Tensor piezoelectric strain

The piezoelectric strain tensor d and the piezoelectric voltage tensor g are classically used for linking the mechanical variables to the electrical ones. The piezoelectric strain tensor links the electric field E to the strain and the piezoelectric voltage tensor links... 

Tables 4.4-3-4.4-21 are arranged according to piezoelectric classes in order of decreasing symmetry (see Table 4.4-2), and alphabetically within each class. They contain a number of columns placed on two pages, even and odd. The following properties are presented for each dielectric material density q, Mohs hardness, thermal conductivity k, static dielectric constant Sij, dissipation factor tanS at various temperatures and frequencies, elastic stiffness Cmn, elastic compliance s n (for isotropic and cubic materials only), piezoelectric strain tensor di , elastooptic tensor electrooptic coefficients r k (the lat-...

General Static dielectric constant Dissipation factor Elastic stiffness tensor Piezoelectric strain tensor Elastooptic tensor... 

From tensor algebra, the tensor property relating two associated tensor quantities, of rank / and rank g, is of rank (/-b g). Hence, the physical property connecting /, and aj is the third-rank tensor known as the piezoelectric effect, and it contains 3 = 27 piezoelectric strain coefficients, dyk. The piezoelectric coefficients are products of electrostriction constants, the electric polarization, and components of the dielectric tensor. 

When written in matrix form these equations relate the properties to the crystallographic directions. For ceramics and other crystals the piezoelectric constants are anisotropic. For this reason, they are expressed in tensor form. The directional properties are defined by the use of subscripts. For example, d i is the piezoelectric strain coefficient where the stress or strain direction is along the 1 axis and the dielectric displacement or electric field direction is along the 3 axis (i.e., the electrodes are perpendicular to the 3 axis). The notation can be understood by looking at Figure 31.19. 

Piezoelectricity links the fields of electricity and acoustics. Piezoelectric materials are key components in acoustic transducers such as microphones, loudspeakers, transmitters, burglar alarms and submarine detectors. The Curie brothers [7] in 1880 first observed the phenomenon in quartz crystals. Langevin [8] in 1916 first reported the application of piezoelectrics to acoustics. He used piezoelectric quartz crystals in an ultrasonic sending and detection system - a forerunner to present day sonar systems. Subsequently, other materials with piezoelectric properties were discovered. These included the crystal Rochelle salt [9], the ceramics lead barium titanate/zirconate (pzt) and barium titanate [10] and the polymer poly(vinylidene fluoride) [11]. Other polymers such as nylon 11 [12], poly(vinyl chloride) [13] and poly (vinyl fluoride) [14] exhibit piezoelectric behavior, but to a much smaller extent. Strain constants characterize the piezoelectric response. These relate a vector quantity, the electrical field, to a tensor quantity, the mechanical stress (or strain). In this convention, the film orientation direction is denoted by 1, the width by 2 and the thickness by 3. Thus, the piezoelectric strain constant dl3 refers to a polymer film held in the orientation direction with the electrical field applied parallel to the thickness or 3 direction. The requirements for observing piezoelectricity in materials are a non-symmetric unit cell and a net dipole movement in the structure. There are 32-point groups, but only 30 of these have non-symmetric unit cells and are therefore capable of exhibiting piezoelectricity. Further, only 10 out of these twenty point groups exhibit both piezoelectricity and pyroelectricity. The piezoelectric strain constant, d, is related to the piezoelectric stress coefficient, g, by... 

Similarly in the converse piezoelectric phenomena each of the 6 components of the strain tensor is related to each of the 3 components of the electric field... 

Since the stress and strain tensors are symmetric. the piezoelectric coefficients can be converted from tensor to matrix notation. Table 13 provides the piezoelectric matrices for a-quartz together with several values rfyk and Cjjk. including the corresponding temperature coefficients [259], [261]. 

The direct effect coefficients are defined by the derivatives (5D/SX) = d (piezoelectric strain coefficient), (5D/5x) = e, -(5E/5X) = g (piezoelectric voltage constant) and -(5E/5x) = h. The converse-effect coefficients are defined by the derivatives (8x/5E) = d, (5x/5D) = g, -(5X/5E) = e, and -(5X/5D) = h. As the piezoelectric coefficients are higher-rank tensors, their mathematical treatment is rather tedious. Fortunately, in higher symmetric crystals the number of tensorial components will be drastically reduced due to symmetry constraints. An example is shown below. 

Electrostriction refers to the elastic deformation of all dielectric materials upwn the application of an electric field. Unlike piezoelectricity, the electrostrictive strain is quadratic to the electric field and reversal of the field doesn t reverse the strain direction. The basic phenomenology of electrostriction in materials is discussed in detail in many texts (Lines and Glass 1977 Jona and Shirane 1962 Mason 1958 Mason 1950). For a dielectric material under isothermal, adiabatic and stress-free conditions, upon the application of an electric field E t, the strain tensor canbe written as ... 

P. Curie and J. Curie discovered the piezoelectric effect in 1880. It was found that, when a compressive or a tensile force was applied on some crystals along some special directions (for example, a quartz) electrical charges could be created on the corresponding surfaces of the crystal and the size of the created charge was proportional to the strength of the apphed force. This phenomenon is called the piezoelectric effect . All ferroelectric crystals show a piezoelectric effect. The piezoelectric effect can be described by piezoelectric equations. On the basis of thermodynamic principles, piezoelectric equations can be derived (e.g., see Xu, 1991). These equations describe linear relationships between the four variables stress tensor [T], strain tensor [S], electrical field vector E and electric displacement vector D. The piezoelectric equations can be expressed as four kinds of equations, depended on the variables. Selecting E and Tas variables, we have ... 

We note that, sometimes the symmetric stress and strain tensors with six components are represented as six element vectors and S, where A = l/2(k + j)5p, + [9- a + j)](l - dp,), i.e., tire 11- 1 22- 2 33->3 23 = 32- 4 13 = 31 5 12 = 21 6 transformations are used. ° In this case, the piezoelectric constants are formally expressed as 3 x 6 element second-rank tensors. These notations are simpler, but much less transparent than the third-rank tensor notation, so in the following, we will keep the mathematically more transparent notation. 

Figure 1 shows tbe busk princ of the two-port piezoelectric transducer 2. The general transducer has an electrical port and a mechanical port, where the parameters at tbe mechanical port are fotoe (stress tensors) or elastic displacement (strain tensors), and tbe parameters at the electrical port are voltage, current, and surface charge. The transducer can be driven bom either port with another port driving a load. The basic equations at the transducer for a finite element of material are given by... 

The mechanical properties of materials involve various concepts such as hardness, stiffness, and piezoelectric constants, Young s and bulk modulus, and yield strength. The solids are deformed under the effect of external forces and the deformation is described by the physical quantity strain. The internal mechanical force system that resists the deformation and tends to return the solid to its undeformed initial state is described by the physical quantity stress. Within the elastic limit, where a complete recoverability from strain is achieved with removal of stress, stress g is proportional to strain e. The generalized Hooke s law gives each of the stress tensor components as linear functions of the strain tensor components as... 

For many problems it is convenient to separate the piezoelectric (i.e., strain induced) polarization P from electric-field-induced polarizations by defining D = P + fi , where s is the permittivity tensor. For uniaxial strain and electric field along the 1 axis, when the material is described by Eq. (4.1) with the E term omitted. 

The piezoelectric constant eik is a third-rank tensor and vanishes when the material has a center of symmetry. When the strain is not uniform, however, a higher order piezoelectricity appears in proportion to the... 

The most useful piezoelectric constant is the tensor which relates electric polarization to the stress causing the polarisation. The d-constant is also identified to the derivative of the resulting strain with respect to the applied electric field [24] ... 

In practice the energy transfer electrical-to-mechanical (or vice versa) occurs in a complex 3-dimensional way. The strains caused by applied electrical or mechanical stresses have components in three orthogonal directions necessitating the description of the piezoelectric effect in terms of tensors, as outlined below. 

S]). The direct piezoelectric effect is the production of electric displacement by the application of a mechanical stress the converse piezoelectric effect results in the production of a strain when an electric field is applied to a piezoelectric crystal. The relation between stress and strain, expressed by Equation 2.7, is indicated by the term Elasticity. Numbers in square brackets show the ranks of the crystal property tensors the piezoelectric coefficients are 3rd-rank tensors, and the elastic stiffnesses are 4th-rank tensors. Numbers in parentheses identify Ist-rank tensors (vectors, such as electric field and electric displacement), and 2nd-rai tensors (stress and strain). Note that one could expand this representation to include thermal variables (see [5]) and magnetic variables. 

In the literature we find not only the piezoelectric constants discussed above but also the inverse tensors, less directly related to experiment, which characterize the stresses resulting from an electrostatic field, as well as the polarization resulting from a strain ... 

---