Formula Expression
Parameters
| Symbol | Name | Unit |
|---|---|---|
| material | material | — |
| temp_C | temp_C | °C |
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Variation of Elastic Modulus $E(T)$ with Temperature
The elastic modulus $E$ decreases as temperature rises, directly causing a proportional reduction in the load $F$ of a disc spring at the same compression. The general formula describing this relationship and the behavior of specific materials are presented below.
1. Core Calculation Formula
The variation of elastic modulus with temperature is typically described using a linear model:
Parameter Description: - $E(T)$: Elastic modulus at temperature $T$ (°C) (MPa) - $E_{20}$: Reference elastic modulus at room temperature (20 °C). For spring steel, $E_{20} \approx 206,000\ \text{MPa}$ - $\beta$: Temperature coefficient of the elastic modulus ($1/\text{K}$). This is a key material constant that determines the rate of modulus decrease - $T$: Operating temperature (°C)
2. Material Temperature Coefficient and Case Verification
Different materials have significantly different $\beta$ values, which directly determine their performance at high temperatures.
| Material | Typical $\beta$ Value (1/K) | $E(T)/E_{20}$ at 200 °C | $E(T)/E_{20}$ at 300 °C |
|---|---|---|---|
| 51CrV4 Spring Steel | Approx. $2.8 \times 10^{-4}$ | ≈ 95% (5% decrease) | ≈ 92% (8% decrease) |
| Inconel 718 High-Temperature Alloy | Approx. $1.0 \times 10^{-4}$ | ≈ 98.2% (1.8% decrease) | ≈ 97.2% (2.8% decrease) |
51CrV4 shows a decrease of approximately 5% at 200°C, which is consistent with the data you provided; for a 10% decrease at 300°C, the corresponding average $\beta$ value is slightly higher, at $3.6 \times 10^{-4}$. In practical applications, it is recommended to use precise values provided by the material supplier or measured data.
3. Direct Impact on Disc Spring Load
At high temperatures, the load capacity of a disc spring decreases proportionally with the reduction in elastic modulus:
This means: - If a 51CrV4 disc spring requires a preload of 10,000 N at 20 °C, at 200 °C, maintaining the same compression, the force will decrease to approximately 9,500 N. - If Inconel 718 is used instead, at 300 °C, the force will only decrease to approximately 9,720 N.
4. Key Points for Engineering Applications
- Load Verification: When designing for high temperatures, the elastic modulus $E(T)$ at the operating temperature must be used to calculate the permissible load and the flat load $F_{flat}$.
- Material Selection: For applications exceeding 250 °C, the modulus and strength of spring steel decrease significantly. It is recommended to use high-temperature alloys with a smaller $\beta$ value (e.g., Inconel 718) to ensure stable force values.
- Precise Calculation: The formula in this document is a linear approximation sufficient for most engineering calculations. For high-precision requirements, more accurate polynomial fitting or direct lookup of $E(T)$ values can be employed.