Crystal Plasticity Finite Element Analysis of Spherical Nanoindentation Stress–Strain Curve of Single-Crystal Copper

Abstract

In this paper, we perform crystal plasticity finite element (CPFE) simulations of spherical nanoindentation to extract the indentation stress–strain (ISS) curve for a single-crystalline copper. The load–displacement curves on the Cu (010) surface at incremental indentation depths are obtained. Surface pile-up topography is explored and characterized by the activated slip systems on the indented surface and stress distribution on the cross-section to reveal the crystal anisotropy. And the effect of indentation depth on the stiffness and surface pile-up height is further analyzed. Finally, the zero point is defined, and the indentation stress–strain (ISS) curve is extracted from load–displacement curves. The validity of the ISS curve is demonstrated for crystalline copper materials by comparing measured results published in the literature.

1. Introduction

Successful engineering applications of crystalline metals require a detailed understanding of their microscale deformation behavior, which is controlled by the mechanical properties of individual grains for both polycrystalline and single-crystalline materials [1,2]. In turn, the mechanical characteristics govern the overall constitutive behavior of crystalline materials undergoing plastic deformation.

Considerable attention has been paid to measuring or estimating stress–strain curves for crystalline materials. As an important mechanical response of material deformation, the stress–strain curve has a heavy dependence on the material itself and works independently of test processing and instruments to a certain extent, which also possesses great significance in the self-explanatory for material constitutive laws. Some common practices have been applied to extract stress–strain curves by calibrating the experiments with corresponding simulations. For example, an inverse approach integrated with finite element (FE) analysis was developed to adjust the stress–strain data obtained from compression experiments, enabling the determination of the material’s true stress–strain behavior [3]. Homogeneity constitutive is usually used to describe the deformation behavior of metallic materials, and the anisotropy effect is neglectful for mechanical properties. The plastic deformation behavior of crystalline materials is typically direction-dependent and non-uniform, resulting in the development of specific deformation textures and internal microstructural features [4,5]. Delaire et al. [6] performed uniaxial tension for a copper sample made of a single layer of grains and found a good agreement between the experiments and FE simulations based on crystal plasticity finite element (CPFE). The CPFE simulation can describe microstructure influence on the deformation behavior of materials, and has been widely used in uniform deformation processes [7,8,9,10]. In particular, to achieve the consistency of process size and material microstructure, a substantial amount of effort is typically needed to generate a sufficient available specimen that provides a robust simulation model. Alternatively, it is possible to grow and test single crystals [11,12] or fabricate micropillars in FIB for the use of uniaxial compression [13,14], but those methods have high requirements for the equipment and are high cost and time-consuming.

Among the different experimental techniques, instrumented indentation exhibits tremendous potential for its low cost and high throughput, and can quickly probe multiple local volumes in a small, specified area [15,16]. It has been widely used to investigate the material properties, which usually measures the indentation load–depth curve at first. The concept of spherical indentation can be traced back to the pioneering work of Meyer in the early 1900s, who introduced it as a means to characterize material hardness. This method was further elaborated and theoretically grounded by David Tabor in his classic monograph The Hardness of Metals (1951), which remains a foundational reference in the field [17]. However, the indentation load–depth curve varies greatly with different conditions even concerning the same single-crystal specimen, such as indenter geometry [18]. So, comparing indentation stress–strain (ISS) curves proves to be a more effective and meaningful method for calibrating material constitutive models than relying solely on load–displacement data. Recently, Kalidindi and Pathak proposed to convert the load–displacement data into a meaningful ISS for spherical nanoindentation, which has shown tremendous potential [19,20].

This study builds upon the methodologies introduced by Kalidindi and Pathak [19,20] to derive the ISS curves for single-crystal copper using CPFE simulations of spherical indentation. A 3D CPFEM model was first established to replicate the load–depth responses on the Cu (010) surface across ten indentation depths. Surface pile-up behavior was investigated and linked to active slip systems and stress distribution to highlight the anisotropic nature of single-crystal copper during nanoindentation. Additionally, the influence of indentation depth on both stiffness and pile-up height was examined. Ultimately, the ISS curve was successfully obtained, offering a concise and reliable representation of the material’s mechanical behavior.

2. Materials and Methods

2.1. Spherical Indentation Process

As presented in Figure 1a, a typical indentation load–displacement curve obtained in spherical nanoindentation consists of two stages, namely, loading and unloading, where P designates the load and h the displacement of the indenter relative to the initial position. For modeling purposes, material deformation during the loading stage is regarded as a result of both a reversible elastic part and an irreversible plastic part. The maximum value of load $\tilde{P}$ appears at the maximum displacement $h_t$. For the unloading stage, it is assumed that only the elastic recovery occurs without the plastic part. The elastic unloading stiffness, S ($=dP/dh$), defined as the slope of the unloading curve, is derived from the initial unloading stage. After the indenter is fully unloaded, the residual penetration depth $h_r$ and the elastic penetration depth $h_e$ can be obtained. Figure 1b illustrates the primary indentation zone formed by a spherical indenter with radius R. In this diagram, $a$ represents the radius of the contact area under the applied load $\tilde{P}$, and $h_c$ denotes the contact depth between the surface and the indenter. Based on Hertzian contact theory [21,22], the indentation-affected region extends to an approximate length of $2.4a$.

2.2. ISS Curve

The obtained load–displacement data from spherical nanoindentation can be transformed into indentation stress–strain curves, facilitating a more precise assessment of the local mechanical behavior. The procedure for analyzing these data is provided in Ref. [19] and can be concisely described in two main stages. The first stage involves precisely locating the effective initial contact point within the dataset—in other words, clearly identifying a zero point that ensures the initial elastic portion of the curve aligns with the theoretical predictions of Hertz’s model. This zero point is typically determined using the following expression, applicable to the initial elastic regime under frictionless spherical contact:

$$S={\displaystyle \frac{3P}{2h_e}}={\displaystyle \frac{3\left(\tilde{P}-P_{\ast }\right)}{2\left(h_e-h_{\ast }\right)}}$$

(1)

where $\tilde{P}$, $h_e$, and S represent the measured load, displacement, and continuous stiffness, respectively. $P_{\ast }$ and $h_{\ast }$ correspond to the load and displacement at the point of effective initial contact. By rearranging Equation (1), a linear relationship $\tilde{P}-{\displaystyle \frac{2}{3}}S{h}_e$ emerges when plotting versus S, where the slope equals and the y-intercept corresponds to $-{\displaystyle \frac{2}{3}}{h}_{\ast }$ and $P_{\ast }$. This linear trend enables precise determination of the effective initial contact point ($P_{\ast }$ and $h_{\ast }$) through linear regression analysis.

In the second step, the indentation stress and strain values are determined by reformulating Hertz’s theory for elastic, frictionless spherical indentation as follows:

$$\begin{array}{l}{\sigma }_{ind}=E_{eff}{\epsilon }_{ind},{\sigma }_{ind}={\displaystyle \frac{P}{\pi a^2}},{\epsilon }_{ind}={\displaystyle \frac{4}{3\pi }}{\displaystyle \frac{h_e}{a}}\approx {\displaystyle \frac{h_e}{2.4a}},\\ a={\displaystyle \frac{S}{2E_{eff}}},{\displaystyle \frac{1}{E_{eff}}}={\displaystyle \frac{1-v_s^2}{E_s}}+{\displaystyle \frac{1-v_i^2}{E_i}},{\displaystyle \frac{1}{R_{eff}}}={\displaystyle \frac{1}{R_i}}+{\displaystyle \frac{1}{R_s}}\end{array}$$

(2)

where ${\sigma }_{ind}$ and ${\epsilon }_{ind}$ represent the indentation stress and strain, respectively. $a$ denotes the contact radius under the applied load P, while $h_e$ is the elastic indentation depth, and S ($=dP/d{h}_e$) refers to the elastic stiffness as previously defined. $R_{eff}$ and $E_{eff}$ correspond to the effective radius and effective modulus of the combined indenter–specimen system. Additionally, $v$ and $E$ are the Poisson’s ratio and Young’s modulus, with subscripts s and i indicating the specimen and indenter, respectively.

3. CPFEM Model of Nanoindentation

3.1. Setup of Nanoindentation

Figure 2a presents the 3D CPFE model used for the nanoindentation simulation, comprising a cylindrical sample and a spherical indenter. The specimen, measuring 6 μm in diameter and 3 μm in height, is discretized with 21,344 C3D8 and 736 C3D6. To minimize mesh symmetry effects, C3D6 elements are placed along the cylinder’s axis. Element sizes transition from 10 nm at the center to 500 nm near the outer edge. The spherical indenter, with a 1 μm radius, is modeled using 1792 rigid quadrilateral elements (R3D4) and 112 rigid triangular elements (R3D3), with the smallest element also sized at 10 nm. The orientation of the crystal is defined as the [010] direction being perpendicular to the surface, which is the orientation used for the nanoindentation simulations.

Table 1. The crystalline structure and mechanical properties of the material of Cu (010).

Parameter	Lattice Structure	Density (gm/cc)	Hardness (GPa)	Young’s Modulus (GPa)
value	FCC	8.33	1.33	125.6

gives the crystalline structure and mechanical properties of the material of Cu (010). Figure 2b illustrates the loading and unloading process during displacement-controlled nanoindentation tests. During loading, the indenter moves into the specimen at a steady speed of 10 nm/s until reaching a specified depth. In the unloading phase, the indenter retracts at a constant rate of 33.3 nm/s. The friction coefficient is set to 0.1. A set of simulations is carried out, and the indentation depth varies from 10 nm to 100 nm and increments by 10 nm for each simulation.

3.2. Crystal Plasticity

Within this investigation, a computational framework employing a three-dimensional finite element model (3D FEM) was established using the implicit finite element platform ABAQUS. The model integrates a crystallographic plasticity-based constitutive framework implemented via the UMAT user-defined subroutine. Crystal plasticity theory, as articulated by Rice and Hill [23,24], posits that crystalline material deformation encompasses three key mechanisms: reversible elastic lattice stretching, rigid-body lattice orientation adjustments (lattice rotations), and irreversible slip deformation driven by dislocation mobility. To quantify these processes, the total velocity gradient is derived from the multiplicative decomposition of the deformation gradient into elastic and plastic components:

$$F=F_e{F}_p$$

(3)

where $F_e$ denotes the elastic deformation gradient and $F_p$ represents the plastic deformation gradient. The stress–strain relationship is described using the second Piola–Kirchhoff stress tensor ($S=\mathrm{C}:E_e$), which depends on the Lagrange Green strain tensor ($E_e={\displaystyle \frac{1}{2}}\left(F_e^T{F}_e-I\right)$):

$$S={\displaystyle \frac{1}{2}}C:\left\{{\left(F{F}_p^{-1}\right)}^T\left(F{F}_p^{-1}\right)-I\right\}$$

(4)

where I represents the second-order identity tensor, and $C$ is the fourth-order elasticity tensor. The plastic deformation gradient rate is defined through the plastic velocity gradient and can be written as

$${\dot{F}}_p=L_p{F}_p$$

(5)

Plastic deformation is considered to be fully accommodated by dislocation slip. Consequently, the plastic velocity gradient can be expressed as

$$L_p={\displaystyle \sum _{\alpha =1}^{N_s}{\dot{\gamma }}^{\alpha }}{\tilde{m}}^{\alpha }\otimes {\tilde{n}}^{\alpha }={\displaystyle \sum _{\alpha =1}^{N_s}{\dot{\gamma }}^{\alpha }}{\tilde{M}}^{\alpha }$$

(6)

where ${\dot{\gamma }}^{\alpha }$, ${\tilde{m}}^{\alpha }$, ${\tilde{n}}^{\alpha }$, and ${\tilde{M}}^{\alpha }$ represent the slip rate, slip direction, slip plane normal for the α slip system, and the Schmid tensor, respectively. The constitutive models described above are applicable to most crystal plasticity analyses, with differences among methods mainly arising from how the slip rate is determined from the resolved shear stresses. In this study, the crystallographic slip rate is defined by the following general equation [25]:

$${\dot{\gamma }}^{\alpha }={\dot{\gamma }}_0{\left|{\displaystyle \frac{{\tau }^{\alpha }}{{\tau }_c^{\alpha }}}\right|}^{m}sign\left({\tau }^{\alpha }\right)$$

(7)

where ${\tau }_c^{\alpha }$ represents the critical resolved shear stress (CRSS) that depends on work hardening, ${\tau }^{\alpha }$ is the resolved shear stress, ${\dot{\gamma }}_0$ denotes the reference shear rate, and $m$ is the power law exponent. The strength of active slip systems evolves with plastic shear strain and is effectively described by the following hardening law [26]:

$${\dot{\tau }}_c^{\alpha }={\displaystyle \sum _{\beta =1}^{N_s}{h}_{\alpha \beta }}\left|{\dot{\gamma }}^{\beta }\right|$$

(8)

where $h_{\alpha \beta }$ is defined as

$$h_{\alpha \beta }=h_0{\left(1-{\displaystyle \frac{{\tau }_c^{\alpha }}{{\tau }_s}}\right)}^n{q}_{\alpha \beta }$$

(9)

The parameters $h_0$ and $n$ are hardening parameters, and ${\tau }_s$ is saturated shear stress. The parameter $q_{\alpha \beta }$ quantifies the latent hardening interaction between slip systems α and β, introducing anisotropy into the hardening model. Its value is set to 1.0 for coplanar slip systems and 1.4 for non-coplanar slip systems [27].

Table 2. CPFE Parameters of Cu (010).

Parameter	C11	C12	C44	Nslip	m	${\dot{\mathit{\gamma }}}_0$	${\mathit{h}}_0$	${\mathit{\tau }}_{\mathit{c}}^{\mathit{\alpha }}$	${\mathit{\tau }}_{\mathit{s}}$	n
value	168.4 d3	121.4 d3	75.4 d3	12	13	1 × 10−9	110	32	100	3

provides the specific parameters applied in the current constitutive model. These values are sourced from previous studies [28,29,30,31] and have been calibrated based on indentation experiments performed on single-crystal copper with three distinct grain orientations.

4. Results and Discussion

4.1. Mechanical Properties

Figure 3 provides simulated load–displacement curves of single-crystalline copper Cu (010) at different indentation depths. Each indentation process has loading and unloading stages when the indenter reaches the pre-determined depth. It can be seen that there is no apparent separation for the loading variation trend among the loading stages, and the loading curve for low indentation depth is just part of that for deep indentation depth. During the unloading, the rapid decline of the force is clear from the maximum value, which can be used to calculate the elastic unloading stiffness S.

4.2. Surface Pile-Up

Figure 4a illustrates the simulated Cu (010) morphology after spherical indentation at 100 nm depth. A persistent indentation scar marks the surface, indicative of irreversible plastic deformation. Additionally, significant material buildup is observed around the residual indentation, forming a surface pile-up with a distinct four-fold symmetric pattern. Figure 4b further illustrates that the distribution of Mises stress aligns closely with the surface pile-up profile. It also shows that the equivalent stress values drop to zero at the specimen boundaries, indicating minimal effects from the chosen specimen size-to-penetration depth ratio in the current CPFE model. The simulation results confirm that the deformation pattern of Cu (010) during indentation exhibits a clear four-fold symmetry, consistent with findings reported in earlier nanoindentation experiments and simulations [32,33,34,35].

As shown in Figure 4a, two dashed lines—one red and one green—are drawn across the surface pile-up regions to quantitatively measure the height. Correspondingly, Figure 5a presents the height–distance profiles along these lines, revealing orientation-dependent differences. Notably, the height along the red line reaches 26.2 nm, significantly higher than the 4.1 nm observed along the green line.

The surface pile-up symmetry is largely dictated by the material’s crystallographic orientation, which determines the activation of specific slip systems. In face-centered cubic (FCC) copper, there are twelve available slip systems, and deformation begins when the shear stress on a given slip plane exceeds the CRSS. In the case of spherical indentation on the Cu (010) surface, which avoids additional symmetries beyond the crystal’s own, the expected slip line patterns on the surface are illustrated in Figure 5b. Analysis of these patterns shows that four slip directions—[110], [−110], [−1–10], and [1–10]—are activated under indentation. Dislocation slip along these directions on the primary glide planes causes the material to accumulate where slip planes intersect the surface, resulting in the four-fold symmetric surface pile-up observed in Figure 4a and Figure 5a.

The two dashed lines shown in Figure 4a also divide the sample vertically into two distinct sections, creating a cross-section for detailed analysis of the crystal interior. Figure 6a presents the simulated distribution of the S23 shear stress component on this cross-section, representing shear stress along the Z direction within the XZ plane. According to Hertzian theory, which assumes the indented material is ideally linear elastic and isotropic, the shear stress is expected to be concentrated along the indentation axis within the inner region. However, the CPFEM simulation of single-crystal copper accounts for anisotropy, which shifts the location of the S23 shear stress away from the indentation axis, as shown in Figure 6a. This offset arises from slip systems influenced by the FCC crystal’s anisotropic nature, leading to asymmetry on either side of the indentation axis.

Furthermore, the total dislocation density generated by the spherical indenter in the CPFEM model indicates that the indentation zone extends to a depth of approximately $2.4a$, as illustrated in Figure 6a, supporting the definition of indentation strain used here (see also Figure 1b). Despite the anisotropic effects, the overall deformation pattern within the indentation zone remains consistent with Hertzian predictions.

4.3. Effect of Indentation Depth

Following the simulation procedures represented in Section 3.1, the elastic unloading stiffness S is calculated for each indentation depth. And Figure 7 illustrates the relationship between the elastic unloading stiffness S and the indentation depth. All the simulated points change well enough along a single unique line, indicating a remarkable linear relationship between the elastic unloading stiffness S and the indentation depth for CPEFM simulation of nanoindentation.

Figure 8 plots the maximum height value variation of the surface pile-up on both sides of the dashed line illustrated in Figure 4a, indicating the orientation–dependence discrepancy of pile-up under the indentation. In Figure 8a, black line 1 and red line 2 represent even maximum height along the red dashed and green dashed lines in Figure 4a, respectively. Although there is a considerable discrepancy in pile-up height along different directions, the change in maximum height value strongly correlates with the indentation depth for both lines, which exhibits a linear relationship. On the other hand, the result in Figure 8b reveals an almost parabola trend of maximum height value variation with the distance of the highest point from the indentation center. Specifically, different surface directions yield significantly different distributions and numerical values of surface pile-up patterns, which is essential for the analysis of the anisotropy properties of materials. Once the indentation is obtained, it can be used to determine the position and the numerical value at the highest point of pile-up under indentation.

4.4. Indentation Stress–Strain Curve

Figure 9 plots the change of $\tilde{P}-{\displaystyle \frac{2}{3}}S{h}_e$ as a function of S in the spherical nanoindentation simulations of single-crystalline copper. According to Equation (2), the values of $P_{\ast }$ and $h_{\ast }$ can be identified by the regression analysis of the line chart. And a linear regression analysis is performed, which can be estimated using the following relationship:

$$y=a+bx$$

(10)

where the y-intercept $a$ and the slope $b$, respectively, are equal to $P_{\ast }$ and $-{\displaystyle \frac{2}{3}}{h}_{\ast }$.

The indentation stress–strain curves obtained from the CPFE simulations are displayed in Figure 10, including one curve derived using Hertzian theory (without a zero point) and another incorporating the correction. It is evident that applying the zero-point determination method proposed here results in a significantly improved indentation stress–strain curve. A regression analysis performed on the curve with the zero point—excluding the first and last data points—reveals a strong linear correlation between indentation stress and strain. Additionally, Figure 10 illustrates the evolution of the plastic zone beneath the indenter through contours of the equivalent plastic strain (PEEQ) at corresponding indentation stresses. At an indentation depth of 10 nm, the indentation stress is relatively low, indicating that both elastic and plastic behaviors influence the stress field during the initial deformation stage. As the indentation depth increases, the deformation zone transitions into an elastic–plastic regime where plasticity becomes the dominant mechanism.

This indentation stress–strain curve is significant in establishing useful statistics of material properties during the deformation process. By this curve, one can determine the availability for each indentation test since the indentation stress depends on the material deformation concerning indentation strain, regardless of the indenter size and indentation parameters. This lends credibility to the simulation results, particularly when materials with comparable mechanical properties are evaluated, and it also facilitates the calibration of the developed material subroutine. For copper, the indentation stress–strain curve derived using the approach proposed in this study shows good agreement with the stress–strain data from both pillar compression tests and spherical indentation experiments reported in Ref [35].

5. Conclusions

CPFE simulations of spherical nanoindentation were conducted to derive the indentation stress–strain curve for single-crystal copper. Load–displacement data were generated for the Cu (010) surface at ten different indentation depths, serving as the foundation for further analysis. At a depth of 100 nm, the surface pile-up exhibits a distinct four-fold symmetrical pattern, consistent with previous experimental and numerical studies, thereby validating the simulation model. This symmetric feature is attributed to the activation of specific slip systems on the Cu (010) surface and is further supported by the stress distribution observed in the cross-sectional view. The influence of indentation depth on stiffness and pile-up height was also systematically investigated. Ultimately, the indentation stress–strain curve was extracted using the proposed method, demonstrating a significantly improved linear correlation between stress and strain when the zero-point correction is applied. The resulting curve shows strong agreement with data from pillar compression and spherical indentation experiments reported in the literature, highlighting its reliability. These findings are of considerable value for characterizing material behavior during deformation and for improving constitutive model calibration.