DIN 2093 Plastic Accumulation / Low Cycle Fatigue (LCF): Coffin‑Manson Strain‑Life Method

1. Definition and Applicability of Low Cycle Fatigue

Low Cycle Fatigue (LCF) typically refers to fatigue failure where the number of cycles to failure $N_f < 10^5$. Unlike High Cycle Fatigue (stress-controlled, where stress amplitude is below the macroscopic yield limit), LCF occurs under high loads, where each cycle produces significant plastic deformation, and damage is dominated by the plastic strain amplitude.

In disc springs, the following situations may enter the LCF regime: - Excessive maximum compression, causing the elastic stress at the OM point and other locations to far exceed the material's yield strength, resulting in cyclic plasticity; - Subjected to impact or overload, where each load cycle produces considerable plastic strain; - High service temperature, reducing the material's yield strength and increasing plastic deformation under the same load; - Intentional use of plastic reserve for finite-life design (e.g., emergency buffers).

LCF assessment must be based on strain amplitude rather than stress amplitude, because the stress-strain relationship is no longer unique in the plastic regime.

2. The Coffin‑Manson Relationship

The Coffin‑Manson law describes the power-law relationship between the plastic strain amplitude $\Delta\varepsilon_p$ and the number of cycles to failure $N_f$ for metallic materials in the LCF regime:

or equivalently:

where: - $\Delta\varepsilon_p$ — plastic strain range (difference between maximum and minimum plastic strain within one cycle); - $\varepsilon_f'$ — fatigue ductility coefficient (approximately equal to the true fracture strain $\varepsilon_f$ from a monotonic tensile test); - $c$ — fatigue ductility exponent, for most metals $c \approx -0.5 \sim -0.7$; - $C_p = \frac{1}{2} \left( \frac{2}{\varepsilon_f'} \right)^{1/|c|}$ — combined constant; - $m_p = \frac{1}{|c|} \approx 0.5 \sim 0.7$ for spring steels.

Common engineering values (quenched and tempered spring steel such as 50CrV4, hardness 45~50 HRC): - $m_p \approx 0.6$ - $C_p$ is approximately $0.2 \sim 0.5$ (depending on the units of $\Delta\varepsilon_p$ being mm/mm, i.e., dimensionless); a more common form is to directly use the material's Coffin‑Manson curve.

If the total strain amplitude $\Delta\varepsilon/2$ is used, the elastic and plastic strain amplitudes must be separated. In the LCF-dominated regime, the plastic strain amplitude is much larger than the elastic strain amplitude, so the plastic part can be used directly as an approximation.

3. Methods for Determining Parameters

1. Material Testing: Conduct a series of symmetric strain-controlled cyclic tests, record the fatigue life at different strain amplitudes, fit to obtain $\varepsilon_f'$ and $c$, and then calculate $m_p, C_p$.
2. Handbook Data: Some material standards or literature provide Coffin‑Manson parameters. For example, for spring steels, $\varepsilon_f' \approx 0.5 \sim 1.0$, $c \approx -0.6$.
3. Approximate Estimation (when testing data is unavailable): Use monotonic tensile plasticity indicators. Manson's universal slopes method gives:
   $$\Delta\varepsilon_p = 3.5 \cdot \frac{\sigma_b}{E} \cdot N_f^{-0.12} + D^{0.6} \cdot N_f^{-0.6}$$

where $\sigma_b$ is the ultimate tensile strength and $D$ is the reduction of area. However, measured data is more recommended.

Recommendation for disc spring applications: Since disc spring materials undergo shot peening and heat treatment, their actual fatigue performance is superior to standard specimens. Using standard data may be conservative; it is best to use LCF test data from disc spring products.

4. Obtaining the Plastic Strain Amplitude $\Delta\varepsilon_p$ from Disc Spring Operating Conditions

Under LCF conditions for disc springs, the OM point is the control point. Due to geometric nonlinearity and elastic-plastic behavior, the plastic strain cannot be directly obtained from elastic stress formulas. Methods for obtaining $\Delta\varepsilon_p$:

4.1 Elastic-Plastic Finite Element Analysis (Recommended)

• Establish a disc spring model incorporating the real material hardening curve;
• Apply the minimum compression $s_{min}$ and maximum compression $s_{max}$, perform static analysis at the two extreme points (or direct cyclic loading);
• Extract the equivalent plastic strain $\varepsilon_{eq}^{p}$ at the OM point for both limit states;
• Plastic strain range $\Delta\varepsilon_p = \varepsilon_{eq,max}^{p} - \varepsilon_{eq,min}^{p}$.

If the material exhibits kinematic hardening, the plastic strain range can be used directly in the Coffin‑Manson formula.

4.2 Local Stress-Strain Method (Neuber Correction)

When FEA is unavailable, elastic stress analysis combined with Neuber's rule can be used to estimate the local elastic-plastic strain:

where $S$ is the nominal elastic stress (elastic value at the OM point), and $K_t$ is the theoretical stress concentration factor (for a smooth disc spring OM point $K_t \approx 1.0$; for notches like teeth, refer to tables). Combined with the material's cyclic stress-strain curve $\varepsilon = \sigma/E + (\sigma/K')^{1/n'}$, the local stress $\sigma$ and total strain $\varepsilon$ can be solved, and the plastic part can be obtained. This method is suitable for cases with significant notches.

Disc springs generally have no notches, and the geometric transition at the OM point is small, so finite element analysis can be used directly.

5. Mean Stress Correction

LCF data is mostly obtained from symmetric cycling ($R=-1$). The OM point of a disc spring typically experiences pulsating compressive stress ($R \approx 0$ or positive compression), with a high mean compressive stress. A mean stress correction to the Coffin‑Manson relationship is necessary.

The most common correction is the Morrow correction:

where $\sigma_m$ is the mean stress (algebraic value, compression is negative). Substituting a compressive stress increases the right-hand side of the equation, indicating that a compressive mean stress increases fatigue life for the same plastic strain amplitude; conversely, a tensile mean stress decreases life. For disc springs, the mean compressive stress at the OM point is high, so the corrected life will be longer than that from symmetric cycling.

A simple criterion for design: if $\sigma_m \le -0.5\,R_{p0.2}$ (sufficiently large compressive stress), the adverse effect of mean stress can be neglected, and symmetric cycling data can be used directly for a conservative design.

6. Combination with High Cycle Fatigue (S‑N)

The entire fatigue life curve can be unified by the Basquin‑Coffin‑Manson equation:

• High cycle regime: dominated by elastic strain, the first term is dominant;
• Low cycle regime: dominated by plastic strain, the second term is dominant.

For disc springs, if the operating stress amplitude lies in the transition zone between high and low cycle fatigue (e.g., $N_f \approx 10^5 \sim 10^6$), both terms should be evaluated, and the more conservative life should be adopted.

7. Low Cycle Fatigue Design Procedure

1. Determine the load cycle: minimum/maximum compression $s_{min}, s_{max}$.
2. Perform elastic-plastic FEA (or use the Neuber method) to obtain $\Delta\varepsilon_p$ and $\sigma_m$ at the OM point.
3. Look up material Coffin‑Manson parameters: $\varepsilon_f', c$ or $C_p, m_p$.
4. Apply the Morrow correction (if necessary).
5. Calculate the predicted life $N_f$.
6. Safety factor: require $N_f \ge N_{req} \cdot S_{LCF}$, with $S_{LCF} \ge 1.5$ (safety factors for LCF are typically higher).
7. If life is insufficient, reduce compression, increase disc spring dimensions, or select a material with better fatigue ductility (e.g., increase tempering temperature for better plasticity).

8. Calculation Example

Disc spring: $D_e=40\ \text{mm}, D_i=20.4\ \text{mm}, t=2.0\ \text{mm}, h_0=1.2\ \text{mm}$, material 50CrV4, $R_{p0.2}=1500\ \text{MPa}, E=206\,000\ \text{MPa}$.
Operating cycle: $s_{min}=0.2\ \text{mm}, s_{max}=0.9\ \text{mm}$.

Elastic-plastic FEA results: - At $s_{max}$, equivalent plastic strain at OM point $\varepsilon_{eq}^{p} = 0.012$ (1.2%); - At $s_{min}$, OM point is still elastic ($\varepsilon_{eq}^{p} \approx 0$); - $\Delta\varepsilon_p = 0.012$.

Coffin‑Manson parameters (material measured): $\varepsilon_f' = 0.6, c = -0.55$.

Morrow correction: Mean compressive stress at OM point $\sigma_m \approx -800\ \text{MPa}$ (algebraic value). After correction:

Precise calculation: $\ln(2N_f) = -\frac{1}{0.55} \ln(0.009936) \approx -1.818 \times (-4.61) \approx 8.38$, $2N_f \approx e^{8.38} \approx 4350$, $N_f \approx 2175$ cycles.

Conclusion: The expected life is approximately 2 200 cycles, which is typical LCF. If a life > 10 000 cycles is required, the maximum compression must be reduced.

9. Important Notes

• Compressive stress and plasticity: Even if the cycle is pulsating compression, local plastic strain can still occur, and the Coffin‑Manson relationship remains applicable.
• Hysteresis loop energy: Energy dissipation from plastic deformation causes temperature rise, which may lead to creep-fatigue interaction; special evaluation is needed at high temperatures.
• Multiaxial effects: The stress state at the OM point of a disc spring is actually multiaxial, but the equivalent plastic strain already integrates this information and can be used directly for analysis.
• Experimental support: LCF parameters have high scatter; for critical applications, strain-controlled fatigue tests should be performed to calibrate the model.

Summary: The Coffin‑Manson relationship is a core tool for LCF design of disc springs, directly linking plastic strain amplitude to fatigue life. By extracting the plastic strain at the OM point via elastic-plastic FEA and applying the Morrow mean stress correction, the finite life under high loads can be predicted with reasonable accuracy, providing direction for optimization.