Piezoelectric Analysis (Static)
Piezoelectric Analysis (Static): Theoretical Foundations
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.
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.
Computational Methods for Piezoelectric Analysis (Static)
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.