Add UserMaterialStrain for small-strain elastic-plastic material formulations

[adtzlr]

The idea is to use only one single function for both the stress and the elasticity evaluation (like UMAT in Abaqus, HYPELA2 in Marc, etc). A tangent=True input could be provided for effective evaluations.

Input Arguments for function: de, e (old), s (old), statevars (old)

Outputs: dsde, s (new), statevars (new)

[adtzlr]

The current implementation is based on the strain increment. Old stress and old strain are also provided (internally as state variables). The additional **kwargs are used for the tangent-flag and may be extended in the future.

def fun(dε, εn, σn, ζn, **kwargs):
    return dσdε, σ, ζ
    
umat = UserMaterialStrain(material=fun, **kwargs)

[adtzlr]

A linear-elastic material formulation could be implemented like this.

from felupe.math import identity, cdya, dya, trace

def linear_elastic(dε, εn, σn, ζn, λ, μ, **kwargs):
    """3D linear-elastic material formulation.

    Arguments
    ---------
    dε : ndarray
        Incremental strain tensor.
    εn : ndarray
        Old strain tensor.
    σn : ndarray
        Old stress tensor.
    ζn : ndarray
        Old state variables.
    λ : float
        First Lamé-constant.
    μ : float
        Second Lamé-constant (shear modulus).
    """

    # change of stress due to change of strain
    I = identity(dε)
    dσ = 2 * μ * dε + λ * trace(dε) * I

    # update stress and evaluate elasticity tensor
    σ = σn + dσ
    dσdε = 2 * μ * cdya(I, I) + λ * dya(I, I)

    # update state variables (not used here)
    ζ = ζn

    return dσdε, σ, ζ

umat = UserMaterialStrain(material=linear_elastic, μ=1, λ=2)

[adtzlr]

Implementation:

felupe/felupe/constitution/_user_materials_models.py

Line 40 in ca0876b

def linear_elastic(de, εn, σn, ζn, λ, μ, **kwargs):