Norton-Bailey Creep Constitutive Model: Calculation and Application of Steady-State Creep Rate

Under high-temperature environments, disc springs undergo creep—a slow, irreversible plastic deformation over time under constant load (or constant displacement). The Norton-Bailey constitutive equation is the most classic and widely used engineering model for describing the steady-state creep stage (secondary creep).

1. Core Formula

The steady-state creep rate (minimum creep rate) is given by the following power-law equation:

Parameter Description: - $\dot{\varepsilon}_c$: Steady-state creep rate (s⁻¹ or h⁻¹), i.e., the increment of creep strain per unit time. - $A$: Material constant (units related to stress and activation energy), determined by experimental fitting. - $\sigma$: Applied nominal stress (MPa). In disc springs, this is typically taken as the maximum compressive stress at the OM point (inner edge of the upper surface). - $n$: Stress exponent (dimensionless). For metallic materials, $n$ is typically in the range of 3 to 10, reflecting the sensitivity of creep to stress. A higher $n$ value indicates more severe creep acceleration with increasing stress. - $Q$: Creep activation energy (J/mol). For creep controlled by dislocation climb, $Q$ is close to the lattice self-diffusion activation energy, typically in the range of 200 to 400 kJ/mol. A higher $Q$ value indicates stronger creep resistance. - $R = 8.314\ \text{J/(mol·K)}$: Universal gas constant. - $T$: Absolute temperature (K).

2. Physical Significance and Disc Spring Application

This formula shows that the creep rate is extremely sensitive to temperature and stress. - Temperature: In the exponential term $\exp(-Q/RT)$, an increase of approximately 20–30 °C in temperature can double the creep rate. - Stress: Due to the high power exponent $n$ (e.g., n=5), a 20% increase in stress can lead to a 2.5-fold increase in creep rate ($1.2^5 \approx 2.49$).

In disc spring design, this means that when the operating temperature exceeds 250 °C, a creep life assessment is mandatory. Without control, the disc spring may experience significant loss of free height (set loss) and preload within weeks or months due to creep.

3. Typical Material Parameter Examples

The Norton-Bailey parameters vary greatly among different materials, directly determining their high-temperature applicability. The following are typical values (units: $\dot{\varepsilon}$ in s⁻¹, $\sigma$ in MPa, $Q$ in kJ/mol):

Note: Inconel 718 has an extremely small $A$ value, along with very high activation energy $Q$ and stress exponent $n$. This explains its excellent creep resistance at high temperatures: even above 600 °C, the creep rate can be maintained at an acceptable level.

4. Creep Strain and Relaxation Life Prediction

4.1 Creep Strain Accumulation

Under constant stress, the creep strain after time $t$ (considering only the steady-state stage) is:

When the creep strain reaches the material's creep rupture ductility or the allowable deformation limit of the disc spring, the service life is reached.

4.2 Relaxation Life Estimation

Disc springs typically operate under constant displacement, where stress gradually decreases due to creep (stress relaxation). The relaxation life $t_{relax}$ can be obtained by integrating the Norton-Bailey constitutive model:

where $\sigma_0$ is the initial stress. This equation indicates that reducing the initial stress $\sigma_0$ (e.g., by increasing the disc spring thickness to lower the working stress) is one of the most effective means to extend the high-temperature relaxation life.

5. Engineering Application Guidelines

• Material Selection:
• $T_{max} \le 250°C$: 51CrV4 creep effect is negligible; can be ignored.
• $250 < T_{max} \le 500°C$: H13 has a low creep rate and is usable, but long-term deformation must be calculated.
• $T_{max} > 500°C$: Inconel 718 must be selected, and a rigorous creep assessment is required.
• Safety Factor: High-temperature creep data exhibit large scatter. A safety factor of 2 to 3 (based on time) is recommended in design.
• Surface Strengthening: Residual compressive stresses introduced by shot peening can be accelerated by creep at high temperatures. Therefore, for applications above 400 °C, the benefit of shot peening is limited.

6. Calculation Example

Given: H13 disc spring, operating temperature 450 °C (723 K), OM point stress $\sigma = 800\ \text{MPa}$. Material parameters: $A = 5.0 \times 10^{-20}$, $n = 5.5$, $Q = 320\,000\ \text{J/mol}$.

Calculation:

• $800^{5.5} = \exp(5.5 \cdot \ln 800) \approx \exp(5.5 \cdot 6.6846) \approx \exp(36.76) \approx 9.1 \times 10^{15}$
• $\frac{Q}{RT} = \frac{320000}{8.314 \times 723} \approx \frac{320000}{6010} \approx 53.2$
• $\exp(-53.2) \approx 1.5 \times 10^{-23}$

Converted to annual creep: One year is approximately $3.15\times10^7$ s, creep strain $\approx 2.1\times10^{-19}$, completely negligible.
If the stress increases to 1000 MPa and the temperature rises to 500 °C (773 K), the rate will increase by 5–6 orders of magnitude, potentially reducing the life to thousands of hours. This fully validates the principle that "a slight increase in stress leads to a dramatic increase in creep rate."

Summary: The Norton-Bailey equation, through the stress exponent $n$ and activation energy $Q$, precisely characterizes the steady-state creep behavior of disc spring materials at high temperatures. It is the core theoretical tool for predicting long-term service life at high temperatures, setting stress limits, and selecting materials.