Page 244 - Introduction to Continuum Mechanics

Young's Modulus, Poisson's Ratio, Shear Modulus, and Bulk Modulus 229

The ratio —Eyi/Eu and -E-^/En, corresponding to the ratio —£d/£ a of the same tensile
test, is the Poisson's ratio. Thus, from Eq. (5.4.4)

Using Eqs. (5,4.6) and (5.4.7) we write Eq. (5.4.1) in the frequently used engineering form

Even though there are three material constants in Eq. (5.4.8), it is important to remember
that only two of them are independent for the isotropic material. In fact, by eliminating A from
Eqs. (5.4.6) and (5.4.7), we have the important relation

Using this relation, we can also write Eq. (5.4.1) as

If the state of stress is such that only one pair of shear stresses is not zero, it is called a
T
simple shear stress state. This state of stress may be described by 7\2 = TI\ — and
Eq. (5.4.8d) gives

Defining the shear modulus G, as the ratio of the shearing stress r in simple shear to the
small decrease in angle between elements that are initially in the ej and e 2 directions, we have