Electric field stress tensor

Equation [19] expresses the condition of equilibrium as a balance between the electrostatic pressure on ions, where the first term is simply the Maxwell electric field stress tensor,and the osmotic pressure with respect to bulk, given by the second term. Only in special cases can the integral in Eq. [17] be evaluated analytically. Several analytical approximations to the planar PB equation for asymmetric electrolytes have been suggested. " We now present three examples possessing exact analytical solutions beginning with the classic Gouy-Chapman solution. 

If analytical evaluation of the electrostatic free energy is not possible, an alternative route to the pressure is to add the electric field stress tensor to the osmotic pressure as in Eq. [19] and evaluate their sum (for a bulk z z electrolyte) ... 

From the field stress tensor components, we may write the electric and magnetic field components as... 

We can now identify the first term in (A.4) with Maxwell s stress tensor, which acts on any dielectric in an electric field. The magnitude of this force Pe is given by... 

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] ... 

Piezoelectric materials are materials that exhibit a linear relationship between electric and mechanical variables. Electric polarization is proportional to mechanical stress. The direct piezoelectric effect can be described as the ability of materials to convert mechanical stress into an electric field, and the reverse, to convert an electric field into a mechanical stress. The use of the piezoelectric effect in sensors is based on the latter property. For materials to exhibit the piezoelectric effect, the materials must be anisotropic and electrically poled ie, there must be a spontaneous electric field maintained in a particular direction throughout the material. A key feature of a piezoelectric material involves this spontaneous electric field and its disappearance above the Curie point. Only solids without a center of symmetry show this piezoelectric effect, a third-rank tensor property (14,15). 

Generally the linear approximation suffices, but, because the refractive index can be measured with considerable precision, the change in the impermeability tensor due to stress and electric field should be written as... 

The state of strain in a body is fully described by a second-rank tensor, a strain tensor , and the state of stress by a stress tensor, again of second rank. Therefore the relationships between the stress and strain tensors, i.e. the Young modulus or the compliance, are fourth-rank tensors. The relationship between the electric field and electric displacement, i.e. the permittivity, is a second-rank tensor. In general, a vector (formally regarded as a first-rank tensor) has three components, a second-rank tensor has nine components, a third-rank tensor has 27 components and a fourth-rank tensor has 81 components. 

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. 

Note that the many-body response functions are in general non-local, implying that the response (i.e. the polarization) at point r depends on the electric field at other locations. This makes sense In a system of interacting molecules the response of molecules at location r arises not only from the field at that location, but also from molecules located elsewhere that were polarized by the field, then affected other molecules by their mutual interactions. Also note that by not stressing the vector forms of and P we have sacrificed notational rigor for relative notational simplicity. In reality the response function is a tensor whose components are derived from the components of the polarization vector, and the tensor product ... / is the corresponding sum over vector components of and tensor components of /. 

It must be stressed that there are fundamental differences in the natures of the different nuclei discussed above. Si and P are spin 1/2 nuclei, and MAS yields particularly simple spectra with complete averaging of the chemical shift tensor components. The average isotropic shift values are field independent and correspond to the solution chemical shifts. AI and O, however, are quadrupolar nuclei with nonintegral spins greater than 1, and their solid-state spectra are often much more complex. Since the quadrupolar interaction depends on local electric field gradients at the nucleus studied, solid-state NMR of quadrupolar nuclei can yield further information about the local environment. More complicated spinning techniques have been introduced to average the anisotropies in the spectra of quadrupolar systems, and these new methods have increased interest in the use of such nuclei as probes of local microstructure. 

The solntion to Equations 4.75 and 4.76 yields the mechanical displacements and the electrical potential in piezoelectric medium. The above mechanical and electrical eqnations are conpled by matrix K ,j,that is represented in terms of the piezoelectric stress tensor e. As e —> 0, K ,j, —> 0 and the two sets (Equations 4.75 and 4.76) then represent pnre mechanical finite element and electrostatic field models, respectively. The sets of eqnation represented by Eqnation 4.75 and 4.76 can be solved using various commercially available packages such as ANS YS [67], PZFLEX, ABAQUS, and so forth (Section 4.7.6), all of which offer excellent postprocessing capabilities. 

Now, find the force acting on the conducting drop. The momentum fiux density in an electric field is defined by Maxwell s stress tensor [77]... 

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... 

For ionic surfactant solution the body force tensor, Pb, is not isotropic - it is the Maxwell electric stress tensor, i.e. Pb = f6bEE - i6jE l2, where E = -V is the electric field (Landau and Lifshitz 1960). The density of the electric force plays the role of a spatial body force, f, in the Navier-Stokes equation of motion (3). In the lubrication approximation the pressure in the continuous phase depends on the vertical coordinate, z, only through its osmotic part generated from the electric potential and the pressure in the middle plane (or the pressure, pn, corresponding to the case of zero potential) ... 

Particle trajectory is the result of the interaction of the particle with the electric field and the flow field. To simulate the particle trajectories, there are two approaches. The first approach is the Lagrangian tracking method, which neglects the finite size of the particles and treats them as point particles and solves the field variables without the presence of the particles [8]. In this case, only the effect of the field variables on the particle is considered. The second approach is the stress tensor approach, which includes the size effect of the particle. In this approach, the field variables are solved with the presence of the finitesized particle, and the particle translates as a result of the interaction of the particle with the electric and flow field [8]. In each incremental movement of the particle, the field variables need to be resolved. The former approach is very simple and works good to some extent, and the latter approach is accurate yet computationally expensive. 

To evaluate the resultant electromechanical force, one needs to utilize the fact that the tangential component of the electrical field at the surface of the liquid droplet vanishes, and accordingly, one may express the components of E on a plane with direction cosines n, as , = Utilizing the Cauchy s theorem relating the traction vector with the stress tensor components, one can write... 

The coupling between the hydrodynamics and the electric field therefore arises through the Maxwell stress tensor. The total electric force density comprises the sum of the Coulombic force arising from the presence of free charges... 

Hoburg and Melcher [6] demonstrated electrohydrodynamic instabilities in macroscale systems at an oil-oil interface with a discrete conductivity change at the interface under the influence of an applied electric field. In the presence of an applied electric field, charge accumulates at the fluid-fluid interface, and the electrical force on the interface is balanced by the fluid interfacial stress tensor. At a critical field strength, the electrical force exceeds the... 

The coupling factor between electrodynamics and translational mechanics is not classically used as such but as a piezoelectric voltage coefficient g (in m C ) divided by a characteristic length. In an anisotropic three-dimensional material, this coefficient is a tensor that links the stress F a to the electric field E and is equivalent to the multiplication of the coupling factor with the spatial integration of the stress (i.e., the lineic density of the force) ... 

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