SSHT Thickness Direction Thermal Gradient Stress

1. Physical Mechanism

When a disc spring has a large thickness (t > 3 mm) and is subjected to rapid heating or one-sided heating, heat conduction from the surface to the interior takes time, resulting in a transient temperature gradient within the cross-section: high temperature on the surface, low temperature in the interior.

• The high-temperature surface layer tends to expand but is constrained by the low-temperature interior layer → Compressive stress on the surface
• The low-temperature interior layer is pulled to expand by the surface → Tensile stress in the interior

This self-equilibrating stress system is the thermal gradient stress. It becomes more significant in disc springs with greater thickness, due to longer heat conduction paths and larger temperature differences.

2. Core Calculation Formulas

2.1 Temperature Difference Under Steady-State Heat Conduction

For two-sided heating or cooling, after reaching steady state, the temperature distribution in the thickness direction is approximately linear. The maximum temperature difference:

• $\dot{q}$: Heat flux density (W/mm²)
• $t$: Disc spring thickness (mm)
• $k$: Material thermal conductivity (W/(mm·K))

2.2 Thermal Gradient Stress (Thermal Bending Stress)

The temperature difference induces a linearly distributed thermal stress within the cross-section, with the maximum value occurring at the surface:

Where: - $\sigma_{th,grad}$: Maximum stress caused by the thickness-direction thermal gradient (MPa), negative for surface compression, positive for core tension - $E(T)$: Elastic modulus at operating temperature (MPa) - $\alpha(T)$: Coefficient of thermal expansion (1/K) - $\nu$: Poisson's ratio (steel ≈ 0.3) - $\Delta T_{th}$: Temperature difference between surface and core (K)

Physical meaning: The denominator $(1-\nu)$ reflects the amplification effect of biaxial constraint (radial and tangential) on thermal stress, which is approximately 40% higher than uniaxial constraint ($E\alpha\Delta T$).

3. Superposition with Mechanical Stress

The OM point originally bears mechanical compressive stress. The thermal gradient also produces compressive stress on the surface. Both are superimposed in the same direction, further increasing the surface compressive stress:

Verification criterion:

Where $S_F \ge 1.3$ is the recommended safety factor.

4. Engineering Criteria for Thick-Section Springs

5. Calculation Example

Given: H13 disc spring, thickness $t = 6\ \text{mm}$, operating temperature 500 °C, rapid heating causes a temperature difference between surface and core $\Delta T_{th} = 40\ \text{°C}$.

, , .

This thermal gradient stress is approximately 62 MPa. Although the absolute value is not large, for fatigue evaluation (stress amplitude on the order of 60 MPa), this is a non-negligible additional stress.

If the mechanical compressive stress is 800 MPa, the superimposed stress is 862 MPa, which is still within the allowable range for H13 (σ_y ≈ 950 MPa), but the safety margin is reduced.

6. Design Countermeasures

• Control heating rate: During equipment startup, limit the heating rate to ≤ 10 °C/min to allow time for heat conduction and reduce the temperature difference.
• Material selection: Choose materials with higher thermal conductivity $k$ (e.g., copper alloys) or lower coefficient of thermal expansion $\alpha$ (e.g., Inconel 718) to directly reduce $\sigma_{th,grad}$.
• Structural optimization: Under the premise of meeting load requirements, prefer thinner disc springs ($t \le 3\ \text{mm}$) or use multiple thinner discs in series instead of a single thick disc.

Summary: For disc springs with thickness greater than 3 mm under rapid heating, the thermal gradient stress $\sigma_{th,grad}$ superimposes with the mechanical stress, increasing the surface compressive stress. The transient temperature difference must be evaluated and incorporated into the strength verification. If necessary, mitigate the risk by controlling the heating rate or optimizing the material.