Abstract The presence of in-homogeneity or defects in materials cannot be ignored. There is great need tounderstand the influence of defects on the mechanical response of nano-materials. In this study, atomistic sim-ulations have been used to investigate the mechanical response of gold nano-wires under twisting. Simulationsshow that nano-wires have different elastic properties when defects are present. Embedded cracks of differentsizes have been created in nano-wires to quantify in-homogeneity. The inter-atomic interactions are representedby employing an embedded-atom potential. The effect of different sizes of crack on potential energy, torqueand stresses for investigating the mechanical response of a nano-wire is part of the whole investigation. It ispredicted from our simulation that the presence of a crack and dimensions of the crack control the torsionalresponse of gold nano-wires. Deviation in the behavior of gold nano-wires from continuum expectations isalso discussed. The comparison of results of atomistic simulations is made with a linear elasticity model (ofhealthy and cracked nano-wires) to get deep insights into the nano-scale behavior of nano-wires.
1 Introduction
Due to dominating mechanical, electrical, optical and catalysis properties, metallic nano-wires have attractedintense research in recent years [1–5]. Metallic nano-wires have been expected to apply in nano-electromechanical systems (NEMS), such as active components of circuits [6,7], sensors [8] and actuatorsin NEMS. So detailed studies regarding the properties of materials are required at scaled down dimensionsbefore going for manufacturing the systems using these nano-wires.
However, when the metallic components are scaled down to nano-sizes [9], it becomes difficult to control theprocess of manufacturing nano-structures or nano-wires. There is great probability of getting in-homogeneityor defect in nano-wires. These defects can be present in the form of crack, void, alloying atoms, other phases,etc. [10], which may create asymmetry in the crystalline structure. Stress concentration (resulting from defects)leads to plasticity at continuum level in the regions around it [11–13]. At nano-scale, the presence of stressconcentration will significantly affect the properties of nano-structures depending on the location, size ofthe stress concentration and the loading conditions [14–16]. For this point, it is very important to know the
K. Saini (B) · N. KumarSchool of Mechanical, Material and Energy Engineering, Indian Institute of Technology Ropar,Rupnagar 140001, Punjab, IndiaE-mail: [email protected].: +91-998-840-0524Fax: +91-1881-223395
N. KumarE-mail: [email protected].: +91-1881-242170Fax: +91-1881-223395
688 K. Saini, N. Kumar
properties of the defected nano-wires [17,18]. This may elaborate the extent of suitability of such defectednano-wires and to decide whether such defected nano-wires are able to perform under particular loadingconditions or not.
When materials are bent [19,20] or twisted [21–24], they experience strain gradients. These strain gradientsare known to alter the response of materials [22,25–27]. The torsion behavior of CNT has been investigatedin detail [28–31]. Depending upon the twist rate and amount of twist, the torsional response [23,32] of nano-wires varies. Molecular dynamics (MD) methods are extensively used to study the mechanical behavior ofnano-structures [33–37], particularly tension [38–40], compression [41] and also thermal behavior [42–45].Early atomistic simulations of fracture were carried out by Ashurst and Hoover [46] in which they comparedthe free energies of a finite element model and a bead-spring model in a triangular lattice. Later Abraham et al.[47] simulated the brittle fracture of a material using molecular dynamics models consisting of million atomsand with Lennard-Jones (LJ) inter-atomic potential. Zhou et al. [48] described fracture as the energy releaseby bond breakage which accumulates in a local phonon field and moves with the crack tip. Xu et al. [49]performed molecular dynamics simulation on ductile material and computed the virial stress field around thecrack tip and its evolution during the crack growth [50]. This naturally leads to the question of what happenswhen defected nano-wires are deformed under torsion. Wires at this size are often single crystals. There arevery few experimental or numerical studies available on elasticity of single crystal defected nano-wires [51]in torsion.
Moreover, experiments at high strain rates lead to shear localization arising from adiabatic heat dissipationand local thermal softening of the material. This results in highly non-equilibrium systems that are difficultto study experimentally and which can be easily done through MD to understand the mechanisms involved.In MD, high strain rates are usually investigated [23,24,52] to differentiate and understand the mechanismsinvolved in the deformation processes which may play a key role in controlling the mechanical properties ofthe nano-structures. From a macroscopic viewpoint, these loading rates are very high for any materials andstructures. However, at nano-scale, this loading rate can be considered as quasi-static for some cases.
In this paper, properties of healthy and cracked nano-wires are compared for torsion loading. The effect ofthe size of the embedded crack (dimensions of crack) on properties such as potential energy, torque and stresses[53] has been investigated. Continuum modeling in combination with atomistic modeling has been used inthe previous studies [54–56] to simulate the material behavior at small length scales. We have developed acontinuum model to understand the elastic behavior of healthy and cracked nano-wires. Efforts have beenmade to quantify the role of different factors on the torsional response of nano-wires. This investigation canbe used as a method to measure the extent of suitability of defected nano-wires for applications where thepresence of a crack like defect can affect its performance. This study can also be extended to quantify [57]and understand the influence of presence of other asymmetries created by alloying atoms, voids, other phases,etc., on the mechanical characteristics of nano-crystalline materials.
2 Material and methods
2.1 Atomistic modeling
In atomistic modeling [58], an embedded-atom model (EAM) potential [59] has been used for defining theinteraction between the atoms of FCC metal Gold (with lattice constant 4.08 Å) for the present study. In asimulation, the potential energy of an atom is given as
Etotal =N∑
i, j
⎛
⎝Fα
⎛
⎝∑
j �=i
ρα(ri j )
⎞
⎠ + 1
2
⎛
⎝∑
j �=i
φαβ(ri j )
⎞
⎠
⎞
⎠ (1)
where F is the embedding energy which is a function of atomic electron density ρ (rho), φ (phi) is a pairpotential interaction, and α (alpha) and β (beta) are the element types of atoms i and j. The multi-body natureof EAM potential is a result of the embedding energy term. Both summations in the formula are over allneighbors j of atom i within the cutoff distance.
The virial stress σi j is derived based on the virial theorem [53] as shown in relation (2)
σi j = − 1
Ω
⎡
⎣mviv j − 1
2
N∑
q=1
(ap − aq) f pq
⎤
⎦ (2)
Torsional deformation behavior of cracked gold nano-wires 689
Fig. 1 Basic geometry of the gold nano-wire with [1 0 0] orientation
where m is the mass of the atom, (i, j) take values of x, y and z (directions), q takes values from 1 to Nneighbors of atom p, ap is the position of atom p, f pq is the force on atom p due to neighboring atom q , andΩ is volume.
For obtaining torque values, the moment of atoms about the axis of rotation is calculated (vector productof distance of atom from axis of rotation with force acting on atom). Vector summation of moments of atomsis calculated as shown in relation (3),
τ =Nt∑
k=1
(rk × Fk) (3)
where τ represents total torque, Nt is total number of atoms, rk is the distance of atom from axis of rotationand Fk is total force acting on the atom.
Figure 1 presents the basic geometry of the nano-wire (having [1 0 0] orientation) with structural boundaryconditions. The length of the nano-wire has been kept parallel to X-axis and the circular cross-section is lyingin Y–Z plane. The nano-wire has been divided into three portions; two extreme ends are moved relative toeach other, keeping one end rotationally constrained (free to move axially) and the other end as free. Beforeapplication of the load, the nano-wire energy was minimized. The free end of the nano-wire has been twisted(in steps of 1◦) about an axis (parallel to X-axis) passing through the center of the circular cross-section. Dueto rotation of the free end, twisting of the rest of the nano-wire takes place. The nano-wire has been keptfree from axial stresses developed (resulting from axial displacement of rotationally constrained end due totwisting) [60]. This has been done by keeping the rotationally constrained end free to move in axial direction.Hence, axial stresses did not influence the stresses induced in the nano-wire and the mechanical propertiesof it. The nano-wire has been well relaxed after each degree the twist to make its energy lowest possiblein the deformed state. This also transferred the twist effect uniformly in the nano-wire (from twisted end torotationally constrained end). Hence, it resulted the most stable twisted state of the nano-wire after every 1◦ oftwist. The twisting rate (5.40×108 radian/sec unit nano-meter length of the nano-wire) has been kept constantduring all simulations. The temperature of the nano-wire has been maintained at 300 K using velocity rescalingmethod [61] which also absorbed the heat generated due to high rate torsional deformation. This balanced thekinetic component and allowed the total system energy to contribute toward the potential energy componentonly. Non-periodic boundary conditions that were employed for the simulation are the same as referred in theliterature depending on the type of loading conditions [62].
Potential energies of healthy and cracked nano-wires at every step of twist have been plotted in Fig. 2.The potential energy of both nano-wires increased as we kept on increasing the angular twist. This was dueto storing of torsional strain energy into the nano-wire. But after a certain angle of twist, a fall in the potentialenergy was observed. The maximum amount of potential energy referred to the elastic limit of the materialafter which it entered in the plastic regime [23,24,27]. Here, the elastic limit represents the maximum twistangle up to which a nano-wire of particular size can be twisted without any permanent deformation in it. Atnano-scale, the elastic limit is a result of slipping of planes of atoms. A certain amount of energy is dissipatedfor this slipping phenomenon. Due to this, the energy stored in the nano-wire suddenly dropped upon furthertwisting. Keeping the cross-sectional dimensions constant, the effect of change in length of the cracked nano-wire on the elastic limit in terms of angular twist is shown in Fig. 3. It is evident from Fig. 3 that the elasticlimit varies linearly with the length of the nano-wire. The ratio of the elastic limit in degrees to twisted lengthin Ångstroms remains almost equal to 0.23 degrees/Å.
The influence of crack like defects on the mechanical response has been investigated by creating rectangularembedded cracks in a nano-wire (as shown in Fig. 4 by a representative volume element). Cracks were createdby removing the interaction between two adjacent layers of atoms. Cracks were located at the center of the
690 K. Saini, N. Kumar
0 5 10 15 20 25 30−3.81
−3.8
−3.79
−3.78
Angular twist (degrees)
Pot
enti
al e
nerg
y (p
er a
tom
) in
eV
For healthy nano−wireFor cracked nano−wire
Elastic limit ofcracked nano−wire
Elastic limit ofhealthy nano−wire
Slipping ofplanes of atoms
Fig. 2 Change in potential energy of healthy and cracked nano-wire (having Type IV crack of width 36.72 Å)
Fig. 3 Effect of length on potential energy of cracked nano-wire (having Type I crack of width 28.56 Å)
Fig. 4 Orientation of crack and position of embedded crack (dimensions in Ångstroms) in atomistic model
nano-wire, placed symmetrically, with the center of the crack lying at the center of the nano-wire. It has beenobserved that due to energy minimization crack surfaces are sufficiently apart and do not interpenetrate.
In the present study, we have investigated cracks of four different lengths (32.64, 40.8, 48.96 and 57.12 Å),and for each length, we further considered four widths (12.24, 20.4, 28.56 and 36.72 Å). Cracks have beenclassified (on the basis of crack length) into Types I, II, III and IV as shown in Table 1. In total, the behaviorof sixteen cracked nano-wires (carrying embedded cracks of such types) has been compared with a healthynano-wire. The elastic limit of the nano-wire has been obtained from potential energy (averaged over time andspace) per atom curve. To understand the behavior of nano-wires and to quantify the influence of the crack onphase transformation and stress concentration, a continuum model is incorporated into the present study.
2.2 Comparison with continuum model
To understand the elastic response of a nano-wire, a continuum model of nano-wire has been developed byusing linear elasticity theory. Linear elasticity theory assumes the material to be linearly elastic, irrelevant ofthe amount of deformation. Hence, the continuum model describes the elastic behavior of the nano-wire andexcludes the influence of plasticity on the behavior. The nano-wire was assumed as a right circular cylinder
Torsional deformation behavior of cracked gold nano-wires 691
Table 1 Various crack dimensions under investigation
Type of crack Crack length (Å) Crack width (Å)
Type I 32.64 12.2432.64 20.432.64 28.5632.64 36.73
Type II 40.8 12.2440.8 20.440.8 28.5640.8 36.73
Type III 48.96 12.2448.96 20.448.96 28.5648.96 36.73
Type IV 57.12 12.2457.12 20.457.12 28.5657.12 36.73
0 5 10 15 20 25 30−4.62
−4.61
−4.6
−4.59x 10
4
Angular twist (degrees)
Pot
enti
al e
nerg
y (e
V)
For atomistic cylinder
For continuum cylinder
(a)
0 5 10 15 20 25 300
200
400
600
800
Angular twist (degrees)
Vir
ial T
orqu
e (e
V)
For atomistic cylinder
For continuum cylinder
(b)
Fig. 5 Comparison of atomistic model with continuum model for healthy nano-wire. a Comparison for total potential energy,b comparison for torque
having dimensions equal to the size of the nano-wire. The length of the nano-wire was not constrained in theatomistic model. This made the loading condition similar to pure torsion which kept the cross-sectional planeperpendicular to the axis of the nano-wire at different twist angles (in the elastic regime). Although materialproperties vary along different crystallographic directions of the nano-wire they have been assumed to be same[63] for modeling purposes. The behavior of both the models for potential energy and torque is shown in Fig. 5.
As evident in Fig. 5, the continuum model of the nano-wire behaves in a similar manner as its atomisticmodel. According to the theory of linear elasticity of cylinders [64] and considering Hooke’s law, the strainenergy can be related to the rotation angle by
U =∫∫
σ.dε.dV = 1
2C
∫γ 2
torsion.dV = C Jφ2
2L(4)
where U is the torsional strain energy (or potential energy), σ and ε are stress and strain, C is torsional shearmodulus, γtorsion is torsional shear strain, V is volume, J represents the polar moment of inertia of the cylinder(solid or hollow), L is the length of the cylinder and φ represents the angle of twist.
692 K. Saini, N. Kumar
Fig. 6 Equivalence of atomistic model with continuum model for cracked nano-wire
0 5 10 15 20 25 300
0.01
0.02
0.03
0.04
0.05
Angular twist (degrees)
Dif
fere
nce
of P
oten
tial
ene
rgy
(eV
)
a/b=1a/b=2a/b=3a/b=4a/b=5a/b=6
Fig. 7 Effect of major to minor axis ratio (a/b) on the difference of energy storing capacity of cracked nano-wire (12.24 Å ×32.64 Å) to healthy nano-wire
For creating a continuum model of a crack in a nano-wire, we assumed a crack in the shape of an ellipticalcylinder at the center as shown in Fig. 6. The total length of this cylinder was kept equal to the length of thecracked nano-wire, i.e., 114.24 Å, having a crack of length (Lc) in it. The major axis of the ellipse was equalto the width of crack. Depending upon the location of the crack, considering loading and boundary conditions,the length of the minor axis (2b) of the ellipse can be considered as a function of the length of the major axis(2a). In atomistic simulations, it is dependent upon the energy minimization procedure used. The influence ofratio (a/b) on the total energy storing capacity of the cracked nano-wire has been elaborated in Fig. 7. It showsthe deviation from the behavior of the healthy nano-wire as a function of this ratio. We assumed this ratio tobe of constant magnitude equal to 3. Theoretical calculations and experimental evidences show that the shearmodulus is a function of the crystal orientation [63,65,66]. In the present study, the shear modulus is assumedas 45.4 GPa for continuum calculations. The value of the modulus of elasticity in shear is obtained fromtorque versus twist plots (in Fig. 5b) is 46.37 GPa for atomistic simulations which resembles with availableexperimental results in the literature. The polar moment of inertia of the healthy (Jh) and cracked (Jc) regionof the cylinder can be described as
Jh = π D4
32, (5)
Jc =[π D4
32
]−
[πab(a2 + b2)
4
], (6)
where D represents the outer diameter of the cylinder, a and b represent the length of semi-major and semi-minor axis of the ellipse, respectively. As shown in Fig. 6, a cracked nano-wire can be considered as a compositecylinder, made of healthy and cracked cylinders, and connected in series. When such a composite cylinder issubjected to twist, the total twist can be written as
Torsional deformation behavior of cracked gold nano-wires 693
whereφcomposite represents the total angle of twist in the composite cylinder, andφhealthy−region andφcracked−regionrepresents twists in the healthy and cracked region of the composite cylinder,
φhealthy–region = T Lh
G Jh, (8)
φcracked–region = T Lc
G Jc, (9)
φcomposite = T Lcomposite
G Jcomposite(10)
where T is torque, Lh and Lc are lengths of healthy region and cracked region in the composite cylinder,Lcomposite represents its total length, G is shear modulus, Jh and Jc represents polar moment of inertia ofhealthy and cracked regions of the cylinder, and Jcomposite is the total polar moment of inertia of the compositecylinder.
Using Eqs. (7), (8), (9) and (10), we get
Lcomposite
Jcomposite= Lc
Jc+ Lh
Jh. (11)
Considering Eq. (11), the total strain energy (or potential energy) stored in the composite cylinder of polarmoment of inertia Jtotal of length L total at a particular angle of twist φ is given as follows:
Ucracked = C Jcompositeφ2
2Lcomposite(12)
where Ucracked is the strain energy stored in the cracked cylinder.
3 Results and discussion
During elastic deformation, the energy of the material increases as work done is stored in form of additionalpotential energy over the initial potential energy of the system. But if energy stored due to deformation inthe material increases beyond a certain limit (elastic limit), it causes slipping. This slipping results in energydissipation which is not recoverable from the material upon unloading. As evident from Fig. 8, there is asignificant change in von-Mises stress distribution on the surface of the nano-wire at various stages of twist.From stage 〈1〉 to 〈5〉, the stresses increased considerably with increasing twist but after crossing the elasticlimit, a fall in stress values can be observed. This fall is due to energy absorbed in slipping of planes of atoms(as visible in stage 〈6〉 of Fig. 8). These planes are almost at 45◦ to the axis of the nano-wire. The cutting plane(section plane) has been used to elaborate the stress distribution in the vicinity of the embedded crack. Thecutting plane has been oriented perpendicular to the cross-section of the nano-wire. The stress distribution in acracked nano-wire is shown [67] in Fig. 9 at each stage of twist (using sectional views). With increasing angleof twist, there have been significant changes in the distribution of the stress in the nano-wire. The changes instress distribution [67] with twisting are shown in Fig. 9 by means of sectional views of the nano-wire. In theinitial stage (stage 〈1〉), the nano-wire has been well relaxed and there is no twisting applied. The stresses havebeen observed higher at surface as compared to the rest of the body of the nano-wire (except in the vicinity ofthe crack). But as we start twisting of the nano-wire, the distribution of stresses starts varying dominantly inradial direction, with higher stresses toward outer surface and decreasing toward the center of the nano-wire.If we increase the twist beyond the elastic range, plasticity in nano-wire commences (as visible in stage 〈6〉 inFig. 9). As mentioned in the previous section, the elastic limit obtained from potential energy has been definedas degrees of twist (for healthy as well as defected nano-wires).
The elastic limit of cracked nano-wire has been found to be different from healthy nano-wire. Crack regionsin the nano-wire acted as separate surfaces as well as stresses are concentrated in these regions. We know thatat very small length scales an increase in surface area improves the elastic properties due to increasing surfacearea to volume ratio [68,69]. For small nano-wires, an area increase in the form of crack size also showeda similar effect. For small sizes of cracks, the elastic limit remained uninfluenced (as shown in Figs. 10a–c).But a larger crack area increased the amount of stress concentration which in turn shortened the elastic limit(as evident in Fig. 10d). As shown in Fig. 9 (at stage 〈6〉), slipping takes place from the corners of the crack.
694 K. Saini, N. Kumar
Fig. 8 Shapes of cylindrical gold nano-wire with increasing angle of twist (from stages 〈1〉 to 〈6〉) along with change in the stressdistribution (stresses in bars)
Figure 10 shows a comparison of potential energy with change in crack length at constant crack widths (equalto 12.24, 20.4, 28.56, 36.72 Å). Figure 11 shows a comparison of potential energy with the change in crackwidth at constant crack lengths, i.e., for different types of cracks. Figure 12a compares the effect of crack widthon the elastic limit for cracks of various lengths (Types I, II, III and IV cracks), and Fig. 12b compares theeffect of a change in crack length for cracks having widths 12.24, 20.4, 28.56 and 36.72 Å. From the plots, itis evident that for nano-wires with embedded cracks (having small crack widths), the elastic limit was almostequal (within 2◦ range) to a healthy one. But when crack widths were large (crack width >28.56 Å), the elasticlimit of nano-wire shortened. This also elaborates that the torsional response of a nano-wire is more sensitivetoward changes in crack width as compared to changes in crack length. This fact is supported by the distributionof stresses which are varying significantly in radial direction after a certain radial distance. As shown in Fig. 9,at initial stage (stage 〈1〉), there is an almost uniform stress distribution in the longitudinal direction and non-uniform stress distribution in the radial direction. Stress gradients increased in radial direction as we increasedthe amount of twist. There is not such a change in stress gradients in longitudinal direction of the nano-wirewith increasing twist.
Torque has been used as a parameter to understand the torsional stiffness of nano-wires. Figure 13 representsthe variation of elastic torque with twist angles for nano-wires having embedded cracks of various dimensionsand their comparison with a healthy nano-wire. There is no considerable difference between the trends followedby the defected nano-wires and healthy nano-wires for torque values. The torsional stiffness remains almostthe same for healthy and cracked nano-wires as represented by the slope of the curves shown in Fig. 13a–dwithin the elastic range. It is also evident from Fig. 13d that when crack widths are high, the torsional stiffnessremains unaffected in the elastic limit.
To understand the response of cracked nano-wires, we investigated the amount of energy stored with theamount of twist. For this, we looked into the continuum model, having the same size of cracks. Figure 14ashows the difference in the behavior of healthy and cracked nano-wires through atomistic modeling. It is clearthat with the presence of a crack in the atomistic model and same amount of twist, the magnitude of potentialenergy stored is lesser than in healthy nano-wires. But similar changes in the potential energy storing capacityhave not been not observed (as shown in Fig. 14b) by the continuum model, even with large width crack inthe nano-wire. Moreover, when we slightly changed the material property (shear modulus) in the continuummodel, the behavior obtained was found approaching the atomistic model behavior. For the continuum model,the magnitude of potential energy stored in the cracked nano-wire with twist is shown in Fig. 15 when changes
Torsional deformation behavior of cracked gold nano-wires 695
Fig. 9 a von-Mises stress distribution at different angles of twist up to the elastic limit (from stages 〈1〉 to 〈3〉). b von-Mises stressdistribution change during twisting from elastic range to plastic range (from stages 〈4〉 to 〈6〉) (stresses in bars)
in material properties are considered. This clarifies the contribution of material properties toward potentialenergy storing capacity for a cracked nano-wire. This also explains that there is not significant effect of geo-metrical weakening (in the form of cracks) on the potential energy storing capacity of nano-wires. Figure 15clarifies the contribution of a change in the overall material properties resulting from a local phase transfor-mation [68–70]. In gold nano-crystals, there is phase transformation (from FCC to BCT) in the vicinity ofthe surface. The presence of a crack in nano-wire serves as additional free surfaces which served in phasetransformation and are resulting in a local change of the material properties of the nano-crystal. The BCTstructure has a higher number of slip planes as compared to FCC which results in a local reduction in shearmodulus of the nano-wire in the vicinity of the crack region and hence in overall shear modulus values. Theabove discussion also clarifies that due to crack surfaces, the total magnitude of initial potential energy in acracked nano-wire is lower than in healthy ones (Figs. 10, 11).
4 Conclusions
The torsional response of healthy and cracked nano-wires is significantly influenced by the distribution ofstresses in it. Stress values are high toward the outer surface as we move away from the center of the nano-wire. In radial direction, the stress gradient magnitude increases with an increase in the amount of twist.
Fig. 10 Effect of change in the crack length (at constant crack width) on the potential energy (per atom) storing capacity ofcracked nano-wires. Plots show nano-wires having cracks of different types, having constant widths of (a) crack-width 12.24 Å,b crack-width 20.4 Å, c crack-width 28.56 Å and d crack-width 36.72 Å
Fig. 11 Effect of change in the crack width (at constant crack lengths) on the potential energy (per atom) storing capacity ofcracked nano-wires. Plots show nano-wires having cracks of a Type I, b Type II, c Type III and d Type IV, having widths as32.64, 40.8, 48.96 and 57.12 Å
Dimensions of defects like crack control the elastic limit of the nano-wire. The presence of large width cracksshortens the elastic limit considerably. This consolidates that in-homogeneity present closer to the surface haslarger effect on the elastic limit than that which is present in the center of the nano-wire. Further, within the
Torsional deformation behavior of cracked gold nano-wires 697
Fig. 12 Effect of crack dimensions on the elastic limit of a nano-wire a. Effect of crack width for various crack lengths (i.e., TypeI, II, III, IV cracks). b Effect of crack lengths for various crack widths (12.24, 20.4, 28.56 and 36.72 Å)
Fig. 13 Effect of change in crack length (at constant crack width) on the torque of cracked nano-wires. Plots show nano-wireshaving cracks of different types, having constant widths of a crack-width 12.24 Å, b crack-width 20.4 Å, c crack-width 28.56 Å,d crack-width 36.72 Å
elastic limit, the amount of potential energy stored is influenced by the presence of a crack in the body of thenano-wire which decreases as compared to healthy nano-wires. The influence of a crack like defect on thepotential enegy storing capacity is dominated by local material property change (due to phase transformation)
Fig. 14 Effect of crack width (for Type I crack) on the total potential energy stored in the nano-wire through a atomistic modeland b continuum model
0 5 10 15 20 25 300
50
100
150
200
Angular twist (degrees)
Pot
enti
al e
nerg
y (e
V)
Same Shear modulusShear modulus changed by 25%Shear modulus changed by 50%Shear modulus changed by 75%
Fig. 15 Effect of change in the shear modulus on total potential energy storing capacity (by continuum model for crackednano-wire)
of the nano-wire. Such change in the material property results in a decrease of the overall shear modulus ofthe nano-wire material. The torsional stiffness remains uninfluenced with the presence of a crack. It can beconcluded that the torsional mechanical response depends on the size of the in-homogeneity, its location andstrain magnitude in the nano-structure.
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