F-SSHT-K107 Creep Strain Accumulation and Life Assessment

1. Three-Stage Creep Curve

A typical creep curve consists of three stages, reflecting the evolution of strain over time in materials under high temperature and constant load:

Stage I (Primary Creep / Initial Creep):
Upon loading, elastic strain occurs instantaneously, followed by a gradual decrease in creep rate over time. At the microscopic level, dislocation motion is hindered by multiplied dislocations and subgrain boundaries, producing a strain-hardening effect that makes the material "increasingly difficult" to creep.

Stage II (Secondary Creep / Minimum Creep Rate):
A dynamic equilibrium is reached between strain hardening and thermal recovery (dislocation climb and annihilation), maintaining a constant and minimum creep rate. This is the longest stage in the service life of disc springs and is the direct subject of the Norton-Bailey equation.

Stage III (Tertiary Creep / Final Fracture):
Microvoids coalesce into microcracks, reducing the effective load-bearing area. True stress increases sharply, accelerating creep until fracture. Once Stage III is entered, the disc spring's life is near its end, and immediate replacement is required.

2. Constitutive Equations for Each Stage

2.1 Stage I: Time-Hardening Model

• $\varepsilon_0$: Initial elastic strain ($\sigma/E$)
• $A_1, n, m$: Material constants ($0 < m < 1$, e.g., $m \approx 0.3 \sim 0.5$)
• $\sigma$: Constant stress (MPa)
• $t$: Time (h)

Creep rate decreases with time: $\dot{\varepsilon}_I = A_1 \cdot \sigma^n \cdot m \cdot t^{m-1}$, since $(m-1) < 0$, the rate decreases.

2.2 Stage II: Steady-State Creep (Norton-Bailey)

This is the core of creep life calculation. The smaller $\dot{\varepsilon}_{II}$, the stronger the disc spring's ability to maintain preload at high temperatures.

2.3 Stage III: Damage-Coupled Accelerated Creep

Stage III cannot be described by a simple power law. It is typically modeled using continuum damage mechanics, coupling the damage variable $D$ with the creep rate:

• $D$: Damage variable, $D=0$ indicates intact, $D=1$ indicates fracture.
• $\phi, \psi, \chi$: Material damage exponents, typically fitted from experiments.

When $D \to 1$, the creep rate tends to infinity, corresponding to macroscopic crack formation. In engineering, this stage is only used for remaining life warning and is not considered a normal design state.

3. Creep Strain Thresholds and Replacement Criteria

For disc springs, creep strain directly affects their geometry and mechanical performance:

• Creep strain ≤ 0.5%: Free height loss is minimal, stiffness change is negligible, and the condition is safe.
• 0.5% < Creep strain ≤ 1.0%: Enhanced monitoring is required to assess whether the preload still meets functional requirements. If the preload margin is near its lower limit, spare parts should be prepared.
• Creep strain > 1.0%: The free cone height $h_0$ is significantly reduced, the flattening force may drop by more than 10%, and the material has accumulated substantial microscopic damage. Immediate replacement is recommended.

Engineering Threshold: $\varepsilon_c > 1\%$ is the critical point at which a disc spring loses its elastic compensation capability due to creep and must be a mandatory replacement indicator in maintenance plans.

4. Cumulative Creep Strain Calculation

When disc springs operate under variable temperature and load conditions, the creep strain from each stage must be accumulated over time segments, considering the coupling effect of stress relaxation.

Segmented Accumulation Formula (applicable to segmented variations in temperature and stress):

• $(T_i, \sigma_i)$: Temperature and OM point stress during the $i$-th time interval.
• $\Delta t_i$: Duration of the $i$-th segment (h).
• $N$: Total number of time segments.

Stress Relaxation Coupling Correction:
Disc springs operate under constant displacement, and creep causes a continuous decrease in stress (relaxation). In this case, the Norton-Bailey equation must be substituted into the relaxation equation and solved iteratively:

Stepwise numerical integration is performed to simultaneously calculate creep strain accumulation and stress decay until the design life is reached or the creep strain exceeds the limit.

5. Calculation Example

Given: H13 disc spring, operating temperature 500 °C (773 K), OM point initial stress $\sigma = 600\ \text{MPa}$.
Norton parameters: $A_2 = 8.0 \times 10^{-20}$, $n = 5.5$, $Q = 320,000\ \text{J/mol}$.
Design requirement: 20,000 hours of operation, creep strain ≤ 1%.

Step 1: Calculate Steady-State Creep Rate

• $600^{5.5} = \exp(5.5 \cdot \ln 600) \approx \exp(35.2) \approx 1.9 \times 10^{15}$
• $\frac{Q}{RT} = \frac{320000}{8.314 \times 773} \approx 49.7$
• $\exp(-49.7) \approx 2.5 \times 10^{-22}$
  $$\dot{\varepsilon}_{II} \approx 8.0 \times 10^{-20} \times 1.9 \times 10^{15} \times 2.5 \times 10^{-22} \approx 3.8 \times 10^{-26}\ \text{s}^{-1}$$

Converted to creep per year ($3.15 \times 10^7$ s): $\approx 1.2 \times 10^{-18}$, 20,000 hours ≈ 2.28 years, cumulative strain is completely negligible.

Step 2: Correct for Stress Effect
If the stress increases to 900 MPa (due to disc spring overload), then:

The rate increases by approximately 9 times, but is still extremely low.
However, if the temperature simultaneously rises to 600 °C (873 K), $\frac{Q}{RT} \approx 44.1$, $\exp(-44.1) \approx 7.0 \times 10^{-20}$, the rate will surge by more than 6 orders of magnitude. The creep strain over 20,000 hours could exceed 5%, requiring a material change to Inconel 718.

Summary: Creep strain accumulation follows a three-stage law. The steady-state creep stage determines the long-term life of disc springs, and the Norton-Bailey equation is the core quantification tool. In engineering, 1% creep strain is set as the mandatory replacement threshold, and a segmented accumulation method for variable temperature and load is used for full-life assessment.