DIN 2093 Fatigue Crack Growth: Paris Formula and Residual Life

1. Basic Concepts of Fatigue Crack Growth

Under cyclic loading, even if the initial stress does not exceed the material's yield strength, small defects on the surface or within a disc spring (such as inclusions, corrosion pits, or machining marks) can gradually propagate under cyclic stress, eventually reaching a critical size and causing unstable fracture. This process is called fatigue crack growth, and its rate is described by the Paris formula.

Fatigue crack growth analysis is used for: - Damage tolerance design: Determining the allowable defect size within periodic inspection intervals. - Residual life prediction: Estimating the number of remaining cycles given a known current crack size. - Material and process selection: Comparing the crack growth resistance of different materials and surface treatments.

2. Paris Formula

In the stable growth stage (where the crack growth rate $da/dN$ and the stress intensity factor range $\Delta K$ exhibit a linear relationship on a log-log scale), the Paris formula is:

• $a$ — Crack depth or length (mm or m);
• $N$ — Number of stress cycles;
• $da/dN$ — Crack growth per cycle (mm/cycle or m/cycle);
• $\Delta K$ — Stress intensity factor range (MPa·√m), $\Delta K = K_{max} - K_{min}$;
• $C$ — Paris constant (dependent on material, stress ratio, and environment);
• $m$ — Paris exponent, for most metallic materials, $m \approx 3 \sim 4$.

This formula indicates that the crack growth rate is dominated by $\Delta K$ and is highly sensitive to changes in $\Delta K$ (due to the large exponent $m$).

3. Calculation of Stress Intensity Factor Range $\Delta K$

For typical crack morphologies in disc springs (surface semi-elliptical cracks, edge corner cracks), $\Delta K$ can be expressed as:

• $\Delta \sigma$ — Stress amplitude at the critical point (MPa). For fatigue problems, this is the difference between the maximum and minimum stress at that point during the working cycle. For example, at point I (lower surface outer edge) of a disc spring, which experiences tensile stress, $\Delta \sigma = \sigma_{I,max} - \sigma_{I,min}$.
• $Y$ — Geometric correction factor (dimensionless), dependent on crack shape, component dimensions, and boundary conditions. For common surface semi-elliptical cracks, $Y \approx 0.65 \sim 1.2$; specific values can be found in fracture mechanics handbooks.
• $a$ — Crack depth (mm or m).

Note: If the critical point is under a compressive stress cycle (e.g., point OM is always in compression), the crack faces will close, the crack growth driving force $\Delta K$ will be extremely small, and the crack will either not propagate or propagate very slowly. Therefore, fatigue crack growth analysis should focus on tensile stress zones, such as points I and II of the disc spring, where tensile stresses may occur.

4. Material Constants $C$ and $m$

For high-strength spring steels (e.g., 50CrV4, C75S) in a room-temperature air environment, typical Paris constants are:

Note: The value of $C$ strongly depends on the units of $\Delta K$ (MPa·√m or N/mm³/²); ensure unit consistency when using. Here, $C$ is given for $\Delta K$ in MPa·√m and $a$ in mm.

Effect of Stress Ratio $R$: The Paris formula does not explicitly include mean stress, but $R = \sigma_{min}/\sigma_{max}$ affects crack closure and thus the effective $\Delta K_{eff}$. For design, data for $R = 0$ can be conservatively used, or the Forman correction formula can be applied.

5. Residual Life Integration

The number of cycles $N_f$ required to propagate from an initial crack size $a_0$ to a critical crack size $a_c$ is obtained by integrating the Paris formula:

Separating variables and integrating:

Assuming the geometric factor $Y$ is constant (or an average value is used) during crack growth, direct integration is possible. For $m \neq 2$, the integration result is:

When $m = 2$ (rare), a logarithmic form applies. For most steels, $m \approx 3$, and the integral becomes:

Parameter Explanation: - $a_0$ — Initial crack size (mm), determined by non-destructive testing capability or an assumed initial defect size (e.g., 0.1 mm); - $a_c$ — Critical crack size (mm), determined by fracture toughness: $a_c = \frac{1}{\pi} \left( \frac{K_{IC}}{Y \cdot \sigma_{max}} \right)^2$; - $\Delta \sigma$ — Stress amplitude (MPa), the range of stress variation at the critical point during the working cycle.

6. Application Steps for Disc Springs

1. Identify Critical Points: Typically tensile stress zones, such as point I (lower outer edge) or point II (upper outer edge). These locations may experience tensile or tensile-compressive cycles during operation, making cracks prone to propagation.
2. Stress Analysis: Perform elastic or elastic-plastic FEA to calculate stresses at the minimum and maximum compression levels, obtaining $\sigma_{max}, \sigma_{min}$ and $\Delta \sigma$.
3. Determine Initial Crack: Based on non-destructive testing sensitivity or a conservative assumption (e.g., $a_0 = 0.1\ \text{mm}$).
4. Determine Critical Crack: Calculate $a_c$ from the material's $K_{IC}$ and $\sigma_{max}$.
5. Select $Y$ Factor: Look up or calculate based on crack morphology and location. For example, for a surface crack at the inner edge of a disc spring, $Y \approx 1.12$ (surface shallow crack correction) can be used as an approximation.
6. Calculate Residual Life $N_f$: Substitute into the integral formula above.
7. Safety Factor: Fatigue crack growth life predictions have high scatter; a safety factor $S_{crack} \ge 2.0$ is recommended, requiring $N_f \ge N_{service} \cdot S_{crack}$.

7. Calculation Example

Given: - Disc spring point I: $\sigma_{max} = 800\ \text{MPa}$, $\sigma_{min} = 100\ \text{MPa}$ → $\Delta \sigma = 700\ \text{MPa}$, $R \approx 0.125$ - Material 50CrV4, $K_{IC} = 65\ \text{MPa·√m}$, Paris constant $C = 3.0 \times 10^{-12}$ (mm/cycle, MPa·√m units), $m = 3.2$ - Surface semi-elliptical crack, depth direction $a$, longer in length direction, simplified with $Y = 0.72$ (from lookup table) - Initial crack $a_0 = 0.1\ \text{mm}$, critical crack $a_c$ from $K_{IC} = Y \cdot \sigma_{max} \cdot \sqrt{\pi a_c}$:

Calculate Residual Life ($m=3.2 \neq 2$):

Step-by-step calculation: - $Y \cdot \Delta \sigma = 0.72 \times 700 = 504\ \text{MPa}$ - $(504)^{3.2} = \exp(3.2 \ln 504) = \exp(3.2 \times 6.2226) \approx \exp(19.912) \approx 4.42\times10^8$ - $\pi^{1.6} = \exp(1.6 \ln \pi) = \exp(1.6 \times 1.1447) \approx \exp(1.8315) \approx 6.24$ - Denominator part: $1.2 \times 3.0\times10^{-12} \times 4.42\times10^8 \times 6.24 \approx 1.2 \times 3.0\times10^{-12} \times 2.76\times10^9 \approx 1.2 \times 8.28\times10^{-3} = 9.94\times10^{-3}$ - Numerator $2 / (9.94\times10^{-3}) \approx 201.2$ - Bracket term: $1/0.1^{0.6} - 1/4.05^{0.6}$
$0.1^{0.6} = \exp(0.6 \ln 0.1) = \exp(0.6 \times (-2.3026)) = \exp(-1.3816) \approx 0.251$

, reciprocal is
$4.05^{0.6} = \exp(0.6 \ln 4.05) = \exp(0.6 \times 1.3987) \approx \exp(0.8392) \approx 2.315$

, reciprocal is
Difference $= 3.98 - 0.432 = 3.548$

cycles.

Interpretation: The expected crack growth life is only about 714 cycles, indicating that this stress level is too high for long-term use. If a life of $10^5$ cycles is required, the stress amplitude must be significantly reduced.

8. Design Recommendations and Limitations

• No crack growth assessment needed for compressive stress zones: Crack growth at point OM under compressive stress cycles is extremely slow and typically does not control life; focus on tensile stress zones like point I.
• Applicability of Paris Formula: Valid for stress intensity factor ranges $\Delta K$ above the threshold $\Delta K_{th}$ and below the rapid fracture region. For spring steels, $\Delta K_{th} \approx 3 \sim 6\ \text{MPa·√m}$; below this value, cracks do not propagate.
• Variable Amplitude Loading: If the actual load spectrum is variable amplitude, an equivalent $\Delta K$ or the strip method must be used; a single $\Delta \sigma$ is not sufficient.
• Safety Factor: Due to high scatter in crack growth data, a high safety factor (≥ 2) should be used in design, combined with periodic non-destructive testing.
• Experimental Validation: For critical components, crack growth rate testing is recommended to determine accurate $C$ and $m$ values.

Summary: The Paris formula is a core tool for damage tolerance design of disc springs. By integration, the residual life from an initial crack to a critical crack can be predicted. Correct selection of the stress amplitude at the critical point, geometric factor, and material constants, along with the application of fracture mechanics criteria, can effectively ensure the safe operation of disc springs within a finite life.