SSHT Temperature Dependence of Tensile Strength $\sigma_u(T)$

In high-temperature environments, the tensile strength $R_m$ (i.e., $\sigma_u$) of disc spring materials decreases with increasing temperature, similar to the yield strength. However, its rate of decrease is typically slightly slower, causing a change in the material's yield ratio ($\sigma_y/\sigma_u$), which affects plastic reserve and fracture behavior.

1. Core Calculation Formula

The variation of tensile strength with temperature can also be described using a thermal activation model or a simplified exponential model:

or in simplified exponential form:

Where: - $\sigma_u(T)$: Tensile strength at temperature $T$ (MPa) - $\sigma_{u,20}$: Tensile strength at room temperature (20 °C) (MPa), e.g., approximately 1600 MPa for 51CrV4 - $Q_u$: Thermal activation energy corresponding to tensile strength (J/mol), typically slightly larger than the $Q$ for yield strength, hence a slower decrease - $\alpha_u$: Empirical attenuation coefficient (1/°C), smaller than the $\alpha$ for yield strength - Remaining symbols are the same as in the yield strength model.

Physical Meaning: Tensile strength represents the maximum load-bearing capacity of the material before fracture. Its temperature sensitivity is slightly lower than that of yield strength because the fracture process is influenced by work hardening and strain rate effects, which have a weaker dependence on thermal activation.

2. Comparison with Yield Strength and Change in Yield Ratio

Since the tensile strength decreases at a slightly slower rate while the yield strength decreases faster, their ratio—the yield ratio $R = \sigma_y/\sigma_u$—changes with increasing temperature:

Where $R_{20}$ is the yield ratio at room temperature, typically 0.85–0.90 for quenched and tempered spring steel.

Engineering Significance of Yield Ratio Change:

For spring steel, the yield ratio may exceed 0.9 at high temperatures, meaning the material becomes relatively brittle and more sensitive to notches, impact, and fatigue. This must be considered in design.

3. Impact on Disc Spring Design

Tensile strength is used for: - Fracture limit in static strength verification - Mean stress correction in Goodman diagrams for fatigue assessment ($\sigma_b = R_m$) - Plastic failure criteria in limit load analysis

A decrease in $\sigma_u(T)$ at high temperatures will lead to: - A reduction in the allowable fracture safety factor, potentially failing to meet design requirements; - A leftward shift of the Goodman fatigue line, reducing the allowable stress amplitude; - An increase in the yield ratio, making the disc spring more prone to brittle fracture at local stress concentrations.

Design Criteria: - If the yield ratio exceeds 0.9, the following should be implemented: - Reduce the working stress level and increase the safety factor; - Avoid any notches, scratches, or corrosion pits; - Select a material with better toughness (e.g., by increasing the tempering temperature or reducing the initial hardness).

4. Typical High-Temperature Tensile Strength Data for Materials

From the table, it can be seen that the yield ratio of common spring steel exceeds 0.9 at 300 °C, increasing the risk of brittleness; Inconel 718, however, maintains a lower yield ratio, offering better safety.

5. Calculation Example

Conditions: 51CrV4 disc spring, room temperature $R_m = 1600\ \text{MPa}$, room temperature $R_{p0.2} = 1450\ \text{MPa}$, $R_{20} \approx 0.906$.
Operating temperature 250 °C, attenuation coefficients $\alpha_y = 7.0 \times 10^{-4}\ \text{K}^{-1}$, $\alpha_u = 4.5 \times 10^{-4}\ \text{K}^{-1}$.

Calculation:

Yield ratio $R(250) \approx 1235 / 1444 \approx 0.855$, still below 0.9, indicating some plastic reserve.

If the temperature rises to 350 °C:

Yield ratio $R(350) \approx 1150 / 1378 \approx 0.835$? This value appears to decrease, which contradicts empirical experience. In reality, due to different attenuation coefficients, the yield ratio may either decrease or increase, depending on the specific material data. Spring steel typically shows an increase. This example is for calculation demonstration only; actual design should be based on measured data.

6. Key Points for Engineering Application

• Use Measured Data: The temperature coefficient of tensile strength should be obtained from the material supplier as a priority.
• Monitor Yield Ratio: When the yield ratio exceeds 0.9, the design safety factor must be increased, and the risk of brittle fracture must be considered.
• Adjust Allowable Stress: In strength verification, replace the room temperature $R_m$ with $\sigma_u(T)$ at the operating temperature and apply an appropriate safety factor.
• High-Temperature Material Selection: For conditions consistently above 300 °C, it is recommended to select nickel-based alloys with a low yield ratio and stable high-temperature strength.

Summary: The temperature dependence of tensile strength is similar to that of yield strength but slightly milder, which can cause the yield ratio to increase at high temperatures, increasing the risk of brittleness. Design should use actual high-temperature data for verification and closely monitor changes in the yield ratio to ensure the disc spring has sufficient plastic reserve and fracture resistance.