Material displacement functions vectors

Where u is the Fourier transformed displacement vector, p is the density of material, V is the three dimensional differential operator and O) is the angular frequency. The complex frequency dependent functions X and p are related to the relaxation functions of the material and //(f). 

The element common to the function of all piezoelectric transducers is particle displacement within a solid. Relative particle displacements cause the generation of restoring stress forces and for piezoelectric materials cause the generation of electrical fields as well. Displacement is defined as the vector indicating the difference between equilibrium and perturbed positions of a solid particle. However, this vector is not invariant to translational motion so that strain is used to represent relative displacement. Strain and displacement are related by the equation ... 

In Eqs. (1-5), the vector x and the scalar t denote spatial coordinates (x ) and time, respectively. The vector u signifies the displacement vector with contravariant components u, the components s denote the contavariant components of the stress tensor, and the vectors , Uo, and vo are prescribed functions. The vectors u(x, f) and t(x, t) represent time dependent prescribed boundary conditions on the parts T and of the boundary F, respectively, and p denotes mass density. Finally, n, signify the components of the outward unit normal to F(. It should be noted that this set of equations is supplemented by the equilibrium of angular momentum (a generalized symmetry condition on the stress tensor), the material law, and the kinematic relationships between strains/rigid rotations and the spatial derivatives of displacements. 

In the linear description of the motion of solid bodies it is assumed that the displacements and their gradients are infinitely small and that the material is linearly elastic. In addition, it is also assumed that the nature of the boundary conditions remains unchanged during the entire deformation process. These assumptions imply that the displacement vector u is a linear function of the applied load vector F, i.e., if the applied load vector is a scalar multiple aF then the corresponding displacements are au. 

The ratio between the dielectric displacement vector (D) and that of the electric field strength (E) is called the dielectric permittivity (e) of a material = f o = (D/E). Here So = 8.8542 lO AsA m indicates the permittivity of vacuum and (Sr) is the so-called relative permittivity of the material. As e, depends on magnirnde and spatial arrangement of all electric charges included in a material, it changes if gas is either adsorbed or desorbed in the material. Indeed, the absolnte value of (Sr) can be considered as measure, i. e. a linear function of number of gas molecules adsorbed in the material [1, Chap. 6,9, 10]. 

Ordinary liquids and liquid crystals are nearly incompressible. In ordinary fluid dynamics the incompressibility approximation under the constraint div v = 0 has frequently been utilized. In a soft elastomer such as vulcanized rubber, where shear modulus is very small as compared with bulk modulus, the incompressibility approximation has also been usefully employed. The constraint of the incompressibility approximation, div v = 0 for ordinary fluids or divergence of displacement vector for elastic (isotropic) materials, does not modify any other terms of the equations of motion div v = 0, or divergence of displacement vector, is a solutirai of the equations of motion, provided that pressure p is chosen as an appropriate harmoiuc function (V p = 0). However, for anisotropic matters, such as liquid crystals or anisotropic solids (crystals), since the div v = 0 or its elastic version cannot be a special solution of equations of motion, the incompressibility approximation requires a careful consideration [12, 18]. 

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