A Nanoindentation-Based Study on the Mechanical Properties of Main Rock-Forming Minerals in Granite

Abstract

Granite is widely used in buildings, stone carvings, and sculptures, where long-term durability is strongly influenced by the micromechanical behavior of its constituent minerals and mineral interfaces. However, conventional rock mechanics tests cannot resolve the mechanical heterogeneity at the mineral scale, particularly at mineral interfaces. To address this limitation, a systematic nanoindentation study was conducted to quantitatively characterize the elastic modulus, hardness, creep behavior, residual deformation, and fracture toughness of both individual minerals and mineral interfaces in granite, and to clarify their mechanical contrasts and interrelationships. The results show that the constituent minerals quartz, feldspar, and biotite exhibit elastic modulus of 121.9 GPa, 115.6 GPa, and 66.3 GPa, respectively. Quartz and feldspar show relatively better mechanical properties, whereas biotite exhibits the weakest mechanical behavior. Hardness shows the same trend. In contrast, creep displacement and residual indentation depth follow the opposite order, i.e., quartz < feldspar < biotite. In addition, the elastic modulus and hardness of mineral interfaces are lower than those of the adjacent minerals, whereas their creep displacement and residual indentation depth are higher. The dispersion of these micromechanical parameters for mineral interfaces is generally greater than that of the adjacent minerals. The fracture toughness values of both minerals and mineral interfaces were also obtained: mineral fracture toughness ranges from 3.1 to 6.2 MPa·m^0.5, while mineral interfaces range from 0.7 to 4.3 MPa·m^0.5. Further analysis of the micromechanical parameters indicates that elastic modulus, hardness, and fracture toughness exhibit clear positive correlations among minerals, mineral interfaces, and the mineral aggregate. Comparatively, the correlations are strongest for minerals and weakest for mineral interfaces.

1. Introduction

Granite has been one of the most important construction materials throughout human history due to its aesthetic value and long-term durability, and it has been extensively used in applications ranging from stone carvings and sculptures to modern architectural structures [1,2,3]. Nevertheless, under the combined influence of natural environmental conditions and anthropogenic activities, granite buildings commonly develop various forms of deterioration, including mechanical damage, surface weathering and exfoliation, as well as cracking. Similar degradation processes, such as spalling and cracking, are also frequently observed in stone carvings and sculptures. These damage phenomena are governed not only by the bulk mechanical properties of granite but also by the mechanical characteristics of its constituent mineral phases and the interfaces between adjacent mineral grains. As a typical multiphase material, granite exhibits significant heterogeneity in both mineral distribution and mechanical properties, which critically influences its macroscopic mechanical behavior, including strength, deformation, and fracture responses [4,5,6]. Consequently, investigating the microstructural features and mechanical properties of individual mineral phases in granite is essential for elucidating the origins of macroscopic failure and the mechanisms of microscopic damage evolution, thereby providing a theoretical foundation for the conservation and reinforcement of granite-based architectural heritage.

At present, the mechanical properties of granite are predominantly investigated using conventional rock mechanics testing methods. Studies focusing on granite used as a building material have shown that both anisotropy and the degree of weathering significantly affect the mechanical performance of granite structures [7,8]. Yuan et al. [9] demonstrated that potassium feldspar and biotite exert a greater influence on the uniaxial compressive strength of granite than quartz and plagioclase. Likewise, Tandon et al. [10] performed uniaxial compression tests on a range of rock types, including granite, and reported that rock strength is largely governed by the content and orientation of mica minerals. However, the mechanical parameters obtained from such conventional rock mechanics tests primarily represent the bulk mechanical response of granite and cannot precisely capture its micromechanical properties.

Nanoindentation provides an effective solution to this challenge and has been widely used to characterize the mechanical properties of thin films and artificial heterogeneous materials, such as concrete, in which constituent boundaries are well defined. In addition, it enables the probing of crystallographic defects within bulk materials [11,12,13,14]. In recent years, with the advancement of microscopic-scale rock mechanics testing, nanoindentation has emerged as a powerful technique for quantifying the micromechanical properties of individual mineral phases owing to its high spatial resolution, minimal sample constraints, and quasi-nondestructive nature. To date, this technique has been applied to a wide range of rock types, including granite, sandstone, shale, and coal rock. Zhang et al. [15] investigated the micromechanical properties of granite constituent minerals using nanoindentation and reported that quartz exhibits the highest mechanical performance, followed by feldspar, whereas mica is significantly softer and characterized by a pronounced porous structure. Xu et al. [16] analyzed the relationship between peak indentation load and mechanical parameters, showing that the elastic modulus and hardness of granite minerals decrease with increasing peak load. Ma et al. [17] employed nanoindentation to investigate the creep mechanisms of mineral phases in granite. Yi et al. [18] conducted nanoindentation tests on various rocks and compared the differences in the micro-mechanical properties exhibited by the same mineral in different rocks. Liu et al. [19] obtained the mechanical properties and fracture toughness of minerals in granite through nanoindentation experiments and combined them with the discrete element method to study the microcrack evolution and crack characteristics of different minerals in granite. Mahabadi et al. [20] conducted grid micro-indentation and micro-scratch tests to obtain the elastic modulus and fracture toughness of the granite constituent mineral phases, and used numerical simulations to study the effects of micro-scale heterogeneity and microcracks on the mechanical response and brittle fracture of crystalline rocks. Zhou et al. [21] obtained micromechanical parameters of sandstone through grid nanoindentation testing. By combining nanoindentation with Brazilian splitting tests, Cong et al. [22] demonstrated that variations in mineral composition and micromechanical properties are key factors governing the geomechanical contrast between thinly interbedded sand–mud layers. Bennett et al. [23] conducted nanoindentation tests on shale at different indentation depths and along different bedding orientations to investigate its mechanical behavior and anisotropy. Cai et al. [24] identified the optimal indentation depth for nanoindentation testing of coal rock. Lecomte et al. [25] investigated the three major mineral phases in granite by combining nanoindentation with SEM–EBSD, and found that the hardness of feldspar decreases and gradually stabilizes when the indentation depth reaches approximately 1000 nm. Lei et al. [26] performed nanoindentation tests on quartz, feldspar, and mica in granite and reported strong linear correlations among elastic modulus, hardness, and fracture toughness of these minerals. Furthermore, homogenization-based approaches have been employed to upscale micromechanical parameters obtained from nanoindentation, and the applicability of nanoindentation data for predicting macroscopic mechanical properties has been systematically discussed [27,28,29].

Although previous studies have provided valuable insights into the micromechanical properties of rocks, the mechanical heterogeneity at mineral interfaces has rarely been explicitly addressed. Nevertheless, intergranular mechanical effects play a critical role in evaluating the mechanical performance of granite, particularly with respect to damage initiation and failure evolution. To address this issue, Liu et al. [30] investigated the mechanical properties of major minerals and their intergranular regions in granite using nanoindentation, and found that the elastic modulus and hardness of the interfaces between quartz–plagioclase and quartz–mica fall between those of the corresponding adjacent minerals. Zhou et al. [31] further proposed a testing approach combining SEM–EDS and nanoindentation, revealing that the mechanical properties of mineral interfaces in granite generally lie between those of the two neighboring mineral grains, while exhibiting greater variability. However, studies on mineral interfaces in granite remain limited, particularly regarding the mechanical heterogeneity at interfaces between identical minerals.

Therefore, this study first conducts systematic nanoindentation tests on both individual minerals and multiple types of mineral interfaces, including identical and dissimilar interfaces, which have rarely been quantitatively investigated in granite studies, and compares the mechanical properties of individual minerals with those of mineral interfaces. Second, fracture toughness values of both minerals and mineral interfaces are estimated, enabling a quantitative comparison of fracture resistance between minerals and mineral interfaces. Third, the micromechanical parameters of minerals, mineral interfaces, and mineral aggregates are analyzed within an integrated framework to reveal their correlations across different structural scales.

2. Materials and Methods

2.1. Preparation of Test Sample

Granite samples were collected from Kunming, Yunnan Province, China. X-ray diffraction (XRD) (D8 Advance X-ray Diffractometer (Bruker, Karlsruhe, Germany)) analysis was conducted to qualitatively and semi-quantitatively determine the mineral composition of the granite. The XRD diffraction patterns (Figure 1a) indicate that the granite is primarily composed of quartz, feldspar, and biotite, with a minor amount of magnetite. Feldspar is the most abundant mineral, accounting for 46% of the total mineral content, followed by quartz (44%), biotite (9%), and magnetite (1%). These petrographic observations also provided the basis for distinguishing mineral phases and identifying mineral interfaces for subsequent nanoindentation testing.

Petrographic characterization of granite thin sections was carried out using a polarized optical microscope (BX53-P Polarizing Microscope (Olympus Corporation, Tokyo, Japan)). The observations indicate that the granite is characterized by a medium- to coarse-grained holocrystalline texture and a massive structure (Figure 1b). Most minerals occur as predominantly subhedral grains, and feldspar exhibits well-developed twinning (Figure 2a–c). Grain-size analysis was performed for quartz, feldspar, and mica, and the corresponding grain-size distribution histograms and cumulative frequency curves are presented in Figure 2d–f. The results show that quartz has the largest average grain size of 1.233 mm, followed by feldspar with an average grain size of 1.063 mm, whereas mica exhibits the smallest average grain size of 0.545 mm.

Subsequently, the granite sample was processed into cylindrical specimens (24 mm in diameter × 7 mm in height) and subjected to stepwise grinding and polishing using an MP-1000 automatic metallographic grinder–polisher (Lihua Instrument Co., Ltd., Laizhou, China). The grinding sequence employed SiC abrasive papers with grit sizes of P180, P800, P1200, P2000, and P3000, followed by fine polishing with diamond suspensions of W5, W3, W1, and W0.5 and a final polish using a W0.05 silica suspension. Specimens were rinsed with anhydrous ethanol after each polishing stage to prevent contamination. Post-polishing surface quality was verified via optical microscopy to ensure sufficient flatness and smoothness, meeting the precision requirements for subsequent nanoindentation testing. The polished surface morphology is illustrated in Figure 3.

2.2. SEM-EDS Testing and Indentation Point Arrangement

To ensure that the measurement points during nanoindentation testing correspond accurately to minerals and mineral interfaces, scanning electron microscopy (SEM) coupled with energy-dispersive spectroscopy (EDS) was first employed to characterize the surface morphology and identify minerals within the target indentation areas. Mineral phases were identified based on petrographic observations and characteristic elemental distributions obtained from SEM–EDS mapping. Quartz was identified by Si-rich regions without Al (Figure 4a,b), biotite by Fe–Mg-rich regions (Figure 4c,d), and feldspar by excluding quartz and biotite zones. Mineral interfaces were defined as clear boundaries between adjacent mineral grains observed in SEM–EDS and optical microscopy images. Indentation points were placed only within clearly identified mineral phases or interface zones to ensure accurate measurement of their micromechanical properties.

Nanoindentation tests were subsequently carried out in regions where the mineral distribution had been precisely identified using SEM–EDS analysis. All tests were performed using a Hysitron TI Premier nanoindenter. Indentation points were systematically arranged on both individual minerals and mineral interfaces, including interfaces between identical minerals and between dissimilar minerals. For each mineral, six indentation points were selected to ensure statistical reliability, while five indentation points were arranged for each type of mineral interface (Figure 4f). Due to the limited thickness of mineral interfaces in granite, it is difficult to completely avoid the mechanical influence of adjacent grains during nanoindentation. Therefore, the measured properties are interpreted as effective micromechanical parameters of interface-dominated regions. Indentation locations were selected in clearly visible boundary zones identified by SEM imaging to minimize mixed-phase effects. The spacing between adjacent indentation points was fixed at 10 μm for both minerals and mineral interfaces, satisfying the requirement of greater than 20 times the maximum indentation depth to avoid mechanical interaction between neighboring indents [32]. Tests were performed using a trapezoidal loading function consisting of loading, holding, and unloading segments. The loading rate was 500 μN/s, the maximum load was 5000 μN, the holding time at peak load was 5 s, and the unloading rate was 500 μN/s (Figure 5). The holding segment was introduced to quantify time-dependent creep displacement under constant load.

2.3. Nanoindentation Procedure

Nanoindentation is a technique in which a diamond indenter tip is pressed into the specimen surface under an ultralow load, while the indentation depth and the corresponding load are continuously recorded [33]. The testing principle is schematically illustrated in Figure 6a, where P_max denotes the peak load, h_c is the contact depth, h_max represents the maximum indentation depth, and h_f is the residual indentation depth after unloading. A representative load–displacement curve obtained from nanoindentation is shown in Figure 6b, comprising three stages: loading, holding, and unloading. The total indentation work can be decomposed into different energy components. U_c is the fracture energy associated with microcrack initiation, U_e is the elastic strain energy induced by elastic deformation, and U_p represents the plastic deformation energy resulting from irreversible plastic flow. Based on the load–displacement response, the elastic modulus, hardness, and fracture toughness are calculated according to established nanoindentation analysis methods.

(1)

Elastic modulus and hardness

Hardness and elastic modulus were derived from the load–displacement curves using the Oliver–Pharr approach, in which the unloading segment of the load–displacement curve is fitted by an exponential function [34]:

$$P=\alpha {(h-h_f)}^m$$

(1)

where $\alpha$ and $m$ are fitting constants, $P$ is the applied load, and $h$ is the indentation depth. The initial unloading stiffness, defined as the contact stiffness $S$, is obtained by differentiating Equation (1) at the maximum indentation depth:

$$S={({\displaystyle \frac{dp}{dh}})}_{h=h_{\max }}$$

(2)

The reduced modulus $E_r$ and hardness $H$ are then calculated as:

$$E_r={\displaystyle \frac{\sqrt{\pi }}{2\beta \sqrt{A_c}}}S$$

(3)

$$H={\displaystyle \frac{P_{\max }}{A_c}}$$

(4)

where $\beta$ is a correction factor related to the indenter geometry and is taken as 1.05 for the Berkovich indenter used in this study. $A_c$ is the projected contact area, which is determined from the contact depth:

$$A_c=24.56h_c^2+{\displaystyle \sum _{i=0}^7{C}_i}{h}_c^{1/2^i}$$

(5)

The contact depth $h_c$ is calculated as:

$$h_c=h_{\max }-\epsilon {\displaystyle \frac{P}{S}}$$

(6)

where $\epsilon$ is a geometry-dependent constant equal to 0.75 for a Berkovich indenter. The elastic modulus of the specimen is obtained from:

$$E={\displaystyle \frac{1-v^2}{\frac{1}{E_r}-\frac{1-v_i^2}{E_i}}}$$

(7)

where $\nu$ is the Poisson’s ratio of the specimen, ${\nu }_i$ is the Poisson’s ratio of the diamond indenter (0.07), $E$ and $E_i$ are the elastic moduli of the specimen and the indenter, respectively, with $E_i$ is typically 1141 GPa.

(2)

Fracture toughness

Fracture toughness is a key parameter characterizing a material’s resistance to crack propagation. However, microscale fracture toughness, particularly at mineral interfaces, has been relatively less explored. In this study, an energy-based method is adopted to evaluate the fracture toughness of mineral phases and mineral interfaces [35]. During nanoindentation, the total input energy $U_t$ is expressed as:

$$U_t=U_c+U_e+U_p$$

(8)

The critical energy release rate associated with crack initiation, $G_c$, is calculated as:

$$G_c={\displaystyle \frac{U_c}{A_{\max }}}$$

(9)

where $A_{\max }$ is the maximum projected contact area. The fracture toughness $K_{IC}$ is then determined as:

$$K_{IC}=\sqrt{G_c{E}_r}$$

(10)

3. Results and Analysis

3.1. Load–Displacement Curve

The nanoindentation load–displacement curves of minerals and mineral interfaces in granite are shown in Figure 7. Figure 7a–c illustrate the load–displacement responses of the three minerals. Quartz exhibits a high degree of curve overlap among different indentation sites, together with smooth loading–unloading profiles, indicating a dense internal structure, good homogeneity, and superior mechanical properties. In contrast, feldspar shows a lower degree of overlap, suggesting the presence of microvoids and other internal defects, which lead to reduced homogeneity and inferior mechanical properties compared with quartz. Mica presents the lowest overlap, and its load–displacement curves contain a highly pronounced “pop-in” phenomenon. Such behavior is characteristic of layered minerals [36] and indicates that slip along structural planes in mica occurs more readily under applied load, resulting in the weakest mechanical performance among the three minerals.

Figure 7d–i illustrate the nanoindentation load–displacement curves of mineral interfaces. In general, the load–displacement curves of mineral interfaces exhibit a substantially lower degree of overlap than those of mineral crystals, indicating inferior mechanical homogeneity at mineral interfaces. Moreover, the loading–unloading curves of Qz–Qz, Qz–Fs, and Fs–Fs interfaces are relatively smooth, whereas those of Bt–Bt, Qz–Bt, and Fs–Bt interfaces show a “pop-in” phenomenon. This contrast suggests that the mechanical response of mineral interfaces is significantly influenced by the mechanical characteristics of the adjacent minerals.

3.2. Analysis of Mechanical Properties of Minerals and Mineral Interfaces

3.2.1. Microscopic Elastic Modulus and Hardness

Based on the experimental results shown in Figure 7, the elastic modulus and hardness of the minerals and mineral interfaces were calculated using Equations (1)–(7), with the results presented in Figure 8a–c, and both the standard deviation and the 95% confidence intervals are included. As shown in Figure 8a, the elastic modulus and hardness of quartz are (121.9 ± 1.3) GPa and (14.5 ± 0.1) GPa, respectively, while those for feldspar are (115.6 ± 7.0) GPa and (11.4 ± 0.3) GPa, respectively, and for biotite, they are (66.3 ± 2.7) GPa and (3.5 ± 0.2) GPa, respectively. Among the minerals, quartz exhibits the highest elastic modulus and hardness, followed by feldspar, with biotite having the lowest values. The values for feldspar and biotite show notable dispersion, primarily due to internal defects.

As shown in Figure 8b,c, the elastic modulus and hardness of mineral interfaces depend strongly on the mineral combinations forming the interface. Among interfaces between identical minerals, the Qz–Qz interface exhibits the highest elastic modulus and hardness, with values of (79.3 ± 3.8) GPa and (9.9 ± 0.6) GPa, followed by the Fs–Fs interface at (56.7 ± 1.7) GPa and (6.4 ± 0.4) GPa. The Bt–Bt interface shows the lowest elastic modulus and hardness, at (24.3 ± 6.1) GPa and (0.8 ± 0.2) GPa, respectively. For interfaces between different minerals, the Qz–Fs interface exhibits relatively high elastic modulus and hardness values of (77.6 ± 5.0) GPa and (9.1 ± 1.5) GPa. In contrast, interfaces involving biotite, including Qz–Bt and Fs–Bt, show significantly lower elastic modulus and hardness, with values of (52.8 ± 5.8) GPa and (2.2 ± 0.9) GPa, and (47.8 ± 4.8) GPa and (2.1 ± 0.5) GPa, respectively.

Comparison of Figure 8a–c reveals that the elastic modulus and hardness of mineral interfaces are consistently lower than those of the adjacent minerals, indicating that mineral interfaces represent mechanically weak zones within the rock. The greater dispersion of interface data relative to mineral crystals further implies a higher defect density and stronger heterogeneity at mineral interfaces.

As shown in

Table 1. Comparison of mechanical parameters between granite minerals and mineral interfaces. E denotes the elastic modulus. H denotes the hardness. h_cr denotes the creep displacement. h_f denotes the residual indentation depth. G_c denotes the critical energy release rate. K_IC denotes the fracture toughness.

Mineral: Mineral Interface	E	H	hcr	hf	Gc	KIC
Quartz: Qz-Qz	1.54	1.46	0.23	0.91	1.34	1.44
Quartz: Qz-Fs	1.57	1.59	0.19	0.86	1.55	1.55
Quartz: Qz-Bt	2.31	6.59	0.09	0.24	3.28	2.82
Feldspar: Fs-Fs	2.04	1.78	0.22	0.95	1.82	1.93
Feldspar: Qz-Fs	1.49	1.25	0.32	0.98	1.31	1.40
Feldspar: Fs-Bt	2.42	5.43	0.15	0.30	2.65	2.67
Biotite: Bt-Bt	2.74	4.38	0.21	0.30	6.02	4.43
Biotite: Qz-Bt	1.26	1.59	0.63	0.64	1.52	1.41
Biotite: Fs-Bt	1.39	1.67	0.60	0.55	1.46	1.48

, the elastic modulus of quartz is 1.54, 1.57, and 2.31 times that of the Qz–Qz, Qz–Fs, and Qz–Bt interfaces, respectively, and its hardness is 1.46, 1.59, and 6.59 times higher. For feldspar, the elastic modulus is 2.04, 1.49, and 2.42 times that of the Fs–Fs, Qz–Fs, and Fs–Bt interfaces, respectively, while the hardness is 1.78, 1.25, and 5.43 times higher. For biotite, the elastic modulus is 2.74, 1.26, and 1.39 times higher than that of the Bt–Bt, Qz–Bt, and Fs–Bt interfaces, respectively, and the hardness is 4.38, 1.59, and 1.67 times higher. These findings demonstrate that, for an individual mineral, the mechanical properties of its interfaces progressively deteriorate as the mechanical properties of the minerals on either side of the interface decrease.

3.2.2. Creep Displacement and Residual Indentation Depth

Based on the experimental results shown in Figure 6, the creep displacement during the holding stage and the residual indentation depth after unloading for minerals and mineral interfaces were determined from the load–displacement curves, with the results presented in Figure 8d–f. Each creep displacement value represents the average of repeated indentation tests conducted under identical conditions, and the error bars indicate the standard deviation, demonstrating acceptable experimental repeatability.

As shown in Figure 8d, quartz exhibits a creep displacement of (1.1 ± 0.5) nm and a residual indentation depth of (53.6 ± 2.3) nm. Feldspar shows higher values of (1.9 ± 0.6) nm and (78.9 ± 1.8) nm, while biotite exhibits the largest creep displacement and residual indentation depth, at (7.4 ± 3.8) nm and (141.8 ± 18.4) nm, respectively. The relatively small creep displacement and residual indentation depth of quartz indicate a high hardness and strong elastic recovery capability. Feldspar displays intermediate behavior, whereas biotite exhibits the strongest time-dependent deformation and plasticity, reflecting its relatively weak mechanical properties among the three minerals.

As shown by the results for mineral interfaces in Figure 8e,f, among identical mineral interfaces, the creep displacement and residual indentation depth are (4.8 ± 0.7) nm and (59.2 ± 15.3) nm for the Qz–Qz interface, (8.8 ± 0.6) nm and (83.3 ± 12.4) nm for the Fs–Fs interface, and (35.2 ± 7.1) nm and (471.7 ± 100.4) nm for the Bt–Bt interface, respectively. For dissimilar mineral interfaces, the creep displacement and residual indentation depth are (5.9 ± 2.1) nm and (80.5 ± 10.9) nm for the Qz–Fs interface, (11.8 ± 8.2) nm and (222.2 ± 121.1) nm for the Qz–Bt interface, and (12.4 ± 6.6) nm and (259.3 ± 51.7) nm for the Fs–Bt interface, respectively. Overall, the creep displacement and residual indentation depth of mineral interfaces are significantly larger than those of the adjacent minerals, indicating that mineral interfaces exhibit stronger plastic deformation and poorer mechanical properties.

As shown in Table 1, the creep displacement of quartz is 0.23, 0.19, and 0.09 times that of the Qz–Qz, Qz–Fs, and Qz–Bt interfaces, respectively, whereas the residual indentation depth reaches 0.91, 0.86, and 0.24 times that of the corresponding interfaces. Similarly, the creep displacement of feldspar is 0.22, 0.32, and 0.15 times that of the Fs–Fs, Qz–Fs, and Fs–Bt interfaces, respectively, with corresponding residual indentation depths of 0.95, 0.98, and 0.30. For biotite, the creep displacement is 0.21, 0.63, and 0.60 times that of the Bt–Bt, Qz–Bt, and Fs–Bt interfaces, respectively, while the residual indentation depth is 0.30, 0.64, and 0.55 times that of the corresponding interfaces. These results indicate that the plastic deformation behavior of mineral interfaces is quantitatively enhanced with increasing plasticity of the minerals on both sides of the interface. Graham et al. [37] have shown that nanoscale structural heterogeneity and differences among mineral phases significantly influence local mechanical response. A similar mechanism can be invoked to interpret the time-dependent deformation observed in granite. In this study, biotite exhibits a layered crystal structure with relatively weak interlayer bonding, which facilitates interlayer sliding and localized plastic deformation under sustained loading. As a result, biotite-rich regions show larger creep displacement and residual indentation depth compared with quartz and feldspar, whose framework structures are more rigid. At mineral interfaces, the mismatch in elastic modulus and hardness between adjacent phases leads to stress redistribution and local stress concentration, promoting plastic accommodation and enhanced creep near compositional boundaries. The pop-in events observed in biotite-rich areas may be associated with sudden interlayer sliding or microcrack initiation along cleavage planes.

3.2.3. Microscopic Critical Energy Release Rate and Fracture Toughness

Based on the experimental results presented in Figure 7, the critical energy release rate and fracture toughness of the minerals and mineral interfaces were calculated using Equations (8)–(10), with the results shown in Figure 8g–i.

As illustrated in Figure 8g, quartz exhibits a critical energy release rate of (316.4 ± 5.2) J·m⁻² and a fracture toughness of (6.2 ± 0.05) MPa·m^0.5. The corresponding values for feldspar are (267.2 ± 6.1) J·m⁻² and (5.6 ± 0.22) MPa·m^0.5, whereas biotite shows lower values of (146.8 ± 22.5) J·m⁻² and (3.1 ± 0.20) MPa·m^0.5. These results indicate that quartz exhibits the highest resistance to fracture among the three minerals, followed by feldspar, while biotite shows the lowest resistance to fracture.

Figure 8h,i illustrate the fracture-related properties of mineral interfaces. Among identical mineral interfaces, the Qz–Qz interface exhibits the highest resistance to fracture, with a critical energy release rate and fracture toughness of (235.3 ± 41.19) J·m⁻² and (4.3 ± 0.45) MPa·m^0.5, respectively. The Fs–Fs interface shows reduced values of (147.0 ± 35.54) J·m⁻² and (2.9 ± 0.34) MPa·m^0.5, whereas the Bt–Bt interface displays the weakest fracture resistance, with a critical energy release rate of only (24.4 ± 19.14) J·m⁻² and a fracture toughness of (0.7 ± 0.36) MPa·m^0.5. For dissimilar mineral interfaces, the Qz–Fs interface maintains relatively high fracture resistance, with a critical energy release rate of (204.2 ± 35.71) J·m⁻² and a fracture toughness of (4.0 ± 0.38) MPa·m^0.5. In contrast, interfaces involving biotite, namely Qz–Bt and Fs–Bt, exhibit substantially lower resistance to fracture, with critical energy release rates of (96.4 ± 53.35) J·m⁻² and (100.7 ± 47.91) J·m⁻², and corresponding fracture toughness values of (2.2 ± 0.41) MPa·m^0.5 and (2.1 ± 0.57) MPa·m^0.5, respectively. These results demonstrate that mineral interfaces possess weaker resistance to fracture than the adjacent mineral crystals.

Table 1 shows that the critical energy release rate of quartz is 1.34, 1.55, and 3.28 times that of the Qz–Qz, Qz–Fs, and Qz–Bt interfaces, respectively, while its fracture toughness is 1.44, 1.55, and 2.82 times higher. For feldspar, the critical energy release rate is 1.82, 1.31, and 2.65 times that of the Fs–Fs, Qz–Fs, and Fs–Bt interfaces, respectively, with corresponding fracture toughness ratios of 1.93, 1.40, and 2.67. In the case of biotite, the critical energy release rate is 6.02, 1.52, and 1.46 times that of the Bt–Bt, Qz–Bt, and Fs–Bt interfaces, respectively, while the fracture toughness ratios are 4.43, 1.41, and 1.48. Comparison of fracture toughness values indicates that the resistance to fracture of mineral interfaces decreases as the fracture toughness of the adjacent minerals decreases.

3.2.4. Data Plausibility

The experimental results of this study are compared with the data obtained from nanoindentation tests on minerals in granite by other scholars, as shown in

Table 2. Compilation of literature data on the micro-mechanical parameters of granite.

Minerals and Mineral Interfaces	E (GPa)	H (GPa)	KIC (MPa·m0.5)
Quartz	121.9 ± 1.3 105.5 ± 1.3 [16] 104.5 ± 0.5 [18] 106.3 ± 2.9 [19] 108.71 ± 7.77 [26] 102.4 ± 2.6 [31] 122.11 [38] 102.2 ± 8.4 [39]	14.5 ± 0.1 14.8 ± 0.5 [16] 12.9 ± 0.2 [18] 14.5 ± 0.9 [19] 13.13 ± 0.85 [26] 12.48 ± 0.44 [31] 14.4 ± 2.0 [39]	6.2 ± 0.05 5.34 ± 0.07 [19] 8.68 ± 0.18 [20] 4.30 [26] 2.01 ± 0.19 [31] 6.7 ± 1.8 [36]
Feldspar	115.6 ± 7.0 100.8 ± 0.8 [16] 81.9 ± 5.4 [18] 100.6 ± 1.2 [19] 87.99 ± 9.06 [26] 60.56 ± 4.26 [31] 96.42 [38] 67.4 ± 6.2 [39]	11.4 ± 0.3 11.7 ± 0.3 [16] 8.6 ± 0.5 [18] 11.4 ± 0.4 [19] 9.22 ± 0.66 [26] 7.46 ± 0.68 [31] 7.7 ± 1.2 [39]	5.6 ± 0.22 4.83 ± 0.16 [19] 4.18 ± 0.09 [20] 3.70 [26] 1.41 ± 0.20 [31] 4.8 ± 1.2 [39]
Biotite	66.3 ± 2.7 46.3 ± 6.7 [16] 59.1 ± 4.2 [18] 45.2 ± 6.3 [19] 44.14 ± 4.76 [31] 72.91 [38] 34.0 ± 12.8 [39]	3.5 ± 0.2 2.0 ± 0.5 [16] 3.6 ± 0.2 [18] 1.8 ± 0.5 [19] 1.33 ± 0.26 [31] 1.9 ± 0.9 [39]	3.1 ± 0.20 2.41 ± 0.34 [19] 3.21 ± 0.05 [20] 2.12 [26] 0.86 ± 0.23 [31] 2.5 ± 1.1 [39]
Qz–Fs interface	77.6 ± 5.0 91.6 ± 9.7 [30] 82.93 ± 4.46 [31]	9.1 ± 1.5 12.1 ± 1.7 [30] 8.36 ± 1.19 [31]	4.0 ± 0.38 4.96 ± 0.5 [30] 1.63 ± 0.57 [31]
Qz–Bt interface	52.8 ± 5.8 56 ± 5.7 [30] 59.16 ± 5.13 [31]	2.2 ± 0.9 2.8 ± 0.8 [30] 4.16 ± 1.57 [31]	2.2 ± 0.41 2.59 ± 0.46 [30] 1.28 ± 0.48 [31]
Fs–Bt interface	47.8 ± 4.8 56.4 ± 5.2 [30] 35.22 ± 2.97 [31]	2.1 ± 0.5 3.6 ± 1.1 [30] 1.61 ± 0.72 [31]	2.1 ± 0.57 2.85 ± 0.48 [30] 0.90 ± 0.30 [31]

. There is a good consistency with the findings of other studies, further validating the rationality and reliability of the data in this research.

4. Discussion

4.1. Microscale Analogy of Mineral Interface Heterogeneity and Large-Scale Stress Patterns

The mechanical contrasts between mineral phases and interfaces in granite can be viewed as a microscale analogue of lithology-controlled stress partitioning observed in Taiwan. Chelungpu Fault Drilling Project (TCDP) boreholes [40]. This analogy is particularly significant for understanding the long-term performance evolution of building stones. In the TCDP study, lithological contrasts between sandstone and siltstone layers created measurable stress heterogeneity, which was further validated by the variation in horizontal stress magnitudes after the 1999 Chi-Chi earthquake. Similarly, in granite, the mechanical differences between minerals such as quartz, feldspar, and biotite at their interfaces may also induce stress redistribution and localized damage, ultimately influencing crack propagation pathways and the material’s degradation over time. These microscale insights are crucial for predicting the durability and mechanical performance of stone materials in both architectural heritage and engineering applications, as they help understand how microstructural heterogeneity can affect material behavior under long-term loading conditions.

4.2. Correlation Between Microscopic Elastic Modulus and Hardness

Figure 9a–c show the relationships between elastic modulus and hardness for minerals (r = 0.98, p < 0.01), mineral interfaces (r = 0.90, p < 0.01), and their macroscopic mineral aggregate (r = 0.92, p < 0.01). As shown, the elastic modulus–hardness relationships for minerals, mineral interfaces, and the mineral aggregate all exhibit an S-shaped distribution. The S-shaped relationship does not represent an intrinsic nonlinear scaling law of granite minerals. Instead, it mainly reflects clustering among different mineral phases and mineral interfaces with distinct mechanical property ranges. In this study, data from quartz, feldspar, biotite, and several interface types occupy different E–H intervals. When combined, these phase-dependent datasets produce the apparent S-shaped distribution rather than a continuous material-dependent relationship. In particular, interfaces such as Qz–Bt and Fs–Bt show a relatively slow increase in hardness with elastic modulus, which cannot be well described by simple power-law or logarithmic models. Therefore, the S-shaped curve should be regarded as a descriptive model reflecting phase-controlled mechanical clustering rather than an intrinsic constitutive law. The correlation coefficients are 0.95 for minerals, 0.87 for mineral interfaces, and 0.87 for the mineral aggregate, indicating strong correlations. This suggests that materials or material boundaries that are resistant to elastic deformation also tend to be resistant to plastic deformation.

4.3. Correlation Between Microscopic Fracture Toughness and Elastic Modulus

Figure 9d–f show the relationships between fracture toughness and elastic modulus for minerals, mineral interfaces, and the mineral aggregate. As shown, fracture toughness exhibits a good linear correlation with elastic modulus for minerals (r = 0.99, p < 0.01), mineral interfaces (r = 0.94, p < 0.01), and the mineral aggregate (r = 0.97, p < 0.01), with correlation coefficients of 0.97, 0.87, and 0.86, respectively. This indicates that materials or material boundaries with a higher elastic modulus have a stronger resistance to fracture.

4.4. Correlation Between Microscopic Fracture Toughness and Hardness

Figure 9g–i show the relationships between fracture toughness and hardness for minerals (r = 0.99, p < 0.01), mineral interfaces (r = 0.87, p < 0.01), and the mineral aggregates (r = 0.92, p < 0.01). A clear linear correlation between fracture toughness and hardness is observed for minerals, mineral interfaces, and the mineral aggregate. This behavior can be attributed to the fact that materials with higher hardness generally exhibit higher elastic modulus, which in turn leads to a stronger resistance to fracture. The correlation for mineral interfaces is relatively weaker, with a correlation coefficient of 0.75.

By comparing the correlation coefficients of the mechanical parameters for minerals, mineral interfaces, and the mineral aggregate in Figure 8, it can be observed that the correlation coefficients for mineral interfaces are consistently lower than those for minerals and the mineral aggregate. This reflects, to some extent, the greater data scatter and stronger heterogeneity associated with mineral interfaces. Among the examined relationships, fracture toughness exhibits the strongest correlation with elastic modulus. Therefore, the fracture toughness of a material can be reasonably predicted using its elastic modulus, which helps to avoid more complex experimental measurements and computational procedures.

5. Conclusions

The primary conclusions of this study are summarized as follows:

(1)

Quartz exhibits the highest elastic modulus, hardness, and fracture toughness, indicating the strongest mechanical performance among the three minerals, followed by feldspar. In contrast, biotite shows the lowest values of these mechanical parameters, together with the largest creep displacement and residual indentation depth, reflecting a pronounced tendency toward plastic deformation.

(2)

The elastic modulus and hardness of all mineral interfaces—including interfaces between identical minerals and between dissimilar minerals—are lower than those of the adjacent minerals, and their measured values exhibit greater overall dispersion. The fracture toughness values of the minerals range from 3.1 to 6.2 MPa·m^0.5, while the range for mineral interfaces is from 0.7 to 4.3 MPa·m^0.5. This suggests that mineral interfaces are preferential sites for the initiation of microscopic damage in granite. Their relatively weak mechanical properties and higher uncertainty are important contributors to the macroscopic heterogeneity and anisotropy of granite.

(3)

The elastic modulus, hardness, and fracture toughness of mineral interfaces decrease with the degradation of the mechanical properties of the adjacent minerals, whereas the creep displacement and residual indentation depth increase as the plasticity of the adjacent minerals becomes more pronounced.

(4)

Across minerals, mineral interfaces, and the mineral aggregates, elastic modulus, hardness, and fracture toughness exhibit consistently strong positive correlations, indicating an intrinsic coupling between resistance to elastic deformation, resistance to plastic deformation, and resistance to fracture. This relationship confirms that materials (or material interfaces) with higher stiffness and hardness generally possess stronger fracture resistance at both the microscopic and macroscopic scales. Comparatively, the correlations for individual minerals are the strongest, with correlation coefficients of 0.95, 0.97, and 0.99, respectively, while those for mineral interfaces are relatively weaker, with correlation coefficients of 0.87, 0.87, and 0.75, reflecting the greater structural complexity, defect density, and mechanical heterogeneity inherent to interfaces. This further demonstrates that mineral interfaces act as mechanically unstable zones, playing a critical role in governing the heterogeneous and anisotropic mechanical behavior of granite.

Finally, it should be noted that the fracture toughness values obtained for mineral interfaces in this study should be interpreted as effective micromechanical parameters representing the indentation-induced fracture resistance of interface-dominated regions, rather than intrinsic material constants. We acknowledge this limitation and plan to explore more refined techniques in future research to more accurately capture the influence of material heterogeneity on fracture behavior.