On the Da Vinci size effect in tensile strengths of nanowires: A molecular dynamics study

In recent decades, size effects caused by grain size, strain gradient, typical defects etc., have been widely investigated. Nevertheless, the dependence of tensile strength on the specimen length, addressed by Da Vinci around 500 hundred years ago, has received rather limited attention, even though it is one unavoidable question to answer if people attempt to bring materials’ amazing nano-scale strengths up to macro-level. Therefore, we make efforts to study tensile behaviors of copper nanowires with a common cross-section and various lengths by employing the molecular dynamics simulations. Surprisingly, a strong size effect of Da Vinci type indeed arises. We have shown the influences of lattice orientation, temperature and prescribed notch on such a Da Vinci size effect. Two different theoretical explanations are briefly proposed for a qualitative understanding. Finally, a simple scaling rule is summarized to cover the tendencies observed. © 2018 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/1.5006078


I. INTRODUCTION
Size effects in terms of stiffness and strength in solids are of great importance in component design and manufacturing, which have been studied intensively. In micron-scaled experiments like wire torsion, film bending and so on, the yield strength is reported to be related to the specimen's characteristic size along which nonhomogeneous plastic deformation happens, which many believe is due to geometrically necessary dislocations (GND), stress gradient 1-3 and/or strain gradient. 4 In experiments on the level of sub-micron and below, for example, nanopillar compression, the strength has been found to depend on the pillar diameter, for which the dominant mechanism is dislocation starvation. 4,5 The well-known Hall-Petch rule tells the dependence of yield strength on grain size, which is proven questionable for nano-polycrystalline. 6 We are revisiting size effects in mechanical strengths from a historic viewpoint. In 1400s, Da Vinci stated that "Among cords of equal thickness the longest is the least strong", indicating that the cord strength is dependent on cord length. However, in 1638, Galileo argued that cutting a long cord should not make the remaining part stronger. Griffith 7 adopted fracture mechanics to study size effect in fracture, saying that the effective strength of engineering materials is crucially dominated by discontinuities and flaws in them. Weibull 8 proposed the later well-known Weibull distribution to explain the same phenomenon, emphasizing the influence of extremely small strength values with extremely small probabilities. How strong the Da Vinci size effect is should be material dependent. 9 There has not been a unified method to check the existence of the Da Vinci size effect, which remains an open topic. Thus, our present aim is to investigate such a size dependence at the nanometer level.
Here, we study the Da Vinci size effect in nano-tensile tests 10-12 by adopting the molecular dynamics simulating technique. Relatively, molecular simulations can spare some challenges.
(1) Failure happens naturally on the atomistic scale, avoiding difficulties like finite element mesh dependence intrinsically existing in the conventional continuum theory. (2) Taking atomistic defects as the "smallest" flaws is usually an acceptable approximation, leading to controllable flaw distribution in molecular simulations.
Investigations on mechanical responses of nanowires has attracted intensive interests. Elastic properties of nanowires are strongly dependent on both specimen size and lattice orientation. [13][14][15] Granberg et al 16 simulated the tensile failure of Fe and FeCr, and confirmed that both atomistic potentials and deformation twinning along preferent orientations play a role in material strength. Sainath et al 11 studied the dependence of BCC iron nanowires' tensile properties on the cross-section width within the range from 1.42nm to 24.27nm. The tensile fracture behavior of cylindrical β-SiC nanowires was found to depend on wire diameter, lattice orientation and temperature. 17 Except tensile properties, the compressive behaviors of nanopillars also have attracted plenty of research efforts. 4,12,18 Notably, besides the above mechanical properties, some unique thermal and electric characteristics of nanowires have also been intensively studied. [19][20][21][22] We have also proposed simple and qualitative theoretical explanations about the Da Vinci size effect, in order to show how the nanowire tensile strength is related to the wire length. Finally, we present a simple scaling rule to summarize the tendencies found in our simulations. Proposing a predictable continuum theory remains challenging, because of the complexity of physical mechanisms involved, and also because the continuum assumption has to be understood in a loose sense due to the discreteness of atomic lattice structures at the nanometer level. The present investigation confirms that it is important for a related continuum theory to be capable of recurring the Da Vinci size effect, which is very strong for nanowires.

II. COMPUTATIONAL METHOD
The molecular dynamics code LAMMPS was used to simulate tensile behaviors of single crystal copper nanowires. The wire has a square cross section with the side length being 2.17nm, 3.61nm, 5.42nm and 7.22nm, respectively. The wire length varies from 21.6nm to 288.8nm. The simulated systems contained 10 4 to 10 6 Cu atoms. To consider the effect of orientation, the x-axis of models was spacing in <100> or <111> orientation. We also calculated a group of examples where the specimens have a notch at the middle.
Shrink-wrapped condition was applied onto six surfaces of each specimen, and the tensile loading conditions were prescribed on two ends in a gradual manner. The simulations were performed in constant NVT ensemble with a velocity-Verlet integrator. 14 Temperature, which was controlled by Nose-Hoover thermostat, 14 was chosen to be 10K, 150K or 300K. Embedded Atom Method (EAM) potential proposed by Adams 23 was adopted. The Ovito 24 software was used for visualization.
Before tests, perfect crystal NWs were relaxed for 100 ps to reach a relatively steady state. Then, uniaxial tension was implemented by displacing the boundary fixed atomic layers at the strain rate of 0.1% ps-1. After each displacement increment, the structure takes 3ps to relax. The axial stress of each increment was approximated as the average over the last 1ps of the relaxation process. The axial stress along the tensile direction, p xx , is calculated in the form of the virial formula, 25 i.e., where, V is the ensemble volume, m i and v ix are respectively the mass and the axial velocity of the ith atom, r ix and f ix are the atom position vector and the axial component of the resultant force on the ith atom.

A. Da Vinci size effect: Analysis of a typical case
As generally stated in textbooks of mechanics of materials, material tensile strength is often perpended not to depend on both cross section size and length of bar-shaped specimens. Nevertheless, recent studies, especially on the nano-and micro-scales, have shown that cross section size actually plays a significant role. However, to date we can hardly find suspicions on the independence of tensile strength on specimen length, which is studied here in the tensile tests on single crystal copper nanowires. Fig. 1 shows the stress-strain curves for tensile tests when the wire cross-section is fixed as 2.17nm and the wire length varies from 21.66 nm to 86.64nm. It is found that the uphill slopes of four curves, i.e. Young's modulus along the tensile direction, are very close to each other, indicating that NW length plays a negligible role on elastic stiffness. For each curve, the yield stress is defined as the first peak stress, corresponding to peak points of strain-stress curves, which is also adopted by Yang. 26 While the strain at which the specimen totally breaks into two is taken as the fracture strain. 27 From Fig. 1 both yield stress and fracture strain are considerably influenced by specimen length. Yield stress decreases with increasing wire length. For example, yield stress for L=21.66nm is 7.54 GPa, while that for L=86.64 nm is only 3.83 GPa. Analogously, the specimen length dependence of fracture strain qualitatively has the same tendency. The results in Fig. 1 suggest that the plastic and fracture properties of nanowires are considerably affected by wire length, which is a very important note when people investigate the dependence on wire diameter or when some amazing nano-level properties are attempted to be extended to larger scales.
The above dependence on wire length, namely the Da Vinci size effect on the nanoscale may be interpreted by some elemental theoretical analysis. Instead of focusing on deformation details such as dislocations, twins and so on, we present analyses based on some phenomenological assumptions, which are believed to be reasonable at least in a qualitative sense.
Theoretical explanation 1: Once the wire breaks into two, the fracture segments released most of stress previously stored up, the elongation is mainly localized into a fracture zone whose thickness is much smaller than the wire length. We assume that the critical elongation, or equivalently the deformed fracture zone thickness is a material constant, noted as l f . This can be taken as a kin of the crack band model. 28 Thus, the yield stress and strain can be written as where E is the Young modulus. Theoretical explanation 2: The specimen keeps deforming elastically and homogeneously until the elastic energy stored in the whole wire reaches a critical value e c = 2AG with A and G are crosssection area and the energy needed to produce unit area of new surface, respectively. Then the yield  happens. The yield stress is obtained as (3) Interestingly, both above theoretical predictions leave room for the wire length dependence of tensile strength. We take Eq. 2 as an example, which assumes a material-dependent fracture zone thickness (FZT). From Table I, we indeed find that when the cross-section size is fixed, FZTs of various wire lengths are close to each other, indicating the reasonability of the fundamental assumption. Fig. 2 shows the deformation patterns at the fracture strain and it is found that FZT is approximately independent of wire length under a fixed cross-section size. The white atoms are boundaries atoms, the red atoms belong to HCP structure and the green atoms belong to FCC structure.
We summarize the deformation mechanisms prevailing in NWs in Table I. Dislocations keep arising and disappearing along with the increasing external loading. Dislocations are hard to be permanently stored in the specimens with nano-sized cross-sections, due to the dislocation starvation mechanism. 29 Phase transformation is the other important factor. 30 As shown in Table I, for example, for the wire with the length of 21.66nm, 14.2% atoms have transformed from the initial FCC structure into the HCP arrangement.

B. Influences of lattice orientation, temperature and notch on Da Vinci size effect
In the following we study how the above Da Vinci size effect is affected by material and environmental factors. For this end, specimens with and without a notch in the middle were subjected to tensile loading along the lattice orientation of either <1 0 0> or <111> under temperature 10K, 150K and 300K, respectively. Fig. 3 shows the tendencies in terms of yield stress versus wire length. In all cases studied here, the yield stress decreases with increasing wire length, which has already been explored in Fig. 1. The fracture strain of each case is plotted in Fig. 4. In brief, under each condition, the fracture strain decreases with increasing wire length and provided various conditions effect on Da Vinci effect. Such a Da Vinci effect is considerably stronger along <1 1 1> than along <1 0 0>. Furthermore, in the present study we also find a considerable anisotropy in elastic properties. That is, the tensile modulus along <1 1 1> is much bigger than that along <1 0 0>. In the temperature range investigated, temperature plays a very weak role in the size effect. The prescribed notch seems to slightly weaken the length dependence.

C. A possible scaling rule
As shown by results in section B and C, both yield stress and fracture strain considerably depend on the wire length. On the other hand, wire strength has been found to be dependent on wire cross-section size by many existing investigations. Therefore, there is a need to figure out some dimensionless shape parameter, which is expressed in terms of both wire cross-section size and wire length. When orientation, temperature and notching setting are kept unchanged, wires with the common shape parameter are expected to have unique yield stress and fracture strain.
We have attempted to take such a shape parameter as √ A/L. Interestingly, Fig. 5  with the same √ A/L, both yield stress and fracture strain are indeed close to each other. This serves as a very important note -tensile strengths of nanowires depend on both cross-section size and wire length.

IV. CONCLUSIONS
With the help of molecular dynamics simulations, we have found the surprisingly strong dependence of NW's tensile strength on wire length, which should be considered as a parallel to the frequently studied cross-section-size-related size effect and is called the Da Vinci size effect here. Such a size effect is found to exist under various conditions including different lattice orientations, temperatures, notching settings and so on. We have attempted to explain it via providing straightforward phenomenological theoretical formulations, clearly showing the dependence of yield stress on NW length. This finding confirms the existence of the Da Vinci size effect in NWs, which needs to be accounted for during designs of nano-devices. Just as originally proposed by Da Vinci himself, such a Da Vinci size effect prevails in nanowire tensile strength. Nevertheless, in the present study it has been found not to exist in the tensile stiffness of nanowires. Under the present parameter setting, the conclusion can be reached that NWs with the same dimensionless shape parameter √ A/L approximately have the common tensile yield stress and fracture strain, which, to be frank, is realized largely be guesswork and certainly calls for further future investigations. This can be taken as a necessary preparatory investigation for developing a predictable continuum constitutive theory. Furthermore, we believe that the above Da Vinci size effect should serve as an important note if people attempt to bring materials' amazing nano-scale strengths up to macro-level.