Materials Science Applications

Curve fitting for mechanical testing, thermal analysis, and relaxation.

Stress-Strain Analysis

Power Law Hardening

import jax.numpy as jnp
from nlsq import fit


def power_law_hardening(strain, K, n):
    """Power law strain hardening: sigma = K * epsilon^n."""
    return K * strain**n


# Fit
popt, pcov = fit(
    power_law_hardening,
    strain_data,
    stress_data,
    p0=[500, 0.2],
    bounds=([0, 0], [2000, 1]),
)

Voce Hardening

def voce_hardening(strain, sigma_y, sigma_sat, theta):
    """Voce hardening law."""
    return sigma_sat - (sigma_sat - sigma_y) * jnp.exp(-theta * strain / sigma_sat)

Hollomon Equation

def hollomon(strain, K, n):
    """Hollomon equation for true stress-strain."""
    return K * strain**n

Thermal Analysis

Arrhenius Rate

def arrhenius(T, A, Ea):
    """Arrhenius rate constant: k = A * exp(-Ea/RT)."""
    R = 8.314  # J/(mol·K)
    return A * jnp.exp(-Ea / (R * T))


# Thermal decomposition fitting
popt, pcov = fit(
    arrhenius,
    temperature_K,
    rate_constant,
    p0=[1e13, 100000],
    bounds=([1e8, 10000], [1e20, 500000]),
)

Kissinger Peak

def kissinger_peak(T, Tm, Ea, A):
    """Kissinger peak shape for DSC."""
    R = 8.314
    x = Ea / (R * T)
    xm = Ea / (R * Tm)
    return A * jnp.exp(xm - x - jnp.exp(xm - x))

Glass Transition

def wlf_viscosity(T, eta_g, C1, C2, Tg):
    """Williams-Landel-Ferry equation for viscosity near Tg."""
    return eta_g * jnp.exp(-C1 * (T - Tg) / (C2 + T - Tg))

Relaxation Phenomena

Kohlrausch-Williams-Watts

def kww_relaxation(t, amplitude, tau, beta):
    """Stretched exponential relaxation."""
    return amplitude * jnp.exp(-((t / tau) ** beta))

Cole-Cole

def cole_cole(omega, eps_s, eps_inf, tau, alpha):
    """Cole-Cole dielectric relaxation.
    Returns complex permittivity (real part)."""
    denom = 1 + (1j * omega * tau) ** (1 - alpha)
    eps = eps_inf + (eps_s - eps_inf) / denom
    return jnp.real(eps)

Havriliak-Negami

def havriliak_negami(omega, eps_s, eps_inf, tau, alpha, beta):
    """Havriliak-Negami dielectric relaxation."""
    denom = (1 + (1j * omega * tau) ** (1 - alpha)) ** beta
    eps = eps_inf + (eps_s - eps_inf) / denom
    return jnp.real(eps)

Creep Analysis

Power Law Creep

def power_law_creep(t, epsilon_0, A, n):
    """Power law creep: epsilon = epsilon_0 + A * t^n."""
    return epsilon_0 + A * t**n

Burgers Model

def burgers_model(t, sigma, E1, eta1, E2, eta2):
    """Burgers viscoelastic model."""
    # Maxwell arm
    maxwell = sigma * t / eta1
    # Kelvin-Voigt arm
    tau2 = eta2 / E2
    kelvin = (sigma / E2) * (1 - jnp.exp(-t / tau2))
    # Instantaneous elastic
    elastic = sigma / E1
    return elastic + maxwell + kelvin

Tips for Materials Fitting

  1. Use appropriate units:

    • SI units for consistency

    • Document units in code comments

  2. Consider measurement uncertainty:

    # Weight by measurement precision
sigma = measurement_uncertainty
popt, pcov = fit(model, x, y, sigma=sigma, absolute_sigma=True)

  3. For rate parameters, use log-transform if spanning orders of magnitude

  4. For temperature-dependent data, use 1/T for Arrhenius analysis

  5. Validate with standards: Test on known materials first

See Also