Page 344 - Microsensors, MEMS and Smart Devices - Gardner Varadhan and Awadelkarim

324 SURFACE ACOUSTIC WAVES IN SOLIDS

Moreover, if we consider the stress tensor T ijto be time-dependent and acting upon
a unit cube (assumed free body), the stress analysis may be extended to deduce the
dynamical equations of motion through the sum of the acting forces. Thus,

= p- 2 (10.8)
dt

where p is the mass density, F i are the forces acting on the body per unit mass, and u i,
represents the components of particle displacements along the i-direction.

10.4.3 The Piezoelectric Effect

Within a solid medium, the mechanical forces are described by the components of the
stress field T ij, whereas the mechanical deformations are described by the components
of the strain field S ij. For small static deformations of nonpiezoelectric elastic solids,
the mechanical stress and strain fields are related according to Hooke's Law (Slobodnik
1976):
Tij=c ijk,S k, (10.9)

2
where T ij are the mechanical stress second-rank tensor components (units of N/m ), S kl
are the strain second-rank tensor components (dimensionless), and c ikl is the elastic
2
stiffness constant (N/m ) represented by a fourth-rank tensor. Taking into account the
symmetry of the tensors, the previous equation can be reduced to a matrix equation using
a single suffix. Thus, the tensor components of T, S, and c are reduced according to the
following scheme of replacement (Auld 1973a; Slobodnik 1976):
(22) 2;
(10.10)
(32) = (23) (13) = (31) (33) 3 5; (21) =

Therefore, the elastic stiffness constant can be reduced to a 6 x 6 matrix. Depending on
the crystal symmetry, these 36 constants can be reduced to a maximum of 21 independent
constants. For example, quartz and lithium niobate, which present trigonal symmetry, have
their number of independent constants reduced to just 6 (Auld 1973b):

c u C 12 C\2 C\4 0 0
C\2 c\\ C\3 —C 14 0 0
C\2 C 13 C 33 0 0 0 (10.11)
C\4 —C 14 0 C44 0 0
0 0 0 0 C44
0 0 0 C\4 C 11 –C 12 ))

In piezoelectric materials, the relation given by Equation (10.8) no longer holds true.
Coupling between the electrical and mechanical parameters gives rise to mechanical
deformation and vice versa upon the application of an electric field. The mechanical
stress relationship is thus extended to

– e kijE k (10.12)