3.1. Initial Microstructure
Quantification of microstructure characteristics was achieved by EBSD. Pole figures show that there is no obvious texture in the material and the statistical results show that the sample mainly consists of equiaxed grains and the average grain size is 11.61 µm. The graphic of the frequency of the grain misorientation distribution of the as-received alloy, presented in
Figure 2d, exhibits a relatively high fraction of low angle grain boundaries (LAGBs,
θ < 15°) [
19] caused by the presence of substructures created by a large number of dislocations. These structures can hinder the slip of dislocations explaining to some extent the high tensile strengths of the superalloy [
20]. Moreover, a significant level of misorientation about 60 deg indicates an array of twins in the as-received materials [
21].
Furthermore, the main composed phase in as-received Inconel 625 superalloy presented in
Figure 3 by TEM images is
γ phase, and the needled morphology distributed within the grains and at their boundaries suggests
δ phase [
22]. Apart from the needled-like
δ phase, a globular precipitate is also be found in the intergranular regions.
To differentiate the identity of these precipitates, two areas, labeled 1 and 2, respectively, in
Figure 3b,c, were selected for the TEM observations and EDS pattern microanalysis. For precipitate 1, based on its chemical analysis result, the average composition of this needled-like precipitation was determined for the major concentrations of Ni, Nb, Fe, Ti, Cr. The results confirm that it is
δ phase [
1,
23,
24,
25]. As to the large blocky precipitate 2, the EDS analysis results of this area revealed that this polygonal shaped precipitate was Laves phase, which is consistent observed by Wang et al. [
26,
27,
28,
29]. To summarize, according to the SEM and TEM observations, it can be claimed that the
δ phase is present in the
γ matrix, and a few brittle phase of Inconel 625, Laves particles can be found at grains boundaries [
30,
31].
3.2. Dependence on Strain Rate of Tensile Behavior
The stress–strain curves under different strain rates were shown in
Figure 4. All the specimens show a similar performance during the whole elastic stage. However, no apparent serrated flow behavior was observed on the micro-plasticity stage. As to the macro-plasticity stage, with continuous deformation, the true stress increases until fracture. It was evident that the yield strength and the ultimate tensile strength increase with increasing strain rate from 5 × 10
−4 s
−1 to 5 × 10
−5 s
−1 as illustrated in
Table 2.
Besides, from
Figure 4, a significant strain rate sensitivity was observed above the elastic limit. To evaluate the strain hardening behavior, the hardening capacity
Hc of Inconel 625 can be described as following [
32],
Considering no obvious yield point in the stress–strain curves,
σY is determined as the engineering stress proof 0.2% plastic deformation, and
σUTS is the ultimate tensile strength. The
Hc value of each specimen was calculated and listed in
Table 3. However, a small gap of
Hc between both strain rates is not sufficient to justify the hardening capacity of the superalloy. To further confirm the strain hardening behavior of the superalloy, the strain hardening exponent (
n) of the alloy was evaluated using several mathematical expressions, which are the most common. However, it is not surprising that these empirical equations can not accurately describe the stress–strain curves for a specified metals. Thus, the aim of this section is to study the applicability of two types (unsaturation extrapolation formula and saturated extrapolation formula) widely used fit functions [
33,
34] for the estimation of the
n exponent, and to suggest an improved stress–strain fitting model.
The unsaturated model is represented by classical Hollomon [
35] and Ludwik equation [
13,
36]. The Hollomon model is a typical full-strain model, which describes that the material strength increases in the form of power of constant hardening coefficient (
n) during the whole process of deformation. The Ludwik model is an evolution of the Hollomon model with a fixed initial value. The extrapolated stress value of the Hollomon and Ludwik model has no upper limit, and they are written, respectively, as the following:
where
n1 and
n2 are the strain hardening exponent;
K1 and
K2 are the strength parameters of the superalloy;
σ and
σy are true stress and yield stress, respectively;
ε is the true strain and
εy used in Equation (3) is the true strain before yielding, which means the Ludwik equation only considers the true plastic deformation stage (i.e., between yield strength and ultimate tensile strength) for the curve fitting.
The variations in true stress (
σ) with the true strain (
ε) and true plastic strain (
ε −
εY) were, respectively presented as double logarithmic plots in
Figure 5. The hardening exponent (
n1,
n2) is determined as the slope of the corresponding curve, as shown in
Table 3. Calculation results of
n1 and
n2 expose a similar rule that hardening exponent increases with increasing strain rate. This distinctly indicates that the strain hardening stages of the Inconel 625 alloy used in this work are related to the strain rate. It must be pointed out that a much higher growth of
n2 at strain rate from 5 × 10
−4 s
−1 to 5 × 10
−5 s
−1, fitted by Equation (3) than that of
n1 was strongly due to Equation (3) exclusive the impact of elastic deformation stage based on Hook’s law which has no contribution for strain hardening. Accordingly, the index
n2 is more sensitive to the strain rate than
n1 fitted by the Hollomon equation.
Referring to the above mentioned, model with the constant initial value, i.e., start from the yield point (0,
σY) seems to be more suitable for the hardening exponent calculation. However, as the stress increases indefinitely with the strain increases, the unsaturated extrapolation model shows deficiencies at the end of the deformation as presented in
Figure 6 (blue dotted line). On this basis, saturated extrapolation model introduced the concept of hardening factor is of great significance in describing the large strain stage, in which Hockett–Sherby (H–S) [
37] is a typical model, as follows:
m1 is material constant; σ∞ is the stress at fracture.
The fitting curves of H–S method were shown in
Figure 6. It is not hard to find that when the material starts to yield, flow stress rises rapidly, and the Ludwik equation is better fit for this stage than the H–S, then as the strain increases, unsaturated extrapolation loss accuracy, in contrast, H–S curves match better with the true case. At this point, a key problem of these above-mentioned fit equations appears: How to find a more appropriate formula, that considers both specialties of the whole variation trend. Hence, a hybrid method consists of the Ludwik and the H–S equation with a strain-dependent factor
Φ is presented as below:
where
Φ =
cε/
ε∞,
ε∞ is the strain at fracture;
n* is the hardening exponent;
m2,
c and
K are the material constant. Relevant parameters of the hybrid model and the relative error are in
Table 4. Error analysis was realized by taking relative error (RE) of curve integral, where the error is defined as:
where,
σFIT,
σEXP are the fitted stress and experimental stress, respectively;
εP is the true plastic strain. The error results show that the curve fitted by the hybrid model has the highest approximation to the measured curve, and its relative error reaches 0.48% and 0.79% at 5 × 10
−4 s
−1 and 5 × 10
−5 s
−1, respectively.
In addition, according to the change in strain, this new hybrid model contributes to harmonizing the proportion of saturated and unsaturated equations. Obviously, the divergence of flow stress control by the Ludwik equation decreases significantly, as H–S gradually takes its effect with the increase in true plastic strain. The fit curves of these models can be seen in
Figure 6, in which Ludwik and H–S fitting curves show their limitations, hybrid model, by contrast, always keeps consistent with the true curve and the hardening exponents obtained by the new method were also listed in
Table 3. The largest gap of
n* from 0.3 to 0.35 between two strain rates indicates the material’s strain rate sensitivity, whereas a dramatic increase in stress at the latter stage of deformation leads to a higher value of
n2 fitted by Ludwik equation. Analysis of exponents also confirms the applicability of this new hybrid model for specified 625 superalloys.
3.3. Fracture Pattern and Microstructure Morphology
According to the above investigation, the tensile properties of Inconel 625 superalloy depend strongly on the strain rates, resulting in different fracture patterns at both strain rates. In that way, the representative SEM micrographs of fracture morphology of the specimens after tensile tests show quite different features, as shown in
Figure 7.
A slight neck can be observed at both strain rates, as shown in
Figure 7a,b, which indicates that local plasticity presents before final failure. There is a flat fracture surface with river patterns and fine and shallow dimples on the fracture surface at 5 × 10
−5 s
−1 presented in
Figure 7d,f, which show ductile fracture mode by the occurrence of dimples on the fracture surface. Comparing to the low strain rate, a large number of transgranular cracks can be observed from the specimens destroyed under 5 × 10
−4 s
−1 in
Figure 7c. However, the tensile fracture of high strain rate is mixed mode of fracture though the failure was predominantly intercrystalline. It should also be indicated that the dimple size increases with the increasement of the strain rate.
To study the deformation mechanism and dislocation configuration, the TEM images of interrupted tensile tests (2.0% plastic strain excluding elastic strain) as shown in
Figure 8 and are gathered for the following analysis. All the deformed samples show planar slip with high dislocation density. Some specific slip bands (
Figure 8a) consisted of dislocation structure of densely packed primary and secondary dislocations was observed in the gauge length [
38,
39]. A few nano deformation twins are occasionally observed in several grains [
40] at 5 × 10
−4 s
−1 without 5 × 10
−5 s
−1, see
Figure 8b Generally, in face-centered cubic (FCC), twinning is facilitated through lower stacking fault energy (SFE) and special deformation conditions (such as low deformation temperatures or high strain rates) [
41,
42]. For Inconel 625, as an FCC metal with low SFE, the deformation mechanism is twinning ({111}<112>) and dislocation slip ({111}<110>) through the tensile deformation modes [
43]. Therefore, slip is the main deformation mode during the initial stages of tensile deformation under both strain rates. After dislocation multiplication and tangle formation, further deformation results in dislocation cross slip being suppressed to that extent the cross slip of Shockley partial dislocations could lead to intrinsic stacking faults on parallel {111} planes, leading to twins [
44]. Thus, twinning is the other deformation mode during the futher stages of tensile deformation, and twin boundaries act as strong obstacles to the dislocation motion, resulting in improvement of alloy strength [
45,
46]. Li et al. [
47] has also found the deformation twins in other alloys, and demonstrate that this critical change of deformation mechanisms from dislocation slips to twinning behavior is responsible for such an increasing of hardening exponent (
n) value from 5 × 10
−5 s
−1 to 5 × 10
−4 s
−1. This is consistent with our experimental results.
In addition, the typical planar slip during deformation is dicided by the Schmid factor and yield stress. With the consideration of the crystal orientation, the critical resolved shear stress (CRSS) can be determined as following with Schmid’s law [
48].
where
M is Schmid factor.
To define the relationship between Schmid factor and slip, a free MATLAB toolbox MTEX [
49] was used to calculate the Schmid factor and visualize the active slip systems of a given EBSD map in
Figure 9. The Schmid factor on different slip systems and CRSS of a typical grain is presented in
Table 5.
The slip direction calculated by MTEX (white arrow) is in accordance with the EBSD map presented in
Figure 9. and the results show possible primary slip system was (−1, 1, −1) [0, −1, −1], the secondary slip system (111) [0, −1, 1] and the third slip system (−1, 1, 1) [1, 0, 1] are activated during tensile tests. Furthermore, the max CRSS value on the strain rate of 5 × 10
−4 s
−1 is 573.72 Mpa, which is higher than 547.91 MPa at 5 × 10
−5 s
−1. It indicates that CRSS value can be affected by the strain rate, which can be explained with the Taylor equation;
where
α is a constant measuring the efficiency of dislocation strengthening,
G is the shear modulus and
b is the Burgers vector, hence it is clear that shear stress (
τ) is directly related to the final dislocation density (
ρ).
3.4. Failure Mechanism
In this section, the work-hardening concepts were induced to explain and predict the stress–strain response of the alloy from the point of dislocation theosries, and the work-hardening rate of 625 superalloy shows an increase with increasing strain rate. As seen in the Kocks–Mecking type plot of strain hardening rate
θ (=
dσ/
dε); vs. net flow stress (
σ −
σY) at two strain rates of the Inconel 625 superalloy, as shown in
Figure 10, work-hardening behaviour of this alloy is characterized by an initial sharp fall in
θ, followed by satge II, i.e., a plateau and then a further gradual fall can be denoted as stage III work-hardening, respectively [
17].
Stage II is characterized by a high initial
θ value that almost stabilised at a constant, and such a behaviour is attributed to an initially linear stage II strain hardening behaviour [
12]. Additionally, the stress reaches the CRSS, one or more slip systems are activated in this stage. Meanwhile, dislocation shear into
δ phase has been observed, as shown in
Figure 11, which plays an important role in controlling the tensile performance of the alloy. As to stage III, the sharp drop in the slope of the sample under 5 × 10
−4 s
−1, as shown in
Figure 10, occurs earlier than that of the low strain rate sample. This indicates a premature recovery process occurs on a high strain rate sample, speculated as once the recovery process starts in the specimen with higher dislocation density, it goes faster than the other [
17].
Based on the above discussion, a model about strain hardening takes into account to explain the strain hardening behavior of this alloy [
50],
where
σ0 is the stress contributed by the friction;
σHP =
kd1/2 is contributed by the Hall–Petch;
σd =
MαGbρ1/2 is contributed by the Taylor dislocation.
During deformation, the gliding of dislocation causes plastic strain in the material. As the strain increases, the material begins to yield, dislocations nucleate, and interact, leading to dislocation density increases. Thus stress contribution caused by dislocation density can be written as the total flow stress subtracting the yield stress,
The applied stress necessary to deformation is obviously proportional to the dislocation density in the material. Thus far, dislocation density is affirmed necessary in this investigation. The magnitude
ρ was determined by the line intersection method [
51,
52,
53] based on the superimposition of a grid consisting of horizontal and vertical test lines on the TEM micrographs that contained dislocations of the specimens at both strain rates. Since here we only need to compare the different influence between the two strain rates qualitatively, we can briefly distinguish the dislocation density of different strain rates by computing the average number of intersections of each test lines with dislocations. To simplify the computing process further, the grid was drawn as a square, as illustrated in
Figure 12. For each strain rate, two pictures were used, and for each picture, we grid two areas that unaffected by the precipitate, then the computing results are listed in
Table 6.
However, the average values of intersection number determined from the TEM micrographs of tensile specimens strained at 5 × 10
−4 s
−1 was relatively higher than the low stain rate ones 5 × 10
−5 s
−1. This can also be verified by examining the TEM micrographs presented in
Figure 11, showing the denser population of dislocations at this strain rate. Essrntially, the number and velocity of dislocations are improved at a high strain rate, which accompanied with the increasing of the dislocation density per unit area. Accordingly, the high initial dislocation density in the high strain rate specimens might have contributed to the initial high and nearly constant strain hardening rate (i.e., stage II linear hardening shown in
Figure 10 of the specimens under 5 × 10
−5 s
−1. This suggests that much higher activation energy is required for the plastic flow due to powerful barriers to the dislocation movement [
54]. As described by reference [
12], a positive work hardening stage II occurs due to continuous reduction in mean free path during dislocation–dislocation interaction and dislocation pileups at the grain boundary. Thus, the Taylor dislocation contribution
σd =
MαGbρ1/2 in Equation (7) dominates this region. The increasing of the number of dislocations leads to the increasing of the resistance to the dislocation movement, and the stress required to deform the materials becomes higher with increasing deformation. This is in accordance with Zhang’s results in nickel-based superalloy [
16].
Moreover, an interesting phenomenon worth to note is that the original shearing direction changes when dislocations slip shear through the
δ phase under a low strain rate, see
Figure 11. In contrast, under a high strain rate, shearing is always in the same direction. The following statement may interpret this strain rate related performance: When experiments were carried out at a high strain rate, with a high per unit time strain, the inside of the material is subjected to more intense deformation per unit time, which means more energy is imported to help dislocations go through the obstacle. On the opposite, low strain rate specimens cannot cross the
δ phase directly. Therefore, shearing direction will change to the pass with the lowest energy cost [
55,
56,
57,
58].