DIN 2093 Continuum Damage Mechanics (CDM): Material Degradation Assessment Under Cyclic Compressive Stress

1. Purpose of Continuum Damage Mechanics

Traditional fatigue life estimation (S‑N curves, Miner accumulation) is based on the "no damage" assumption and cannot describe the progressive degradation of materials under cyclic loading. Continuum Damage Mechanics (CDM) introduces an internal variable—the damage variable $D$—to quantitatively describe the initiation and propagation of microcracks and microvoids, thereby directly linking the degradation of material mechanical properties to external loads, enabling more accurate life prediction and health monitoring.

For disc springs, the OM point (upper surface inner edge) experiences pulsating compressive stress. Although no tensile stress appears macroscopically, cyclic compressive stress can still drive damage accumulation through mechanisms such as microplasticity and microcrack closure effects. CDM can capture this early damage, providing a theoretical basis for setting inspection intervals and replacement thresholds.

2. Definition of Damage Variable $D$

In CDM, the damage variable $D$ is a scalar (0 ≤ $D$ ≤ 1), typically defined by the reduction in effective load-bearing area or elastic modulus:

• $\tilde{E}$ — Effective elastic modulus of the currently damaged material (MPa)
• $E$ — Elastic modulus of the undamaged (virgin) material (MPa)

Physical meaning: - $D = 0$: Material is intact, no microscopic defects. - $0 < D < 1$: Internal microcracks and microvoids gradually increase, stiffness decays. - $D = 1$: Material completely loses load-bearing capacity (macrocrack formation or fracture).

During the fatigue process of a disc spring, damage first appears in the region near the OM point, manifested as a local decrease in stiffness. This local softening alters the force‑deflection characteristics of the disc spring (force reduction), which can serve as an online monitoring indicator.

3. Effective Stress and Strain Equivalence Principle

Damage reduces the effective area bearing the external load, so the effective stress $\tilde{\sigma}$ is greater than the nominal stress $\sigma$:

Similarly, according to the strain equivalence principle, the stress‑strain relationship for damaged material can be written as:

This means that, under the same deformation, the axial force provided by a damaged disc spring will decrease proportionally by $(1-D)$.

4. Fatigue Damage Evolution Equation

To describe the growth rate of damage with the number of cycles $N$, a power-law type damage evolution law based on stress (or strain) is commonly used. For the pulsating compressive stress condition of disc springs, the following simplified model is often employed (referencing Lemaitre / Chaboche theory):

Where: - $dD/dN$ — Damage increment per cycle (dimensionless) - $\sigma_{a}$ — Stress amplitude at the OM point (MPa), calculated using absolute values; in a disc spring pulsating compression cycle, $\sigma_{a} = (|\sigma_{OM,max}| - |\sigma_{OM,min}|)/2$ - $A$ — Material damage resistance coefficient (MPa), determined by experiment - $m$ — Damage exponent, reflecting the sensitivity of damage to load; for spring steel $m \approx 4 \sim 8$

This equation assumes that the damage rate is proportional to the $m$-th power of the effective stress amplitude and accelerates with damage accumulation (denominator $1-D$ decreases). When $D$ approaches 1, the damage rate tends to infinity, representing macrocrack instability.

4.1 Mean Stress Correction

The OM point experiences compressive mean stress, which has a certain inhibitory effect on damage (crack closure effect). For precise consideration, a mean stress influence factor $f(\sigma_m)$ can be introduced to modify the damage driving force:

Where $f(\sigma_m) = (1 - b \cdot \frac{\sigma_m}{R_{p0.2}})$, $b$ is a material constant (0.3 ~ 0.5). For compressive mean stress ($\sigma_m < 0$), $f(\sigma_m) > 1$, reducing the damage driving force, indicating that compressive stress has a mitigating effect on damage.

5. Damage Accumulation and Life Assessment

Integrating the damage evolution equation from initial damage $D_0$ (usually assumed to be 0) to a critical value $D_c$ yields the fatigue life $N_f$. Assuming initial $D=0$, integration gives:

Solving for the explicit relationship of damage $D$ with cycle count (for $m \neq 1$):

When $D = 1$, the theoretical number of cycles to failure $N_f$ is obtained:

6. Damage State Classification and Replacement Criteria

Based on the value of $D$, the damage state of a disc spring can be classified into three levels to guide maintenance decisions:

Engineering Threshold: When $D > 0.8$, the residual load-bearing capacity of the disc spring has dropped below 20% of its original value, and sudden fracture may occur at any time; replacement is mandatory.

7. CDM Analysis Procedure for Disc Springs

1. Determine the working stress cycle: From the minimum compression $s_{min}$ and maximum compression $s_{max}$, calculate the elastic stresses at the OM point $\sigma_{OM,min}, \sigma_{OM,max}$, then obtain $\sigma_a, \sigma_m$.
2. Obtain material damage parameters: Fit $A$ and $m$ through material fatigue tests (pulsating compression at controlled stress levels), or refer to existing data.
3. Calculate damage for the design life: If the required number of cycles $N_{req}$ is known, substitute into the $D(N)$ equation to predict $D$. If $D < 0.5$, the design is acceptable; if $D > 0.5$, reduce stress or increase inspection frequency.
4. Real-time monitoring (optional): Infer $D$ by measuring the reduction ratio of the disc spring force $(F_{current}/F_0 \approx 1-D)$, enabling condition-based maintenance.
5. Critical replacement: When $D$ reaches 0.8, enforce replacement.

8. Calculation Example

Given: Disc spring OM point stress amplitude $\sigma_a = 400\ \text{MPa}$, mean stress $\sigma_m = -1200\ \text{MPa}$ (compressive).
Material damage parameters: $A = 1200\ \text{MPa}$, $m = 6$, mean stress correction omitted.

Calculate theoretical failure life:

(This example has a high stress amplitude, resulting in a very short life; for illustration only)

Predict damage after $N = 50$ cycles:

, within the safe operating zone.

If the stress amplitude increases to 500 MPa, then $\sigma_a/A = 500/1200 \approx 0.4167$, $0.4167^6 \approx 0.00523$, 7×0.00523×50 ≈ 1.83, which exceeds 1, indicating $N_f < 50$, D has reached 1, premature failure.

9. Important Notes

• Specificity of compressive stress damage: The OM point is under compressive stress; microcracks tend to close, so the actual damage rate may be lower than the model prediction. The model provides a conservative estimate.
• Material parameter acquisition: $A$ and $m$ must be determined through pulsating compression fatigue tests; tensile data should not be directly applied.
• Multiaxial effects: The OM point is actually under multiaxial compressive stress; damage evolution may be slower due to hydrostatic pressure effects. The simplified model is conservative.
• Relationship with Miner accumulation: CDM provides a micro-mechanistic complement to Miner's rule, can account for load sequence effects, and offers higher prediction accuracy.

Summary: Continuum Damage Mechanics uses the damage variable $D$ to quantify the material degradation process of disc springs under cyclic compressive stress, predicts life through evolution equations, and provides decision support for the safe use and condition-based maintenance of disc springs based on the criteria of accelerated damage for $D > 0.5$ and mandatory replacement for $D > 0.8$.