pyvisco.prony

Collection of function to pre-process the master curve and perform the Prony series parameter identification.

Functions

E_freq_norm(omega, alpha_i, tau_i)

Calculate normalized storage and loss modulus values.

E_relax_norm(time, alpha_i, tau_i)

Calculate normalized relaxation modulus values.

GMaxw_temp(shift_func, df_GMaxw, df_coeff, df_aT)

Calculate Gen.

calc_GMaxw(E_0, df_terms, f_min, f_max, ...)

Calculate the Generalized Maxwell model data from the Prony series parameter.

discretize(df_master[, window, nprony])

Discretizes relaxation times over time or frequency axis.

fit(df_dis[, df_master, opt])

Generalized function to call the domain dependent curve fitting routine.

fit_freq(df_dis, df_master[, opt])

Fit Prony series parameter in frequency domain.

fit_time(df_dis, df_master[, opt])

Fit Prony series parameter in time domain.

ls_res(func)

Wrapper function that calculates the least squares residual.

plot_GMaxw(df_GMaxw, units)

Plot Generalized Maxwell model data for the reference temperature.

plot_GMaxw_temp(df_GMaxw_temp, units)

Plot Generalized Maxwell model data for varies temperature and frequencies.

plot_dis(df_master, df_dis, units)

Plot relaxation times on top of master curve.

plot_fit(df_master, df_GMaxw, units)

Plot the master curve and corresponding Prony fit (Gen.

plot_param(prony_list[, labels])

Plot illustrating the Prony series parameters of one or more fits.

split_x0(func)

Wrapper that splits array x0 of the minimization routine into two arrays.
pyvisco.prony.discretize(df_master, window='round', nprony=0)

Discretizes relaxation times over time or frequency axis.

Discrete relaxation times are required for Prony parameter curve fitting routine. This function spaces the relaxation times over the experimental characterization window.

Parameters:

  • df_master (pandas.DataFrame) – Contains the master curve data.

  • window ({'round', 'exact', 'min'}) – Defines the location of the discretization of the relaxation times. - exact : Use whole window of the experimental data and logarithmically space the relaxation times inbetween. - round : Round the minimum and maximum values of the experimental data to the nearest base 10 number and logarithmically space the remaining relaxation times inbetween the rounded numbers - min : Position of relaxation times is optimized during minimization routine to reduce the number of Prony terms.

  • nprony (numeric, default = 0) – Number of Prony terms to be used for the discretization. The number of Prony terms and the number of relaxation times is equal. If no number or 0 is specified, the default behavior of one Prony term per decade is used to automatically calculate the number of Prony terms.

Returns:

df_dis – Contains discrete point, equal to the relaxation times, of the master curve data (df_master).

Return type:

pandas.DataFrame

References

Kraus, M. A., and M. Niederwald. “Generalized collocation method using Stiffness matrices in the context of the Theory of Linear viscoelasticity (GUSTL).” Technische Mechanik-European Journal of Engineering Mechanics 37.1 (2017): 82-106.

pyvisco.prony.plot_dis(df_master, df_dis, units)

Plot relaxation times on top of master curve.

Parameters:

  • df_master (pandas.DataFrame) – Contains the master curve data.

  • df_dis (pandas.DataFrame) – Contains the discrete relaxation times and corresponding data.

  • units (dict of {str : str}) – Contains the names of the physical quantities as key and the corresponding names of the units as item.

Returns:

fig – Plot showing the relaxation times on top of the master curve.

Return type:

matplotlib.pyplot.figure

pyvisco.prony.ls_res(func)

Wrapper function that calculates the least squares residual.

Parameters:

func (function) – Time domain: prony.E_relax_norm Frequency domain: prony.E_freq_norm

Returns:

residual – Calculates least squares residual for specified domain.

Return type:

function

pyvisco.prony.split_x0(func)

Wrapper that splits array x0 of the minimization routine into two arrays.

Splits the the first argument x0 into two arrays alpha_i and tau_i and forwards both arrays to the called function. A single array x0 is necessary to optimize both alpha_i and tau_i at the same time. However, typically, only alpha_i is optimized and tau_i is kept constant. This wrapper allows to use the same function in both scenarios.

Parameters:

func (function) – Function that calculates least squares residual.

Returns:

split

Return type:

function

See also

prony.ls_res

Function to be wrapped during minimization of Prony terms.

pyvisco.prony.E_relax_norm(time, alpha_i, tau_i)

Calculate normalized relaxation modulus values.

Parameters:

  • time (array-like) – Time in s.

  • alpha_i (array-like) – Normalized relaxation moduli (unitless).

  • tau_i (array-like) – relaxation times in s.

Returns:

Relaxation modulus values.

Return type:

numpy.ndarray

pyvisco.prony.fit_time(df_dis, df_master, opt=False)

Fit Prony series parameter in time domain.

A least-squares minimization is performed using the L-BFGS-B method from the scipy package. The implementation is similar to the optimization problem described by [1] for a homogenous distribution of discrete times.

Parameters:

  • df_dis (pandas.DataFrame) – Contains the discrete relaxation times and corresponding data.

  • df_master (pandas.DataFrame) – Contains the master curve data.

  • opt (bool, default = False) – Flag indicates wether the Prony term minimization routine should be executed or not.

Returns:

prony – Contains the Prony series parameters of the fit.

Return type:

dict

References

[1] Barrientos, E., Pelayo, F., Noriega, Á. et al. Optimal discrete-time Prony series fitting method for viscoelastic materials. Mech Time-Depend Mater 23, 193-206 (2019). https://doi.org/10.1007/s11043-018-9394-z

pyvisco.prony.E_freq_norm(omega, alpha_i, tau_i)

Calculate normalized storage and loss modulus values.

Parameters:

  • omega (array-like) – Angular frequency in rad/s.

  • alpha_i (array-like) – Normalized relaxation moduli (unitless).

  • tau_i (array-like) – relaxation times in s.

Returns:

Concatenated array of normalized storage and loss modulus values.

Return type:

numpy.ndarray

pyvisco.prony.fit_freq(df_dis, df_master, opt=False)

Fit Prony series parameter in frequency domain.

A least-squares minimization is performed using the L-BFGS-B method from the scipy package. The implementation is similar to the optimization problem described by [1] for a homogenous distribution of discrete times.

Parameters:

  • df_dis (pandas.DataFrame) – Contains the discrete relaxation times and corresponding data.

  • df_master (pandas.DataFrame) – Contains the master curve data.

  • opt (bool, default = False) – Flag indicates wether the Prony term minimization routine should be executed or not.

Returns:

prony – Contains the Prony series parameters of the fit.

Return type:

dict

References

[1] Barrientos, E., Pelayo, F., Noriega, Á. et al. Optimal discrete-time Prony series fitting method for viscoelastic materials. Mech Time-Depend Mater 23, 193-206 (2019). https://doi.org/10.1007/s11043-018-9394-z

pyvisco.prony.calc_GMaxw(E_0, df_terms, f_min, f_max, decades, modul, **kwargs)

Calculate the Generalized Maxwell model data from the Prony series parameter.

Parameters:

  • E_0 (numeric) – Instantaneous storage modulus. Same variable name is used for either tensile (E_0) or shear (G_0) loading.

  • df_terms (pandas.DataFrame) – Contains the Prony series parameters tau_i and alpha_i.

  • f_min (numeric) – Lower bound frequency for calculation of physical quanitities.

  • f_max (numeric) – Upper bound frequency for calculation of physical quanitities.

  • decades (integer) – Number of decades spanning the frequency window. Is used to calculate the necessary number of data points spanning the frequency range for an appropriate resolution.

  • modul ({'E', 'G'}) – Indicates wether tensile (E) or shear (G) modulus data are provided.

Returns:

df_GMaxw – Contains the calculated Generalized Maxwell model data for the fitted Prony series parameters with the specified boundaries and parameters.

Return type:

pandas.DataFrame

pyvisco.prony.GMaxw_temp(shift_func, df_GMaxw, df_coeff, df_aT, freq=None)

Calculate Gen. Maxwell model for different loading frequencies and temperatures.

This function showcases the temperature and rate-dependence of the visco- elastic material. The specified shift function is used to calculate the material response at different temperatures and different loading rates.

Parameters:

  • shift_func ({'WLF', 'D4', 'D3', 'D2', 'D1'}) – Specifies the shift function to be used for calculations.

  • df_GMaxw (pandas.DataFrame) – Contains the Generalized Maxwell model data for the reference temperature at different loading rates.

  • df_coeff (pandas.DataFrame) – Contains the coefficients and parameters for the specified shift function.

  • df_aT (pandas.DataFrame) – Contains the shift factors. The shift factors are used to identify the Temperature range for the calculation.

  • freq (array-like, default = [1E-8, 1E-4, 1E0, 1E4]) – Loading frequencies for which the calculations are performed.

Returns:

Contains the Generalized Maxwell model data for a wide range of temperatures at the specified frequencies.

Return type:

df_GMaxw_temp

See also

shift.fit_WLF

Returns WLF shift functions.

shift.fit_poly

Returns polynomial shift functions of degree 1 to 4.

pyvisco.prony.plot_GMaxw(df_GMaxw, units)

Plot Generalized Maxwell model data for the reference temperature.

Parameters:

  • df_GMaxw (pandas.DataFrame) – Contains the Generalized Maxwell model data for the reference temperature at different loading rates.

  • units (dict of {str : str}) – Contains the names of the physical quantities as key and the corresponding names of the units as item.

Returns:

fig – Plot of calculated storage, loss, and relaxation modulus.

Return type:

matplotlib.pyplot.figure

pyvisco.prony.plot_GMaxw_temp(df_GMaxw_temp, units)

Plot Generalized Maxwell model data for varies temperature and frequencies.

Parameters:

  • df_GMaxw_temp (pandas.DataFrame) – Contains the Generalized Maxwell model data for various temperatures and different loading rates.

  • units (dict of {str : str}) – Contains the names of the physical quantities as key and the corresponding names of the units as item.

Returns:

fig – Plot of showing the temperature and rate dependence of the storage, loss, and relaxation modulus.

Return type:

matplotlib.pyplot.figure

pyvisco.prony.plot_param(prony_list, labels=None)

Plot illustrating the Prony series parameters of one or more fits.

Parameters:

  • prony_list (list) – List of prony dictionaries containing the Prony series parameters.

  • labels (list of str) – List of strings to be used as legend label names.

Returns:

fig – Plot showing the relaxation moduli over the relaxation times.

Return type:

matplotlib.pyplot.figure

See also

prony.fit

Returns the prony dictionary to be used in prony_list.

pyvisco.prony.fit(df_dis, df_master=None, opt=False)

Generalized function to call the domain dependent curve fitting routine.

Parameters:

  • df_dis (pandas.DataFrame) – Contains the discrete relaxation times and corresponding data.

  • df_master (pandas.DataFrame, default = None) – Contains the master curve data. Not required for the initial fit in the frequency domain (opt = False).

  • opt (bool, default = False) – Flag indicates wether the Prony term minimization routine should be executed or not.

Returns:

  • prony (dict) – Contains the Prony series parameters of the fit.

  • df_GMaxw (pandas.DataFrame) – Contains the calculated Generalized Maxwell model data for the fitted Prony series parameters.

pyvisco.prony.plot_fit(df_master, df_GMaxw, units)

Plot the master curve and corresponding Prony fit (Gen. Maxwell model).

Parameters:

  • df_master (pandas.DataFrame) – Contains the master curve data.

  • df_GMaxw (pandas.DataFrame) – Contains the calculated Generalized Maxwell model data for the fitted Prony series parameters.

  • units (dict of {str : str}) – Contains the names of the physical quantities as key and the corresponding names of the units as item.

Returns:

fig – Domain dependent plot of master curve and Prony fit.

Return type:

matplotlib.pyplot.figure