Symbol	Name	Unit
material	material	—
temp_C	temp_C	°C

Yield Strength σ_y(T) Exponential Decay with Temperature

At high temperatures, the yield strength of disc spring materials decreases significantly, leading to reduced allowable loads and increased risk of plastic deformation. The most commonly used physical model to describe this relationship is the thermally activated Arrhenius law, exhibiting exponential decay characteristics.

1. Core Calculation Formula (Arrhenius Thermal Activation Model)

The variation of yield strength with temperature can be expressed as:

\boxed{\sigma_{y}(T) = \sigma_{y,20} \cdot \exp\left[ \frac{Q}{R} \left( \frac{1}{T} - \frac{1}{293} \right) \right]}

Or using a simplified exponential form (within a narrow temperature range):

\boxed{\sigma_{y}(T) \approx \sigma_{y,20} \cdot \exp\left( -\alpha \cdot (T - 20) \right)}

Parameter Description: - $\sigma_{y}(T)$ : Yield strength at operating temperature $$T$$ (K) (MPa) - $\sigma_{y,20}$ : Initial yield strength at room temperature (20 °C) (MPa), e.g., for 51CrV4 approximately 1500 MPa - $$Q$$ : Thermal activation energy (J/mol), representing the energy required for dislocations to overcome obstacles - $R = 8.314\ \text{J/(mol·K)}$ : Universal gas constant - $$T$$ : Absolute temperature (K), $$T = t + 273.15$$ - $\alpha$ : Empirical decay coefficient (1/°C), considered constant within a narrow temperature range

This model indicates that as temperature increases, the thermal vibration energy of atoms increases, making it easier for dislocations to overcome obstacles, resulting in an exponential decrease in yield strength.

2. Typical Data and Model Parameters for 51CrV4 Spring Steel

Based on the reference data you provided: - 200 °C: Yield strength decreases by approximately 8%, retention rate 92% - 300 °C: Yield strength decreases by approximately 20%, retention rate 80%

We can back-calculate the corresponding model parameters:

Temperature (°C)	Absolute Temperature (K)	Measured Retention Rate	Back-calculated $$Q$$ from Arrhenius (kJ/mol)	Back-calculated $\alpha$ from Simplified Exponential (1/K)
200	473	0.92	≈ 45	$4.6 \times 10^{-4}$
300	573	0.80	≈ 48	$8.0 \times 10^{-4}$

Note: The parameters back-calculated from the two temperature points are not identical, indicating that a single activation energy or constant decay coefficient cannot precisely describe the behavior across the entire temperature range. In engineering, piecewise linear interpolation or direct reference to high-temperature performance curves provided by material suppliers is typically used. The above formulas are mainly for understanding trends and quick estimation.

3. Sharp Decline Beyond the Tempering Temperature

For quenched and tempered spring steel, when the temperature exceeds its original tempering temperature, the microstructure undergoes further tempering, carbide coarsening, and even phase transformation, leading to a sharp accelerated decline in yield strength, far deviating from the predictions of the thermal activation model.

The common tempering temperature for 51CrV4 is approximately 400～460 °C.
Once the operating temperature approaches or exceeds this value, strength and hardness will be significantly lost in a short time, accompanied by substantial permanent deformation (relaxation).

Therefore, the above exponential decay formula is only applicable for temperatures approximately 50 °C below the original tempering temperature. For 51CrV4, it is generally safe to use this model at temperatures ≤350 °C.

4. Direct Impact on Disc Spring Design

The decrease in yield strength directly reduces the disc spring's allowable load and maximum elastic compression. Design must be verified using $\sigma_y(T)$ at the operating temperature:

F_{zul}(T) \approx F_{zul,20} \cdot \frac{\sigma_{y}(T)}{\sigma_{y,20}}

At 200 °C, the allowable load of a 51CrV4 disc spring will decrease by approximately 8%.
Using room temperature data erroneously could lead to unacceptable plastic deformation (exceeding Set loss limits) in the disc spring at high temperatures.

5. Comparative Advantage of High-Temperature Alloys

When the operating temperature exceeds 300 °C, the performance degradation of ordinary spring steel becomes non-negligible, and high-temperature alloys should be selected. For example, Inconel 718 retains over 95% of its yield strength at 300 °C and approximately 90% at 500 °C, far superior to spring steel.

Material	Retention Rate at 200 °C	Retention Rate at 300 °C	Retention Rate at 500 °C
51CrV4 Spring Steel	≈ 92%	≈ 80%	< 50% (unusable)
Inconel 718 High-Temperature Alloy	≈ 98%	≈ 95%	≈ 90%

Design Principles: - ≤ 250 °C: Alloy spring steels such as 50CrV4, 51CrMoV4 can be used. - 250 °C ~ 500 °C: Stainless steel or nickel-based high-temperature alloys (e.g., Inconel 718) should be selected. - Above 500 °C: Cobalt-based or nickel-based high-temperature alloys must be used, with detailed creep and relaxation checks.

6. Calculation Example

Given: 51CrV4 disc spring, room temperature yield strength $\sigma_{y,20} = 1500\ \text{MPa}$ , operating temperature 200 °C.
Using the simplified exponential model with an average decay coefficient $\alpha = 6.0 \times 10^{-4}\ \text{1/K}$ :

\sigma_{y}(200) = 1500 \cdot \exp\left( -6.0 \times 10^{-4} \times (200 - 20) \right) \approx 1500 \cdot \exp(-0.108) \approx 1500 \times 0.897 \approx 1346\ \text{MPa}

The retention rate is approximately 89.7%, slightly higher than the expected 92%, which is conservative.

Conclusion: When designing disc springs for high-temperature applications, the yield strength at the operating temperature must be used as the basis, and the temperature must not exceed the material's tempering temperature. For long-term high-temperature applications, selecting high-temperature alloys is the fundamental solution.

Yield Strength σ_y(T)

Formula Expression

Parameters

Need to compute this formula?

Detailed Calculation Guide