Application of Lagrangian effective material constants to the study of the thermal behavior of SAW propagation in piezoelectric crystals

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Abstract
The consistent equations of infinitesimal vibrations in piezoelectric crystals submitted to slow homogeneous temperature variations are shortly recalled. They involve so called effective material constants in Lagrangian description and their thermal derivatives up to any order. The boundary conditions of the temperature dependent problem are referred to the fixed geometry and orientation of the crystal in a well defined and known stress free static state. Then, it is no longer necessary to analytically know the influence of anisotropic thermal expansion on the actual shape and crystalline orientation of the external free surface, provided the temperature variations be homogeneous. Nevertheless, the effective coefficients have lower symmetries and different thermal derivatives than the usual coefficients simply defined in the framework of linear piezoelectricity. It has already been demonstrated that this method gives identical results with the “classical” one whenever it is possible to analytically obtain the temperature dependence of the orientation and shape of the boundary surface with respect to the crystalline axes of symmetry. The effective constants have already proved their usefulness in the field of BAW contoured resonators. We present updated values of their thermal derivatives for quartz and we show how they should be handled in the calculation of the velocity of surface waves up to the third order in terms of temperature. We give some numerical results for a few crystal orientations commonly encountered in practice