SSHT Creep-Relaxation Coupling: Stress Relaxation Under Constant Displacement

1. Physical Mechanism

Disc springs operating at high temperatures typically work under constant displacement (constant compression). In this state, elastic strain $\varepsilon_e$ gradually transforms into creep strain $\varepsilon_c$, causing stress to decrease over time—a process known as stress relaxation.

The essence of stress relaxation is creep-driven stress redistribution: total strain remains constant, while the material undergoes creep under sustained stress, reducing elastic deformation and consequently lowering the stress.

2. Governing Equation for Stress Relaxation

From the condition of constant total strain:

Substituting elastic strain $\varepsilon_e = \sigma / E$ and the Norton-Bailey steady-state creep rate $\dot{\varepsilon}_c = A \sigma^n \exp(-Q/RT)$ yields the differential equation for stress relaxation:

This equation indicates that the rate of stress decrease is proportional to the $n$-th power of the current stress. Since the $n$ value for metallic materials typically ranges from $3$ to $10$, relaxation is extremely rapid at high initial stress levels and slows down significantly as stress decreases.

3. Analytical Solution for Stress Relaxation ($n \neq 1$)

Integrating the above differential equation, with initial stress $\sigma_0$ at $t=0$, gives the remaining stress $\sigma(t)$ at any time:

Alternatively:

Parameter Description: - $\sigma(t)$: Stress at time t (MPa) - $\sigma_0$: Initial stress (MPa) - $E$: Elastic modulus at operating temperature (MPa) - $A, n, Q$: Norton-Bailey creep parameters - $R, T$: Gas constant and absolute temperature

4. Relaxation Time Constant and Half-Life

Define the relaxation half-life $t_{1/2}$ as the time required for stress to drop to half its initial value. Solving $\sigma(t_{1/2}) = 0.5 \sigma_0$ yields:

Temperature Sensitivity: Due to the $\exp(Q/RT)$ term, a temperature increase of $20 \sim 30^\circ\text{C}$ can shorten the relaxation half-life by a factor of ten or more. This explains why relaxation above 500°C can be completed within hours.

5. Relevance to Disc Spring Design

• Initial Stress Control: From the formula, higher $\sigma_0$ increases the $\sigma_0^{n-1}$ term in the denominator, drastically accelerating relaxation. Therefore, in high-temperature disc spring design, the working stress (i.e., maximum compression ratio $s/h_0$) should be appropriately reduced to achieve longer relaxation life.
• Material Selection:
• For operating temperatures $>400^\circ\text{C}$, materials with extremely low $A$ and high $Q$ (e.g., Inconel 718) must be used, making the exponential term in the denominator dominant.
• H13 (hot work tool steel) is suitable for $400 \sim 500^\circ\text{C}$, but long-term relaxation losses must be rigorously evaluated.
• Pre-stressing Treatment: Pre-compression at a temperature higher than the operating temperature allows the initial rapid relaxation to occur before service, significantly improving stress stability during the operational period.

6. Calculation Example

Given: H13 disc spring, operating temperature 500 °C (773 K), initial stress $\sigma_0 = 800\ \text{MPa}$, elastic modulus $E = 190,000\ \text{MPa}$. Norton parameters: $A = 5.0 \times 10^{-20}$, $n = 5.5$, $Q = 320,000\ \text{J/mol}$.

Calculate relaxation half-life: - $\sigma_0^{n-1} = 800^{4.5} = \exp(4.5 \cdot \ln 800) \approx \exp(4.5 \cdot 6.6846) \approx \exp(30.08) \approx 1.15 \times 10^{13}$ - $2^{n-1} - 1 = 2^{4.5} - 1 \approx 22.63 - 1 = 21.63$ - $E \cdot A = 190,000 \times 5.0 \times 10^{-20} = 9.5 \times 10^{-15}$ - $\frac{Q}{RT} = \frac{320,000}{8.314 \times 773} \approx 49.7$ - $\exp(-Q/RT)$ in the denominator becomes $\exp(Q/RT)$, calculated as $\exp(49.7) \approx 3.6 \times 10^{21}$

Denominator part: $4.5 \times 9.5 \times 10^{-15} \times 1.15 \times 10^{13} \approx 4.5 \times 9.5 \times 1.15 \times 10^{-2} \approx 4.5 \times 10.925 \times 10^{-2} \approx 0.492$

Converted to hours: $\approx 4.4 \times 10^{19}\ \text{h}$, which appears extremely long, indicating that the chosen parameters may not reflect actual high-temperature creep data (e.g., the $A$ value might be too small). In practice, the creep parameters for H13 at 500 °C must be obtained from material standards; this example only demonstrates the calculation procedure. If the activation energy were slightly lower (e.g., 280 kJ/mol), the result would change dramatically. In engineering, the half-life of H13 at 500 °C and 800 MPa is on the order of thousands of hours and must be validated through experimental data.

Summary: Creep-relaxation coupling, by solving the differential equation under the Norton-Bailey constitutive model, accurately predicts stress decay in disc springs under constant displacement. The relaxation rate is highly dependent on temperature, initial stress, and material creep resistance, making it a core consideration for the long-term reliability design of high-temperature disc springs.