Calculator
Formula used
This calculator uses the Paris–Erdogan crack growth relation:
- da/dN = C(ΔK)^m
- ΔK = Y · Δσ · √(πa) (when loading and geometry are used)
If a threshold is provided, growth is set to zero when ΔK ≤ ΔKth. If KIC and σmax are provided, the tool also estimates Kmax = Y·σmax·√(πa) for a safety advisory.
How to use this calculator
- Select a unit system and enter the Paris constants C and m.
- Choose From loading and geometry to compute ΔK(a), or Direct constant ΔK for constant driving force.
- Enter a0 and af. For geometry-based mode, provide Δσ and geometry details.
- Optional: add ΔKth and KIC for threshold and fracture advisories.
- Press Calculate. Use the buttons to export the profile as CSV or PDF.
Example data table
| System | C | m | Mode | Δσ | Y | a0 | af | Expected outcome |
|---|---|---|---|---|---|---|---|---|
| SI | 1e-12 | 3.0 | Geometry | 100 MPa | 1.12 | 0.001 m | 0.005 m | Cycles decrease as Δσ or a increase. |
| Imperial | 1e-10 | 2.7 | Direct | — | — | 0.05 in | 0.20 in | Growth is linear in N for constant ΔK. |
1) Why Paris law matters in fatigue design
Paris–Erdogan crack growth links crack extension per cycle to the stress-intensity range, bridging laboratory data and field life prediction. When parts contain small flaws, the growth model helps choose inspection intervals, repair timing, and conservative retirement limits safely.
2) Inputs this calculator uses and what they represent
Enter Paris constants C and m, plus crack limits a0 and af. In geometry mode, Δσ and a selected factor Y generate ΔK(a). In direct mode, you supply constant ΔK for quick comparison studies.
3) Stress intensity range and the geometry factor Y
For many cases, ΔK = Y·Δσ·√(πa). The dimensionless Y captures finite-width and specimen effects. The calculator includes constant Y, single-edge, center-crack, and compact-tension options. A mismatched geometry can shift ΔK and strongly bias predicted cycles.
4) Typical parameter magnitudes and unit consistency
In metals, m often lies near 2–4, while C spans orders of magnitude and depends on units and environment. For example, with C=1e-12 and m=3, a constant ΔK=10 gives da/dN≈1e-9 length per cycle. Keep a, W, and C consistent within one system.
5) Threshold and fracture limits add realism
If you provide ΔKth, growth is set to zero when ΔK ≤ ΔKth, which can dominate at low loads; typical values for many steels are a few MPa·√m but vary widely. If you also supply KIC and σmax, the tool estimates Kmax and warns when unstable fracture may occur before af.
6) Cycles from a0 to af are an integral
With constant ΔK, cycles follow N=(af−a0)/(C·ΔK^m). With variable ΔK(a), cycles require integrating 1/(C·ΔK(a)^m) over crack length. This calculator uses a trapezoidal numerical integration and reports a profile table including cumulative cycles across the interval.
7) Converting cycles into an inspection plan
Choose a0 as the smallest reliably detectable flaw and af as an allowable size from design or fracture criteria. Apply a safety factor to the cycle count, then convert to time using frequency: at 10 Hz, 2,000,000 cycles correspond to about 55.6 hours of loading.
8) Practical data checks and common pitfalls
Use crack-growth data measured for the same temperature, environment, and stress ratio when possible. Ensure a/W remains within the validity range of your selected geometry, especially near large cracks. Extremely large or infinite results often indicate ΔK falls below threshold or inputs are inconsistent.
1) What is Paris law used for?
It estimates fatigue crack growth per load cycle using measured material constants. Engineers use it to predict remaining life between detectable and critical crack sizes under repeated loading.
2) Should I enter C in SI or Imperial?
Enter C in units consistent with your chosen system and crack-length units. If you change from meters to inches, you must also use a C value fitted for that unit system.
3) What does the stress ratio R do here?
When you provide R, the calculator can estimate σmax from Δσ. Paris growth primarily depends on ΔK, but σmax is useful for the optional KIC advisory.
4) When should I use direct ΔK mode?
Use it when ΔK is known from a separate analysis or when you want a quick sensitivity study. It assumes ΔK does not change with crack length, so it is not suitable when geometry effects are strong.
5) Why do I get infinite cycles?
Infinite or extremely large cycle counts usually appear when ΔK is at or below ΔKth somewhere in the interval, or when the computed growth rate becomes effectively zero. Recheck loads, geometry, and thresholds.
6) Is the geometry factor valid for any a/W?
No. Many closed-form factors are intended for specific ranges, such as edge-crack polynomials often used around a/W ≤ 0.6. If your crack is too large, switch to a more appropriate factor or a standards-based solution.
7) Can I use this for variable-amplitude loading?
This calculator assumes a representative constant range per cycle. For variable amplitude, you typically need cycle counting (like rainflow) and an accumulation model, or you can approximate using an effective ΔK spectrum derived from your load history.