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Gopalkrishna Advanced Rural Research Foundation, India Dr. APJ Abdul Kalam Zero Funding Research Initiative

Annamdas – Soh Coupled 3D Electromechanical Impedance (EMI) Formulation & SHM Solver

Virtual Smart Materials Laboratory • Multi-Scale 1D/2D/3D FEM Elastodynamic Stress Tensor Analysis

Three.js 3D WebGL
1

Stage 1: Healthy Specimen & Material

3D Solid Block
Modulus E 69.0 GPa
Density ρ 2700 kg/m³
Poisson ν 0.33
Loss η 0.002
Length ($L$) 240 mm
Width ($W$) 60 mm
Thickness ($H$) 20 mm
PZT Sensor Location X: 60 mm, Y: 30 mm
Position $X_p$ (along length) 60 mm
Position $Y_p$ (across width) 30 mm
Healthy Baseline Computed
Static Capacitance: 1.82 nF
Units: $ (Siemens)
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Step 2: Investigate Structural Damage

Click the toggle below to inject a fracture into the specimen and observe the resulting peak shift and attenuation in the Annamdas–Soh impedance spectrum!

Crack Length ($a$)
mm
1.0 mm (Micro-defect) Full Specimen: 240 mm
Crack Width ($w_c$) 15 mm
Depth / Thick ($h_c$) 8 mm
Position along Length ($X_c$) 140 mm
Position across Width ($Y_c$) 30 mm
RMSD Metric
0.00%
Root Mean Sq Dev
MAPD Metric
0.00%
Mean Abs % Dev
CCM Metric
1.000
Correlation Coeff

3D Solid Hexahedral Continuum FEM Field

3D Brick Mesh
PZT Transducer Patch (Au Electrodes)
Hexahedral 8-Node Solid Continuum Elements
0.0 (Min Stress)
Max Stress Tensor
Left Drag: Rotate • Right Drag: Pan • Scroll: Zoom
Unit: $ (Siemens / mho / Ω⁻¹)
Excitation Sweep: 10 kHz – 400 kHz
Modal Shift ($\Delta f$): 0.00 kHz (Healthy Baseline)
Transducer-to-Defect: 80 mm

Theoretical Framework & Student Engineering Notes: Annamdas–Soh Coupled 3D EMI Formulation

Coupled 3D PZT-Structure Elastodynamic Constitutive Interaction Tensor • Units shown as $ (Siemens / inverted Ohm $\mho$)

1. Coupled Complex Admittance $Y(\omega)$

The electrical admittance across the bonded piezoceramic patch is governed by electro-mechanical coupling:

Y(ω) = G(ω) + j B(ω)
= j ω (A_p / h_p) · [ ε₃₃ᵀ (1 - j δ) - 2 d₃₁² / (s₁₁ᴱ (1 - j η_p)) · (Z_a / (Z_s + Z_a)) · (tan(k l_p) / (k l_p)) ]

Units are expressed in $ (Siemens / mho), where real part $G$ represents active power dissipation and imaginary part $B$ represents reactive susceptance.

2. 1D vs. 2D vs. 3D Impedance Tensors

  • 1D Slender Beam: Pure unconstrained axial-flexural coupling with scalar mechanical impedance $Z_s(\omega) = \rho A c \tanh(\gamma L)$.
  • 2D Thin Plate: Bi-axial multi-directional planar impedance coupling scaled by $1/(1-\nu)$, showing Chladni 2D nodal grid patterns.
  • 3D Solid Block: Full tri-axial tensorial coupling scaled by $(1-\nu)/((1+\nu)(1-2\nu))$, capturing volumetric shear and Rayleigh wave interactions.

3. Fracture Damage Indexing

The crack introduces localized loss of structural stiffness $(\Delta k_s)$ and damping perturbation $(\Delta c_s)$:

RMSD (%) = √[ Σ (G_damaged - G_healthy)² / Σ (G_healthy)² ] × 100%

Internal core cracks cause volumetric wave scattering, while surface cracks induce dominant Rayleigh surface acoustic wave distortions.