Temperature Dependence of Thermal Expansion Coefficient α(T) and Thermal Stress Sensitivity

The coefficient of thermal expansion (CTE) is an indispensable material parameter for high-temperature mechanical analysis of disc springs (such as thermomechanical coupling and preload variation calculations). It is not a constant but a function of temperature $\alpha(T)$.

1. Physical Definition and Calculation Formulas

In engineering calculations, the average coefficient of thermal expansion $\alpha_{avg}$ or the instantaneous coefficient of thermal expansion $\alpha_{inst}$ is typically used to describe the expansion behavior of materials.

• Average linear expansion coefficient (commonly used for simplified engineering calculations):
  $$\alpha_{avg} = \frac{L - L_0}{L_0 \cdot (T - T_0)}$$

Used to calculate the total dimensional change of a material from a reference temperature $T_0$ to the current temperature $T$.

• Instantaneous linear expansion coefficient (used for precise differential equations and FEA input):
  $$\boxed{\alpha_{inst}(T) = \frac{1}{L} \cdot \frac{dL}{dT}}$$

This is the tangent slope of the thermal strain-temperature curve, more accurately reflecting the material's thermal sensitivity at the current temperature point.

Temperature-Dependent Empirical Formula: Over a wide temperature range, $\alpha(T)$ increases with temperature and is typically fitted using a quadratic or cubic polynomial:

where $a_0, a_1, a_2$ are material constants obtained by fitting experimental data.

2. Typical α(T) Numerical Ranges for Materials

Your description is very accurate. The thermal expansion coefficients of common disc spring materials generally range between $10 \times 10^{-6} \sim 16 \times 10^{-6} / \text{°C}$.

Note: The CTE of austenitic stainless steel is significantly higher than that of ferritic spring steel. This must be precisely accounted for in thermal stress calculations when designing connections involving dissimilar materials (e.g., stainless steel bolts with spring steel disc springs).

3. Sensitivity Analysis of CTE Error on Thermal Stress Calculations

Your observation that "a 10% CTE error can lead to a 10-15% thermal stress error" is fully consistent with mechanical principles. This can be quantitatively analyzed using the thermal stress formula.

For a fully constrained component, the thermal stress $\sigma_{th}$ is:

Assuming the elastic modulus $E$ and temperature difference $\Delta T$ are measured without error, taking the logarithm and differentiating the above formula yields the relative error propagation relationship for thermal stress:

This means that the relative error in CTE is linearly transferred 100% to the thermal stress.

Error Amplification Mechanism

However, in disc spring systems, we are more concerned with the preload variation caused by thermal expansion differences:

When the CTEs of two materials are subtracted, the relative error of the difference itself is amplified. For example, if the true values of $\alpha_P$ and $\alpha_S$ are 16 and 11 (units 10⁻⁶/°C), the difference is 5. If both CTEs have a 1% error, the error in the difference can be as high as 27%. This is the fundamental reason why "a 10% CTE error can lead to a 10-15% or even higher thermal stress error," as you mentioned.

Therefore, in applications involving high temperatures and high-precision preload control, it is strongly recommended to: - Request batch-measured thermal expansion curves from material suppliers. - Define $\alpha(T)$ as a temperature-dependent field variable in finite element analysis (FEA), rather than a constant.

4. Applications in Disc Spring Design

• Zero Expansion/Low Stress Design: By matching the CTE of the bolt (e.g., ferritic steel, low CTE) and the clamped component (e.g., austenitic steel, high CTE) such that $\alpha_P l_P \approx \alpha_S l_S$, thermally induced preload changes are eliminated at the source.
• Compensation Calculation: When performing VDI 2230 R5 (minimum preload) verification, the accurate $\alpha$ value at the operating temperature must be substituted to calculate the thermally induced preload loss $F_{Z,th}$.
• Coating Effects: Thick anti-corrosion coatings (e.g., Dacromet, hot-dip galvanizing) themselves have a thermal expansion coefficient and may hinder heat conduction; this should be considered in precision analyses.

Summary: An accurate coefficient of thermal expansion $\alpha(T)$ is critical for high-temperature disc spring design. It not only directly affects the magnitude of thermal stress, but its small measurement error can be drastically amplified in thermal expansion difference calculations, potentially undermining the entire reliability assessment of the connection.