We use indifferently the intrinsic (with no indices) notation or the indexed Cartesian tensor notation. Here the symbol or a superimposed dot denotes the partial time derivative. The symbol stands for the gradient (e.g., in components,); div means the divergence of second order tensors (e.g.,). provides a system of rectangular coordinates and the time parametrization by the Newtonian time t. Symbol u will denote the elastic displacement. Accordingly, in any regular material point of the considered piezoelectric body the local balance of linear momentum and Gauss equation read:

.(2.1)

Here is the linear momentum, σ is Cauchy’s (symmetric) stress tensor, D is the electric displacement, is the constant matter density, and u is the elastic displacement. Any body force is discarded. Only small strains and weak electromagnetic fields are considered. The theory is linear so that both electromagnetic ponderomotive force and couple that are basically quadratic in the fields are discarded (for these see Maugin, 1988 [16]). The electric framework is that of quasi-electrostatics (no electromagnetic inertia, Maxwell’s equations reduced to (2.1 2) and, so that the electric field vector E derives from the potential i.e., , but all fields still depend on time). The electric displacement vector D is such that

where is the vacuum electric permeability, and P is the electric polarization vector per unit volume. LorentzHeaviside units are used (no factor 4π). Natural boundary conditions associated with Equation (2.1) read

These hold for a mechanically free surface, and a connection to an external electric field in the vacuum outside the body, the symbolism indicating the finite jump of the enclosed quantity at the bounding surface, i.e., , where denotes the uniform limit of the function A in approaching the limit surface from the positive and negative sides of the surface, respectively, and n is the unit normal to the boundary oriented from the minus to the plus side. Whenever this surface is electroded fixing the electric potential on it, say (zero potential), then (2.3a) are replaced by

,(2.3b)

where w is an imposed surface density of electric charges. This is the case mostly considered in the present work. Type (2.3a) is briefly considered in Section 4 below.

In the presence of dissipation of the viscous and electric-relaxation type the constitutive equations for σ and D are given in Cartesian tensor components by

,(2.4)

The nondissipative contributions here derivable from the volume energy are the standard ones given by the theory of linear piezoelectricity (cf.Maugin, 1988 [16]; Chapter 4):

,(2.5)

,(2.6.1)

where (quadratic energy)

,(2.7)

with the following symmetries:

,(2.8)

for the tensorial coefficients of elasticity, piezoelectricity and dielectricity, respectively. The field e of components stands for the small strain tensor, and parentheses around a set of indices indicate the operation of symmetrization.

Simple examples of dissipative contributions in the context of Bleustein-Gulyaev waves are given by (cf. Maugin et al, 1992 [17]); a superimposed dot is the same as the partial time derivative)

,(2.9)

with positive viscosity and relaxation constant. A symmetry class (no center of symmetry) allowing for the existence of piezoelectricity must be selected for (2.8). Simple isotropy has been considered for the dissipative effects, bearing no restriction for the application in this paper.

For the case of Bleustein-Gulyaev surface acoustic waves (SAWs) with elastic displacement polarized orthogonally to the sagittal plane spanned by the propagation direction and the in-depth coordinate, the only surviving components of (2.3) are given by (compare the nondissipatif case in Maugin and Rousseau, 2010 [2])

,(2.10)

with

.(2.12)

Here, and are the only intervening elasticity, piezoelectric and dielectric constants (in the socalled Voigt’s notation commonly used in piezoelectricity).

Of course, the corresponding wave problem becomes dispersive since the polynomials of differentiation are no longer homogeneous.

If we multiply (2.1.1) by and sum over indices, we obtain

,(2.13)

or, on account of (2.4),

.(2.14)

But (2.1.2) yields

Subtracting the (vanishing) right-hand side of (2.15) from (2.14) yields the (non)-conservation of energy in the form

Remark: Equation (2.16) has a remarkable symmetric structure for mechanical and electric effects. Quite often, however, the Poynting vector for quasi-electrostatic fields is written as

,(2.17)

[cf. Maugin, 1988, Equation (4.6.14), p.238 [16]; or Eringen and Maugin, 1990, Equation (7.3.15) [18], p. 246]. This can be accommodated by Equation (2.16) by a re-definition of the energy W. For instance, we can rewrite (2.16) as

With

Obviously, (2.18) is less convenient than (2.16) for our purpose. While the SAW problem is based on an exploitation of Equation (2.1) and accompanying boundary conditions, that of the formulation of the mechanics of associated quasi-particles (subsequent work) is based on an exploitation of Equation (2.16) and of an analogous spatial co-vectorial equation known as the conservation (or non-conservation) of wave momentum. (general concept in Maugin, 2011 [19]; Chapter 12), once the SAW solution is known, just like in a post-processing procedure. This completes the thermo-electromechanical modeling per se.

In this case, after introduction of an effective scalar electric potential, the surviving Equation (2.1) for the fields are

.(3.1)

with

,(3.2)

where K is the so-called electromechanical coupling factor. The boundary conditions (2.3b1,3) at the mechanically free, but electrically grounded surface, yield

For the half-space, the SAW solution generally reads

.(3.6)

From (3.1.1) and (3.1.2) there follows that

and (3.1.2) is not a propagation equation

.(3.8)

That is,

.(3.9)

The boundary conditions (3.4) yield a nontrivial solution for

.(3.10)

The first of these has to be substituted in (3.7.1) on account of (3.10)₂. This yields

from which there follows the “dispersion relation” of Bleustein-Gulyaev surface waves for the present electric boundary condition:

.(3.11)

Noting that, the real BG SAW for can be written as the solution

,(3.12)

with

.(3.13)

For a vanishing electromechanical coupling coefficient, the surface wave degenerates into a face shear wave (cf. Equation (3.12.1) for). Consistently with (3.11), we note and the wave parameters (velocity, wave number and wavelength) of this solution. Those corresponding to a dissipatively perturbed solution will be denoted with an additional subscript d, e.g., , etc.

For the sake of simplicity we discard dielectric relaxation. Constitutive Equations (2.10) and (2.11) reduce to

with.

We follow the same strategy as for the nondissipative case recalled in the preceding paragraph. The ansatz SAW solution is like in Equations (3.5)-(3.6) but with all k’s now possibly complex. The dimensionless parameter defined by

,(3.16)

that compares the viscous relaxation time to the time scale of the wave motion, is considered as an infinitesimally small quantity of the first order, so that in the sequel. Relation (3.1.3) is still valid, so that together with (3.1) and (3.2) Equations (2.1) reduce to the following system:

,(3.17)

for, with conditions (2.3.b1,3) at, i.e.,

.(3.18)

Equations (3.7) are replaced by the following ones:

and

.(3.20)

Whence,

.(3.21)

Finally, (3.11.1) is replaced by the following—still exact—complex (true) dispersion relation

,(3.22)

with defined in (3.11.2). Let the complex wave-number solution of (3.22). We have thus

where.

Now we look for approximations of in terms of. We write for the left-hand side of (3.23)

,(3.24)

or at order,

.(3.25)

At the same order of approximation the right-hand side of (3.23) yields

.(3.26)

Identifying the like powers of from (3.25) and (3.26), we can draw the following conclusions.

•     At order zero in we obviously have the solution provided by (3.11);

•     At order one in, we have (K being small by itself) :

;(3.27)

•     At order two in, we obtain:

,(3.28)

with

This solution is completed by applying the same approximation to the relation given by (3.9).

That is, we can write

This manipulation yields

We also show that

.(3.32)

The SAW solution finally reads

,(3.33)

,(3.34)

where superscripts I and R denote imaginary and real parts, respectively. Summing up, we have up to order:

,(3.35)

Globally, we see that at order:

•     yields attenuation in the propagation direction. This is of order of.

•     yields the expected exponential attenuation in depth for a surface wave.

•     yields a superimposed oscillation in depth (due to the viscous behavior).

We also remark that at order, describes dispersion in the propagation direction. This dispersion that varies like, results from the viscous behavior.