Compute LAOS harmonic terms from one-cycle stress samples. Review odd, even, normalized, and waveform metrics. Use responsive inputs, exports, examples, formulas, FAQs, and graphs.
The calculator fits one steady LAOS cycle with a truncated Fourier series. The modeled response is σ(θ) = c0 + Σ[an sin(nθ) + bn cos(nθ)], where θ is the cycle angle and n is the harmonic number. The constant term c0 is the mean stress over the cycle.
For each harmonic, the reported amplitude is In = √(an² + bn²). The phase angle is φn = atan2(bn, an). When strain amplitude γ0 is provided, the harmonic moduli are reported as G′n = an / γ0 and G″n = bn / γ0.
Normalized intensity is calculated as In / I1. Total harmonic distortion is calculated as THD = √(Σ In² for n ≥ 2) / I1 × 100. The third harmonic ratio I3 / I1 and the even-harmonic percentage help identify nonlinear distortion strength and any symmetry deviations in the measured response.
| Cycle fraction | Example stress (Pa) |
|---|---|
| 0.0000 | 25.0000 |
| 0.0625 | 95.6802 |
| 0.1250 | 125.8650 |
| 0.1875 | 124.6965 |
| 0.2500 | 107.0000 |
| 0.3125 | 79.4311 |
| 0.3750 | 62.2254 |
| 0.4375 | 38.6623 |
Large amplitude oscillatory shear analysis helps reveal nonlinear behavior that small-amplitude testing can miss. A single harmonic pair is often enough in linear response, but nonlinear materials generate higher harmonics that carry important structural information.
This calculator is designed for practical waveform analysis. It accepts sampled stress data from one cycle, estimates harmonic coefficients with a least-squares fit, reconstructs the waveform, and organizes the result into usable engineering outputs. That makes it useful for quick screening, laboratory checks, educational demonstrations, and reporting workflows.
The odd harmonics usually dominate in symmetric strain-controlled LAOS responses, while even harmonics can act as a diagnostic signal. A strong third harmonic commonly indicates growing nonlinearity. The normalized ratios and total harmonic distortion let you compare samples even when the absolute stress level changes.
The waveform plot helps you see whether the selected harmonic order captures the measured response. The amplitude spectrum shows how much each harmonic contributes. Together with the exported tables, these outputs support repeatable interpretation and easier comparison across amplitudes, formulations, temperatures, or frequencies.
Paste one complete steady-state cycle of measured stress values. Add matching cycle positions only if your samples are not equally spaced.
Strain amplitude lets the tool convert fitted sine and cosine coefficients into harmonic moduli G′n and G″n. Leave it nonzero for modulus outputs.
It is the third harmonic amplitude divided by the fundamental amplitude. Larger values usually indicate stronger nonlinear distortion in the LAOS response.
Even harmonics are useful diagnostics. They can appear because of asymmetry, wall slip, structure evolution, or measurement imperfections during testing.
Use enough harmonics to capture the waveform without overfitting. More samples support higher harmonic orders. The calculator reduces unsafe values automatically.
It compares your measured points with the fitted Fourier reconstruction. A close match means the retained harmonics represent the cycle well.
Yes. Choose the cycle position mode that matches your data. Fractions represent 0 to 1 of a full cycle.
THD is total harmonic distortion. It combines all higher harmonic amplitudes relative to the fundamental and expresses the result as a percentage.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.