Cubic materials under the asymmetric transverse stresses can exhibit another interesting phenomenon: negative Poisson's ratio along principal directions. In addition to the study of failure modes in nanowires, we investigate the effect of the cross-sectional shape on Poisson's ratio between the nanowires. 140 3.10 The lateral strains εy and εz as functions of the applied strain of Au bulk at different.
Introduction
Scientific topics
- Elastic stability of bulk materials
- Surface stress calculation
- Nanoscale materials
According to this criterion, the stability state of a structure is solely a function of the material property. The Hill and Milstein condition for the stability of a crystal is the general case of the Born condition. The elastic stiffness was introduced by Wallace in the study of stress propagation28, is used for predicting the stability of crystal structure instead of the elastic constantC.
Research objective and organization of this dissertation
In Section 3.2 we investigate how the ideal strength of the materials changes with different loading methods. In Section 3.3, the effect of asymmetry of superimposed biaxial stresses on Poisson's ratio along principal directions of the materials is given. In addition, we present how the cross-sectional shape of the nanowires affects the mechanical properties.
Elastic Stability Theory and Simulation Methods
Elastic stability theory
The system in an arbitrary deformed state X undergoes a deformation with a deformation gradient F, it is moved to state x. If the system is deformed again from state x to state x' by the green strain Eij'. In addition, the system changes under load and becomes unstable if the condition (2.22) is broken or detB 0.
Atomic simulations
- Molecular dynamics simulation
- MD simulations with other ensembles
- Molecular statics (MS) simulation
- Interatomic potential models
For example, Berendsen thermostat78 rescales volume of the system is rescaled at each time step with a proper time scale. However, extended Lagrangian method is more reliable because it can capture the dynamics of the system. In an EAM potential model, the total energy of the system, Etot, is given by.
Stress and elastic moduli calculations
In this study, mainly EAM potentials are used, and we use the EAM potential developed by Foiles et al.87, the EAM potential developed by Cai and Ye91, and the EAM potential developed by Liu et al. While typical EAM potentials are usually generated in tabular form, the EAM potential of Cai and Ye provides an analytical form of the potential energy, so that the calculation of elastic moduli by Eq. 2.80) can be applied more easily. In the second method, we calculate the elastic stiffness of a structure at finite strain using Eq.
Bulk Materials
Cubic materials under uniaxial loading along [100]-direction
- Introduction
- Simulation methods
- Uniaxial tensile stress
- Compression
- Temperature effects on elastic instability of Au
- Summary
The non-positivity of incremental voltages (as shown in Figure 3.2, voltage drop at the load of 0.77) can be considered as a signal of instability. Therefore, the critical instability strain of Au under uniaxial compressive strain along [100] direction is 0.085. This is why the stress-strain curve in Figure 3.6a is smooth and the slope of the curve at the critical instability strain is 0.
Ideal strength of cubic material
- Introduction
- Simulation details
- Symmetric loading
- Asymmetric loading
We further investigated the effect of the symmetry of the transverse stresses on the ideal strength of the Au crystal. As shown in Figure 3.19, it is clear that the asymmetry of the transverse stresses lowers the ideal strength of the material. The ideal strengths of the six FCC materials depend mainly on the shear stresses.
Negative Poisson’s ratio in cubic material
- Introduction
- Simulation method
- Loading condition: xx 0 , yy const , others 0
- Loading condition: xx 0 , yy const , zz k yy , others 0
- Other materials
- Summary
Poisson's ratios in the z direction as a function of the applied strain in the x direction are shown in Figure 3.23a. Even with higher tensile or compressive stress in the y direction, Poisson's ratio does not become negative. Our results reveal another interesting property of cubic materials, namely the negative Poisson's ratio in the principal directions.
Surface Stress Calculation at Finite Strain
- Introduction
- Surface stress calculation
- Surface stress of Cu, Ag and Cu (001) surfaces under uniaxial strain condition
- Discussions
- Conclusions
Therefore, taking the first derivative of Eq. 4.1), the surface tension can be rewritten as The surface tensions of the three (001) metal nanoplates at different deformations are shown in figure 4.3. The component of the surface tension along the lateral direction (f22) is linearly proportional to the applied strain in the Cu (001) nanoplate, but is almost constant in the Au (001) nanoplate.
Unlike f22, the surface tension component along the deformed direction (f11) changes slightly in Cu (001) nanoplates, but drastically decreases in Au (001) nanoplates as the strain increases, e.g. For the Ag (001) nanoplate, the surface tension behavior is between that of Cu and Au nanoplates. Due to the greater relaxation in the Au nanoplate, the surface tension on the Au nanoplate changes more during relaxation than on the Cu nanoplate.
The shortening of the distance between the first and second layers with increasing load causes the component of the surface tension along the stressed direction (f11) to decrease significantly, while the component of the surface tension along the in-plane lateral direction (f22) remains constant . On the other hand, the surface tension component in the stretched direction of the Cu surface is less sensitive to strain than the Au surface. The surface tension behavior of the Ag surface is between that of the Cu and Au surfaces.
However, for Au (001) surfaces, the atoms in the first and second layers move in opposite directions and become closer as strain increases, leading to the different response of the surface tension to strain.
Nanoplates
Negative Poisson’s ratios in metal (001) nanoplates
- Introduction
- Simulation methods
- Negative Poisson’s ratio in Al (001) nanoplates
- Surface effects on Poisson’s ratio
- Poisson’s ratio in other cubic (001) nanoplates
- Comparison with other surfaces
- Discussion
In addition, the initial stress at which the Poisson's ratio becomes negative in the thickness direction also changes: it increases as the thickness increases. In bulk, the Poisson's ratios in both the thickness and the in-plane lateral directions are equal and positive. The Poisson's ratio converges to about 1.0 in the in-plane lateral direction and about -0.3 in the thickness direction for large strain (>9%).
Greater compressive stress induces greater changes in Poisson's ratio until they reach extrema. Poisson's ratio in the thickness direction of the Fe (001) nanosheet with a thickness of 3a0 becomes negative at ca. a strain of 2.9%, while bulk Fe (001) never shows a negative Poisson's ratio. The change in Poisson's ratio of silicon (Si) (001) nanoplates with strain is illustrated in Figure 5.10.
Although bulk Si (001) has the same cubic symmetry as FCC and BCC metals, and thus undergoes a definite phase transformation under uniaxial stretching in the [100] direction, Si nanoplates do not show the same drastic change in Poisson's ratio . It is noteworthy that Si (001) nanoplates also exhibit a negative Poisson's ratio when we deliberately applied a large compressive stress (> . 1GPa) in the in-plane lateral direction. It has been reported that 69% of the cubic metals have a negative Poisson's ratio along the [110] direction, when stretched along the [110] direction31,32.
We show that FCC (001) and BCC (001) nanoplates have a negative Poisson's ratio when stretched (FCC) or contracted (BCC).
Controlling auxeticity of metal nanoplates
- Introduction
- Simulation details
- Effects of thickness
- Effect of mechanical stress
- Effect of temperature
- Poisson’s ratios of (001) nanoplates of different metals
- Summary
First, we examined the effects of nanoplate thickness on the behavior of Poisson's ratios. Second, we investigated the effects of mechanical loading on Poisson's ratio behavior. Third, we studied the effects of temperature on the behavior of Poisson's ratios of metal nanoplates.
Therefore, in principle, one of Poisson's ratios (eg νxy) approaches positive infinity, while the other (eg νxz) approaches negative infinity at the critical instability. The Poisson's ratio of the nanoplate does not change slightly under load or sharply at the critical instability load. The behavior of the Poisson's ratio for Au (001) nanoplates with different thicknesses is compared in Figure 5.13b.
This strong dependence of the Poisson's ratios on the thickness arises from the effects of the surfaces of the nanoplates. In this section, we investigate the effects of mechanical stress in the lateral direction on the Poisson's ratio behavior of metal nanoplates. Therefore, surface tension cannot explain the linear dependence of the Poisson's ratio on temperature.
The differences in the Poisson's ratios of the different metal nanoplates were also significant, even though their bulk values are relatively similar.
Nanowires and Nanotubes
Mechanical failure modes of FCC [100]/(001) nanowires
- Introduction
- Calculation details
- Global deformation
- Global deformation versus local deformation
- Transient temperature
- Discussions
Consequently, the cross-section of the bulk becomes rectangular at the onset of instability, as shown in Figure 6.4a. All investigated metallic nanowires showed the same global deformation as their failure modes. Then a number of questions arise: why do the simulations show global deformation as the failure mode instead of local deformation as many other reports?
This means that, at low temperature (~10 K), the nucleation stress of the dislocations at the ends of the nanowires lies in a similar stress range for the global deformation. First, at 0.01 K, the circular nanowire failed with a global deformation at the critical strain of 0.108, which was the same as the critical strain of the square nanowire as shown in Figure 6.6. We believe that the elastic modulus of circular and square nanowires are almost the same at a given temperature and load.
As discussed earlier, the nucleation stress of dislocations on the edges of a nanowire is lower than on the surfaces of the nanowire. At low temperatures (< 12 K), the global deformation takes place first and thus the mechanical properties of the nanowire change significantly during the deformation. On the other hand, it is noteworthy that the transient temperatures of the square Au nanowires were extremely low (<15 K), whereas the temperatures of the square Ni nanowires were much higher (~120 K).
At near-transient temperatures, nanowires fail due to the simultaneous occurrence of global and local deformations.
Designing metal nanowires with negative Poisson’s ratio
- Introduction
- Methods
- Rectangular nanowires
- Stress fields in rectangular nanowires
- Hollow nanowires
- Negative Poisson’s ratio in other nanowires
- Conclusions
As a result, there is a negative Poisson's ratio along the out-of-plane direction of the metal (001) nanosheets. Poisson's ratio components of a square nanowire are larger than that of the bulk counterpart (0.46) due to surface effect. The tensile zone is quite important for the authenticity of rectangular nanowire and it is a unique property of the rectangular nanowire.
Therefore, even with an aspect ratio of about 2, the Poisson's ratios of rectangular nanowires are comparable to those of "rectangular nanowires" of the same thickness with an aspect ratio of infinity (nanoplate). As shown in Figure 6.17, the slopes of the contour lines of y are quite large. We illustrate the design by considering the change of Poisson's ratio of nanotubes with a = 19.6 nm, b = 9.8 nm and different c and d.
The structure with the voids in these studies may show a negative Poisson's ratio due to rotation of part of the structure. We compare the Poisson's ratio behavior of six RMNTs of the following six metals: Cu, Ag, Au, Ni, Pd and Pt. However, the Poisson's ratio of the materials when they are at the nanoscale are different from each other.
All Poisson's ratio components xzof the various materials are reduced when they are at the nanoscale.
Conclusions
Changes of ideal strengths under asymmetric transverse loadings
Lattice parameters of a single Al crystal under various transverse loadings
Poisson’s ratios of a single Al crystal under various transverse loadings
Phase transformation for a single Al crystal under uniaxial stress in the [100] direction
Negative Poisson’s ratio in a single Al crystal under multi-axial loadings
Poisson’s ratio of a single Al crystal for various loading directions
Poisson’s ratios of six FCC metals and a single Si crystal under constant transverse
Comparison of MS and DFT results for single Cu and Si crystals
Schematic diagram of crystallographic orientations and loading direction of a metal
Bulk stress vs. bulk strain curves for the Cu, Ag, and Au nanoplates
Mechanical response of surface stress to applied strain
Bulk stress profile along the thickness direction
Movement of atoms in the two outermost layers during relaxation
Schematic diagram of crystallographic orientations, loading direction and compressive
Poisson’s ratio of Al (001) nanoplates
Phase transformation for bulk Al (100) under uniaxial loading
Stress distribution of Al (001) nanoplates
Relationship between compressive stress and the thickness of Al and Au (001)
Effect of compressive stress on Poisson’s ratio
Comparison of DFT and MS results in the effect of compressive stress
Poisson’s ratio of Fe (001) nanoplates with thickness of 3a 0 and 5a 0 as examples of BCC
Stress profiles in cubic nanoplates with different elements and surface
Poisson’s ratio of Al (111) and (110) nanoplates
Poisson’s ratios of Au (001) nanoplates of different thicknesses
Compressive stress induced within Au (001) nanoplates of different thicknesses at 0K
Effect of a constant stress applied in the y-direction
Dependency of the Auxeticity of Au (001) nanoplates on their thickness and applied
Surface stresses of fcc (001) nanoplates of six metals as functions of the temperature
Poisson’s ratios of the six fcc (001) nanoplates under uniaxial stress
Auxetic strains and critical instability of the six fcc (001) nanoplates and the bulk
Poisson’s ratios of the bulk counterparts and the six metal (001) nanoplates in the
Schematic diagram of crystallographic orientations and loading direction of metal
Stress and energy curves of a square Au[100]/(001) nanowire
Deformation of the Au [100]/(001) nanowire under tensile loading
Details in the deformation during a global failure mode
Effects of temperature on the mechanical response of a Au[100]/(001) nanowire
Tensile loading test of a circular Au (001) nanowire
Tensile loading test of a circular Au (001) nanowire: Deformation at point A during a
Comparison of transient temperatures for (001) nanowires
Comparison of a Poisson’s ratio among different nanoscale materials under loading.
Effect of nanowire thickness on the Poisson’s ratios of the nanoplate
![Figure 3.2: Response of Au bulk under uniaxial tensile stress along [100]- direction. a:](https://thumb-ap.123doks.com/thumbv2/123dokinfo/10540725.0/153.892.198.689.109.927/figure-response-under-uniaxial-tensile-stress-along-direction.webp)
![Figure 3.6: Response of Au bulk under uniaxial compressive stress along [100]- direction](https://thumb-ap.123doks.com/thumbv2/123dokinfo/10540725.0/157.892.197.686.116.933/figure-response-under-uniaxial-compressive-stress-along-direction.webp)
