Page 236 - Introduction to Continuum Mechanics

Linear Elastic Solid 221

where T is the Cauchy stress tensor, E is the infinitesimal strain tensor, with T (0) = 0. If in
addition, the function is to be linear, then we have, in component form

The above nine equations can be written compactly as

Since TJy and EJJ are components of second-order tensors, from the quotient rule discussed
in Sect, 2B14, we know that CJJM are components of a fourth-order tensor, here known as the
elasticity tensor. The values of these components with respect to the primed basis e/ and the
unprimed basis e/ are related by the transformation law

(See Sect. 2B14). If the body is homogeneous, that is, the mechanical properties are the same
for every particle in the body, then C,^/ are constants (i.e., independent of position). We shall
be concerned only with homogeneous bodies.
There are 81 coefficients in Eq. (5.2.2). However, since Ey = £y/, we can always combine
ne
the sum of two terms such as C\niE\i + C\\2\ £21 mto ° term, (C\n2 + Cmi )En so tnat
+
(£-1112 £1121) becomes one independent coefficient. Equivalently, we can simply take
=
£1112 ^1121- Thus, due to the symmetry of strain tensor, we have

Eqs. (5.2.4) reduce the number of independent C,y# from 81 to 54.
We shall consider only the cases where the stress tensor is symmetric, i.e.,

as a consequence,

Eqs. (5.2.6) further reduce the number of independent coefficient by 18. Thus, we have for
the general case of a linear elastic body a maximum of 36 material coefficients.