Page 235 - Introduction to Continuum Mechanics

220 The Elastic Solid

In a suitable experiment, we measure the relation between o, the applied stress and e, the
change in volume per initial volume (also known as dilatation, see Eq. (3.10.2)). For an elastic
material, a linear relation exists for small e and we define the bulk modulus k, as

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A typical value of A: for steel is 138 GPa (20x 10 psi).
A torsion experiment yields another elastic constant. For example, we may subject a
cylindrical steel bar of circular cross-section of radius r to a torsional moment M t along the
cylinder axis. The bar has a length / and will twist by an angle 9 upon the application of the
moment M t. A linear relation between the angle of twist 0 and the applied moment will be
obtained for small 0. We define a shear modulus w

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where l p = n r /2 (the polar area moment of inertia). A typical value of ft for steel is 76 GPa
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(Ilxi0 psi).
For an anisotropic elastic solid, the values of these material coefficients (or material
constants) depend on the orientation of the specimen prepared from the block of material.
Inasmuch as there are infinitely many orientations possible, an important and interesting
question is how many coefficients are required to define completely the mechanical behavior
of a particular elastic solid. We shall answer this question in the following section.

5.2 Linear Elastic Solid

Within certain limits, the experiments cited in Section 5.1 have the following features in
common:
(a) The relation between the applied loading and a quantity measuring the deformation is
linear
(b) The rate of load application does not have an effect.
(c) Upon removal of the loading, the deformations disappear completely.
(d) The deformations are very small.
The characteristics (a) through (d) are now used to formulate the constitutive equation of an
ideal material, the linear elastic or Hookean elastic solid. The constitutive equation relates
the stress to relevant quantities of deformation. In this case, deformations are small and the
rate of load application has no effect. We therefore can write