## 1 Introduction  
The applicability of anisotropic failure criteria in layered rock formations and their associated engineering applications constitute a critical research focus in rock mechanics and hydrocarbon reservoir evaluation [1, 2, 3, 4]. Characterized by distinct bedding structures, layered rocks exhibit significant mechanical anisotropy that challenges the predictive accuracy of conventional isotropic failure criteria in determining strength parameters and fracture propagation patterns [5, 6, 7, 8, 9, 10]. As an interdisciplinary field investigating rock stress-strainrelationships, strength characteristics, and failure mechanisms under multifactorial stress conditions, rock mechanics provides fundamental theoretical support for geological exploration, petroleum engineering, mining operations, and related industrial practices [11; 12; 13; 14; 15; 16]. Within this domain, shale formations have garnered particular research attention due to their complex laminated textures and pronounced directional dependence of mechanical properties [9; 17; 18; 19]. This structural anisotropy manifests as orientation-dependent variations in shale's physical and mechanical responses to external loading, making the development of appropriate strength criteria and failure models for anisotropic rocks both theoretically valuable and practically essential for optimizing engineering designs in hydrocarbon extraction and underground storage applications [20; 21; 22; 23; 24; 25]. In geological engineering, the anisotropic characteristics of shale can significantly impact aspects such as formation pressure, wellbore stability, and reservoir modification [9; 26; 27; 28; 29]. Therefore, research into the strength criteria and failure mechanisms of anisotropic shale holds crucial theoretical significance and practical application value. Current understanding of anisotropic rock mechanics reveals significant advancements and limitations in strength criterion development. Foundational work by [PERSON] [7] established the single plane of weakness model for sedimentary rocks, later challenged by experimental evidence showing overestimation of shale strength at low loading angles [12; 30; 31; 32; 33]. [PERSON] et al. [24] demonstrated the critical dependence of laminated rock failure on initial stress states and structural orientations, prompting subsequent revisions to conventional criteria. The Hoek-Brown framework has been adapted for anisotropy, notably through [PERSON] et al. [9]'s introduction of a directional strength parameter and [PERSON] et al. [17]'s integration of empirical anisotropic functions that account for curvature variations in failure envelopes. These modified criteria incorporate piecewise functions describing uniaxial compressive strength (UCS) variations with bedding angles (\(\beta\)), particularly addressing minimum strength observed between \(\beta=45\)-\(60^{\circ}\) while preserving boundary values at \(\beta=0^{\circ}\) and \(90^{\circ}\). Complementary studies by [PERSON] et al. [23] quantified pore water loss effects, showing substantial shale strengthening during preservation. Comparative analyses highlight the modified Hoek-Brown criterion's superior predictive capability over [PERSON]'s model [31; 33], though persistent discrepancies arise from heterogeneous rock matrices and bedding plane irregularities. [PERSON] et al. [30] advanced this field by introducing an anisotropy-driven strength reduction parameter, outperforming both modified Hoek-Brown and Ramamurthy criteria in transversely isotropic rock predictions. Recent innovations include [PERSON] et al. [34]'s \"patchy weakness\" concept, reconciling transverse isotropy with coupled failure modes through localized bedding plane defects. While numerous criteria exist, their judicious application in drilling and wellbore engineering necessitates careful consideration of material brittleness-ductility transitions and rigorous physical interpretation of underlying failure mechanisms. Experimental validations by [PERSON] et al.