2.1 Fabrication of Multi-Sheet Stacked Ni-Cr-Al Superalloy Foam

A multi-sheet stacked Ni-Cr-Al superalloy foam was fabricated through the following process (Fig 1(a)). As described in previous studies [¹⁹-²¹,²³], a pure Ni pre-foam sheet was coated with a binder and alloying powder to achieve a target composition of Ni-11.45% Cr-4.56% Al (wt.%). After de-binding, transient liquid phase sintering was performed to obtain porous sheets with a thickness of 4.8–5 mm. These sheets were then stacked in 20 layers and subjected to high-temperature hot rolling, yielding a block-type Ni-Cr-Al metallic foam with dimensions of 300 mm (WD) × 500 mm (RD) × 60 mm (ND) and a compression ratio of 37.5%.

2.2 Structural and Microstructural Characterization

The pore structure and strut surface morphology of the block Ni-Cr-Al metallic foam were analyzed using scanning electron microscopy (SEM, TESCAN, VEGA II). The average pore size, strut thickness, and wall thickness were measured using an image analyzer. Phase analysis was conducted using X-ray diffraction (XRD, RIGAKU D2000). For microstructural observations, standard metallographic techniques were applied. The specimens were ground using SiC emery paper and mirror-polished with a 1 μm Al₂O₃ slurry. Electro-etching was performed for a few seconds using a solution of 85 ml H₂O, 10 ml HNO₃, and 5 ml glacial acetic acid, followed by a SEM analysis (Jeol, JSM-6700F). Microstructural parameters were quantitatively analyzed using image analysis software.

2.3 High-Temperature Creep Testing Procedure

To evaluate time-dependent deformation characteristics, specimens with dimensions of 30 mm (WD) × 30 mm (RD) × 60 mm (ND) (Fig 1(b)) were prepared and tested at 923 K and 1073 K. Creep tests were conducted under constant compressive load conditions using a direct creep tester (R&B RB306-DW), with applied stresses maintained below the yield strength at each temperature. The heating rate was set to 15 K/min, followed by 15 min of holding at the target temperature before starting the compressive time-dependent tests, which lasted for more than 130 hours. Specimen displacement was measured over time using an extensometer capable of detecting changes as small as 0.001 mm. After deformation, a SEM analysis was conducted to examine the effects of compression and time-dependent deformation on the pore structure.

3.1 Structural Features and Porosity of the Multi-Sheet Stacked Ni-Cr-Al Foam

Figure 2 presents the structure of the block Ni-Cr-Al metallic foam fabricated using the multi-sheet stacking process. The measured structural parameters are summarized in Table 1. The average pore sizes were 2569.6 μm (WD), 2988.1 μm (RD), and 2493.2 μm (ND), while the strut thickness and cell wall thickness were 240.8 μm and 27.7 μm, respectively. The pore morphology indicates that the foam underwent compression along the normal direction and elongation along the rolling direction due to the hot rolling process applied during fabrication. The relative density of the foam was measured as 4.55% (compared to 2.7% for a Ni-Cr-Al sheet), corresponding to a porosity of 95.45%. Unlike conventional porous structures, the block Ni-Cr-Al foam exhibited distinct structural features, with sheet interface regions visible, as indicated by arrows in Fig 2(b). These regions contained intersected struts or struts undergoing bending deformation due to contact interactions.

3.2 Phase Identification and Microstructure of γ and γ' Phases

The phase analysis of the Ni-Cr-Al metallic foam was conducted using X-ray diffraction (XRD). The results (shown in Fig 3) confirmed that the foam consists of a γ (Ni, Cr) matrix and γ' (Ni₃Al) precipitates. Representative microstructures in Fig 4 reveal that most regions exhibit a cuboidal phase morphology In Ni-based superalloys, γ' (Ni₃Al) precipitates generally appear in spherical (<100 nm) or cuboidal (>100 nm) shapes, depending on their size [²⁵-²⁶]. Microstructural analysis suggests that the cuboidal precipitates observed in the block Ni-Cr-Al struts correspond to γ' (Ni₃Al) (bright phase), while the dark regions represent the γ (Ni, Cr) matrix. Table 2 summarizes the quantitative microstructural characteristics. The volume fraction of γ' was 60.7%, with an average size of 0.943 μm, a maximum size of 3.985 μm, and a minimum size of 0.112 μm. The γ' contiguity was calculated using Fan et al.’s method [²⁷], yielding a value of 0.053, which indicates that most γ' precipitates are individually dispersed within the γ matrix.

where C_γ' represents γ' contiguity, N L γ ' γ ' denotes the number of intercepts at γ'/γ' interfaces, and N L γ ' γ indicates the number of intercepts at γ'/γ interfaces. When two γ′ phases are in contact, they are identified as γ′/γ′ interfaces, whereas when a γ channel and γ′ are distinguished, they are defined as γ′/γ interfaces. The calculated value of 0.053 suggests that most γ' precipitates are individually dispersed, indicating that they are surrounded by the γ matrix.

3.3 Compressive Time-Dependent Deformation at Elevated Temperatures

Since block Ni-Cr-Al foams are expected to be used in high-temperature catalytic supports and filters, their time-dependent deformation behavior should be investigated. Figure 5 presents the compressive time-dependent deformation behavior of the foams at 923 K and 1073 K. Under the highest stress condition at 923 K (0.29 MPa, 0.387σ_y), the deformation reached 0.123, whereas at 1073 K under a similar stress ratio (0.14 MPa, 0.389σ_y), the deformation doubled to 0.246. Even at lower stress conditions, significant deformation was observed. Generally, high-temperature time-dependent deformation consists of instantaneous deformation, primary creep, and secondary creep. The time-dependent strain (or creep strain) over time can be described using the following equation [²⁸]:

where ε denotes the total time-dependent deformation, ε₀ is the instantaneous strain, ε_t refers to the transient creep limit, r is the ratio of the transient creep rate to the transient creep strain, ε ˙ s s represents the steady-state deformation (creep) rate, and t is the time. The authors focused on, ε ˙ s s which has the most significant influence on creep time, to interpret the creep behavior of the superalloy foam. The time-dependent deformation mechanism can be further analyzed in relation to applied stress using the following equation [²⁷]:

where C is a material constant, σ denotes applied stress, n is the stress exponent, Q represents activation energy for time-dependent deformation, k is the Boltzmann constant, and T is the homologous temperature. Using these equations, the steady-state deformation (creep) rate and stress exponent were determined and are presented in Fig 6. The constant-load time-dependent deformation behavior varied significantly with temperature. At 1073 K, as shown in Fig 6, the stress exponent divides into two distinct regions, showing a decreasing trend instead of the typical increase with applied stress. At 923 K, the stress exponent was measured as 2.86. Generally, the stress exponent provides insight into time-dependent deformation mechanisms, where diffusion creep exhibits an exponent of ~1, grain boundary sliding is around 2, and dislocation movement ranges between 3 and 8 [²⁸]. Previous studies [¹⁵,²³] reported stress exponent values between 3 and 5 for Ni-based superalloy foams, indicating that dislocation movement typically governs time-dependent deformation. However, in this study, despite significant strain, the stress exponent values at 1073 K (1.8 and 0) indicate anomalous deformation behavior, while the value at 923 K (2.86) was relatively low. The low stress exponent can be attributed to densification of the metallic foam, which greatly reduces the effect of stress, and the following section discusses the underlying causes.

4.1 Effects of Applied Stress, Densification, and Sheet Interfaces

The time-dependent deformation behavior of the multi-sheet stacked block Ni-Cr-Al superalloy foam at 1073 K was further examined through post-deformation macroscopic analysis, as shown in Fig 7. The results indicate that the final deformation varied significantly depending on the initial applied stress. One notable observation was that the bulk sample retained its initial cross-sectional area, exhibiting primarily uniaxial deformation rather than significant lateral expansion. Additionally, as the initial stress increased, the total amount of time-dependent deformation also increased. In particular, when the applied stress exceeded 0.14 MPa, the deformation surpassed 25% and densification became evident. These findings emphasize the necessity of considering densification effects in long-term time-dependent deformation analyses, as densification can influence fluid flow characteristics, oxidation behavior and mechanical properties. All these factors must be considered when predicting the service life of metallic foams in practical applications. Accordingly, this study focuses on the influence of densification on the time-dependent deformation rate.

4.2 Anomalous High-Temperature Creep and Stress Exponent Correction

Under constant load conditions, the complete time-dependent strain (creep strain) vs. time curve typically progresses through three distinct stages, as shown in Fig 8: primary, secondary (steady-state), and tertiary deformation. The primary deformation stage is characterized by a gradual decrease in strain rate, while the secondary stage maintains a constant strain rate and is considered the most critical stage for determining the total duration of time-dependent deformation. The tertiary stage exhibits an increasing strain rate, ultimately leading to failure. This behavior is commonly observed in both tensile and compressive loading conditions, particularly in the low-strain regime (≤ 15%). However, the results of this study suggest that the multi-sheet stacked block Ni-Cr-Al superalloy foam does not fully conform to this conventional deformation trend (this can be seen in Figure 9).

Figure 9 presents the creep strain-log(t), strain rate-time, and strain rate-creep strain curves obtained at 923 K and 1073 K. At 923 K, the log(t)-strain curve exhibits a gradual increase in creep strain, following a typical time-dependent deformation pattern. When analyzing the strain rate variations, a saturation region similar to that described in Fig 8 is observed. However, the strain rate continuously decreases at a constant rate, contrary to the expected behavior where lower stresses should lead to more rapid strain rate saturation. Instead, a more rapid saturation was observed at higher stresses. In contrast, the material exhibited unexpected deformation behavior at 1073 K compared to 923 K. The log(t)-strain curve at 1073 K does not show a rapid increase in strain over time, and the strain rate exhibits an inverse decrease. This trend is further confirmed in the strain ratetime curve, where the strain rate unexpectedly reverses between 0.18 MPa and 0.14 MPa. Additionally, in the strain rate-creep strain curve, a sharp decrease in strain rate is observed beyond a certain strain threshold. These results suggest that total deformation at 1073 K was significantly greater than at 923 K, yet the strain rate rapidly decreased as the creep strain increased. Although 1073 K is a higher temperature than 923 K, a lower creep rate was observed, indicating that both the microstructural characteristics of the alloy and the structural features of the foam contributed to the observed deformation behavior.

A previous study by the authors [²³] reported that in single-sheet compression tests, where a plateau region is observed, the reduction in strain rate was not detected in multi-sheet stacked foams. The plateau region is generally associated with constant flow stress, as deformation primarily occurs through pore collapse without significant interactions between struts. The results of this study indicate that time-dependent deformation behavior is significantly affected by flow stress, which is a crucial factor in determining the service life and reliability of components.

Figure 10 presents the compressive stress-strain curve at 923 K, along with images of deformed specimens under low- and high-stress conditions. The data indicate that at the maximum applied stress at 923 K, deformation was constrained within the plateau region, which is typically associated with pore collapse. In conventional metallic foams, when plateau strength remains constant, stress effects are expected to remain uniform, leading to a saturated time-dependent deformation rate. However, in the multi-sheet stacked block Ni-Cr-Al superalloy foam investigated in this study, strain hardening was observed within the plateau region. According to Mangipudi et al. [²⁴], this hardening effect can be attributed to interactions such as friction and anchoring between intersecting struts. Unlike single-layered foams, the multi-sheet stacked foam used in this study consists of stacked sheets, resulting in distinct sheet interfaces, as seen in Fig 2(b). These interfaces contain noncontact regions with intersecting struts, which increasingly interact under compressive deformation and lead to hardening. A similar phenomenon was also observed in time-dependent deformation tests, where deformation concentrated around the interface regions, as shown in Fig 10 (yellow arrows). The same effect was identified at 1073 K under low-stress conditions, as indicated in Fig 11 (yellow arrows). Although the hardening effect diminished with increasing temperature, it remained significant due to interactions at the sheet interfaces. Under higher stress conditions, deformation occurred in both the interface regions and the surrounding areas, ultimately leading to densification. As a result, the strength of the metallic foam increased, and the effect of the initially applied stress was not maintained consistently throughout the deformation process.

In bulk materials, the effects of stress on mechanical properties are often analyzed by normalizing the strain rate against yield stress to generate a strain rate-normalized stress curve for evaluating temperature effects [²⁸]. However, for the metallic foam used in this study, strain hardening was induced by interactions at sheet interfaces. As deformation progressed, strain localization occurred due to densification, resulting in a lower effective stress than initially applied. To better interpret the time-dependent deformation behavior, this study did not apply yield strength normalization but instead determined effective normalized stress based on flow stress at the same strain levels obtained from the compressive stressstrain curve. In previous studies on metallic foams, creep analyses have generally adopted yield-stress normalization; however, because the present foam exhibited interfacial interactions and densification absent in bulk materials, the initial yield stress could not represent the operative stress. Therefore, the stress exponent was reevaluated using flow stress at equivalent strain levels. The results are presented in Fig 12.

The stress exponent at 923 K, initially measured as 2.86, increased to 3.76 after correction. At 1073 K, the originally measured values of 1.8 and 0 converged to a single corrected value of 2.31. The corrected stress exponent of 3.76 at 923 K closely aligns with previously reported values for Ni-based superalloy foams, where time-dependent deformation is typically governed by dislocation movement. Furthermore, observations of the deformed foam confirmed the presence of plastic deformation at the interfaces, supporting the validity of this interpretation. In contrast, at 1073 K, deformation progressed at a relatively higher temperature and led to an increased contribution of diffusion mechanisms. Consequently, despite the significant amount of deformation observed, the stress exponent was determined to be 2.31. Structural analysis confirmed that the multi-sheet stacked block Ni-Cr-Al superalloy foam contained structural discontinuities and defects at sheet interfaces. While these structural instabilities might be considered limitations in quasi-static deformation, their effects on high-temperature time-dependent deformation resulted in flow stress hardening and densification, which reduced the deformation rate and ultimately enhanced high-temperature durability.