Constitution

felupe.constitution.linear_elastic(dε, εn, σn, ζn, λ, μ, **kwargs)

3D linear-elastic material formulation.

1. Given state in point x (σn) (valid).

2. Given strain increment dε, so that ε = εn + dε.

3. Evaluation of the stress σ and the algorithmic consistent tangent modulus dσdε.

   dσdε = λ 1 ⊗ 1 + 2μ 1 ⊙ 1

   σ = σn + dσdε : dε

Parameters

• dε (ndarray) – Strain increment.

• εn (ndarray) – Old strain tensor.

• σn (ndarray) – Old stress tensor.

• ζn (list) – List of old state variables.

• λ (float) – First Lamé-constant.

• μ (float) – Second Lamé-constant (shear modulus).

felupe.constitution.linear_elastic_plastic_isotropic_hardening(dε, εn, σn, ζn, λ, μ, σy, K, **kwargs)

Linear-elastic-plastic material formulation with linear isotropic hardening (return mapping algorithm).

1. Given state in point x (σn, ζn=[εpn, αn]) (valid).

2. Given strain increment dε, so that ε = εn + dε.

3. Evaluation of the hypothetic trial state:

   dσdε = λ 1 ⊗ 1 + 2μ 1 ⊙ 1

   σ = σn + dσdε : dε

   s = dev(σ)

   εp = εpn

   α = αn

   f = ||s|| - sqrt(2/3) (σy + K α)

4. If f ≤ 0, then elastic step:

   Set y = yn + dy, y=(σ, ζ=[εp, α]),

   algorithmic consistent tangent modulus dσdε.

   Else:

   dγ = f / (2μ + 2/3 K)

   n = s / ||s||

   σ = σ - 2μ dγ n

   εp = εpn + dγ n

   α = αn + sqrt(2 / 3) dγ

   Algorithmic consistent tangent modulus:

   dσdε = dσdε - 2μ / (1 + K / 3μ) n ⊗ n - 2μ dγ / ||s|| ((2μ 1 ⊙ 1 - 1/3 1 ⊗ 1) - 2μ n ⊗ n)

Parameters

• dε (ndarray) – Strain increment.

• εn (ndarray) – Old strain tensor.

• σn (ndarray) – Old stress tensor.

• ζn (list) – List of old state variables.

• λ (float) – First Lamé-constant.

• μ (float) – Second Lamé-constant (shear modulus).

• σy (float) – Initial yield stress.

• K (float) – Isotropic hardening modulus.