DIN 2093 Fracture Mechanics Assessment: Linear Elastic Fracture Mechanics (LEFM)

1. Scope and Basic Concepts

Linear Elastic Fracture Mechanics (LEFM) is applicable for assessing the safety of disc springs containing cracks or crack-like defects. When stress levels are high, or material toughness decreases after heat treatment, or fatigue cracks propagate to a certain size under cyclic loading, LEFM can be used to determine whether a crack will become unstable and lead to sudden fracture.

Basic premise: The plastic zone size at the crack tip is much smaller than the crack length and component dimensions, and the overall response remains linear elastic.

2. Stress Intensity Factor $K_I$

2.1 Definition

The stress intensity factor $K_I$ characterizes the intensity of the stress field near the crack tip and is the driving force for crack propagation. For opening mode (Mode I) cracks, the general formula is:

Where: - $K_I$ — Mode I stress intensity factor (MPa·√m or N·mm⁻³/², common unit MPa·√m) - $\sigma$ — Remote applied stress (MPa). For disc springs, take the maximum principal stress at the critical point (if the OM point is under compressive stress, the crack may close; use the minimum principal stress or shear stress for assessment; if the crack is in a tensile region, such as the I point, use the tensile stress) - $a$ — Crack depth (for surface cracks) or half-length (for embedded cracks) (m or mm) - $Y$ — Geometric correction factor (dimensionless), dependent on crack shape, location, component geometry, and loading method

2.2 Typical $Y$ Factors for Disc Spring Cracks

Disc spring cracks may occur at: - Surface semi-elliptical cracks (most common, originating from surface defects or corrosion pits) - Edge corner cracks (at the inner or outer edge) - Embedded cracks (internal inclusions, etc.)

For plate or shell components of thickness $t$, geometric factors for common cracks can be obtained from handbooks. For example, for a center through-crack in an infinite plate $Y = 1.0$; for an edge crack in a finite-width plate, $Y$ varies with $a/t$.

The inner edge (OM point) of a disc spring is under compressive stress, causing crack faces to close; pure Mode I loading does not drive propagation. If a tensile stress component exists at this point (e.g., during unloading), or if shear (Mode II/III) is present on the crack plane, assessment is still required. In practice, disc spring fatigue cracks often originate in tensile stress regions (e.g., the I point), where $K_I$ verification is applied.

3. Fracture Toughness $K_{IC}$

Fracture toughness $K_{IC}$ is an inherent material property representing resistance to unstable crack propagation. In the LEFM context, the safety condition is:

For common disc spring materials:

Note: Specific values are influenced by heat treatment, tempering temperature, and grain size; higher strength implies lower toughness.

Plane Strain vs. Plane Stress

The above $K_{IC}$ is the plane strain fracture toughness. If the disc spring thickness is small, the crack tip may be in a plane stress state, and the apparent toughness may be slightly higher than $K_{IC}$; using $K_{IC}$ is conservative.

4. Critical Crack Size $a_{crit}$

When $K_I = K_{IC}$, the crack reaches the critical state for unstable propagation. Solving the stress intensity factor formula inversely gives the critical crack size:

• $\sigma$ — Maximum working stress (MPa)
• Other symbols as before.

Significance: If the actual crack (or defect) size $a < a_{crit}$, brittle fracture will not occur, and the crack propagates slowly under fatigue loading; once $a \ge a_{crit}$, the crack will propagate rapidly and unstably, leading to component fracture.

5. Application in Disc Spring Design

5.1 Allowable Initial Defect Size

Given the disc spring stress level and material $K_{IC}$, the maximum allowable initial defect size $a_{allow}$ can be derived (divided by a safety factor $S_F \ge 1.5$):

This serves as an acceptance criterion in non-destructive testing (e.g., magnetic particle, penetrant, or ultrasonic inspection).

5.2 Fatigue Crack Propagation Life

For disc springs under cyclic loading, cracks undergo subcritical propagation under cyclic stress. Use the Paris formula to estimate the propagation rate:

Where $\Delta K = K_{I,max} - K_{I,min}$, and $C$ and $m$ are material constants. Integrating from the initial defect size to $a_{crit}$ yields the fatigue crack propagation life $N_f$. This is crucial for determining inspection intervals.

5.3 Characteristics of Compressive Stress State

The OM point of a disc spring operates under a compressive stress state; cracks here often close, resulting in $\Delta K$ being zero or very low, and propagation is extremely slow. This is why disc spring fatigue cracks typically do not initiate from compressive stress regions but from tensile stress regions (e.g., the I point). Therefore, LEFM verification should focus on tensile stress regions.

6. Calculation Example

Given: - Disc spring material 50CrV4, $K_{IC} \approx 65\ \text{MPa·√m}$ - I point working tensile stress $\sigma = 800\ \text{MPa}$ (maximum) - Surface semi-elliptical crack, depth $a = 0.3\ \text{mm}$, geometric factor $Y \approx 1.12$ (approximation for shallow surface crack)

Calculation:

Using MPa·√m units, $a = 0.3\ \text{mm} = 3\times10^{-4}\ \text{m}$:

, safe.

Critical crack size:

With an existing 0.3 mm crack, the safety margin is approximately 1.67/0.3 ≈ 5.6 times.

7. Design Recommendations

• High-strength disc springs (hardness > 48 HRC) should undergo LEFM assessment, especially when surface damage or corrosion pits are present.
• Regular non-destructive testing: Set inspection thresholds based on critical crack size to ensure replacement before cracks reach dangerous dimensions.
• Safety factor: A fracture safety factor of ≥ 1.5 (for $K_I$ or $a$) is recommended.
• Compressive stress region treatment: For the OM point compressive stress region, requirements can be moderately relaxed, but attention must still be paid to microcracks that may be generated during manufacturing presetting.

Summary: LEFM, through the comparison of the stress intensity factor $K_I$ and material fracture toughness $K_{IC}$, provides a quantitative assessment of brittle fracture and fatigue crack propagation in disc springs, forming the basis of damage tolerance design.