Constitutive model of an additively manufactured combustor material at high-temperature load conditions

Figures & data

Figure 1. Illustration of how the test specimens were printed in the AM machine, where $\mathit{v}$ is the building direction, i.e. the normal to the building platform.

Figure 2. Utilised specimen geometries for a) ${625}^{\circ }$C, b) ${700}^{\circ }$C, ${750}^{\circ }$C and ${800}^{\circ }$C creep, c) tensile, d) LCF and e) TMF tests.

Table 1. Summary of the performed creep tests.

Temperature [${ }^{\circ }$C]	Building direction [${ }^{\circ }$]	Stress levels [-]
625	0	0.289, 0.347, 0.376
	90	0.289, 0.347, 0.376
700	0	0.145, 0.174, 0.246
	90	0.145, 0.188, 0.260
750	0	0.087, 0.116, 0.145
	90	0.116, 0.130, 0.159
800	0	0.058, 0.072, 0.087
	90	0.087, 0.101, 0.116

Figure 3. Experimental creep strain vs. time at a) ${625}^{\circ }$C, b) ${700}^{\circ }$C, c) ${750}^{\circ }$C and d) ${800}^{\circ }$C.

Figure 4. Strain rate vs. applied strain in the monotonic tensile tests.

Figure 5. Experimental true stress ($\hat{\sigma }$) vs. true strain ($\hat{\epsilon }$) for the monotonic tensile tests at a) ${600}^{\circ }$C, b) ${700}^{\circ }$C and c) ${800}^{\circ }$C.

Figure 6. The experimental minimum creep rate vs. stress from the creep tests at ${800}^{\circ }$C, a fitted Norton expression for the $0^{\circ }$ and ${90}^{\circ }$ specimens, respectively, and the obtained strain rates and stress plateaus from the tensile tests.

Table 2. Summary of the performed LCF tests at ${700}^{\circ }$C.

Building direction [${ }^{\circ }$]	$R_{\epsilon }$	Strain ranges [%]
0	0	0.43, 0.65, 0.9, 1.5
0	−1	0.43, 0.5, 0.65, 0.75, 1.0
90	0	0.35, 0.55, 0.75, 1.05, 1.5
90	−1	0.43, 0.5, 0.8, 1.1, 1.5

Figure 7. Experimental mid-life hysteresis loops for the LCF tests at ${700}^{\circ }$C for a) $R_{\epsilon }=0$ and b) $R_{\epsilon }=-1$.

Figure 8. Loading sequence for the temperature and mechanical strain during a) in-phase and b) out-of-phase TMF loading.

Figure 9. Experimental stress vs. mechanical strain for the TMF tests with $T_{\mathit{max}}={800}^{\circ }$C, showing cycle 1, 2 and the mid-life cycle for test a) IP800:0:0.4, b) OP800:0:0.6, c) IP800:90:0.3, and d) OP800:90:0.4.

Table 3. Summary of the performed TMF tests with $T_{\mathit{max}}={800}^{\circ }$C.

Specimen	$R_{\epsilon }$	Building direction [${ }^{\circ }$]	$\mathrm{\Delta \epsilon }$ [%]
IP800:0:0.4	0 (IP)	0	0.4
OP800:0:0.6	$-\mathrm{\infty }$ (OP)	0	0.6
IP800:90:0.3	0 (IP)	90	0.3
OP800:90:0.4	$-\mathrm{\infty }$ (OP)	90	0.4

Figure 10. Experimental inelastic strain rate vs. stress from the hold-times during the first cycle and in the mid-life cycle for IP and OP TMF tests with $T_{\mathit{max}}={600}^{\circ }$C and OP TMF tests with $T_{\mathit{max}}={800}^{\circ }$C.

Figure 11. Schematic overview of the cycle jumping procedure, where the initial virgin parameters are changed after the first onloading and hold-time to mid-life parameters.

Figure 12. Illustration of the stress update procedure, where the elastic trial stress is updated with a creep corrector. If $\mathit{f}>0$ after the creep corrector, the stress is updated with a plastic corrector to obtain $\mathit{f}=0$.

Figure 13. Illustration of how the saturated backstress was based on the accumulated creep strain in the beginning of the secondary creep stage and the double-Norton terms evaluated from the minimum creep strain rates.

Table 4. Summary of which tests the material parameters were extracted from.

	Parameter
LCF RT-${700}^{\circ }$C	$E_{\theta }$, ${\sigma }_{\theta }^Y$, $h_{\theta }$, ${\alpha }_{\mathit{sat},\mathit{\theta }}$
Creep $625-{800}^{\circ }$C	$A_{90}$, $A_0$, $n_{90}$, $n_0$
Tensile ${800}^{\circ }$C
Creep $700-{800}^{\circ }$C	$\mathit{\gamma }$, $\mathit{m}$
Tensile ${800}^{\circ }$C
TMF $450-{600}^{\circ }$C	$A_{90}$, $A_0$, $n_{90}$, $n_0$, $\mathit{\gamma }$, $\mathit{m}$, ${\kappa }_{90,\mathit{mid}}$, ${\kappa }_{0,\mathit{mid}}$, ${\chi }_{\mathit{mid}}$
TMF ${800}^{\circ }$C	${\kappa }_{90,\mathit{mid}}$, ${\kappa }_{0,\mathit{mid}}$, ${\chi }_{\mathit{mid}}$, ${\beta }_{\theta }$

Figure 14. The obtained parameters for the rate-dependent terms and the temperature interpolation for a) $A_{90}$, $A_0$ and $\mathit{\gamma }$, and b) $n_{90}$, $n_0$ and $\mathit{m}$.

Table 5. Summary of the parameters that are dependent on instant and maximum temperature.

	Parameter
Instant temperature dependent	$E_{\theta }$, ${\sigma }_{\theta }^Y$, $A_{90}$, $A_0$, $n_{90}$, $n_0$, $\mathit{\gamma }$, $\mathit{m}$
Maximum temperature dependent	${\kappa }_{90}$, ${\kappa }_0$, $\mathit{\chi }$, ${\beta }_{\theta }$, $h_{\theta }$, ${\alpha }_{\mathit{sat},\mathit{\theta }}$

$\mathit{\theta }$=$0^{\circ }$, ${45}^{\circ }$, ${90}^{\circ }$

Figure 15. Experimental and simulated creep response for a) ${625}^{\circ }$C, b) zoomed in to strains up to 1% and the first 2000 hours for ${625}^{\circ }$C, c) ${700}^{\circ }$C, d) zoomed in ${700}^{\circ }$C, e) ${750}^{\circ }$C, f) zoomed in ${750}^{\circ }$C, g) ${800}^{\circ }$C and h) zoomed in ${800}^{\circ }$C.

Figure 16. Experimental and simulated stress vs. mechanical strain for the TMF tests with $T_{\mathit{max}}={800}^{\circ }$C, showing cycle 1, 2 and the mid-life cycle for the experiments and cycle 1 and mid-life for the simulation without the aging term, for test a) IP800:0:0.4, b) OP800:0:0.6, c) IP800:90:0.3, and d) OP800:90:0.4.

Figure 17. Experimental and simulated stress vs. mechanical strain for IP TMF tests with $T_{\mathit{max}}={600}^{\circ }$C for $0^{\circ }$ specimen with $\mathrm{\Delta \epsilon }$ a) $0.4\mathrm{\%}$, b) $0.55\mathrm{\%}$, c) $0.7\mathrm{\%}$ and d) $1.0\mathrm{\%}$ and ${90}^{\circ }$ specimen with $\mathrm{\Delta }\mathit{\epsilon }$ e) $0.7\mathrm{\%}$ and f) $0.9\mathrm{\%}$.

Figure 18. Experimental and simulated stress vs. mechanical strain for OP TMF tests with $T_{\mathit{max}}={600}^{\circ }$C for $0^{\circ }$ specimen with $\mathrm{\Delta \epsilon }$ a) $0.7\mathrm{\%}$ and b) $1.0\mathrm{\%}$ and ${90}^{\circ }$ specimen with $\mathrm{\Delta }\mathit{\epsilon }$ c) $0.7\mathrm{\%}$ and d) $1.0\mathrm{\%}$.

Figure 19. Experimental and simulated stress vs. mechanical strain for TMF tests with $T_{\mathit{max}}={450}^{\circ }$C, for a) IP TMF ${90}^{\circ }$ specimen with $\mathrm{\Delta \epsilon }=0.7\mathrm{\%}$ and b) OP TMF $0^{\circ }$ specimen with $\mathrm{\Delta }\mathit{\epsilon }=0.83\mathrm{\%}$.

Figure 20. Experimental and simulated mid-life response of the LCF tests at ${700}^{\circ }$C for $0^{\circ }$ specimens with a) $R_{\epsilon }=0$ and b) $R_{\epsilon }=-1$ and ${90}^{\circ }$ specimens with c) $R_{\epsilon }=0$ and d) $R_{\epsilon }=-1$.

Figure 21. Simulated vs. experimental mid-life results for the TMF tests with $T_{\mathit{max}}$ up to ${800}^{\circ }$C and LCF tests at ${700}^{\circ }$C for a) maximum stress, b) minimum stress, and c) inelastic strain range.

Figure 22. Experimental and simulated stress vs. mechanical strain for the TMF tests with $T_{\mathit{max}}={800}^{\circ }$C, showing cycle 1, 2 and the mid-life cycle for the experiments and cycle 1 and mid-life for the simulation with the aging term, for test a) IP800:0:0.4, b) OP800:0:0.6, c) IP800:90:0.3, and d) OP800:90:0.4.

Table 6. $\%\mathit{Error}$ between the experimental and simulated values of ${\sigma }_{\mathit{max}}$, ${\sigma }_{\mathit{min}}$ and $\mathrm{\Delta }{\epsilon }_{\mathit{in}}$ for the TMF tests with $T_{\mathit{max}}={800}^{\circ }$C with and without using an aging term $\mathit{\beta }$.

		$\%\mathit{Error}$
		IP800:0:0.4	OP800:0:0.6	IP800:90:0.3	OP800:90:0.4
Without $\mathit{\beta }$	${\sigma }_{\mathit{max}}$	14.34	−11.53	31.52	−15.62
	${\sigma }_{\mathit{min}}$	−2.22	2.57	4.60	8.15
	$\mathrm{\Delta }{\epsilon }_{\mathit{in}}$	35.19	13.80	100.00	55.76
With $\mathit{\beta }$	${\sigma }_{\mathit{max}}$	8.17	−6.32	8.76	0.54
	${\sigma }_{\mathit{min}}$	3.33	−7.26	11.85	3.14
	$\mathrm{\Delta }{\epsilon }_{\mathit{in}}$	13.06	6.79	46.57	19.56

Data availability statement

The data required to reproduce the results cannot be shared due to confidentiality with regard to the partner company intellectual properties.