import lmfit
import numpy as np
from numpy import pi
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("alpha", value=25, min=0, max=90, 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_conical(E, delta, alpha, nu, contact_point=0, baseline=0):
r"""Hertz model for a conical indenter
.. math::
F = \frac{2\tan\alpha}{\pi}
\frac{E}{1-\nu^2}
\delta^2
Parameters
----------
E: float
Young's modulus [N/m²]
delta: 1d ndarray
Indentation [m]
alpha: float
Half cone angle [degrees]
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:
- infinitely sharp probe
References
==========
Love (1939) :cite:`Love1939`
"""
aa = 2*np.tan(alpha*pi/180)/pi * E/(1-nu**2)
root = contact_point-delta
pos = root > 0
bb = np.zeros_like(delta)
bb[pos] = root[pos]**2
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_conical(E=params["E"].value,
delta=x,
alpha=params["alpha"].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_conical.__doc__
model_key = "hertz_cone"
model_name = "conical indenter (Hertz)"
parameter_keys = ["E", "alpha", "nu", "contact_point", "baseline"]
parameter_names = ["Young's Modulus", "Half Cone Angle",
"Poisson's Ratio", "Contact Point", "Force Baseline"]
parameter_units = ["Pa", "°", "", "m", "N"]
valid_axes_x = ["tip position"]
valid_axes_y = ["force"]