Wave Theory In Pyroelectric Medium

Pyroelectricity is polarization in dielectrics due to the uneven temperature distribution. There are many applications of pyroelectrics in MEMs. Since, with the scale decreasing, heat has an big influence on electricity and elasticity, it is necessary to take account of the pyroelectricity and study the temperature field and interactions with electric-elastic field.It is necessary to consider mechanical, electric and temperature fields in the research of pyroelectric medium. Though there are so many researches and applications of pyro-electric/piezoelectric medium, the reflection/transmission problems with quasi-static electric field is unsolved by far. At the same time, the temperature field and its relationships with elastic and electric fields in pyroelectrics are unsystematic and unclear. In previous researches, generally, the effects of temperature are only taken account with "stiffened method" and analysis about temperature wave is ignored, especially in reflection/transmission problems. An Indian researcher published one paper on bulk waves in pyroelectrics, but some unreasonable analysis exists in his works. Therefore, it is believed that papers we presented are the pioneering works in this field. In this thesis, the fundamental problems are investigated systematically that includes:1. Based on the generalized heat conduction theory and quasi-static electric assumption, the waves in the unbounded pyroelectric medium are studied. We use three slightly different versions of the governing equations to study the problem. It is found that there are four wave modes in the pyroelectric medium, three of which are in the form of strains or displacements and one of which is in the form of temperature disturbances. Because the electric field is quasi-static, there is no independent electric wave mode. However, the electric wave can propagate with other modes through constitutive relationship. In anisotropic plane, phase velocity surface and slow surface are related to wave propagation direction as well as material constants; in isotropic plane, only material constants are concerned. Piezo/pyroelectricitic constants have bigger influence on elastic wave modes, while only a little effects on temperature wave. The relaxation time, in our discussed range, plays big role on temperature wave. The L-S theory as well as inertial entropy theory are investigated at the same time. Though the physical meanings of temperature terms having twice derivative with time are different, they can turn into the same mathematical form by ignoring the displacement and electric potential terms containing relaxation time. In inertial entropy theory, when coefficient exceeds some value, the heat inertial term can amplify elastic wave, at the same time, the temperature is still an attenuating wave.2.The general theory of inhomogeneous waves in pyroelectric medium is addressed firstly. The character of inhomogeneous wave lies in that its propagation direction is different from the biggest attenuation direction, between which attenuation angle is defined. The complex-valued wave vector is determined by four parameters. The attenuation angle is a material constant, however, so far, there is no method to determine its value. By specifying the propagation angle and attenuation angle as independent parameters and guaranteeing the acquired wave is not amplifying, the range of attenuation angle is found to be confined in (-90°,90°). Further analysis shows that in anisotropic plane, the positive and negative attenuation angle have different effect on waves, while, in isotropic plane, they are the same. When relaxation time is as small as enough, the inertial theory and L-S theory almost have the same results.3. The reflection/transmission problems are formulated mathematically for pyroelectric mediums and illustrative computation is presented for the reflection case. Unlike the classical reflection/transmission problems with equal number of wave modes and boundary conditions, pyroelectric mediums have their special character on this topic. There are only four wave modes in the pyroelectric medium, and the electric wave doesn't have independent wave modes only propagating with other wave modes through constitutive relations. We proposed four methods to solve this problem. Method 1: By adopting method Burkov used in piezoelectric medium, we introduce an unknown parameter to electric potential and make the induced electric field to satisfy boundary conditions. Whereas, it is found that such obtained reflected/transmitted wave don't satisfy motion equation; Method 2: Imposing free charges on boundary: the reflected/transmitted waves satisfy motion equations and mechanical and thermal boundary conditions. Because the electric charges appear on the boundary face, generally the waves can't satisfy the prescribed electric boundary conditions; Method 3: By slightly changing the Gauss' equation and introducing a tiny number, the electric field can have its wave mode and have ignored effects on other wave modes. The reflected and transmitted waves satisfy both motion equations and boundary conditions; Method 4: Surface wave is introduced in this problem and there is one unknown parameter for surface wave. Therefore, surface wave, together with reflected and transmitted waves, satisfies boundary conditions. It is seen that surface wave method can solve the reflection/transmission prob- lems with satisfactory requirements both physically and mathematically. It is seen from comparison results that surface wave method is different from others in reflected temperature wave amplitude coefficient, and has a smooth curve in propagation angle domain; small parameter method is very close to surface wave method in reflected elastic wave amplitude coefficients. With these methods, we can formulate a solvable system of equations in amplitude coefficients, by which the generalized Snell laws can be deduced. This equation set does not include any space coordinate and thus agrees with semi-infinite space assumption. According to this equation set, operator L is defined. The regularity requires that there must be three mode waves generated at the interface when some wave falls on the interface in two dimensional case. The prerequisite of regularity in L is used throughout the analysis. In the case of selection of incidence, reflection and transmission waves, two kinds of criteria are used, one of which is wave vector+radiation law and the other is energy flux vector+radiation law. We used the first but tested the results with the second criterion. We found that, generally, both criterions are in agreement with each other in the case of mechanical wave modes, but contradict in the case of temperature wave. On the basis of inhomogeneous wave theory, all phenomena are considered as functions defined in the domain of propagation angle and attenuation angle. We found that the attenuation play a big role on the regularity on L. The incident waves of mechanical modes can generate temperature mode wave, but this mode wave is very tiny even to be ignored, which can be seen from amplitude or energy coefficient plots. It is also found that the incident temperature mode wave can't generate any reflected waves.4. The fundamentals of energy process in the pyroelectric medium with generalized heat conduction theory is proposed. The energy transformation in an arbitrary instant is described explicitly by the energy conservation relation in integral form. From this relation, it is found that the energy density consists of several parts: the electric energy, the heat energy, the mechanical potential energy and the kinetic energy. The dissipation energy is attributed by the heat conduction, relaxation time and pyroelectric effect. The energy flux is also obtained explicitly. The heat generated or dissipation energy is found to equal to the reduction of the entire energy within a fixed volume plus the reduction of this energy flux outward the surface bounding this volume. Further, some applications are performed. By using the definition of energy flux and assuming the boundary conditions of stress free, electric charge free and heat insulated, we arrived at the conservation law of energy flux component normal to boundary surface. In the framework of inhomogeneous wave, we expressed the individual energy quantities explicitly and defined the energy velocity. Results in the domain of propagation angle and attenuation angle are presented with plots. It is found that the averaged energy velocity is a little more than the corresponding phase velocity, attenuation angle plays more role on energy velocity and positive and negative attenuation angles have different effects.