docs(book): Add NeoHookean hyperelastic implementation guide

- Compressible Neo-Hookean strain energy function - Automatic differentiation for stress and tangent computation - Dual constructor: Lamé (μ,λ) or engineering (E,ν) - Total Lagrangian formulation with 2nd Piola-Kirchhoff stress - Zero-allocation AD via Tensors.jl - When to use: rubber, large deformation, contact mechanics - 571 lines: Complete AD-based material model documentation
2026-08-06 04:21:33 +00:00 · 2025-11-12 01:05:49 +02:00
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+---
+title: "NeoHookean Hyperelastic Material with Automatic Differentiation"
+date: 2025-11-11
+author: "JuliaFEM Contributors"
+status: "Authoritative"
+last_updated: 2025-11-11
+tags: ["materials", "hyperelasticity", "automatic-differentiation", "finite-strain"]
+---
+
+## Overview
+
+The NeoHookean material model represents the simplest hyperelastic constitutive law for finite strain elasticity. This implementation uses **automatic differentiation** to compute stress and material tangent directly from the strain energy function, eliminating manual derivative errors and enabling rapid prototyping of complex material models.
+
+**Key Features:**
+
+- Compressible Neo-Hookean strain energy
+- Automatic differentiation via Tensors.jl (no external dependencies)
+- Zero allocations (suitable for FEM assembly loops)
+- Dual constructor: Lamé parameters (μ, λ) OR engineering constants (E, ν)
+- Type-stable implementation with AbstractMaterial hierarchy
+
+**When to Use:**
+
+- Rubber-like materials (polymers, elastomers, biological tissues)
+- Large deformation problems (>10% strain)
+- Research/prototyping of hyperelastic models
+- Contact mechanics (naturally produces unsymmetric tangent)
+
+**When NOT to Use:**
+
+- Small strain problems (use LinearElastic - 40× faster)
+- High-performance production code with millions of evaluations
+- Materials with complex loading history (use plasticity models)
+
+## Mathematical Foundation
+
+### Strain Energy Function
+
+The compressible Neo-Hookean model is defined by the strain energy density:
+
+$$
+\psi(C) = \frac{\mu}{2}(I_1 - 3) - \mu \ln(J) + \frac{\lambda}{2} \ln^2(J)
+$$
+
+Where:
+
+- $C = F^T F$ - Right Cauchy-Green deformation tensor
+- $I_1 = \text{tr}(C)$ - First invariant
+- $J = \sqrt{\det(C)} = \det(F)$ - Volume ratio (Jacobian determinant)
+- $\mu$ - Shear modulus (resistance to distortion)
+- $\lambda$ - Lamé parameter (resistance to volume change)
+
+**Physical Interpretation:**
+
+- Term 1: $\frac{\mu}{2}(I_1 - 3)$ - Energy from shape change
+- Term 2: $-\mu \ln(J)$ - Coupling between shear and volume
+- Term 3: $\frac{\lambda}{2} \ln^2(J)$ - Energy from volume change
+
+### Stress Computation (Total Lagrangian)
+
+The 2nd Piola-Kirchhoff stress (energy conjugate to Green-Lagrange strain) is:
+
+$$
+S = 2 \frac{\partial \psi}{\partial C} = \mu(I - C^{-1}) + \lambda \ln(J) C^{-1}
+$$
+
+**Key Property:** Symmetric for elastic materials, but becomes **unsymmetric** in contact!
+
+### Material Tangent
+
+The material tangent (elasticity tensor) required for Newton's method:
+
+$$
+\mathbb{D} = 4 \frac{\partial^2 \psi}{\partial C \partial C}
+$$
+
+This is a 4th-order tensor with major and minor symmetries. Computing it manually is **error-prone** (81 components, complex chain rule). Automatic differentiation computes it **exactly** from the strain energy.
+
+## Implementation
+
+### Struct Definition
+
+```julia
+struct NeoHookean <: AbstractElasticMaterial
+    μ::Float64  # Shear modulus [Pa]
+    λ::Float64  # Lamé parameter [Pa]
+end
+```
+
+**Design Decisions:**
+
+1. **Immutable struct** - Thread-safe, cache-friendly
+2. **Float64 only** - No generic types (performance)
+3. **Inherits from AbstractElasticMaterial** - Type hierarchy for dispatch
+4. **Minimal fields** - Only material constants (no history)
+
+### Dual Constructor
+
+```julia
+# Option 1: Lamé parameters (direct)
+rubber = NeoHookean(μ=1e6, λ=1e9)
+
+# Option 2: Engineering constants (convenience)
+rubber = NeoHookean(E_mod=3e6, nu=0.45)
+```
+
+**Implementation Strategy:**
+
+```julia
+function NeoHookean(; μ::Real=NaN, λ::Real=NaN, E_mod::Real=NaN, nu::Real=NaN)
+    if !isnan(μ) && !isnan(λ)
+        # Direct Lamé parameters
+        return NeoHookean(Float64(μ), Float64(λ))
+    elseif !isnan(E_mod) && !isnan(nu)
+        # Convert engineering constants
+        μ_val = E_mod / (2(1 + nu))
+        λ_val = E_mod * nu / ((1 + nu) * (1 - 2nu))
+        return NeoHookean(Float64(μ_val), Float64(λ_val))
+    else
+        throw(ArgumentError("Must provide either (μ, λ) or (E_mod, nu)"))
+    end
+end
+```
+
+**Why Unified Constructor?**
+
+Julia does NOT support multiple keyword-only methods with different parameter names. The unified constructor with NaN defaults checks which parameters were provided.
+
+### Strain Energy Implementation
+
+```julia
+function strain_energy(material::NeoHookean, C::SymmetricTensor{2,3})
+    # Extract material parameters
+    μ = material.μ
+    λ = material.λ
+    
+    # Compute invariants
+    I₁ = tr(C)
+    J = √(det(C))
+    
+    # Validate deformation
+    J > 0 || throw(DomainError(J, "Invalid deformation: det(C) ≤ 0"))
+    
+    # Strain energy density
+    ψ = μ/2 * (I₁ - 3) - μ * log(J) + λ/2 * log(J)^2
+    
+    return ψ
+end
+```
+
+**Critical Details:**
+
+1. **Domain check:** $J > 0$ (negative Jacobian = inverted element)
+2. **Symmetric tensor input:** Uses SymmetricTensor{2,3} (6 components, not 9)
+3. **No allocations:** Pure function, stack-allocated tensors
+
+### Stress Computation (The Magic!)
+
+```julia
+function compute_stress(material::NeoHookean, 
+                       E::SymmetricTensor{2,3},
+                       state_old::Nothing=nothing,
+                       Δt::Float64=0.0)
+    # Convert Green-Lagrange strain to Right Cauchy-Green
+    C = 2E + one(E)
+    
+    # Automatic differentiation for stress
+    S = 2 * Tensors.gradient(C_arg -> strain_energy(material, C_arg), C)
+    
+    # Automatic differentiation for tangent
+    𝔻 = 4 * Tensors.hessian(C_arg -> strain_energy(material, C_arg), C)
+    
+    # Stateless material (no history)
+    state_new = nothing
+    
+    return S, 𝔻, state_new
+end
+```
+
+**How It Works:**
+
+1. **Tensors.gradient()** - Computes $\nabla_C \psi$ using forward-mode AD
+2. **Tensors.hessian()** - Computes $\nabla^2_C \psi$ using nested forward-mode AD
+3. **Factor of 2 and 4** - Chain rule for stress and tangent definitions
+4. **Zero allocations** - All tensors stack-allocated via Tensors.jl
+
+**Why This Is Powerful:**
+
+- **Correctness:** Derivatives are exact (machine precision)
+- **Maintainability:** Change strain energy → stress/tangent update automatically
+- **Extensibility:** Easy to add new hyperelastic models (just change ψ function)
+
+## Usage Examples
+
+### Example 1: Simple Uniaxial Tension
+
+```julia
+using Tensors
+include("src/materials/neo_hookean.jl")
+
+# Create material (rubber-like)
+rubber = NeoHookean(E_mod=3e6, nu=0.45)  # Nearly incompressible
+
+# Uniaxial extension: λ = 1.5 (50% stretch)
+λ₁ = 1.5
+λ₂ = 1/√λ₁  # Lateral contraction (incompressible assumption)
+
+# Deformation gradient
+F = Tensor{2,3}((λ₁, 0.0, 0.0, 0.0, λ₂, 0.0, 0.0, 0.0, λ₂))
+
+# Green-Lagrange strain: E = ½(C - I)
+C = symmetric(transpose(F) ⋅ F)
+E_GL = (C - one(C)) / 2
+
+# Compute stress and tangent
+S, 𝔻, _ = compute_stress(rubber, E_GL)
+
+println("2nd PK Stress (S₁₁): ", S[1,1], " Pa")
+println("Tangent norm:         ", norm(𝔻))
+```
+
+**Expected Results:**
+
+- $S_{11} > 0$ (tensile stress)
+- $S_{22} < 0$ (lateral compression from Poisson effect)
+- Tangent is positive-definite (stable material)
+
+### Example 2: Simple Shear
+
+```julia
+# Simple shear: F = I + γ·e₁⊗e₂
+γ = 0.5  # Shear angle (radians)
+F = one(Tensor{2,3}) + γ * Tensor{2,3}((0.0, 1.0, 0.0, 
+                                         0.0, 0.0, 0.0,
+                                         0.0, 0.0, 0.0))
+
+# Green-Lagrange strain
+C = symmetric(transpose(F) ⋅ F)
+E_GL = (C - one(C)) / 2
+
+# Compute stress
+S, _, _ = compute_stress(rubber, E_GL)
+
+println("Shear stress (S₁₂): ", S[1,2], " Pa")
+```
+
+### Example 3: Small Strain Validation
+
+For small strains, Neo-Hookean should match linear elasticity:
+
+```julia
+# Very small strain
+ε_small = 1e-6
+E_small = SymmetricTensor{2,3}((ε_small, 0.0, 0.0, 0.0, 0.0, 0.0))
+
+# Compare models
+S_neo, _, _ = compute_stress(rubber, E_small)
+
+# Linear elastic approximation: S ≈ λ·tr(E)·I + 2μ·E
+μ = rubber.μ
+λ = rubber.λ
+I = one(E_small)
+S_linear = λ * tr(E_small) * I + 2μ * E_small
+
+# Should be very close
+relative_error = norm(S_neo - S_linear) / norm(S_linear)
+println("Relative error: ", relative_error)  # Should be < 1e-4
+```
+
+## Performance Analysis
+
+### Benchmark Results
+
+Performance measured on a typical workstation (benchmarks/neo_hookean_analysis.jl):
+
+| Metric | LinearElastic | NeoHookean | Overhead |
+|--------|---------------|------------|----------|
+| **Single evaluation** | 26 ns | 1,057 ns | **40×** |
+| **1000 evaluations** | 10.1 μs | 1.05 ms | **103×** |
+| **Memory** | 0 bytes | 0 bytes | **0×** |
+| **Allocations** | 0 | 0 | **0** |
+
+### Performance Breakdown
+
+Where does the time go?
+
+- **Strain energy:** 1.7% (17 ns)
+- **AD gradient (stress):** ~30%
+- **AD hessian (tangent):** ~68%
+
+**Key Insight:** Almost all time is in automatic differentiation (98.3%), not the energy function itself.
+
+### Scaling Characteristics
+
+**Strain-Independent Performance:** ✅
+
+Time variation across strain magnitudes (1e-6 to 0.5): **0.5%**
+
+This is crucial for Newton solvers - consistent iteration times regardless of deformation state.
+
+**Zero Allocations:** ✅
+
+All operations use stack-allocated Tensors.jl types. No garbage collection overhead.
+
+### Production Recommendations
+
+**Research/Prototyping:** ⭐⭐⭐⭐⭐
+
+- Correctness guaranteed
+- Rapid implementation (minutes, not days)
+- Easy experimentation with new models
+
+**Production FEM (< 100K DOF):** ⭐⭐⭐⭐
+
+- Acceptable overhead for moderate problems
+- Profile first, optimize if needed
+
+**Production FEM (> 1M DOF):** ⭐⭐⭐
+
+- 40× overhead may dominate runtime
+- Consider manual derivatives for critical hot paths
+- AD still recommended for validation
+
+**Contact Mechanics:** ⭐⭐⭐⭐⭐
+
+- Unsymmetric tangent required (AD handles naturally)
+- Complex derivatives (stick-slip, friction)
+- Correctness critical (convergence issues hard to debug)
+
+## Comparison: Manual vs Automatic Derivatives
+
+### Manual Implementation (Traditional)
+
+```julia
+# Stress - must derive by hand
+C_inv = inv(C)
+J = √(det(C))
+S = μ * (I - C_inv) + λ * log(J) * C_inv
+
+# Tangent - 81 components, complex chain rule ��
+𝔻 = zeros(SymmetricTensor{4,3})
+for i in 1:3, j in 1:3, k in 1:3, l in 1:3
+    𝔻[i,j,k,l] = (... pages of algebra ...)
+end
+```
+
+**Problems:**
+
+1. **Error-prone:** Easy to make sign errors, index mistakes
+2. **Maintenance:** Change energy → must rederive everything
+3. **Time:** Days to weeks for complex models
+4. **Validation:** How to verify? Finite differences (slow, inaccurate)
+
+### Automatic Differentiation (This Implementation)
+
+```julia
+# Stress - one line
+S = 2 * Tensors.gradient(C_arg -> strain_energy(material, C_arg), C)
+
+# Tangent - one line
+𝔻 = 4 * Tensors.hessian(C_arg -> strain_energy(material, C_arg), C)
+```
+
+**Advantages:**
+
+1. **Correctness:** Machine precision (no human errors)
+2. **Maintainability:** Change ψ → done
+3. **Time:** Minutes
+4. **Validation:** Automatic
+
+**Trade-off:**
+
+- **Speed:** 40× slower than manual
+- **Worth it?** Almost always YES (unless profiling proves otherwise)
+
+## Advanced Topics
+
+### Nearly Incompressible Materials
+
+For rubber-like materials (Poisson's ratio → 0.5):
+
+```julia
+# Nearly incompressible (ν = 0.499)
+rubber = NeoHookean(E_mod=3e6, nu=0.499)
+
+# This gives: λ >> μ (large bulk modulus)
+println("μ = ", rubber.μ)  # ~1e6
+println("λ = ", rubber.λ)  # ~1e9 (1000× larger!)
+```
+
+**Numerical Note:** For ν > 0.49, consider mixed formulations (pressure as separate variable) to avoid volumetric locking.
+
+### Incompressibility Constraint
+
+For perfectly incompressible materials (det(F) = 1), use Lagrange multiplier:
+
+$$
+\psi(C, p) = \frac{\mu}{2}(I_1 - 3) + p(J - 1)
+$$
+
+Where $p$ is the hydrostatic pressure (unknown field). **Not implemented** - requires mixed FEM formulation.
+
+### Extending to Other Hyperelastic Models
+
+Want to try Mooney-Rivlin? Just change the strain energy!
+
+```julia
+function strain_energy(material::MooneyRivlin, C::SymmetricTensor{2,3})
+    C₁₀ = material.C₁₀
+    C₀₁ = material.C₀₁
+    
+    # Invariants
+    I₁ = tr(C)
+    I₂ = (tr(C)^2 - tr(C ⋅ C)) / 2
+    J = √(det(C))
+    
+    # Mooney-Rivlin energy
+    ψ = C₁₀ * (I₁ - 3) + C₀₁ * (I₂ - 3) - (C₁₀ + C₀₁) * log(J) + λ/2 * log(J)^2
+    
+    return ψ
+end
+
+# Stress and tangent: SAME CODE (just call compute_stress)!
+```
+
+This is the power of automatic differentiation!
+
+### Integration with FEM Assembly
+
+Typical usage in element stiffness computation:
+
+```julia
+function assemble_element(element::Tet10, material::NeoHookean, u_nodal::Vector)
+    K_elem = zeros(30, 30)  # 10 nodes × 3 DOF
+    f_elem = zeros(30)
+    
+    for (ξ, w) in quadrature_points(element)
+        # Kinematics
+        ∇N = shape_gradients(element, ξ)
+        F = deformation_gradient(∇N, u_nodal)
+        E_GL = green_lagrange_strain(F)
+        
+        # Material response (automatic differentiation here!)
+        S, 𝔻, _ = compute_stress(material, E_GL)
+        
+        # Tangent stiffness
+        K_elem += geometric_tangent(∇N, S, w) + material_tangent(∇N, 𝔻, w)
+        
+        # Internal forces
+        f_elem += internal_forces(∇N, S, w)
+    end
+    
+    return K_elem, f_elem
+end
+```
+
+**Performance Note:** The `compute_stress` call is typically 1-5% of element assembly time (most time in matrix operations).
+
+## Testing
+
+Comprehensive test suite (test/test_neo_hookean.jl): **41/41 tests passing** ✅
+
+### Test Coverage
+
+1. **Construction:** Valid inputs, invalid inputs, both constructor variants
+2. **Strain Energy:** Reference state, uniaxial, shear, invalid deformations
+3. **Stress:** Small strain, large strain, pure shear, symmetry
+4. **Tangent:** Structure, finite difference validation, positive definiteness
+5. **AD Verification:** Consistency between stress and energy gradient
+6. **Limits:** Small strain → linear elastic, incompressibility
+7. **Performance:** Zero allocation, type stability
+
+### Running Tests
+
+```bash
+cd /path/to/JuliaFEM.jl
+julia --project=. test/test_neo_hookean.jl
+```
+
+Expected output:
+
+```text
+Test Summary:        | Pass  Total  Time
+Neo-Hookean Material |   41     41  1.8s
+```
+
+## References
+
+### Theoretical Background
+
+1. **Holzapfel (2000)** - "Nonlinear Solid Mechanics" - Definitive reference for hyperelasticity
+2. **Bonet & Wood (2008)** - "Nonlinear Continuum Mechanics for Finite Element Analysis"
+3. **Wriggers (2008)** - "Nonlinear Finite Element Methods"
+
+### Implementation References
+
+1. **Tensors.jl documentation** - <https://github.com/Ferrite-FEM/Tensors.jl>
+2. **Automatic Differentiation (Griewank & Walther, 2008)** - "Evaluating Derivatives"
+3. **JuliaFEM Architecture** - `docs/book/element_architecture.md`
+
+### Benchmarking References
+
+1. **BenchmarkTools.jl** - <https://github.com/JuliaCI/BenchmarkTools.jl>
+2. **Performance Tips** - Julia manual: <https://docs.julialang.org/en/v1/manual/performance-tips/>
+
+## Appendix: Material Parameter Selection
+
+### Typical Values
+
+| Material | E [Pa] | ν | μ [Pa] | λ [Pa] |
+|----------|--------|---|--------|--------|
+| **Rubber (soft)** | 1e6 | 0.49 | 3.4e5 | 1.6e7 |
+| **Rubber (hard)** | 1e7 | 0.48 | 3.4e6 | 8.2e6 |
+| **Biological tissue** | 1e5 | 0.45 | 3.4e4 | 1.5e5 |
+| **Polymer (soft)** | 1e9 | 0.40 | 3.6e8 | 6.7e8 |
+
+### Parameter Relationships
+
+From engineering constants to Lamé parameters:
+
+$$
+\mu = \frac{E}{2(1 + \nu)}, \quad \lambda = \frac{E \nu}{(1 + \nu)(1 - 2\nu)}
+$$
+
+From Lamé parameters to engineering constants:
+
+$$
+E = \frac{\mu(3\lambda + 2\mu)}{\lambda + \mu}, \quad \nu = \frac{\lambda}{2(\lambda + \mu)}
+$$
+
+**Constraint:** For physical materials, require:
+
+- $\mu > 0$ (positive shear stiffness)
+- $\lambda > 0$ (for small strain stability)
+- $-1 < \nu < 0.5$ (thermodynamic constraint)
+
+### Calibration from Experiments
+
+1. **Uniaxial tension:** Measure stress-stretch curve → fit E and ν
+2. **Simple shear:** Measure shear stress-strain → verify μ
+3. **Hydrostatic compression:** Measure bulk modulus → verify λ
+
+**Note:** Neo-Hookean is accurate only for strains < 50%. For larger strains, use Ogden or Arruda-Boyce models.
+
+## Changelog
+
+### v1.0.0 (2025-11-11)
+
+- ✅ Initial implementation with automatic differentiation
+- ✅ Dual constructor (Lamé parameters or engineering constants)
+- ✅ Comprehensive test suite (41 tests)
+- ✅ Performance benchmarks (40× overhead vs LinearElastic)
+- ✅ Zero-allocation implementation
+- ✅ Complete documentation
+
+### Future Enhancements
+
+Potential improvements (not yet implemented):
+
+1. **Nearly incompressible formulation** - Mixed pressure-displacement
+2. **Ogden model** - Better large-strain accuracy
+3. **Anisotropic extension** - Fiber-reinforced materials
+4. **Visco-hyperelasticity** - Rate-dependent behavior
+5. **Manual derivatives option** - For high-performance production use
+
+---
+
+**Author:** JuliaFEM Contributors  
+**License:** MIT  
+**Last Updated:** November 11, 2025  
+**Version:** 1.0.0