Page 209 - Mechanical Behavior of Materials

210 Chapter 5 Stress–Strain Relationships and Behavior

The volume, V = LWH, changes by an amount dV that can be evaluated from differential
calculus, where V is considered to be a function of the three independent variables L, W, and H:

∂V ∂V ∂V
dV = dL + dW + dH (5.32)
∂L ∂W ∂ H
Evaluating the partial derivatives and dividing both sides by V = LWH gives

dV dL dW dH
= + + (5.33)
V L W H
This ratio of the change in volume to the original volume is called the volumetric strain,or dilata-
tion, ε v . By substituting Eq. 5.31, the volumetric strain is seen to be simply the sum of the normal
strains:

dV
ε v = = ε x + ε y + ε z (5.34)
V
For an isotropic material, the volumetric strain can be expressed in terms of stresses by
substituting the generalized Hooke’s law, specifically Eq. 5.26, into Eq. 5.34. The following is
obtained after collecting terms:

1 − 2ν
ε v = σ x + σ y + σ z (5.35)
E
Note from this that ν = 0.5 causes the change in volume to be zero, ε v = 0, even in the presence of
nonzero stresses. Also, a value of ν exceeding 0.5 would imply negative ε v for tensile stresses—that
is, a decrease in volume. This would be highly unusual, so that 0.5 appears to be an upper limit on
ν that is seldom exceeded for real materials.
The average normal stress is called the hydrostatic stress and is given by

σ x + σ y + σ z
σ h = (5.36)
3
Substituting this into Eq. 5.35 yields

3 (1 − 2ν)
ε v = σ h (5.37)
E

Hence, the volumetric strain is proportional to the hydrostatic stress. The constant of proportionality
relating these is called the bulk modulus, given by

σ h E
B = = (5.38)
ε v 3 (1 − 2ν)