F-SSHT-K109 Thermal Expansion Constraint Stress

1. Physical Mechanism

When a disc spring operates at high temperatures, if its radial or axial free thermal expansion is constrained by external structures (e.g., installed in a rigid sleeve, wedged between tightly fitted shaft shoulders), significant thermal expansion constraint stress $\sigma_T$ will be generated.

2. Constraint Stress Calculation Formula

Under ideal conditions of complete constraint (zero expansion space), the constraint compressive stress is obtained by superimposing Hooke's law and thermal strain:

Where: - $\sigma_T$: Thermal expansion constraint stress (MPa), compressive stress. - $E(T)$: Elastic modulus at operating temperature (MPa), decreasing with increasing temperature. - $\alpha(T)$: Coefficient of thermal expansion at operating temperature (1/K). - $\Delta T = T_{work} - T_{assembly}$: Temperature rise (K).

3. Stress-Gap Relationship and Clearance Criteria

In practical design, providing adequate radial expansion clearance $\Delta r$ is the fundamental means to release thermal stress.

• Stress-Gap Quantitative Relationship:
  $$\sigma_T \propto E \cdot \left( \alpha \cdot \Delta T - \frac{\Delta r}{r_0} \right)$$

where $r_0$ is the fit radius at room temperature, and $\Delta r$ is the reserved radial clearance. As long as $\frac{\Delta r}{r_0} \ge \alpha \cdot \Delta T$, the thermal expansion stress can be completely eliminated.

• Minimum Clearance Estimation (based on your suggestion):
  $$\boxed{\Delta r_{min} \ge 0.1\% \cdot D_e}$$

i.e., for a disc spring with an outer diameter of 100 mm, the inner diameter of the mounting sleeve should be at least 0.1 mm larger than the disc spring outer diameter.

4. Material Constraint Stress Comparison

5. Strength Verification Criteria

It must be ensured that the superposition of thermal expansion constraint stress and mechanical stress does not exceed the material's allowable value:

where $\sigma_{mech}$ is the stress caused by mechanical load, and $S_F \ge 1.5$ is the recommended safety factor at high temperatures.

6. Design Guidelines

• Reserve Clearance: When designing sleeves, guides, or shaft shoulders, strictly implement a minimum radial expansion clearance of 0.1% $D_e$.
• Material Matching: If clearance cannot be achieved, select materials with a lower coefficient of thermal expansion $\alpha$ (e.g., Invar alloy) for the constraining components to fundamentally reduce the $\Delta\alpha$ mismatch.
• FEA Verification: For complex constraint conditions, a thermomechanically coupled finite element analysis must be performed to check for stress singularities caused by local contact.

Core Conclusion: For high-temperature disc spring applications under constraint, the design must include sufficient expansion clearance (≥ 0.1% $D_e$). Otherwise, the enormous thermal expansion constraint stress will rapidly lead to disc spring collapse failure.