Page 203 - Mechanical Behavior of Materials

204 Chapter 5 Stress–Strain Relationships and Behavior

ν
ε y =− σ x (5.25)
E
This linear relationship is also shown in Fig. 5.8. The same situation also occurs along any other
diameter, such as the z-direction.

5.3.2 Discussion
Values of the elastic modulus vary widely for different materials. Poisson’s ratio is often around
0.3 and does not vary outside the range 0 to 0.5, except under very unusual circumstances. Note
that negative values of ν imply lateral expansion during axial tension, which is unlikely. As will be
seen subsequently, ν = 0.5 implies constant volume, and values larger than 0.5 imply a decrease in
volume for tensile loading, which is also unlikely. Values of ν for various materials are included in
Table 5.2.
It should be noted that no material has perfectly linear or perfectly elastic behavior. Use of
elastic constants such as E and ν should therefore be regarded as a useful approximation, or model,
that often gives reasonably accurate answers. For example, most engineering metals can be modeled
in this way at relatively low stresses below the yield strength, beyond which the behavior becomes
nonlinear and inelastic. Also, original dimensions and cross-sectional areas are used in the present
discussion to determine stresses and strains. Such an approach is appropriate for many situations
of practical engineering interest, where dimensional changes are small. Except where otherwise
indicated, this assumption, called small-strain theory, will be used.
If a given metal is alloyed (melted together) with relatively small percentages of one or more
other metals, the effect on the elastic constants E and ν is small. Hence, where specific values of
these elastic constants are not available for a given alloy, they can be approximated as being the
same as the corresponding pure metal values, as from Table 5.2. For example, this applies for all
common aluminum alloys and titanium alloys, where the total alloying is in most cases less than
10%. But a contrary example is 70Cu-30Zn brass, where the 30% zinc causes the values to differ
significantly from those for pure copper. For low-alloy steels, which are iron with total alloying less
than 5%, the values are close to those for pure iron. However, for some high-alloy steels, such as
stainless steels, which contain at least 12% chromium and also other alloying, the values may be
affected to a modest degree. Materials handbooks, such as those listed in the references for Chapters
3 and 4, can be consulted to obtain E and ν for specific alloys. For polymers and ceramics, there
may be significant batch-to-batch variation in the elastic constants, as the values are affected by
processing.

5.3.3 Hooke’s Law for Three Dimensions
Consider the general state of stress at a point, as illustrated in Fig. 5.9. A complete description
consists of normal stresses in three directions, σ x , σ y , and σ z , and shear stresses on three planes,
τ xy , τ yz , and τ zx . Considering normal stresses first, and assuming that small-strain theory applies, the
strains caused by each component of stress can simply be added together. A stress in the x-direction
causes a strain in the x-direction of σ x /E.This σ x also causes a strain in the y-direction, from
Eq. 5.25, of −νσ x /E, and the same strain in the z-direction. Similarly, normal stresses in the y- and
z-directions each cause strains in all three directions.