Page 241 - Introduction to Continuum Mechanics

226 The Elastic Solid

It can be shown that any isotropic fourth order tensor can be represented as a linear
combination of the above three isotropic fourth order tensors (we omit the rather lengthy proof
here. In part B of this chapter, we shall give the detail reductions of the general Cp/ to the
isotropic case). Thus, for an isotropic linearly elastic material, the elasticity tensor C,yw can
be written as a linear combination of A^\, 8^, and //p/.

where A , a, and ft are constants. Substituting Eq. (5.3.5) into Eq. (i) and since

we have

Or, denoting a + ft by 2^ , we have

or, in direct notation

where e = EM = first scalar invariant of E. In long form, Eqs. (5.3,6) are given by

Equations (5.36) are the constitutive equations for a linear isotropic elastic solid. The two
material constants A and ft are known as Lame's coefficients, or, Lame's constants. Since Ejy
are dimensionless, A and/* are of the same dimension as the stress tensor, force per unit area.
For a given real material, the values of the Lame's constants are to be determined from suitable
experiments. We shall have more to say about this later.