Role of crystallographic orientation in material behaviour under nanoindentation: Molecular Dynamics study

As a result of its low density, aluminium (Al) and its alloys are commonly used materials in various lightweight applications. As such, they are often subjected to processing during metal-forming operations, like rolling, forging, or extrusion [1]. However, the higher the demand for final products with more sophisticated properties, the more fundamental research on alloy behaviour is required. Various aspects of aluminium behaviour are currently being studied during laboratory tests. The influence of temperature, strain rates, or deformation levels in elastic and plastic regimes is primarily investigated with the use of standard macroscopic plastometric tests [2–4]. However, when local material behaviour is of interest, then the micro- nano- or picoindentation tests are most often used [5, 6]. These tests have the capability to measure the load-displacement response during deformation at a small length scale and identify local material properties, including Young’s modulus and nano-hardness [7]. It is also possible to identify the stress–strain characteristics of particular grains that can be further used in advanced full-field numerical simulations [8]. Unfortunately, many unexpected factors can affect the identified local material response significantly at this length scale. Particular attention has to be paid to the surface hardening during sample preparation stages, the parallel surface alignment with respect to loading directions, the selection of subsequent indentation locations, the selection of indentation load, or even the removal of the debris from the indenter tip’s surface. Details about this test, along with its advantages and limitations, can be found in Nanoindentation by Fischer-Kripps [9]. Additionally, in Madej et al. [10], during the investigation of porous materials, the authors pointed out the role of the microstructure state directly under the indentation imprint, which can also significantly affect the test results.

Therefore, this research is focused on the evaluation of the material response from nano/picoindentation testing with respect to the grain orientations in the direct vicinity of the imprint. Computational material science capabilities were employed during the research to improve a fundamental understanding of these mechanisms not affected by the real material heterogeneities nor the setup of the laboratory tests. The molecular dynamics (MD) technique was selected in the current work in particular to point out the role of the local grain crystallographic orientation during nano/picoindentation testing. The aluminium sample previously mentioned was selected as a case study. For the sake of clarity, two distinctively different crystallographic orientations, cube {100}<001> and hard {110}<011> (Figure 1), were investigated in a set of arrangements: monocrystalline, bicrystalline, and polycrystalline.

Fig. 1.

Generally, it is expected that when a load is applied to a single crystal, the atoms in the lattice will experience local forces and deformations. In a cube orientation, the crystal lattice has equal symmetry in all directions, which means that the atoms are arranged in a regular and symmetric manner. This arrangement allows for easier slip and movement of dislocations within the crystal lattice when an external load is applied. On the other hand, in a hard orientation, the crystal lattice has a specific direction that is less symmetric than the cube orientation. This means that the crystal structure is more resistant to deformation and dislocation movement compared to the cubic orientation. The presence of crystallographic planes or directions with low atomic packing densities can lead to increased resistance to slip and dislocation movement. Because of these differences in crystal structure and symmetry, monocrystals in hard orientations tend to exhibit higher strength and resistance to deformation compared to monocrystals in cube orientations. As a result of this increased resistance to slip and dislocation motion in hard orientations, higher forces are required to initiate plastic deformation.

Molecular dynamics [11–13] is a powerful tool for evaluating materials’ behaviour and predicting their properties. MD is a well-established research tool in various fields, e.g., biology [14], chemistry [15], physics [16], and also materials science [17]. It is a computational technique that has the capability to evaluate direct atoms’ and molecules’ interactions over time. Therefore, MD simulations allow us to study the material’s behaviour in great detail, including point, linear, and two- and three-dimensional defects like grain boundaries. As such, it provides an insight into the fundamental slip activities at particular crystalline planes.

Molecular dynamics is based on Newton’s equation of motion [18, 19] in the following form: 1Fi(t)=mi⋅d2ridt2$${F_i}(t) = {m_i} \cdot {{{d^2}{r_i}} \over {d{t^2}}}$$ where: F_i(t), the force acting on the i-th atom; m_i, the mass of the i-th atom; d2ridt2$${{{d^2}{r_i}} \over {d{t^2}}}$$, the acceleration of the i-th atom; and r_i = (x_i,y_i, z_i), the position of the i-th atom.

Then with the Verlet algorithm [20] and Taylor approximation [21], the position of the i-th atom is evaluated by means of the following formula: 2ri(t+Δt)=2ri(t)−ri(t−Δt)+Fi(t)miΔt2$${r_i}(t + \Delta t) = 2{r_i}(t) - {r_i}(t - \Delta t) + {{{F_i}(t)} \over {{m_i}}}\Delta {t^2}$$ where: r_i is the displacement of the i-th atom, t is the time, F_i is the force acting on the i-th atom, and m_i is the mass of the i-th atom.

The relationship between force and potential energy (U) is expressed as: 3Fi=∂U∂ri$${F_i} = {{\partial U} \over {\partial {r_i}}}$$ where: r_i is the displacement of the i-th atom, U is the potential energy, and F_i is the force acting on the i-th atom.

The key element of the MD model is the potential, which determines the interaction between subsequent atoms. As a result, selecting the appropriate atomic potential is important in this procedure [22]. The embedded atom method (EAM) [23, 24] interatomic potential was selected for the current investigation with the parameters taken for the Al from available datasets [25].

The MD model was developed with the use of a large-scale atomic/molecular massively parallel simulator (LAMMPS) [26, 27] in 3D computational space to reflect the complexity of the real material behaviour, while initial atom arrangements in the samples were developed with Atomsk software [28, 29].

First, the unit cell of the sample atomic structure being investigated was selected. The face-centred cubic lattice (FCC) structure [30] was used for Al with a lattice constant equal to 4.02Å. To capture the sample constraints imposed by the surrounding material volume, a set of periodic boundary conditions was set on the side walls. A shrink-wrapped boundary condition was defined on the top surface, while the bottom surface was fixed. Based on such input data, a series of Al samples, namely, two monocrystals, two bicrystals, and three polycrystals, were generated with the Voronoi tessellation algorithm [31, 32]. The sample dimensions were the same and equal to 150 × 150 × 120Å. At the same time, two crystallographic orientations representing the cube and hard orientations previously mentioned were assigned to monocrystalline and bicrystalline samples. The bicrystalline sample alignment is presented in Figure 2.

Fig. 2.

The baseline material behaviour under loading can be investigated with the presented monocrystalline sample alignment. At the same time, the bicrystalline samples allow studying the effect of the substrate on the applied indentation load.

The polycrystalline samples were also generated to investigate complex material behaviour under loading conditions. Three samples with five grains were constructed within the 150 × 150 × 150Å computational domain, as seen in Figure 3.

Fig. 3.

The first two case studies were created on the basis of the cube and hard orientations. In each case, the central grain is in the cube Figure 3a or hard Figure 3b orientation. At the same time, the remaining grains represent the cube or hard orientations, respectively.

The last digital model of the sample (Fig. 3c) represents a fully random alignment of the grain’s crystallographic orientations. In each case, the indentation was always carried out in the central grain during the simulation. As a result, the influence of the central grain neighbourhood can be clearly described.

As mentioned, two commonly used types of indenter tips were used during the investigation. The first is a sharp tip type indenter [33, 34], as presented in Figure 4a. This indenter imitates the shape of an inverted pyramid with a square base. The model was created on the basis of the carbon atoms with a diamond lattice and lattice parameter of 3.567Å. The second is a spherical type of indenter from Figure 4b. The sphere radius defines the size of this indenter and is equal to 20Å. In each case, the indenter speed was set as 2Å/femtosecond. Additionally, the spherical indenter and monocrystalline cube samples were used to evaluate the hardness value (7.14 GPa), to validate the model predictions against the experimental data presented by Liu et al. [35], and to successfully prove its robustness.

Fig. 4.

As presented, the molecular dynamics technique helped to evaluate the role of the local grain crystallographic orientation during the nanoindentation of the two distinctively different crystallographic orientations cube {100}<001> and hard {110}<011>. Based on the investigations with digital models representing monocrystalline, bicrystalline, and polycrystalline samples, the following conclusions can be drawn:

The force used during indentation is affected by both the substrate and the environment of the grain into which it was indented.