Page 202 - Materials Science and Engineering An Introduction

174 • Chapter 6 / Mechanical Properties of Metals

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mechanics-of-materials principles, it is possible to develop equations for s and t
in terms of s and u, as follows:
1 + cos 2u
s = s cos u = sa b (6.4a)
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2
sin 2u
t = s sin u cos u = sa b (6.4b)
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These same mechanics principles allow the transformation of stress components from
one coordinate system to another coordinate system with a different orientation. Such
treatments are beyond the scope of the present discussion.

Elastic Deformation

6.3 STRESS–STRAIN BEHAVIOR

The degree to which a structure deforms or strains depends on the magnitude of an
imposed stress. For most metals that are stressed in tension and at relatively low levels,
Hooke’s law— stress and strain are proportional to each other through the relationship
relationship between
engineering stress
and engineering s = EP (6.5)
strain for elastic
deformation (tension This is known as Hooke’s law, and the constant of proportionality E (GPa or psi) is
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and compression) the modulus of elasticity, or Young’s modulus. For most typical metals, the magnitude
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modulus of elasticity of this modulus ranges between 45 GPa (6.5 * 10 psi), for magnesium, and 407 GPa
(59 * 10 6 psi), for tungsten. Modulus of elasticity values for several metals at room
elastic deformation
temperature are presented in Table 6.1.
Deformation in which stress and strain are proportional is called elastic deformation; a
plot of stress (ordinate) versus strain (abscissa) results in a linear relationship, as shown in
Figure 6.5. The slope of this linear segment corresponds to the modulus of elasticity E. This
modulus may be thought of as stiffness, or a material’s resistance to elastic deformation.
: VMSE The greater the modulus, the stiffer the material, or the smaller the elastic strain that results
Metal Alloys

Table 6.1
Modulus of
Room-Temperature Elasticity Shear Modulus
Elastic and Shear 6 6
Moduli and Poisson’s Metal Alloy GPa 10 psi GPa 10 psi Poisson’s Ratio
Ratio for Various Aluminum 69 10 25 3.6 0.33
Metal Alloys
Brass 97 14 37 5.4 0.34
Copper 110 16 46 6.7 0.34
Magnesium 45 6.5 17 2.5 0.29
Nickel 207 30 76 11.0 0.31
Steel 207 30 83 12.0 0.30
Titanium 107 15.5 45 6.5 0.34
Tungsten 407 59 160 23.2 0.28

5 See, for example, W. F. Riley, L. D. Sturges, and D. H. Morris, Mechanics of Materials, 6th edition, Wiley, Hoboken,
NJ, 2006.
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6 The SI unit for the modulus of elasticity is gigapascal (GPa), where 1 GPa = 10 N/m = 10 MPa.
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