Piezoelectric materials are fascinating: they generate electricity when squeezed and deform when subjected to electric fields. This two-way coupling between mechanical and electrical fields makes them indispensable in devices like:

• Ultrasound transducers
• MEMS sensors and actuators
• Energy harvesters
• Precision control systems

In this post, we derive the strongly coupled PDE system that governs linear piezoelectricity — where displacement and electric potential are fully interdependent.

Field Variables

We work with the following:

• $\mathbf{u}$: Displacement vector
• $\phi$: Electric potential
• $\boldsymbol{\varepsilon}$: Strain tensor
• $\boldsymbol{\sigma}$: Cauchy stress tensor
• $\mathbf{E}$: Electric field ($E_i = -\partial \phi / \partial x_i$)
• $\mathbf{D}$: Electric displacement (flux density)

Material constants:

• $\mathbf{C}$: Elasticity (stiffness) tensor
• $\mathbf{e}$: Piezoelectric tensor
• $\boldsymbol{\varepsilon}^S$: Permittivity tensor (at constant strain)

Strain–Displacement and Electric Field–Potential

Strain tensor (small deformation assumption):

\[\varepsilon_{ij} = \frac{1}{2} \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right)\]

Electric field vector:

\[E_i = -\frac{\partial \phi}{\partial x_i}\]

Coupled Constitutive Laws

Stress–Strain–Electric Field:

\[\sigma_{ij} = C_{ijkl} \varepsilon_{kl} - e_{kij} E_k\]

Electric Displacement–Strain–Electric Field:

\[D_i = e_{ikl} \varepsilon_{kl} + \varepsilon^S_{ik} E_k\]

This defines the bidirectional electromechanical coupling:

• Mechanical strain creates electric displacement
• Electric field induces mechanical stress

Governing PDEs

Mechanical Equilibrium

No body forces or inertia assumed:

\[\frac{\partial \sigma_{ij}}{\partial x_j} = 0\]

Substitute stress:

\[\frac{\partial}{\partial x_j} \left( C_{ijkl} \varepsilon_{kl} - e_{kij} E_k \right) = 0\]

Gauss’s Law for Dielectrics

No free charges:

\[\frac{\partial D_i}{\partial x_i} = 0\]

Substitute electric displacement:

\[\frac{\partial}{\partial x_i} \left( e_{ikl} \varepsilon_{kl} + \varepsilon^S_{ik} E_k \right) = 0\]

Final Coupled PDE System

Mechanical PDE:

\[\frac{\partial}{\partial x_j} \left( C_{ijkl} \varepsilon_{kl} - e_{kij} \frac{\partial \phi}{\partial x_k} \right) = 0\]

Electrical PDE:

\[\frac{\partial}{\partial x_i} \left( e_{ikl} \varepsilon_{kl} - \varepsilon^S_{ik} \frac{\partial \phi}{\partial x_k} \right) = 0\]

With strain defined by:

\[\varepsilon_{kl} = \frac{1}{2} \left( \frac{\partial u_k}{\partial x_l} + \frac{\partial u_l}{\partial x_k} \right)\]

Boundary Conditions

Mechanical:

• Dirichlet (Displacement): $\mathbf{u} = \bar{\mathbf{u}}$
• Neumann (Traction): $\boldsymbol{\sigma} \cdot \mathbf{n} = \bar{\mathbf{t}}$

Electrical:

• Dirichlet (Voltage): $\phi = \bar{\phi}$
• Neumann (Charge Flux): $\mathbf{D} \cdot \mathbf{n} = \bar{q}$

These conditions are applied on respective boundary segments depending on device geometry and loading.

Practical Implementation: Voigt Notation

For numerical implementation, we typically rewrite these in matrix (Voigt) form:

• $\boldsymbol{\varepsilon} \in \mathbb{R}^6$
• $\boldsymbol{\sigma} \in \mathbb{R}^6$
• $\mathbf{E}, \mathbf{D} \in \mathbb{R}^3$

Then the constitutive relations become:

\[\boldsymbol{\sigma} = \mathbf{C} \boldsymbol{\varepsilon} - \mathbf{e}^T \mathbf{E}\] \[\mathbf{D} = \mathbf{e} \boldsymbol{\varepsilon} + \boldsymbol{\varepsilon}^S \mathbf{E}\]

Where:

• $\mathbf{C}$ is the $6 \times 6$ stiffness matrix
• $\mathbf{e}$ is the $3 \times 6$ piezoelectric matrix
• $\boldsymbol{\varepsilon}^S$ is the $3 \times 3$ dielectric matrix

This is the form used in most FEM solvers and PINN loss functions.

Real-World Applications

• Actuators – voltage → precision displacement
• Sensors – strain/stress → voltage
• Energy harvesters – vibration → power
• Ultrasound devices – bidirectional transduction
• MEMS and NEMS – integrated electromechanical control

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Thanks and Happy Learning
Team Elastropy