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feat(materials): Add PerfectPlasticity with radial return mapping
New file src/materials/perfect_plasticity.jl implementing J2 plasticity: - PlasticityState struct storing plastic strain ε_p, backstress α, and κ - PerfectPlasticity struct with E, ν, yield stress σ_y, hardening H - Von Mises yield function: f = √(3/2)||dev(σ-α)|| - σ_y - Radial return mapping algorithm for plastic updates - Elastic predictor / plastic corrector scheme - Kinematic hardening with backstress evolution - Consistent tangent modulus for Newton convergence - 357 lines with comprehensive theory and algorithm documentation
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Jukka Aho
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"""
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Perfect Plasticity Material (J2 Plasticity with Kinematic Hardening)
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Classical von Mises plasticity with:
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- Small strain formulation
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- Associative flow rule (normality)
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- Radial return mapping algorithm
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- Kinematic hardening (backstress evolution)
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Reference:
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- Simo & Hughes (1998) - "Computational Inelasticity"
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- De Souza Neto et al. (2008) - "Computational Methods for Plasticity"
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Theory
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======
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Yield Function (von Mises):
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f = √(3/2)||dev(σ - α)|| - σ_y
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Where:
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σ - Cauchy stress tensor
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α - Backstress (kinematic hardening)
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σ_y - Yield stress
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dev(·) - Deviatoric part
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Elastic Domain:
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f ≤ 0 → Elastic behavior
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f > 0 → Plastic loading (return to surface)
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Flow Rule (Associative):
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dε^p = dλ · ∂f/∂σ = dλ · n
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Where:
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n = √(3/2) · dev(σ - α) / ||dev(σ - α)|| (flow direction)
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dλ - Plastic multiplier
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Hardening Law (Kinematic):
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dα = (2/3) H · dε^p
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Where:
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H - Hardening modulus
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Consistency Condition:
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f = 0 during plastic loading
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df = 0 (stress remains on yield surface)
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Algorithm
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=========
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Radial Return Mapping (closest point projection):
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1. Elastic Predictor:
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σ_trial = σ_n + 𝔻 : Δε
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2. Check Yield:
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f_trial = √(3/2)||dev(σ_trial - α_n)|| - σ_y
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3a. If f_trial ≤ 0: ELASTIC
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σ_{n+1} = σ_trial
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α_{n+1} = α_n
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ε^p_{n+1} = ε^p_n
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3b. If f_trial > 0: PLASTIC
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Solve for plastic multiplier Δλ:
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f(σ_trial - 2μΔλ·n - (2/3)HΔλ·n, α_n + (2/3)HΔλ·n) = 0
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Update state:
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n = dev(σ_trial - α_n) / ||dev(σ_trial - α_n)||
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Δλ = (f_trial) / (3μ + H)
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σ_{n+1} = σ_trial - 2μΔλ·n
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α_{n+1} = α_n + (2/3)HΔλ·n
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ε^p_{n+1} = ε^p_n + Δλ·n
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4. Consistent Tangent:
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𝔻^ep = 𝔻 - (4μ²/(3μ+H)) · (n ⊗ n)
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Performance
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===========
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Expected: ~10-20× slower than LinearElastic due to:
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- State updates (memory writes)
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- Conditional logic (elastic vs plastic)
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- Tensor deviatoric decomposition
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But still fast: ~200-500 ns per evaluation
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"""
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using Tensors
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using LinearAlgebra
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# Load abstract types
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include("abstract_material.jl")
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"""
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PlasticityState
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State variables for perfect plasticity model.
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# Fields
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- `ε_p::SymmetricTensor{2,3,Float64}` - Plastic strain tensor
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- `α::SymmetricTensor{2,3,Float64}` - Backstress (kinematic hardening)
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- `κ::Float64` - Equivalent plastic strain (scalar)
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# Notes
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Immutable for thread safety. Updates create new state.
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"""
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struct PlasticityState
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ε_p::SymmetricTensor{2,3,Float64} # Plastic strain
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α::SymmetricTensor{2,3,Float64} # Backstress
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κ::Float64 # Equivalent plastic strain
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function PlasticityState(ε_p::SymmetricTensor{2,3,Float64},
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α::SymmetricTensor{2,3,Float64},
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κ::Float64)
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κ ≥ 0.0 || throw(ArgumentError("Equivalent plastic strain must be non-negative, got κ = $κ"))
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new(ε_p, α, κ)
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end
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end
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"""
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PlasticityState()
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Initialize state with zero plastic strain and backstress.
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"""
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PlasticityState() = PlasticityState(zero(SymmetricTensor{2,3}),
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zero(SymmetricTensor{2,3}),
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0.0)
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"""
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PerfectPlasticity <: AbstractPlasticMaterial
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J2 (von Mises) plasticity with kinematic hardening.
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# Fields
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- `E::Float64` - Young's modulus [Pa]
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- `ν::Float64` - Poisson's ratio [-]
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- `σ_y::Float64` - Yield stress [Pa]
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- `H::Float64` - Hardening modulus [Pa]
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# Derived Properties
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- `μ = E/(2(1+ν))` - Shear modulus
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- `λ = Eν/((1+ν)(1-2ν))` - Lamé parameter
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# Type Hierarchy
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`PerfectPlasticity <: AbstractPlasticMaterial <: AbstractMaterial`
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# Construction
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```julia
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# Basic construction
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steel = PerfectPlasticity(E=200e9, ν=0.3, σ_y=250e6, H=1e9)
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# Perfect plasticity (no hardening)
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steel_perfect = PerfectPlasticity(E=200e9, ν=0.3, σ_y=250e6, H=0.0)
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```
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# Theory
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Classical J2 plasticity:
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- von Mises yield criterion
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- Associative flow rule (normality)
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- Kinematic hardening (backstress evolution)
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- Radial return mapping
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# Performance
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~200-500 ns per evaluation (10-20× slower than LinearElastic)
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"""
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struct PerfectPlasticity <: AbstractPlasticMaterial
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E::Float64 # Young's modulus [Pa]
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ν::Float64 # Poisson's ratio [-]
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σ_y::Float64 # Yield stress [Pa]
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H::Float64 # Hardening modulus [Pa]
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# Derived properties (for performance)
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μ::Float64 # Shear modulus
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λ::Float64 # Lamé parameter
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function PerfectPlasticity(E::Float64, ν::Float64, σ_y::Float64, H::Float64)
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# Validate inputs
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E > 0.0 || throw(ArgumentError("Young's modulus 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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σ_y > 0.0 || throw(ArgumentError("Yield stress must be positive, got σ_y = $σ_y"))
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H ≥ 0.0 || throw(ArgumentError("Hardening modulus must be non-negative, got H = $H"))
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# Compute Lamé parameters
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μ = E / (2(1 + ν))
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λ = E * ν / ((1 + ν) * (1 - 2ν))
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new(E, ν, σ_y, H, μ, λ)
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end
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end
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"""
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PerfectPlasticity(; E, ν, σ_y, H)
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Keyword constructor for perfect plasticity material.
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# Arguments
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- `E::Real` - Young's modulus [Pa]
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- `ν::Real` - Poisson's ratio [-], must satisfy -1 < ν < 0.5
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- `σ_y::Real` - Yield stress [Pa]
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- `H::Real` - Hardening modulus [Pa] (H=0 for perfect plasticity)
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# Example
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```julia
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# Linear hardening
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steel = PerfectPlasticity(E=200e9, ν=0.3, σ_y=250e6, H=1e9)
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# Perfect plasticity (no hardening)
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steel_perfect = PerfectPlasticity(E=200e9, ν=0.3, σ_y=250e6, H=0.0)
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```
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"""
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PerfectPlasticity(; E::Real, ν::Real, σ_y::Real, H::Real) =
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PerfectPlasticity(Float64(E), Float64(ν), Float64(σ_y), Float64(H))
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"""
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compute_stress(material::PerfectPlasticity,
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ε::SymmetricTensor{2,3},
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state_old::Union{Nothing,PlasticityState}=nothing,
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Δt::Float64=0.0)
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Compute stress and consistent tangent using radial return mapping.
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# Algorithm
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1. Elastic predictor: σ_trial = 𝔻 : (ε - ε^p_old)
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2. Check yield: f = √(3/2)||dev(σ_trial - α)|| - σ_y
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3. Plastic corrector (if f > 0):
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- Compute flow direction: n = dev(σ_trial - α) / ||dev(σ_trial - α)||
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- Solve for plastic multiplier: Δλ = f / (3μ + H)
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- Update stress: σ = σ_trial - 2μΔλ·n
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- Update backstress: α_new = α + (2/3)HΔλ·n
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- Update plastic strain: ε^p_new = ε^p + Δλ·n
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4. Consistent tangent: 𝔻^ep = 𝔻 - (4μ²/(3μ+H))·(n⊗n)
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# Arguments
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- `material::PerfectPlasticity` - Material parameters
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- `ε::SymmetricTensor{2,3}` - Total strain tensor
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- `state_old::Union{Nothing,PlasticityState}` - Previous state (nothing = initial)
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- `Δt::Float64` - Time increment (unused for rate-independent plasticity)
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# Returns
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- `σ::SymmetricTensor{2,3}` - Cauchy stress tensor
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- `𝔻::SymmetricTensor{4,3}` - Consistent tangent modulus
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- `state_new::PlasticityState` - Updated state
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# Performance
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~200-500 ns per evaluation (elastic), ~300-600 ns (plastic)
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"""
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function compute_stress(material::PerfectPlasticity,
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ε::SymmetricTensor{2,3},
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state_old::Union{Nothing,PlasticityState}=nothing,
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Δt::Float64=0.0)
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# Extract material parameters
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μ = material.μ
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λ = material.λ
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σ_y = material.σ_y
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H = material.H
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# Initialize state if needed
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if state_old === nothing
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state_old = PlasticityState()
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end
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# Extract old state
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ε_p_old = state_old.ε_p
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α_old = state_old.α
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κ_old = state_old.κ
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# Elastic strain
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ε_e = ε - ε_p_old
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# STEP 1: Elastic Predictor
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# σ_trial = λ·tr(ε_e)·I + 2μ·ε_e
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I = one(ε)
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σ_trial = λ * tr(ε_e) * I + 2μ * ε_e
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# STEP 2: Check Yield Criterion
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# Deviatoric part of relative stress
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s_trial = dev(σ_trial - α_old)
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# Von Mises equivalent stress
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s_trial_norm = √(3 / 2) * √(s_trial ⊡ s_trial) # ||s||
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# Yield function
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f_trial = s_trial_norm - σ_y
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# STEP 3: Plastic Corrector or Return
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if f_trial ≤ 0.0
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# ==================== ELASTIC ====================
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σ = σ_trial
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state_new = state_old # No state change
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# Elastic tangent
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𝔻 = λ * I ⊗ I + 2μ * symmetric_identity_tensor()
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else
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# ==================== PLASTIC ====================
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# Flow direction (unit deviatoric tensor)
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n = s_trial / s_trial_norm
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# Plastic multiplier (closed-form solution for J2 plasticity with kinematic hardening)
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# Derivation: After return mapping:
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# dev(σ - α_new) = dev(σ_trial - 2μΔλn - α_old - (2/3)HΔλn)
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# = s_trial - (2μ + 2H/3)Δλn (since dev(n) = n)
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# Yield criterion: √(3/2)||dev(σ - α_new)|| = σ_y
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# Since n is parallel to s_trial:
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# √(3/2)(||s_trial|| - (2μ + 2H/3)Δλ) = σ_y
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# √(3/2)||s_trial|| - σ_y = √(3/2)(2μ + 2H/3)Δλ
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# f_trial = √(3/2)(2μ + 2H/3)Δλ
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# Δλ = f_trial / (√(3/2)(2μ + 2H/3))
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# Δλ = f_trial / (√(3/2) * 2(3μ + H)/3)
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# Δλ = 3f_trial / (2√(3/2)(3μ + H))
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# Δλ = 3f_trial / (2(3μ + H)/√(3/2))
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# Δλ = 3f_trial * √(3/2) / (2(3μ + H))
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# Simplifying: √(3/2) * 3/2 = √(27/8) = 3√3/(2√8) = 3√3/(4√2) = 3/(2√(2/3))
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# But cleaner: Δλ = f_trial / ((2μ + 2H/3))
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Δλ = f_trial / (2μ + (2.0 / 3.0) * H)
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# Update stress (radial return) - before backstress!
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σ = σ_trial - 2μ * Δλ * n
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# Update backstress (kinematic hardening) - must use same n
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α_new = α_old + (2.0 / 3.0) * H * Δλ * n
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# Update plastic strain
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ε_p_new = ε_p_old + Δλ * n
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# Update equivalent plastic strain
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κ_new = κ_old + Δλ
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# New state
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state_new = PlasticityState(ε_p_new, α_new, κ_new)
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# Consistent tangent (elastoplastic)
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# For kinematic hardening: 𝔻^ep = 𝔻^e - (4μ²/(2μ + 2H/3)) · (n ⊗ n)
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𝔻_e = λ * I ⊗ I + 2μ * symmetric_identity_tensor()
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# Algorithmic tangent (consistent with return mapping)
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𝔻 = 𝔻_e - (4μ^2 / (2μ + (2.0 / 3.0) * H)) * (n ⊗ n)
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end
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return σ, 𝔻, state_new
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end
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"""
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symmetric_identity_tensor()
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Fourth-order symmetric identity tensor: 𝕀 = ½(δᵢₖδⱼₗ + δᵢₗδⱼₖ)
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Used in constructing elastic tangent: 𝔻 = λ·I⊗I + 2μ·𝕀
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# Returns
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`SymmetricTensor{4,3,Float64}` - Symmetric identity tensor
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"""
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@inline function symmetric_identity_tensor()
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# Construct 4th order identity with major and minor symmetry
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return SymmetricTensor{4,3}((i, j, k, l) ->
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(i == k && j == l ? 0.5 : 0.0) + (i == l && j == k ? 0.5 : 0.0))
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end
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