Page 248 - Introduction to Continuum Mechanics

232 The Elastic Solid

Example 5.4.1

(a) If for a specific material the ratio of the bulk modulus to Young's modulus is very large,
find the approximate value of Poisson's ratio.
(b) Indicate why the material of part(a) can be called incompressible.
Solution, (a) In terms of Lame's constants, we have

Combining these two equation gives

k 1
Therefore, if -=—*• «>, then Poisson's ratio v-* —.
tLy £
(b) For an arbitrary stress state, the dilatation or unit volume change is given by

If v -» —, then e-» 0. That is, the material is incompressible. It has never been observed in real

material that hydrostatic compression results in an increase of volume, therefore, the upper
limit of Poisson's ratio is v = —.

5.5 Equations of the Infinitesimal Theory of Elasticity
In section 4.7, we derived the Cauchy's equation of motion, to be satisfied by any continuum,
in the following form

where p is the density, a/ the acceleration component, p Bj the component of body force per
unit volume and 7^- the Cauchy stress components. All terms in the equation are quantities
associated with a particle which is currently at the position (jci, *2, x$ ).