Page 243 - Introduction to Continuum Mechanics

228 The Elastic Solid

Example 5.3.3

For an isotropic material
(a) Find a relation between the first invariants of stress and strain.
(b) Use the result of part (a) to invert Hooke's law so that strain is a function of stress
Solution, (a) By adding Eqs. (5.3.6c,d,e), we have

(b) We now invert Eq. (5.3.6b) as

5.4 Young's Modulus, Poisson's Ratio, Shear Modulus, and Bulk Modulus
Equations (5.3.6) express the stress components in terms of the strain components. These
equations can be inverted, as was done in Example 5.3.3, to give

We also have, from Eq. (5.3.7)

If the state of stress is such that only one normal stress component is not zero, we call it a
uniaxial stress state. The uniaxial stress state is a good approximation to the actual state of
stress in the cylindrical bar used in the tensile test described in Section 5.1. If we take the ej
direction to be axial with TU *0 and all other 7/j = 0, then Eqs. (5.4.1) give

The ratio TU/EU, corresponding to the ratio o/e a of the tensile test described in
Section 5.1, is the Young's modulus or the modulus of elasticity E Y. Thus, from Eq. (5.4.3),