import lmfit
import numpy as np
from . import weight
def get_parameter_defaults():
# 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(E, delta, 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
----------
E: float
Young's modulus [N/m²]
delta: 1d ndarray
Indentation [m]
R: float
Tip radius [m]
nu: float
Poisson's ratio
contact_point: float
Indentation offset [m]
baseline: float
Force offset [N]
negindent: bool
If `True`, will assume that the indentation value(s) given by
`delta` are negative and must be mutlitplied by -1.
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
References
----------
Sneddon (1965) :cite:`Sneddon1965`,
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
def model(params, x):
if x[0] < x[-1]:
revert = True
else:
revert = False
if revert:
x = x[::-1]
mf = hertz_sneddon_spherical_approx(
E=params["E"].value,
delta=x,
R=params["R"].value,
nu=params["nu"].value,
contact_point=params["contact_point"].value,
baseline=params["baseline"].value)
if revert:
return mf[::-1]
return mf
def residual(params, delta, force, weight_cp=5e-7):
""" Compute residuals for fitting
Parameters
----------
params: lmfit.Parameters
The fitting parameters for `model`
delta: 1D ndarray of lenght M
The indentation distances
force: 1D ndarray of length M
The corresponding force data
weight_cp: positive float or zero/False
The distance from the contact point until which
linear weights will be applied. Set to zero to
disable weighting.
"""
md = model(params, delta)
resid = force-md
if weight_cp:
# weight the curve so that the data around the contact_point do
# not affect the fit so much.
weights = weight.weight_cp(cp=params["contact_point"].value,
delta=delta,
weight_dist=weight_cp)
resid *= weights
return resid
model_doc = hertz_sneddon_spherical_approx.__doc__
model_key = "sneddon_spher_approx"
model_name = "spherical indenter (Sneddon, approximative)"
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"]