50
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nanocrystalline and amorphous state.
For the current binary metallic system, it needs only 6 cycles for the bi-layered structure, with 5 nm in each layer thickness, to transform into a fully amorphous state. The current result on ARB is very similar to the MA case observed by Lund and Schuh [176, 177].
Meanwhile, with further rolling (to 14 ARB cycles), the amorphous structure was maintained until the end of simulation processing, though there are slight local reversions back to the partial nanocrystalline phases (i.e. slight local cyclic transformation between the 8th and 10th ARB cycle). This conclusion is also supported by the results of Fourier transformation.
Figure 5-3 (a) reveals the associated Fourier transform of the pure FCC Ni in the [111] zone axis. The Ni [111] plane texture tends to be stress induced and lie along the rolling plane. The re-crystallized Ni phase size is very close to the observed critical phase size of ~2 nm before vitrification. The corresponding morphologies at the 8th and 9th cycles shown in Figs. 5-3 (a) and (b) indicate that the crystalline structure, surrounded by circles, reappear during ARB cycles. And a phase with such a small size cannot maintain its crystalline structure upon further rolling cycles. Thus, the nano-sized FCC Ni re-states to collapse from the 9th F&R cycle and returns back to amorphous structure at the end of the 10th F&R cycle.
The MD simulated microstructural evolution over the final ARB stage and the associated two-dimensional Fourier transformation of the bi-layered Zr50Ti50 subjected to various ARB cycles is revealed in Fig. 5-4. In the Zr50Ti50 systems, the nanocystalline structure is fully vitrified after 6 and 4 F&R cycles, respectively. Unlike the observations in the Zr50Ni50
system, a faster amorphization process is revealed in Zr50Ti50, consistent with the experimental findings.
5-1-2 Morphologies of pure Zr during ARB cycles
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The microstructural evolutions of bi-layered pure Zr alloy under strain rate of 8.35×108 s-1 after second, 7th, and 8th cycles are respectively shown in Figs. 5-5, 5-6, and 5-7. The pronounced close-packed structures, namely 1421 and 1422 geometries as shown in Figs. 5-5 (a) and (b), are appearing in the matrix. Through the aid of 1321 index in Fig. 5-5 (c), it could be inferred that a number of deformation twinning or twin boundaries are produced by a simple movement of atoms as a result of shear stress parallel to the x-y plan during ARB cycles, either the occurrence of partial dislocations or stacking-fault. Figure 5-5 (d) shows the a few amorphous structures existing in the grain boundaries among different orientation domains. Those deformation mechanisms are induced by large shear load repeatedly during first and 6th cycles with increasing folding and rolling steps. However, the dis-match plans accompanied with those defects are annihilated at 7th cycle especially for amorphous structures (see Fig. 5-6) but resume happening after 8th cycle, as shown in Fig. 5-7.
While applying high strain rate at 9.25×109 s-1, the vitrification would happen quickly after a few cycles. Although one could still find some apparent close-packing structures after the finishing of first F&R cycle in the Figs. 5-8 (a) and (b), most of the amorphous structures have existed in the matrix of pure Zr, as demonstrated in Fig. 5-8 (c) and (d). Subsequently, the corresponding morphologies are found in the later ARB cycles and it is almost to be a full amorphous structure in the matrix from second to 13th cycles, as observed in Fig. 5-9, in addition to 14th cycle. In the 14th cycle, an uncertain re-crystallized Zr phase whose size approach to 2 nm is identified in the matrix, as indicated in Fig. 5-10.
5-1-3 The results of radial distribution function (RDF) calculations for Zr-Ni, Zr-Ti, and pure Zr
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Figure 5-11 presents the partial radial distribution function (PRDF) after the ARB cycles indicated for the Zr-Ni system. From the first three PRDF profiles of each pair, it is evident that the microstructure have gradually transformed from the crystalline to amorphous phase.
In the Zr-Ni alloys, the Ni atoms have fully transferred to amorphous till sixth cycles (see Fig.
5-11 (a)) but Zr atoms just need three cycles, as shown in Fig. 5-11 (b). The Zr side appears to vitrify faster is not surprising, since the Ni atoms are the more dominant moving species [63]. It appears that the FCC Ni crystals are relatively more reluctantly to lose their FCC packing nature. This trend has been observed experimentally by the X-ray diffraction and transmission electron diffraction that the FCC Ni crystal structure appears to the most stable phase and needs more ARB cycles to force it to transform into fully amorphous state [37-39].
Figure 5-12 reveals the variations of RDF from 6th to 12th cycles and their relative packing density of Zr32Ni68 alloy subjected to various F&R cycles. The RDF in Fig. 5-12 (a) exhibits sharper peaks at the 8-9th cycles, and the packing density is seen to increase as the appearance of nanocrystalline Ni phase at the 8-9th and 12th F&R cycles in Fig. 5-12 (b).
Figure 5-13 presents the partial radial distribution function (PRDF) after the ARB cycles indicated for the Zr-Ti system. Compared with Zr-Ni system, both Zr and Ti atoms need three cycles only to render the peaks of crystalline become smooth. The simulated RDF for the pure Zr at two rolling speeds is shown in Fig. 5-14. For the higher simulation rolling speed of 0.025 nm/fs (strain rate of 9.25×109 s-1), and still maintaining the ambient temperature, the RDF peaks for the ordered HCP structure is seen to gradually become more and more smooth, and eventually similar to the RDF similar to amorphous materials, as depicted in Fig. 5-14 (a).
However, for the lower simulation rolling speed of 0.001 nm/fs (strain rate of 8.35×108 s-1), the HCP structure remains nearly unchanged from 1 to even 15 ARB cycles, as seen in Fig.
5-14 (b).
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5-1-4 The average coordination number and potential energy for Zr-Ni, Zr-Ti, and pure Zr
The variations of the MD-simulated alloy potential energy (representing the atomic interaction enthalpy energy) with increasing F&R cycles are shown in Fig. 5-15. In Fig. 5-15 (a), the system potential energy for Zr32Ni68 starts to decline, or becomes more stable with unlike atom mixing, from the third F&R cycle due to the severe structure change during the transition period and becomes saturated at the sixth cycle. This results is consistent with the strongly negative mixing enthalpy of Zr-Ni (ΔHm = -49 kJ/mol) that decreasing potential enthalpy energy leads to the lower Gibbs free energy of the mixed amorphous phase.
In contrast, the potential energy variation for Zr50Ti50 in Fig. 5-15 (b) does not show the similar trend. No obvious variation of potential energy in Zr50Ti50, because of the unique characteristic of the completely dissolubility between Zr and Ti atoms together with the near zero mixing enthalpy. The variation of potential energy of pure Zr alloys subjected to different F&R cycles for the lower simulation rolling speed of 0.025 nm/fs is shown in Fig.
5-16. The potential energy of pure Zr was raised to the level that for the metastable amorphous state after finishing the first F&R cycle and has since maintained the state during ARB processes.
Figure 5-17 presents the variation of average coordination number (CN) of Zr-Ni system for every cycle. For example, the profile of the CN for the Zr-Ni pair is referred to the case that Ni is the referenced atom and Zr is the first neighbor atom surrounding Ni. The coordination numbers of all pairs in Fig. 5-17 show a continuously decreasing or increasing trend until around the 6th cycle. When the mixed microstructure becomes finer and finer, the mixing and thus the interaction between Zr and Ni atoms become stronger, leading to the
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more apparent drop of potential energy from the 4th (-4.8 eV) to 6th ARB cycle (-4.9 eV), as indicated in Fig. 5-15.
In Fig. 5-18, the profile presents the variation of average CN of the Zr-Ti pair for every cycle is referred to the case that Ti is the referenced atom and Zr is the first neighbor atom surrounding Ti. The coordination numbers of all pairs also show a continuously decreasing or increasing trend until around the 6th cycle similar to the observations in Zr-Ni system. The drop in overall system energy is not apparent in the Zr-Ti alloy system (see Fig. 5-18), as compared with the previous Zr-Ni simulation, since the mixing enthalpy of Zr-Ti is intrinsically low. The variation of average coordination number (CN) of pure Zr system at two strain rates for every ARB cycles is shown in Fig. 5-19. There are slight cyclic fluctuations in CN of the higher strain rate condition but relatively smooth for lower one.
However, the amorphous Zr shows a higher average coordinate number than crystalline.
5-1-5 The results of HA analysis for Zr-Ni, Zr-Ti, and pure Zr
For the Zr-Ni system
The more detailed local pairing variations with increasing ARB cycles for the Zr-Ni alloy are depicted in Fig. 5-20, respectively. Figure 5-20 (a) shows the evolution of the close-packed pairs of 1421 FCC-like and 1422 FCC/HCP-combined bond pairs in the Zr-Ni system. The 1421 pairs continue to decrease from the initial 54% to 25% until the 4th cycle and 1422 pairs does not reduce their population as fast as 1421 over this stage but also drop from 25% to 20%. Above the 5th cycle, both the 1421 and 1422 pairs appear to drop and then scatter at a low value of ~5%. It appears that the close-pack structures cannot be sustained in the Zr-Ni alloys after the 5th cycle.
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In Fig. 5-20 (b), the 1541 icosahedra-defect and 1431 FCC-defect pairs in the Zr-Ni system have a finite amount existing in the first stage (~10% and 5%, respectively) and continue to rise or scatter to 15% and 20%, respectively. The 1541 icosahedra-defect pairs show a particularly wide scattering during the intermediate stage, from the lowest of 8% to the highest of 25% and with an overall average of ~15%. In contrast, the 1551 icosahedra pairs continue in their evolution from the initial less than 1% to 8% at the 4th cycle, and to 30% at the later stage. It seems that the icosahedra-defect and FCC-defect local structures are easier to form during the initial stage of the crystalline-to-amorphous transition processes.
From the above evolutions, it appears that the 1541 icosahedra-defect pairs are the intermediate transition atomic arrangement, and would transform gradually into the 1551 icosahedra pairs. The fluctuation between 1541 and 1551 pairs tend to approach to a more steady state till the 9th cycle, thus 1551 icosahedra pair structure is the more stable atomic local arrange in the current amorphous phase of the Zr-Ni system. Furthermore, an interesting phenomenon is evident in Fig. 5-20 (c) that the BCC related 1441 pairs will increase and then decrease during the amorphization transition processes. The other BCC type 1661 pairs and the rhombohedra-related 1321 pairs are also found, but the fractions of 1661 and 1321 pairs are consistently present with a low value ~5%.
For the Zr-Ti system
The detailed local pairing variations with increasing ARB cycles for the Zr-Ti alloys with an ARB rolling speed of 0.025 nm/fs are depicted in Fig. 5-21. The pairing variation of Zr-Ti shows basically smooth and steady evolution during the ARB cycles. The close-packed initial structures characterized by the 1421 or 1422 pairs continue to decrease from 50% to
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less than 10% in Fig. 5-21 (a). The overall percentage adding both 1421 and 1422 is about 20%. In contrast, the icosahedra and icosahedra-defect 1431, 1541 and 1551 pairs, characteristic of the amorphous phase increase lastingly from initial 0% to an overall 60% in Fig. 5-21 (b) for the Zr-Ti system. As shown in Fig. 5-21 (c), the BCC-related 1441 and 1661 pairs consistently occupy minimum amounts; while the icosahedra-accompanied 1321 pairs smoothly increase to 20%. The evolution sequence as a function of ARB cycles exhibits a simple trend. There seems no intermediate atomic pairing formed during the transformation from the HCP to icosahedra (i.e., the quasi-amorphous) structure, unlike the Zr-Ni system where a BCC-like intermediate pairing was seen.
For the pure Zr system
The detailed atomic pairing evolution of the pure Zr during ARB at two simulation speeds are shown in Figs. 5-22 and 5-23, respectively, for the higher and lower speeds. For the case of pure Zr at the high rolling speed, as shown in Fig. 5-22, the HA evolution trends resemble the trends seen in the Zr-Ti binary alloy. From Fig. 5-22 (a), it can be seen that the overall close-packed and ordered 1421 plus 1422 pairs continue to decrease from the initial 100% to overall 40% after 13 ARB cycles. At the same time, the amorphous natured icosahedra and icosahedra-defect 1431, 1541 and 1551 pairs increase continue to from the initial 0% to an overall 45% in Fig. 5-22 (b). The rest minor 1321, 1441 and 1661 pairs consistently occupy an overall 10% in Fig. 5-22 (c). The above result for pure Zr was simulated at a faster rolling speed of 0.025 nm/fs, the situation for the pure Zr at the lower simulation rolling speed of 0.001 nm/fs reveals a different story.
From the RDF curves shown in Fig. 5-14 (b), the HCP structure is resistant to be altered.
This can be confirmed from the extracted HA index, shown in Fig. 5-23. Now, the overall
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close-packed and ordered 1421 plus 1422 pairs decrease from the initial 100% to only overall 70% after 15 ARB cycles, as presented in Fig. 5-23 (a). It means that the crystal structure of the pure Zr after 15 ARB cycles is still predominantly HCP. The amorphous-like icosahedra and icosahedra-defect 1431, 1541 and 1551 pairs together can only occupy 20%, as shown in Fig. 5-23 (b). The rest 1321, 1441 and 1661 pairs are basically negligible. The amorphization degree is very limited in this case.
5-2 The simulation results of vitrification transformation in Mg-Cu Thin Film
5-2-1 Morphologies of Mg-Cu during different temperatures
A series of atomic position projections in the interfacial layers during the structural evolution, corresponding to at 300, 413 and 500 K, are illustrated in Figs. 5-24 (a), (b), and (c), respectively. An inter-diffusion behavior occurred in the interfacial layers caused by the energy difference between Mg and Cu, resulting in the disorder-like structures in the Mg-Cu interfacial layers. The Mg and Cu atoms far away from their interfaces, which are not presented in the figure, are still retained the crystalline state during the processes. Compared with the Mg atoms which display a relatively uniform distribution in the interfacial layers, the Cu atoms are seen likely to congregate together forming a network distribution due to their higher binding energy than that of the Mg atoms, as shown in Fig. 5-24 (a). Subsequently, the regularly ordered clusters of the Cu atoms are gradually formed in the Mg matrix and approach toward the Cu-rich sides in Figs. 5-24 (b) and (c). The interfacial diffusion originated in the Mg/Cu disordered interlayer, however, does not further grow into their crystalline matrix even when the simulation is finished.
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5-2-2 The results of density distribution profiles for the Mg-Cu
To define whether the amorphous structures have been formed in the interfacial layer in this sandwich model, the density profile of each species α along the z direction ρα(z), which can measure the number of α atoms within a certain distance interval along the z axis, is calculated to describe and quantify the mixing circumstances at each of interfacial region in this model. The density profiles ρMg(z) of the Mg atoms (red line) and ρCu(z) of the Cu atoms (green line) are shown in Fig. 5-25 for three different simulation states corresponding to Fig.
5-24. Figure 5-25 shows that the density profiles of the Mg and Cu atoms merge at the interfacial region, indicating the occurrence of mixing phenomena on the atomic scale. Some different species atoms in the interfacial region diffuse, indeed, in the areas over each other.
The intermixing of Mg and Cu atoms at the interfacial region is not accompanied with the obvious local loss of crystalline order, although the thickness of the mixing areas present a well mixed and smooth distribution with increasing temperature.
5-3 The simulation results of cyclic loading fatigue in Zr-Cu binary amorphous alloy
5-3-1 The simulation results for Zr50Cu50 under different cooling rates
The volume versus temperature curves at different cooling rates, 1, 2.5, 5, and 10 K/ps obtained by quenching the sample model from 2000 to 300 K are shown in the Fig. 5-26. The glass transition temperature can be identified through the changes in the volume slope during cooling process. Figure 5-26 reveals only a slight difference among the Tg points (around 650 K) for four different quenching rates. The potential energies at their as-quenched state are about -5.237, -5.235, -5.233, and -5.232 eV and the average densities are 7.294, 7.293, 7.29,
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7.287 g/cm3, respectively. Because the difference of densities between 1 K/ps and 10 K/ps is 0.096%, the density is considered insensitive to the cooling rate for the Zr-Cu system. The similar trend is observed in their respective RDF curves. The PRDF of cooling rate 5 K/ps is shown in Fig. 5-27. The distinctly splitting phenomena occurring for the second peak among the gCu-Cu(r), gZr-Zr(r),and gCu-Zr(r) curves of PRDF are similar to the results of Duan et al.
[156], in which an amorphous phase was claimed to be produced under a cooling rate of 5 K/ps. For referring to literatures of other Zr-Cu simulations, the results for cooling rate of 5 K/ps would be used as the initial state for subsequent mechanical test simulation.
5-3-2 The mechanical properties of monotonic tension tests at different strain rates
Variation of potential energy
Figure 5-28 reveals the variations of potential energies for three conditions: one is using 5 × 109 s-1 strain rate to reach to strain of 20% (case 1), the second is using 5 × 109 s-1 to reach to strain of 30% (case 2), and the other is using 2.5 × 109 s-1 to reach 20% strain (case 3). The energies of case 1 and 2 increase from -5.233 to -5.227 eV when their reaching to 20% strains and that of case 2 keeps on rising to -5.226 eV at strain of 30%. The energy of case 3 increases from -5.233 to – 5.228 eV when reaching 20% strain. Apparently, high strain rate would induce a large steep slop of potential energy curve and a high energy state in Fig.
5-28. All three energy curves alter their slope to become smooth after passing through the 5%
strain.
Variation of density
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The density variations of three strain conditions are shown in Fig. 5-29. The densities of three cases decreases rapidly with increasing their strains from as-quenched state (7.29 g/cm3 ) but approach to the respective saturation states after passing through 5% strains. The density in case 3 (7.26 g/cm3) is slightly higher than those of cases 1 and 2 (7.255 g/cm3). The decrease in densities from the initial 7.29 g/cm3, is the evidence that the free volumes in three Zr-Cu amorphous models would increase during deformation. The amount of increase in total free volume in the simulation models is about 0.4% for case 3 and 0.48% for cases 1 and 2.
Stress-strain curves and local atomic strains
The stress versus strain curves for the above three conditions are displayed in Fig. 5-30.
The σy is measured 2.2 GPa for the case 3 and about 2.4 GPa for the case 1 and 2, respectively. The Young’s modulus for Zr50Cu50 metallic glass is fitted to be about 70 GPa, and the Poisson’s ratio is about 0.43. There is a strain softening phenomenon appearing in the three cases after occurring yielding. Figures 5-31 (a) to (c) show the excessive atomic strain distribution of a slice parallel to the yz plane of the Zr-Cu amorphous alloy during and after plastic strains of 5, 10, and 30% under the monotonic tensile mode at strain rate of 5 × 109 s-1. The strain of 5% corresponding to the position of σy on the strain-stress curve in Fig. 5-30 (a) shows that the initiation of STZs were induced by the irreversible local atomic strains (the green colors in Fig. 5-31) in the matrix and the total macroscopic strain of model has not depended on those irreversible local atomic movements yet. When the deformation of model gets into plastic deformation range, the STZ groups grow quickly and stochastically but not show an apparent shear localization (even under 30% plastic strain), as show in Figs. 5-31 (b) and (c).
The development of STZs in the condition of case 3 (see Fig. 5-32) is consistent with the
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observation in Fig. 5-31, and net-work organization composed of individual STZ groups is clearly indicated in Fig. 5-32 (c). The distribution of both figures of local atomic strains indicates no shear band forming in the current models even though they are applied under strain loading of 20 and 30%.
5-3-3 The results of structural analysis for different cyclic loading conditions
After 100 cycles of the tensile fatigue test under the stress-control mode at σmax = 2 GPa, their PRDF curves for the Zr50Cu50 specimen in Fig. 5-33 did not show an apparent change during the whole deformation process, still similar to the as-quenched PRDF in Fig.
5-27. The case of the more severe strain-control tension-compression fatigue (even to 10%) still shows the same results. These results are interesting, since the specimen has been subjected to the severe local shear deformation under numerous fatigue cycles. The PRDF results indicate that structures of the Zr50Cu50 amorphous alloy still maintains in the liquid-like state, at least from the viewpoint of a medium range order, even when the alloy is subjected to the more severe strain-control tests.
The different HA indices for the compression and tension cyclic loadings under 1 GPa are shown in Figs. 5-34 (a)-(c) and 5-35 (a)-(c), respectively. Clearly, both the close-packed and liquid-like indices have an apparent fluctuation with the cyclic loading but do not show any continuously decreasing or increasing trend (like the trend shown in Fig. 5-20) throughout cycling, irrespective of the tensile or compressive mode. This trend means that the local structures with short-range ordering were indeed changed in response to the cyclic loading, but could be relaxed back to the amorphous state upon unloading or reversed loading.
These detailed HA-pair analysis shown in Figs. 5-34 and 5-35 explains why there is no major
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change in the PRDF curves of Fig. 5-38. Although shear deformations are accumulated with increasing cyclic loading, it does not cause permanent structure transitions in the current Zr-Cu amorphous alloy. Similar behavior is also seen in Fig. 5-36, for the HA evolution in the most severe strain-control mode at εmax = 10%.
However, few crystal-like atomic clusters, such as close-packed or B2 structures, are indeed detected during the simulation process, as shown in Figs. 5-37 (a)-(c). Figures 5-37 (a) and (b) record the superimposed projections of close-packed atoms in the compression and tension at σmax = 1 GPa during cyclic loading process and Fig. 5-37 (c) show those of the B2 atoms during the stress-control tensile mode at σmax = 2 GPa. A blue particle indicates a FCC atom composed of its 12 near-neighboring atoms following -A-B-C- proper stacking sequence as shown on the top left corner of Fig. 5-37 (a), and a HCP atom is identified as green particle in the same figure similar to above method. The superimposed projections of B2 cluster is shown in Fig. 5-37 (c).
It is found that these crystal-like atoms always appeared near or within the shear transition zones (STZs), but not necessarily appeared at the places with the highest local strains. The appearance of the close-packed or B2 atoms is very interesting, because the formation of crystal-like clusters can be regarded as the precursors for the subsequent crystallization or the candidate of crystal nuclei [178]. Unlike the nucleation in quenched liquids, such ordered atoms in the simulated specimen did not congregate together to form larger clusters or keep on growing in the amorphous matrix. Instead, they simply appeared as an unstable state, and would appear and, then, disappear during cyclic loading. Throughout the simulation, such crystal-like atoms occurrences are rather rare, and did not have the ability to grow and form larger crystalline nuclei during the course of cyclic loading. The occasional occurrence of unstable local structures cannot result in the fatigue-induced
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crystallization in the current Zr-Cu amorphous alloy. This finding is in agreement with the fatigue studies of Zr-based BMGs.
5-3-4 Potential energies for different cyclic loading conditions
The comparison of potential energies between tension and compression of cyclic loading under stress-control at 1 GPa is shown in Fig. 5-38. Both energy curves show the continuously decreasing tendency with the evolution of time steps in Fig. 5-38 (a) that indicates the cyclic loading will influence Zr-Cu metallic glass going to a new steady state from the as-quenched condition. Although both decreasing amplitudes for tension and compression are very slight, the values of potential energy between tension and compression are approximately coincidence which indicates the same route occurring in this energy change between them. In Fig. 5-38 (b), the amplitude of potential energy of tensile mode has a higher fluctuation than that of compressive mode shows a better loading endurance for the compressive mode than the tensile one.
Figures 5-39 (a) and (b) are similar comparison of potential energies for the stress-control at 2 GPa. The potential energy for tensile mode is raised to - 5.231 eV from as-quenched state (- 5.233 eV) upon the first tensile loading cycle as a result of the stored deformation energy but gradually relaxes to - 5.235 eV at the end upon unloading (only 0.04% decrement as compared with the as-quenched state) with continuous cycling loading, as seen in Fig. 5-39 (a). Similarly, the amplitude of energy fluctuation for the compressive mode is smaller than that for the tensile mode but more evident than that for the case of stress-control at 1 GPa.
The comparison of potential energies between long and short periods for the cyclic
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stress-control at 2 GPa is shown in Fig. 5-40. The time steps for relaxation during the cyclic loading process for the case of long period is twice as much as the case of short one, as seen in Fig. 5-40 (b). Form Fig. 5-40, the variation of potential energy for the case of long period has the same way with short one, but just needs fewer loading cycles. In Fig. 5-40 (a), both energy curves show a change of their slope at the point corresponding to 500,000 time steps, and then become gradually saturated in later stages.
The comparison of potential energies between tension and compression of cyclic loading under strain-control at strain rate of 2.5 × 1010 s-1 is shown in Fig. 5-41. In Fig. 5-41 for the strain control mode, both energy curves show more apparent fluctuations due to the more severe strain amplitude. The energy suddenly increases to -5.224 eV for tensile and -5.226 eV for compressive mode upon the first loading cycle, and then gradually recovers to -5.234 eV as well as -5.232 eV, respectively, after cycle 12th and then keeps on this steady-state till the end upon unloading (only 0.02% decrement as compared with the as-quenched state). Unlike the observation on the stress-control cases, the energy curve of compressive mode reveals a higher value than tensile which indicates an more stable state could be easy raised under tensile mode.
Figure 5-42 is the comparison of potential energies of cyclic loading under strain-control among three different strain ranges. The loading condition of strain of 4% could cause larger fluctuation of energy amplitude than that of 2.5%, but the energy level of them is basically at the same range during the process of cyclic loading. In contrast, the energy state would be raise to a high level entirely when the loading condition for the strain-control was stretched to 10%.
5-3-5 Variations of density (dilatation) analysis for different cyclic loading
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conditions
The results of density variation for the stress-control mode at σmax = 1 GPa under tension and compression are shown in Figs. 5-43 (a) and (b). Generally, the average of density variation in Fig. 5-43 (a) does not show the pronounced change during the cyclic loading regardless of cyclic tension or compression. The density curves will increase or decrease with applying compressive or tensile stress but return to original density level when the applied stresses are released, as indicated in Fig. 5-43 (b). While applied tensile stress reaches to σmax = 1 GPa, the density will decrease from 7.29 g/cm3 to 7.30 g/cm3 (about 0.14% decrement as compared with the as-quenched state). Similar behavior is shown in the compressive mode instead of increment in density.
While the maximum stress is, respectively, added to σmax = 2 GPa for the tensile and compressive mode, both the amplitude of density variation for two these modes in Fig. 5-44 are raised higher than that for the maximum stress at 1 GPa in Fig. 5-43. The amplitude of density fluctuation for the tensile mode at σmax = 2 GPa is about twice of that at σmax = 1 GPa.
For the compressive mode at σmax = 2 GPa, the amplitude of density fluctuation approach to triple of that at σmax = 1 GPa. However, the variation of average density does no show the difference with the results of σmax = 1 GPa. Compared with the results of density variation between two loading periods in Fig. 5-45, there is almost not any difference existing between long and short loading periods under the tensile mode at 2 GPa.
Figure 5-46 is the comparison of density variation under strain-control mode at εmax = 10% between tension and compression. The density for the tensile mode shows a greater decrease to 7.24 g/cm3 at the first cycle but then gradually increase to 7.32 g/cm3 at the end upon unloading (about 0.4% increment), suggesting a slight decrease in free volumes via