2.2.1 Norton model

The equation of $\ln ({\dot{\epsilon }}^{\text{*}})=\ln (C_1)+C_2\ln ({\sigma }_{\text{eq}})$ can be obtained by taking the natural logarithm of both sides of Eq. (1). The equivalent stress σ_eq in the Norton model is equal to the applied stress σ. The linear fitting curve is shown in Fig. 2, and the coefficient of determination R-squared between the fitted values and experimental data is greater than 0.99. The two obtained material constants are C₁ = 1.98×10^-18 and C₂ = 5.79, respectively.

Fig. 2Full-screen View

Fitting curve of the Norton model

2.2.2 K-R model

In order to fit the material constants of the K-R model, the expressions of the creep strain and creep damage are derived in this section. Integrating Eq. (4) with the initial condition of $\omega |{}_{t=t_{\text{r}}}=1$, the equation of the creep fracture time (t_r) and creep damage (ω) under the multi-axial stress state can be obtained.

Substituting Eqs. (5) and (6) into Eq. (3), the equation of the multi-axial creep strain (${\epsilon }_{ij}^{\text{c}}$) can be derived by integrating the variable t.

Then, the creep fracture strain ${\epsilon }_{\text{r}}={\epsilon }_{ij}^{\text{c}}|{}_{t=t_{\text{r}}}$ can be deduced:

where, $K=(q+\text{1}-n)/\left(q+\text{1}\right)$; $f({\sigma }_{ij})=1.5B{\sigma }_{\text{eq}}^{n-1}{S}_{ij}$.

Under the uniaxial stress state, Eqs. (5) and (8) are simplified with ${\sigma }_{\text{max}}={\sigma }_{\text{eq}}={\sigma }_{\text{r}}=\sigma$, $f({\sigma }_{ij})=B{\sigma }^n$, and ${\epsilon }_{ij}^{\text{c}}\text{=}{\epsilon }_{\text{c}}$. Using the natural logarithm for Eqs. (5) and (8) after transformation, two linear equations $\ln (1/t_{\text{r}})=\ln \left[A(q+1)\right]+p\ln (\sigma )$ and $\ln ({\epsilon }_{\text{r}}/t_{\text{r}})=\ln \left[B/K\right]+n\ln (\sigma )$ are obtained and then fitted by the uniaxial creep test data. Hence, p = 6.97, n = 5.69, $\text{ln}\left(A\left(q+1\right)\right)=-44.815$, and $\text{ln}\left(B/K\right)=-40.083$ are obtained. Furthermore, the equation of ${\epsilon }_{\text{c}}/{\epsilon }_{\text{r}}=1-{(1-t/t_{\text{r}})}^K$ is deduced by dividing Eq. (7) and Eq. (8). Nonlinear curve fitting is performed and shown in Fig. 3. The coefficient of determination R-squared is greater than 0.98. Thus, K = 0.57 and q = 12.23 are obtained. Substituting q and K into $\text{ln}\left(A\left(q+1\right)\right)$ and $\text{ln}\left(B/K\right)$, the constants A and B are also obtained.

Fig. 3Full-screen View

Nonlinear fitting curve for the K-R model

The fitted material constants p, q, n, A, and B of the K-R creep damage model are listed in Table 4. In this study, the material constant α, which is unknown, is assumed to be 0.15 according to a nickel-base alloy (Waspaloy) in a previous study [23].

2.3.1 Uniaxial stress state

(1) Creep strain analysis

The Norton and K-R models can describe the creep behavior of the UNS N10003 alloy under the uniaxial stress state. By integrating Eq. (1), the uniaxial Norton creep strain equation is obtained.

By substituting the material constants (in Table 4) into Eqs. (5)–(7), the uniaxial K-R creep strain equation can also be derived as follows.

The prediction curves of the creep strain obtained by these two equations (Eqs. (9) and (10)) are compared with the creep test data, as shown in Fig. 4(a) and Fig. 4(b), respectively. The prediction curves from the Norton model are a linear function of time, and the curves from the K-R model are a power function of time under the constant stress state.

Fig. 4Full-screen View

Comparison of the uniaxial creep strain between the theoretical results and test data: the (a) Norton model and (b) K-R model

In order to analyze and evaluate the accuracy of the two theoretical models quantitatively, the residual sum of squares (RSS) and the coefficient of determination (R-square) of the creep strain between the theoretical prediction values and test data for the Norton and K-R models were calculated, as listed in Table 5. The RSS values of the K-R model were closer to zero, and the R-square values were closer to 1 than those of the Norton model, except for the higher stress level (σ = 380 MPa). These results indicated that the K-R model is more consistent with the test data and hence more suitable for describing the creep behavior of the UNS N10003 alloy than that of the Norton model.

Furthermore, the theoretical predicted values of the creep fracture time (t_r) and creep fracture strain (ε_r) of the K-R model are calculated and listed in Table 6. The maximum errors between the theoretical values and the test data are 7.27% for t_r and 4.20% for ε_r, which are within the allowable range. Consequently, the K-R model is more suitable than the Norton model for describing the creep characterization of the UNS N100003 alloy.

(2) Creep damage analysis

The K-R theoretical creep damage equation can be obtained by substituting the material constants in Table 4 into Eq. (6) under the uniaxial stress state.

As mentioned, the Norton creep damage value in Eq. (2) can be considered as the tested damage value [12]. Thus, the K-R creep damage curve (theory results) with t/t_r are compared with the Norton creep damage values (test values), as shown in Fig. 5. The theoretical results are slightly higher than the test values at the same t/t_r. In engineering applications, the conservative theoretical results will be obtained by the K-R model. Therefore, the true damage of the structure will not reach the predicted values during the operation period. Thus, the K-R theory model is reliable for the structural safety design. According to Eqs. (5) and (6), the creep damage ω correlates with the stress σ, time t, and the material constant q under the uniaxial stress state. For the same t/t_r, the ω value is related to the material constant q, which is obtained by fitting the tested creep strain. In addition, Fig. 5 shows that creep damage occurs when t > 0.9t_r. When t < 0.9t_r, ω < 0.2, which is less than the critical creep damage value of ω_cr = 1.0. When t > 0.9t_r, the ω value increases significantly. Thus, the operating time of the elevated temperature structure designed with the UNS N10003 alloy should not exceed 90% of the creep life t_r.

Fig. 5Full-screen View

Comparison between the K-R creep damage and Norton creep damage under the uniaxial stress state

2.3.2 Multi-axial stress state

(1) Creep strain analysis

The equivalent creep strain ε_eq is adopted to assess the creep behavior under the multi-axial stress state. Three components of the principal creep strain are assumed as ε₁₁, ε₂₂, and ε₃₃. The equation for ε_eq is as follows.

In this study, a common multi-axial stress state in nuclear power engineering is analyzed and discussed. The three principal stress components are σ₁ = 0, σ₂ = 2σ, and σ₃ = σ. The curves of ε_eq versus t are plotted under the stress level of σ = 100 MPa, as shown in Fig. 6. The α values in the K-R model are 0, 0.5, and 1.0, respectively.

Fig. 6Full-screen View

Curves of the equivalent creep strain under the multi-axial stress state

From Fig. 6, the Norton creep curves overlap with the K-R creep curves. The time of the intersection point is t_o. When t = t_o, the predicted values (ε_eq) of the two theoretical models are equal. When t < t_o, the ε_eq values of the Norton model are larger than those of the K-R model. When t > t_o, the ε_eq values of the Norton model are smaller than those of the K-R model. Based on these calculations, the results of the accumulated equivalent creep strain by the K-R model would be more conservative than those of the Norton model during long-term service. In addition, the value of α directly determines the variation of the K-R creep curves. When α varies from 0 to 1, the t_r and ε_r values predicted by the K-R model are decreased by 65%.

(2) Creep damage analysis

The theoretical prediction values of the creep damage can be obtained using the K-R creep damage model under the multi-axial stress state. According to Eqs. (5) and (6), the K-R creep damage ω bears on the reference stress σ_r, which is related to the α value. The curves of ω versus time are plotted with α = 0, 0.5, and 1.0 and are shown in Fig. 7. When α = 0, σ_r = σ_eq = 1.732σ, and the ω value depends on σ_eq. When α = 0.5, σ_r = 0.5(σ₁+σ_eq) = 1.867σ, and the ω value depends on σ_eq and σ₁. When α = 1.0, σ_r = σ₁=2σ, and the ω value depends on σ₁. Thus, the ω value depends on σ_eq, σ₁, and α under the multi-axial stress state. Furthermore, the accumulated creep damage gradually increases with the increase of α at the same time. When t = 3000 h, the ω value is equal to 0.0331 (α = 0), 0.0691 (α = 0.5), and 0.334 (α = 1.0). The ω value with α = 1.0 is approximately ten times than that with α = 0. The α value affects the accumulated creep damage and time of the creep rupture.

Fig. 7Full-screen View

Curves of the K-R creep damage ω versus α under the multi-axial stress state

The above analysis shows that the multi-axial creep damage is more complicated than the uniaxial creep damage. The material constant α influences the theoretical prediction values of the creep strain and creep damage. Thus, the α value of the UNS N10003 alloy should be measured by multi-axial creep tests.

3.2.1 Uniaxial stress state

The Norton and K-R models are adopted to analyze the creep behavior of a rectangular block model under the uniaxial stress state. A comparison between the creep strains of the FEA results and the theoretical solution is shown in Fig. 10. The FEA results of the Norton and K-R models are consistent with their theoretical prediction values.

Fig. 10Full-screen View

Comparison of the creep strain between the FEA results and theoretical solution under the uniaxial stress state

The FEA results of the creep strain at t = 540 h are listed in Table 7. The results indicate that the maximum errors of the creep strain between the FEA results and theoretical solution are approximately 1.69% for the Norton model and 0.84% for the K-R model. Therefore, the K-R model is more accurate than the Norton model.

In order to determine the convergence of the numerical calculation by the UMAT subroutine, the creep strain increments $\Delta {\epsilon }_{\text{c}}{}^{(n)}$ at iteration step n and the creep strain increments $\Delta {\epsilon }_{\text{c}}{}^{(n+1)}$ at iteration step n+1 are calculated. The ratio of $\Vert \Delta {\epsilon }_{\text{c}}{}^{(n+1)}-\Delta {\epsilon }_{\text{c}}{}^{(n)}\Vert /\Vert \Delta {\epsilon }_{\text{c}}{}^{(n+1)}\Vert$ is obtained as the relative norm error [22]. During the iterations, the relative norm error gradually decreases and is stable. When the number of iteration reaches 2702, the minimum relative norm error is equal to 1.73 × 10^-4 and lower than the admissible error (2 × 10^-4 is defined). Therefore, the subroutine is adequate, and the calculation results are convergent.

3.2.2 Multi-axial stress state

The thick-walled cylinder subjected to an internal pressure leads to the multi-axial stress (radial stress σ_r, circumferential stress σ_θ, and axial stress sigma σ_z). The theoretical solutions of the stress components are available under the elastic state as well as the steady creep state for the thick-walled cylinder [21]. The FEA results by the Norton and K-R models are obtained and compared with the theoretical solution. The circumferential stress distribution curves of the thick-walled cylinder along wall thickness direction r are shown in Fig. 11. Fig. 11(a) shows the elastic stress at t = 0. Fig. 11(b) shows the creep stress at t =1000 h. From the diagram, the redistribution of the circumferential stress is caused by creep relaxation. The FEA results of the creep stress are consistent with the theoretical solutions.

Fig. 11Full-screen View

Circumferential stress distribution curves of the thick-walled cylinder along the r direction: (a) t = 0 and (b) t = 1000 h

The FEA results of the creep stresses at the inner wall (t = 1000 h) are listed in Table 8 and compared with the theoretical solutions. The errors of the equivalent stress between the FEA results and the theoretical solutions are approximately 1.69% for the Norton model and 0.69% for the K-R model, which are within the permissible deviation range. Therefore, the multi-axial creep behavior can also be accurately simulated by the calculation method presented in this study. In addition, the FEA results of the K-R model are more accurate than that of the Norton model.