Strength Envelope Research Articles

The Influence of the Root Diameter of Cunninghamia lanceolata (Chinese Fir) on the Strength and Deformation Behavior of Sand

This study used triaxial tests to examine the impact of the root diameter of Cunninghamia lanceolata (Chinese fir) on the mechanical behavior of sand, including stress–strain development, strength, volumetric strain, and failure envelope. It also revealed the reinforcement mechanisms of roots with different diameters based on root–soil interactions. The results showed the following: (1) The addition of roots significantly enhanced sand strength and reduced volumetric deformation. The average peak strength increased by 31.8%, while the average peak volumetric strain decreased by 34.3%. (2) Roots provided additional cohesion and increased the friction angle of the sand, causing the failure envelope to shift upward and deflect. (3) Smaller-diameter roots improved the mechanical properties of sand more significantly, leading to higher peak strength, shear strength parameters, and smaller volumetric deformation. As the root diameter increased from 1 mm to 5 mm, the peak strength ratio decreased from 1.78 to 1.13, and the peak volumetric strain increased from 0.48 to 0.79. (4) Smaller-diameter roots, which form denser networks, allowing more roots to resist loads, and have a higher elastic modulus providing greater tensile stress, also possess higher tensile strength and critical sliding tensile stress, making them less likely to fail, thereby making the mechanical reinforcement of sand more significant.

Shallow Foundation Load Testing on Natural Florida Limestone: Single Rock Layer and Rock-Over-Sand Subsurface

This article presents three shallow foundation load tests under various layer conditions (single rock layer and rock-over-sand subsurface), rock strengths, and limestone formations. Rock coring with laboratory triaxial testing, standard penetration tests (SPT), seismic shear tests, and measuring while drilling (MWD) tests were performed to identify the representative mass bulk dry unit weight (γ<sub>dt</sub>) and the spatial variability of γ<sub>dt</sub>. The geometric mean and median γ<sub>dt</sub>, together with seismic shear results, were used to characterize rock strength and stiffness. Each footing test was sized at 1.07–1.83 m to fit the 12.20 m load frame (8.01 MN capacity) using laboratory-assessed bilinear strength envelopes of the rock. Predicted bearing capacities with Florida bearing capacity equations provided reasonable estimates for three load tests. Mean and differential settlement, including spatial variability for a single footing on a heterogeneous rock layer, were captured with the Fenton and Griffiths method. For the settlement prediction of rock-over-sand subsurface, a parametric study was conducted for typical footing size, rock thickness, and mass properties with the finite element method (FEM), and a Winkler approach with a stress-dependent weighted harmonic mean modulus (E<sub>eq</sub>). The proposed method achieved good agreement with the FEM results and was validated with load tests.

Time-Dependent Displacement Analysis of Anchored Slope Subjected to Seismic Loading Considering Nonlinear Mohr-Coulomb Failure Criterion

ABSTRACT A time-dependent yield acceleration is adopted to estimate the seismic permanent displacement of anchored slope by establishing a dynamic failure mechanism and an instant displacement mode. A real-time equivalent tangential strength envelope is adopted to determine the time-dependent yield acceleration while considering a nonlinear failure criterion of slope soil. The recursion formulas of slope and anchor parameters versus seismic displacement are established to decouple the relationship between the time-dependent yield acceleration and the seismic displacement. The result indicates that the anchored slope becomes more vulnerable to earthquake when the nonlinear parameter of soil increases.

Limit analysis of earthquake-induced landslides considering two strength envelopes

Abstract. Stability analysis of soil slopes undergoing earthquake remains an important research aspect. The earthquake may have some different effects on slope stabilities associated with nonlinear and linear criteria, which need to be further investigated. For homogeneous soil slopes undergoing earthquakes, this paper established the three-dimensional (3D) failure mechanisms with the power-law strength envelope. The quasi-static method was employed to derive the work rate done by the earthquake in limit analysis theory. The critical heights and critical slip surfaces associated with nonlinear and linear criteria were obtained for four slope examples undergoing different seismic loads. Comparisons of the nonlinear and linear results illustrated that two critical inclinations (resulting from the overlap of nonlinear and linear results) both decrease as the seismic force increases, but their difference is almost constant. For steep slopes, the use of linear strength envelope can lead to the non-negligible overestimation of slope critical height. This overestimation will become significant with the increase in seismic force, especially for the steeper slope with a narrow width. Since the seismic force has a positive influence on equivalent internal friction angle, the critical slip surface for the slope-obeying nonlinear envelope tends to be slightly deeper as the earthquake becomes stronger. For steep soil slopes undergoing the earthquake, the development of 3D stability analysis with a nonlinear yield criterion is necessary and significant. These findings can provide some references for the risk assessment and landslide disaster reduction of soil slopes.

An Efficient Strength Evaluation Method Based on Shell-Fastener Model for Large Hybrid Joint Structures of C/SiC Composites.

C/SiC composites are widely used in aerospace thermal structures. Due to the high manufacturing complexity and cost of C/SiC composites, numerous hybrid joints are required to replace large and complex components. The intricate contact behavior within these hybrid joints reduces the computational efficiency of damage analysis methods based on solid models, limiting their effectiveness in large-scale structural design. According to the structure characteristic, a fractal contact stiffness model considering bonded behaviors is established in this paper. By introducing this model, it is proved that the bonded layer can affect the interface strength between plates but not the bearing strength of the specimen for the bolt/bonded hybrid joint structure. Furthermore, by introducing the strength envelope method, this paper overcomes the problem wherein the shell-fastener model cannot accurately describe the complex stress field. Validation through experimental comparison confirms that this approach can accurately predict both the failure mode and strength of multi-row hybrid joint structures in C/SiC composites at a detailed level with an error of 5.4%, including the shear failure of bolts. This method offers a robust foundation for the design of large-scale C/SiC composite structures.

The rheological structure of East Asian continental lithosphere

The rheological structure of the East Asia continent is the key to understanding its broad, heterogeneous, and intense Cenozoic deformation. Based on a refined three-dimensional thermal structure of the lithosphere in this region and the latest strain rate data, we derived a model of the rheological structure of the East Asian continental lithosphere. The strength envelopes, defined by the yield strength of frictional, fractural, and plastic creep, are constrained by the lithological stratification based on previous studies and the depth distribution of earthquakes. The results show large vertical and lateral variations of lithospheric strength in the East Asian continent. A weak lower crust with low effective viscosity is ubiquitous. The rheological structure agrees with the jelly sandwich model in cratons, where the mantle lithosphere is relatively strong. The Tibetan Plateau has the weakest lower crust, with its effective viscosity ranging from 1019 to 1020 Pa∙s. Its mantle lithosphere is weakened by relatively high temperature; hence, its rheological structure can be described by the crème brûlée model. The lithospheric scale faults and suture zones in and around the Tibetan Plateau, with low strength or viscosity, correspond to the banana split model. The strength of the lithosphere in the Tibetan Plateau and other zones of active Cenozoic tectonics mainly derive from the crust, while the strength of the cratonic lithosphere is dominated by that of the mantle lithosphere. The rheological heterogeneity controls the lateral growth of the Tibetan Plateau and the widespread and differential deformation in the East Asian continent.

Experimental study on the mechanical properties of non-reinforced UHPC element under biaxial stress states

The ultra high performance concrete (UHPC) has been gradually applied in bridge and building engineering to replace normal concrete as a new type of cementitious material. The biaxial strength criteria of UHPC play an essential role in engineering design and research. However, the experimental research on biaxial strength of UHPC is still lacking. In this study, four groups of biaxial loading tests on UHPC planar elements were carried out, using the Planar Bi-directional Element Tester (PBET) developed by Tsinghua University, including biaxial compressive test, biaxial tensile test, biaxial tension-compression sequential test, and biaxial tension-compression proportional test. The biaxial stress ratio is also studied as a key parameter. Based on the test results, the typical failure modes under different biaxial loading conditions as well as the biaxial strength criteria of UHPC are revealed. Taking the biaxial strength criteria of normal concrete as a reference, the complete biaxial strength envelope equations of UHPC are finally derived.

Dynamic strength criterion of concrete utilizing dynamic coordinates

AbstractThe effects of strain rate on the strength of concrete should be considered when analyzing the dynamic responses of concrete structures subjected to earthquakes or explosions. This paper shows that the effect of strain rate on strength characteristics can be attributed to an increase in cohesion. Notably, the effects of friction, hydrostatic pressure, and intermediate principal stress tend to remain rate‐independent under the appropriate reference system. Consequently, a dynamic coordinate system is established to account for the effects of strain rate on isotropic tensile strength. In this dynamic coordinate system, the strength envelopes for concrete closely resemble those in quasi‐static conditions under varying strain rates, as defined in the unified strength criterion. Using this proposed dynamic strength criterion, this paper explores the dynamic characteristics of different stress paths, including uniaxial, biaxial, and triaxial compression and tension. The predictions, both in terms of tendencies and magnitudes, are consistent with test results. The proposed method enables the extension of most strength criteria to dynamic scenarios by introducing two additional parameters with clear physical interpretations. This advancement enhances the current understanding of dynamic strength characteristics and provides a theoretical foundation for dynamic response analysis.

Development of a dynamic cumulative damage model and its application to underground hydropower caverns under multiple blasting

Underground infrastructures are crucial for resource extraction, energy storage, and space utilisation. The geomaterials that make up these structures, such as rock and concrete, are subjected to multiaxial stress conditions and are frequently exposed to dynamic and extreme loadings caused by both natural disasters and human activities. These stresses are particularly significant during the construction phase, which involves operations such as drilling and blasting excavation, as well as during the operational phase, which may include events like explosions. For instance, while drilling and blasting induce rock breakage within the excavated profile as designed, they inevitably lead to the formation of a damage zone in the surrounding rock mass. Moreover, the cumulative effects of sequential excavations and multiple blasts can cause significantly greater damage, thereby threatening the stability of tunnel structures during the operation phase. This paper highlights the importance of thoroughly analysing these phenomena during both static and dynamic loadings to ensure the stability of underground infrastructures. To address these challenges, a rate-dependent damage constitutive model is proposed for geomaterials to assess the impacts of blasting loads and the cumulative damage resulting from repeated blasts. The model is conceptualised using the strength envelope of loading-unloading curves to represent the progressive accumulation of damage under repeated impacts. Through theoretical derivation, a dynamic cumulative damage model is developed, based on a modified Mohr-Coulomb strain-softening model incorporating rate-dependent parameters, and is validated against dynamic experimental data. The model captures the transition between static strain-softening and dynamic cumulative damage, triggered by a critical strain-rate threshold. The applicability of the model is demonstrated through simulations of tunnel excavation, emphasising the impact of blasting loads and the accumulation of damage zones. To assess its practical feasibility, the developed model is applied to simulate different excavation scenarios for an underground hydropower cavern. Damage in the surrounding rock mainly results from static unloading and/or dynamic disturbances. Blasting construction, in particular, causes significant damage in tunnel intersection zones and the connecting areas of two benches, leading to increased displacement and higher damage levels compared to static excavation. To mitigate excessive damage while maintaining the construction timeframe, it is recommended to consider alternating cycles of dynamic loading and static excavation unloading continuously, which helps understand damage formation in critical zones without significantly delaying project completion.

Structure design and ultimate strength envelope of a modular carbon fiber-reinforced polymer (CFRP) laminate tube box on a 5000 ft deep-sea ultra large unmanned underwater vehicle

Deep-sea tube box is a crucial and basic structural component of various deep-sea equipment such as deep-sea submarine and unmanned underwater vehicles. The deep-sea ultra large unmanned underwater vehicle (ULUUV) is the most representative one of them. Tube boxes are the basic modular members on the ULUUV, which are widely used as the load-bearing hull in torpedo launch, load transportation, pipeline protection and etc. Due to the extremely harsh operation condition, deep-sea tube box simultaneously suffers extremely high lateral water pressure and tremendous axial compression induced by water pressure. Considering the strict requirements for lightweight, high strength, and corrosion resistance of deep-sea tube boxes, carbon fiber-reinforced polymers (CFRP) have become the most potential choice for the construction materials of future deep-sea tube box. Thus, the aim of this paper is to make a structure design of a deep-sea CFRP tube box which can operate at the water depth of 5000 ft. In this paper, a detailed design process of the deep-sea CFRP tube box was introduced, including geometrical configuration, critical buckling load prediction method, FE modeling technique and experimental validation. Corresponding to 5 ply schemes, 495 cases of FE analyses were performed to obtain the ultimate strength envelopes of the CFRP tube box subjected to combined axial compression and external lateral water pressure. Based on the obtained results, the best ply scheme for the deep-sea CFRP tube box was figured out. This paper could provide design method and data references for deep-sea CFRP structures which are subjected to combined loads.

A Virtual Assembly Technology for Virtual–Real Fusion Interaction of Ship Structure Based on Three-Level Collision Detection

With the rapid advancement of new-generation information technology, the virtual–real fusion interaction has increasingly become a crucial technique for structural analysis to determine the strength envelope of hulls. A high-precision assembly of the experimental devices in a virtual environment is vital. A virtual assembly method for a structural virtual–real fusion test based on the oriented bounding box (OBB) algorithm, the Devillers and Guigue algorithm, and differential triangle facets algorithm are proposed in this paper. The experiment of the connector in a typical offshore floating platform is performed as a case, which indicates that the virtual assembly method proposed in this paper enables the assessment of assembly virtually prior to the actual experiment, and the assembly accuracy can reach 0.01 mm. The digitalization and virtual–real fusion interaction for the mechanical experiment of hulls are advanced to ensure efficiency, safety, and economy.

Particle morphology and principal stress direction dependent strength anisotropy through torsional shear testing

The major principal stress direction angle (ασ) experienced by granular soils varies widely in engineering, causing different strengths. However, how particle morphology affects the strength anisotropy behavior under different ασ remains unclear. To address this gap, this study performed drained hollow cylinder torsional shear tests under different ασ on six granular materials with distinct morphologies. Results highlight the significant dependence of peak strengths of granular materials on both particle morphology and ασ. Increasing particle shape irregularity and surface roughness leads to a considerable enhancement in peak strength, while this peak strength significantly degrades with increasing ασ. Materials with more irregular shapes were found to have a more pronounced strength anisotropy. Furthermore, the initial fabric of particle packings, derived from three-dimensional X-ray microtomography, was used to interpret microscopic mechanisms behind the morphology-dependent strength anisotropy. Irregular-shaped materials display broader preferred particle orientations and higher initial fabric anisotropy compared to relatively regular-shaped materials. This higher morphology-induced fabric anisotropy contributes to strength anisotropy, and a correlation was established for describing this trend. Additionally, an anisotropic failure criterion incorporating fabric anisotropy was developed to characterize the strength envelope for granular materials with diverse shapes.

Research on mechanical properties of HTPB propellants

Abstract Slat propellants with different sizes and shapes are designed to investigate the strength criteria of HTPB propellants. According to the calculation and simulation results, with the increase of multiple, the designed size slat specimens can achieve equivalent biaxial tensile stress ratios of 1:2, 1:2.5, 1:3, 1:3.5, and 1:4 under uniaxial tensile conditions. The biaxial mechanical properties of the slab with a stress ratio of 1:2 were tested at low temperatures (25°C, −30°C, −50°C, and 0.4 s−1, 4.0 s−1, 14.29 s−1, 42.86 s−1). It was found that as the temperature and the strain rate rise, an increase is observed in the maximum tensile strength and the initial modulus, a decrease is observed in the maximum elongation of the propellant, and the strength envelope is constructed according to the typical mechanical parameters: the strength envelope of HTPB propellant increases as the temperature decreases. The research findings are expected to provide statistical support for the analysis of the structural integrity of HTPB propellants under low-temperature ignition.

Mechanical mechanism of dip effect on bearing capacity of the pillar strength

Abstract The dip effect on pillar strength in underground ore-body mining is well established, but the variation in stress path (magnitude and direction of stress) due to changing inclination angles requires further study. Using elasticity theory, the Euclidean mean stress tensor characterizes the stress state in pillar zones. Numerical simulations provided the second-order tensor of peak stress for each pillar unit. Through tensor statistical analysis, the Euclidean mean stress tensor matrix was calculated, and its eigenvalues and eigenvectors, representing the magnitude and direction of the principal stress, were derived. This analysis explained the intrinsic dip effect on pillar strength through principal stress characteristics. Finally, the pillar strength envelope function for varying width-height ratios at any dip angle was obtained using the random gradient descent algorithm. Results indicate that in the peak stress state, the average principal stress directions of the pillar change with orebody dip angle, affecting the stress path. The average principal stress increases with pillar size due to increased constraints. These findings offer theoretical insights for pillar design and stability analysis.

A multiscale modeling for progressive failure behavior of unidirectional fiber-reinforced composites based on phase-field method

The effective macroscopic properties of composites are determined by the intricate interactions among the individual components within their microstructure. Preserving these microscopic details during the failure simulation of macrostructures presents significant challenges. This work proposes a multiscale modeling framework to numerically predict the macroscopic fracture properties of unidirectional fiber-reinforced composites based on micromechanical analysis. In this study, 2D representative volume elements (RVEs) combined with the phase-field method are utilized to simulate fiber-reinforced composites under transverse loadings. A series of representative loading conditions are employed to investigate cracking patterns and to construct failure strength envelopes of the composites subjected to different multiaxial proportional loadings. By extracting the softening curve from the uniaxial tensile simulation of the RVE and fitting it with a tenth-order polynomial, the homogenized cohesive law, combined with the phase-field method, is applied to the damage analysis of macroscopic heterogeneous materials. The homogenized model of unidirectional fiber reinforced composites is numerically validated through simulations of a 2D flat plate. The simulation results demonstrate the excellent potential of the proposed multiscale modeling framework to accurately and efficiently predict the progressive failure and fracture behavior of fiber-reinforced composites in engineering applications.

Triaxial compressive behaviour of ultra-high performance geopolymer concrete (UHPGC) and its applications in contact explosion and projectile impact analysis

Ultra-high performance geopolymer concrete (UHPGC) is increasingly recognised as a promising construction material in protective engineering owing to its exceptional material strength and eco-friendly characteristics. However, there is currently limited research data on the behaviour of UHPGC under complex stress states, and the applicability of some common concrete dynamic constitutive models to UHPGC requires calibration and evaluation. This paper presents an experimental study on the triaxial compression (TXC) behaviour of UHPGC under varying confining pressures with a range of 0–40 MPa. The stress-strain curves under various confinement levels and failure patterns of UHPGC are discussed. The results indicate that the peak axial stress and toughness of UHPGC increase with higher confining pressure levels, but it still exhibits noticeable softening behaviour. Under uniaxial compression, UHPGC typically fails with inclined shear and axial tensile cracks, while under TXC, particularly at confining pressures above 20 MPa, the failure mode transitions to shear failure with prominent oblique shear cracks. Additionally, existing failure criteria are calibrated to match the strength envelop of UHPGC, with the Power-law failure criterion providing the closest fitting results. Subsequently, utilising both current and pre-existing TXC data, three commonly used concrete constitutive models, including HJC, K&C and CSC, are calibrated with respect to the dominant strength parameters for UHPGC in the finite element (FE) hydrocode ANSYS/LS-DYNA. Further, numerical simulations are conducted to evaluate the effectiveness of such three calibrated concrete models in replicating the response of UHPGC panels exposed to the contact explosion or projectile impact. The simulation results demonstrate that the K&C and CSC models effectively replicate the localised damage of UHPGC panels under contact explosions, while the HJC model more accurately simulates the depth of penetration (DOP) for thick UHPGC panels under projectile impact.

Numerical prediction of beam capacity using the new constitutive model based on concrete biaxial behavior

The biaxial strength of concrete exhibits intriguing nonlinearity and scattering, posing challenges to the assessment of building structural safety. Experiments have confirmed that biaxial strength envelopes can manifest concave, quasi-linear, and convex shapes. Despite various empirical formulas being proposed to describe this behavior, they cannot be applied to finite element simulations due to the absence of a constitutive perspective. To study the mechanisms forming the shape of strength envelopes, a novel three-invariant model constitutive model is proposed, which reveals the characteristics of the envelopes resulting from the coupling effect of hydrostatic pressure and Lode angle. By adjusting the model parameters, it effectively describes concave, quasi-linear, and convex envelopes. The model's accuracy and compatibility were validated against experimental data spanning various concrete types, with compression strengths ranging from 19.29 to 96.5 MPa and tension-compression ratios form 0.056 to 0.095. To investigate how biaxial strength impacts the capacity of beams, numerical experiments were conducted on notched and intact beams based on the proposed model. The results underscored that beams prepared using concrete with convex envelopes exhibited higher capacity compared to those with quasi-linear envelopes. Therefore, conducting biaxial tension-compression experiments is recommended to accurately evaluate beam bearing capacity. The proposed model is expected to have further applications in the safety evaluation of building structures.

Stress-dependent instantaneous cohesion and friction angle for the Mohr–Coulomb criterion

The strength criterion of rock is essential for stability control and safety design of geotechnical engineering constructions. Due to its widespread adoption, the Mohr–Coulomb (M-C) criterion is prominent among strength criteria. However, the M-C criterion is constrained by three significant limitations: it fails to capture the nonlinear strength response, overlooks the critical state, and disregards σ2. This study introduces a novel Stress-dependent Instantaneous Friction angle and Cohesion (SIFC) model for the M-C criterion to represent the convex strength envelope of intact rock, covering the spectrum from non-critical to critical states. In pursuit of this objective, an innovative approach for calculating these instantaneous shear parameters at each corresponding σ3 is initially introduced. By examining the confining pressure dependency of the instantaneous friction angle and cohesion, the SIFC model is derived and introduced to the M-C criterion. The SIFC-enhanced M-C criterion, utilizing parameters obtained from triaxial tests under lower σ3, delineates the complete non-linear strength envelope in (σ1, σ3) space, covering brittle to ductile behavior. This criterion is then extended to polyaxial stress conditions. Validation through triaxial test data confirms that the SIFC-enhanced M-C criterion accurately reflects the strength characteristics of the tested rocks.

Strength characteristics of different stress spaces of red sandstone under loading and unloading tests

Red sandstone tests were conducted on the rock mechanics test system to clarify the strength and failure characteristics of red sandstone under triaxial loading and different unloading confining pressure rates. Strength change rules of the red sandstone in various stress spaces were analyzed. Results show that a large initial confining pressure indicates a long yield section and large axial strain. A small unloading confining pressure rate indicates a long yield section and large axial strain at failure under the identical initial confining pressure. The confining pressure reduction increases with the unloading confining pressure rate increasing when the red sandstone fails. The stress path of unloading confining pressure weakens the red sandstone cohesion and strengthens its internal friction angle. The general octahedral shear stress formula for judging whether the red sandstone fails under loading and different unloading rates was acquired on the basis of the octahedral shear stress characteristics of the red sandstone. In addition, a 3D empirical criterion for evaluating the red sandstone strength was constructed through the deviator plane function of Eekelen and the strength envelope curve on the meridian plane. A great initial confining pressure indicates serious internal damage of the red sandstone at failure under the identical unloading rate of confining pressure. A low unloading rate of confining pressure indicates internal serious damage of the red sandstone at failure when the initial confining pressure is the same. The analysis of macroscopic failure characteristics revealed that the stress path significantly influences the failure mechanism of the red sandstone. The results are instructive for determining the stability of underground engineering of the red sandstone.

Stress-dependent Mohr–Coulomb shear strength parameters for intact rock

Rock strength is imperative for the design and stability analysis of engineering structures. The Mohr–Coulomb (M-C) criterion holds significant prominence in geotechnical engineering. However, the M-C criterion fails to accurately capture the nonlinear strength response and neglects the critical state of rocks, potentially leading to inaccuracies in the design phase of deep engineering projects. This study introduces an innovative stress-dependent friction angle and cohesion (SFC) for the M-C criterion to capture the nonlinear strength responses of intact rocks, spanning from non-critical to critical states (brittle to ductile regions). A novel method for determining these stress-dependent parameters at each corresponding σ3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sigma_{3}$$\\end{document} is initially introduced. Subsequently, an examination of the confinement dependency of the friction angle and cohesion is conducted, leading to the derivation of the SFC model. The SFC-enhanced M-C criterion, utilizing parameters obtained from triaxial tests under lower σ3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sigma_{3}$$\\end{document}, demonstrates the capability to delineate the complete non-linear strength envelope from brittle to ductile regions. Validation through triaxial test data confirms that the predictions of the SFC-enhanced M-C criterion accurately correspond to the strength characteristics of the tested rocks.