Page 206 - Materials Science and Engineering An Introduction

178 • Chapter 6 / Mechanical Properties of Metals

Figure 6.9 Axial (z) elongation z
(positive strain) and lateral (x and y) l z
contractions (negative strains) in 2 l
response to an imposed tensile stress. 0 x l x
Solid lines represent dimensions after 2
stress application; dashed lines, before.

l 0 z

z

z y
z = l z /2
2 l
0 z
x l x /2
– =
2 l x
0 x

will be of opposite sign; therefore, the
For virtually all structural materials, P x and P z
7
negative sign is included in the preceding expression to ensure that v is positive.
1
Theoretically, Poisson’s ratio for isotropic materials should be ; furthermore, the maxi-
4
mum value for n (or the value for which there is no net volume change) is 0.50. For many
metals and other alloys, values of Poisson’s ratio range between 0.25 and 0.35. Table 6.1
shows v values for several common metallic materials.
For isotropic materials, shear and elastic moduli are related to each other and to
Poisson’s ratio according to
Relationship among
elastic parameters—
modulus of elasticity, E = 2G(1 + n) (6.9)
shear modulus, and
Poisson’s ratio
In most metals, G is about 0.4E; thus, if the value of one modulus is known, the other
may be approximated.
Many materials are elastically anisotropic; that is, the elastic behavior (i.e., the
magnitude of E) varies with crystallographic direction (see Table 3.4). For these ma-
terials, the elastic properties are completely characterized only by the specification
of several elastic constants, their number depending on characteristics of the crystal
structure. Even for isotropic materials, for complete characterization of the elastic
properties, at least two constants must be given. Because the grain orientation is
random in most polycrystalline materials, these may be considered to be isotropic;
inorganic ceramic glasses are also isotropic. The remaining discussion of mechanical
behavior assumes isotropy and polycrystallinity because this is the character of most
engineering materials.

7 Some materials (e.g., specially prepared polymer foams) when pulled in tension actually expand in the transverse
direction. In these materials, both P x and P z of Equation 6.8 are positive, and thus Poisson’s ratio is negative. Materials
that exhibit this effect are termed auxetics.