Gopalkrishna Advanced Rural Research Foundation

Comprehensive Electromechanical Impedance (EMI) SHM Theory

1. Coupled Electromechanical Modeling Across Substrates

When a surface-bonded PZT patch is excited by a high-frequency harmonic alternating voltage \(V\), it interacts directly with the host structure's mechanical impedance \(Z_s(\omega)\) and the PZT patch's internal mechanical impedance \(Z_a(\omega)\). The substrate material properties—such as Young’s modulus \(E\) and mass density \(\rho\)—dictate baseline resonance frequencies (\(f_r \propto \sqrt{E/\rho}\)). The total complex electrical admittance \(Y(\omega)\) [\(\text{mS}\)] measured by an impedance analyzer is formulated as:

$$Y(\omega) = G(\omega) + j B(\omega) = j\omega \varepsilon_{33}^T \left(1 - j\delta\right) \frac{A}{h} + \omega k_{31}^2 \frac{Z_s(\omega)}{Z_s(\omega) + Z_a(\omega)} \tan\left(\frac{\theta}{2}\right)$$

Where:

• \(G(\omega)\) [\(\text{mS}\)] = Conductance (real part of admittance), highly sensitive to structural resonant shifts and damage across aluminum, steel, concrete, or composites.
• \(B(\omega)\) [\(\text{mS}\)] = Susceptance (imaginary part of admittance), dominated by the baseline capacitive properties of the PZT sensor.
• \(Z_s(\omega)\) & \(Z_a(\omega)\) [\(\text{N}\cdot\text{s}/\text{m}\)] = Mechanical impedances of the host structure and PZT actuator respectively.

2. Damage Detection and Statistical Metrics (RMSD)

Any localized structural change alters local stiffness and damping, shifting the resonant peaks in the conductance \(G(\omega)\) spectrum. Quantifying this damage is commonly performed using the Root Mean Square Deviation (RMSD) index:

$$\text{RMSD} = \sqrt{\frac{\sum_{i=1}^{N} \left[ G_{\text{damage}}(\omega_i) - G_{\text{baseline}}(\omega_i) \right]^2}{\sum_{i=1}^{N} \left[ G_{\text{baseline}}(\omega_i) \right]^2}} \times 100\%$$