Piezoelectric Analysis (Static)
Theory and Physics
Piezoelectric Effect
Professor, is the piezoelectric effect about converting mechanical force into voltage?
There are both the direct piezoelectric effect (force → voltage) and the converse piezoelectric effect (voltage → deformation).
Constitutive equations:
$$ \begin{cases} \{T\} = [c^E]\{S\} - [e]^T\{E\} \\ \{D\} = [e]\{S\} + [\varepsilon^S]\{E\} \end{cases} $$
$\{T\}$: Stress, $\{S\}$: Strain, $\{E\}$: Electric field, $\{D\}$: Electric displacement, $[c^E]$: Elastic constant (constant electric field), $[e]$: Piezoelectric constant, $[\varepsilon^S]$: Permittivity (constant strain).
So structural and electromagnetic fields are coupled.
Correct. In FEM, it becomes a piezoelectric coupled analysis where displacement $u$ and electric potential $\phi$ are solved simultaneously as unknowns.
Major Piezoelectric Materials
Material $d_{33}$ [pC/N] Application PZT (Lead Zirconate Titanate) 300–600 Actuators, Sensors BaTiO₃ 190 Ceramic Capacitors PVDF -33 Flexible Sensors AlN 5 MEMS Resonators LiNbO₃ 6 SAW Filters
Summary
- Mechanical-Electrical Coupling — Stress↔Electric field interact bidirectionally
- Solve for $u$ and $\phi$ simultaneously in FEM — Piezoelectric coupled analysis
- PZT is the most widely used — $d_{33} = 300$–600 pC/N
Coffee Break Trivia
Discovery of the Piezoelectric Effect—Pierre Curie and Paul Curie's 1880 Experiment
The piezoelectric effect was discovered in 1880 by French physicists Pierre Curie (husband of Marie Curie) and his brother Paul-Jacques Curie using quartz crystals. They demonstrated the "direct piezoelectric effect" where electric charges appear on the surface when force is applied to quartz. The following year, in 1881, Gabriel Lippmann theoretically predicted the converse "converse piezoelectric effect (deformation occurs when an electric field is applied)." Industrial application of the piezoelectric effect began with early 20th-century sonar (underwater sound wave detection), and during World War I, Paul Langevin developed underwater detection devices using piezoelectric quartz. Modern smartphone camera optical image stabilization actuators, medical ultrasound diagnostic devices, and inkjet printer heads are all descendants of the Curie brothers' discovery.
Physical Meaning of Each Term
- Electric field term $\nabla \times \mathbf{E} = -\partial \mathbf{B}/\partial t$: Faraday's law of electromagnetic induction. Time-varying magnetic flux density generates electromotive force. 【Everyday Example】A bicycle dynamo (generator) produces voltage in a nearby coil by rotating a magnet—a direct application of this law that a changing magnetic field induces an electric field. Induction cooking (IH) heaters also use the same principle, where high-frequency magnetic field changes induce eddy currents in the pot bottom, heating it via Joule heating.
- Magnetic field term $\nabla \times \mathbf{H} = \mathbf{J} + \partial \mathbf{D}/\partial t$: Ampère–Maxwell law. Current and displacement current generate a magnetic field. 【Everyday Example】When current flows through a wire, a magnetic field is created around it—this is Ampère's law. Electromagnets operate on this principle, flowing current through a coil to create a strong magnetic field. Smartphone speakers also apply this law: current → magnetic field → force on the diaphragm. At high frequencies (e.g., GHz-band antennas), the displacement current $\partial D/\partial t$ cannot be ignored and describes electromagnetic wave radiation.
- Gauss's Law $\nabla \cdot \mathbf{D} = \rho_v$: Shows that electric charge is the divergence source of electric flux. 【Everyday Example】Rubbing hair with a plastic sheet creates static electricity, making hair stand up—the charged sheet (electric charge) radiates electric field lines outward, exerting force on the light hair. Capacitor design calculates the electric field distribution between electrodes using this law. ESD (electrostatic discharge) countermeasures are also based on electric field analysis following Gauss's law.
- Magnetic flux conservation $\nabla \cdot \mathbf{B} = 0$: Indicates that magnetic monopoles do not exist. 【Everyday Example】Even if you cut a bar magnet in half, you cannot create a magnet with only an N pole or only an S pole—they always exist as an N-S pair. This means magnetic field lines form "closed loops with no start or end points." In numerical analysis, to satisfy this condition, the formulation uses a vector potential $\mathbf{B} = \nabla \times \mathbf{A}$, automatically guaranteeing magnetic flux conservation.
Assumptions and Applicability Limits
- Linear material assumption: Permeability and permittivity do not depend on magnetic/electric field strength (nonlinear B-H curve needed in saturation region)
- Quasi-static approximation (low frequency): Displacement current term can be ignored ($\omega \varepsilon \ll \sigma$). Common in eddy current analysis
- 2D assumption (cross-section analysis): Effective when current direction is uniform and end effects can be ignored
- Isotropic assumption: Direction-specific property definitions needed for anisotropic materials (e.g., rolling direction of silicon steel sheets)
- Non-applicable cases: Additional constitutive laws needed for plasma (ionized gas), superconductors, nonlinear optical materials
Dimensional Analysis and Unit Systems
Variable SI Unit Notes / Conversion Memo Magnetic Flux Density $B$ T (Tesla) 1T = 1 Wb/m². Permanent magnets: 0.2–1.4T Magnetic Field Strength $H$ A/m Horizontal axis of B-H curve. Conversion with CGS unit Oe (Oersted): 1 Oe = 79.577 A/m Current Density $J$ A/m² Calculated from conductor cross-sectional area and total current. Note non-uniform distribution due to skin effect Permeability $\mu$ H/m $\mu = \mu_0 \mu_r$. In vacuum $\mu_0 = 4\pi \times 10^{-7}$ H/m Electrical Conductivity $\sigma$ S/m Copper: approx. 5.96×10⁷ S/m. Decreases with temperature rise
Professor, is the piezoelectric effect about converting mechanical force into voltage?
There are both the direct piezoelectric effect (force → voltage) and the converse piezoelectric effect (voltage → deformation).
Constitutive equations:
$\{T\}$: Stress, $\{S\}$: Strain, $\{E\}$: Electric field, $\{D\}$: Electric displacement, $[c^E]$: Elastic constant (constant electric field), $[e]$: Piezoelectric constant, $[\varepsilon^S]$: Permittivity (constant strain).
So structural and electromagnetic fields are coupled.
Correct. In FEM, it becomes a piezoelectric coupled analysis where displacement $u$ and electric potential $\phi$ are solved simultaneously as unknowns.
| Material | $d_{33}$ [pC/N] | Application |
|---|---|---|
| PZT (Lead Zirconate Titanate) | 300–600 | Actuators, Sensors |
| BaTiO₃ | 190 | Ceramic Capacitors |
| PVDF | -33 | Flexible Sensors |
| AlN | 5 | MEMS Resonators |
| LiNbO₃ | 6 | SAW Filters |
- Mechanical-Electrical Coupling — Stress↔Electric field interact bidirectionally
- Solve for $u$ and $\phi$ simultaneously in FEM — Piezoelectric coupled analysis
- PZT is the most widely used — $d_{33} = 300$–600 pC/N
Discovery of the Piezoelectric Effect—Pierre Curie and Paul Curie's 1880 Experiment
The piezoelectric effect was discovered in 1880 by French physicists Pierre Curie (husband of Marie Curie) and his brother Paul-Jacques Curie using quartz crystals. They demonstrated the "direct piezoelectric effect" where electric charges appear on the surface when force is applied to quartz. The following year, in 1881, Gabriel Lippmann theoretically predicted the converse "converse piezoelectric effect (deformation occurs when an electric field is applied)." Industrial application of the piezoelectric effect began with early 20th-century sonar (underwater sound wave detection), and during World War I, Paul Langevin developed underwater detection devices using piezoelectric quartz. Modern smartphone camera optical image stabilization actuators, medical ultrasound diagnostic devices, and inkjet printer heads are all descendants of the Curie brothers' discovery.
Physical Meaning of Each Term
- Electric field term $\nabla \times \mathbf{E} = -\partial \mathbf{B}/\partial t$: Faraday's law of electromagnetic induction. Time-varying magnetic flux density generates electromotive force. 【Everyday Example】A bicycle dynamo (generator) produces voltage in a nearby coil by rotating a magnet—a direct application of this law that a changing magnetic field induces an electric field. Induction cooking (IH) heaters also use the same principle, where high-frequency magnetic field changes induce eddy currents in the pot bottom, heating it via Joule heating.
- Magnetic field term $\nabla \times \mathbf{H} = \mathbf{J} + \partial \mathbf{D}/\partial t$: Ampère–Maxwell law. Current and displacement current generate a magnetic field. 【Everyday Example】When current flows through a wire, a magnetic field is created around it—this is Ampère's law. Electromagnets operate on this principle, flowing current through a coil to create a strong magnetic field. Smartphone speakers also apply this law: current → magnetic field → force on the diaphragm. At high frequencies (e.g., GHz-band antennas), the displacement current $\partial D/\partial t$ cannot be ignored and describes electromagnetic wave radiation.
- Gauss's Law $\nabla \cdot \mathbf{D} = \rho_v$: Shows that electric charge is the divergence source of electric flux. 【Everyday Example】Rubbing hair with a plastic sheet creates static electricity, making hair stand up—the charged sheet (electric charge) radiates electric field lines outward, exerting force on the light hair. Capacitor design calculates the electric field distribution between electrodes using this law. ESD (electrostatic discharge) countermeasures are also based on electric field analysis following Gauss's law.
- Magnetic flux conservation $\nabla \cdot \mathbf{B} = 0$: Indicates that magnetic monopoles do not exist. 【Everyday Example】Even if you cut a bar magnet in half, you cannot create a magnet with only an N pole or only an S pole—they always exist as an N-S pair. This means magnetic field lines form "closed loops with no start or end points." In numerical analysis, to satisfy this condition, the formulation uses a vector potential $\mathbf{B} = \nabla \times \mathbf{A}$, automatically guaranteeing magnetic flux conservation.
Assumptions and Applicability Limits
- Linear material assumption: Permeability and permittivity do not depend on magnetic/electric field strength (nonlinear B-H curve needed in saturation region)
- Quasi-static approximation (low frequency): Displacement current term can be ignored ($\omega \varepsilon \ll \sigma$). Common in eddy current analysis
- 2D assumption (cross-section analysis): Effective when current direction is uniform and end effects can be ignored
- Isotropic assumption: Direction-specific property definitions needed for anisotropic materials (e.g., rolling direction of silicon steel sheets)
- Non-applicable cases: Additional constitutive laws needed for plasma (ionized gas), superconductors, nonlinear optical materials
Dimensional Analysis and Unit Systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Magnetic Flux Density $B$ | T (Tesla) | 1T = 1 Wb/m². Permanent magnets: 0.2–1.4T |
| Magnetic Field Strength $H$ | A/m | Horizontal axis of B-H curve. Conversion with CGS unit Oe (Oersted): 1 Oe = 79.577 A/m |
| Current Density $J$ | A/m² | Calculated from conductor cross-sectional area and total current. Note non-uniform distribution due to skin effect |
| Permeability $\mu$ | H/m | $\mu = \mu_0 \mu_r$. In vacuum $\mu_0 = 4\pi \times 10^{-7}$ H/m |
| Electrical Conductivity $\sigma$ | S/m | Copper: approx. 5.96×10⁷ S/m. Decreases with temperature rise |
Numerical Methods and Implementation
Piezoelectric FEM Formulation
After discretization:
$[K_{uu}]$: Mechanical stiffness, $[K_{\phi\phi}]$: Dielectric stiffness, $[K_{u\phi}]$: Piezoelectric coupling term.
So the structural and electrical degrees of freedom are integrated into a single matrix.
Each node has displacement DOFs ($u_x, u_y, u_z$) and electric potential DOF ($\phi$), totaling 4 (in 3D). Elements like Abaqus piezoelectric element C3D8E support this.
Summary
- Coupled Matrix — Solves mechanical and electrical simultaneously
- Abaqus C3D8E / COMSOL Piezoelectric — Commercial implementations
Piezoelectric FEM Coupling Setup—Weak Formulation of Mechanical-Electrical Coupled Equations and Handling of Material Tensors
FEM analysis of piezoelectric materials is a "mechanical-electrical coupled problem," solving displacement u (mechanical field) and electric potential phi (electric field) simultaneously. In the weak formulation, the piezoelectric constitutive equations (stress tensor = elastic constant × strain – piezoelectric constant × electric field) are incorporated into the virtual work principle, expressed as a 3-block matrix: stiffness matrix K (mechanical), dielectric matrix (electrical), and coupling matrix (piezoelectric). A common implementation pitfall is the coordinate transformation of the piezoelectric tensor e—a mismatch between the crystal axis direction and the analysis coordinate system weakens the coupling and underestimates voltage output. COMSOL's Piezoelectric Devices Physics and ANSYS Mechanical's PIEZO module handle this transformation automatically, but for user-defined materials, the transformation matrix must be set manually.
Edge Elements (Nedelec Elements)
Elements specialized for electromagnetic field analysis. Automatically guarantee tangential component continuity and eliminate spurious modes. Standard for 3D high-frequency analysis.
Nodal Elements
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