2024 Volume 65 Issue 5 Pages 494-501
The nanoindentation test is a widely adopted technique for characterizing the mechanical properties of materials. In this study, a dislocation density-based and a phenomenological crystal plasticity hardening model are employed to investigate the evolution of plastic anisotropy and pile-up of a single-crystal aluminum specimen with varying crystallographic orientations during nano-indentation. Utilizing crystal plasticity finite element (CPFE) simulations, we delve into the influence of crystal orientations on key factors such as depth-load curves, stress distributions, shear strains across different slip systems, and dislocation density evolution. Our analysis highlights the plasticity anisotropy inherent in the material, elucidated through the evolving shear strain exhibited by activated slip systems. Furthermore, we gain insights into the pile-up phenomenon by examining the evolution of shear strains within slip systems and the associated dislocation density, employing various modeling approaches. The height of pile-up evolution is determined by the localized cumulative shear strains and evolution of dislocation density.
Fig. 8 Evolution of pile-up height and dislocation density in the selected element during the nanoindentation on (011) surface.
Nanoindentation serves as a crucial tool for characterizing the mechanical properties of materials at a small scale, enabling the assessment of various parameters such as elastic modulus,1–5) hardness,6–8) size effect,9–13) plastic anisotropy.14–16) Throughout nanoindentation studies, the impact of crystallographic orientation in a wide range of metals has been thoroughly investigated.6,17,18) For example, Khan et al.19) observed a pronounced influence of crystal orientation on the indentation crack patterns while examining the deformation behavior and hardness of (100), (011), and (111) surfaces of a single crystal MgO, employing both Vickers and spherical indenters. Lim and Chaudhri7) reported the indentation hardness values of copper single crystals with different orientations are quite similar using a spherical indenter. Yoshida et al.20) investigated the effects of crystallographic orientation on nanoindentation of Al single crystal. Yield load, critical resolved shear stress, hardness, and shape of slip line/pile-up around the indentation have been systematically analyzed. Additionally, Filippov et al.21) explored the effects of crystal orientation and azimuthal indenter orientation on indentation hardness and modulus through Vickers indentation testing on a single crystal of aluminum.
Traditional macroscopic plasticity theories, like J2 plasticity, fall short of representing the anisotropy and orientation effects observed in crystalline materials during deformation. The influences of grain size and crystallographic orientation become pronounced at and below the grain length scale, particularly the scale of interest in nanoindentation.22) The crystal plasticity theory has been widely used to understand the macroscopic and microscopic mechanical behavior of polycrystalline materials.22–26) Wang et al.18) employed a 3D elastic-viscoplastic crystal plasticity finite element model to explore how grain orientations in single-crystal copper influence nanoindentation pile-up patterns and local texture. Liu et al.27) studied the plastic behavior of single crystal copper by crystal plasticity model and indentation with a spherical indenter. To enhance the precision and physical fidelity of predicting mechanical responses, several authors have introduced models grounded in dislocation density. For example, Han et al.28) reported the indentation pile-up behavior of Ti-6Al-4V alloy with Berkovich nanoindentation and predicted the pile-up patterns using crystal plasticity finite element. The results show that the evolution of pile-up patterns is determined by the cumulative shear strains and dislocation density distributions. Recently, Zhou et al.29) investigated the nanoindentation induced lattice rotations in TC6 single crystal though a combination of experimental tests and CPFE simulations.
The objective of this study is to explore the evolution of plastic anisotropy and pile-up of an aluminum single crystal during nanoindentation using phenomenological and dislocation density-based crystal plasticity model. The structure of this paper is outlined as follows: Section 2 offers a concise introduction to the fundamentals of crystal plasticity theory. Section 3 presents the simulation results and ensuing discussions. Finally, Section 4 encapsulates the study’s conclusions.
According to the theory of crystal plasticity,30) the total deformation gradient F can be divided into two portions: along the slip system sliding plastic deformation Fp and describes the stretching and rotation of the crystal lattice Fe
| \begin{equation} \mathbf{F} = \mathbf{F}^{e}\mathbf{F}^{P} \end{equation} | (1) |
According to the eq. (1), the definition of the velocity gradient L can be written as
| \begin{equation} \begin{split} \mathbf{L} & = \dot{\mathbf{F}}\mathbf{F}^{-1}\\ \mathbf{L} & = \mathbf{L}^{e} + \mathbf{L}^{p} \end{split} \end{equation} | (2) |
Since the plastic deformation is occurred primary due to dislocation slip, the plastic velocity gradient tensor of the slip systems is given by
| \begin{equation} \mathbf{L}^{p} = \sum_{\alpha = 1}^{N}\dot{\gamma}^{\alpha} \mathbf{s}_{\alpha}^{*} \otimes \mathbf{m}_{\alpha}^{*} \end{equation} | (3) |
$\dot{\gamma }^{\alpha }$ is the shear strain rate of the slip system and superscript α denotes the slip system number in the crystallographic system according to the eq. (4), which can be expressed as a rate-dependent power-law:31)
| \begin{equation} \dot{\gamma}^{\alpha} = \dot{\gamma}^{\alpha}_{0}\mathop{\text{sgn}}\nolimits (\tau^{\alpha})\left| \frac{\tau^{\alpha}}{g^{\alpha}} \right|^{\frac{1}{m}} \end{equation} | (4) |
where $\dot{\gamma }_{0}^{\alpha }$ is the reference shear strain rate and gα denotes the deformation resistance on slip system α, the superscript m is the rate sensitivity exponent, when m is achieving for 0, which approaches rate-independent material.
$\mathbf{m}_{\alpha }^{\ast}$ and $\mathbf{n}_{\alpha }^{\ast}$ represent the slip direction and slip plane normal on slip systems in the deformed configuration respectively, which can be regarded as:
| \begin{equation} \begin{split} \mathbf{m}_{\alpha}^{*} & = \mathbf{m}_{\alpha} \cdot (\mathbf{F}^{e})^{-1}\\ \mathbf{n}_{\alpha}^{*} & = \mathbf{F}^{e} \cdot \mathbf{n}_{\alpha} \end{split} \end{equation} | (5) |
where $\mathbf{m}_{\alpha }^{\ast}$ and $\mathbf{n}_{\alpha }^{\ast}$ are the slip direction and the normal to the slip-plane before deformation.
Taking symmetric and anti-symmetric parts to plastic Dp and Lp, called inelastic stretching and spin tensors, the velocity gradient of plasticity (identical as elastic) can be decomposed into:
| \begin{equation} \begin{split} \mathbf{L}^{e} & = \mathbf{D}^{e} + \mathbf{W}^{e}\\ \mathbf{L}^{p} & = \mathbf{D}^{p} + \mathbf{W}^{p} \end{split} \end{equation} | (6) |
Inelastic stretching tensor Dp and spin for plastic tensor Wp can be written as, respectively:
| \begin{equation} \mathbf{D}^{p} = \frac{1}{2}(\mathbf{L}^{p} + \mathbf{L}^{P^{T}}) = \frac{1}{2}\sum_{\alpha = 1}^{N}\dot{\gamma}^{\alpha}(\mathbf{m}_{\alpha}^{*} \otimes \mathbf{n}_{\alpha}^{*} + \mathbf{n}_{\alpha}^{*} \otimes \mathbf{m}_{\alpha}^{*}) \end{equation} | (7) |
| \begin{equation} \mathbf{W}^{p} = \frac{1}{2}(\mathbf{L}^{p} - \mathbf{L}^{P^{T}}) = \frac{1}{2}\sum_{\alpha = 1}^{N}\dot{\gamma}^{\alpha}(\mathbf{m}_{\alpha}^{*} \otimes \mathbf{n}_{\alpha}^{*} - \mathbf{n}_{\alpha}^{*} \otimes \mathbf{m}_{\alpha}^{*}) \end{equation} | (8) |
The elastic deformation of the single crystal can be described by deformation gradient Fe in a state of stress-free from the middle of the configuration to the current configuration, which reflects the lattice distortion and the rigid body rotation of the crystal. Therefore, the Green strain in the middle configuration can be written as:
| \begin{equation} \mathbf{E}^{e} = \frac{1}{2}(\mathbf{F}^{e^{T}}\mathbf{F}^{e} - \mathbf{I}) \end{equation} | (9) |
The elastic constitutive equation is given by:
| \begin{equation} \overset{\nabla}{\boldsymbol{\sigma}^{e}} = \mathbf{C}:\mathbf{D}^{e} \end{equation} | (10) |
$\overset{\nabla}{\boldsymbol{\sigma}^{e}}$ is the Jaumann rate of Cauchy stress σ, which is the co-rotational stress rate in terms of the coordinate system that rotates with the crystal lattice. $\overset{\nabla}{\boldsymbol{\sigma}}$ is the co-rotational stress on the coordinate system that rotates with the material.30)
| \begin{equation} \overset{\nabla}{\boldsymbol{\sigma}^{e}} = \overset{\nabla}{\boldsymbol{\sigma}} + \mathbf{W}^{p} \cdot \boldsymbol{\sigma} + \boldsymbol{\sigma} \cdot \mathbf{W}^{p} \end{equation} | (11) |
Combing eqs. (10), (12) and (15) can obtain the constitutive equation of crystal plasticity in the form of:
| \begin{equation} \overset{\nabla}{\boldsymbol{\sigma}^{e}} = \mathbf{C}:\mathbf{D}^{e} - \sum_{\alpha = 1}^{Ns}\dot{\gamma}^{\alpha}[(\mathbf{W}^{p} \cdot \boldsymbol{\sigma} - \boldsymbol{\sigma} \cdot \mathbf{W}^{p}) + \mathbf{C}:\mathbf{P}^{\alpha}] \end{equation} | (12) |
In the equation, $\mathbf{P}^{\alpha } = \frac{1}{2}(\mathbf{m}_{\alpha }^{\ast} \otimes \mathbf{n}_{\alpha }^{\ast} + \mathbf{n}_{\alpha }^{\ast} \otimes \mathbf{m}_{\alpha }^{\ast})$. Ns is the number of slip system of the crystal.
Then, the relationship for updating the stress at the integration point when the shear strain rate $\dot{\gamma }^{\alpha}$ are solved at the slip system, can be described as:
| \begin{equation} \boldsymbol{\sigma}_{n + 1} = \mathbf{R}_{n + 1} \cdot \boldsymbol{\sigma}_{n} \cdot \mathbf{R}_{n + 1}^{T} + \Delta t\overset{\nabla}{\boldsymbol{\sigma}} \end{equation} | (13) |
where R is the rotation tensor of lattice, which is obtained by the polar decomposition of the elastic deformation gradient, using the equation R = FeU−1.
2.2 Hardening model 2.2.1 Phenomenological hardening model (Model 1)The strain hardening evolves with the accumulation of dislocations on active slip systems and can be constructed with the increment relation as follow:32)
| \begin{equation} \dot{g} = \sum_{\beta = 1}^{N}h_{\alpha \beta}\dot{\gamma}^{\beta} \end{equation} | (14) |
where hαβ represents the slip hardening moduli that ranges over all activated slip systems, consisting of self-hardening moduli (α = β) and latent hardening moduli (α ≠ β).
In this paper, a simple form for the self-hardening moduli is taken as:32)
| \begin{equation} h_{\alpha \alpha} = h_{0}\sec h^{2}\left| \frac{h_{0}\gamma}{\tau_{s} - \tau_{0}} \right| \end{equation} | (15) |
where τ0 the initial value of current strength, h0 is the initial hardening modulus. γ is the cumulative shear strain on all slip systems:
| \begin{equation} \gamma = \sum_{\alpha}\int_{0}^{t} | \dot{\gamma}^{(\alpha)} | dt \end{equation} | (16) |
The latent hardening moduli hαβ in eq. (14) is given in the format:
| \begin{equation} h_{\alpha \beta} = h_{\alpha \alpha}(\mathrm{q} + (1 - \mathrm{q})\delta^{\alpha \beta}) \end{equation} | (17) |
δαβ is Kronecker symbol and q is a constant value.
2.2.2 Dislocation density-based hardening model (Model 2)The plastic deformation arising from the slipping and hardening of three slip systems on each of the four {111} planes is a consequence of dislocation evolution and interactions, as demonstrated in previous studies.33) The resistance to slip on each plane $\tau_{c}^{\alpha }$ is postulated to be contingent upon dislocation density and encompasses both frictional stress $\tau_{0}^{\alpha }$ and the stress arising from interactions with forest dislocations $\tau_{\textit{for}}^{\alpha }$.
| \begin{equation} g^{\alpha} = \tau_{c}^{\alpha} = \tau_{0}^{\alpha} + \tau_{\textit{for}}^{\alpha} \end{equation} | (18) |
The initial slip resistance $\tau_{0}^{\alpha }$ is dependent on the Peierls stress and the initial content of dislocation debris. According to the Taylor law,34,35) the evolution of slip resistance $\tau_{\textit{for}}^{\alpha }$ is described by the changes in forest dislocation densities.
| \begin{equation} \tau_{\textit{for}}^{\alpha} = \mu b\sqrt{\chi^{\alpha \alpha'}\rho^{\alpha'}} \end{equation} | (19) |
where b is the value of Burgers vector, μ is the shear modulus, and χαα′ is a dislocation interaction matrix. According to the research of FCC polycrystalline by Arsenlis,36) there are six parameters in dislocation interaction matrix, χ0 = 0.1, χ1 = 0.22, χ2 = 0.3, χ3 = 0.38, χ4 = 0.16, χ5 = 0.45. The coefficients χ0 and χ1 are associated with in-plane interactions, where χ0 characterizes the interaction between dislocations sharing the same Burgers vector, and χ1 describes interactions between dislocations with different Burgers vectors on the same gliding plane. The remaining coefficients, χ2-χ5, correspond to out-of-plane interactions and are linked to four distinct types of junctions or locks: the Hirth junction, collinear lock, glissile junction, and Lomer junction, respectively.36) The first two coefficients address in-plane interactions, while the remaining four coefficients pertain to out-of-plane interactions.
The evolution of forest dislocation density is governed by the competition between dislocation propagation and annihilation.37)
| \begin{equation} \dot{\rho}^{\alpha} = \left(k_{1}\sqrt{\sum_{\alpha = 1}^{N}\rho^{\alpha}} - k_{2}\rho^{\alpha}\right) \end{equation} | (20) |
where k1 represents a rate-insensitive coefficient governing dislocation storage. k2 is a coefficient accounting for annihilation. By combing eqs. (18)–(20), the evolution of strain hardening, incorporating dislocation density, can be expressed as:
| \begin{equation} \dot{g}^{\alpha} = \dot{\tau}_{c}^{\alpha} = \sum_{\alpha' = 1}^{N}\frac{\mu b\chi^{\alpha \alpha'}}{2\sqrt{\displaystyle\sum_{\beta = 1}^{N}\chi^{\alpha \beta}\rho^{\beta}}} \left(k_{1}\sqrt{\sum_{\beta = 1}^{N}\rho^{\beta}} - k_{2}\rho^{\alpha'} \right) | \dot{\gamma}^{\alpha'} | \end{equation} | (21) |
The Jacobian matrix plays a pivotal role in determining the rate of global convergence of the finite element governing equation, which is established on the principle of satisfying global equilibrium. This matrix is defined as (∂σ/∂ΔE), where ΔE represents the logarithmic strain increment tensor provided by ABAQUS. During Newton-type implicit finite element iterations, the Jacobian matrix is employed to adjust the deformation field at each finite element integration point (F) within the current strain increment. This iterative process continues until the current stress field aligns with the principle of virtual work. The Jacobian matrix for the two different model encompasses the relationship between $\dot{\tau }^{\alpha }$ and Δγα, showing in eq. (14) and eq. (21).
2.3 Crystal plasticity finite element implementationThe crystal plasticity constitutive model described above has been integrated into the implicit finite element code ABAQUS/Standard using the user material subroutine (UMAT). At the beginning of each time increment, denoted as time t, the following information: time increment Δt, global strain increment ΔE, global stress σ, and solution-dependent state variables (SDVs), is provided to the subroutine from the primary ABAQUS program. Within the UMAT subroutine, the strain increment is first transformed from the global coordinate system to the local coordinate system. Subsequently, a series of calculations are performed for all slip systems, leading to the determination of specific SDVs, including the current resolved shear stress and shear strain rate.
The incremental shear strength is computed according to eq. (14) for the phenomenological model, utilizing the hardening modulus matrix described in eq. (17). In the dislocation density-based model, the incremental shear strength is computed by eq. (21). Furthermore, updates to the crystal orientation are computed and stored within the SDVs. Upon calculating the Jacobian matrix, the incremental stress is computed, and a transformation back to the global coordinate system is executed. Upon completing the incremental step, both the global stress and SDVs undergo updates before being returned to the primary ABAQUS program. The displacement is then increased incrementally to calculate the subsequent step.
The nano-indentation specimen was geometrically modeled as a cylindrical section with a height equal to 30 µm and a radius equal to 25 µm with 8276 CAX4 stress elements and 8398 nodes, as shown in Fig. 1. Knezevic et al.38) previously validate the 2D axisymmetric and 3D finite element-visco-plastic self-consistent model in axisymmetric compression simulation with ABAQUS, with results indicating virtually indistinguishable texture evolution. Because the maximum depth of indentation is 200 nm, the thickness of the specimen is large enough to avoid the influence from the substrate. In the simulation setup, the vertical displacement of the Berkovich indenter progressed along the axis of symmetry. The frictionless contact was applied between the indenter and the material sample. This is considered reasonable, since Liu et al.27) found that friction between the indenter and the material affects the height of pile-ups but hardly the load-displacement curves. Meanwhile, according to the conclusions of Liu et al.,26) the indentation depth and the load-displacement curve are not affected by the coefficient of friction. The diamond indenter was characterized by a Young’s modulus of 1000 GPa and a Poisson’s ratio of 0.007.39) Boundary conditions stipulated that the y-axis displacement at the bottom of the specimen was fixed, while the x-direction was defined as the free boundary.
Mesh of the built model for nanoindentation simulations.
The single crystal elastic constants for FCC structure Al are used in the literature,40) C11 = 108.2 GPa, C12 = 61.3 GPa and C44 = 28.5 GPa. Reference strain rate is taken as 1.0 and rate sensitivity exponent n = 1/m as 20. In the numerical algorithm, the rate sensitivity exponent is tuned to comply with Schmid’s law. n → ∞ means an exact satisfaction of Schmid’s law, but will encounter the numerical instability. To address this issue, it is recommended to set m slightly larger than zero, in this manuscript is m = 0.05 (n = 20).41) In model 1, the initial resolved shear stress τ0 was taken as 50.5 MPa, the initial hardening modulus h0 and the saturation yield stress τs are given as 146 MPa and 60 MPa. In model 2, the initial dislocation density for each slip system was set by 4.0 × 105 mm−2, the initial resolved shear stress τ0 was taken as 50.5 MPa, the coefficient k1 for dislocation storage is 1.0 × 104 mm−1 and the coefficient k2 accounts for annihilation is given as 10. The computational parameters mentioned above were established through a fitting process involving the numerical and experimental load-displacement curves, as reported in Ref. 17), for the (111) surface of a single crystal aluminum specimen. Following multiple iterations and trials, a commendable level of concordance between the numerical simulation and the experimental load-displacement curve was achieved, as visually depicted in Fig. 2. The parameters for the numerical analysis are listed in Table 1.
Comparison of the load-displacement curve on the (111) single crystal surface between the experimental data11) and simulated results.
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In this section, we will present the nanoindentation calculation results on single crystals with various crystallographic orientations. Figure 2 illustrates the comparison between the displacement-load curves obtained from the simulation models and the corresponding experimental data points (depicted as dots).11) The accuracy of strain rate sensitivity used in numerical simulations and the simplification of the axisymmetric model may cause differences between the experimental and numerical simulation profiles.
Figure 3 displays the load-displacement curves at a depth of 200 nanometers for different values of K1 and K2. The accuracy of dislocation density evolution is dependent on these two important parameters. In the dislocation density-based hardening model, K1 plays a pivotal role in determining dislocation generation, essentially controlling the material’s hardening behavior. As dislocations accumulate due to dislocation generation, they become entangled, requiring higher stress levels for continued slip, ultimately enhancing the material’s hardening capacity. Consequently, an increase in K1 leads to more pronounced dislocation pile-up under elevated stress conditions. On the other hand, K2 represents a material parameter governing the annihilation of dislocations. When dislocation line segments with matching characteristics but opposite polarity come into close proximity, these segments are eliminated, leading to a reduction in the overall dislocation density. The result elucidates the significance of K1 and K2 in the context of the dislocation density-based hardening model, shedding light on their respective roles in dislocation generation, accumulation, and annihilation within the material.
Load-displacement curves calculated at different K1 and K2.
Figure 4 illustrates the numerical load-displacement curves for various crystal orientations, revealing the presence of plastic anisotropy. At a depth of 200 nanometers, the nanoindentation force on the (111) surface measures 543.7 µN, slightly higher than the 542.8 µN observed on the (011) surface, while the (100) surface exhibits a force of 517.5 µN. This results in a ratio of 1.1 between the highest and lowest indentation forces. Notably, the indentation forces on the (111) and (011) surfaces are quite similar at the same depth. Moreover, when applying the dislocation density-based model, the computational results indicate indentation forces of 549.8 µN and 546.5 µN on the (011) and (111) surfaces, respectively. The lowest force of 511.1 µN is registered on the (100) surface at the maximum indentation depth. The steep unloading curves signify that there is minimal elastic recovery during unloading for the Al single crystal. These findings underscore the plastic anisotropy evident in the material’s response to nanoindentation, with variations in forces and unloading behaviors across different crystal orientations. It’s worth noting that a similar analysis of plastic anisotropy in single crystal copper was also conducted by Liu et al.27) This calculation provides a clear interpretation of the results, highlighting the trends in indentation forces and elastic recovery while emphasizing the plastic anisotropy in the material’s behavior during nanoindentation.
Numerical load-depth curves for three single crystal orientations.
Figure 5 shows the distribution of Von Mises stress in various single crystals at a depth of 200 nanometers, which represents the maximum indentation depth. The stress field’s distribution clearly highlights a significant reliance on crystal orientation during the nanoindentation process, and notably, the computational results generated by both models exhibit a high degree of similarity. In all orientations, the maximum stress concentration is observed directly beneath the indenter tip, but the patterns and magnitudes of stress in the vicinity of the indenter differ based on the crystallographic orientation. This emphasizes the significant influence of crystal orientation on the stress field during the nanoindentation process. These results underscore the distinct stress distributions observed around the indenter tip for different crystal orientations and reaffirms the consistency of results between the two modeling methods.
Simulation results showing the stress field in the three different orientations: (011) surface, (100) surface and (111) surface. (a) Model 1 (b) Model 2.
In the context of pyramidal indentation, it is generally observed that quasi-isotropic polycrystals tend to exhibit pile-up when the material has a low hardening rate. In contrast, materials with a high strain hardening rate typically exhibit sink-in behavior.1) The occurrence of pile-ups can be attributed to the anisotropic plastic shear deformation, and the contributions of different slip systems to this anisotropic plasticity are visually depicted in Fig. 6. Figure 6 presents the distribution of shear strains on {111} ⟨110⟩ slip systems following nanoindentation on (011) surface with a depth of 200 nm. By examining Figs. 6(b) and (c), it becomes apparent that the shear strains on two specific slip systems, namely (111) [1 0 -1] and (111) [-1 1 0], play a predominant role in driving the observed plastic anisotropy. In contrast, two other slip system, (111) [0 -1 1] and (-111) [0 -1 1], were not active in this situation. It provides a clear explanation of the pile-up phenomenon observed during pyramidal indentation and highlights the key slip systems responsible for the plastic anisotropy observed in this material.
Simulated shear strain distribution on {111} ⟨110⟩ slip systems after nanoindentation on (011) surface with depth of 200 nm. (a) (111) [0 -1 1] (b) (111) [1 0 -1] (c) (111) [-1 1 0] (d) (-111) [1 0 1] (e) (-111) [110] (f) (-111) [0 -1 1] (g) (1-11) [011] (h) (1-11) [110] (i) (1-11) [1 0 -1] (j) (11-1) [011] (k) (11-1) [101] (l) (11-1) [-1 10].
The progression of shear strains on various slip systems at a specific selected element during the loading phase of nanoindentation is shown in Fig. 7. The reason for choosing this element is it locates in the pile-up regions. Incipient plasticity occurs after some indentation displacement of about 70 nm, eight slip systems are active simultaneously. At the end of the loading stage, it is noteworthy that the shear strains on the (111) and (-111) slip systems exceed 15%. This observation indicates that these two slip systems make the most significant contributions to the pile-up phenomenon when compared to the other slip systems. This description provides a insight into the evolution of shear strains on different slip systems during the loading phase of nanoindentation, highlighting the predominant role played by specific slip systems in pile-up formation.
Evolution of shear strain on 12 slip systems in the selected element during the nanoindentation on (011) surface.
Figure 8 provides an insight into the evolution of dislocation density and the height of the pile-up at a selected element (specifically, at the top of the pile-up, matching the same element shown in Fig. 7) during both the loading and unloading stages of nanoindentation. At depths less than 75 nm, the pile-up height decreases with increasing indentation depth, as this stage corresponds to elastic deformation. Subsequently, plastic deformation initiates, leading to a gradual increase in the pile-up height due to the accumulation of localized shear strains in the slip systems.
Evolution of pile-up height and dislocation density in the selected element during the nanoindentation on (011) surface.
During the unloading stage where elastic recovery occurs, the pile-up height still increases as matrix materials beneath indenter rebound at the unloading stage. The total initial dislocation density is 4.8 × 106 mm−2, it increases with the nanoindentation process and the value is 2.19 × 107 mm−2 when the depth at 200 nm. During the unloading stage, characterized by elastic recovery, there is a slight increase in dislocation density, eventually stabilizing at a value of 2.32 × 107 mm−2. Remarkably, the evolution of dislocation density during the unloading stage has minimal impact on the pile-up height. The results provide an understanding of the interplay between dislocation density and pile-up height throughout both loading and unloading phases of nanoindentation, revealing their intricate relationship.
In this study, we conducted an in-depth investigation into the indentation behavior of single-crystal aluminum using crystal plasticity finite element simulations. We employed two distinct crystal plasticity models: a phenomenal hardening model (Model 1) and a dislocation density-based hardening model (Model 2). These models provided insights into the dynamic evolution of plastic anisotropy and pile-up formation during the nanoindentation process on single-crystal aluminum. The main conclusions are as follows:
This work is supported by Zhejiang Provincial Natural Science Foundation of China (grant number LQ23A020007) and the National Natural Science Foundation of Shanxi Province (grant number 202203021221066). The financial contributions are gratefully acknowledged.
