Energy conservation is a fundamental principle in physics. When modeling devices or systems, it can be used as a first principle for validation; furthermore, the various energy terms and their proportion gives good insight into dominant processes. In all cases, it is more natural to define energies for unit volume – we will see that this directly follows from the field quantities. Afterwards, the total energy can be obtained by integrating over the whole model/domain of interest.
Non-piezoelectric example for energy conservation
Imagine you are in a bell tower. When the bell is struck, it starts to vibrate, meaning that within the structure, there are two types of energies that convert periodically into each other: the elastic (or strain/potential) and kinetic energies. The elastic energy is due to the deformed shape, much like a spring in a lumped system. The kinetic energy is due to the movement of the elemental parts of the structure, similarly to a moving car. While these convert back and forth, there are losses within the material and radiation into the surrounding air. We can consider the energetically closed system as follows: the sum of the elastic, kinetic energies, acoustic losses (both in the structure and the surrounding air), plus any mechanical power flow through the windows/walls of the bell tower must equal the energy that is supplied to the bell with the initial mechanical strike:
E IN = E elastic + E kinetic + E mechanical loss + E mechanical boundary flow
And to be able to describe these energies, the constitutive equation between stresses and strains is required and is given using the stiffness tensor:
T = C : S
with T, C, and S corresponding to stress, stiffness tensor, and strain. (A good detailed overview of these quantities is given in B. A. Auld: Acoustic waves and fields in solids.)
In a piezoelectric device the above description still holds, as the device follows mechanical equations; however, electrical and coupling terms have to be defined to account for the piezoelectric effect.
In this case the constitutive equations are the following:
T = C E: S – e : E
D = ε · E + e T : S
with E, D, ε, and e being the electric field, electric displacement field, permittivity, and piezoelectric coupling coefficient, respectively.
The total energy balance is
E IN = Eelastic + Ekinetic + Edielectric + Emechanical loss + Emechanical boundary flow + Edielectric loss + Eelectric boundary flow
with additional terms for the electric domain.
Note that the elastic and dielectric energies for piezoelectric materials can be defined in two different ways – these will be discussed after reviewing all of these terms individually.
Equation of motion
The link between forces and the resulting motion is given by the equation of motion. The full derivation is given in B. A. Auld as before, resulting in
∇ · T + F = a2u ⁄ at2
which is analogous to Newton’s second law in the infinitesimally small limit. The particle displacement, u, is directly related to the strain: the symmetric part of the displacement gradient is strain, the constitutive equation closes the circle and links stress to strain to be able to have a closed system of equations to solve.
Note: In OnScale the losses are also incorporated into the equation of motion rather than the constitutive equation; an additional force term proportional to au ⁄ at is added.
Elastic (potential/strain) energy
Again, this is a very similar concept to the energy stored in a spring. However, in this case the stress and strain are used to define the energy density. If we assume a block (with sides Lx, Ly, and Lz) made of a certain material, stressed along x, there will be strain in x, denoted by Sxx , resulting from the stress, Txx . (For now, put the y deformation to the side; this will be treated later.)
First, for convenience, convert these quantities to force and the deformation length. The face normal to x has area Ax = LyLz and therefore the force acting on it is TxxAx = TxxLyLz, whereas the absolute elongation in x is Δx = SxxLx.Generally, the energy stored in a spring is Espring = 1/2 F(Δx)2, where k is the spring stiffness. This can be reformulated into Espring = 1/2 FΔx using Hooke’s law. Now, substituting the above, and normalizing for energy density: EstrainΔV = 1/2ΔVTxxLyLzSxxLx = 1/2TxxSxx, since . Moreover, clearly all the different contributions of stress and strain need to be summed, so a tensor product is required: EstrainΔV = 1/2 T : S. Finally, from the constitutive equation, stress can be substituted, and the stored elastic energy expressed by the strain and stiffness tensor: EstrainΔV = 1/2 (C : S): S. (Note that this is valid for a non-piezoelectric material. See below for the general case.)
This is a direct equivalent to the lumped, macroscopic kinetic energy. The macroscopic one is calculated as Ekinetic = 1/2 mv2, with m being the mass, v the velocity magnitude (for derivation refer to A. Hudson, R. Nelson: University Physics or any other textbook on fundamentals). Averaging this over a small elemental volume of the structure, with constant density, p: EkineticΔV = 1/ΔV (1/2mv2) = 1/ΔV (1/2pΔVv2) = 1/2pv2 = 1/2pv · v, where the inner product is used to obtain the magnitude of the velocity. This is the kinetic energy density.
The derivation is again omitted here but can be found in introductory physics or electrical engineering textbooks (such as Z. and B. D. Popovic: Introductory Electromagnetics). In its most general form it is given as
Edielectric = 1/2E · D
which can be re-written for a non-piezoelectric dielectric as
Edielectric = 1/2E · ε · E
(Note that this is valid for a non-piezoelectric material. See below for the general case.)
This is actually a virtual energy but is very useful in resolving the conflict between defining energies for piezo- and non-piezo materials.
First, let’s approach this from the elastic energy. For piezo materials its constitutive equation has an extra term, T’ = -e:E. Now, if we were to calculate the contribution to elastic energy only due to this term, we have
Ecoupling elastic= 1/2 T’ : S = -1/2 (e:E):S
where the prime signifies the extra piezoelectric contribution only. Similar investigation can be done for the electric displacement field contribution, D’ = eT : S. The dielectric energy now has an extra term
Ecoupling dielectric = 1/2E · D= 1/2E · (eT :S)
On rearranging the terms (defining them as matrices and using the commutativity of transpose and general associativity), these are equal in magnitude and different in sign.
To model lossy materials, the stiffness tensor usually has a strain-rate (derivative of strain) dependent addition (T” = ηdS / dt), and that term results in a loss. However, this cannot be obtained directly, but first the power has to be calculated. As work and energy are the same, and both are time integral of the power, at constant force, the power is generally given as P = dE /dt = F · dx/dt = Fv, where v is velocity. Now, for the field quantities we have F” = AxTxx“(as before) and vx = Lx · dSxx/dt, therefore
Pacoustic loss = Fv = AxTxx“Lx · dSxx/dt = ΔV · T” : dS/dt = ΔV (η: dS/dt): dS/dt
or normalized for volume
Pacoustic lossΔV =∫ Pacoustic lossΔVdt = ∫ (η: dS/dt) : dS/dt dt
which usually does not have a closed analytical form.
Note: In OnScale we model acoustic losses through the equation of motion; the results are nevertheless in excellent agreement with the detailed method.
The concept is very similar as for the acoustic loss – the permittivity having an electric-field-rate proportional term (D” = ηE : dE/dt). Afterwards similar considerations can be made to arrive at
Edielectric lossΔV = ∫ Pdielectric lossΔVdt = ∫ (ηE : dE/dt): dE /dt dt
Note: For both the acoustic and dielectric losses for sinusoidal excitation the derivative is proportional to , resulting in complex stiffness and permittivity.
Boundary power flow
Here again two domains must be distinguished: mechanical and electrical. We touched on the mechanical power before, as the product of velocity and stress; this is defined as the acoustic Poynting vector:
Smechanical = -v · T
and afterwards power flow through a surface can be obtained from a surface integral; total energy flow is the time integral of that quantity.
For the electrical domain, the well-known electrical Poynting vector can be applied:
Selectrical = E x H
and the same surface and time integrals result in total energy flow. In piezoelectric device simulation, the magnetic field is de-coupled using an electrostatic approximation (dB / dt = 0), leading to a form
Selectrical = Φ · dD / dt
with Φ the electric potential, and D the electric displacement field.
(For boundary power flow detailed derivation, refer to B. A. Auld as noted above.)
Notes for piezoelectric materials – generalized electrical and mechanical energy
The reader can see that defining kinetic energy for piezoelectric materials, two terms appear, one that is the “standard”, present for all types of materials, and a second term, specific to piezoelectric coupling:
Eelastic piezo = 1/2 (CE:S) :S – 1/2 (e:E):S
To avoid confusion, some authors (see Zaitsev and Kuznetsova: “The energy density and power flow of acoustic waves propagating in piezoelectric materials”) define this total elastic energy as Generalized Mechanical energy, comprising the usual pure mechanical term and the electromechanical coupling.
Similar steps can be done for the dielectric energy:
Edielectric piezo = 1/2 E · ε · E + 1/2 (e:E): S
The full dielectric energy is referred to as Generalized Electrical energy, with a pure electrical term and the coupling term.
Therefore, it is sensible to always define these energies using strain and the E field to avoid confusion, rather than any terms having the stress or displacement field.
Yes, calculation of all these quantities can be cumbersome – but the rewards are invaluable! Identifying energy storage and loss mechanisms in piezoelectric resonators, sensors, or actuators can inform device design and significantly boost performance.