Introduction

Nanocrystalline (NC) metals grab significant attention of both material scientists and engineers due to their high mechanical strength and unique deformation nature with respect to their coarse-grained polycrystalline counterparts and accordingly they become a comparatively better choice for structural application [1,2,3,4,5,6,7]. However, in order to enhance their applicability as structural materials, it is necessary to improve their creep resistance property. In this perspective, thorough understanding of high-temperature deformation mechanism of NC metals at the atomic scale is highly crucial for achieving improved creep properties and several relevant majorly experimental based investigations have been started over the last few decades [8,9,10,11,12,13,14].

In addition to uniaxial creep studies, several investigations on biaxial creep behaviour by experimentations for coarse-grained metals have been performed [15,16,17,18,19,20,21,22,23,24,25,26,27,28]. It is reported that the dislocation creep mechanism is the major cause for deformation at different stress conditions [19, 20]. Tung et al. have also studied the biaxial creep for coarse grained materials by experimentally and it is found that dislocations, as well as grain boundary sliding with diffusional accommodation creep mechanisms, are active during biaxial creep deformation [18].

In case of ultrafine nanocrystalline metal having grain size in the order of 10 nm, conventional intra-granular dislocation governed deformation mechanisms become subdued [29, 30]. Plastic deformation of such ultrafine grained NC metal is controlled by grain boundary sliding [31,32,33], grain rotation [34,35,36], and grain boundary migration [37,38,39]. Simultaneously activated multiple mechanisms, such as grain rotation and grain boundary migration, can be operative and responsible for the plasticity NC metal [40]. All the mechanisms of plastic deformation of NC metal basically represent movement of groups of atoms collectively, which in turn change the internal grain architecture [41]. Rotation of two adjacent grains causes variation in orientation angle between the grains and features of grain boundary present between them. In this perspective, molecular dynamics (MD) simulation is found to be effective to get insight of operative deformation mechanisms for ultrafine grained NC metal. Hasnaoui et al. [42] have performed MD simulation and revealed that the appearance of shear planes in nanocrystalline Ni due to the arrangement of GB planes by dint of grain rotation, grain boundary sliding and grain boundary migration and that phenomenon causes localized shear deformation along with grain coalescence and growth. In this line, Rupert [43] has also reported that grain boundary network connectivity is highly responsible for strain localization. At this juncture, it is understood that grain boundary network connectivity, grain boundary sliding and grain boundary migration may play an important role for creep deformation of NC metals and accordingly the study of the evolution of grain rotation and the internal grain architecture during creep deformation process is highly essential. Through experimentation, very limited information has been extracted on atomic-scale structure evolution during creep deformation of NC metals as it is known that atomic-level experimental techniques along with in-situ characterization techniques are very difficult to implement for capturing the dynamics of deformation [29, 44,45,46,47,48,49,50]. Recent trends of research in this area reveals that molecular dynamics (MD) simulation based study of atomic-scale mechanisms responsible for creep deformation of ultrafine NC metals having grain size (diameter, d) in the order of 10 nm, is reliable [7, 51,52,53,54,55,56,57,58,59,60,61]. Numerous researchers have done creep study for NC metals by MD simulations previously and it is reported that grain boundary (GB) diffusional (Coble creep) creep mechanism is the operative and major factor for deformation of ultrafine grained NC metals under low and medium level of stress [7, 51,52,53,54,55, 61,62,63,64,65]. The uniaxial tensile creep behaviour study for ultrafine-grained NC Ni metal using MD simulation has been performed previously and Coble creep is found to be dominant creep mechanism for NC Ni having less than 10 nm grain size [7, 54].

The investigation of biaxial creep for ultrafine-grained NC metals (grain size in the order ~ 10 nm) is essential, but the study of biaxial creep deformation behaviour and internal grain structure evolution dynamics during creep deformation for such ultrafine grained NC metals is not reported until date in the literature as per author knowledge. Therefore, we have performed MD simulation-based investigation of biaxial creep deformation behaviour of NC Ni. The objective of the current study is to investigate the effect of the biaxial state of stress on the creep behaviour NC Ni (having grain size ~ 6 nm) and provide rational insight of the underlying biaxial creep deformation mechanism at the atomic scale. In order to do so, grain rotation and evolution of internal grain structure of NC specimen during biaxial creep deformation have been investigated. Diffusional creep mechanism is dominated during uniaxial creep deformation process [7], while dislocations, as well as grain boundary diffusion, creep mechanisms are operative during biaxial creep deformation for same grain size NC Ni specimen. Thus, the deformation mechanism is different for biaxial as compared with uniaxial creep deformation on nanocrystalline metal. It is well known that the state of stress has a huge influence on deformation behaviour of metals so only understanding of uniaxial creep behaviour will not be sufficient. Therefore, biaxial creep deformation simulation is necessary to perform. The motivation of the work is to understand the influence of stress state on creep properties of NC metals and its associated underlying mechanistic rationality, because such knowledge will help as well as create interest to find out the ways to achieve better creep resistance under biaxial stress condition.

Molecular dynamics simulation details

A nanocrystalline (NC) cubic Ni specimen with an average grain size ~ 6 nm is generated by Voronoi construction algorithm [66] under full three-dimensional periodic boundary conditions. The full specimen of NC Ni contains 14 uniform grains with random position and crystallographic orientations. The dimension of cubic NC Ni specimen is taken as 14.08 nm × 14.08 nm × 14.08 nm for performing MD simulation and the cubic simulation box contains 240,678 atoms. Before initiating the biaxial creep process, energy minimization of NC Ni specimen is carried out using the conjugate gradient algorithm with 2 fs time step. Then, the NC Ni specimen is equilibrated at 1210 K (~ 0.72*Tm) temperature for 30 ps using NVT ensemble to control temperature by Nose-Hoover thermostat method [67, 68]. Calculated melting temperature (Tm) of NC Ni having average grain size 6 nm is ~ 1674 K and the same is represented in Fig. 1. After that, NC Ni specimen is allowed to deform along the X- and Y-direction under constant biaxial applied load/stress state in the form of pressure using the Berendsen barostat method [69] and no stress is applied in Z-directions. The equivalent biaxial creep strain is calculated using the following formula.

Fig. 1
figure 1

Plot of potential energy vs. temperature to determine the melting point for NC Ni having average grain size 6 nm specimen

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Calculated strain for X-direction is given below,

$$ {\varepsilon}_x=\frac{l_x-{l}_{0x}}{l_{0x}} $$
(1)

where lx and l0x are the instantaneous length of the specimen along X-direction under applied stress and initial length of the specimen along X-direction respectively.

Calculated strain for Y-direction is given below,

$$ {\varepsilon}_y=\frac{l_y-{l}_{0y}}{l_{0y}} $$
(2)

where ly and l0y are the instantaneous length of the specimen along Y-direction under applied stress and initial length of the specimen along Y-direction respectively.

Effective strain (εeff) formula from von Mises criterion is given below.

For von Mises criterion, the effective strain is

$$ {\varepsilon}_{eff}=\frac{\sqrt{2}}{3}{\left\{{\left({\varepsilon}_x-{\varepsilon}_y\right)}^2+{\left({\varepsilon}_y-{\varepsilon}_z\right)}^2+{\left({\varepsilon}_z-{\varepsilon}_x\right)}^2\right\}}^{1/2} $$
(3)

where εx, εy, and εz are the strain in the X-, Y-, and Z-direction respectively.

If εz = 0, then the modified formula from Eq. (3) for equivalent biaxial creep strain calculation is given below

$$ {\varepsilon}_{eff}=\frac{\sqrt{2}}{3}{\left\{{\left({\varepsilon}_x-{\varepsilon}_y\right)}^2+{\left({\varepsilon}_y-0\right)}^2+{\left(0-{\varepsilon}_x\right)}^2\right\}}^{1/2} $$
(4)
$$ {\varepsilon}_{eff}=\frac{\sqrt{2}}{3}{\left\{{\left({\varepsilon}_x-{\varepsilon}_y\right)}^2+{\left({\varepsilon}_y\right)}^2+{\left(-{\varepsilon}_x\right)}^2\right\}}^{1/2} $$
(5)
$$ {\varepsilon}_{eff}=\frac{\sqrt{2}}{3}{\left\{{\left({\varepsilon}_x\right)}^2+{\left({\varepsilon}_y\right)}^2-2{\varepsilon}_x{\varepsilon}_y+{\left({\varepsilon}_y\right)}^2+{\left({\varepsilon}_x\right)}^2\right\}}^{1/2} $$
(6)
$$ {\varepsilon}_{eff}=\frac{\sqrt{2}}{3}{\left\{2{\left({\varepsilon}_x\right)}^2+2{\left({\varepsilon}_y\right)}^2-2{\varepsilon}_x{\varepsilon}_y\right\}}^{1/2} $$
(7)
$$ {\varepsilon}_{eff}=\frac{{\left(\sqrt{2}\right)}^2}{3}{\left\{{\varepsilon}_x^2+{\varepsilon}_y^2-{\varepsilon}_x{\varepsilon}_y\right\}}^{\frac{1}{2}} $$
(8)
$$ {\varepsilon}_{equi}=\frac{2}{3}{\left({\varepsilon}_x^2+{\varepsilon}_y^2-{\varepsilon}_x{\varepsilon}_y\right)}^{1/2} $$
(9)

where εequi is the equivalent strain for bi-axial creep, εx is the strain in the X-direction and εy is the strain in Y-direction. According to von Mises criterion, the effective stress (σeff) is

$$ {\sigma}_{eff}=\frac{1}{\sqrt{2}}\ {\left\{\left[{\left({\sigma}_x-{\sigma}_y\right)}^2+{\left({\sigma}_y-{\sigma}_z\right)}^2+{\left({\sigma}_z-{\sigma}_x\right)}^2\right]\right\}}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.} $$
(10)

where σx, σy, and σz are the stress in the X-, Y-, and Z-direction respectively.

If σz = 0, and σx = σy then the modified formula from Eq. (10) for the effective stress for equivalent biaxial creep is given below

$$ {\sigma}_{eff}=\frac{1}{\sqrt{2}}\ {\left\{\left[{\left({\sigma}_x-{\sigma}_x\right)}^2+{\left({\sigma}_x-0\right)}^2+{\left(0-{\sigma}_x\right)}^2\right]\right\}}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.} $$
(11)
$$ {\sigma}_{eff}=\frac{1}{\sqrt{2}}\ {\left\{\left[{(0)}^2+{\left({\sigma}_x\right)}^2+{\left(-{\sigma}_x\right)}^2\right]\right\}}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.} $$
(12)
$$ {\sigma}_{eff}=\frac{1}{\sqrt{2}}\ {\left\{2{\left({\sigma}_x\right)}^2\right\}}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.} $$
(13)
$$ {\sigma}_{eff}={\sigma}_x $$
(14)

Thus, it is cleared from Eq. (14) that no any effective value of stress.

Simulated biaxial creep curves are performed under different constant biaxial stress state such as 1, 1.4, 2, 2.5, and 3 GPa at 1210 K temperature. Now, equivalent biaxial creep rate and minimum creep rate curves are calculated from the simulated biaxial creep curves. Time step of 2 fs is taken and periodic boundary conditions (PBCs) are applied in three directions for performing all the biaxial creep simulations, and meting point calculation. PBCs are a set of boundary conditions which are often chosen for approximating a large (infinite) size system by using a finite size system. Hence, specimen consists of 14 randomly oriented grains is fine to some extent for predicting the macroscopic features like the Taylor factor for materials by using PBCs in all three directions. All the atomistic calculations such as biaxial creep and meting point are executed using LAMMPS [70] with EAM FS (embedded atom model Finnis-Sinclair) potential developed by Mendelev et al. [71] which is applicable for Ni systems. Atomic configuration snapshots capturing, atomic trajectory, CSP analysis, atomic strain, and dislocation extraction have performed using OVITO [72]. Grain tracking algorithm (GTA) is the post-processing technique, which is used to convert the positions of the atoms in the simulation cell into the structural measurements [41, 73] during biaxial creep process. The measured crystallographic orientation of all grains as well as the overall specimen texture are analysed in nanocrystalline materials such as inverse pole figures, grain rotation and orientation maps via GTA [41, 73].

Results and discussions

Discussion on simulated biaxial creep curve and corresponding creep rate

Effective creep strain vs. time curves for various applied biaxial stresses (i.e. 1, 1.4, 2, 2.5, and 3 GPa) are presented in Fig. 2 to study the effect of applied biaxial stress on the biaxial creep behaviour at 1210 K creep temperature. The biaxial creep curves (refer Fig. 2) constitute a short primary stage of creep (denoted as R1) followed by a longer secondary stage of creep (denoted as R2) and subsequently very short tertiary stage of creep (denoted as R3). At the initial stage of creep curve, a sharp rise in effective creep curve is observed for all the three cases studied here. This kind of sharp rise in effective creep strain is known as primary creep and creep rate is observed to be decreased in this stage with increasing creep time. Duration and rise in effective strain in primary creep stage is increased with the increasing applied biaxial stress. Then the secondary stage of creep initiates, which prevalent for a comparatively longer period and exhibit almost constant creep rate. It is observed from Fig. 2 that secondary stage of creep is decreased with increasing applied biaxial stress. Just after secondary stage of creep, tertiary stage of creep is observed only when the ultrafine-grained NC Ni specimen is subjected to higher biaxial load. Tertiary stage of creep is observed to be increased with increasing applied biaxial stress. In case of creep deformation occurring under 1 GPa applied biaxial stress, tertiary creep regime is absent up until 500 ps time period (refer Fig. 2). As the applied biaxial stress on ultrafine-grained NC Ni specimen increases, the effective creep strain curves are shifted upward for 1210 K creep temperature (refer Fig. 2). Similar trend of shifting of creep curves for biaxial creep phenomena have also been observed previously through experimentations for coarse-grained metals [18, 20, 23]. Calculated effective creep rate-time curves for NC Ni at different biaxial applied stresses (i.e. 1, 1.4, 2, 2.5 and 3 GPa) are represented in Fig. 3. Biaxial creep strain rate in the primary stage of creep (R1) is decreased with increasing the creep time. It is observed that the effective creep rate in the primary stage of creep is shifted towards upward with increasing biaxial applied stress for particular creep temperature (i.e. 1210 K). In the secondary stage of creep (R2), as expected, effective creep rate is found to be almost constant. It can be seen from Fig. 3 that creep rate is increased in the tertiary stage of creep (R3) for 1.4 GPa, 2 GPa, 2.5 GPa, and 3 GPa applied stress. Effective creep rate in the tertiary stage of creep is found to be increased with increasing applied stress. Creep rate curve in the tertiary stage of creep is observed to be shifted towards upward with increasing biaxial applied stress.

Fig. 2
figure 2

Biaxial effective creep-time curves of NC Ni for five different stresses at 1210 K temperature. (R1 → Primary stage of creep, R2 → Secondary stage of creep, and R3 → Tertiary stage of creep)

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Fig. 3
figure 3

Biaxial effective creep rate-time curves of NC Ni for five different stresses at 1210 K temperature. (R1 → Primary stage of creep, R2 → Secondary stage of creep, and R3 → Tertiary stage of creep)

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Calculation of stress exponent during biaxial creep process

Minimum creep rate verse applied stress (σ) for NC Ni at 1210 K temperature is represented in Fig. 4(a). It is observed that the minimum creep rate is increased linearly with increasing applied stress. Plot of logarithmic of minimum creep rate vs. logarithmic applied stress for NC Ni is represented in Fig. 4(b) to determine the stress exponent (m) under biaxial creep process. Biaxial creep characteristics for minimum (or steady state) creep rate can be explained by the Norton’s power law expression [18].

$$ {\dot{\varepsilon}}_{\mathrm{min}}=A{\sigma}^m $$
(15)

where \( {\dot{\varepsilon}}_{\mathrm{min}} \) is the minimum creep rate, σ is an external stress, m is a stress exponent, and A is a constant. The analysis of stress exponent for NC Ni using Eq. (15) is shown in Fig. 4(b). At 1210 K, m = 1.30, may imply that the diffusion control creep mechanism is operative [74, 75]. However, only on the basis of stress exponent analysis, it is impossible to identify an exact creep mechanism.

Fig. 4
figure 4

Plots a minimum creep rate vs. applied stress (σ) and b logarithmic of minimum creep rate vs. logarithmic σ

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Atomic structure analysis during biaxial creep deformation process

Atomic configuration snapshots of NC Ni during biaxial creep process for different creep time period such as 0 ps, 100 ps, 200 ps, 300 ps, 400 ps, and 500 ps at 1210 K and 3 GPa stresses are represented in Fig. 5 and colour code of atomic snapshots is taken according to centrosymmetry parameter (CSP) (colour code range value is taken from 0 to 6 for better visualization). It can be seen that microstructure of ultra-fine grained NC Ni is almost unchanged in the primary stage of creep. But grain boundary (GB) diffusion is observed in the primary stage of creep (refer Fig. 5(b)). After the primary stage of creep, GB diffusion (called Coble creep [76]) as well as grain boundary sliding (called Lifshitz sliding [77]) mechanisms for NC Ni are observed in the secondary and tertiary stage of creep (refer Fig. 5(c–f)). The underlying GB diffusion and grain boundary sliding mechanisms, which are responsible for the deformation of NC Ni under biaxial creep process, are observed. It indicates that the diffusion-mediated creep deformation mechanism is active during biaxial creep process. Figure 5 shows that the lattice diffusion mechanism is slightly initiated during biaxial creep deformation process at 3 GPa applied stress. During biaxial creep deformation process, it can be seen that GB diffusion and grain boundary sliding mechanism may assist the grain growth. Stress-induced grain growth for NC Ni is also observed during biaxial creep process previously by Gianola et al., which also influences deformation behaviour [78].

Fig. 5
figure 5

Atomic snapshots (colour according to CSP) of NC Ni after different time period of creep during biaxial creep process for 3 GPa applied stress and 1210 K temperature

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Dislocation analysis during biaxial creep deformation process

Figure 6 shows the dislocation density vs. time curves during biaxial creep deformation process for different applied biaxial stress. Dislocation density is found to be increased with increasing creep time. It is observed that with increasing applied stress, dislocation density is shifted upward for NC Ni. It indicates the dislocation nucleation, formation and movement in NC Ni are increased with the increase of applied stress during biaxial creep deformation. It can be seen from Figs. 6 and 2 that the behaviour of dislocation density curves supports the nature of biaxial creep curves. Representative snapshots of dislocation, which is contained in the specimen (shown as inset in Fig. 6), are shown after 2 ps, 250 ps and 500 ps creep time period at 3 GPa applied stress. Different types of dislocations such as Shockley partial (green colour), Stair-rod (pink colour), Hirth partial (yellow colour), perfect dislocation (blue colour) and Frank partial (cyan colour) with Burger vector (blue colour with arrowhead) are shown inside the simulation cell. It is evident from Fig. 6 that dislocation is also assisted to deform of NC Ni materials during biaxial creep process and it is indicated by simulation that dislocation-mediated creep deformation mechanism is also operative [79, 80].

Fig. 6
figure 6

Plots of dislocation density vs. time during biaxial creep process for different applied stress at 1210 K temperature. An inset shows dislocation snapshots at three different positions of creep time for 3 GPa applied stress

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Atomic strain and atomic trajectory analysis during biaxial creep deformation process

Atomic snapshots of shear strain after 2 ps, 200 ps, 300 ps and 500 ps time period of creep at 3 GPa applied stress during biaxial creep process is represented in Fig. 7. It is observed that higher shear strains accumulation occurs at grain boundary region with the progress of biaxial creep deformation. Figure 8 shows the atomic displacement vector trajectory snapshots for NC Ni after 2 ps, 100 ps, 300 ps and 500 ps time period of creep at 1210 K creep temperature and 3 GPa applied stress. It is found that the movement of the atoms mainly occurs at grain boundary region during creep deformation. The trajectory of atoms indicates that atomic diffusion is more intensified in grain boundary regions during biaxial creep deformation of ultrafine grained NC Ni. At this juncture, it can be inferred that grain boundary diffusion as well as dislocation controlled deformation both mechanisms are operative during biaxial creep deformation process ultrafine-grained NC Ni (refer Figs. 5, 6, 7, and 8).

Fig. 7
figure 7

Atomic snapshots of shear strains after different time period of creep during biaxial creep process for 3 GPa applied stress and 1210 K temperature

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Fig. 8
figure 8

Atomic displacement vector snapshots after different time period of creep during biaxial creep process for 3 GPa applied stress and 1210 K temperature

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Grain structure evolution during biaxial creep deformation process

Overall grain structure evolution during biaxial creep deformation is investigated using grain tracking algorithm (GTA) and it is represented in Figs. 9, 10, and 11. The stress dependence on grain rotation during biaxial creep deformation process is analysed in this study using the GTA. The grain rotation during creep process for all grains as a function of creep time for different applied stresses (i.e. 1 GPa, 2 GPa and 3 GPa) is represented in Fig. 9. The grain rotation for NC Ni has been measured in terms of the orientation after different creep deformation times with respect to the original orientation just before the creep process. It is observed from Fig. 9 that with increasing creep deformation process as a function of time, grain rotation for almost all grains is significantly increased for almost all cases of applied stresses (i.e. 1 GPa, 2 GPa and 3 GPa). Similar pattern of average grain rotation with respect to the number of cycle for NC material is reported in literature under cyclic deformation condition [41]. On the other hand, grain rotation curves for some specific grains such as G2, G5, G6, G12, G13 and G14 are found to be shifted downward with increasing applied stress during biaxial creep deformation process (refer Fig. 9). It is also found that the rotation magnitude of the G1, G3, G8, and G13 grains are found to be increased up to certain creep time and then decreased with the progress of creep deformation (refer Fig. 9). It can be seen from Fig. 9 that multiple numbers of grains are experienced much larger grain rotation during biaxial creep deformation.

Fig. 9
figure 9

Plots of grain rotation vs. time during biaxial creep process for each grain at a 1 GPa, b 2 GPa and c 3 GPa applied stress and 1210 K temperature

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Fig. 10
figure 10

Inverse pole stereographic projections for a 1 GPa, b 2 GPa and c 3 GPa applied stress at 1210 K temperature to show the evolution of the orientations and distribution of random texture for all grains during biaxial creep process

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Fig. 11
figure 11

Orientation map snapshots of the random NC Ni after a 0 ps, b 258 ps and c 500 ps of creep time at 3 GPa applied stress and 1210 K temperature, showing the distribution random texture for all grain during biaxial creep process

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The inverse pole figures during biaxial creep process, having grains are labelled on the basis of the creep deformation time, are represented in Fig. 10. The inverse pole figures are assisted to visualize specific types of textures. It gives the information related to the orientation of grains in the specimen with respect to reference of crystallographic directions. The inverse pole figures plots are clearly demonstrated effective grain rotation in the specimen at different applied biaxial load. During biaxial creep deformation process, all the grains contained in the specimen are tracked as a function of creep time for three different applied stresses and subsequently analysed. It can be seen that grain rotation is occurred during biaxial creep deformation process and exhibits almost distinct distribution (refer Fig. 10). It is found that all the grain rotations are clearly visualized in inverse pole figures (texture map) by displacement of each point of grains during creep deformation process at all applied stresses. It is indicated that the inverse pole figures are influenced by applied stress. Stress-induced grain rotation happens during creep process. Grain rotation during deformation is assisted as a result of atomic shuffling within the grain boundaries [40, 41, 81, 82]. It can be inferred from Fig. 10 that there is slightly altered in texture evolved with the increase in applied stress during biaxial creep deformation process. Crystallographic orientation map for NC Ni specimen after 0 ps (initial sample), 258 ps and 500 ps creep time during creep deformation process (colours code according to inverse poles for number of grains) are represented in Fig. 11. Orientation map distribution for all grains of NC Ni is parallel to the simulation Z [0 0 1] direction. Small triangular colour code, which is shown in Fig. 11 (below lower side), is used to identify the crystallographic orientation of each grain. In this study, atomic snapshots for NC Ni during biaxial creep process are analysed by GTA. Grain growth, small grain disappearance and grain boundary migration during creep deformation process are observed to be occurred simultaneously with increasing creep time (refer Fig. 11) and it is analysed using GTA. Grain growth is a common type of the structural evolution for NC materials and is reported in literature during deformation [41, 78, 83]. However, the changes of crystallographic orientation for NC Ni during biaxial creep process are achieved using GTA to extract final structure of texture.

Conclusions

In summary, molecular dynamics (MD) simulation is used for the first time to identify the biaxial creep behaviour of NC Ni for various applied stress at 1210 K temperature. Structural evaluation as well as underlying deformation mechanism is also investigated during biaxial creep process. Overall, significant findings related to biaxial creep deformation of ultrafine NC Ni are summarized below.

  1. 1.

    Duration of primary creep stage and rise in effective strain are observed to be increased with the increasing applied biaxial stress.

  2. 2.

    It is observed that comparatively longer period of secondary creep stage occurs during biaxial creep process for ultrafine grained NC Ni as compared with the primary and tertiary stages of creep.

  3. 3.

    Tertiary creep regime of ultrafine grained NC Ni is absent up until 500 ps time period for the simulated biaxial creep test performed under 1 GPa constant load. Only in case of higher biaxial load, the tertiary stage of creep is observed within 500 ps creep duration.

  4. 4.

    During biaxial creep deformation process of ultrafine grained NC Ni, grain boundary diffusion, grain sliding and dislocation governed deformation mechanisms are found to be operative.

  5. 5.

    Grain rotation in ultrafine-grained NC Ni specimen is observed during biaxial creep deformation and exhibits almost distinct distribution. It is happened due to the atomic shuffling within the GBs. It is found that there is slightly changed in texture evolved with the increase in applied stress during biaxial creep deformation.

  6. 6.

    Stress-induced grain growth for ultrafine-grained NC Ni occurs during biaxial creep deformation process and it assists due to the GB diffusion and GB sliding mechanism.

In one sentence, the scientific contribution of the work is to provide detail underlying physics of biaxial creep deformation behaviour of NC metals, which will help to determine the strategy for widening application domain of NC metals with respect to range of service temperature, stress, and most importantly state of stress.