Chapter 2 Bulk Metals
2.8 Stability of tensioned structure with additional transverse deformation
In Chapter 2.5, we have obtained the critical strains of instability by elastic and phonon criteria, 0.11, 0.10, and 0.08, for Cu, Ag, and Au structures under uniaxial tensile loading, respectively. For further study, to observe the effect of transverse loading with the constraint of the uniaxial tensile loading along the [100]-direction, we have focused on the stabilities of structure with loading condition, as shown in Figure 2.16a. In this Section, we have calculated whether the trend of instability under tensile loading along the x-direction varies by additional loading in the transverse directions ([010]- and/or [001]- directions).
First, in Figure 2.16 (elastic stability conditions) and Figure 2.17 (phonon stability), elastic and phonon stability conditions for FCC Cu, Ag, and Au structures are shown. Here, we applied different transverse loading to structures that are under pre-imposed uniaxial tensile loading in the x-direction around 10%, 9%, and 7% of tensile strain for Cu, Ag, and Au structures, respectively. We note that the resulting tensile strains from the pre-imposed uniaxial tensile loading are slightly smaller than the critical strains where the structures lose their stability. Under symmetrical transverse deformation () along the [010]- and [001]-directions, the elastic stability conditions, from equation (1.50) to (1.54), are utilized for tetragonal symmetry. We observed that the elastic stability condition term of T2 turned from positive to negative value at –0.0341 (–0.0311 before applying additional transverse strain), –0.0308 (–0.0298), and –0.0284 (–0.0274) of transverse strains by the increase of compressive transverse deformation for Cu, Ag, and Au bulk, respectively. In contrast, as the tensile transverse strain increases, the elastic stability condition term (T2 ) increases for Cu, Ag, and Au. For other elastic stability conditions, transverse loading from additional compression to tension results in the decrease of stability excepting the elastic stability condition term of T5 (almost constant) for Cu, Ag, and Au.
Using phonon stability for uniaxial tensile loading with additional external loading in the [010]- and [001]-directions, Figure 2.17a-c presents the trend of frequency with different wave vectors. With the
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increase of compressive transverse loading, the phonon branches become softer, indicating decrease in frequencies (in THz). We found that one of acoustic phonon is negative value at –0.0421 (–0.0311 before applying additional transverse strain), –0.0428 (–0.0298), and –0.0484 (–0.0274) of transverse strains for Cu, Ag, and Au, respectively. For other possible wave-vectors of phonon tendency (they would be not presented here.), the instability would be not found under compressive transverse strain.
As applying the tensile transverse deformation, for all phonon branches of Cu, Ag, and Au, the instabilities would be initiated along wave-vector q(x x 0)-q(0 0 0) shown in the right side of Figure 2.17a-c. For both criteria, under 10%, 9%, and 7% of tensile strains along the x-direction to Cu, Ag, and Au, respectively, the instability criteria could occur due to additional compressive deformation along the [010]- and [001]-directions even before the critical strain of instability (0.11, 0.10, and 0.08 for Cu, Ag, and Au, respectively) due to uniaxial tensile loading. In this case, the instability by elastic criterion predicted unstable configuration, under loading condition shown in Figure 2.16a, earlier than by phonon dispersion in terms of transverse strain. In addition, these enhanced stabilities of materials as applying additional strain to transverse directions could change the ideal strength of Cu bulk as shown in Figure 2.17d. In case of Cu bulk under 10% of tension along loading direction, the ideal strength is changed from 9.49 to 20.17 GPa with the increase of transverse tensile stress. We show that the ideal strength of material could be increased up to approaching theoretical strength by transverse loading condition.
As same way, under 11%, 10%, and 8% of tensile deformation, which is previously obtained from uniaxial loading condition (primary path), the effect of transverse is checked as shown in Figure 2.18 and 2.19, using elastic and phonon stabilities. For elastic stability conditions, we obtained the transverse strains for instability around –0.0305 (–0.0335 before applying transverse strain), –0.0273 (–0.0323), and –0.0276 (–0.0306) for Cu, Ag, and Au bulk, respectively, based on the term of T2 shown in Figure 2.18. It means that under 0.11, 0.10, and 0.08 of tensile in [100]-direction for Cu, Ag, and Au, respectively, additional tensile transverse deformation recovers the stability of structure. Also, using phonon stability, the critical strains are around –0.0305 (–0.0335 before applying transverse strain), –0.0273 (–0.0323), and –0.0286 (–0.0306) for Cu, Ag, and Au, respectively, shown in Figure 2.19a-c.
Compared to those results between before and after applying additional tensile transverse deformation, the phonon instability is also enhanced. As mentioned above, the transverse strain also affected the stability of configuration, indicating that as the amount between loading and transverse direction increase, the status of material stability could be changed. So far, we have seen that as applying tensile transverse loading under uniaxial tensile loading along the [100]-direction (that is, the signal of strains is same between loading and transverse directions), the stability of structure is able to be transited from unstable to stable configuration in terms of elastic and dynamic criteria.
As discussed in Section 1.2.1, according to Hill’s study, the bifurcation is the interesting phenomenon
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resulting from the violation of Hill’s criterion, which is the elastic stability condition term of T3 shown in Eq. (1.52). We have investigated the change of elastic stability condition terms under asymmetrical loading along the transverse directions as already applied tensile deformation around 10%, 9%, and 7%
along the x-direction, before the initiation of instability induced by uniaxial tension, as shown in Figure 2.20a-d. To introduce the Hill’s theory under the loading condition (Figure 2.20a), the elastic stability condition terms for orthorhombic symmetry are expressed as
1 11> 0
O =B (2.5)
2 2 11 22 12 > 0
O =B B −B (2.6)
2 2 2
3 = 11 22 33 + 2 23 12 13 11 23 22 13 33 12 0
O B B B B B B −B B −B B −B B (2.7)
4 44 0
O =B (2.8)
5 55 > 0
O =B (2.9)
6 66 > 0
O =B . (2.10)
As the asymmetrical transverse loading increases ( : the amount of applied strain in transverse directions to structure, respectively), before the instability occurs due to tensile loading along the [100]- direction, there is the transition from stable to unstable configuration. The structures become unstable at 0.0210, 0.0410, and 0.0110 of the additionally applied transverse strains to Cu, Ag, and Au bulk, respectively. As shown in Figure 2.20b-d, we have observed that the one of elastic stability condition terms, O3 = B B B11 22 33+ 2B B B23 12 13−B B11 232 −B B22 132 −B B33 122, reaches zero, as increasing the amount of asymmetrical transverse loading (), indicating unstable structure. In other words, after structural change from a tetragonal to an orthorhombic configuration, the structures become unstable in terms of elastic stability condition term O3. Additionally, for Au, the other elastic stability condition term of O5 is also transited to unstable state around = 0.3 along the transverse directions, respectively.
However, by using phonon dispersion shown in Figure 2.21, the structure shows stable configuration along q(0 x x)-q(0 0 0)-q(x x 0) pathway in wave vector. Here, we can see the both criteria show different results. In case of asymmetrical transverse deformation with tensile loading in x-direction which instability occurs (0.11, 0.10, and 0.08 of tensile strains to Cu, Ag, and Au bulk, respectively), Figure 2.22 and 2.23 show the elastic and phonon instabilities, respectively. Obviously, in Figure 2.22a-c, the elastic condition term of O3 becomes zero. Even though the increase of transverse deformation to reach phase transformation from a tetragonal to an orthorhombic symmetry as predicted by Hill’s theory, the structures are still unstable. In contrast, in phonon dispersion shown in Figure 2.23a-c, the stability
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of structure is enhanced with the increase of transverse loading, which becomes stiffer in phonon branch, around 0.001, 0.011, and 0.011 of strains for Cu, Ag, and Au, respectively. These results show that after phase-transformation to be another stable configuration, the both criteria could differently predict the status of stability.