Strain Rate Effects on Characteristic Stresses and Dynamic Strength Criterion in Granite Under Triaxial Quasi-Static Compression

Abstract

To investigate the effects of the strain rate and confinement on characteristic stresses and strength criterion in granite under static to quasi-static loading, triaxial compression tests were systematically conducted across strain rates of 10⁻⁶ to 10⁻² s⁻¹ and confining pressures of 0–40 MPa. Stress–strain curves, characteristic stresses, macro-fracture patterns, and dynamic strength criterion were analyzed. The experimental results indicate the following: (1) crack damage stress (σ_cd) and peak stress (σ_p) show strong linear correlations with logarithmic strain rate, while crack initiation stress (σ_ci) exhibits weaker rate dependence; (2) linear regression establishes characteristic stress ratios σ_ci = 0.58σ_p and σ_cd = 0.85σ_p; (3) macroscopic fractures transition from Y-shaped shear patterns under low confinement and strain rate conditions to X-shaped shear failures at higher confinement and strain rate; (4) the Mohr–Coulomb criterion effectively characterizes dynamic strength evolution in granite, with cohesion increasing 22% across tested strain rates while internal friction angle remains stable at around 50°; (5) variations in microcrack activity intensity during rock deformation stages result in the dynamic increase factor for characteristic stresses (CSDIF) of σ_ci being lower than σ_cd and σ_p. More importantly, σ_cd and σ_p exhibit CSDIF reductions as confining pressure increases. This differential behavior is explained by confinement-enhanced shear fracturing dominance during crack propagation stages, combined with the lower strain rate sensitivity of shear versus tensile fracture toughness.

1. Introduction

The mechanical properties of rock masses exhibit significant variations depending on burial depth, dynamic disturbances, and loading paths [1]. Dynamic loading intensities can be categorized into six regimes based on strain rate levels [2,3], as shown in Figure 1. Under both creep (<10⁻⁷ s⁻¹) and static loading (10⁻⁷–10⁻⁴ s⁻¹) conditions, the compressive strength of rocks typically exhibits strain rate independence, with engineering fields encompassing fault slip, slope stability, and long-term stability of surrounding rock in underground caverns. In quasi-static regimes (10⁻⁴–10⁻¹ s⁻¹), there is a gradual increase in compressive strength with strain rate increases (distinct from inertia-dominated mechanisms), correlating with engineering scenarios such as seismic loads (10⁻³–10⁻¹ s⁻¹) [4], rockburst (1.5 × 10⁻³–1.7 × 10⁻¹ s⁻¹) [5], and mining tremors (1.6 × 10⁻³–1.6 × 10⁻¹ s⁻¹) [5]. When the strain rate exceeds 10⁻¹ s⁻¹, transitional rate dependency governed by inertial effects becomes prominent in rock compressive strength, observed in blasting engineering, impact loading, and explosions. This analysis reveals that the compressive strength of rocks under quasi-static loading exhibits distinct deviations from both conventional static loading conditions and dynamic loading regimes (e.g., explosive/shock wave scenarios), highlighting their strain rate dependency [6]. Nevertheless, current engineering practices predominantly employ static mechanical parameters or empirically adjusted values (typically increased by 30%) based on early foreign experimental data to determine dynamic parameters [7]. This oversimplified approach, neglecting project-specific dynamic loading environments, substantially elevates the risk of rock mass instability.

The strain rate effects on rock mechanical properties under quasi-static loading have been extensively validated through experimental studies. For instance, Li et al. [8] performed triaxial compression tests on granite at strain rates of 10⁻⁴–10⁰ s⁻¹ and confining pressures of 20–170 MPa, revealing that peak strength increased synchronously with both the strain rate and confining pressure, whereas the elastic modulus and Poisson’s ratio remained rate-insensitive. Similarly, Liang et al. [6,9] identified 5 × 10⁻⁴ s⁻¹ as the threshold strain rate: strength remained constant below this value, while both strength and the elastic modulus increased above it. In contrast, Hokka et al. [10] conducted dynamic compression tests at confining pressures up to 225 MPa and reported that the strain rate sensitivity of strength is influenced by the confinement at low pressures. Kumar [11] demonstrated strain rate dependence in basalt/granite peak strength: under quasi-static conditions, brittle fracture persisted, while strain rate effects mirrored temperature-induced strengthening, supporting the thermal activation theory. Li et al. [12] and Zhao et al. [13] found that the total absorbed energy, elastic strain energy, and the dissipated energy of rock specimens at the peak strength point increased as the strain rate increased under uniaxial compression or true triaxial compression. Qin et al. [14] reported that the maximum energy and cumulative energy of acoustic emission increase continuously with the strain rate. In addition, the strain rate effect on compressive strength of rock was investigated across lithologies: shale [15] showed lower strength growth rates compared to marble [12], while sandstone [13,16] and coal [17] exhibited different rate dependencies. Characteristic stress, mainly including crack initiation stress (σ_ci), crack damage stress (σ_cd), and peak stress (σ_p), is a mechanical parameter used to describe the failure process of rock. Zhai et al. [18] and Liu et al. [19] reported consistent increases in σ_cd and σ_p for granite with the increase in strain rate, a result corroborated by Li et al. [20] using the grain-based finite-discrete element method. Experimental studies on uniaxial and triaxial conditions have also explored strain rate effects on σ_ci for various rock types [21,22]. While existing studies have analyzed the strain rate effects on characteristic stresses, the effects of the strain rate and confinement in quasi-static compression across varying strain rates have not been fully understood. In addition, the evolutionary patterns of characteristic stresses and their underlying mechanisms under combined strain rate and confining pressure conditions have not been systematically investigated.

Rock strength criteria establish functional relationships between principal stresses at failure. Over 100 static criteria have been proposed to date, with widely adopted formulations including the Mohr–Coulomb and Hoek–Brown criteria [23]. Recent research has prioritized adapting these classical frameworks for dynamic loading scenarios. Sato et al. [24] revealed a logarithmic growth of cohesion (c) with the strain rate, while the internal friction angle (φ) remained rate-independent when applying the Mohr–Coulomb criterion to dynamic rock failure. The numerical simulations [25] verified the Mohr–Coulomb framework’s applicability across dynamic loading regimes. Huang et al. [26] and Xia et al. [27] extended applications to dynamic tensile and punching shear failure. Ma et al. [28] demonstrated that the m-value in the Hoek–Brown criterion remains unaffected by the strain rate, established a correlation between parameter a and strain rate, and proposed a generalized Hoek–Brown dynamic strength criterion. Although the mechanical response of rocks under dynamic loading conditions has been extensively characterized, the development of strength criteria within quasi-static loading regimes has received comparatively limited attention. Furthermore, existing criteria for quasi-static conditions often exhibit excessive mathematical complexity, highlighting the need for simplified criteria applicable to the quasi-static loading regime.

After reviewing the literature, existing studies exhibit three main limitations: (1) the variation trends of σ_ci and σ_cd under combined confining pressure and strain rates have not been clearly defined; (2) the underlying mechanisms governing confining pressure’s influence on strain rate effects of different characteristic stresses have not been fully explained; (3) practical dynamic strength criteria for rock under quasi-static loading are still lacking. Based on this, triaxial compression tests on granite specimens were conducted over a wide range of strain rates (10⁻⁶ to 10⁻² s⁻¹) and confining pressures (0–40 MPa) to characterize quasi-static mechanical behaviors. Stress–strain curves, characteristic stresses, and macro-fracture patterns were analyzed, with a dynamic strength criterion established based on the Mohr–Coulomb failure theory. The values, ratios, and correlations of characteristic stresses were systematically investigated in relation to the strain rate and confining pressure. Finally, strain rate dependence discrepancies between σ_ci and σ_cd under varying confinement were discussed in detail, accompanied by proposed governing mechanisms.

2. Rock Specimens and Experiment Preparation

2.1. Rock Specimens and Testing System

The granite specimens (Figure 2a) were sourced from intact bedrock formations in Liuyang City, Hunan province, China. The detailed procedure for extracting the specimens as follows: (1) core drilling is performed in designated areas using a 50 mm diameter bit, with core lengths controlled within 300 mm; (2) immediate paraffin sealing is applied to obtained cores to minimize moisture loss-induced damage, followed by storage in core boxes; (3) transportation packaging is implemented through a layered configuration (inner to outer): core specimen → plastic membrane → core box → wooden boards → foam padding; (4) upon arrival at the lab, secondary machining is conducted according to ISRM-suggested methods [29]. Cylindrical specimens (Φ50 × 100 mm) are fabricated through precision grinding, satisfying geometric requirements of <20 μm end flatness and <50 μm parallelism deviation (Figure 2a). Prepared specimens are stored in hermetic containers for subsequent testing. To minimize potential disturbances to rock cores during sampling, we implemented the following measures: (1) the selection of intact and structurally stable granite formations as sampling locations; (2) strictly control drilling speed to minimize core disturbance during coring; (3) borehole spacing exceeding five times the drilling diameter to prevent stress interference between adjacent cores; (4) immediate paraffin sealing upon core extraction to minimize moisture loss, coupled with a multi-layered anti-vibration transport system to mitigate mechanical disturbances during transportation; (5) a single standard specimen preparation per 300 mm core segment to eliminate end effects. The physical parameters of the specimens are systematically listed in

Table 1. The basic physical parameters of rock under triaxial compression (CP—confining pressure; D—diameter; H—height; M—mass; V—volume; D—density).

Specimen	Strain Rate (s−1)	CP (MPa)	D (mm)	H (mm)	M (g)	V (cm3)	D (g.cm−3)
GS-S-1	10−6	10	49.81	100.62	524.90	196.07	2.68
GS-S-2		20	49.92	100.26	524.90	196.23	2.67
GS-S-3		30	49.91	100.67	523.90	196.95	2.66
GS-S-4		40	49.95	100.72	526.00	197.37	2.67
GS-S-5	10−5	10	49.94	100.71	526.00	197.27	2.67
GS-S-6		20	49.97	100.45	524.30	197.00	2.66
GS-S-7		30	49.89	100.62	524.80	196.70	2.67
GS-S-8		40	49.94	100.32	524.60	196.51	2.67
GS-S-9	10−4	10	49.99	100.31	525.60	196.88	2.67
GS-S-10		20	49.92	100.33	525.40	196.37	2.68
GS-S-11		30	49.99	100.51	525.30	197.27	2.66
GS-S-12		40	49.91	100.19	524.10	196.02	2.67
GS-S-13	10−3	10	49.99	100.44	526.50	197.13	2.67
GS-S-14		20	49.98	100.28	525.70	196.74	2.67
GS-S-15		30	50.03	100.34	525.80	197.25	2.67
GS-S-16		40	49.92	100.47	524.40	196.64	2.67
GS-S-17	10−2	10	50.01	100.59	526.00	197.59	2.66
GS-S-18		20	49.87	100.44	525.10	196.19	2.68
GS-S-19		30	50.03	100.18	524.70	196.94	2.66
GS-S-20		40	49.88	100.43	525.60	196.25	2.68

. A tandardized nomenclature (“GS-S-N”) was adopted, where GS denotes the granite specimen, S represents triaxial compression, and N indicates the specimen number.

Scanning electron microscopy (SEM) observations at varying magnifications (Figure 2b–e) revealed microstructural features of the granite specimen, including densely packed mineral grains, small pores, closed microcracks, and smooth surfaces. XRD results show that the granite is predominantly composed of quartz, K-feldspar, and biotite, consistent with its classification as biotite granite.

2.2. Testing System

Triaxial compression testing was conducted using an MTS 815 rock mechanics testing system at Zhejiang Provincial Key Laboratory of Rock Mechanics and Geohazards (Figure 3e,f). The maximum axial loading capacity is 4600 KN, and the maximum confining pressure is up to 140 MPa. The specimen ends are mirror-polished and oil-lubricated to minimize end friction. Acoustic emission (AE) monitoring utilized a Physical Acoustics Corporation PCI-2 system with 8 active channels to capture microfracture activities.

2.3. Procedures for Testing

Triaxial compression tests were conducted across five strain rate levels ranging from 10⁻⁶ to 10⁻² s⁻¹, as specified in Table 1. Each strain rate level incorporated five confining pressures: 0 MPa, 10 MPa, 20 MPa, 30 MPa, and 40 MPa. The testing protocol was initiated with confining pressure application at 0.1 MPa/s [30]. Upon reaching the target confining pressure, axial loading commenced under displacement control, with loading rates corresponding to predefined strain rate levels [29]. Loading termination occurred when the stress–strain curve entered the post-peak residual deformation phase. The MTS testing system and AE monitoring system were then deactivated, followed by specimen extraction for macroscopic fracture pattern documentation. This procedure was iterated until the completion of all planned tests.

3. Experimental Results and Analysis

3.1. Stress–Strain Curves

Figure 4 presents the stress–strain curves of granite specimens under varying strain rates and confining pressures. The results reveal three fundamental characteristics: (1) Distinct from the concave-upward shape of uniaxial compression curves [12], triaxial specimens demonstrated pronounced linear elasticity in the initial loading phase. This mechanical response stems from the confining pressure preloading stage in triaxial testing, where hydrostatic compression induces the closure and compaction of pre-existing fissures and structural discontinuities. The resultant densification elevates the material’s initial integrity, thereby establishing a quasi-homogeneous state conducive to linear elastic behavior at the onset of axial loading. (2) While both the strain rate (10⁻⁶ to 10⁻² s⁻¹) and confining pressure (0–40 MPa) enhanced triaxial compressive strength (TCS), their efficiencies differed fundamentally. Taking the 10 MPa confining pressure dataset as an example, when the strain rate escalates from 10⁻⁶ to 10⁻² s⁻¹ (four orders of magnitude increase), TCS rises from 228 MPa to 257 MPa, representing a 12.72% enhancement. Contrastingly, under a constant strain rate of 10⁻⁶ s⁻¹, elevating the confining pressure from 10 MPa to 40 MPa (fourfold increase) elevates TCS from 172 MPa to 313 MPa—an 81.98% improvement. This 6.4-fold disparity in strengthening efficiency (81.98%/12.72%) quantitatively confirms the dominant role of the confining pressure over strain rate in TCS enhancement. (3) An analysis of stress–strain curves under combined confining pressure (0–40 MPa) and strain rate (10⁻⁶ to 10⁻² s⁻¹) conditions revealed pronounced brittle failure characteristics in granite specimens: an average post-peak stress drop gradient of −5.97 × 10⁵ MPa, residual strength ratios consistently below 10%, and the absence of strain hardening, even at maximum loading parameters (strain rate: 10⁻⁶ s⁻¹; σ₃: 40 MPa). Experimental evidence confirms that within the tested σ₃ range (≤40 MPa), specimens predominantly maintained brittle failure modes regardless of strain rate reduction to quasi-static levels (10⁻⁶ s⁻¹), indicating that confinement thresholds for brittle-ductile transition were not attained under these experimental constraints.

3.2. Strain Rate Effects on Characteristic Stresses

The compressive deformation and failure of rock materials intrinsically represent a continuum of progressive evolution, characterized by the nucleation, propagation, and coalescence of internal microcracks [6]. Classical theoretical models [31] delineate the stress–strain response into five characteristic phases, as illustrated in Figure 5. The initial compaction stage manifests a concave–upward curve morphology where the tangent modulus progressively increases with axial loading, indicative of the gradual closure of pre-existing fractures. This transitions into a linear elastic deformation phase marked by a near-ideal proportional relationship between stress and strain, the slope of which quantifies the rock’s elastic modulus [32]. Upon reaching σ_ci, the system enters a stable crack propagation regime characterized by randomly distributed nascent microcracks lacking dominant growth trajectories. Exceeding σ_cd triggers unstable crack growth, exhibiting preferential orientation alignment and concomitant volumetric dilation. Macroscopic fracture plane formation occurs subsequent to attaining σ_p of the stress–strain curve. These critical thresholds—σ_ci governing crack nucleation, σ_cd signifying damage acceleration, and σ_p demarcating ultimate compressive strength—collectively define brittle behavioral attributes. Additionally, the characteristic stress ratios serve as pivotal indices for evaluating rock brittleness and failure mode transitions.

The σ_cd corresponds to the peak stress on the axial stress–volumetric strain curve, where the volumetric strain is determined by Equation (1):

$${\mathrm{\epsilon }}_{\mathrm{v}}={\mathrm{\epsilon }}_1+{2\mathrm{\epsilon }}_3$$

(1)

where ε_v denotes volumetric strain, ε₁ represents axial strain, and ε₂ is lateral strain.

There are many methods to ensure the value of σ_ci, including the crack volumetric strain method [32], AE method [16], and lateral strain response method (LSR method) [33]. In this paper, the LSR method was applied to determine σ_ci, and the detailed procedures were as follows:

• The peak point of the axial stress–volumetric strain curve is the volumetric dilantancy point (point A in Figure 6), and the corresponding axial stress value is σ_cd;
• Assuring the point of σ_cd in the curve of the axial stress-–ateral strain and connecting it to the curve’s origin as the reference line, as shown in Figure 6;
• Drawing the curve based on the difference value between the curve of axial stress–lateral strain and the reference line, and the peak point of this curve corresponding to the axial stress is σ_ci, as shown in Figure 6.

The characteristic stresses calculated using the aforementioned methodology are systematically presented in

Table 2. Characteristic stresses within different strain rates and confining pressures (CP—confining pressure).

Scheme 1.	Strain Rate (s−1)	CP (MPa)	σci (MPa)	σcd (MPa)	σp (MPa)	σci/σp	σcd/σp	σci/σcd
GS-S-1	10−6	10	101.43	172.30	228.10	0.44	0.76	0.59
GS-S-2		20	161.63	220.12	289.38	0.56	0.76	0.73
GS-S-3		30	179.28	282.83	356.78	0.50	0.79	0.63
GS-S-4		40	207.14	313.10	399.84	0.52	0.78	0.66
GS-S-5	10−5	10	111.66	174.73	211.90	0.53	0.82	0.64
GS-S-6		20	151.43	225.78	301.60	0.50	0.75	0.67
GS-S-7		30	201.57	302.40	372.48	0.54	0.81	0.67
GS-S-8		40	235.18	328.94	411.37	0.57	0.80	0.71
GS-S-9	10−4	10	125.83	188.48	237.52	0.53	0.79	0.67
GS-S-10		20	157.54	251.04	318.16	0.50	0.79	0.63
GS-S-11		30	188.19	295.80	371.28	0.51	0.80	0.64
GS-S-12		40	229.33	349.96	433.78	0.53	0.81	0.66
GS-S-13	10−3	10	138.72	210.61	269.64	0.51	0.78	0.66
GS-S-14		20	164.19	254.99	328.23	0.50	0.78	0.64
GS-S-15		30	200.82	315.18	379.26	0.53	0.83	0.64
GS-S-16		40	248.45	371.26	454.48	0.55	0.82	0.67
GS-S-17	10−2	10	115.59	195.80	257.07	0.45	0.76	0.59
GS-S-18		20	196.51	282.44	354.03	0.56	0.80	0.70
GS-S-19		30	213.29	332.89	412.68	0.52	0.81	0.64
GS-S-20		40	247.12	382.10	468.68	0.53	0.82	0.65

. Figure 7 illustrates the relationships between the logarithm of strain rate and three characteristic stresses. The results demonstrate positive correlations between these characteristic stresses and the strain rate across the investigated strain rate range of 10⁻⁶ to 10⁻² s⁻¹ under various confining pressures. For instance, at a confining pressure of 20 MPa, increasing the strain rate from 10⁻⁶ to 10⁻² s⁻¹ elevates σ_ci from 161.63 MPa to 196.51 MPa, σ_cd from 220.12 MPa to 282.44 MPa, and σ_p from 289.39 MPa to 354.03 MPa, corresponding to relative increases of 21.58%, 28.31%, and 22.34%, respectively. Notably, the rate of increase in σ_ci relative to the logarithm of strain rate is lower than that observed for σ_cd and σ_p. This pattern is consistent with previously reported strain rate dependency characteristics of σ_ci under uniaxial loading conditions [19], suggesting that the strain rate exerts minimal influence on σ_ci across the quasi-static strain rate regime.

Table 3. The fitted equations related to the relationship between characteristic stresses and strain rate under triaxial loading (CP—confining pressure).

CP (MPa)	σci		σcd		σp
	Fitting Equation	R2	Fitting Equation	R2	Fitting Equation	R2
10	${\mathrm{\sigma }}_{\mathrm{ci}}=5.54\lg \dot{\mathrm{\epsilon } }+140.80$	0.38	${\mathrm{\sigma }}_{\mathrm{cd}}=8.29\lg \dot{\mathrm{\epsilon }}+221.54$	0.69	${ \mathrm{\sigma }}_{\mathrm{p}}=11.57\lg \dot{\mathrm{\epsilon } }+287.11$	0.64
20	${\mathrm{\sigma }}_{\mathrm{ci}}=8.25\lg \dot{\mathrm{\epsilon }}+199.27$	0.55	${\mathrm{\sigma }}_{\mathrm{cd}}=15.39\lg \dot{\mathrm{\epsilon }}+308.42$	0.94	${ \mathrm{\sigma }}_{\mathrm{p}}=15.59\lg \dot{\mathrm{\epsilon } }+380.65$	0.99
30	${\mathrm{\sigma }}_{\mathrm{ci}}=6.73\lg \dot{\mathrm{\epsilon }}+223.54$	0.65	${\mathrm{\sigma }}_{\mathrm{cd} }=11.29\lg \dot{\mathrm{\epsilon }}+350.98$	0.87	${ \mathrm{\sigma }}_{\mathrm{p}}=11.86\lg \dot{\mathrm{\epsilon } }+425.93$	0.81
40	${\mathrm{\sigma }}_{\mathrm{ci}}=9.32\lg \dot{\mathrm{\epsilon }}+270.74$	0.77	${\mathrm{\sigma }}_{\mathrm{cd}}=18.03\lg \dot{\mathrm{\epsilon }}+421.20$	0.99	${ \mathrm{\sigma }}_{\mathrm{p}}=18.08\lg \dot{\mathrm{\epsilon }}+206.39$	0.99

provides the linear regression equations between characteristic stresses and the logarithm of the strain rate, while Figure 8 illustrates the relationship between linear correlation coefficients and the confining pressure. The results reveal three key patterns: (1) the linear correlation coefficients for σ_ci, σ_cd, and σ_p progressively increase with elevated confining pressure; (2) under identical confining pressure conditions, σ_ci exhibits significantly lower correlation coefficients compared to σ_cd and σ_p; (3) the correlation coefficients of σ_cd and σ_p demonstrate consistent variation trends with confining pressure and maintain comparable magnitudes across the experimental pressure range.

Figure 9a demonstrates a negligible correlation between characteristic stress ratios (σ_ci/σ_p, σ_cd/σ_p) and the logarithm of the strain rate, with respective fluctuation ranges of 0.41–0.56 and 0.75–0.84. The σ_ci/σ_p ratio exhibits a 25% greater dispersion amplitude compared to σ_cd/σ_p. Statistical analysis under triaxial quasi-static conditions (confining pressure: 0–40 MPa; strain rate: 10⁻⁶–10⁻² s⁻¹) reveals strong linear correlations between σ_ci-σ_p and σ_cd-σ_p pairs, as shown in Figure 9b, with determination coefficients R² = 0.96 and 0.99, respectively. The derived empirical slopes (σ_ci/σ_p = 0.58; σ_cd/σ_p = 0.85) approximate the upper bounds of their corresponding fluctuation ranges in Figure 9a. Previous studies report σ_ci/σ_p and σ_cd/σ_p under uniaxial compression in quasi-static regime: 0.60 and 0.80 for granite [19] and 0.55 and 0.86 for soil–rock mixtures [34]. These results indicate that stress ratios do not change with the increase in strain rate, suggesting they may represent intrinsic material properties that are independent of loading conditions. In addition, the established empirical relations σ_ci = 0.58σ_p and σ_cd = 0.85σ_p enable a practical estimation of critical stress thresholds directly from σ_p measurements, providing theoretical foundations for the rapid field assessment of the rock mass failure potential.

The related investigations have demonstrated that under triaxial quasi-static loading conditions, the strain rate sensitivity of σ_p exhibits a decreasing trend with increasing confining pressure, a phenomenon commonly observed in brittle rocks such as granite and sandstone [13,21]. However, the underlying mechanisms governing this confining pressure-induced suppression of rate-dependent strength remain inadequately understood. Furthermore, the potential existence of similar confining pressure dependencies in strain rate effects on characteristic damage thresholds—specifically σ_ci and σ_cd—requires experimental verification. To address these problems, this study proposes dynamic strength factors for characteristic stresses (CSDIF), enabling quantitative comparison of rate effects across varying confining pressures through normalization methodology.

$$\mathrm{CSDIF}={\displaystyle \frac{{\mathrm{\sigma }}_{\min }}{{\mathrm{\sigma }}_{\max }}}$$

(2)

where σ_min and σ_max represent characteristic stresses at strain rates of 10⁻⁶ s⁻¹ and 10⁻² s⁻¹, respectively.

The CSDIF values for granite under different confining pressures are systematically presented in

Table 4. The relationship between the CSDIF of characteristic stresses and confining pressures.

Confining Pressure (MPa)	CSDIF (σci)	CSDIF (σcd)	CSDIF (σp)
0	1.15	1.30	1.41
10	1.14	1.14	1.13
20	1.22	1.28	1.22
30	1.19	1.18	1.16
40	1.19	1.22	1.17

and Figure 10, and the result demonstrates that (1) under unconfined conditions (0 MPa), σ_p exhibits the highest CSDIF (1.41), followed by σ_cd (1.30), with σ_ci demonstrating the lowest amplification (1.15); (2) at a 10 MPa confining pressure, the CSDIF for σ_cd and σ_p decrease to 1.14 and 1.13, respectively, while the value of σ_ci remains stable at 1.14. Comparative analysis reveals that under triaxial loading conditions (σ₃ = 10 MPa), the values of CSDIF corresponding to three characteristic stresses exhibit a significant reduction trend compared with those under uniaxial loading. Existing studies [35] have confirmed a pronounced positive correlation between the heterogeneity degree of rock materials and their strain rate sensitivity characteristics. The confining pressure effectively attenuates the rate sensitivity of characteristic strains in triaxial conditions through facilitating microcrack closure and compaction effects, which significantly reduces structural heterogeneity in rock; (3) beyond a 10 MPa confining pressure, all three factors display oscillatory convergence with diminishing inter-parameter differences, ultimately approaching the same value. Comparative analysis in Figure 10b reveals that the CSDIF of σ_cd and σ_p demonstrate significant pressure-dependent attenuation, whereas the crack initiation factor maintains remarkable stability across the full confining pressure range. In addition, Figure 10b reveals that the CSDIF of σ_cd and σ_p exhibit significant confining pressure-dependent reductions with increasing confining pressure (0–40 MPa), while the CSDIF of σ_ci maintains exceptional stability throughout the pressure loading spectrum.

3.3. Macro-Fracture Patterns

Figure 11 illustrates the macroscopic failure morphology of granite specimens under confining pressures over a wide range of strain rates (10⁻⁶ to 10⁻² s⁻¹). The experimental results demonstrate three principal findings: (1) At the strain rate of 10⁻⁶ s⁻¹, specimens under a 0 MPa confining pressure exhibit vertical fracture planes perpendicular to the end surfaces, representing typical splitting failure. When the confining pressure increases to 10 MPa, Y-shaped fracture patterns initiate from the specimen mid-height with subsequent bifurcation, maintaining geometric stability under elevated confining pressures, indicative of a shear failure mechanism. (2) Under uniaxial conditions (0 MPa) at a strain rate of 10⁻² s⁻¹, the rock specimen develops a single inclined shear plane, with fracture planes wider than those observed under static loading conditions (10⁻⁶ s⁻¹). Additionally, the quasi-static regime (10⁻² s⁻¹) produces a higher volume of pulverized rock debris observed on the fractured surfaces compared to static loading, demonstrating fundamental morphological differences compared to the static loading condition (10⁻⁶ s⁻¹). This result aligns with existing research [6] showing strain rate-induced transition from axial splitting to shear-dominated macro-fracture pattern with increasing strain rate in granite. (3) At a 10⁻⁶ s⁻¹ strain rate with confining pressures increasing from 10 MPa to 40 MPa, a progressive failure pattern evolution is observed: Y-shaped fractures persist at 10–20 MPa, multi-branching fractures emerge at 30 MPa, and characteristic X-conjugate shear bands form at 40 MPa. Systematic analysis reveals that within the triaxial quasi-static loading regime, the macroscopic failure modes of granite are governed by strain rate-confining pressure coupling effects, following a consistent transition rule: increasing strain rates and confining pressures promote gradual evolution from hybrid tensile–shear failure to shear-dominated failure mechanisms.

3.4. Dynamic Strength Criterion

3.4.1. Mohr–Coulomb Strength Criterion

The Mohr–Coulomb strength criterion, equivalent to the third strength theory, posits that material failure occurs when the maximum shear stress at any point reaches the material’s shear strength. This criterion incorporates two parameters: the internal friction angle and cohesion. The shear strength of rock comprises cohesive forces and internal friction on the failure surface, expressed as

$$\mathrm{\tau }={\mathrm{\sigma }}_{\mathrm{n}}\tan \mathrm{\phi }+\mathrm{c}$$

(3)

where τ denotes shear stress, σ_n represents normal stress, φ is the internal friction angle, and c indicates cohesion.

Under conventional triaxial loading, normal stress (σ_n) and shear stress (τ) can be derived from principal stresses (σ₁ and σ₃):

$${\mathrm{\sigma }}_{\mathrm{n}}={\displaystyle \frac{1}{2}}\left({\mathrm{\sigma }}_1+{\mathrm{\sigma }}_3\right)+{\displaystyle \frac{1}{2}}\left({\mathrm{\sigma }}_1 - {\mathrm{\sigma }}_3\right)\cos 2\mathrm{\beta }$$

(4)

$$\mathrm{\tau }={\displaystyle \frac{1}{2}}\left({\mathrm{\sigma }}_1 - {\mathrm{\sigma }}_3\right)\sin 2\mathrm{\beta }$$

(5)

Here, β represents the angle between the fracture plane and the lateral specimen surface (Figure 12).

Combining Equations (3)–(5) yields the critical stress condition for arbitrary β [36]:

$${\mathrm{\sigma }}_1={\displaystyle \frac{2\mathrm{c}+{\mathrm{\sigma }}_3\left\{\sin \left(2\mathrm{\beta }\right)+\tan \mathrm{\phi }1 - \cos \left(2\mathrm{\beta }\right)\right\}}{\sin \left(2\mathrm{\beta }\right) - \tan \mathrm{\phi }1+\cos \left(2\mathrm{\beta }\right)}}$$

(6)

The geometric relationship between β and φ (Figure 13) gives

$$2\mathrm{\beta }={\displaystyle \frac{\mathrm{\pi }}{2}}+\mathrm{\phi }$$

(7)

Substituting Equation (7) into Equation (6) simplifies to

$${\mathrm{\sigma }}_1=2\mathrm{c}·{\displaystyle \frac{\cos \mathrm{\phi }}{1 - \sin \mathrm{\phi }}}+{\displaystyle \frac{1+\sin \mathrm{\phi }}{1 - \sin \mathrm{\phi }}}·{\mathrm{\sigma }}_3$$

(8)

Letting the first term in Equation (8) be A and the coefficient of σ₃ be B, the Mohr–Coulomb parameters are derived as

3.4.2. Dynamic Mohr–Coulomb Strength Criterion

Figure 14 presents the Mohr stress circles and corresponding strength envelopes under varying strain rates. The results demonstrate that with an increasing strain rate, the inclination angle of the strength envelopes (corresponding to the internal friction angle) remains relatively stable, while the intercept (representing cohesion) exhibits a marked increasing trend. This observation aligns with findings in [37], confirming that rock strength enhancement during quasi-static loading primarily originates from cohesion improvement, whereas the internal friction angle demonstrates rate-insensitive characteristics.

Figure 15 delineates the evolution of minor principal stress and major principal stress relationships under varying strain rates. Experimental data demonstrate that the compressive strength exhibits significant enhancement with increasing strain rates and confining pressures, where the confining pressure constitutes the predominant contribution to strength improvement. Linear regression analysis of σ₁-σ₃ curves (fitting equations shown in Figure 15) reveals strict linear dependence between principal stresses across the quasi-static loading regime, supported by correlation coefficient values (R² > 0.98) at all tested strain rates. Parameter calculations (

Table 5. Results of internal friction angle and cohesion under different strain rates.

Strain Rate (s−1)	Parameters of Fitting Equation		Internal Friction Angle (°)	Cohesion (MPa)
	A	B
10−6	152.24	7.52	49.93	27.76
10−5	146.52	8.04	51.15	25.84
10−4	169.23	7.77	50.53	30.36
10−3	187.15	7.70	50.36	33.72
10−2	193.28	8.15	51.39	33.85

) via Equations (9) and (10) show that cohesion increases from 27.76 MPa to 33.85 MPa (Δc = 6.09 MPa) as the strain rate elevates from 10⁻⁶ to 10⁻² s⁻¹, while the internal friction angle shows minimal variation from 49.93° to 51.39° (Δφ = 1.46°). When normalized to the reference strain rate of 10⁻⁵ s⁻¹, cohesion demonstrates a 31% increase (Δc = 8.01 MPa), contrasting with the stable internal friction angle, as shown in Figure 16. These quantitative findings confirm that dynamic strength enhancement principally originates from cohesion evolution, with subsequent sections focusing on establishing the relationship between cohesion and the strain rate.

Qian et al. [38] derived the relationship between dynamic strength and strain rate under quasi-static loading based on thermal activation theory, expressed as

$${\mathrm{\sigma }}_{\mathrm{d}}=\mathrm{C}+\mathrm{D}·\ln {\displaystyle \frac{\dot{\mathrm{\epsilon }}}{\dot{{\mathrm{\epsilon }}_0}}}$$

(11)

where σ_d is dynamic strength, and C or D is the fitting parameter.

Given the established dominance of cohesion enhancement in strain rate effects, we extend this formulation to characterize dynamic cohesion evolution. Defining static strain rate as 10⁻⁵ s⁻¹ and dynamic cohesion increase (Δc_d) as the difference between dynamic and static cohesion, Figure 17 reveals nonlinear growth patterns: Δc_d increases rapidly at 10⁻⁵–10⁻³ s⁻¹ before plateauing. The fitted relationship is

$${\Delta \mathrm{c}}_{\mathrm{d}}=0.99+1.19·\ln ({\displaystyle \frac{\dot{\mathrm{\epsilon }}}{\dot{{\mathrm{\epsilon }}_0}}})$$

(12)

where $\dot{\mathrm{\epsilon }}$ is loading strain rate, and $\dot{{\epsilon }_0}$ is strain rate of 10⁻⁵ s⁻¹.

Through trigonometric transformation of the Mohr–Coulomb criterion,

$${\mathrm{\sigma }}_1={\mathrm{\sigma }}_3·{\tan }^2\mathrm{\theta }+2\mathrm{c}·\tan \mathrm{\theta }$$

(13)

$$\mathrm{\theta }={\displaystyle \frac{\mathrm{\pi }}{4}}+\mathrm{\phi }$$

(14)

When subjected to varying strain rates, the dynamic compressive strength of rock can be expressed as

$${\mathrm{\sigma }}_{\mathrm{dc}}={\mathrm{\sigma }}_3·{\tan }^2\mathrm{\theta }+2{\mathrm{c}}_{\mathrm{d}}·\tan \mathrm{\theta }$$

(15)

The relationship between dynamic cohesion and static cohesion can be expressed as

$$c_d=c_s+{\Delta c}_d$$

(16)

where c_s denotes the static cohesion under reference strain rate conditions, specifically defined as the cohesion at 10⁻⁵  s⁻¹ in this study.

By synthesizing Equations (12), (15), and (16), the dynamic compressive strength of granite under quasi-static loading conditions can be expressed as

$${\mathrm{\sigma }}_{\mathrm{d}}={ \mathrm{\sigma }}_3·{\tan }^2\mathrm{\theta }+2{\mathrm{c}}_{\mathrm{s}}·\tan \mathrm{\theta }+\left[1.98+2.38·\ln ({\displaystyle \frac{\dot{\mathrm{\epsilon }}}{\dot{{\mathrm{\epsilon }}_0}}})\right]·\tan \mathrm{\theta }$$

(17)

From Equation (15), the uniaxial static compressive strength (σ_us) at zero confining pressure and a static strain rate is derived as

$${\mathrm{\sigma }}_{\mathrm{us}}=2{\mathrm{c}}_{\mathrm{s}}·\tan \mathrm{\theta }$$

(18)

Thus, the dynamic compressive strength can be simplified to

$${\mathrm{\sigma }}_{\mathrm{d}}={\mathrm{\sigma }}_{\mathrm{us}}+{\mathrm{\sigma }}_3·{\tan }^2\mathrm{\theta }+\left[1.98+2.38·\ln ({\displaystyle \frac{\dot{\mathrm{\epsilon }}}{\dot{{\mathrm{\epsilon }}_0}}})\right]·\tan \mathrm{\theta }$$

(19)

As delineated by Equation (19), the dynamic compressive strength of granite under triaxial quasi-static loading conditions can be characterized as a linear superposition of three components: uniaxial static strength, confining pressure enhancement, and strain rate effect amplification. This constitutive formulation explicitly elucidates the synergistic strengthening mechanisms of confining pressure and strain rate effects on the dynamic strength of granite.

4. Discussion

The failure process of brittle rock (such as granite) is characterized by several distinct deformation stages, which include crack initiation, crack propagation, and coalescence [39]. The microcrack evolution characteristics before the peak stress point (Figure 18) reveals three progressive stages: During the OA phase (prior to σ_ci), rock deformation primarily manifests as recoverable elastic behavior dominated by the closure of pre-existing pores and fractures, with minimal micro-fracturing activity observed. As stress transitions into the AB phase (stable crack growth stage between σ_ci and σ_cd), newly generated microcracks preferentially propagate along the boundaries of inherent defects, exhibiting stochastic spatial distribution patterns that demonstrate a negligible influence on peak strength. Upon exceeding σ_cd (BC phase, the unstable crack propagation stage), the microcrack network achieves self-organized criticality, where branching-coalescence mechanisms govern the formation of dominant failure pathways, ultimately leading to a through-going macroscopic fracture surface at peak stress. By comparing the evolution characteristics of microcracks (Figure 18) with strain rate effects on characteristic stresses (Figure 10b) during rock deformation, it is found that a significant correlation exists between them. This correlation suggests that the differences in strain rate effects on characteristic stresses can be explained through microcracking evolution. Under unconfined conditions (0 MPa), the CSDIF of σ_ci is lower than that of σ_cd or σ_p. This discrepancy arises because σ_ci marks the terminal point of elastic deformation in brittle rocks, during which micro-fracturing processes are limited and material behavior remains quasi-homogeneous. Conversely, a substantial quantity of microcracks is generated during the AC phase (Figure 18), which progressively interconnect and coalesce into continuous fracture networks. The intensity of microcrack activity within the rock demonstrates a positive correlation with the associated strain rate effect [40]. Notably, the deformation phases corresponding to σ_cd or σ_p exhibit significantly heightened microcrack activity levels compared to those associated with σ_ci. Consequently, the CSDIF of σ_ci remains less pronounced than those of σ_cd or σ_p.

The strain rate effects on σ_cd and σ_p exhibit systematic attenuation with increasing confining pressure. The underlying physics involves confining pressure suppressing lateral dilatancy, thereby reducing tensile microcrack proportion while enhancing shear-dominated micro-fracturing [41]. Given the lower strain rate sensitivity of fracture toughness in shear failure compared to tensile failure [42,43], this transition mechanism establishes a pronounced negative correlation between confining pressure and strain rate effects on σ_cd and σ_p. The disparity in the strain rate sensitivity of fracture toughness between tensile and shear failure modes constitutes an important mechanism governing the differential dynamic responses of rock strength parameters. This mechanism is corroborated by the distinct strain rate effects observed in rock tensile versus compressive strengths. Figure 19 presents comparative data on the dynamic strength factors of granite under tensile and compressive loading. The dynamic increase factor (DIF) is defined as the ratio of the strength at a given strain rate to the reference value at a strain rate of 10⁻⁵ s⁻¹. The DIF of tensile strength reaches 1.61 at the strain rate of 10⁻¹ s⁻¹. Both semi-circular bend tests and Brazilian disc tests demonstrate dynamic tensile strength factors exceeding 1.3. Comparative analysis reveals a consistent hierarchy under identical strain rate conditions: tensile dynamic intensity factor > uniaxial compressive dynamic intensity factor > triaxial compressive dynamic intensity factor, which aligns precisely with the proportional dominance of tensile fracture components in their respective failure modes and minimizes the confining pressure’s influence on strain rate sensitivity.

5. Conclusions

In this paper, a series of triaxial compression tests covering a wide range of strain rates from 10⁻⁶ to 10⁻² s⁻¹ were systematically conducted. The values of characteristic stresses and the ratios of macro-fracture characteristics with different strain rates and confining pressures were analyzed. The dynamic strength criterion within the strain rate were established. More importantly, the causes of strain rate effect discrepancies between σ_ci and σ_cd or σ_p were discussed, and the mechanism underlying confining pressure sensitivity of strain rate effects on σ_cd or σ_p was proposed. The main conclusions can be summarized as follows:

(1)

Under varying confining pressures, characteristic stresses of granite exhibit positive correlations with the logarithm of strain rate and confining pressure. Specifically, σ_cd and σ_p demonstrate significant linear positive correlations with the strain rate logarithm, while σ_ci shows a relatively weaker correlation.

(2)

Under triaxial quasi-static loading conditions, the value of σ_ci/σ_p ranges from 0.41 to 0.56, and the value of σ_cd/σ_p ranges from 0.75 to 0.84. Linear regression analysis demonstrates strong linear correlations between σ_ci/σ_p and σ_cd/σ_p, expressed as σ_ci = 0.58σ_p and σ_cd = 0.85σ_p.

(3)

Macroscopic fracture patterns of granite are controlled by the confining pressure and strain rate: Y-shaped shear fractures predominantly form under low confining pressures and strain rates. With increasing confining pressure and strain rate, fracture patterns progressively transition to X-shaped shear-dominated failures.

(4)

The strain rate-induced enhancement of dynamic compressive strength primarily originates from the rate-strengthening effect on cohesion, while the internal friction angle remains stable across varying strain rates.

(5)

The disparity in microcrack activity intensity during different deformation stages of rocks leads to the CSDIF of σ_ci being lower than those of σ_cd and σ_p.

(6)

The CSDIF of σ_cd and σ_p decreases with elevated confining pressures. This differential behavior is explained by confinement-enhanced shear fracturing dominance during crack propagation stages combined with a lower strain rate sensitivity of shear versus tensile fracture toughness.