# Relation between strength and local stresses in nanostructures as a function of their size and shape

Laurent Pizzagalli, Riccardo Gatti, Jonathan Amodeo, Julien Godet, Julien Durinck, Sandrine Brochard

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Laurent Pizzagalli, Riccardo Gatti, Jonathan Amodeo, Julien Godet, Julien Durinck, et al. Relation between strength and local stresses in nanostructures as a function of their size and shape. Journal of the Mechanics and Physics of Solids, 2025, 205, pp.106306. 10.1016/j.jmps.2025.106306. hal-05232618

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# Relation between strength and local stresses in nanostructures as a function of their size and shape

L. Pizzagalli<sup>a ID,*</sup>, R. Gatti<sup>b</sup>, J. Amodeo<sup>c</sup>, J. Godet<sup>a</sup>, J. Durinck<sup>a</sup>, S. Brochard<sup>a</sup>

## ARTICLE INFO

### Keywords:

Nanostructure

Strength

Molecular dynamics

## ABSTRACT

The aim of this work is to understand how and why the strength of nanostructures depends on their size and shape. We performed extensive molecular dynamics simulations of the uniaxial compression of 2D FCC models with various sizes and shapes. It is found that the strength varies as a power law with size, with an exponent critically depending on shape. Plastic deformation is triggered by a unique mechanism, which is the nucleation of Shockley dislocation at local stress gradients in the vicinity of contact corners. This allows to unravel the effect of size and shape by examining the relation between local stresses and applied stress. Our analyses reveal that spatial variations of local stresses as a function of size and shape are complex and not simply related to the stress concentration as predicted by the elasticity theory. However, it is found that these local stresses linearly depend on the applied stress. Finally, we propose that the observed size effect on strength is caused by a screening interaction between the stress gradients occurring at all contact corners.

## 1. Introduction

Most material properties change as their characteristic dimensions decrease to the nanoscale, and mechanical properties are no exception to this rule. Perhaps the most famous consequence is that materials are much stronger, especially at the nanoscale (Uchic et al., 2004; Zhu et al., 2009). The strength variation versus size is usually described by a power law $D^{-n}$ (Dou and Derby, 2009; Greer and Hosson, 2011; Korte and Clegg, 2011; Dunstan and Bushby, 2013; Yilmaz and Derby, 2024). In several cases, it has even been shown that the measured strength could reach the theoretical limit (Tang et al., 2012; Wang et al., 2013; Wheeler et al., 2016; Nie et al., 2019). Nanoscaled materials are also often tougher, i.e. they can be elastically deformed up to much larger strains than bulk systems (Han et al., 2007; Gordon et al., 2009; Stan et al., 2012; Banerjee et al., 2018). Finally, bulk brittle materials become ductile at low scales (Kang and Cai, 2007; Östlund et al., 2009; Abed El Nabi et al., 2015).

Reducing the dimensions of a material has three main consequences: (i) the decrease of the amount of intrinsic structural defects (Mompiou et al., 2012), (ii) the hindering of several common plasticity mechanisms due to the confined volume, and (iii) the increasing role of surfaces. Their combination explains why materials at nanoscale can be brought in a high stress/strain regime with no plastic deformation or failure. For instance, initially present or nucleated dislocations can leave the material due to the proximity of surfaces, preventing multiplication mechanisms or the formation of junctions (Greer and Nix, 2006; Uchic et al., 2009; Kiener and Minor, 2011). In high stress/strain conditions, specific plasticity mechanisms can be activated (Kraft et al., 2010; Park et al.,

<footnote><sup>a</sup> Institut P', CNRS UPR 3346, Université de Poitiers, SP2MI, Boulevard Marie et Pierre Curie, TSA 41123, 86073 Poitiers Cedex 9, France</footnote>

<footnote><sup>b</sup> Université Paris-Saclay, ONERA, CNRS, Laboratoire d'étude des microstructures, Châtillon 92322, France</footnote>

<footnote><sup>c</sup> Aix-Marseille Université, Université de Toulon, CNRS, IM2NP, Marseille 13013, France</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: Laurent.Pizzagalli@univ-poitiers.fr (L. Pizzagalli).</footnote>
<figure><img src="image_4.png" /><figcaption>**Fig. 1.** 2D models used in compression simulations. They are characterized by their lateral extension $dx$ along $X = [110]$ axis and height $dz$ along $Z = [001]$ axis, as well as the angle $\gamma$ between the top/bottom surfaces and the lateral facets. For $\gamma \ge 136.7^\circ$, the models are trimmed to the right and to the left, far from the contact corners, to avoid unconventional pointed tips.</figcaption></figure>

2009), like spiral dislocation sources (Takeuchi et al., 2021) and heterogeneous dislocation nucleation from surfaces (Oh et al., 2009; Mordehai et al., 2011; Roos et al., 2014; Abed El Nabi et al., 2015; Mordehai et al., 2018). Twinning can also be favored in specific cases (Seo et al., 2011; Sedlmayr et al., 2012; Weinberger and Cai, 2012; Lee et al., 2014; Kong et al., 2016). Furthermore, diffusive plasticity is reported at very small scales (Tian et al., 2013; Sun et al., 2014; Kong et al., 2018; Comtet et al., 2019; Mathesan and Mordehai, 2023), as well as the formation of amorphous (Ovid'ko, 2012; Tang et al., 2012) or alternative crystalline (Vincent et al., 2018) phases. Finally, unconventional slip mechanisms are also predicted to be activated in high stress/strain regime (Guénolé et al., 2011; Kilymis et al., 2018; Merabet et al., 2018; Pizzagalli et al., 2020).

The above-mentioned studies mostly concerned 1D systems like nanowires, nanoneedles, or nanopillars. Nanoparticles are another class of nanomaterials which are also interesting for mechanical properties investigation at the nanoscale. Uniaxial compression tests performed using flat-punch indenters are usually the method of choice. Most previous studies report an increase of strength when the dimensions are reduced (Amodeo and Pizzagalli, 2021). This effect has been reported for FCC metals (Mordehai et al., 2011; Bel Haj Salah et al., 2017; Yang et al., 2017; Sharma et al., 2018, 2019; Zimmerman et al., 2021; Griesbach et al., 2021; Pizzagalli et al., 2024), and their alloys (Maaß et al., 2012; Amodeo et al., 2014), BCC metals (Han et al., 2015; Sharma et al., 2020; Pizzagalli et al., 2025b), HCP metals (Yu et al., 2015), and covalent materials (Gerberich et al., 2003; Kilymis et al., 2019; Vidable et al., 2022; Pizzagalli and Godet, 2023; Pizzagalli et al., 2025a). However, it has also been suggested that the strength value could saturate below a certain threshold size (Zimmerman et al., 2021; Han et al., 2015), or even a possible decrease of strength (Vidable et al., 2022; Laurens et al., 2023). Nanoparticles are also tougher than their corresponding bulk materials (Beaber et al., 2011; Gerberich et al., 2009). Finally, the plastic deformation of nanoparticles can involve various mechanisms like heterogeneous dislocation nucleation (Wagner et al., 2015), amorphization (Laurens et al., 2023) or alternative crystalline phase transformation (Valentini et al., 2007; Hale et al., 2011), diffusion (Azadehranjbar et al., 2023), twinning (Bian and Wang, 2013) and single-arm source dislocation (Kiani et al., 2019).

It has been proposed that the nanoparticle strength could depend on its dimension according to a power law, like for 1D systems (Amodeo and Pizzagalli, 2021). Mordehai and co-workers proposed that such a dependence could be explained by the buildup of stress gradients in the vicinity of nanoparticle corners in contact with a flat-punch indenter (Mordehai et al., 2011). They found an exponent $n \approx 0.77$ for gold nanoparticles with a Winterbottom-like shape. The relation between the spatial variation of local shear stress and the size dependence of the strength is however not firmly established in this paper. Another questioning concerns the exponent value itself, and its dependence on the shape of the nanoparticle. For instance, a value of about 0.5 was determined from atomistic calculations of FCC nanoparticles with Wulff shapes (Feruz and Mordehai, 2016). The strength of cubic nanoparticles also seems to be weakly dependent on dimensions (Shreiber and Mordehai, 2015; Wagner et al., 2015). Numerical simulations also suggested that the influence of the shape on the strength is more significant than the influence of size (Amodeo and Lizoul, 2017; Kilymis et al., 2018).

In this work we aim at understanding how the strength of a nanoparticle depends on its size and shape, by determining the relation between the applied stress and local stresses. The paper is structured as follows. The models and molecular dynamics calculations are first detailed (Section 2). Next we describe how these models yield during uniaxial compression (Section 3). The effect of shape and size on strength is reported in Section 4. We analyze the relation between local stresses and applied stress in Section 5. It is then shown how the size effects are closely linked to the presence of several stress gradients (Section 6). These results are discussed in Section 7 and summarized in Section 8.

## 2. Calculation methods

We consider FCC aluminum 2D models with different shapes and sizes as characterized by $dx$, $dz$ and $\gamma$ values (Fig. 1). The use of 2D conditions allows for exploring an extensive set of models, up to large sizes, and for easily tuning the shape of the models. The $X = [110]$, $Y = [\bar{1}10]$, and $Z = [001]$ orientations are chosen so as to easily identify dislocation nucleation in {111} planes. A large set of models is employed, with $dx$ and $dz$ ranging from about 20 Å to 1000 Å, and eight different $\gamma$ values. Particular cases are the rectangular shape ($\gamma = 90.0^\circ$), and the hexagonal shape with $\gamma = 125.3^\circ$ which exhibits {111} lateral surfaces. For all other cases, the lateral surfaces are vicinal, thus including steps. In order to save computational time, and to avoid highly pointed
<figure><img src="image_5.png" /><figcaption>**Fig. 2.** (a) Applied stress (GPa) variations as a function of strain $\epsilon$, during the compression of a 2D model with $\gamma = 125.3°$, $dx = 172$ Å, and $dz = 154$ Å. (b–d) Sequence showing different stages during compression. HCP-like atoms, which are indicative of the passing of Shockley partial dislocations, are colored in yellow. At a strain of $\epsilon = 0.046$ (b), a Shockley dislocation has nucleated from the bottom left corner. At $\epsilon = 0.047$ (c), this dislocation has further propagated, reached the upper surface, and rebounded in a different {111} plane. A supplementary alike dislocation nucleated in an adjacent plane, thus forming a nano-twin. Other Shockley dislocations have nucleated and propagated from the top right corner. At $\epsilon = 0.053$, two well formed nano-twins can be observed.</figcaption></figure>

configurations with low stability, the models are trimmed to the left and to the right for $\gamma \ge 136.7°$. Tests were made to ensure that this had no influence on mechanical properties investigations. Models with high $dz$ and very small $dx$ are also discarded in the analyses because their compression leads to the rotation of the system at low strains.

Molecular dynamics simulations at 1 K are performed using LAMMPS (Thompson et al., 2022). Such a low temperature allows for greatly mitigating the computed yield stress overestimation associated with high MD strain rates. A standard EAM interatomic potential is used to describe aluminum. This potential reproduces with a good accuracy the elastic constants, the surface energies, and the gamma surfaces (Mishin et al., 1999). The calculated lattice parameter of FCC bulk Al is $a_0 = 4.05$ Å. The models are placed in a supercell with dimensions along X and Z set as to prevent spurious interactions between consecutive replicas due to periodic boundary conditions. Along Y, the supercell dimension is set to a fixed value of $a_0/\sqrt{2}$, thus defining 2D systems. Two planar repulsive force fields are used to achieve compression, as in previous investigations (Amodeo and Pizzagalli, 2021; Pizzagalli and Godet, 2023). These force fields are initially positioned on top and below the model along Z. The top one is lowered during the compression while the bottom is held fixed. The compression rate is set to $1 \times 10^8$ s$^{-1}$ and the rigidity parameter in the repulsive force field (Amodeo and Pizzagalli, 2021) is equal to 1000 eV Å$^{-3}$ for all simulations.

The atomic stress tensor is computed for all atoms according to the virial stress formalism implemented in LAMMPS (Plimpton, 1995), using a fixed atomic volume equal to $a_0^3/4$, at regular steps during compression. After the simulation, resolved shear stresses (RSS) for all atoms are calculated from the atomic stress tensor for all possible slip systems. For a [001] compression, the maximum RSS is obtained for $[1\bar{1}2]\langle\bar{1}\bar{1}\bar{1}\rangle$ and $\bar{1}\bar{1}\bar{2}\langle\bar{1}\bar{1}\bar{1}\rangle$, with a Schmid factor $m = 0.471$.

Finally, the applied compressive stress $\sigma$ is calculated as the ratio of the applied force divided by the contact surface. The latter is determined according to the following procedure. First we compute the distance between the two outermost atoms along X belonging to the contact surface. A correction of $a_0/\sqrt{2}$ is added to take into account the spatial contribution of these two atoms. Then the contact surface is obtained by multiplying this distance by the system length along Y, i.e. $a_0/\sqrt{2}$.

### 3. Plastic deformation: general behavior

The analysis of the simulations reveals a similar scenario for all investigated systems, irrespective of size and shape. An illustrative example is shown in Fig. 2. An elastic stage is first obtained, with a seemingly homogeneous deformation and a steady quasi linear increase of stress and load. At a given strain, hereinafter defined as the yield strain, sharp and significant stress/load drops are observed. They are concomitant to the formation of Shockley dislocations, i.e. partial dislocations with a Burgers vector equal to $a_0/\sqrt{6}\langle112\rangle$. These dislocations glide in {111} planes. The activation of these slip systems is in agreement with a Schmid factor analysis, since they correspond to a maximum value of 0.471. The dislocations nucleate in the close vicinity of the contact corners formed at the intersection of top or bottom contact surfaces, and lateral facets. By construction, the dislocation nucleation is equiprobable in these four sites. Due to the low temperature in the simulations, we often observe the quasi simultaneous nucleation of dislocations from two or more contact corners.

Following the nucleation of the first dislocations, the plastic deformation proceeds by the combination of various mechanisms. Nucleated dislocations often propagate until they emerge at the opposite surface. Both dislocation nucleation and exit locally modify the surfaces, creating low height steps. This makes easier the formation of new dislocations in adjacent {111} planes (Brochard et al., 2000; Hirel et al., 2008). The repetition of this process, largely documented in previous investigations (Godet et al., 2004a; Hirel et al., 2007), leads to the formation of nano-twins, as seen in Fig. 2. We also observe at a lesser frequency the nucleation of a partial dislocation in a different {111} slip plane than the initial dislocation. Finally, dislocations can cross and interact together, forming
<figure><img src="image_6.png" /><figcaption>Fig. 3. (a-h) Colormaps representing the yield stress $\sigma_y$ (color levels, in GPa) as a function of $dx$ and $dz$, obtained by cubic interpolation of a discrete set of simulations (Each black dot corresponds to a compression simulation). Only values for $dx \le 250$ Å and $dz \le 200$ Å are shown for clarity. There is one colormap for each $\gamma$ value, as represented in Fig. 1. The dashed lines in (e) show the paths used to extract the yield stress profiles shown in Fig. 4.</figcaption></figure>

junctions inside the systems. The succession of these different mechanisms is associated with several stress drops in the stress-strain curve, albeit at much lower levels than for the first initial dislocation nucleation marking the onset of plasticity.

Following the majority of previous published investigations, the strength of each system is defined as the maximum stress $\sigma_y$ at the yield strain, i.e. it is the stress preceding the first large stress drops associated with the nucleation of a Shockley dislocation. Since we find that yield is initiated by the same mechanism for all investigated systems, the comparison of $\sigma_y$ values allows for a thorough analysis of the influence of size and shape.

Finally, note that it is difficult to determine precisely the exact location of the dislocation nucleus, although visual inspection suggests the nucleation occurs in the three first atomic surface layers. It seems then reasonable to qualify this process as heterogeneous, as a truly homogeneous nucleation should probably occur in deeper layers. This result confirms that heterogeneous nucleation is favored in the case of nanoparticles with sharp corners, in contrast to blunt corners for which homogeneous dislocation nucleation is reported (Amodeo and Lizoul, 2017; Sharma et al., 2018).

## 4. Influence of size and shape

We first examine the relation between the strength, size and shapes for the 2D systems. Fig. 3 shows the variation of $\sigma_y$ as a function of $dx$, $dz$ and $\gamma$. Results can be classified according to three groups. For $\gamma = 90^\circ$ and $\gamma = 95.53^\circ$ (Fig. 3-(a–b)), i.e. rectangular and quasi-rectangular shapes, the strength values are high, with little dependence on dimensions. It is particularly true for the rectangular shape, where the computed yield stresses range from 6.8 GPa to 8.4 GPa. A value of 8.1 GPa is obtained for the largest model with $dx = 1008.05$ Å and $dz = 1008.45$ Å. A second category includes models with $\gamma = 106.4^\circ$ and $\gamma = 115.24^\circ$ (Fig. 3-(c–d)). For those the stress is overall smaller, with values lower than 6 GPa for $dz \gtrsim 50$ Å. Below this threshold, there is a marked strength increase when $dz$ decreases. Finally, the last group gathers models with $\gamma \ge 125.26^\circ$ (Fig. 3-(e–h)), for which the yield stress markedly increases above 6 GPa when either $dx$ or $dz$ become small.

These results confirm the significant influence of shape on the mechanical strength of nano-objects, in agreement with previous investigations (Amodeo and Lizoul, 2017; Kilymis et al., 2018). It is noteworthy that the yield stress was associated to the activation of different plasticity mechanisms in these referenced studies, depending on nanoparticle shape. Conversely, in the present work, the plastic deformation is always initiated by the heterogeneous nucleation of a Shockley dislocation from contact corners.

To better quantify the yield stress variations as a function of size and shape, we extract values from the colormaps in Fig. 3 according to three profiles: $dx = dz$ (a), $dx = 100$ Å (b), $dz = 100$ Å (c). The results are plotted in Fig. 4. For profile (a), the system dimensions are conjointly modified, i.e. the size changes with a constant shape. For a rectangular shape ($\gamma = 90^\circ$), $\sigma_y$ is high for small systems, about 7.5 GPa, and slowly increases to a constant value of 8 GPa for larger sizes. However, drastic changes are observed even with only a slight departure from the rectangular shape. For $\gamma = 95.53^\circ$, the yield stress at the smallest size is similar to the one for $\gamma = 90^\circ$ but it decreases when the size grows. The lowering remains moderate, since $\sigma_y$ is about 5.5 GPa for $dx, dz \approx 1000$ Å. For larger $\gamma$ values, we observe a sharp $\sigma_y$ decrease starting at the lowest sizes. Overall the larger $\gamma$, the larger the lowering rate.

One can reasonably consider that models with $dx = dz$ and $\gamma = 90^\circ$ are 2D projections of 3D cubic nanoparticles. We found high yield stress and a weak size variation, in agreement with previous investigations (Shreiber and Mordehai, 2015; Wagner et al., 2015; Kilymis et al., 2018; Chen et al., 2022) showing no or a weak size effect on the strength of small cubic nanoparticles. 2D systems with $dx = dz$ and $\gamma = 125.26^\circ$ can be compared to 3D Wulff-shaped nanoparticles including {111} and {001} facets. Fig. 4 shows that the strength is high for small sizes, and largely decreases when the latter increase. These stress-size relations are in agreement with previous studies (Amodeo and Pizzagalli, 2021). Note that the use of 2D systems allows for a detailed investigation
<figure><img src="image_7.png" /><figcaption>**Fig. 4.** Yield stress profiles as a function of size ($dx = dz$) (a), of height ($dx = 100$ Å) (b), and of width ($dz = 100$ Å) (c), extracted from the yield stress maps shown in Fig. 3. Each color corresponds to a different $\gamma$ value.</figcaption></figure>

of the influence of the $\gamma$ value on the strength. Hence, our calculations reveal that only small deviations from 90° are enough to recover the strength decrease as a function of size.

For the second profile, the height is varied while the width is constant (Fig. 4-(b)). It appears that $\sigma_y$ variations are similar to the $dx = dz$ case, except for the rectangular shape ($\gamma = 90°$). For the latter, $\sigma_y$ is equal to 8 GPa at the smaller size, then decreases to about 7.5 GPa as height is increased. Finally, we focus on the last profile (Fig. 4-(c)), corresponding to a constant height and a variable width. High and weakly increasing $\sigma_y$ values are obtained for low $\gamma$ angles (90°, 95.53°). Variations similar to profile (a) and (b) are obtained for $\gamma \ge 125.26°$. Differences are observed for $\gamma = 106.4°$ and $\gamma = 115.24°$. For the former $\sigma_y$ is only 4.5–5 GPa for small $dx$ values, and decreases weakly as width increases. For the other the stress at the smaller width is 1 GPa higher, and its decrease with decreasing width is more pronounced. To our knowledge width and height are always varied conjointly in previous studies, therefore no comparison with the literature is possible for constant-width and constant-height profiles.

The curves for profiles (a–c) in Fig. 4 are next fitted using the general expression $\alpha d^{-n}$ for each $\gamma$ value, as it is commonly done in the literature (Amodeo and Pizzagalli, 2021). $d = dx = dz$ for profile (a), $d = dz$ for profile (b), and $d = dx$ for profile (c). Computed $n$ values are reported in Fig. 5. For profile (a), we found that $n$ is close to zero for $\gamma = 90°$ value, which was expected. In fact, the greater the value of $n$, the stronger the size effect, and $n = 0$ means a constant strength. $n$ then increases to reach a maximum value of 0.40 for the hexagonal systems ($\gamma = 125.26°$), then slowly decreases. Note that $n$ values reported in the literature for FCC metals are approximately 0.5 for Wulff-shaped and spherical nanoparticles (Feruz and Mordehai, 2016; Bel Haj Salah, 2018), and 0.74 for Winterbottom-shaped nanoparticles (Mordehai et al., 2011). However, a direct comparison with our 2D systems is probably not meaningful. The $n$ variation clearly demonstrates that the size effect critically depends on the shape variation, and more precisely the angle $\gamma$ between the contact surface and lateral facets. Fig. 5 also shows $n$ exponents for profiles (b) (constant width) and (c) (constant height). For the former $n$ is overall smaller than for profile (a) but with similar variations. For profiles (c), $n$ is slightly negative at low angles, then increases continuously as a function of $\gamma$ up to values remaining smaller than those for profile (a). These results suggest that the observed size effect could be considered as the addition of constant-width and constant-height contributions.

In a recent article, Zimmerman and co-workers proposed to model the size and the shape variation of platinum strength by using the expression $\sigma_y = \beta D^{-p}G^{-q}$ (Zimmerman et al., 2021). $G$ is defined as the ratio between the contact surface effective diameter and the diameter $D$ of the projected area of the nanoparticle on a plane perpendicular to the applied loading. In the present work, we assume that the system height is equal to $D$, i.e. $D = dz$. The ratio $G$ is then equal to $dx/dz$. The expression becomes $\sigma_y = \beta dx^{-q}dz^{q-p}$. With this expression, we perform 2D fits of the strength variations shown in Fig. 3 for each $\gamma$ value. The right side of Fig. 5 shows the values of $p$ and $q$ as a function of $\gamma$. We find that the $p$ is close to $n$ values, except for minor differences at low angles. The shape exponent $q$ is negative at low angle, and increases as a function of $\gamma$ up to 0.06. These values are drastically different from the reported $q = -1.66$ for platinum nanoparticles (Zimmerman et al., 2021).
<figure><img src="image_8.png" /><figcaption>Fig. 5. Left: power law exponent *n* as a function of the contact corner angle *γ*, calculated using the expression *αd*⁻ⁿ for the three sets of profiles shown in Fig. 4 (blue curve for Fig. 4-(a) (*dx* = *dz*), orange curve for Fig. 4-(b) (*dx* = 100 Å), green curve for Fig. 4-(c) (*dz* = 100 Å)). Right: power law exponent *p* (blue curve) and *q* (purple curve) versus *γ*, from a 2D fit of stress maps shown in Fig. 3 using the expression *σ*<sub>*y*</sub> = *βd*<sup>*q*</sup>*dx*<sup>*q*−*p*</sup>.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 6. Sequences showing the heterogeneous nucleation of a Shockley dislocation from the corner of a rectangular (a–d) and hexagonal (e–h) system at different strains. The atoms are colored according to their local RSS, for the slip system [112](111). This figure is generated using Ovito Stukowski (2009).</figcaption></figure>

The main conclusion of this section is that the yield stress $\sigma_y$ critically depends on the angle of the contact corners, i.e. on the shape of the system. It is also shown to follow a power law dependence as a function on both the height and the width, except for the rectangular case ($\gamma = 90^\circ$). From this perspective, the yield stress depends on both the shape (aspect ratio) and the size (width equal to height). The size effect can be thought as the combination of height and width contributions.

**5. Local stress analysis at a contact corner**

In this section, we investigate the relation between the local stress, and the size and shape during uniaxial compression. In particular, we focus on conditions favoring the heterogeneous formation of dislocations at contact corners. Fig. 6 shows the local stress state in the region where a dislocation nucleates, at strains before and after the onset of plasticity. Previous atomistic calculations correlated the nucleation of the first dislocations to the stresses gradients in the vicinity of the surface contact corners, during the compression of nanoparticles (Mordehai et al., 2011; Amodeo et al., 2021; de la Rosa Abad et al., 2021). This phenomenon is similar to the nucleation of dislocation from steps during the traction/compression of surfaces (Brochard et al., 2000; Godet et al., 2004b, 2006; Hirel et al., 2008; Brochard et al., 2010). In that case, the step acts as a stress concentrator, and dislocations are observed to form and glide in the plane passing through the step. The nucleation is then favored by both the presence of high stresses or stress gradients, and the step geometry.

Note that there were considerable efforts by several groups to identify a suitable dislocation nucleation criterion in relation with local information in the context of homogeneous nucleation of dislocation during indentation. For instance, it was put forward that the lowest eigenvalue of the atomic-level acoustic tensor could be used to determine the nucleation location (Li et al., 2004; Zhu, 2004). However, this proposition was challenged by Miller and Rodney who showed that it could fail on several occasions (Miller and Rodney, 2008). More recent attempts to find a robust criterion were made by Garg and Maloney (Garg and Maloney, 2016). We think that these analyses are especially relevant in the case of homogeneous nucleation. In the present work, all our calculations unambiguously reveal that heterogeneous nucleation of dislocation occurs at locations where the resolved shear stress, or an averaged stress like the von Mises stress, is maximum. It is therefore only necessary to examine the variation of local stress as a function of applied stress in the vicinity of contact corners, to determine the respective influence of size and shape on the onset of plasticity.

We first analyze the distribution of local stress in a crystalline system of finite size, subjected to a compressive stress according to the setup described in the previous section. If the geometry is such that the cross-section along the compression axis is identical by translation, as is the case for cubic or cylindrical shapes, then the distribution of internal stress is globally uniform. Differences lie in
<figure><img src="image_10.png" /><figcaption>Fig. 7. Resolved shear stress $\tau$ (normalized by the applied compressive stress $\sigma$) as a function of the distance from the top left corner, along the (111) plane (shown as a white dashed line in insets), for cubic (left) and hexagonal (right) shapes, for four different strain values (before the nucleation of a dislocation). Thin dotted black lines in right graph are fits of $\tau/\sigma$ data using the expression $a r^{-b} + c$ with a, b and c as fitting coefficients. The horizontal dashed purple lines correspond to $\tau/\sigma = 0.471$, the Schmid factor for a [001] compression and the [112](111) slip system.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Fig. 8. Variation of the power law exponent $\lambda$ as a function of the angle $\gamma$ from Eq. (3). The inset sketch shows an infinitely hard ($\mu = \infty$) flat punch indenter in contact with one corner of the 2D system, characterized by the angle $\gamma$.</figcaption></figure>

the vicinity of surfaces, where stress variations are observed. The theory of surface stress is well established (Müller and Saül, 2004). It is short-ranged (Pizzagalli and David, 2021), and as such it is expected to be important for the first surface layers. Fig. 6-(a-d) shows the spatial distribution of the RSS for the slip system [112](111), in a rectangular system. We can see that before dislocation nucleation it is fairly uniform, with significant deviations for the first three surface layers only. Extracting $\tau$ variation along a {111} slip plane (Fig. 7, left graph), confirms that the RSS distribution is spatially uniform except for the first surface layers. In average, the RSS far from the surfaces is equal to $m\sigma$ with a Schmid factor $m = 0.471$, independently of the compressive deformation up to the onset of plasticity. We can then write that

$$ \tau(\mathbf{r}, \sigma) = m\sigma + \Gamma(\mathbf{r}) \qquad (1) $$

with $\Gamma(\mathbf{r})$ the excess local stress associated with the presence of the surface. This quantity is then expected to be significant only in the vicinity of surfaces. Note that if one integrates $\Gamma(\mathbf{r})$ over the volume of the system, divided by its surface, the surface stress is obtained.

If the cross-section is not identical by translation along the compression axis, the distribution of internal stress is not homogeneous anymore, even if the material is free of initial defects. In fact local increases of stress can be present depending on the system geometry and the conditions of mechanical loading (Mura, 1987; Murakami, 2017). For instance stress concentrations are predicted to occur in the vicinity of the four corners in contact with planar indenters in the case of a 2D hexagonal shape under compression. We show in Fig. 6 the spatial distribution of the RSS for a compressed hexagonal shaped system, which confirms this scenario. The local stress appears to be maximum close to the surface, and decreases when going deeper in the system (Fig. 7). A similar behavior is anticipated for 3D systems exhibiting sharp corners, like Wulff shapes (cuboctahedron, ...). These local stress concentrations are referred to as stress gradients in Ref. Mordehai et al. (2011).

Local stress concentrations can be predicted using isotropic elasticity theory if shapes and contacts are idealized (Williams, 1952; Bogy and Wang, 1971; Dundurs and Lee, 1972), like represented in the inset of Fig. 8. In these descriptions, stress concentrations are often called stress singularities since the local stress can be unrealistically large at the edge of corners. In the general case, the stress tensor components can be expressed in polar coordinates as:

$$ \tau_{ij}^{s}(r, \theta) = \sum_{l} k_{l} r^{-\lambda_{l}} f_{ij}^{(l)}(\theta) \qquad (2) $$
<figure><img src="image_12.png" /><figcaption>Fig. 9. Power exponent b values determined by fitting the expression $ar^{-b} + c$ with the computed RSS, for three different strain values (hexagonal system, $\gamma = 125.3^\circ$, Fig. 2). The shaded areas show the standard error value for each fit. r is the distance along the {111} plane passing through the corner edge, with an origin defined at the surface. The fits are performed using RSS values in variable range, from the 2nd atom (left graph) relatively to the corner edge (see the inset in Fig. 10), or from the 3rd atom relatively to the corner edge (right graph). The black dashed line shows the exponent value 0.41 predicted by isotropic elasticity theory.</figcaption></figure>

The sum in Eq. (2) implies that there are several contributions, referred by $l$, to the stress. $k_l$ are stress intensity factors, which are commonly used to describe the stress distribution in the vicinity of cracks (Sinclair, 2004). $f_{ij}^{(l)}(\theta)$ are angular functions. $\lambda_l$ are coefficients quantifying the asymptotic variation of stress at the corner ($r \to 0$), with $0 < \lambda_l < 1$.

Both $\lambda_l$ and $f_{ij}^{(l)}(\theta)$ depend on the geometry of the system, and of the conditions of mechanical loading. Dundurs and Lee solved the case of a sharp corner in frictionless contact with a half plane (Dundurs and Lee, 1972). When the latter is infinitely hard, this configuration is equivalent to the compression of our variable shaped 2D systems by a planar repulsive force fields. In this case, there is a unique $\lambda$ coefficient, which is obtained as the root of the equation

$$ (1 - \lambda) \sin(2\gamma) + \sin(2\gamma - 2\gamma\lambda) = 0 \qquad (3) $$

with $\gamma$ the contact corner angle as represented in Fig. 8. This figure shows $\lambda$ values as a function of $\gamma$, which range from 0.0 up to 0.5. Clearly, the stress singularity variation in the vicinity of the corner edge critically depends on the angle. Note that if $\gamma \to 90^\circ$, $\lambda \to 0$ and the stress concentration vanishes in agreement with atomistic simulations for a rectangular system.

For the hexagonal shape represented in Fig. 6, $\gamma = 125.26^\circ$ and $\lambda = 0.41$. The stress is then predicted to vary as $r^{-0.41}$ when $r \to 0$. Atomistic simulations for the hexagonal shape reveal a seemingly power law decrease of the RSS (Fig. 7). However, an important difference is that values do not become unrealistically large at the contact edge, as predicted for an elastic singularity. This is due to the influence of surface and the discrete nature of crystals. Finally, far from the surface, one recovers the $m\sigma$ value. Therefore, in the general case ($\gamma \neq \pi/2$), one can write:

$$ \tau(\mathbf{r}, \sigma) = m\sigma + \Gamma(\mathbf{r}) + g(\theta)r^{-\lambda} \qquad (4) $$

In this expression, $g(\theta) = k f(\theta)$ by comparison with Eq. (2). As it stands, the description of the local stress with Eq. (4) includes a system shape dependence hidden in the value of $\lambda$, related to the corner angle. However, it does not explicitly depend on the size of the system and on the aspect ratio, i.e. $dx$ and $dz$. Actually extra dependences are hidden in the stress intensity factor $k$. The latter is often expressed as

$$ k = \nu\sigma L^{\lambda} \qquad (5) $$

$k$ is linearly proportional to the applied stress $\sigma$ and to $\nu$ which is a geometric factor. The last factor $L^\lambda$ is necessary to ensure the dimensionality of Eq. (2), with $L$ a characteristic dimension of the system. In their pioneering work, Mordehai and co-workers proposed to use the nanoparticle diameter for $L$ (Mordehai et al., 2011). Combining a nucleation model and an equivalent expression for the local stress, they showed that it yields a $\alpha D^{-\lambda}$ relation between the yield stress and the nanoparticle size. Zhang and co-workers used the height of their 2D systems as characteristic dimension instead, and proposed that the geometric factor $\nu$ in Eq. (5) depends on the aspect ratio (Zhang et al., 2006).

We now use the output of our calculations, and in particular the RSS distributions calculated for an hexagonal system ($\gamma = 125.3^\circ$, Fig. 2), to assess the validity of this model. We consider the largest systems ($dx, dz \approx 1000$ Å) in order to decorrelate as much as possible the local stress concentration associated to the contact corner geometry and the size effect. According to Eq. (4), the spatial variation of the local stress is of the form $ar^{-b} + c$ for a given applied stress and a fixed direction (i.e. $\theta$ is constant), and neglecting the contribution of surface stress. We fit the calculated RSS at three different strain values, using values associated with the atoms belonging to the {111} slip plane passing through the corner edge (as represented in Fig. 7). The fits are performed over a variable range of atoms, in order to estimate the influence on the results, and in particular on the power exponent $b$. The first atom in the range is either the 2nd or the 3rd atom with respect to the surface, i.e. the atoms for which the RSS values are high. For both cases, ranges including from 10 to 200 atoms are considered. Note that only coefficients $a$ and $b$ are calculated, and that $c$ is taken to be equal to the RSS values averaged over 100 atoms along the same plane and far from the surface. The results for $b$ are shown in Fig. 9. Obviously the power exponent strongly depends on the range used in the fitting process, with no visible convergence despite the
<figure><img src="image_13.png" /><figcaption>Fig. 10. RSS τ as a function of the contact applied stress σ, calculated at atoms belonging to a (111) plane, the proximity of which the dislocations are observed to be nucleated. These atoms are marked with consecutive numbers, starting from the corner edge (inset graph), corresponding to the labels of the legend. The full curves show data for a selection of these atoms. Dashed colored lines are linear regressions of these data. The thick dashed black line is characterized by a slope of 0.471, the Schmid factor for a [001] compression and the [112](111) slip system.</figcaption></figure>

large size of the compressed system. We also find that the resulting value is largely dependent on the strain, and therefore on the applied stress. Finally, it is also observed that excluding the 2nd atom from the fitting ranges significantly increases the b value, as well as the scattering. Therefore we conclude that it is possible to find any values between 0.3 to 1.1 for the power exponent, depending on the conditions selected for the fit. For comparison, typical values published for 3D nanoparticles are 0.77 (Mordehai et al., 2011), 0.5 (Feruz and Mordehai, 2016), 0.16 and 0.28 (Pizzagalli and Godet, 2023), and 0.83 (Sharma et al., 2018).

Our analyses then show that it is not possible to unambiguously determine a power exponent coefficient describing the RSS variations in a stress concentration region, at the scale of our models. We believe that a possible explanation is the influence of surface. In fact, the elasticity theory of stress singularity states that the power law variation with the theoretical λ value is essentially recovered at the origin of the singularity, i.e. at the surface (Sinclair, 2004). However, in real systems as modeled using atomistic calculations, both the surface stress and structural relaxation at the surface cannot be neglected. The surface stress is short ranged but its influence can be felt at least for the first surface layers. This stress contributes to the local RSS value and modifies the shape of the RSS spatial variations. This result is in agreement with a comparison between atomistic and finite elements calculations made by Li and Picu (2010). These authors show that the power exponent predicted by the theory is not obtained in atomistic calculations, and asymptotically approached in finite element calculations only at surface separations much lower than 1 Å. Furthermore, there is an elastic interaction between the surface and the stress singularity, which also changes the RSS profile in the vicinity of the surface. The power exponent of the RSS spatial variation included in Eq. (4) is therefore an ill-defined quantity. We conclude that the model associated with Eq. (4) is too simple to quantitatively model the local variation of RSS during the compression of systems with corner angles different from 90°.

We now examine how the RSS τ depends on the applied stress σ. Combining Eqs. (4) and (5), τ can be written

$$ \tau(\mathbf{r}, \sigma) = \left[ m + \nu f(\theta) \left( \frac{L}{\Gamma} \right)^{\lambda} \right] \sigma + \Gamma(\mathbf{r}) \quad (6) $$

Assuming a linear relationship between surface stress and σ, which is reasonable for a FCC metal like aluminum (Pizzagalli and David, 2021), we find that τ should linearly depend on σ. Fig. 10 shows the relation between τ and σ for the hexagonal system with large dimensions (dx, dz ≈ 1000 Å). Although small fluctuations are present, there is clearly a linear relationship between τ and σ for all atoms belonging to the {111} slip plane. The lowest slope corresponds to the first atom at the contact edge, and the largest to the 2nd atom. Away from the surface, the curves tend to converge to the same slope value. According to Eq. (6), this proportionality coefficient should be equal to the Schmid factor m. Focusing on extrapolated τ at the origin, we find significant values for the first atoms only, close to the surface, while they become zero for deeper atoms. This is a reasonable result since the value at the origin is the surface stress according to Eq. (6).

Note that a similar analysis was performed for systems with different sizes and different shapes, and that a linear relationship between τ and σ is obtained in all cases. This demonstrates that the linear relationship between applied stress and RSS as included in Eq. (6) is a general property of nanoscaled systems. We plot in Fig. 11 the slope ∂τ/∂σ and the intercept values computed from the linear regression of τ-σ relations for atoms along the {111} slip plane, for various systems. For the large hexagonal system (B in Fig. 11), ∂τ/∂σ is almost zero for the 1st atom, at the surface. Then it becomes maximal for the next atom, and slowly decreases when moving away from the surface. The power law dependence as shown in Fig. 7 is clearly recovered, which was expected since it contributes to ∂τ/∂σ (Eq. (6)). According to Eq. (6), the intercept value is Γ, i.e. it yields information about the magnitude and range of local surface stress. For the hexagonal system, we find that it is non negligible only for the first three atoms relatively to the surface. This confirms the short range of surface stress contributions. We emphasize that the intercept values reported in Fig. 11 likely depend on the definition of the atomic volume.

Fig. 11 also shows ∂τ/∂σ and intercept variations for a rectangular system (A in Fig. 11). Due to the absence of stress singularity, Eq. (6) becomes τ(r, σ) = mσ + Γ(r) in this case. Our results show that the ∂τ/∂σ is constant except for the four first atoms close to
<figure><img src="image_14.png" /><figcaption>**Fig. 11.** Top graph: slope coefficients $\partial\tau/\partial\sigma$ of the $\tau-\sigma$ curves, for all atoms belonging to the {111} slip plane passing through the corner edge (as represented in the inset in Fig. 10). The inset graph is a zoom of the domain for atoms close to the corner. The black dashed line marks the value 0.471 of the Schmid factor for the [11̄2](11̄1) slip system and a [001] compression. Middle graph: Variation of the intercept values for the same $\tau-\sigma$ curves. The six curves labeled A–F correspond to different systems, represented in the bottom at the same scale.</figcaption></figure>

the surface, and that its value is equal on average to the Schmid factor $m = 0.471$. In regions where dislocation nucleation occurs, $\partial\tau/\partial\sigma$ is also much lower than for the hexagonal system. This is in agreement with the analysis made in the previous section. For a given applied stress value, a larger proportionality coefficient means that the local RSS will be higher for the hexagonal system compared to the rectangular one. In others words, the stress transfer from the indenter to the stress concentration region is more efficient in the case of the hexagonal system. The influence of size/shape can then be detected in the analysis of the local RSS. We also observe that the intercept curve for the rectangular system is different than for the hexagonal system, with overall higher values, and also a longer range of surface stress contributions. This difference is probably related to atomic structures of the contact edge, which involves a {111} surface for the hexagonal system, but a {001} surface for the rectangular one.

We now consider hexagonal systems with the same contact angle, but with a different height (C in Fig. 11), width (D), or size (E) compared to B. In the region where dislocation occurs, i.e. close to the corner edge, we can see that the magnitude of $\partial\tau/\partial\sigma$ is the largest for B, followed by C and D, and then by E. This is in agreement with the ordering of the calculated yield stresses (Fig. 3). This shows that the influence of size can also be detected in the applied stress dependence of the local RSS. Note also that the intercept coefficient exhibits the same variation for B,C, D, and E, which confirms that it is a signature of surface stress that depends only on the atomic configurations at the edge corner.

To conclude for this section, the analysis of the spatial and applied stress dependence of the local stress reveals that the spatial dependence is complex when stress singularities occur, and that a simple model based on elasticity theory and a single power law is not satisfactory. In particular, this implies that the size dependence of yield stress is not directly related to the power law variation of the local resolved shear stress. However our results clearly show that the local stress linearly depends on the applied stress, and that regression coefficients bear the signature of the shape and size effects, and of surface contributions.
<figure><img src="image_15.png" /><figcaption>Fig. 12. Stress concentration regions during compression and before plastic deformation (indicated by the dashed ellipse): for an hexagonal system (B in Fig. 11) on the left, for a quarter-truncated rectangle (F in Fig. 11) on the right.</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>Fig. 13. Colormaps representing the yield stress $\sigma_y$ (color levels) as a function of $dx$ and $dz$, obtained by cubic interpolation of a discrete set of simulations (Each black dot corresponds to a compression simulation), for (a) rectangular ($\gamma = 90.0^\circ$, A in Fig. 11), (b) hexagonal ($\gamma = 125.3^\circ$, B in Fig. 11), (c) quarter-truncated rectangular systems (F in Fig. 11). Only values for dimensions lower than 250 Å are shown for clarity. The dashed lines show ($dx$, $dz$) values used to extract the yield stress profiles shown in Fig. 14.</figcaption></figure>

## 6. Considering multiple stress singularities

As originally proposed by Mordehai and co-workers, the onset of plasticity in crystalline pristine nanoparticles is linked to the nucleation of dislocations from contact corner edges, where stress concentrations develop (Mordehai et al., 2011). Subsequent investigations have confirmed this point (Amodeo and Pizzagalli, 2021). Note that in these studies only one stress concentration region is taken into account. For instance, the stress gradient model describes the dislocation model mechanism from a single singularity (Mordehai et al., 2011), and recovers a size effect by introducing a characteristic length ($L$ in Eq. (5)), required by the theory of stress singularities. It is important to note that this length is not related to the presence of other stress concentration regions, and is needed in Eq (5) for dimensional homogeneity even if only one stress singularity is present. However it is noteworthy that in a 2D hexagonal system like that studied in the present work, there are four equivalent stress concentrations forming during compression (Fig. 12). A first consequence is that the probability to nucleate a dislocation and to initiate plastic deformation is higher than for a single one, at least at temperatures higher than the one considered here. A second important consequence is that there is an elastic interaction between stress concentration regions. We believe that this aspect has been overlooked until now, despite being hinted by some authors (Zhang et al., 2006; Amodeo and Lizoul, 2017). In this section we examine whether those elastic interactions could play a role in the observed size effect.

A careful examination of the slope coefficients $\partial\tau/\partial\sigma$ in Fig. 11 brings interesting and valuable insights. According to Eq. (6), the coefficients should converge to the Schmid factor $m$ far from the stress singularity at the corner edges. This is not the case for B and D for which the slope coefficients are lower. This could be the fingerprint of a long range elastic interaction, which would modify the variation of local RSS as a function of the applied compressive stress. Values lower than $m$ suggest a screening interaction, which could explain why larger applied stresses are needed to reach the yield point when the systems are downsized. To provide decisive evidence we build a new system with a specific geometry. Starting from a rectangular shape, we trim one corner so as to create a lateral {111} surface, now forming a contact corner with an angle $\gamma = 125.26^\circ$, and intersecting the lateral {110} surface at midheight (see F in Fig. 11 and right side of Fig. 12). This system includes only one contact corner with $\gamma \neq 90^\circ$, and therefore stress concentration is expected to occur in one region during compression, and not in four as in hexagonal systems (Fig. 12). Our analysis of the local RSS distributions during compression confirms this point.

Fig. 13 shows a colormap of the yield stress as a function of $dx$ and $dz$ for the quarter-truncated system, as well as for the rectangular and hexagonal ones for comparison. The first observation is that lower yield stress values are associated to the quarter-truncated system compared to the two others for similar dimensions. It is especially striking relatively to the rectangular model,
<figure><img src="image_17.png" /><figcaption>Fig. 14. Yield stress profiles as a function of size ($dx = dz$) (top graph), of height ($dx = 100$ Å) (left bottom graph), and of width ($dz = 100$ Å) (right bottom graph), extracted from the yield stress maps represented in Fig. 13, for the systems A, B, and C shown in Fig. 11.</figcaption></figure>

which is geometrically close to the new system. It is also surprising that the values are 1–2 GPa lower compared to the hexagonal shape, although stress concentration occurs in four regions in the latter. The second interesting feature is the absence of size effect when only the width $dx$ varies, which strikingly contrasts with the hexagonal system. However, an influence of dimension remains present when only the height ($dz$) is changed, albeit with lower magnitude than for the hexagonal shape. Extrapolated yield stress profiles, presented in Fig. 14, confirm these results, such as the absence of size effect when only width is varied. The profiles for homothetic variation ($dx = dz$) also suggest that the yield stress for the quarter-truncated system exhibits a power law variation similar to the hexagonal system, but with overall lower values. For the largest investigated size ($dx, dz \approx 1000$ Å), the yield stress is approximately 0.5 GPa lower.

These results hint that the size effect revealed in previous molecular dynamics calculations is due to the long-range interactions between the regions where stress concentrations occur. Lowering the distance between these regions decreases the slope coefficient of the applied stress dependence. As a consequence, a higher applied stress is needed to reach the critical RSS, and smaller systems appear stronger than larger ones. In the case of quarter-truncated system, there is a single region with a stress singularity, and therefore no repulsive elastic interaction with another region. Therefore for the RSS dependence on applied stress, one might expect a larger slope coefficient for the quarter-truncated than for the hexagonal system. This is in fact what is obtained, with a coefficient which is about 30% higher (F in Fig. 11). There is probably an elastic interaction between the stress concentration area and the opposite lateral {110} surface for the quarter-truncated system, but if any it is probably small since it does not clearly appear in our results. Finally we note that the proposed scenario does not explain the apparent lack of elastic interaction for system C in Fig. 11, since the slope coefficient converges to the Schmid factor $m$ in that case. Future investigations are needed to better characterize the elastic interaction between stress concentration regions and its influence on dislocation nucleation.

For systems depicted in Fig. 1 except the ones with rectangular shape ($\gamma = 90°$), there are four stress singularities forming at contact corners. There is an elastic interaction between all of them, which explains that two apparently similar size dependencies are obtained when the width and the height are varied independently. The size effect observed when both dimensions are changed is then the addition of these two effects. It is however puzzling that we observe an increase of the yield stress when the height of the quarter-truncated system is very small, since there is only one stress concentration region. We believe that it is related to the mechanical constraints associated with the uniaxial compression. Top and bottom {001} surfaces being in contact with a very hard planar indenter, they behave like fixed surfaces. It is therefore more difficult for a propagating dislocation to leave the nanoparticle from these surfaces, compared to the lateral ones, which are free to relax. This phenomenon is especially present for systems with low height and $dx \gg dz$.
<figure><img src="image_18.png" /><figcaption>Fig. 15. Wulff-shaped FCC nanoparticle (left). A cross section (right) exhibits a shape equivalent to the hexagonal 2D models investigated in the present work.</figcaption></figure>

## 7. Discussion

Our main objective is to understand how the strength of a nano-object like a nanoparticle depends on its size and shape. At the nanoscale, the strength is usually correlated to the onset of plastic deformation, which is caused by the heterogeneous nucleation of dislocations. The triggering conditions depend on the local stress distributions in the nanoparticle. It is therefore interesting to examine both local and applied stresses as a function of nanoparticle size and shape. In the specific context of the uniaxial compression of 2D systems of finite depth, the results of our investigations bring original insights, which we now gather to propose a synthesis of the whole process.

In the system at rest, the local stress $\tau$ is zero except close to the surface, where surface stress contributions are present. Upon the application of a compressive load, a local stress builds up throughout the system. If the cross-section along the compression axis is constant, the angle between the surface in contact with the indenter and lateral surfaces is 90°, and the local stress contribution due the load is simply $m\sigma$. The local stress field is therefore homogeneous except in the vicinity of surfaces. When the angle of the contact corner is different from 90°, there is a local stress/strain concentration. According to elasticity theory, it is assumed that it could be described by a stress singularity at the corner edge, and a power law decrease when moving away from the singularity with an exponent depending on the corner angle.

The local stress $\tau$ then depends on the applied stress $\sigma$ and on the position inside the system. Our investigations show that the spatial dependence is intrinsically complex. In fact, the stress concentration is largely influenced by the presence of the surfaces. Surface stress and stress relaxation at the surface modify the local stress values in regions where the heterogeneous nucleation of dislocation occurs. Finally, the local stress distribution is also modified by the long range elastic interaction between all stress inhomogeneities. This explains why the spatial dependence of the local stress in the vicinity of a corner cannot be described using a simple decreasing power law. We show that it is meaningless to determine an exponent by fitting, since its value depends on the fitting range and on the applied stress. Conversely, our analyses show that the local stress linearly depends on the applied stress up to the onset of plasticity, at any points in the system. The intercept value of the linear relation (Eq. (6)) allows to recover surface stress contributions. The proportionality coefficient is a good indicator for understanding the influence of shape and size on the yield stress. In fact the larger this coefficient, the higher the local stress for a given $\sigma$ value. And then the larger the coefficient the lower the yield stress. The proportionality coefficient depends on the corner angle which determines the stress concentration magnitude. It also depends on the width and height of the system, due to the interaction between stress concentration regions. There are then two different shape factors, the corner angle and the ratio between width and height, and an apparent size factor, when the width and the height are proportionally changed.

In all of the above, we focused on 2D systems for the sake of simplicity, and also because analytical elastic solutions for stress concentration at corners are available. It also allows for the calculation of a large and numerous models, with different shapes and dimensions. Nonetheless we believe that our conclusions are not restricted to 2D systems, since all important factors exist in 3D systems. For instance, the concentration of stress also occurs at 3D corners, although the spatial dependence of stress is more complicated than in 2D and can only be solved numerically to our knowledge. In addition, the local stress will depend on surface stress, and on the interaction between stress concentration and surface. It will also be modified by the elastic interaction between all stress concentration regions, which are necessarily more numerous than in 2D systems. For instance, there are 8 for a truncated octahedron nanoparticle, and 16 for a truncated cuboctahedron (Fig. 15). Both shapes are obtained using the Wulff construction using the known surface energies of FCC metals, and are often used as model in molecular dynamics calculations of the mechanical properties of nanoparticles (Feruz and Mordehai, 2016). It is therefore likely that the spatial dependence of the local stress is complex and difficult to formalize for 3D nanoparticles. However, the linear relationship between local and applied stresses is certainly preserved.

With these conclusions in mind, we can discuss some key results of the literature. A first one is the prediction of a universal size effect exponent for the yield stress (Feruz and Mordehai, 2016). Our results show that the power law exponent in the applied stress variation only depends on the nanoparticle shape. Therefore as long as the shape is preserved, the exponent will not change as a function of size. This could explain why the same exponent is obtained for different metals if almost similar Wulff-shape nanoparticles are considered (Feruz and Mordehai, 2016).
A second point concerns the comparison with experimental results. FCC metal nanoparticles used for mechanical properties testing are usually larger than our models, and they have also significantly different shapes. In particular, they appear to be roundier, with blunted edges and small facets of limited areas (Mordehai et al., 2011; Sharma et al., 2018; Zimmerman et al., 2021). This largely modifies the concentration of stress at the contact edges, and therefore on the onset of plasticity. The roughness at the nanoparticle surface could also influence the results (Iteney et al., 2024b,a). From this perspective our findings cannot be expected to provide a quantitative description of experiments. They are however useful to improve our fundamental knowledge of plastic deformation in nanoparticles, and they can be seen as the basis for building more realistic and ambitious models.

In this study, we restrict ourselves to hexagonal shapes, which are 2D equivalents of Wulff-like shapes like truncated octahedron. Spherical nanoparticles are also often employed in previous investigations. Models used in uniaxial compression simulations are typically small, with a surface composed of multiple steps and facets of limited area (Bel Haj Salah et al., 2017). Each step under stress can act as a stress concentrator (Brochard et al., 2000; Hirel et al., 2007), thus being the source for dislocation nucleation. Conversely, their efficiency could be limited by their interaction. This screening may explain why a transition towards a Hertz-like regime and the homogeneous nucleation of dislocation could occur for large sizes. This problem is an interesting perspective to continue the present work.

Finally, it is noteworthy that our calculations are performed at 1 K. It is beneficial for reducing the influence of thermal vibrations during the local stress analysis, minimizing the stochastic character of the yield stress determination, and also largely canceling the effect of the large strain rates used in simulations. One may wonder whether the conclusions exposed here would remain valid at room temperature. We believe that it would be the case, but only additional calculations could bring a decisive confirmation. An important aspect to consider is that a stress only criterion for dislocation nucleation would not be appropriate. In fact it would be important to determine the activation energy associated with dislocation nucleation (Hirel et al., 2008; Brochard et al., 2010; Jennings et al., 2013). It is likely that this activation energy would depend on the local geometry, thus making the relation between the nanoparticle shape/size and its strength even more complex.

## 8. Conclusions

In this paper we report investigations of the relation between an externally applied stress and local stresses in nanostructures, up to the initiation of plastic deformation, in order to understand the influence of shape and size on the strength. We performed an extensive set of molecular dynamics simulations of the flat punch compression of 2D aluminum models. It is first observed that the plastic deformation of these models is always initiated by the nucleation of Shockley dislocations from the surface corners being in contact with the flat punch indenters. The yield stress associated with this event depends on both the width and the height of models, according to a power law variation. The size effect on strength can be seen as a combination of the height and width factors. Furthermore the yield stress also critically depends on the surface corner angles, which greatly determine the power law exponent.

The analysis of local stress variations during compression confirms in most cases the presence of stress gradients in the vicinity of contact corner edges, where dislocations nucleate. However we find that spatial variations of the local stresses as a function of the applied stress, the shape and size of the models are complex, and cannot be represented by simple power laws as predicted by elasticity theory. In particular, it is crucial to take into account the short range influence of the surfaces. Our investigations reveal that local stresses are linearly dependent on the applied stress, with a proportionality coefficient depending on the shape and size of the system. Finally, using specific model shapes, we show that the size and shape dependence on the yield stress can be physically explained by a screening interaction between stress concentrations forming at the different contact corner edges. We emphasize that all our conclusions are based on calculations and analyses in the elastic regime and at the onset of plasticity. Then we believe that they could be generalized to other FCC metals than aluminum.

This study opens new perspectives to understand the size and shape dependence of nanostructure strength. In particular it would be particularly interesting to perform similar investigations for 3D nanoparticle models. Work in this direction is currently underway.

## CRediT authorship contribution statement

L. Pizzagalli: Writing – review & editing, Writing – original draft, Investigation, Conceptualization. R. Gatti: Writing – review & editing. J. Amodeo: Writing – review & editing. J. Godet: Writing – review & editing. J. Durinck: Writing – review & editing. S. Brochard: Writing – review & editing.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Acknowledgments

Jérôme Colin is gratefully acknowledged for his help and discussions about elasticity theory. Computer time for this study was partially provided by the MCIA (Mésocentre de Calcul Intensif Aquitain). This work pertains to the French Government program "Investissements d'Avenir" (EUR INTREE, reference ANR-18-EURE-0010, and LABEX INTERACTIFS, reference ANR-11-LABX-0017-01). The authors also acknowledge support by the Agence Nationale de la Recherche, France, grant no. ANR-20-CE09-0015 (ANR SASHA).
# Data availability

Data will be made available on request.

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