Equation electromechanical

Apparently low values of breakdown strength of many rubbery materials are described quantitatively by Equation (6.15), and most plastics fail by the electromechanical mechanism at high temperatures. 

Koh et al. [6] have rigorously modeled the electromechanics of this interaction for the simplified case of uniform biaxial stretching of an incompressible polymer film including many important effects such as the nonlinear stiffness behavior of the polymer film and the variation in breakdown field with the state of strain. With regard to the latter effect, Pelrine et al. [5] showed the dramatic effect of prestrain on the performance of dielectric elastomers (specifically silicones and acrylics) as actuators. We would expect the same breakdown enhancement effects to be involved with regard to power generation. There are many additional effects that may be important, such as electrical and mechanical loss mechanisms, interaction with the environment or circuits, frequency, and temperature-dependent effects on material parameters. The analysis by Koh provides the state equations... 

By using equations 12.3 and 12.4, the efficiency of energy conversion, representing the electromechanical coupling factor, is given by ... 

In attempting to use these equations to solve problems, a complete and consistent set of electromechanical boundary conditions are needed. Tiersten [5] provides such a derivation, though solution of anything beyond simple problems requires computation. Fortunately, the mechanical and electrical boundary crmditions are usually separable , = u or M,Ty = niTij and n,D, = iiiD or p = p. The starred terms represent constants, time-varying values, or values defined within another media adjacent to the piezoelectric material. The outward vector normal to the boundary surface is given by... 

So, the process for a polypyrrole film in a solution containing CP anions can be envisaged as stated above by Equation 16.1. The oxidized polymer is a gel and the electrons are transferred from, or toward, the metal in contact with the conducting polymer. The oxidized polymer [(PPy" )s(Cl ) (H20)m]gei is a nonstoichiometric compound the Cl content can be increased (or decreased) under control of anodic (or cathodic) charges (ne in Equation 16.1). The electrochemical equipment allows a continuous, reversible, and infinitesimal control of the oxidized material composition (a unique fact related to electromechanical actuators) by the flow of constant anodic or constant cathodic currents (constant charge per time unit), by reversing the direction of the current flow, or by the flow of infinitesimal charges, respectively. 

Now, using the presented propositions and electromechanical analogies, an approach to non-Newtonian behaviors and to electroviscoelasticity is to be introduced. If Equation (15.13) is applied to the droplet when it is stopped, for example, as a result of an interaction with some periodical physical field, the term on the left-hand side becomes equal to zero. 

Now, if the electromagnetic force is assumed to be the incident (external) force, which initiates the mechanical disturbance, then the oscillation of the continuum particles (molecules surrounding the droplet—film structure) is described by the differential equation (15.19), where o> is the frequency of the incident oscillations. After a certain time, the oscillations of the free oscillators wo (molecules surrounding the droplet—film stiuctuie) tune with the incident oscillator frequency ft. This process of tuning between free oscillations of the environmental oscillators and the incident oscillations of the electromechanical oscillator can be expressed as... 

Equations (2.24)-(2.30) provide a fundamental guideline to predict the direction of bending depending upon the operational voltage in connection with the surface reactions involved. The CV results reflect the associated surface reactions and the related electromechanical behavior of IPMCs. 

The electromechanical coupling within IPMCs is captured with the following equation [Nemat-Nasser and Li (2000)]... 

These normalized electromechanical equations are applicable to any chemistry, and allow you tailor the optimal single-crystal intrinsic response. Furthermore, any improvement on the chemistry of a ferroelectric material will asymptotically converge to an optimal orientation of 54.76° for materials with weak anisotropy, or to zero in the limit of A.. = 2/3. 

The electromechanical coupling factor ksi is calculated from the v value and the antiresonance frequency /a through Eq. 34. Especially in low-coupling piezoelectric materials, the following approximate equation is available ... 

Book content is otganized in seven chapters and one Appendix. Chapter 1 is devoted to the fnndamental principles of piezoelectricity and its application including related histoiy of phenomenon discoveiy. A brief description of crystallography and tensor analysis needed for the piezoelectricity forms the content of Chap. 2. Covariant and contravariant formulation of tensor analysis is omitted in the new edition with respect to the old one. Chapter 3 is focused on the definition and basic properties of linear elastic properties of solids. Necessary thermodynamic description of piezoelectricity, definition of coupled field material coefficients and linear constitutive equations are discussed in Chap. 4. Piezoelectricity and its properties, tensor coefficients and their difierent possibilities, ferroelectricity, ferroics and physical models of it are given in Chap. 5. Chapter 6. is substantially enlarged in this new edition and it is focused especially on non-linear phenomena in electroelasticity. Chapter 7. has been also enlarged due to mary new materials and their properties which appeared since the last book edition in 1980. This chapter includes lot of helpful tables with the material data for the most today s applied materials. Finally, Appendix contains material tensor tables for the electromechanical coefficients listed in matrix form for reader s easy use and convenience. 

The description of the transfer characteristic of solid-state actuators can be generalised if the system equations which have been introduced in Sect. 6.9.2 are not interpreted as electromechanical equivalent circuit diagrams but as signal flow charts [335]. The result is shown in Fig. 6.134. 

The final determining factor for a material s degree of piezoelectric response is the ability of the polymer to strain with applied stress. Since the remanent polarization in amorphous polymers is lost in the vicinity of Tg, the use of these piezoelectric polymers is limited to temperatures well below Tg. This means that the polymers are in their glassy state, and the further away from Tg the use temperature is, the stiffer the polymer. This also means that measurement of the bulk physical properties is crucial both for identifying practical applications and for comparing polymers. The electromechanical coupling coefficient, kai, is a measure of the combination of piezoelectric and mechanical properties of a material (refer to Table III). It can be calculated using the equation below ... 

Ref. 18 further elaborates on the meaning of the load impedance and the electromechanical analogy. SLA is applicable to a wide range of samples. That certainly includes the Sauerbrey film. For the Sauerbrey film, the stress is mf —a/uo), where njf is the mass per unit are, uo is the oscillation amplitude, and -a> uo is acceleration. The speed, u, is given as i >uo. Inserting these relations, one finds the Sauerbrey equation (A/ = recovered. 

An alternative is to use flexural waves, excited by transverse motions of an electromechanical driver. A flexural (bending) deformation of a rod or strip measures Young s modulus because one side of the sample is stretched and the other compressed at a bend. For traveling flexural waves, the analogs of equations 39 and 40 of Chapter 5 are (for small damping, r 1) ... 

The subsequent characterization of electromechanical coupling covers the various classes of piezoelectric materials. Details with respect to definition and determination of the constants describing these materials have been standardized by the Institute of Electrical and Electronics Engineers [f04]. Stresses flux density D and field strength E on the electrostatic side, may be arbitrarily combined into four forms of coupled constitutive equations ... 

When d is substituted as outlined above and the compliance coefficients associated with the induced strain coefficients d are used, then the formulation turns into the upper part of the constitutive equations given on the right-hand side of Eqs. (4.10a). In addition, the actual thermal coefficients may be taken into consideration by the vector a. Thus, supplying specialized finite elements also capable of capturing anisotropic thermal effects with the constitutive coefficients and electric field strength of the electromechanically coupled problem, as given by Eq. (4.16), is a convenient procedure for the case of static actuation. 

Ionic Polymer-Metal Composite (IPMC) Ionic Polymer Electrodes Characterization Electromechanical model Nemst-Planck equation Onsager relation Electrochemical model Impedance Cyclic voltammetry Mechanical model... 

A fimdamental physics-based model of IPMCs is shown below in this model, the phenomena of electromechanical and mechanoelectrical transduction are induced in the IPMC by an ionic current. This results in a nonzero spatial charge in the vicinity of the electrodes. For both cases, the ionic current in the polymer can be described using the Nemst-Planck equation (Pugal et al. 2011, 2013) ... 

These basic equations form the so-called Poisson-Nemst-Planck (PNP) model for IPMCs and describe the fundamental physics wifliin the polymer membrane. A number of aufliors have developed electromechanical (actuator) and mechanoe-lectrical (sensor) models based on the PNP model as well as modified PNP models (Nemat-Nasser 2002 Nemat-Nasser and Zamani 2006 Wallmersperger et al. 2007 Zhang and Yang 2007 Porfiri 2008 Chen and Tan 2008 Aureli et al. 2009). This model will be described further in subsequent chapters of this book. 

The fundamental theory of IPMC electromechanical transduction (actuation) and mechanoelectrical transduction (sensing) will be presented using the same governing equations. Differences between actuation and sensing phenomena will be highlighted throughout. 

Although the provided equations described both transduction types of ff MCs, some terms are more prevalent than others for different transductions. For instance, it is reasonable to neglect the third flux term in Eq. 1 in the case of electromechanical transduction as it is very small compared to the second term. Also, while the local anion concentration can be expressed as a constant in the case of electromechanical transduction, it plays an important role in mechanoelectrical transduction where field gradients are significantly smaller. 

In the following example, coupling of theNavier equations and the Nemst-Planck equation is presented based on a simplified model where Ca = Co is a constant and AVAP 0. This simplification is generally assumed acceptable in the case of electromechanical transduction. 

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