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