Shear Stress and Strain

The third kind of stress-strain situation is called shear. The ribbon in Fig. 11.12c
is under shear stress: One part of the ribbon is being pushed up while an
adjacent part is being pushed down, producing a deformation of the ribbon.
Figure 11.18 shows a body being deformed by a shear stress. In the figure, forces
of equal magnitude but opposite direction act tangent to the surfaces of opposite
ends of the object. We define the shear stress as the force FΠacting tangent to the
surface divided by the area A on which it acts:

Shear stress, like the other two types of stress, is a force per unit area.
Figure 11.18 shows that one face of the object under shear stress is displaced
by a distance x relative to the opposite face. We define shear strain as the ratio
of the displacement x to the transverse dimension h:

In real-life situations, x is typically much smaller than h. Like all strains, shear
strain is a dimensionless number; it is a ratio of two lengths.
If the forces are small enough that Hooke’s law is obeyed, the shear strain
is proportional to the shear stress. The corresponding elastic modulus (ratio of
shear stress to shear strain) is called the shear modulus, denoted by S:

Table 11.1 gives several values of shear modulus. For a given material, S is usually one-third to one-half as large as Young’s modulus Y for tensile stress. Keep in mind that the concepts of shear stress, shear strain, and shear modulus apply to solid materials only. The reason is that shear refers to deforming an object that has a definite shape (see Fig. 11.18). This concept doesn’t apply to gases and liquids, which do not have definite shapes

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