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p
p
∆t
∆t
∆x
F
A o
l o l o
y

F

p p
S = p / A o S = p / A o τ = F / A o
e = ∆l / l o e = ∆l / l o γ = ∆ x / y
(a) (b) (c)

FIGURE 19.30 Elastic stress and strain for: (a) uniaxial tension; (b) uniaxial compression; (c) simple shear [1].

elastic region, the atoms are temporarily displaced but return to their equilibrium positions when the
load is removed. Stress (S or τ) and strain (e or γ ) in the elastic region are defined as indicated in Fig. 19.30.

n = – e 2 (19.41)
----
e 1
Poisson’s ratio (v) is the ratio of transverse (e 2 ) to direct (e 1 ) strain in tension or compression. In the
elastic region, v is between 1/4 and 1/3 for metals. The relation between stress and strain in the elastic
region is given by Hooke’s law:

S = Ee ( tension or compression) (19.42)
t = Gg ( simple shear) (19.43)

where E and G are the Young’s and shear modulus of elasticity, respectively. A small change in specific
volume (∆Vol/Vol) can be related to the elastic deformation, which is shown to be as follows for an
isotropic material (same properties in all directions):

∆Vol
------------ = e 1 12n–( ) (19.44)
Vol
The bulk modulus (K = reciprocal of compressibility) is defined as follows:

K = ∆p /  ∆Vol  (19.45)
------------
Vol 
where ∆p is the pressure acting at a particular point. For an elastic solid loaded in uniaxial compression
(S):

S
E
K = S /  ∆Vol  = ------------------------ = --------------- (19.46)
------------
Vol 
(
e 1 12n)
–
–
12n
Thus, an elastic solid is compressible as long as ν is less than 1/2, which is normally the case for metals.
Hooke’s law, Eq. (19.42), for uniaxial tension can be generalized for a three-dimensional elastic condition.
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