Modern Engineering Thermodynamics

114 CHAPTER 4: The First Law of Thermodynamics and Energy Transport Mechanisms
4.6.3 Elastic Work
Whenever we compress or extend an elastic solid (like a spring), we perform elastic work. Consider a force ± F !
applied on the end of an elastic rod (see Figure 4.5c). The normal stress σ in the rod is
σ =± j F! j
A
(4.35)
where j F ! j is the magnitude of the force and A is the cross-sectional area of the rod. Since the force F ! and its
corresponding displacement d x ! are always in the same direction, the vector dot product F ! .d x ! always reduces
to Fdx, whereF = j F ! j and dx = jd x ! j, and when the force is applied on the system from the surroundings
rather than being produced by the system, the work is negative and its increment is
The strain ε in the rod is defined as
dW = − F ! .d x ! = − Fdx = − σAdx (4.36)
dε = dx
L = Adx
AL = Adx
V
= dV V
(4.37)
where L is the length of the rod and AL is its volume V: Then,
and Eq. (4.36) becomes
Adx = dV = Vdε (4.38)
Therefore, for elastic work,
dW = −σAdx = −σVdε (4.39)
Elastic work
ð1W 2 Þ elastic
= −
Z 2
1
σVdε
(4.40)
EXAMPLE 4.6
Determine an expression for the work involved in deforming a constant volume elastic solid that obeys Hooke’s law of elasticity
(see Figure 4.13).
L
State 1
L +ΔL
State 2
1 W 2 = ?
F = EA(ΔL/L)
FIGURE 4.13
Example 4.6.
Solution
Here we have V = constant. Also, from strength of materials we can write Hooke’s law as σ = Eε, where E is Young’s modulus
of elasticity. Then, Eq. (4.40) becomes
Z 2
Z 2
ð 1
W 2 Þ elastic
= − σV dε = − EVε dε = − EV
1
1
= − EV ε2 2 −
ε2 1
= − V 2 2E ðσ2 2 − σ2 1 Þ
Thus, if ε 2 2 > ε2 1 , then ð 1 W 2Þ elastic
is negative and work is being put into the system; and if ε 2 2 < ε2 1 , then ð 1 W 2Þ elastic
is positive
and work is being produced by the system. Note that both tensile strains (ε > 0) and compressive strains (ε < 0) are possible
here. But, the resulting work formula deals only with ε 2 and consequently gives the correct result regardless of the strain
direction.
Z 2
1
ε dε