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Formability analysis in materials science.

Project description

PyPI version

formable provides tools for formability analysis in materials science.


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:

# 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


which prints:


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:


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:


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

[0.1.21] - 2023.11.09


  • Resolve numpy deprecation.

[0.1.20] - 2022.08.08


  • Constrain LM fitting process
  • Relabel attribute true_stress to stress in LoadResponse class

[0.1.19] - 2021.11.09


  • Add cyclic_uniaxial load case method to task generate_load_case
  • Add mixed load case method to task generate_load_case

[0.1.18] - 2021.08.06


  • Add option include_yield_functions to LoadResponseSet.show_yield_functions_2D and LoadResponseSet.show_yield_functions_3D, which is a list of fitted yield function indices to include in the visualisation.
  • Add get_load_case_planar_2D load case function.
  • Add option strain_rate_mode to get_load_case_plane_strain, which determines if the load case is defined by deformation gradient (F_rate), velocity gradient (L) or an approximation excluding the stress condition (L_approx), which is useful when we want to avoid using mixed boundary conditions.


  • Functions get_load_case_uniaxial, get_load_case_biaxial and get_load_case_plane_strain have been refactored, documented and generalised where applicable. The returned dict from these functions now includes passing through direction and rotation. A new key rotation_matrix is the matrix representation of the rotation specified, if specified.

[0.1.17] - 2021.05.11


  • Add animation widget for yield func evolution: animate_yield_function_evolution(load_response_sets, ...).
  • Ability to add sheet direction labels in yield function plots.

[0.1.16] - 2021.04.23


  • 2D yield function plotting now use scikit-image to compute the zero-contour, which can then be plotted as scatter-plot data, instead of a Plotly contour plot. Partial fix (2D only) for #9. The old behaviour can be switched on with use_plotly_contour=True in methods: YieldFunction.compare_2D, YieldFunction.show_2D and LoadResponseSet.show_yield_functions_2D.
  • The YieldFunction.yield_point attribute is saved in the dict representation of a each LoadResponseSet.fitted_yield_function, and loaded correctly when loading from a dict, via a change to YieldFunction.from_name.
  • Parameter fitting using the LMFitter class now scales the fitting parameters to one.

[0.1.15] - 2021.04.10


  • Bug fix in LoadResponseSet.to_dict if an associated yield function was not fitted.

[0.1.14] - 2021.04.10


  • Add ability to specify fitting bounds and other optimisation parameters in YieldFunction.from_fit and LoadResponseSet.fit_yield_function.


  • LoadResponseSet.yield_functions attribute renamed LoadResponseSet.fitted_yield_functions.

[0.1.13] - 2021.03.28


  • Do not modify input dict to levenberg_marquardt.LMFitter.from_dict.
  • Fix bug in stress scale.


  • Add to_dict and from_dict methods to LoadResponseSet.

[0.1.12] - 2020.12.16


  • Add LMFitter.from_dict


  • Add single_crystal_parameters to returned dict of LMFitter.to_dict.

[0.1.11] - 2020.12.16


  • Set float values in get_new_single_crystal_params.

[0.1.10] - 2020.12.15


  • Add new module, levenberg_marquardt for fitting single crystal parameters.

[0.1.9] - 2020.11.18


  • Add missing import to formable.utils.

[0.1.8] - 2020.11.18


  • Include tensile_test module from tensile_test package.

[0.1.7] - 2020.09.17


  • Fix plot line colouring for many traces (more than Plotly default colour list)

[0.1.6] - 2020.08.22


  • Add dump_frequency to load case generators.

[0.1.5] - 2020.08.18


  • Default tolerance for LoadResponse.is_uniaxial check loosened to 0.3.

[0.1.4] - 2020.07.01


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

[0.1.3] - 2020.06.09


  • 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.


  • 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 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 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.


  • 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


Image URLs in README

[0.1.0] - 2020.04.12

Initial release.

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