ z =
[(1 − νε ) z + νε ( x + ε y)]
(1 + ν)(1−
2 v) (13.19)
Similarly, Equation (13.17) can be solved for the stresses in terms of the strains as
τ = Gγ τ = Gγ τ = Gγ
(13.20)
xy xy yz yz zx zx
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