DIN 2093 Ratchetting Assessment: Cumulative Plastic Deformation under Cyclic Loading

1. Definition of Ratchetting

Ratchetting refers to the phenomenon where, under asymmetric cyclic loading, each stress cycle produces cumulative plastic deformation in the same direction, leading to continuous geometric changes and eventual failure due to excessive deformation or low-cycle fatigue. In disc springs, ratchetting manifests as:

• Continuous reduction in free height ($h_0$ decreases progressively with cycles), i.e., Set loss accumulates far beyond the allowable value from a single flattening;
• Continuous decay of preload, with gradual loss of elastic compensation capability;
• Progressive plastic flow of material near the OM point, eventually initiating cracks or fracture.

Unlike elastic/plastic shakedown, ratchetting is a progressive failure mode that must be strictly avoided.

2. Conditions for Ratchetting

Ratchetting in disc springs is primarily caused by the combined effect of high mean stress and high stress amplitude. In the stress cycle at the OM point (inner edge of the upper surface):

• The larger the mean stress $\sigma_m$ (absolute value), the longer the material remains under high stress;
• The larger the stress amplitude $\sigma_a$, the stronger the driving force for cyclic plastic deformation.

When the combination exceeds the material's ratchetting limit, each cycle produces an irrecoverable increment of plastic strain, and these increments do not decay with the number of cycles.

3. Engineering Criteria for Ratchetting Limit

For disc springs, there is no simple analytical formula to directly provide the ratchetting limit, but engineering judgment criteria can be summarized based on shakedown theory and extensive testing.

3.1 Criterion Based on Shakedown Factor

In shakedown analysis, a load amplification factor $\lambda_{SD}$ is defined. If $\lambda_{SD} \le 0$, the structure will undergo ratchetting (see the "Shakedown Analysis" section). Therefore, through finite element or shakedown limit analysis, the critical ratchetting load can be quantitatively determined.

3.2 Qualitative Risk Classification Based on Stress State

During the preliminary design phase, risk classification can be performed based on the elastic calculated stress (absolute value) at the OM point.

Risk Level	Condition	Expected Behavior
Low Risk	$\sigma_{max} = \sigma_m + \sigma_a \le R_{p0.2}$	Fully elastic, no plastic accumulation
Medium Risk	$\sigma_{max} > R_{p0.2}$ but $\sigma_{min} = \sigma_m - \sigma_a > 0.2\,R_{p0.2}$	Peak yielding, but high mean compressive stress may lead to plastic shakedown; if stress amplitude is too large, there is a tendency towards ratchetting
High Risk	$\sigma_{max} \gg R_{p0.2}$ and $\sigma_a > 0.5\,R_{p0.2}$	High mean stress + high stress amplitude, very prone to ratchetting

High-Risk Range: When the mean stress $\sigma_m$ exceeds $0.7\,R_{p0.2}$ and the stress amplitude $\sigma_a$ exceeds $0.3\,R_{p0.2}$, the ratchetting risk increases significantly. This qualitative criterion can be used for initial design screening; accurate determination requires FEA.

3.3 Numerical Assessment – Cumulative Plastic Strain Method

Through elastic-plastic finite element analysis, directly simulate several cycles (typically 5~10) and extract the equivalent plastic strain $\varepsilon_{eq}^{p}$ at the OM point as a function of cycles. If the increment per cycle $\Delta \varepsilon_{eq}^{p}$ stabilizes to a non-zero value, ratchetting is occurring.

Criteria: - If $\Delta \varepsilon_{eq}^{p} \to 0$ (increment decreases as cycle number increases) → Plastic shakedown; - If $\Delta \varepsilon_{eq}^{p} \approx \text{constant}$ or increases → Ratchetting.

4. Design Countermeasures

Once a ratchetting risk is identified, the following measures must be taken:

1. Reduce Load Level: Decrease the maximum working deflection to lower $\sigma_{max}$ and $\sigma_a$.
2. Increase Disc Spring Cross-Section: Increase thickness $t$ or outer diameter $D_e$ to reduce stress levels at the same deflection.
3. Reduce Cone Height Ratio $h_0/t$: Flatter disc springs have smaller stress amplitudes for the same compression stroke.
4. Use Higher Strength Material: Increase $R_{p0.2}$ to reduce the tendency for plasticity at the same stress level.
5. Improve Heat Treatment and Shot Peening: Optimize residual stress distribution; introducing compressive stress on the surface can inhibit ratchetting crack initiation.

5. Conclusion

Ratchetting is a severe failure mode for disc springs under asymmetric cyclic high compressive stress. Designers should initially screen for high-risk points through elastic stress analysis, then confirm using finite element shakedown or cyclic plasticity analysis to ensure no continuous cumulative plastic deformation occurs throughout the entire service life. Safe design should ensure the disc spring operates in the elastic shakedown region, or only allows limited plasticity from a single presetting operation, while completely eliminating ratchetting.