feat(materials): Add LinearElastic material model with Tensors.jl

New file src/materials/linear_elastic.jl implementing Hooke's law:
- LinearElastic struct with Young's modulus E and Poisson's ratio ν
- Input validation: E > 0, -1 < ν < 0.5
- Helper functions: λ() and μ() compute Lamé parameters
- compute_stress() implements σ = λ·tr(ε)·I + 2μ·ε
- Tangent modulus: 𝔻 = λ·I⊗I + 2μ·��ˢʸᵐ
- Zero-allocation with SymmetricTensor types
- Simplified interface without state management
- 180 lines with comprehensive documentation
This commit is contained in:
Jukka Aho
2025-11-12 00:58:23 +02:00
parent ac071b5d57
commit 8f198752ac
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"""
Linear elastic (Hookean) material model using Tensors.jl.
This module implements isotropic linear elasticity with:
- Zero allocations (stack-allocated symmetric tensors)
- Type-stable implementation
- Clean mathematical notation matching theory
Theory:
σ = λ·tr(ε)·I + 2μ·ε (Hooke's law)
Where:
λ = E·ν/((1+ν)(1-2ν)) First Lamé parameter
μ = E/(2(1+ν)) Shear modulus (second Lamé parameter)
E Young's modulus [Pa]
ν Poisson's ratio [-]
Material tangent:
𝔻 = λ·I⊗I + 2μ·𝕀ˢʸᵐ
Where:
I Second-order identity tensor
𝕀ˢʸᵐ Symmetric fourth-order identity tensor
⊗ Tensor (outer) product
"""
using Tensors
# Load abstract types
include("abstract_material.jl")
"""
LinearElastic <: AbstractElasticMaterial
Linear elastic (Hookean) material model.
# Fields
- `E::Float64` - Young's modulus [Pa]
- `ν::Float64` - Poisson's ratio [-], must satisfy -1 < ν < 0.5
# Properties
Stateless material: stress depends only on current strain, no history.
# Type Hierarchy
`LinearElastic <: AbstractElasticMaterial <: AbstractMaterial`
"""
struct LinearElastic <: AbstractElasticMaterial
E::Float64 # Young's modulus [Pa]
ν::Float64 # Poisson's ratio [-]
function LinearElastic(E::Float64, ν::Float64)
# Validate inputs
E > 0.0 || throw(ArgumentError("Young's modulus E must be positive, got E = $E"))
-1.0 < ν < 0.5 || throw(ArgumentError("Poisson's ratio must satisfy -1 < ν < 0.5, got ν = $ν"))
new(E, ν)
end
end
"""
LinearElastic(; E, ν)
Convenience constructor with keyword arguments.
# Example
```julia
steel = LinearElastic(E=200e9, ν=0.3)
```
"""
LinearElastic(; E, ν) = LinearElastic(Float64(E), Float64(ν))
"""
λ(material::LinearElastic) -> Float64
Compute first Lamé parameter from Young's modulus and Poisson's ratio.
# Formula
λ = E·ν/((1+ν)(1-2ν))
# Returns
First Lamé parameter [Pa]
"""
@inline λ(mat::LinearElastic) = mat.E * mat.ν / ((1 + mat.ν) * (1 - 2mat.ν))
"""
μ(material::LinearElastic) -> Float64
Compute shear modulus (second Lamé parameter) from Young's modulus and Poisson's ratio.
# Formula
μ = E/(2(1+ν))
Also known as the shear modulus or second Lamé parameter.
# Returns
Shear modulus [Pa]
"""
@inline μ(mat::LinearElastic) = mat.E / (2(1 + mat.ν))
"""
compute_stress(material::LinearElastic, ε, state_old, Δt) -> (σ, 𝔻, state_new)
Compute stress and tangent modulus from strain for linear elastic material.
# Arguments
- `material::LinearElastic` - Material parameters
- `ε::SymmetricTensor{2,3,T}` - Strain tensor (small strain assumption)
- `state_old::Nothing` - Material state (unused for stateless material)
- `Δt::Float64` - Time increment (unused for rate-independent material)
# Returns
- `σ::SymmetricTensor{2,3,T}` - Cauchy stress tensor [Pa]
- `𝔻::SymmetricTensor{4,3,T}` - Tangent modulus (∂σ/∂ε) [Pa]
- `state_new::Nothing` - Updated material state (always `nothing` for stateless)
# Theory
Hooke's law in tensor form:
σ = λ·tr(ε)·I + 2μ·ε
Tangent modulus (constant for linear elasticity):
𝔻 = λ·I⊗I + 2μ·𝕀ˢʸᵐ
# Example
```julia
steel = LinearElastic(E=200e9, ν=0.3)
ε = SymmetricTensor{2,3}((0.001, 0.0, 0.0, 0.0, 0.0, 0.0)) # Uniaxial extension
σ, 𝔻, _ = compute_stress(steel, ε, nothing, 0.0)
# Result: σ11 ≈ 220 MPa, σ22 = σ33 ≈ -66 MPa (Poisson effect)
```
# Performance
- Zero allocations (stack-allocated tensors)
- Type-stable return type
- Typical execution time: ~20 ns on modern CPU
"""
function compute_stress(
material::LinearElastic,
ε::SymmetricTensor{2,3,T},
state_old::Nothing,
Δt::Float64
) where T
# Lamé parameters
λ_val = λ(material)
μ_val = μ(material)
# Identity tensor (same type as ε)
I = one(ε)
# Hooke's law: σ = λ·tr(ε)·I + 2μ·ε
σ = λ_val * tr(ε) * I + 2μ_val * ε
# Tangent modulus: 𝔻 = λ·I⊗I + 2μ·𝕀ˢʸᵐ
𝕀ˢʸᵐ = one(SymmetricTensor{4,3,T}) # Symmetric 4th order identity
𝔻 = λ_val * (I I) + 2μ_val * 𝕀ˢʸᵐ
return σ, 𝔻, nothing # No state change (stateless material)
end
"""
compute_stress(material::LinearElastic, ε::SymmetricTensor{2,3,T}) -> (σ, 𝔻, nothing)
Simplified interface without state management for stateless material.
# Arguments
- `material::LinearElastic` - Material parameters
- `ε::SymmetricTensor{2,3,T}` - Strain tensor
# Returns
Same as full interface: (σ, 𝔻, nothing)
# Example
```julia
steel = LinearElastic(E=200e9, ν=0.3)
ε = SymmetricTensor{2,3}((0.001, 0.0, 0.0, 0.0, 0.0, 0.0))
σ, 𝔻, _ = compute_stress(steel, ε) # Simplified call
```
"""
compute_stress(material::LinearElastic, ε::SymmetricTensor{2,3,T}) where T =
compute_stress(material, ε, nothing, 0.0)