The theory of effective properties of piezoelectric composites

The Piezoelectric Medium and Its Electromechanical Properties

“…An analogous set of adiabatic constants (measured at S = const) can also be derived using the thermodynamic functions and a combination of three arguments including ΔS (for example, ξ jr , E i and ΔS). According to experimental data, the difference between the related isothermic and adiabatic constants of the piezoelectric SCs is approximately 1 % or less [3,5]. Based on this observation, the effect of thermal fields on the elastic and electric responses of the piezoelectric SCs is often neglected, and the description of the piezoelectric effect is carried out in terms of (1.1) and (1.2) at ΔT = 0.…”

“…The first argument can be either mechanical stress σ kl or mechanical strain ξ jr , the second argument can be electric field E or electric displacement (electric flux density) D, and the third argument can be temperature T or entropy S. Obviously, these arguments are represented as second-rank (σ kl and ξ jr ), first-rank (E k and D j ) and zeroth-rank (scalars T and S) tensors. It is also possible to develop additional arguments in terms of magnetic field H or magnetic induction B [5]. Based on the thermodynamic functions and relations between the arguments of the mechanical, electric and thermal states [2,3], one can describe the linked "response -actions" in terms of ξ jr , E i and T as follows: In (1.1)-(1.3) the constants written in the tensor form are elastic moduli at constant electric field (c E kl jr ), piezoelectric coefficients (e ikl ), coefficients of thermal stress (β kl ), dielectric permittivities at constant mechanical strain (ε ξ i f ), pyroelectric coefficients ( p i ).…”

“…Based on this observation, the effect of thermal fields on the elastic and electric responses of the piezoelectric SCs is often neglected, and the description of the piezoelectric effect is carried out in terms of (1.1) and (1.2) at ΔT = 0. Equations (1.1) and (1.2) are written in the matrix form [2,3] as follows: 5) and the conventional transition from two subscripts to one subscript follows Vogt's rule [2][3][4]12]. Equations (1.4) and (1.5) represent the first pair of piezoelectric equations that link two variables, mechanical strain ξ and electric field E. The first term in the right part of (1.5), P k = e kl ξ l , describes the piezoelectric polarisation produced by an external mechanical strain as a result of the direct piezoelectric effect.…”

“…Each piezoelectric coefficient characterises the relationship between components of two fields that are described by first-and secondrank tensors, and all the piezoelectric coefficients from (1.4)-(1.11) are defined to be components of the third-rank tensors, but written in the matrix form (i.e., with two subscripts) [2][3][4][5] only. These individual piezoelectric coefficients can be used to aid in the selection of piezoelectric materials for specific applications.…”

“…The piezoelectric response of poled ferroelectric ceramic (FC) and composite samples are even more complicated in comparison to ferroelectric SCs due to microstructural, domain, orientation, intrinsic and extrinsic contributions, etc. [5][6][7][8][9][10][11].Piezoelectricity was initially discovered as a physical phenomenon and has led to a range of new adapted materials and technologies in the past 130 years. It was a long and sometimes difficult path to improve the initially minor effect to a point where is was a real and even superior competitor to the electro-dynamic magnetic principal, which for a long time was the only effective way to transform electrical energy or signals into mechanical ones or vice versa [11].…”