Physics-Informed Deep Learning for Thermodynamically Consistent Material Modeling

A neural architecture that learns constitutive material behavior while rigorously enforcing thermodynamic consistency through built-in physical constraints.

Key Features

🔬 Material-Specific Formulation

• Transversely Isropic Behavior:
  • Structural tensor formulation (A = a₀ ⊗ a₀)
  • Specialized invariant set for TI materials:
    • I₁ = tr(ε)
    • I₂ = tr(ε²)
    • I₃ = tr(Aε)
    • I₄ = tr(Aε²)
    • I₅ = tr(ε³)
    • I₆ = -2√det(2ε + I)
  • Frame-indifferent stress response through invariant formulation

🌐 Invariant-Based Architecture

• Processes strain invariants instead of full tensor
• Avoids tensor operations through scalar invariant formulation
• Automatic derivative calculations for:
  • ∂I₁/∂ε = I
  • ∂I₂/∂ε = 2ε
  • ∂I₃/∂ε = A
  • ∂I₄/∂ε = Aε + εA
  • ∂I₅/∂ε = 3ε²
  • ∂I₆/∂ε = -J·C⁻¹

🧠 Network Components

• LSTM layers for history-dependent behavior in TI materials
• PICNN architecture for convex free energy in invariant space
• Internal variables capturing anisotropic hardening
• Stress computation via invariant chain rule: $$S = 2∑_{i=1}^6 (∂ψ/∂I_i)(∂I_i/∂ε)$$

📊 Data Integration

• Processes experimental stress-strain data from Excel files
• Automatic calculation of 6 strain invariants (I₁-I₆)
• Derivative computation for stress relationships
• Comprehensive data normalization/visualization pipeline
• Batch training with adaptive learning rate scheduling

Usage

1. Data Preparation
   Format experimental data in Excel with columns:
   • Strain components: E11, E12, E13, E22, E23, E33
   • Stress components: S11, S12, S13, S22, S23, S33
   • Timestep: DT

Results

The model generates:

• Stress-strain predictions vs ground truth comparisons
• Thermodynamic consistency validation plots
• Training loss curves (stress error, dissipation, energy)
• Model checkpoints for deployment
• Detailed visualizations of all network outputs