Introduction

More recently, nano/micro- structure machining has been attracted more and more attentions, due to the increasing need in the fields such as micro-electromechanical systems [1]. Atomic force microscopy (AFM) tip-based nano-mechanical machining has been widely adopted to fabricate nano-structures on different materials, such as metals [2], polymers [3] and semi-conductors [4], due to its relatively low cost and ease of operation. The machining process is the same as a scratching process that a tip scratches the specimen surface with a predefined normal load and scratching path. Different deformation behavior would lead to different scratching depths and topographies. Therefore, it is meaningful to conduct a thorough research on deformation behavior of target material under scratching, so as to improve machining precision.

Copper is one of the materials commonly adopted to fabricate nano/micro- structures, and it is usually composed of a lot of fine grains which exhibit anisotropic mechanical behavior at nano/micro- scale. Since nano-mechanical machining is implemented at grain-scale (single or several grains), the effect of anisotropic behavior of single crystalline copper cannot be ignored. Crystallographic orientation has a significant effect on anisotropic behavior which has been studied by many research works. Zambaldi et al. [5] investigated the plastic anisotropy behavior of γ-TiAl and predicted the indented surface pile-ups of γ-TiAl with different crystallographic orientations. Wang et al. [6] studied the evolution of crystallographic textures of (001)-, (011)- and (111)-oriented coppers under nano-indenting. Experimental test and simulation were employed by Liu et al. [7] to investigate the mechanical properties of single crystalline aluminum with different crystallographic orientations. However, crystallographic orientation is not the only factor that affects anisotropic behavior significantly. Single crystalline copper would also exhibit different mechanical behavior when a tip scratches along different directions, even if the crystallographic orientation of copper does not change. For example, the scratching force of (001)-oriented copper along the [110] direction is greater than that of the [100] direction [8].

Nano-scratching, a test technique that making a specimen plastically deformed along a given scratching direction [9], is an effective testing method which has been widely adopted to study scratching behavior of anisotropic materials [10,11,12,13,14]. Numerical simulation, commonly recognized as an important supplementary of experimental investigation, is an effective method to investigate deformation mechanism at nano/micro- scale. Crystal plasticity finite element method (CPFEM) is a method proposed to model micro-structure of crystalline material, which has been widely employed to simulate deformation processes, such as nano-indentation [15], uniaxial tension [16], compression [17] and torsion [18]. More recently, several publications introduced the CPFEM models to study the machining and scratching behavior of crystalline materials. Wang et al. [19] developed a CPFEM model simulating the cutting process of polycrystalline copper to investigate the effect of grain boundaries on the microscopic deformation behavior, comparing to the macroscopic machining results. Wang et al. [20] employed the CPFEM to investigate the pile-up behavior of (010)-, \((\mathop 1\limits^{ - } 01)\)- and \((\mathop 1\limits^{ - } 11)\)-oriented copper by simulating the Berkovich nano-scratching process, but the scratching direction was not taken into consideration in their work. The simulating results in these works are in good agreements with the corresponding experimental results.

The present work aims to investigate the surface deformation of single crystalline copper on different scratching paths, and the CPFEM model simulating the nano-scratching process is developed. Further, the scratching depth, surface topography and subsurface deformation induced by a tip that scratches along different directions are studied in detail for the cases of (001)-, (101)- and (111)-oriented coppers.

Methods

Phenomenological constitutive model

The CPFEM constitutive model adopted in this work is a phenomenological model which considers dislocation slip as the only deformation mechanism and takes critical resolved shear stress (CRSS) as the state variable for each slip system. As the resolved shear stress on a slip plane reaches the CRSS, dislocation slip will be triggered. Phenomenological model is a kind of effective constitutive models that has been employed by many research works [21,22,23,24,25] to describe the deformation behavior of crystalline material. More details about it can be found in Ref. [26].

CPFEM model implementation

ABAQUS (Simulia 2017) and Düsseldorf Advanced Material Simulation Kit (DAMASK) [27] are employed to implement CPFEM simulation in the present work. The utilized CPFEM model simulating nano-scratching process has a total of 19,422 elements and 22,680 nodes, as shown in Fig. 1a. The dimensions of single crystalline copper specimen are 25 × 20 × 5 μm3 (X × Y × Z), and the mechanical parameters of copper are listed in Table 1 [28]. To ensure both simulation accuracy and computational efficiency, the copper specimen is meshed by using a transition mesh technique, and the meshes around contact region are much finer than those of the other regions. The diamond spherical tip with a radius of 5 μm is simplified as a rigid body due to its far greater hardness than that of single crystalline copper to improve computational efficiency.

Figure 1
figure 1

Configuration of nano-scratching simulation. a Nano-scratching model, schemes of scratching directions in the cases of b (001)-, c (101)- and d (111)-oriented coppers

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Table 1 Mechanical properties of single crystal copper
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In this work, the direction paralleled to the x-axis is set as initial scratching direction which is designated as Direction 0° here. The [100], \([0\overline{1} 0]\) and \([1\overline{1} 0]\) directions are therefore the corresponding initial scratching directions for (001)-, (101)- and (111)-oriented coppers, respectively, shown in Fig. 1b–d. The nano-scratching simulations for (001)-, (101)- and (111)-oriented coppers are all performed along 25 different scratching directions which increase counterclockwise from 0° to 360° with an interval of 15°. The selected typical nano-scratching is composed of loading stage, scratching stage and unloading stage. In the first stage, the tip is penetrated into the specimen by applying a normal force of 6 mN with a constant rate of 3 mN/s. Then the tip scratches 15 μm with a specific speed of 1 μm/s along one scratching direction. Finally, unloading the tip in 2 s at a constant rate in the last stage. The bottom surface of specimen is fixed, and the tip can only scratch along the x and z axes (shown in Fig. 1a). The friction coefficient between specimen surface and indenter is not taken into consideration in this work.

Experimental configuration of nano-scratching

Mechanically and chemically polished high-purity (001)- and (111)-oriented coppers with surface roughness of less than 10 nm (supplied by the Hefei Kejing Materials Technology Co., Ltd) are adopted to implement nano-scratching tests. Both the specimens have cuboid sizes of 10 mm in length × 10 mm in width × 1 mm in height. The employed spherical indenter with a tip radius of 5 μm is made of diamond. Nano-scratching tests are all implemented by a nano-indentation instrument (Micro Materials Ltd, Wrexham, UK) under a room temperature (25 °C). Except for the loading stage, the experimental parameters are consistent with those of simulation. In experimental test, the specimen is first accelerated to 1 μm/s, and then a normal force of 6 mN is applied promptly on the indenter to make it penetrate into the specimen while keeping the specimen velocity unchanged, due to the restricted loading mode of the employed instrument. The scratched surface topographies are measured by a Bruker Contour GT-K Optical Profile.

Verification of simulation models

Figure 2a, b show the scratching depth-distance curves of (001)- and (111)-oriented coppers obtained from both experiments and simulations. In general, the scratching depths of simulations are in good agreement with those of experimental tests. The depth differences between experimental tests and simulations at the initial stage are attributed to the difference of loading mode, as stated in Sect. 2.3. The residual topographies formed on (001)- and (111)-oriented coppers are presented in Figs. 3 and 4. The simulation topographies are also in well agreement with the experimental results. The experimental topography induced by nano-scratching in the case of Direction 45° in Fig. 3 shows a lower level of symmetry than that of the simulation one, which may be attributed to the positioning accuracy in experiment. The simulation model employed by the present work is generally accurate enough to study the deformation behavior of single crystalline copper.

Figure 2
figure 2

Scratching depth-distance curves of a (001)- and b (111)-oriented coppers produced by experimental tests and CPFEM simulations

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Figure 3
figure 3

Topographies of (001)-oriented copper induced by a tip which scratches along different nano-scratching directions. a Direction 0° (experiment), b Direction 0° (simulation), c Direction 45° (experiment), d Direction 45° (simulation)

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Figure 4
figure 4

Topographies of (111)-oriented copper induced by a tip which scratches along different nano-scratching directions. a Direction 0° (experiment), b Direction 0° (simulation), c Direction 30° (experiment), d Direction 30° (simulation)

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Results and discussions

Deformation behavior of (001)-oriented copper

Figure 5a illustrates the scratching depth variations for (001)-oriented copper with respect to scratching direction produced by simulations. Scratching depth changes periodically with a period of 90°. It decreases gradually with the increase of direction angle in the first half period, then increases in the next half period. Besides, the maximum and minimum depths are observed on the < 100 > and < 110 > directions, respectively, and the maximum value is about 22.8% larger than the minimum one, indicating that scratching direction has a significant effect on the scratched depth.

Figure 5
figure 5

a Scratching depths when a tip scratches along different scratching directions. b Schematic illustration of slip systems of (001)-oriented copper

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The periodic variation of scratching depth can be attributed to the projected symmetric slip systems of (001)-oriented copper, as shown in Fig. 5b. There are four symmetrical axes for the slip systems, and the corresponding scratching directions paralleled to the four symmetrical axes are Directions 0° (or 180°), 45° (or 225°), 90° (or 270°) and 135° (or 315°), respectively. As the tip scratches along two directions which are symmetrical to each other, the deformations of slip systems in the two cases are also identical. For example, the shear strains of slip systems shown in Fig. 6 reveal that the maximum strain values in the cases of Directions 0° and 90° are equal to each other, and their corresponding slip systems (the \((111)[01\overline{1} ]\) and \((1\overline{1} \overline{1} )[\overline{1} 0\overline{1} ]\) slip systems) are symmetrical to each other along the symmetrical axis. Therefore, the two scratching depths are equal to each other when the two directions are symmetrical.

Figure 6
figure 6

Evolution of accumulated shear strain of slip systems of when a tip scratches along a Direction 0°, and b Direction 90°

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Figure 7a, b shows the resolved shear stresses of 12 slip systems of reference node (shown in Fig. 1a) when a tip scratches along Directions 90° and 45°, respectively. The resolved stresses in the case of Direction 45° are much greater than those of Direction 90°, which means that the slip systems tend to form greater reaction forces. Since the resultant force along the normal direction (the indent direction) produced by deformation of slip systems is a constant (6mN), the (001)-oriented copper tends to produce less deformation as the tip scratches along Direction 45°. Therefore, the scratching depth in the case of Direction 45° is smaller than that of Direction 90°.

Figure 7
figure 7

Evolution of resolved shear stress of the reference node when a tip scratches along a Direction 90°, and b Direction 45°

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Figure 8a–d illustrates the pile-ups induced by a tip that scratches along Directions 0°, 15°, 30° and 45°, respectively. Significant differences can be observed on the pile-ups of the right ends (where the scratching ends). For the cases of Directions 0° and 45°, the induced pile-ups are symmetric with themselves along the scratching directions. While the pile-ups of the other two cases are asymmetric, and the pile-up heights on the left-side of scratching direction are greater than those of the right-side. Since the slip systems laid symmetrically with respect to Directions 0° and 45°, the slip systems in the two cases tend to deform symmetrically along the scratching direction. Further, when a tip scratches along Direction 15° or 30°, the slip systems on the left-side of scratching direction (shown in Fig. 5b) are more prone to be triggered than those of the right-side, due to the relatively higher density of slip systems and smaller angles between scratching direction and slip systems. Thus, the pile-up heights on the left-side are larger than those of the right-side.

Figure 8
figure 8

Simulated topography pile-ups after scratching along different directions, a Direction 0°, b Direction 15°, c Direction 30°, d Direction 45°

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Figure 9a–d shows the sections vertically cutting along the dash lines AB, CD, EF and GH in Fig. 8a–d, respectively. Apparently, positive displacements along the scratching direction in subsurface are observed for the regions on the left- and right-sides of the tip, indicating that materials at specific local regions around the tip tend to flow upwards. For the cases from Direction 0° to Direction 45°, the main uplifted region decreases gradually. It is clear that scratching direction is important to plastic flow in the subsurface of (001)-oriented copper.

Figure 9
figure 9

Subsurface deformation induced by a tip that scratches along a Direction 0°, b Direction 15°, c Direction 30° and d Direction 45°

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Deformation behavior of (101)-oriented copper

The simulated scratching depths when a tip scratches along different directions on (101)-oriented copper are shown in Fig. 10a. Similar to the case of (001)-oriented copper, the scratching depth also changes periodically with the variation of scratching direction. However, the period in the present case is 180° instead of 90°. Further, the maximum and minimum depths are observed in the cases of < 110 > and < 100 > directions, respectively, which are the opposite of the case of (001)-oriented copper, indicating that a (101)-oriented copper tends to produce larger plastic deformation as a tip scratches along the < 110 > directions. The difference between the maximum and minimum depths is about 27.7%, which is a bit larger than that of (001)-oriented copper case (22.8%).

Figure 10
figure 10

a Scratching depths when a tip scratches along different scratching directions. b Schematic illustration of slip systems of (101)-oriented copper

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The slip systems of (101)-oriented copper with respect to the scratched surface are demonstrated in Fig. 10b. There are two symmetrical axes of slip systems for (101)-oriented copper, and they are parallel to Directions 0° (or 180°) and 90° (or 270°), respectively. The scratching depths when a tip scratches along the directions on the right-side of the symmetrical axes (such as Direction 45°) are equal to those of directions (such as Direction 135°) on the left-side. Therefore, the two symmetrical axes lead to the periodical variation of scratching depth, and the period is 180°.

Figure 11a, b shows the resolved shear stresses when a tip scratches along Directions 0° and 90°, respectively. Compared with the case of Direction 90°, the shear stresses of the other case are generally larger so that a (101)-oriented copper deforms less in the normal direction when a tip scratches along Direction 0°.

Figure 11
figure 11

Evolution of resolved shear stress of the reference node when a tip scratches along a Direction 0°, and b Direction 90°

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The surface topographies of (101)-oriented copper fabricated by a tip which scratches along Directions 0°, 30°, 60° and 90° are shown in Fig. 12a–d, respectively. Similar to the case of (001)-oriented copper, there are two types of topographies, i.e., topographies symmetrical and asymmetric along the scratching direction. Since Directions 0° and 90° are parallel to the symmetrical axes of slip systems, the pile-ups are symmetrical along the scratching directions in the two cases. Moreover, the greater pile-up heights observed on the right-side in the cases of Directions 30° and 60° can be attributed to the denser located slip systems on the right-side (shown in Fig. 10b).

Figure 12
figure 12

Simulated topology pile-ups after scratching along different directions, a Direction 0°, b Direction 30°, c Direction 60°, d Direction 90°

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The subsurface deformation of (101)-oriented copper induced by scratching is shown in Fig. 13. Similar to the case of (001)-oriented copper, there are also regions with positive displacement on the left- and right- sides beneath the surface. Further, with the increase of scratching direction angle, the effected regions become larger instead of smaller, compared to the cases of (001)-oriented copper. However, a larger scratched depth results in a larger uplifted region for both the cases of (001)- and (101)-oriented coppers.

Figure 13
figure 13

Subsurface deformation induced by a tip which scratches along a Direction 0°, b Direction 30°, c Direction 60° and d Direction 90°

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Deformation behavior of (111)-oriented copper

The scratching depth for the case of (111)-oriented copper shown in Fig. 14a reveals that the depth changes periodically with a period of 120°, and both the maximum and minimum depths arise when the scratching direction is parallel or near to the < 112 > directions. The maximum depth is about 10.5% greater than the minimum value, which is smaller than those of (001)-oriented copper (22.8%) and (101)-oriented copper (27.7%) cases. Figure 14b illustrates the slip systems of (111)-oriented copper beneath the scratching surface. By computing the angles between the normal direction (the [111] direction) and slip directions, it is easy to observe that all the [101], [011] and [110] slip directions have an angle of 35.26° with respect to the normal direction. While the angle is 144.74° when the slip directions are \([\overline{1} \overline{1} 0]\), \([0\overline{1} \overline{1} ]\) and \([\overline{1} 0\overline{1} ]\). Therefore, the slip systems are asymmetrical along Directions 0° (or 180°), 60° (or 300°) and 120° (or 240°), but symmetrical along Directions 30° (or 210°), 90° (or 180°) and 150° (or 330°), instead. Therefore, according to the discussion in Sect. 3.1, the period becomes 120° instead of 60° in the present case.

Figure 14
figure 14

a Scratching depths when a tip scratches along different scratching directions. b Schematic of slip systems of (111)-oriented copper

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The resolved stress variations of the reference node in the cases of Directions 30° and 90° are plotted in Fig. 15a, b. There has little difference between the shear stresses of the two cases. The stress values of the latter case are a little larger than those of the former case. Therefore, the corresponding scratching depth is greater when a tip scratches along Direction 30°, but the difference between the maximum and minimum depths is relatively small compared to those of the (001)- and (101)-oriented coppers.

Figure 15
figure 15

Evolution resolved shear stress of the reference node when a tip scratches along a Direction 30° and b Direction 90°

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The pile-ups of (111)-oriented copper are shown in Fig. 16a–e. Similar to (001)- and (101)-oriented coppers, the induced pile-ups are symmetrical along the scratching directions when the scratching directions are parallel to the symmetrical axes of the slip systems. While the pile-ups are asymmetrical for the other two cases, and the pile-up heights located on the left-side of scratching direction are larger than those of the right-side.

Figure 16
figure 16

Simulated topography pile-ups of (111)-oriented copper after scratching along different directions, a Direction 30°, b Direction 45°, c Direction 60°, d Direction 75° and e Direction 90°

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Figure 17a–e shows the subsurface deformation of (111)-oriented copper scratched along different directions. The size difference for the main uplifted regions of the selected five cases is relatively smaller. Similarly, a larger region is yielded when the scratched depth is larger.

Figure 17
figure 17

Subsurface deformation induced by a tip which scratches along a Direction 30°, b Direction 45°, c Direction 60°, d Direction 75° and e Direction 90°

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Conclusions

The deformation behavior of (001)-, (101)- and (111)-oriented coppers under different scratching paths is investigated by CPFEM simulation. The main conclusions can be drawn as follows:

  1. (1)

    The simulated scratching depth and surface pile-up topography are in good agreement with experimental results. The developed CPFEM model is therefore validated.

  2. (2)

    Scratching depth is significantly affected by scratching path. The scratching depth changes periodically with respect to the scratching direction, and the periods for (001)-, (101)- and (111)-oriented coppers are 90°, 180° and 120°, respectively.

  3. (3)

    Various topographies generated in the cases with different scratching paths. The pile-ups are symmetrical when a tip scratches along the directions paralleled to the symmetrical axes of slip systems.