Atomistic and finite element study of nanoindentation in pure aluminum

Journal Pre-proof Atomistic and Finite Element Study of Nanoindentation in pure aluminum Satyajit Mojumder (Conceptualization) (Methodology) (Software) (Validation) (Formal analysis) (Writing - original draft) (Visualization) (Investigation), Monon Mahboob (Supervision) (Writing - review and editing) (Supervision) (Project administration) (Resources), Mohammad Motalab (Supervision) (Resources)

PII:

S2352-4928(19)31149-3

DOI:

https://doi.org/10.1016/j.mtcomm.2019.100798

Reference:

MTCOMM 100798

To appear in:

Materials Today Communications

Received Date:

22 October 2019

Revised Date:

23 November 2019

Accepted Date:

23 November 2019

Please cite this article as: Mojumder S, Mahboob M, Motalab M, Atomistic and Finite Element Study of Nanoindentation in pure aluminum, Materials Today Communications (2019), doi: https://doi.org/10.1016/j.mtcomm.2019.100798

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Atomistic and Finite Element Study of Nanoindentation in Pure Aluminum Satyajit Mojumder1,2, Monon Mahboob2, Mohammad Motalab2 1

Theoretical and Applied Mechanics Program, Northwestern University, Evanston, IL-60208, USA 2

Department of Mechanical Engineering, Bangladesh University of Engineering and

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Technology, Dhaka-1000, Bangladesh.

Abstract:

Nanoindentation is a useful technique to measure mechanical properties of a material such as the

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elastic modulus and hardness etc. In this paper, the effects of crystallographic orientation, indentation speed, indentation depth and indenter size has been studied for pure Aluminum (Al)

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using dislocation nucleation and propagation mechanism in atomistic simulation. The materials properties like hardness and reduced modulus are also calculated from the atomistic simulations.

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Nanoscale finite element (FE) simulations in Al are carried out using molecular dynamics tensile test data as input parameters and compared with atomistic results. The proposed methodology

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reduces the computational time and cost and reproduces the material properties with reasonable accuracy. The investigations of atomistic simulations include the load-displacement analysis, dislocation density and dislocation loops nucleation and propagation, von-Mises stress

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distribution and surface imprint. This study provides a pathway to obtain the materials properties for nanoscale materials without performing large scale atomistic simulation and can be applied for other mechanical properties such as fatigue, creep simulation of materials.

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Keywords: Nanoindentation, Aluminum, Atomistic simulation, Finite Element, Dislocation

1. Introduction Nanoindentation is a powerful technique to measure the localized properties of different

materials and has been an active research topic since last couple of decades. As the thin film technology is emerging with application in nanoelectronics, nanodevices, solar cells, etc., nanoindentation becomes a major way to characterize the mechanical properties for such

materials [1]. Primarily used to study hardness of materials, nanoindentation can offer insight into the incipient plasticity associated with the dislocation nucleation, propagation and the failure mechanism of the materials. Atomistic study has a long tradition of implementation for the nanoindentation. Landman et al. [2] studied the nanoindentation using molecular dynamics approach in Gold surface using Nickel as an indenter and suggested that atomistic approach can be a useful technique to measure material properties and identified the incipient plasticity. Szlufarska et al. [3] studied the

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nanoindentation for SiC and concluded that SiC can go through phase transformation during the loading process and become amorphous beneath the indenter surface. Similar phase

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transformation is also observed in Si [4–6]. Ma and Yang [7] studied nanoindentation in nanocrystalline Cu and compared it with single crystal Cu using LJ potential. They concluded

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that burst and arrest of dislocation in nanocrystalline Cu significantly affects the plastic properties of materials. Li et al. [8] studied the nanoindentation both experimentally and using

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molecular dynamics simulations with an indentation depth up to 50 nm in Aluminum. In experimental results, they visualized the dislocation burst pattern and studied the dislocation loop formation and propagation with applied load using molecular dynamics to explain their

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experimental results. Lee et al.[9] studied the dislocation pattern in Al (111) surface for different types of inter-atomic potential and elucidated the nucleation sites, dislocation locks and loops formation just underneath the indenter tip and the prismatic dislocation loops far away from

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the contact surface. Wagner et al. [10] studied the dislocation nucleation for single crystal Al using MD simulations for temperature of 0 − 300K and suggested that the temperature helps to pre-nucleate the dislocations. They also suggested that the smooth indenter nucleates the dislocation below the contact surface, but the rough indenter can nucleate dislocation both at surface and beneath the surface. Begau et al.[11] studied the nanoindentation for Cu crystals

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and tried to explain how the first dislocation is generated during the pop-in event and how they multiply themselves later through atomistic study. Saraev and Miller [12] investigated the nanoindentation in multi-layers of Cu and found softening behavior which is due to the reverse Hall-Petch effect generally found in nanometer thickness. Catoor et al.[13] studied the nanoindentation on different surface of single crystal Mg using spherical indenter experimentally and identified the slip system involved during failure and pop-in event. Somekawa et al.[14] studied nanoindentation in Mg using experimental and MD approach and found that

indentations on the basal plane have higher pop-in load and higher displacement than in prismatic plane. Similar work on Mg and its alloys are done recently[15,16]. Peng and Zeng[17]

studied the deformation behavior in polyethylene using MD simulations and

calculated the hardness using nanoindentation. Minor et al. [18] studied the nanoindentation in Al experimentally and suggested that theoretical shear strength can be found for material though there may be some defects. Shibutani and Tsuru [19–23] performed several investigations on different FCC materials like Copper (Cu), Aluminum (Al) using molecular dynamics simulation

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to understand the dislocation behavior which affects material properties significantly. They suggested that the indentation direction as well as indentation depth, indenter size and substrate

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have significant effects on the anisotropy of the crystal and the dislocation behaviors are significantly affected by these parameters. Ziegenhain et al.[24] studied the crystal anisotropy onset of plasticity using nanoindentation in Al and Cu and suggested that crystal orientation

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has more profound impact on the elastic deformation but in the plastic deformation crystal orientation plays a trivial role. Fu et al. [25] studied the Vn (001) crystals plasticity using

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nanoindentation and concluded that evolution of partial dislocation is the main mechanism at initial stage of plasticity. Fu et al. [26] performed nanoindentation simulations in nanotwineed

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Cu/Ni multilayer and found that twinning partial slip and partial slip parallel with twin boundary can reduce the hardness of the materials. Hu et al. [27] studied the surface orientation effects on the nanoindentation in Ni substrate and suggested that {111} plane is the dominant plane for

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gliding of the dislocation. Sun et al. [28] studied the formation of prismatic loops in single crystal 3C − SiC under nanoindentation and elucidated the plastic deformation mechanism. Xiang et al.[29] (2017) studied AlN thin film using MD simulations and found that load drop in the load-displacement (P − h) curve forms amorphous structure in the materials. Slip system activated during the nanoindentation governs the plastic deformation of the materials. For the

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FCC materials, the {111} (110) is the activated slip system. Among the 12 slip planes possible, 8 slip planes become active during the indentation in FCC metals like Al, Cu etc. The other four inactivated slip planes are parallel to the indenter[30].

There are different types of indenter available in the nanoindentation experiment. For example, the indenter can be conical, spherical, flat, cubic, cylindrical, Berkovich etc. In the MD study, the most common type of indenter used is spherical. Spherical indenter is easy to model and can be

easily verified for theoretical prediction. The indenter can be any hard materials such as diamond, SiC or can be modeled as an analytical rigid body. The indenter shape has significant impact on the indentation force and hardness of the materials as the contact area is highly dependent on the indenter contact with the substrate. The indentation depth for the MD study is limited due to the computational limitations. In MD studies, the indentation depth is varied up to few nanometers, whereas hundreds of nanometers are feasible for the experiment purpose. Knap and Ortiz [31] studied the indenter size effect for nanoindentation in Au(001) and analyzing the P

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− h diagram, they suggested that indenter force, which is different for different indenter size, is not a reliable indicator for dislocation activity and plastic deformation. Verkhovtsev et al. [32]

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studied nanoindentation in Ti crystal using MD simulation and studied the effect of indenter shape. In their study they used square, conical and spherical shape of indenter and found the P − h curve. From the P − h curve, they have calculated the hardness and elastic modulus for the

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Ti crystal. Their indentation depth was 3nm and the loading velocity was 40 m/s. Yaghoobi and Voyiadjis [33] studied size effect for the Ni substrate and concluded that with the increment of

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indentation depth both the dislocation length and density increases and consequently the material hardness is reduced. Fang et al.[34] studied the dynamic characteristic of nanoindentation in Al

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and found that plastic energy and adhesive force increases with the indentation depth. They also attribute that dislocations nucleate, glide interact along the (111) slip plane for Al. Yu and Shen [35] studied the incipient plasticity for very small radius indenter and concluded that the

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yielding load are not constant. Despite several atomistic simulations of FCC metals like Al and Cu in the literature, the link between dislocation motion and the plasticity of these materials remain unclear.

On a larger bulk-scale, Finite element method (FEM) has been used for nanoindentation

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simulations to characterize the mechanical properties. Using FEM, the materials properties such as elastic modulus and hardness values are obtained previously which are in good agreement with the experimental results[36]. One of the major benefits of the FEM simulation of nanoindentation over experimentation is that it is possible to investigate the complex stressstrain field beneath the indenter from FEM easily[37]. This stress-strain field provides some basic insights of the mechanical properties for the materials. Bressan et al.[38] modeled nanoindentation for bulk and thin film using FEM and identified the effects of mesh size,

indenter tip radius, and hardening law effects for indentation in Cu, Ti, and Fe. They have used the axisymmetric CAX4R element for the modeling purpose with a substrate size of 36 µm radius and 18 µm height. For their study, the indenter radius was 400 nm and the penetration depth was 500 nm. Warren and Guo[39] characterized the surface properties using nanoindentation experiment with FEA and studied the surface integrity and tip geometry effects. They found that P − h curves are affected by the residual stress, modulus and yield strength of the substrate. Chang et al. [40] used FEM to study nanoindentation in Cu nanowires

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and found that indenter tip radius is the key source of error in results. Martın et al. [41] studied

plasticity simulations to observe the twinning activity.

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the deformation mechanism using Atomic Force Microscopy (AFM) and finite element crystal

However, FEM simulations of nanoindentation based on bulk-scale material data may not be

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reliable since, mechanical properties such as yield stress, fracture strain etc. at nanoscale differ from those at bulk-scale due to presence of fewer defects. So, in this article tensile test data

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collected from MD simulations in nanoscale are utilized for FEM modeling. Previously, a similar approach was used by Vodenitcharova et al.[42] to validate their nanoindentation

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results of FEM for the few nm indentation in Al. They found that the P − h diagram from the MD and FEM simulations are in good agreement with each other. The objectives of this article are to study the nanoindentation of pure Al systematically for

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crystallographic orientation, indentation depth, indentation speed and indenter size and to investigate the dislocation nucleation, propagation mechanism, and materials properties such as hardness and reduced modulus using both MD and FE. The rest of the article is organized as follows. In section 2, the computational methodology, both for atomistic and FE has been

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discussed. The results of atomistic simulations of the nanoindentation has been discussed in section 3. Section 4 comprises the FE results and comparison with the MD results. Finally, summary has been presented with some key observation for pure Al nanoindentation.

2. Methodology

In this work, we applied both atomistic and FE simulations for the nanoindentation in pure Al. To compare the atomistic results with the FE simulations we used molecular dynamics tensile test data as the input parameters of the FE simulations.

2.1.Atomistic simulation procedure To perform the quasi-static nanoindentation test, molecular statics simulations were performed using LAMMPS software[43] package adopting MPI/OpenMPI hybrid parallel code. A box of Al

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atoms was created for three different crystallographic orientation as for loading in <0 0 1> direction (<1 0 0> <0 1 0> <0 0 1>) , <1 1 0> direction (<1 1 1> <1 1 -2> <-1 1 0>), and <1 1 1> direction ( <-1 1 0> <-1 -1 2> <1 1 1>). The details simulation parameters for these three

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different crystallographic orientations are presented in Table 1. As shown in Fig. 1, the simulation box is divided into two regions: bottom region, where the atoms are fixed (also called

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Newtonian atom) and the upper region, where the indenter penetrates. The 2 nm bottom region has twofold functions: one is providing rigid support for the substrate and the other as heat bath

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during the penetration of indenter in the upper region. A rigid spherical indenter (virtual indenter in LAMMPS) was then set up over the substrate and the indenter was pushed into the material

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(see Fig. 1). The loading step was followed by an unloading step adopting displacement control of the indenter and the system was minimized using conjugate gradient method after every time step to maintain the quasi-static loading process at 0K temperature. The speed of the indentation

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was varied to 10-1000 m/s while applying the minimization after each time step. The indenter sizes were varied from 3-7 nm and indentation depths were varied from 0.5-2 nm. This range covered both the elastic and plastic region of the indentation process. The indenter force constant was taken as 1ev/Ao3 for all the simulations. EAM potential [44,45] has been used for the present simulations which was used extensively for the Al previously [10,20]. To verify the result

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obtained from the atomistic simulations, the load-displacement curves of atomistic simulations are plotted along with the Hertzian contact theory for all three direction as shown in Fig. 2. The present simulations results are in good agreement within the elastic limit for all three direction of loading. The hardness and the indentation modulus were then calculated using the Oliver-Pharr method [46]. For a typical load-displacement curve, where the load increases up to Pmax and the critical depth of the indentation is hc, the hardness and indentation modulus is calculated from the slope

drawn in the unloading curve. The slope (S) from the unloading portion of the load-displacement (P-h) curve can be expressed as 𝑑𝑝

𝑆 = 𝑑ℎ

(1)

From this slope, the critical depth can be calculated, where the load is zero. 𝑃 = 𝑆ℎ + 𝐶

(2)

Here, h=hc at P=0. The critical contact area (Ac) for a spherical indenter can be obtained from previously calculated

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critical depth using the following relations for a spherical indenter. 𝐴𝑐 = 𝜋(2𝑅ℎ𝑐 − ℎ𝑐 2 )

(3)

𝑃𝑚𝑎𝑥 𝐴𝑐

And the reduced modulus (Er) of the indentation is 1 √𝜋 𝑆

𝐸𝑟 = 𝛽

2 √𝐴𝑐

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𝐻=

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The hardness (H) of the material is now expressed as

(4)

(5)

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Where, β=1 for the spherical indenter. The relation between the elastic modulus of the substrate 1 𝐸𝑟

=

1−𝜈𝑖 2 𝐸𝑖

+

1−𝜈𝑠 2 𝐸𝑠

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and the reduced modulus is given by the following relation

(6)

Where νi and νs are the Poisson’s ratio of the indenter and substrate materials and Ei and Es is the

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elastic modulus of the indenter and substrate materials. As the indenter pushes through the materials, plastic deformation occurs, and dislocation loops are formed. We calculated dislocation density using OVITO [47] as a measure of plastic deformation in the indentation process using DXA algorithm. The dislocation segments were identified using trial circuit of 14 by 9 and total dislocation length is then divided by the total

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volume of the substrate to obtain the dislocation density.

To compare the atomistic results with the FEM result, the materials properties input parameter was obtained from molecular dynamics tensile test. The uniaxial tensile simulations were performed for pure Al nanowire having square cross section and a dimension of 5 nm × 5 nm × 50 nm in X, Y, and Z direction, respectively. We kept the orientation and loading direction similar to the nanoindentation simulation. The aspect ratio (height to width) of all the nanowire is

kept constant as 10:1 and the tensile load is applied in the Z axis of the co-ordinate system (crystal directions of <0 0 1>, <1 1 0> and <1 1 1>). First, the initial geometry of the nanowire was created and relaxed sufficiently (for 100 ps) under the NPT dynamics to remove any residual stress present. Later, a tensile load is applied along the Z direction of the Al nanowire at a temperature 300K (see Fig. S1) of the simulation box at a strain rate of 108 s−1. The timestep chosen for all the simulations was 1fs. From the tensile simulations, the stress-strain curve is obtained and the elastic modulus, yield stress and Poisson’s ratio are identified which was further

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used as the input properties in the finite element simulations (see Fig. S18). The simulation results are discussed in the Supplementary Section 1.

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2.2.FE simulation procedure

The purpose of the FE simulation was to get a reasonable estimation for the materials

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properties such as hardness and reduced modulus with a lower computation time than atomistic simulation. We used Abaqus/Standard for the FE simulations considering a deformable

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axisymmetric material model with rigid indenter. The dimension of the substrate was considered similar (10 nm radius, 15 nm height) to the atomistic simulation and the input materials

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properties (elastic modulus, Poisson’s ratio, yield stress) were obtained from molecular dynamics tensile test as described in section 2.1. The material was modeled using the four-node axisymmetric element with reduced integration (CAX4R) and the indenter was considered as

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analytically rigid. The indenter was pushed into the materials substrate using a displacement control boundary condition. The left vertical edge of the materials was considered the symmetric edge and the bottom of the substrate was fixed as shown in Fig. 3. The top and right vertical edge was kept free to resemble the free surface. A grid independency test was performed for different number of elements for the current study (see Fig. S3). From that grid independency test, we

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chose 13299 elements as an independent grid and performed all finite element simulations for that. We used a fine mesh in the contact region of the substrate and indenter and a coarse mesh where the contact not occurred. The detail of the mesh is shown in Fig. S4. A validation study of the present code was also performed for the finite element simulations and the present code was reliable in consideration of the accuracy with the previous literature (see Fig. S5). From the loaddisplacement curve obtained from the FE simulation we used the similar procedure as atomistic calculation for the calculation of the materials properties such as hardness and reduced modulus.

3. Results of Atomistic Simulation Through atomistic nanoindentation simulations we have investigated the effects of crystallographic orientation, indentation speed, indentation depth and indenter size on the P-h curves, materials properties and the dislocation loops formation and propagation.

3.1.Effects of crystallographic orientation

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Crystallographic orientations play pivotal role on the material properties as the atomic positions are different for different orientations. For the FCC material like Al, Cu etc., most common orientation for loading in nanoindentation is the <0 0 1> crystallographic direction.

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Along with this direction, <1 1 0> and <1 1 1> directions were also investigated in this study. In Fig. 4 (a) the load displacement curves are shown for three different loading direction (<0 0 1>,

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<1 1 0>, and <1 1 1>) of nanoindentation. Here, <1 1 1> direction of loading shows the maximum force corresponding to the yield point as well as the highest indentation depth. The

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yield point (where the load drops with the displacement) is considered where the plasticity is introduced inside the material during the loading. For <0 0 1> and <1 1 0> direction the yield

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point is around displacement of 0.6 nm and for <1 1 1> direction yield point is found at displacement of 0.8 nm. The indentation force increases smoothly up to the elastic limits and after that there are some fluctuations in the load displacement curves which results from the

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dislocation nucleation and interaction. When the loading process is completed at maximum indentation depth of 1.5 nm, the unloading process starts. In the unloading curves the load drops gradually and returns to the initial point with an impression of the indenter on the material surface after the indentation depth of 0.5 nm. The fluctuation in the load-displacement (P − h) curve can be further explained from the Fig. 4(b) where the dislocation densities are shown with

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indentation depth for different orientations. From the figure, it is evident that the dislocation density initiates at the same place where the yielding occurs. For <1 1 1> direction, the dislocation density curve jumps at slightly later than <0 0 1> and <1 1 0> direction due to existence of some initial dislocations.

In Fig. 5, contributions of different types of dislocations on the total dislocation of the materials are represented. In the Fig. 5(a), the dislocation analysis for <0 0 1> direction of indentation is

shown. The dominant type of dislocation for this direction is Shockley partial dislocation. Partial dislocations are more energetically favorable compared to the perfect type of dislocation in FCC crystals and create a stacking fault inside the materials. However, the fluctuations in the P-h curve can be explained as other dislocations interact with the partial dislocations and total dislocation density remains constant in the range of 0.8-1.0 nm indentation depth. In the regime of 1.0-1.5 nm depth of indentation, a sudden jump is observed in dislocation since some dislocation locks themselves during the interaction of partial and Hirth dislocation. These

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dislocations also get multiplied by loop formation. Moreover, it is evident that some perfect (Frank-Reed and Stair-rod type) dislocations still form in the high plastic zone of indentation (1 to 1.5 nm). Similar trend can be found for <1 1 0> direction indentation and for this direction of

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loading, there are less amount of lock formation. Here also, Shockley partial is the dominant type of dislocation. Similarly, in case of <1 1 1> orientation, the Shockley partial dislocation is the

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dominant mode of dislocation as depicted in Fig. 5(c). The formation of the dislocation loops during the indentation process in different direction has been elucidated in Fig. S6-S8 (see

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section S3).

The von-Mises stress distribution signifies the stress anisotropy inside the materials and this

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stress field distribution is the measure of yielding in the materials. In Fig. 6, von-Mises stress distribution is shown for <0 0 1>, <1 1 0>, and <1 1 1> directions at the mid position of loading (at depth 0.75 nm), maximum loading (at depth 1.5 nm), and after complete unloading. From the

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figure, it is clearly observed that the stress distribution is symmetric for <0 0 1>direction at mid loading position of the indenter at the maximum loading depth, the symmetric pattern of this distribution is distributed which changes the behavior of dislocation significantly. After the unloading, there is almost no residual stress on the substrate for this direction. Similar pattern can be seen for the <1 1 0> and <1 1 1> directions of indentation. However, there are some

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residual stresses observed after the unloading of these directions of indentation. The reminiscent surface imprint after the unloading process of indentation can be directly verified by the experiment thus an important measure of plastic deformation. In experiment, this surface imprint plays a critical role as the experimentalist directly measures the critical depth of indentation from microscopic image from this imprint. In Fig. 7, the surface imprints after indentation in different directions are shown. It is clear from the figure that different directions of indentation create different types of surface imprint. In all case, the pile up nature of the

indentation surface is conspicuous. It is evident from the figure that the <1 1 1> direction surface imprint is comparatively shallow in nature while the <1 1 0> shows more depth in the residual imprint due to lowest dislocation density and hardness. <0 0 1> shows similar behavior like <110> direction of indentation for surface imprint.

3.2. Effects of indentation speed Indentation speed is a key parameter that is varied during the indentation experiment and it has significant influence on the plastic deformation thus materials properties. Therefore, it is

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very pertinent to study the indentation speed in detail both on P − h curve and the dislocation behavior. In Fig. 8, the load-displacement curves are shown for different speed and indentation

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in different crystallographic direction of loading. The speed of the indentation is varied from 10 m/s to 1000 m/s. From the figure it is quite obvious that higher indentation velocity creates

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higher fluctuation in the plastic zone of indentation. As the velocity is reduced from 1000 m/s to 10 m/s, there is a converging trend, which means there are very trivial differences between the

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curve obtained for 10 m/s and 50m/s indentation speed velocity. However, the computational time required for 10 m/s is 5 times higher than that required for 50 m/s.

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Fig. 9 shows the variation of dislocation density with the indentation speed for different crystallographic orientations. For <0 0 1> and <1 1 0> directions, the dislocation density are significantly lower for higher velocities while the pattern is opposite in the <1 1 1> direction. It

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is also observed that before yielding dislocation density is highest for 1000 m/s velocity but after yielding there is no clear observable trend for <1 1 1> direction. The formation of dislocation loops for different indentation speed is shown in the Fig. 10. From the figure, it can be depicted that there is formation of the two Shockley partials pair in <0 0 1>direction at low speed (10 m/s) which repels each other and send each other away. If the speed

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is increased up to 100 m/s, similar pattern in the dislocation loop formation prevails. At much higher speed (1000 m/s), there are some blunting effects. The loops do not propagate as much as in case of lower speed. As a result, the dislocation density reduces significantly at the higher speed of indentation. The pattern for <1 1 0>direction is quite different from the <0 0 1>direction of indentation. Here, at low speed formations of the prismatic loop can be seen which becomes separated from the main loop and moves towards the bottom surface. At 50 and 100 m/s speed, the dislocation loops are formed along the edge of the indenter. Shockley partial type dislocation

loops are mainly visible. At 1000 m/s, the dislocation loops are not fully formed. It seems that they are arrested inside the materials for this higher speed. Similar trend is found for the <1 1 1> direction of indentation. At low speed the dislocation loops form and propagate. Here cross slips are more prominent and there are many partial dislocation loops with cross slip. At 1000 m/s, the loops are not fully formed and make a lock in their way. The von Mises stress distribution for different indentation speed has been shown in Fig. S8-S10 (see section S3). In Fig. 11, the variation of hardness and reduced elastic modulus in different directions are

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shown with indentation speed where it is observed that the hardness increases with speed significantly. Though the dislocation density is low for <0 0 1> and <1 1 0> direction at higher velocity the hardness of the materials is higher. The dislocation pattern in the higher speed is

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quite different from the low speed. At high speed the dislocations loops locked each other and make it harder for the indenter to penetrate inside the materials. The hardness value is lower for

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the <1 1 0> direction compare to the other direction as the total dislocation inside the material is less. The reduced modulus also shows a similar pattern. However, the reduced modulus increases

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3.3. Effects of indentation depth

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significantly for the <1 1 1> direction at the high speed.

The material properties such as hardness and elastic modulus vary significantly with the variation of the indentation depth. For properly characterizing the material, it is also important to

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find out the indentation depth where the materials start to show plastic properties. In Fig. 12, the effects of variation of indentation depth for different indentation directions are shown. As the indentation depth is low (0.5 nm), the unloading curves follow the loading curve. Therefore, the loading is completely elastic. For further indentation depth deviation in unloading curve with loading curve can be seen. With the increment of indentation depth, the materials become plastic

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and show more fluctuation in the P−h curve. Moreover, the unloading curve is comparatively steeper for <1 1 1> direction than the other two directions. Fig. 13 shows the dislocation loops formation for different indentation depth and loading directions. For the <0 0 1> direction, the tetrahedrons are visible for 0.5 nm indentation depth which further propagates up to 1 nm depth. At 1 nm depth, the partial dislocation loops are now visible which tries to separate each other by repelling and as a result, at higher indentation depth of 1.5 and 2 nm, the partial dislocation loops are expanded and creates higher stacking fault zone

inside the material. The dislocation pattern for <1 1 0> direction is quite different and interesting. For this direction, there are no dislocation formations at 0.5 nm indentation and as the depth increases, dislocation loops are formed along the edge of the spherical indenter. With further indentation depth, this partial dislocation loops form in the prismatic shape which moves toward the bottom of the substrate. For the <1 1 1> direction, dislocation loops are formed at the 1 nm depth and all the loops show some cross-slip type dislocation behavior. In Fig. 14, the variation of hardness and reduced elastic modulus are shown with different

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indentation depths. From the figure, it can be seen that for all the directions, the hardness increases up to a critical indentation depth of 1.5 nm and after that, the hardness is plummeted. This is because at early stage there are few dislocations present. As the loading increased, this

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dislocation tries to glide across the slip plane. For further propagation, they need to overcome the energy barrier by exceeding the critically resolved shear stress. Therefore, the materials show

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increased hardening behavior. However, at higher indentation depth, the dislocation loops spread across the material, so it requires less amount of energy to propagate. As a result, the hardness

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decreases with the indentation depth. Moreover, the hardness in <0 0 1> and <1 1 1> directions are higher than the <1 1 0> direction of the indentation. On the other hand, the reduced modulus

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is increasing with the indentation depth for all the three directions with the highest change in the <1 1 1> direction. Effects of indentation speed on von Mises stress distribution and surface

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imprint has been discussed in section S4.

3.4. Effects of indenter size

With the increase of indenter size, the contact area of the indentation increases which shows noteworthy effect on the results of hardness and elastic modulus. In Fig. 15, the P-h curves are shown for three different indenter size and in different crystallographic orientation. When the

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indenter radius is increasing from 3 to 7nm, the contact area increases significantly. As a result, the load in the load displacement curve also increases. This is true for <0 0 1>, <1 1 0> and <1 1 1> directions of indentation. However, in <1 1 1> direction the P−h curves after yielding show a significant load drop. Moreover, with the 7nm radius indenter yielding occurs at higher indentation depth compared to the 3 and 5nm radius indenter. This pattern of P-h curve is further illustrated by the dislocation density plot. In Fig. 16, the variation of dislocation density for different indenter radius is shown. The dislocation densities are lower for the 3nm indenter and

higher for the 7 nm indenter. It is higher since higher contact area causes more stacking fault zone inside the material and the sizes of the dislocation loops are also bigger compared to the small indenter. Fig. 17 shows the dislocation loop formation at the maximum loading for different indenter size. It is observed that for all three directions partial dislocations are formed while for <1 1 0> direction the pattern is a bit different. It is also observed that dislocation loops are increased with indenter size. Moreover, there are more cross slips for the 7 nm indenter which causes the

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highest load value in the load- displacement curve. The <1 1 1> direction of indentation shows no exception from our previous finding. The dislocation loops are larger and reached almost to the bottom of the indenter surface as seen from the Fig. 17(i). The hardness and reduced modulus

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increase with the indenter radius which is expected (see Fig. S13). The indenter size also has

4. Results of Finite Element Simulation

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profound impact on the von Mises stress distribution and the surface imprint (see section S5).

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The primary motivation of performing the FEM analysis is to obtain the material properties with less computational time and cost. MD tensile data is used as input parameter of FEM to

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carry out the nanoindentation simulation. From the MD tensile simulations stress-strain curves (see section S6) are obtained and used for the FEM simulations input material property. The elastic modulus, Poisson’s ratio, yield stress obtained from the MD simulations are presented in

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the Table 2. From the nanoindentation simulations of the FEM, the load displacement curve is obtained and after analyzing the load displacement curve the material properties such as hardness and reduced modulus are calculated. To validate the approach, results obtained by the FEM are compared with previous results of the MD. In Fig. 18, the load-displacement curve is shown for <0 0 1>, <1 1 0> and <1 1 1> directions of

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the indentation both for MD and FEM. Here we used eam/alloy potential for Al which shows good agreement between MD and FEM results for the loading curve, however, it cannot predict very well the unloading portion of the curve. It is also observed from these figures that there are some fluctuations in the MD indentation curves. These fluctuations are due to the plastic event such as dislocation nucleation, interactions and propagation inside the materials. However, the curves obtained from the FEM do not show such variation as the material is modeled considering an isotropic material. In MD, the material yielding, and plastic phenomenon can be more

accurately predicted than the FEM modeling. However, since MD takes longer time and limited in length scale, applying FEM for nanoindentation problems are useful, especially when just considering the material properties such as hardness and elastic modulus and not the dislocation mechanism. Similar analysis is done for Al in <0 0 1>, <1 1 0>, and <1 1 1>, directions using the eam/fs potential for the MD simulations of the nanoindentation and the tensile test of the Al nanowire which is shown in Fig. 19. It is observed that the MD and FEM results are again in good agreement for this potential and the unloading curve is well predicted than the eam/alloy

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potential. If the eam/alloy and eam/fs potential are compared, eam/alloy potential can predict more accurate loading portions of the load-displacement curves for the pure Al using FEM while eam/fs is more accurate for unloading curve and the properties calculation. The hardness and the

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reduced modulus calculated using eam/fs potential and fem are shown in Table 3.

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In Fig. 20, von Mises stress contour and effective plastic strain are shown for the different stages of the indentation. Von Mises stress is an indication of the material yielding. The effective

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plastic strain, on the other hand, is a scalar quantity which increases monotonically as the plastic component of the rate of deformation tensor changes. As the material is modeled considering

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isotropic property, the stress distribution becomes symmetric. At the maximum indentation depth (1.5 nm), the deformation is maximum, and the yielding zone can be found from the von Mises stress distribution. At unloading, the residual stress becomes visible. The material shows a

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tendency to pile up at the edge of the indenter. The effective plastic strain distributions are also symmetric and increase with the indentation depth. The zone of plastic deformation is confined just beneath the indenter. Similar results are obtained for the <1 1 0> direction of the indentation. From Fig. 21, It can be seen that the von Mises stress distribution is more spread inside the material for <1 1 0> direction of the loading. After the unloading, the pile up is also visible. The

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plastic strain for this direction is not that prominent compared to the <0 0 1> direction of indentation. The plastic zone increases with the loading and covers a portion of the material beneath the indenter. Similar trend is found for the <1 1 1> direction (See Fig. 22).

5. Summary Nanoindentation simulations has been carried out for pure Al using atomistic and FE approach. While the atomistic approach gives more detailed information about the incipient plasticity,

dislocation nucleation and propagation as well as materials properties such as hardness and reduced modulus, FE on the other hand can predict those properties with lower computational cost with reasonable accuracy. The following conclusion can be made: 

Crystallographic orientation has profound impact on Nanoindentation in pure Al. The load carrying capacity is highest for the <1 1 1>direction of indentation in pure Al. The dislocation density also increases with the <1 1 1>direction of the indentation. The

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governing dislocation pattern for all the directions of loading is Shockley-partial



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dislocations.

Higher indentation speed can blunt the dislocation loop front and can increase the indentation force. At lower speed (10-100 m/s) the variations in load-displacement curves

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show a converging pattern. The hardness and reduced modulus of the materials increase with the indentation speed.

Indentation depth can change the material behavior from elastic to plastic regime. The

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

hardness and modulus of the Al increase up to a sudden indentation depth and then the

properties. 

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stress field created by the dislocation itself plays a critical role to reduce the values of the

As the indenter radius goes up, the indentation force and the dislocation density increase

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for higher contact area of the indenter. The hardness and the reduced modulus of the materials also increase with the higher indenter radius. 

Using MD tensile test data as input parameters in FE study, the materials properties such as hardness and reduced modulus can be calculated within reasonable accuracy. And this approach saves the computation time and cost significantly.

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The results described in this study can be further useful to study the dislocation behavior of other FCC metals. The authors aim to extend the current study of nanoindentation for Al alloy to understand the mechanism for such a complex system.

Conflict of Interest The authors declare no conflict of interest for this work.

CRediT authorship contribution statement Satyajit Mojumder: Conceptualization, Methodology, Software, Validation, Formal analysis, Writing - original draft, Visualization, Investigation.

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Monon Mahboob: Supervision, Writing - re-view & editing, Supervision, Project administration, Resources

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Mohammad Motalab: Supervision, Resources

Acknowledgement

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The authors acknowledge the high-performance computing facilities provided by the IICT, BUET during this study. The authors also thankful to the department of Mechanical Engineering,

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BUET for their support. The authors are grateful to Professor Dibakar Datta and Professor H. M.

References

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Mamun Al Rashed for many useful discussions on the results.

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Fig. 1. Physical modeling of the nanoindentation problem. The indenter is analytically rigid as modeled in LAMMPS.

Fig. 2. Load-displacement curves for indentation in (a) <0 0 1>, (b) <1 1 0>, and (c)<1 1 1>direction and comparison with Hertz contact theory prediction.

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Fig. 3. Schematic of the axisymmetric modeling of the nanoindentation in the FEM.

Fig. 4. (a) Load-displacement curve, and (b) dislocation density for 1.5nm indentation in <0 0 1>, <1 1 0>,<1 1 1> crystallographic orientation.

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Fig. 5. Contribution of different types of dislocations on the total dislocation with displacement of the indenter for (a) <0 0 1>, (b) <1 1 0>, and (c) <1 1 1> direction of indentation

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Fig. 6. Von mises stress distribution in (a-c) <0 0 1>, (d-f) <1 1 0> and (g-i) <1 1 1> directions of indention at mid loading at 0.75 nm depth ((a),(d), (g)), max loading at 1.5 nm depth ((b), (e), (h))and after complete unloading((c),(f),(i)).

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Fig. 7. Surface imprint after unloading in (a) <0 0 1>, (b) <1 1 0>, (c)<1 1 1> direction.

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Fig. 8. Load displacement curve for different speed during indentation in (a)<0 0 1>, (b) <1 1 0>, (c)<1 1 1> direction.

Fig. 9. Variation of dislocation density for different speed during indentation in (a) <0 0 1>, (b) <1 1 0>, (c)<1 1 1> directions.

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Fig. 10. Dislocation formation at the maximum loading for different indentation speed during indentation in (a) <0 0 1>, (b) <1 1 0>, (c)<1 1 1> direction.

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Fig. 11. Variation of (a) hardness and (b) reduced modulus with indentation speed on different loading direction.

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Fig. 12. Load displacement curve for different indentation depth during indentation in (a) <0 0 1>, (b) <1 1 0>, (c)<1 1 1> direction.

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Fig. 13. Dislocation formation at the maximum loading for different indentation depth during indentation in (a-d) <0 0 1>, (e-h) <1 1 0>, (i-l)<1 1 1> direction.

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Fig. 14. Variation of (a) hardness and (b)elastic modulus with indentation depth on different loading direction.

Fig. 15. Load displacement curve for different indenter size during indentation in (a) <0 0 1>, (b) <1 1 0>, (c)<1 1 1> direction.

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Fig. 16. Variation of dislocation density for different indenter size during indentation in (a) <0 0 1>, (b) <1 1 0>, (c)<1 1 1> direction.

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Fig. 17. Dislocation formation at the maximum loading for different indenter size during indentation in (a) <0 0 1>, (b) <1 1 0>, (c)<1 1 1> direction.

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Fig 18. Comparison of FEM and MD results of nanoindentation load displacement curve in (a) <0 0 1>, (b) <1 1 0>, (c)<1 1 1> direction of Al using eam/alloy potential.

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Fig. 19. Comparison of FEM and MD results of nanoindentation load displacement curve in (a) <0 0 1>, (b) <1 1 0>, (c)< 1 1 1> direction of Al using eam/fs potential.

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Fig. 20. Von-Mises stress contour and effective plastic strain for indentation depth of (a,d)0.75 nm (b,e) 1.5 nm and (c,f) after unloading when indenting in <0 0 1> direction of pure Al.

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Fig. 21. Von-Mises stress contour and effective plastic strain for indentation depth of (a,d)0.75 nm (b,e) 1.5 nm and (c,f) after unloading when indenting in <1 1 0> direction of pure Al.

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Fig. 22. Von-Mises stress contour and effective plastic strain for indentation depth of (a,d)0.75 nm (b,e) 1.5 nm and (c,f) after unloading when indenting in <1 1 1> direction of pure Al.

Table 1: Simulation parameters used for the atomistic simulations

Crystallographic Indentation Indenter Indentation orientation depth size speed (nm) (nm) (m/s)

Simulation box dimension

Number of atoms

0.5,1, 1.5, 2

3,5,7

10, 50,100, 22.2 nm ×22.2 nm × 484000 1000 16 nm

<110>

0.5,1, 1.5, 2

3.5.7

10, 50,100, 24.55nm ×23.14nm 488364 1000 ×14.31 nm

<111>

0.5,1, 1.5, 2

3,5,7

10, 50,100, 20.05nm ×23.18nm 487363 1000 × 17.53nm

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<001>

Elastic modulus (GPa) 70.05

<110>

82.18

<111>

76.83

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<001>

Yield stress (GPa)

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Crystallographic orientation

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Table 2: FEM simulations parameters obtained from the MD tensile simulations. MD simulations are performed at 300K and at a strain rate of 108 S-1.

Yield strain

Poisson’s ratio

3.60

0.060

0.33

3.56

0.069

0.33

3.94

0.068

0.33

Table 3: Comparison of FE and MD hardness and reduced modulus data FEM Hardness (GPa) Er (GPa) 11.29 97.68 10.88 157.28 10.33 164.41

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Loading direction <001> <110> <111>

Atomistic Hardness (GPa) Er (GPa) 13.86 211.00 12.47 204.04 13.93 220.00