Source code for nanite.model.model_hertz_paraboloidal

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_paraboloidal(E, delta, R, nu, contact_point=0, baseline=0): r"""Hertz model for a paraboloidal indenter .. math:: F = \frac{4}{3} \frac{E}{1-\nu^2} \sqrt{R} \delta^{3/2} 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 ----- The original model reads .. math:: F = \frac{4}{3} \frac{E}{1-\nu^2} \sqrt{2k} \delta^{3/2}, where :math:`k` is defined by the paraboloid equation .. math:: \rho^2 = 4kz. 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: - radius of spherical cell is larger than the indentation References ---------- Sneddon (1965) :cite:`Sneddon1965` """ 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) 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_paraboloidal(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_paraboloidal.__doc__ model_key = "hertz_para" model_name = "parabolic indenter (Hertz)" 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"]