import lmfit
import numpy as np
def get_parameter_defaults():
"""Return the default model parameters"""
# The order of the parameters must match the order
# of ´parameter_names´ and ´parameter_keys´.
params = lmfit.Parameters()
params.add("E", value=3e3, min=0)
params.add("R", value=10e-6, min=0, vary=False)
params.add("nu", value=.5, min=0, max=0.5, vary=False)
params.add("contact_point", value=0)
params.add("baseline", value=0)
return params
[docs]
def hertz_sneddon_spherical_approx(delta, E, R, nu, contact_point=0,
baseline=0):
r"""Hertz model for Spherical indenter - approximation
.. math::
F = \frac{4}{3} \frac{E}{1-\nu^2} \sqrt{R} \delta^{3/2}
\left(1
- \frac{1}{10} \frac{\delta}{R}
- \frac{1}{840} \left(\frac{\delta}{R}\right)^2
+ \frac{11}{15120} \left(\frac{\delta}{R}\right)^3
+ \frac{1357}{6652800} \left(\frac{\delta}{R}\right)^4
\right)
Parameters
----------
delta: 1d ndarray
Indentation [m]
E: float
Young's modulus [N/m²]
R: float
Tip radius [m]
nu: float
Poisson's ratio
contact_point: float
Indentation offset [m]
baseline: float
Force offset [N]
Returns
-------
F: float
Force [N]
Notes
-----
These approximations are made by the Hertz model:
- The sample is isotropic.
- The sample is a linear elastic solid.
- The sample is extended infinitely in one half space.
- The indenter is not deformable.
- There are no additional interactions between sample and indenter.
Additional assumptions:
- no surface forces
Truncated power series approximation:
This model is a truncated power series approximation of
:ref:`sec_ref_model_sneddon_spher`. The expected error is more
than four magnitues lower than the signal (see e.g.
:ref:`example_model_spherical_indenter`). The Bio-AFM analysis software
by JPK/Bruker uses the same model.
References
----------
Sneddon (1965) :cite:`Sneddon1965` (equations 6.13 and 6.15),
Dobler (personal communication, 2018) :cite:`Dobler`
"""
aa = 4/3 * E/(1-nu**2)*np.sqrt(R)
root = contact_point-delta
pos = root > 0
bb = np.zeros_like(delta)
bb[pos] = (root[pos])**(3/2)*(
+ 1
- 1/10*(root[pos]/R)
- 1/840*(root[pos]/R)**2
+ 11/15120*(root[pos]/R)**3
+ 1357/6652800*(root[pos]/R)**4)
return aa*bb + baseline
model_doc = hertz_sneddon_spherical_approx.__doc__
model_func = hertz_sneddon_spherical_approx
model_key = "sneddon_spher_approx"
model_name = "spherical indenter (Sneddon, truncated power series)"
parameter_keys = ["E", "R", "nu", "contact_point", "baseline"]
parameter_names = ["Young's Modulus", "Tip Radius",
"Poisson's Ratio", "Contact Point", "Force Baseline"]
parameter_units = ["Pa", "m", "", "m", "N"]
valid_axes_x = ["tip position"]
valid_axes_y = ["force"]