formable 0.1.4

Formability analysis in materials science.

formable provides tools for formability analysis in materials science.

Installation

pip install formable

To support showing visualisations within a Jupyter notebook, you will also need to make sure Plotly is set up to work within the notebook environment:

pip install "notebook>=5.3" "ipywidgets>=7.2"

Getting Started

LoadResponse and LoadResponseSet

The response of a material to a load is represented by the LoadResponse class. Use the following code snippet create a LoadResponse, where the arguments passed represent incremental data (i.e. data for each of the "steps" in the loading):

from formable import LoadResponse

load_response = LoadResponse(true_stress=true_stress, equivalent_strain=equivalent_strain)

true_stress and equivalent_strain are Numpy arrays of shapes (N, 3, 3) and (N,), respectively, for N increments within the load response.

A collection of load responses that contain the same incremental data are represented by the LoadResponseSet class:

from formable import LoadResponse, LoadResponseSet

all_responses = [LoadResponse(...), LoadResponse(...), ...]
load_set = LoadResponseSet(all_responses)

Yield functions

A number of yield functions as defined in the literature can be fitted and visualised. As an example, let's visualise the difference between the Von Mises and the Tresca yield criteria:

from formable.yielding.yield_functions import YieldFunction, VonMises, Tresca

von_mises = VonMises(equivalent_stress=70e6)
tresca = Tresca(equivalent_stress=70e6)

YieldFunction.compare_3D([von_mises, tresca])

If run within a Jupyter environment, this code snippet will generated a 3D visualisation of the yield surfaces in principal stress space:

To look at a single plane within principal stress space, we can do this:

YieldFunction.compare_2D([von_mises, tresca], plane=[0, 0, 1])

which generates a figure like this:

We can choose any plane that intercepts the origin. For instance, we can also look at the π-plane (σ₁ = σ₂ = σ₃):

YieldFunction.compare_2D([von_mises, tresca], plane=[1, 1, 1])

which generates a figure like this:

Yield function fitting

Using experimental or simulated yielding tests, we can fit yield functions to the results. Consider a LoadResponseSet object that has a sufficiently large number of increments of true_stress and equivalent_strain data to enable such a fit. Using the Barlat "Yld2000-2D" anisotropic yield function as an example, we can perform a fit:

from formable import LoadResponse, LoadResponseSet
from formable.yielding import YieldPointCriteria

# First generate a LoadResponseSet, using the results from experiment/simulation:
all_responses = [LoadResponse(...), LoadResponse(...), ...]
load_set = LoadResponseSet(all_responses)

# Then define a yield point criterion:
yield_point = YieldPointCriteria('equivalent_strain', 1e-3)

# Now calculate yield stresses according to the yield point criteria:
load_set.calculate_yield_stresses(yield_point)

# Now we can fit to the resulting yield stresses:
load_set.fit_yield_function('Barlat_Yld2000_2D', equivalent_stress=70e6)

Choosing the fitting parameters and initial guesses

We can specify which of the yield function parameters we would like to fit, and which should remain fixed. We can also pass initial values to the fitting procedure. A least squares fit is employed to fit yield functions in formable.

To fix a parameter during the fit, just pass it as a keyword argument to the fit_yield_function method, as we did in the above example, where we fixed the equivalent_stress parameter. To pass initial values for some of the parameters, we can pass a initial_params dictionary:

load_set.fit_yield_function('Barlat_Yld2000_2D', initial_params={'a1': 1.4})

We can see the available parameters of a given yield function by using the PARAMETERS attribute of a YieldFunction class:

from formable.yielding.yield_functions import Barlat_Yld2000_2D

print(Barlat_Yld2000_2D.PARAMETERS)

which prints:

['a1',
 'a2',
 'a3',
 'a4',
 'a5',
 'a6',
 'a7',
 'a8',
 'equivalent_stress',
 'exponent']

Alternatively, if we have created a yield function object (from a fitting procedure, or directly), we can use the get_parameters method to get the parameters and their values:

print(von_mises.get_parameters())

which prints:

{'equivalent_stress': 70000000.0}

Visualising the fit

Once a yield function has been fit to a load set, we can visualise the fitted yield function like this:

load_set.show_yield_functions_3D()

or, in a similar way to above, we can visualise the fitted yield functions in a given principal stress plane, using:

load_set.show_yield_functions_2D(plane=[0, 0, 1])

Change Log

[0.1.4] - 2020.07.01

Changed

• Print out the degree to which the stress state is uniaxial in LoadResponse.is_uniaxial.

[0.1.3] - 2020.06.09

Added

• Add a method to estimate the Lankford coefficient via the tangent of the yield surface at a uniaxial stress state: YieldFunction.get_numerical_lankford
• Add options to YieldFunction.show_2D, YieldFunction.compare_2D and LoadResponseSet.show_yield_functions_2D to visualise the tangent and normal to the yield function at a uniaxial stress state.
• Add incremental data: equivalent_plastic_strain and accumulated_shear_strain, and associated YieldPointCriteria mappings for getting the yield stress (using the same method as that used for equivalent_stress [total]).
• Add show_stress_states to LoadResponseSet.show_yield_functions_3D and LoadResponseset.show_yield_functions_2D to optionally hide stress states.
• Add option to pass Plotly layout parameters to yield function visualisation methods.
• Add property num_increments to LoadResponse.
• Add repr to LoadResponse and LoadResponseSet.
• Add YieldFunction.from_name() class method for generating a yield function from a string name and parameters.
• Add LoadResponse.incremental_data property to return all incremental data as a dict.

Changed

• Check each incremental_data array passed to LoadResponse has the same outer shape (i.e. same number of increments).
• AVAILABLE_YIELD_FUNCTIONS and YIELD_FUNCTION_MAP have been replaced with functions get_available_yield_functions and get_yield_function_map, respectively.
• Number of excluded load responses is printed when performing yield function fitting.

[0.1.2] - 2020.05.09

Fixed

• Fixed an issue when visualising yield surfaces in 3D (via YieldSurface.compare_3D()) (and also 2D) where, if the value of the yield function residual was already normalised (e.g. by the equivalent stress), then the isosurface drawn by Plotly was defective (showing spikes beyond the bounds of the contour grid), since the values that were being contoured were of the order 10^-8. This was because we normalised by the equivalent stress again when calculating the contour values. This was fixed by normalising by the absolute maximum value in the values that are returned by the residual function, rather than always normalising by the equivalent stress, so the contour values should be of the order 1 now, regardless of whether a given yield function residual value is normalised or not.
• Fixed yield function residual for Barlat_Yld91, where hydrostatic stresses would return np.nan.
• Check for bad kwargs in LoadResponseSet.fit_yield_function.
• Added an equivalent_stress parameter to Hill1948 to make it fit and visualise like the others. Not sure if this is the correct approach.

Added

• Added an option to show the bounds of the 3D contour grid when visualising yield functions in 3D.
• Added an option to associate additional text in visualising yield functions (for the legend): legend_text.
• Added module load_cases for generating load cases for simulations.
• Added hover text in YieldFunction.compare_2D that shows the value(s) of the yield function at each grid point.
• Added lankford property to Hill1948 that returns the Lankford coefficient, as determined by the values of the anisotropic parameters.

Changed

• The tolerance for checking if a uniaxial_response passed to LoadResponseSet.fit_yield_function is in fact uniaxial has been loosened, since this way failing when it shouldn't have.
• Normalise all yield function residuals by their equivalent stress parameter.

[0.1.1] - 2020.04.12

Changed

Image URLs in README

[0.1.0] - 2020.04.12

Initial release.