Enter Data and Options
Formula Used
The generalized Kelvin chain represents creep compliance as a sum of delayed compliance terms:
- J(t) is creep compliance (1/Pa).
- J0 is instantaneous compliance.
- Ji are delayed compliance magnitudes for each element.
- τi are characteristic times you provide.
Because the model is linear in J0 and Ji, the calculator estimates them by least squares once τ values are chosen.
How to Use This Calculator
- Select Input Mode: strain under constant stress, or direct compliance.
- If using strain, enter the constant Applied Stress in pascals.
- Enter 1–6 values for τ in seconds, separated by commas.
- Paste your dataset as rows: time then strain or compliance.
- Click Fit Kelvin Chain to compute J0 and Ji.
- Download CSV or PDF for parameters and pointwise predictions.
Example Data Table
Sample creep strain under a constant stress of 100,000 Pa (values are illustrative).
| Time (s) | Strain |
|---|---|
| 0 | 0.00000 |
| 1 | 0.00012 |
| 2 | 0.00018 |
| 5 | 0.00028 |
| 10 | 0.00036 |
| 20 | 0.00044 |
| 50 | 0.00054 |
| 100 | 0.00060 |
Generalized Kelvin Chain Fit: Practical Guide
1) Purpose of this fit
This calculator helps you fit a generalized Kelvin chain to creep data so you can summarize time-dependent viscoelastic response with a compact, interpretable parameter set. You can paste time–strain measurements under constant stress, or time–compliance points directly. The output reports instantaneous compliance J0, delayed compliances J1…JN, and diagnostics such as RMSE and R².
2) What the Kelvin chain captures
A Kelvin chain models creep as a sum of exponential approach terms. Each element contributes a compliance increment Ji that evolves toward its steady value with time constant τi. Short τ values describe rapid early creep, while large τ values describe slow drift. The total long-time compliance trends toward J0 + ΣJi.
3) Data requirements and preparation
For stable estimation, use at least (N + 1) valid points, and preferably 5–10× more than the parameter count. Include early and late times; for example, spanning 0.1τ to 10τ across your τ list improves identifiability. Remove obvious outliers and confirm that time is nonnegative and consistently scaled.
4) Selecting τ values
The calculator treats τ values as fixed design choices and solves only the linear coefficients. A common starting set is logarithmically spaced times such as 0.5, 2, 10, 50, and 200 seconds, adjusted to match your experiment duration. If the fit is poor at early times, add a smaller τ; if the tail is biased, add a larger τ.
5) Least squares fitting in practice
Once τ values are selected, the model becomes linear in J0 and Ji. The calculator builds the design matrix using (1 − e−t/τ) basis functions and computes a least squares solution. This approach is fast, deterministic, and well-suited for repeated what-if runs while you refine τ choices.
6) Interpreting fitted parameters
J0 represents the immediate elastic compliance (inverse stiffness). Each Ji represents an additional recoverable compliance that activates over its τi. If you supplied strain under stress σ, the calculator converts ε(t) to compliance using J(t)=ε(t)/σ, so ensure σ is accurate and constant.
7) Fit quality and diagnostics
RMSE measures average compliance error in 1/Pa. R² summarizes explained variance when the data vary enough to define it. Inspect residuals for structure: systematic early-time residuals often indicate missing short τ, while late-time curvature suggests the longest τ is too small. Use exports to compare runs side-by-side.
8) Reporting, traceability, and exports
For publication or QA, report the τ list, parameter values, stress level, and the time window used. The CSV export provides fitted parameters plus pointwise measured, fitted, and residual values for auditing. The PDF report is convenient for quick sharing in lab notebooks and project documentation.
FAQs
1) What if I do not know good τ values?
Start with 3–5 logarithmically spaced τ values spanning your test duration, such as 1, 10, 100 seconds. Adjust by adding smaller τ for early-time mismatch and larger τ for long-time bias.
2) How many Kelvin elements should I use?
Use the smallest N that captures the trend. Three elements often work for smooth creep curves; complex materials may need 4–6. More elements require more data and can amplify noise sensitivity.
3) Why are some fitted J values negative?
Negative coefficients can occur when τ choices are poorly conditioned or the data are noisy. Try reducing N, changing τ spacing, or increasing data coverage across early and late times to improve stability.
4) Can I fit compliance directly without stress?
Yes. Choose “Compliance input” and paste time–J(t) points in consistent units. The fit then estimates J0 and Ji directly without any strain-to-compliance conversion.
5) Does the calculator estimate viscosities or moduli?
It estimates compliance coefficients and uses τ values you provide. You can convert compliance to modulus only under additional assumptions. For rheology workflows, keep parameters in compliance form for clarity.
6) How do I handle time units like minutes or hours?
Enter time values in one consistent numeric unit and provide τ in the same unit. The “Time Unit Label” is a display label; it does not rescale the underlying numbers.
7) What should I include when sharing results?
Share the τ list, stress level (if used), dataset range, fitted J0 and Ji, and RMSE/R². Exporting CSV provides traceable pointwise predictions and residuals for reviewers.