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Scientific Reports
2045-2322
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39256416
69795
10.1038/s41598-024-69795-5
Article
Non-coaxial plasticity of similar weakly cemented soft rock under directional shear stress path
http://orcid.org/0000-0002-7512-5188
Liu Jiashun liujiashun000@163.com

123
Zheng Zhiyong 1
Zhou Hui 2
Zhu Kaixin 14
Wang Yang 1
Zhou Ni 1
Wang Siyu 1
Sun Mengyao 1
1 https://ror.org/01n2bd587 grid.464369.a 0000 0001 1122 661X School of Civil Engineering, Liaoning Technical University, Fuxin, 123000 People’s Republic of China
2 grid.9227.e 0000000119573309 State Key The indoor of Geomechanics and Geotechnical Engineering, Institute of Rock and Soil Mechanics, Chinese Academy of Sciences, Wuhan, 430071 Hubei People’s Republic of China
3 https://ror.org/01xt2dr21 grid.411510.0 0000 0000 9030 231X School of Mechanics and Civil Engineering, China University of Mining & Technology, Beijing, 100083 People’s Republic of China
4 https://ror.org/023rhb549 grid.190737.b 0000 0001 0154 0904 School of Civil Engineering, Chongqing University, Chongqing, 400044 People’s Republic of China
10 9 2024
10 9 2024
2024
14 211083 4 2024
8 8 2024
© The Author(s) 2024
2024
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The plastic flow behavior of soft rock exhibits non-coaxial features under complex stress paths, while traditional plasticity theories are ill-equipped to adequately represent this, which leads to the mechanism of soft rock failure still unclear. To investigate the evolution law of strain increments and non-coaxial characteristics of weakly cemented soft rock, the directional shear tests are conducted using the hollow cylinder apparatus (HCA). The results show that non-coaxiality does not occur when α is distinct from 0° or 90°. The oscillation of the non-coaxial angle is significantly more variable in soft rock experiencing combined tension–torsion (45° < α < 90°), as opposed to those under the influence of combined compression-torsion (0° < α < 45°). The non-coaxiality swiftly dissipates when the sample is approaching the failure state. The stress rate is decomposed into stress magnitude and direction to describe non-coaxial features of plastic strain. And a new method for non-coaxial stress rate is proposed which can express the plastic strain increment directions. The spherical interpolation coefficient method is utilized to describe the continuous change in non-coaxial plastic flow direction between tangential and normal directions of the yield surface. The non-coaxial parameter (Δ) is introduced to quantify the non-coaxial characteristics of soft rock and its validity is confirmed through test results. This method effectively captures the principal stress direction influence on non-coaxial behavior of soft rock and have significance for rock mechanics.

Keywords

Similar weakly cemented soft rock
Non-coaxial stress rate
Strain increment direction
Principal stress direction
Directional shear test
Subject terms

Energy science and technology
Engineering
http://dx.doi.org/10.13039/501100001809 National Natural Science Foundation of China 52104088 Liu Jiashun Open Research Fund of State Key The indoor of Geomechanics and Geotechnical EngineeringSKLGME022021 http://dx.doi.org/10.13039/501100002858 China Postdoctoral Science Foundation 2022M713383 Liu Jiashun issue-copyright-statement© Springer Nature Limited 2024
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pmcIntroduction

Excavation disturbances will alter the original stress state and stress path of the surrounding rock, leading to adjustments in both the magnitude and direction of the stress1–3. The previous results show that the principal stress magnitude influences the crack depth, and the principal stress direction influences the direction and density of crack propagation. Therefore, the rock damage caused by roadway excavation is the result of the combined action of stress magnitude and stress direction4,5.

The principal stress magnitude influence has been widely researched, while the influence of principal stress direction has often been neglected due to limitations of experimental conditions and the complexity of three-dimensional stress paths6. However, large number of experimental results on soil show that the principal stress rotation will cause the strain increment direction deviate from the principal stress direction, which defines as non-coaxiality7. This phenomenon aggravates the strength damage and plastic deformation of soft rock, affecting the reliability of engineering design8. However, it is still assumed that the plastic strain rate and stress rate are coaxial, which consequently is unable to accurately describe the non-coaxial behavior of soft rock9. It is also not applicable to the mechanical analysis of weakly cemented soft rock subjected to excavation loads10. These deficiencies also lead to the unreliability of the surrounding rock, resulting in underground engineering disasters such as uneven deformation, fragmentation, collapse, and roof falls in soft rock roadways. Therefore, it is urgent to establish a non-coaxial plastic theoretical model that considers the principal stress axis rotation, and to reveal the influence mechanism of deformation of weakly cemented soft rock under complex stress environments.

The principal stress rotation and directional shear tests were conducted using the hollow cylinder apparatus (HCA) to investigate the non-coaxial behavior of geotechnical materials11,12. The evolution rule of the strain increment direction and its influencing factors were investigated13,14. Ma et al.15 conducted true triaxial tests and analyzed the non-coaxial behavior of stacked stone materials under true triaxial stress conditions using a discrete–continuous coupling approach. Feng et al.16 analyzed the laws of tangential displacement, three-dimensional mechanical properties, and the non-coaxial angle of contact surfaces through round-trip stress-controlled shear stress paths. Li et al.17 conducted a study on the dilatation and non-coaxial properties of sandstone under different stress Lode angles. Li et al.18 explored the effect of stress-induced anisotropy on non-coaxial behavior and its structural evolution, and investigated the response of inter-particle friction to non-coaxial behavior. In theoretical research on the non-coaxial behavior of rock and soil, Getierrez19 and Yu et al.20 established a non-coaxial plastic flow law. Qian et al.21,22 developed a non-coaxial principal model in three-dimensional stress space, emphasizing the importance of non-coaxial effects in predicting shear zones in geomaterials. Gao et al.23 introduced a novel three-dimensional constitutive model to describe the non-coaxial behavior of sandy soils, incorporating the explicit expression of the plastic increment direction through the grouping tensor of soil structure anisotropy and its evolutionary law. Tian et al.24 formulated an anisotropic uniform hardening model to depict the mechanical response of geotechnical materials during the rotation of the principal stress axis. Wen et al.25 and Chang et al.26 utilized generalized potentials and micropolar theory to establish a non-coaxial elastic–plastic constitutive model. Li et al.27 integrated stress tensor into non-coaxial theory, exploring the differences in stress principal axis changes induced by conditions of fixed principal stress and fixed stress principal axis. Wang et al.28,29 introduced an improved corner point theory and a generalized non-coaxial theory to resolve the discrepancy between the plastic strain rate and the linear assumption regarding the relationship between stress rates. Du et al.30 integrated a three-dimensional non-coaxial flow law into an existing constitutive model designed for the cyclic loading of clays. Chen et al.31 developed an elastic–plastic constitutive model grounded in the existing non-coaxial theory, utilizing the angle-point method for the description of the yielding surface. Additionally, they introduced a semi-implicit stress-integration algorithm to refine the non-coaxial flow law and develop a corresponding non-coaxial constitutive model31,32.

In summary, the current research on non-coaxial predominantly focuses on soils, similar issues also emerge in soft rock masses subjected to principal stress rotation paths33,34. Notably, weakly cemented soft rock, as compared to hard rocks, display lower strength, inadequate cementation, heightened sensitivity to disturbance, and a pronounced rheological behavior. During excavation and unloading processes, the alteration in the principal stress direction within the surrounding rock of soft rock tunnels intensifies these non-coaxial characteristics, thereby posing significant engineering risks related to large deformations in soft rock.

Consequently, it is imperative to examine the mechanical behavior and non-coaxial characteristics of weakly cemented soft rock under different stress directions. This study involves the execution of directional shear tests utilizing the HCA to investigate the behavior of strain incremental direction and the evolution of the non-coaxial angle in weakly cemented soft rock. Furthermore, a novel formula is devised to depict the non-coaxial stress rate within the principal stress space, assuming a transition of the surrounding rock from a three-dimensional stress state to either unidirectional or bidirectional unloading states following excavation. This findings can provide scientific insights for the construction and disaster prevention in weakly cemented soft rock engineering.

Tests scheme and results analysis

Specimen preparation

Weakly cemented soft rock exhibits low strength and is prone to muddying and disintegration upon contact with water, making the preparation of hollow cylinder specimens challenging. Consequently, similar weakly cemented soft rock specimens are employed for directional shear tests. In this study, aeolian sand serves as the aggregate, bentonite as the filler, gypsum as the binder, talc as the conditioner, and a precise amount of water is used to create these similar weakly cemented soft rock specimens. The composition ratio is set at 1:0.66:0.44:0.28:0.36 for aeolian sand, bentonite, gypsum, talc, and water, respectively. The preparation method is detailed in Fig. 1, and the dimensions of the hollow cylinder specimens are 60 mm × 100 mm × 200 mm35.Figure 1 The preparation of similar soft rock hollow cylindrical specimens.

Test scheme

Directional shear stress path tests are performed using the Hollow Cylinder Apparatus (HCA), which enables independent control of axial load (W), torque (MT) outer cell pressure (Po), and inner cell pressure (Pi). These four loading parameters induce axial stress (σz), circumferential stress (σθ), radial stress (σr), and torsional shear stress (τzθ) on the hollow thin-walled specimen. This configuration also correlates to the major, intermediate, and minor principal stresses (σ1, σ2, and σ3) and the major principal stress rotation angle (ασ) within the stress coordinate system, as depicted in Fig. 2.Figure 2 Test equipment and schematic diagram of stress state of the specimen.

In our research, the stress path module of the LAB is utilized to achieve principal stress axis rotation by controlling the parameters of mean principal stress (p), intermediate principal stress coefficient (b), deviatoric stress (q), and major principal stress direction angle (ασ). Initially, the axial force is set to 0.05 kN, and the inner and outer cell pressures are set to Pi = Po = 0.03 MPa. The p, q, and b are set to 0.5 MPa, 0.03 MPa, and 0.1, respectively. The α is adjusted to 0°, 15°, 22.5°, 30°, 45°, 60°, 67.5°, 75°, and 90°. Subsequently, q is incrementally increased until specimen failure, with a loading rate of 0.01 MPa/min. The test program is detailed in Table 1.Table 1 Experimental scheme.

Test number	α/°	q/MPa	
D1	0°	Load at 0.01 MPa/min until destruction	
D2	15°	
D3	22.5°	
D4	30°	
D5	45°	
D6	60°	
D7	67.5°	
D8	75°	
D9	90°	

The stress path of the directional shear test is illustrated in Fig. 3.Figure 3 The stress paths of directional shear tests35.

As shown in Fig. 3, the stress path forms an angle of 2α with the positive axis of (σz-σθ) exhibiting a linear increase as q increments. Furthermore, as α increases, the stress path undergoes a counterclockwise rotation in the (σz-σθ) ~ 2τzθ plane.

Strain increment evolution

The major principal stress angle (ασ) is calculated as Eq. (1).1 ασ=12arctan2τzθσz-σθ,

where τzθ is the shear stress.

The principal strain increment direction angle (αdε) is calculated as Eqs. (2a), (2b). If dεz ≥ dεθ,2a αdε=12arctandγzθdεz-dεθ,0∘≤αdε≤45∘.

Else if dεz < dεθ,2b αdε=12arctandγzθdεz-dεθ+-1nπ2,45∘<αdε≤90∘,

While, dγzθ ≤ 0, n = 1; dγzθ > 0, n = 2.

where dεz, dεθ, dγzθ are the increment of axial strain, circumferential strain and shear strain respectively.

During shearing processes following changes in the principal stress direction, particles within soft rock undergo rearrangement and mutual friction, resulting in nonlinear variations (non-coaxiality) between the principal stress direction and the principal strain direction. Additionally, soft rock is a typical heterogeneous material, with mechanical properties differing in various directions. This anisotropy intensifies the extent of non-coaxial behavior, particularly under complex stress paths involving changes in the principal stress direction. Non-coaxial behavior in soft rock is particularly pronounced under such conditions.

By setting up directional shear tests or applying Eq. (1), ασ values of 0°, 15°, 22.5°, 30°, 45°, 60°, 67.5°, 75°, and 90° are derived under different stress directions. Subsequently, αdε is calculated using Eqs. (2a), (2b). The evolution of the principal strain increment angle of weakly cemented soft rock under different principal stress direction angle is illustrated in Fig. 4.Figure 4 Evolution of strain increment direction under various α conditions in directional shear tests. (a) α = 0°; ( b) α = 15°; (c) α = 22.5°; (d) α = 30°; (e) α = 45°; (f) α = 60°; (g) α = 67.5°; (h) α = 75°; (i) α = 90°.

In the context of stress loading subsequent to a shift in the principal stress direction, when the deviatoric stress ratio (q/p) falls below 0.4 under low stress conditions, the soft rock specimen enters the compaction phase. During this phase, the shifts in the strain increment direction are pronounced and swift, with the boundaries of the compaction phase dynamically adapting to alterations in the direction of the principal stresses. Consequently, in the experimental analysis, it is imperative to overlook the initial low stress phase and its associated elastic deformation, concentrating rather on the plastic deformation phase to elucidate the rock's deformation dynamics and its non-coaxial attributes under elevated stress conditions.

Figure 4 illustrates that the αdε experiences significant fluctuations during the initial stage, gradually aligning with the principal stress direction from a position initially divergent from it. This phenomenon arises due to the specimen's pores undergoing further compaction and unstable internal structural changes as deformation progresses. As the specimen enters the elastic deformation stage, the αdε aligns closely with the α, indicating a tendency towards coaxiality. However, as stress levels increase, the specimen transitions from an elastic to an elastic–plastic state. Consequently, in experimental analysis, it is advisable to disregard the initial low stress stage and its associated elastic deformation, focusing instead on the plastic deformation stage to accurately capture the rock's deformation behavior and non-coaxial characteristics under high stress conditions.

Figure 4a and i demonstrate that the direction of principal strain increment aligns with the major principal stress in both triaxial compression and tension tests when α is set to 0° and 90°, respectively. In Fig. 4b–d, the specimen undergoes compression-torsion shear when 0° < α < 45°. Initially, the application of the q results in the αdε that exhibits weak non-coaxial behavior. As the q increases, the strain increment direction deviates more significantly from the principal stress direction. Upon reaching critical damage, the specimen rapidly aligns with the principal stress direction. This process involves increased mutual extrusion and shearing between particles, leading to higher friction and wear. Consequently, αdε becomes progressively larger than ασ, intensifying non-coaxial behavior. As the sample nears failure, the mutual extrusion and shearing between internal particles further increase friction and wear, causing αdε to exceed ασ and significantly enhance non-coaxial behavior. Ultimately, as the specimen approaches the failure stage, particle shear failure reduces the material's load-bearing capacity, leading to a rapid convergence of the strain increment direction towards the principal stress direction. When α = 45°, as depicted in Fig. 4e, the sample is subjected to a torsional shear state. In this phase, the sample is predominantly influenced by torque, which results in a substantial discrepancy between the initial αdε and ασ, leading to significant non-coaxial behavior. As the q value and torque increase, the relative dislocation between particles and structural yielding cause αdε to gradually align with ασ. Ultimately, upon sample failure, αdε aligns with the ασ. When the angle α is between 45° and 90°, as shown in Fig. 4f–h, the sample undergoes a tensile-torsion shear state. Comprising primarily of accumulated particles, the soft rock sample experiences lateral compression and torsional shear stress under loading. Initially, the αdε value is smaller than ασ. As the q increases, αdε gradually converges towards ασ. Eventually, with continued stress increase, αdε surpasses ασ. Upon sample damage, αdε rapidly approaches ασ.

When the α ranges from 0° to 45°, the amplitude of strain increment change remains within 10°. Conversely, when α ranges from 45° to 90°, the fluctuation amplitude of strain increment change significantly increases, reaching up to 20°. This observation suggests that when a sample is subjected to tensile-torsional stress, the change in αdε is notably greater than under compressive and torsional stress conditions. This phenomenon highlights the distinct deformation characteristics and responses of weakly cemented soft rock under different stress states.

Non-coaxial angle evolution law

The traditional elastic-plasticity theory, which assumes isotropy and alignment of stress direction with strain increment, is limited in explaining the strain characteristics of homogeneous materials. However, weakly cemented soft rock, being a non-homogeneous material, exhibits anisotropic properties due to its complex formation process and type of cementation. In underground engineering, the strong sensitivity of weakly cemented soft rock to stress perturbations results in non-coaxial behavior under complex stress paths. This non-coaxial behavior changes significantly with varying principal stress directions. The non-coaxial angle β is defined as the difference in direction between the principal stress and the increment of principal strain, as shown in Eq. (3).3 β=αdε-ασ.

The unit length of principal stress increment is calculated as shown in Eq. (4); the unit length of principal strain increment is calculated as shown in Eq. (5).4 AC→=ds=dσz-σθσz+σθ2+d2τzθσz+σθ2,

5 AB→=dε=1dsdεz-dεθ2+dγzθ2.

In the (σz-σθ) ~ 2τzθ plane, a conventional stress point a is chosen for study. Point a signifies the stress state direction, characterized by an angle of 2α in the horizontal direction. The direction of the principal stress increment is orthogonal to this. The plastic principal strain increment direction is indicated by an angle of 2αdε in the horizontal direction. The angle between the principal stress direction and the principal strain increment direction, denoted as 2β, is twice the angle between the principal stress direction and the direction of the principal strain increment, as depicted in Fig. 5.Figure 5 Schematic diagram of non-coaxial features in the (σz-σθ) ~ 2τzθ plane.

In the plastic deformation of weakly cemented soft rock, when the direction of the largest principal stress aligns with the direction of the strain increment (β = 0), the specimen exhibits coaxial deformation characteristics. However, when the direction of the largest principal stress increment does not align with the direction of the strain increment (β ≠ 0), a non-coaxial phenomenon emerges. A positive non-coaxial angle is observed when αdε is larger, whereas a negative non-coaxial angle occurs when the largest principal stress direction angle is larger.

To analyze the non-coaxial evolution rules of weakly cemented soft rock under varying stress conditions, a three-dimensional model is developed using the parameters q, α, and β. Figure 6a illustrates the non-coaxial evolution features in the compression-torsion state, while Fig. 6b highlights the non-coaxial evolution characteristics in the tension–torsion state.Figure 6 Evolution law of non-coaxial characteristics of soft rock in three-dimensional space (q/p, α and β). (a) 0°–45°; (b) 45°–90°.

Figure 6 shows that weakly cemented soft rock exhibits a distinct evolution pattern when subjected to various stress states, including triaxial compression, compression-torsion, tension, tension–torsion, and triaxial torsion. Specifically, the specimen exhibits coaxial behavior under triaxial compression and triaxial torsion conditions, with its final failure following a coaxial failure mode. However, in compression-torsion, tension and tension–torsion, a noticeable deviation between ασ and αdε indicates clear non-coaxial behavior. The β emerges as a critical parameter under these conditions, signaling a shift in the specimen's failure mode towards non-coaxial failure.

In the context of the evolution trend of the non-coaxial angle, a continuous increase is observed. As the specimen approaches the failure point, its internal structure loses its bearing capacity, leading to a rapid shift from a non-coaxial state to a coaxial state. In a compression-torsion state, following an initial disturbance, the sample initially exhibits a roughly coaxial state. Subsequently, with the increase in q, the β gradually rises, indicating a progressive increase in non-coaxial behavior. In a torsion state, the sample displays significant non-coaxial at the onset of loading, with β representing a negative non-coaxial angle. As q continues to rise, the non-coaxial diminishes gradually, and β approaches zero, signaling a shift towards a coaxial state. In a tension–torsion state, β transitions from a negative to a positive non-coaxial angle, with non-coaxial gradually rising after an initial weakening phase.

Specifically, under directional shear stress paths, weakly cemented soft rock exhibit significant non-coaxial behavior due to their heterogeneity. The particles and media inside soft rock experience varying stress, strain change rates, and flow directions influenced by different principal stress directions, leading to distinct anisotropic characteristics in fracture, deformation, and plastic behavior. For α angles from 0° to 45°, the non-coaxial angle remains positive and continues to increase. At 45°, it transitions to a negative value, gradually diminishing non-coaxial. As α ranges from 45° to 90°, the non-coaxial angle shifts from negative to positive, with non-coaxial initially weakening and then strengthening.

Theory of non-coaxial plastic flow

Non-coaxial stress rate

The early non-coaxial theory, as proposed by Rudnicki and Rice36, divides the strain increment dε into coaxial strain increment dεcp and non-coaxial strain increment dεnp.

The flow direction of non-coaxial strain rate is influenced not only by the current stress state but also by factors such as stress path, magnitude, direction, and non-coaxial stress rate. Directional shear tests conducted under fixed α conditions have shown that different principal stress action directions can induce non-coaxial plastic deformation in rock masses when p and b are fixed.

The stress redistribution caused by excavation and unloading in underground engineering can change the rock mass from a three-dimensional stress state to a one-way unloading state, simplifying the problem from a complex three-dimensional one to a two-dimensional stress state. The excavation disturbance stress alters the principal stress direction, as illustrated in Fig. 7.Figure 7 Assumption of unloading of surrounding rock units in tunnels.

In Fig. 7, a unit of the surrounding rock is taken, where σ1 is collinear with the σz direction, σ2 is collinear with the σy direction and points in the unloading direction, and σ3 is collinear with the σθ direction. The excavation disturbance in tunnel engineering disrupts the initial stress equilibrium of the surrounding rock, leading to stress readjustment. During this process, shear displacement occurs between rock layers, resulting in the disappearance of stress in the unloading direction and the emergence of shear stresses on the contact surface of the unit. Consequently, σ1 deviates from the σz direction, and σ3 deviates from the σθ direction.

The transition of surrounding rock from a three-way stress state to a one-way unloading state can be represented by a stress tensor, as depicted in Eq. (6).6 σ^ij=σxxσxyσxzσyxσyyσyzσzxσzyσzz⟶load disappearsσxx0σxz000σzx0σzz⟶rotation of stress directionσxxσxzσzxσzz.

In order to generalize the specific problem after unloading in the Y direction, it is essential to construct the stress transformation matrix M. The stress change matrix is built from the transformation σ1 and σ3. By converting the non-coherent σ1-σ3 principal stress plane to a general stress plane, the axial stress rate problem is transformed, and the stress matrix is represented in Eqs. (7a), (7b).

To generalize the specific problem encountered post-unloading in the Y direction, it is essential to formulate the stress transformation matrix My(α). This matrix is derived from the transformations of σ1 and σ3. By converting the non-coherent σ1-σ3 principal stress plane into a general stress plane, the problem of axial stress rate is transformed, and the stress matrix is depicted in Eqs. (7a), (7b).7a Myα=cosα0-sinα010sinα0cosα,

7b σ^=Myασ1000σ2000σ3MyαT.

It assumes that stress state changes for directional shear tests, a new method for representing non-coaxial stress rates is developed. This method ensures the axial and circumferential stress rotate around the radial stress action surface in the principal stress space. Initially, e^1, e^2, and e^3 are employed to denote the stress directions of the major principal stress σ1, the intermediate principal stress σ2, and the minor principal stress σ3 in the Cartesian coordinate system. The stress rotation diagram can be observed in Fig. 8.Figure 8 Diagram of major intermediate and minor principal stress rotation.

Then the stress tensor σ^ can be expressed as follows.8 σ^=∑i=13σie^i×e^i,

Taking increment Δσ, the stress tensor σ^ changes to σ′ after increment Δσ. A new direction vector n^ associated with σ^ is defined according to limit theory as follows9 n^=limσ′→σ^σ′-σ^σ′-σ^.

In the directional shear test loading process, the p remains constant, while the b is fixed to maintain a constant stress Lode angle (θ). The specimen undergoes damage as the q is gradually increased. The effects of deviatoric stress increments under this loading condition can be mathematically expressed as Eq. (10).10 σ′=p+23q+Δqsinθ+23πe^1×e^1+p+23q+Δqsinθe^2×e^2+p+23q+Δqsinθ-23πe^3×e^3.

Substituting Eq. (10) into (9) so that Δq tends to 0+ yields the deviatoric stress direction vector as shown in Eq. (11).11 n^q=23sinθ+23πe^1×e^1+23sinθe^2×e^2+23sinθ-23πe^3×e^3.

In the directional shear test, the different major principal stress direction angles are examined. The radial stress direction remained constant while the axial stress and hoop stress directions varied, with σ1 and σ3 rotating around the σ2 stress action surface. Additionally, as σ1 and σ3 underwent synchronous rotation during the test, this led to the condition where α equals β.

Substitute Eqs. (7a), (7b) into Eq. (9) by taking the angular increment in the principal stress direction as Δα, and make Δα→0+ the amount of the principal stress direction, n^1, and the amount of the direction of the min-principal stress, n^3, that can be sought, as shown in Eqs. (12a), (12b).12a n^1=22e^2×e^3+22e^3×e^2,

12b n^3=22e^1×e^2+22e^2×e^1.

The association of Eq. (11) with Eqs. (12a), (12b) yields the symmetric transformation tensor I^ in the principal stress space with n^q、n^1 and n^3 relations as shown in Eq. (13).13 I^=n^q×n^q+n^1×n^1+n^3×n^3.

Using Eq. (13) the non-coaxial stress rate σ^ can be expressed as Eq. (14).14 σ^=σ^:I^=σ^q+σ^1+σ^3,

where σ^q=σ^:n^qn^q、σ^1=σ^:n^1n^1、σ^3=σ^:n^3n^3。

The derivation reveals that σ^q is influenced by variations in the principal stress value, while σ^1 and σ^3 are influenced by the direction of the principal stress. Additionally, the determination of the total plastic stress rate under the directional shear stress path is contingent upon the stress magnitude and directional factor. To facilitate a comprehensive understanding, a stress rate space will be delineated, utilizing geometric methodologies to elucidate the non-coaxial characteristics of stress rates, as visually represented in Fig. 9.Figure 9 Coaxial and non-coaxial stress planes in principal stress space.

Non-coaxial strain rate

The elastic–plastic theoretical model primarily emphasizes the role of stress magnitude factors in the context of non-coaxial strain rates, thereby overlooking the plastic deformation effects induced by stress directionality. The definition of the loading strength index (L) is as follows.15 L=1Kp∂f∂σ:σ^,

where the Kp represents plastic modulus, and the ∂σ represents the yield function.

This study aims to elucidate the dual impact of both magnitude and directional factors on the flow direction of non-coaxial strain rates, and to delineate distinct nonlinear loading mechanisms for each constituent stress rate. Specifically, one mechanism will account for variations attributable to fluctuations in principal stress magnitude, whereas the other will capture alterations stemming from changes in principal stress direction.

The first type of loading mechanism refers to the loading mechanism representation method proposed by Zienkiewicz OC9 based on generalized plastic mechanics. By directly defining the loading intensity index, plastic modulus and plastic flow direction, the loading mechanism of the principal stress can be expressed as follows.16a ε^qp=Lqn^q+13Dq23,

16b Lq=1Kpqn^q:σ^,

16c Dq=Dqσ,ψ,C,

16d Kpq=Kpqσ,ψ,C.

whereε˙qp is the strain rate induced by the magnitude of the stress (σ^q); Lq is the loading strength index of n^q; Dq is the shear expansion coefficient corresponding to σ^q; Kpq is the plastic modulus corresponding to σ^q; and both Dq and Kpq can be expressed as a function of the stress σ, the state parameter ψ, and the characteristic parameter C, which includes the anisotropy of the material.

The non-coaxial behavior resulting from stress magnitude is observed. Similarly, the non-coaxial behavior of a loading mechanism due to the rotation of axial and hoop stress around the radial stress action surface can be defined once the direction of radial stress action is established.17a ε^ip=Lin^i+13Di23,

17b Li=1Kpin^i:σ^,

17c Di=Diσ,ψ,C,

17d Kpi=Kpiσ,ψ,C,

where the angular scale i takes 1 and 3 respectively; ε˙ip is the strain rate induced by the factor of stress direction; Li is the loading strength index of n^i; Di is the shear expansion coefficient corresponding to σ^i; Kpi is the plastic modulus corresponding to σ^i; Di and Kpi can also be expressed as a function of the stress σ, the state parameter ψ, and the characteristic parameter C including the anisotropic of the material.

To examine the evolution of non-coaxial behavior in weakly cemented soft rock under directional shear stress, the spherical interpolation method is utilized to represent the direction of the non-coaxial plastic deviator stress rate. This approach facilitates the depiction of the direction of non-coaxial plastic flow at the yield surface. As demonstrated in Fig. 10, the transition between the primary and secondary loading mechanisms is illustrated through continuous variations.Figure 10 The flow direction of non-coaxial deviator stress rate.

The non-coaxial parameter (Δ) is introduced to define the non-coaxial stress rate n^ in the dual loading regime as follows18a Δ=dεijpcdεijpn,

18b n^r=sin1-Δθsinθn^1+sinΔθsinθn^3,

18c n^=sin1-Δφsinφn^q+sinΔφsinφn^r,

where the Δ is the non-coaxial coefficient, and 0 ≤ Δ ≤ 1. Theoretically, when Δ = 0, it is a coaxial state; when Δ ≠ 0, it is a non-coaxial state; θ is the angle between n^1 and n^3; φ is the angle between n^q and n^r.

A dual nonlinear loading mechanism is formulated by explicitly defining plastic index variables to characterize the plastic deformation engendered by the non-coaxial stress rate component. This formulation is realized through the application of geometric mapping rules within the boundary surface theory, in conjunction with the spherical direction interpolation coefficient method. This integration enables a seamless transition between the non-coaxial stress rate attributable to stress magnitude and the non-coaxial stress rate arising from stress direction within the context of non-coaxial plastic flow direction. Consequently, this theoretical framework provides enhanced versatility in delineating non-coaxial plastic deformation in weakly cemented soft rock, surpassing the descriptive limitations of traditional non-coaxial theories.

Model validation

Referring to the yield function f established by considering anisotropic as follows37.19a f=q-Mgθp-σ0,

19b M=6sinφ03-sinφ0,

19c gθ=32L2cosδθ+6ξsinφ03-sinφ032L2cosδ0+6ξsinφ03-sinφ0,

19d σ0=ccotφ0,

where M is the coefficient of friction as shown below, gθ is the shape function, and σ0 is the triaxial tensile strength.

The calculation method for plastic modulus (Kp) is as follows.20 Kp=hGenψG02.97-e2ppa1+eMcgθe-nψη-1,

where G0 is the material constant. Stress ratio η = q/p; n is the material constant.21 D=d0McgθMcgθemψ-η,

where, d0 and m are material constants, ψ = e-ec.

As the deviatoric stress ratio (q/p) increases, the direction of plastic strain increment flow in weakly cemented soft rock under various directional shear stress paths is shown in Fig. 11.Figure 11 The flow direction of non-coaxial plastic strain rate in weakly cemented soft rock.

In Fig. 11, the vertical axis represents the q/p, while the horizontal axis denotes the major principal stress direction angle for different shear paths. As q/p increases, each point on the stress path corresponds to a specific stress state, with arrows indicating the direction of plastic strain increment flow emanating from these points. The angle between these arrows and the stress paths, known as the β, serves to characterize the degree of non-coaxiality in soft rock under stress loading.

When the principal stress direction coincides with the axial direction (α = 0° or 90°), the principal strain increment aligns accordingly. In this coaxial condition of soft rock, the phenomenon ceases to repeat. For α angles from 0° to 45°, the initial application of deviatoric stress induces axial compaction in weakly cemented soft rock, leading to the closure of internal pores. During this phase, the direction of the strain increment exhibits significant variability. Within the axial compaction stage, the angle of the principal strain increment direction surpasses that of the principal stress direction. As deviatoric stress intensifies, the direction of the flow strain increment gradually rotates clockwise, converging towards the principal stress direction. At α = 45°, the initial angle of the principal strain increment is smaller than the angle of the principal stress increment. With escalating stress, the strain increment direction aligns more closely with the principal stress increment direction. For α angles from 45° to 90°, the initial direction of the principal strain increment is opposed to the principal stress direction, gradually rotating clockwise to achieve a coaxial state. This pattern continues even post-coaxiality, with the strain increment direction evolving in the counterclockwise direction relative to the principal stress increment. As the weakly cemented soft rock specimen approaches its bearing capacity limit, the principal strain increment rapidly aligns with the principal stress increment direction.

The coaxial model, isotropic non-coaxial model, anisotropic non-coaxial model, and the newly proposed model are employed to simulate the variation in the trend of plastic principal strain rate flow direction with respect to the direction angle of the major principal stress, as illustrated in Fig. 12.Figure 12 Verification of non-coaxial models considering the direction of plastic strain increment.

While the traditional anisotropic non-coaxial model can partially simulate the incremental direction of the plastic principal strain rate, it exhibits notable fluctuations and room for improvement. Despite some deviation from experimental results, this model generally aligns with the experimental trends in the direction of the plastic principal strain rate relative to the major principal stress angle. Overall, it offers greater stability compared to the traditional non-coaxial model. Furthermore, the model demonstrates better representation of non-coaxial characteristics in compression torsion, torsion, and triaxial tensile states, supporting its validity, reliability, and accuracy.

Figure 12 demonstrates that the plastic principal strain rate in weakly cemented soft rock deviates from the principal stress direction during shear. The model's findings suggest that the direction angles of principal stress and principal strain rate as predicted by the coaxial model are consistently equivalent, indicating a coaxial alignment. Conversely, the isotropic non-coaxial model exhibits a substantial delay in the direction angle of the major principal stress relative to the plastic principal strain rate, thereby failing to accurately represent the non-coaxial behavior of weakly cemented soft rock. Although the anisotropic non-coaxial model can partially replicate the incremental direction of the plastic principal strain rate, it shows significant fluctuations and areas for enhancement. Despite slight discrepancies from experimental outcomes, this model generally corresponds with the experimental trends observed in the direction of the plastic principal strain rate with respect to the major principal stress angle. Overall, it provides greater stability compared to the conventional non-coaxial model. Additionally, the model effectively captures the non-coaxial characteristics under compression-torsion, torsion, and triaxial tension states, thereby substantiating its validity, reliability, and accuracy.

Despite not achieving optimal performance during the tension–torsion phase, the model's overall trend closely corresponds with the experimental results, adeptly delineating the variation in plastic strain increments across various principal stress directions. This observation substantiates the enhanced capabilities of the model detailed in this paper.

Discussion

The stress adjustment of near-field surrounding rock caused by excavation in mining engineering encompasses both changes in the stress magnitude and the rotation of the principal stress axes. However, the existing mining design theory predominantly focuses on designing and calculating based on the stress magnitude and stress influence zone, while the impact of principal stress axis rotation is significantly overlooked.

The plastic principal strain increment direction is inconsistent with the principal stress direction when principal stress direction rotates, which is defined as non-coaxial. However, this will lead to changes in the crack propagation density and direction of geotechnical materials, exacerbating rock mass damage. Traditional plasticity theory cannot objectively describe the non-coaxial characteristic of geotechnical materials. The hollow torsional shear test of continuous cyclic rotation, principal stress directional, and pure rotation of the principal stress axis were conducted to investigate the anisotropic and non-coaxial characteristics of soft clay, sand and silt38 using the HCA. Matsuoka et al.39 established the relationship between stress increment and strain increment that can reflect the rotation of the principal stress axis. Qian et al.40 introduced the non-coaxial theory to analyze the instability phenomenon and instability state of soil deformation bifurcation. Yu et al.41 established the non-coaxial plastic flow law. On the basis of a large number of hollow torsional shear tests, scholars have successively established non-coaxial plastic constitutive models of sand, coarse-grained soil and soft clay based on generalized potential theory42,43. However, unlike soil, weakly cemented soft rock has natural defects such as cracks and joints, a certain degree of damage, and strong rheology and sensitivity of stress disturbances. The non-coaxial theory of rock that considers the rotation of the stress principal axis must fully consider the characteristics of the rock material.

In this research, the principal stress directional shear tests on weakly cemented soft rock were conducted to investigate the evolution of strain increments and non-coaxial characteristics under stress loading. A new method is proposed to express non-coaxial stress rates by rotating around the radial stress action surface. The stress rate is decomposed into stress magnitude and stress direction, with a directly defined plasticity index. Dual nonlinear loading mechanisms were established to describe plastic deformation induced by each stress rate component. The spherical interpolation coefficient method is utilized to describe the continuous change in non-coaxial plastic flow direction between tangential and normal directions of the yield surface. The non-coaxial parameter Δ is introduced to quantify the non-coaxial characteristics of soft rock.

Nevertheless, the non-coaxial theory developed in this study presents certain unavoidable limitations. As demonstrated in the research by Liu et al., numerical simulations using ABAQUS were conducted on weakly cemented soft rock roadways. The findings revealed that during the excavation process, the stress magnitudes and directions in the roof, floor, and sidewalls exhibited concurrent variations, as depicted in the accompanying figure.

Figure 13 shows that the direction of the major, intermediate, and minor principal stresses in the roof, floor, and sidewalls continuous changing during the roadway excavation. This transformation is a three-dimensional process, with the rotation trajectory of the principal stresses occurring throughout the entire space. The proposed non-coaxial theory in this study can consider this three-dimensional principal stress rotation state. However, in this study, we assume that the principal stress rotation fixes the intermediate principal stress direction, which is the unloading direction. The major and minor principal stresses rotate occurs. Then the non-coaxial behavior of weakly cemented soft rock with differ principal stress directions was investigated.Figure 13 The principal stress rotation during roadway excavation7.

The variation in non-coaxial angles during roadway excavation, as shown in Fig. 14.Figure 14 The variation law of non-coaxial angle during roadway excavation. (a) Roof; (b) floor; (c) sidewalls.

Figure 14 illustrates that the principal stress changes in the roof, floor, and sidewalls during roadway excavation lead to non-coaxial phenomena. Initially, the state remains nearly coaxial. As the excavation approaches the monitoring location, the non-coaxial angle experiences a sharp fluctuation. While non-coaxial behavior persists away from the monitoring point, the angle stabilizes. Notably, for a given monitoring section, the trend of non-coaxial angle variation resembles an "N" or "V" shape. Throughout the excavation disturbance, the maximum non-coaxial angles for the tunnel floor, roof, and sidewalls were 52.8°, − 22.3°, and 30.6°, respectively.

In summary, the excavation of tunnels induces rotation of principal stress directions in the surrounding rock, leading to varying degrees of non-coaxiality. The non-coaxial theory proposed in this study demonstrates significant advantages in analyzing the plastic flow characteristics of weakly cemented soft rocks, accurately depicting the deformation behavior of soft rocks under complex stress states. Compared to traditional theories, the non-coaxial theory presented herein better describes the plastic flow behavior of soft rocks.

Conclusion

The directional shear tests were conducted to explore the non-coaxial behavior of weakly cemented soft rock with differ principal stress direction. And a novel formula was formulated to characterize the non-coaxial stress rate of weakly cemented soft rock under directional shear stress paths. The main conclusions are as follows:The non-coaxial behavior of weakly cemented soft rock is evident under directional shear stress paths where α ≠ 0° or 90°. When 0° < α < 45°, the αdε gradually divergence from α with the increasing of q values, which intensify the non-coaxial behavior. At α = 45°, the initial αdε exhibits evident hysteresis from α. As q increases, αdε converges towards α, resulting in a reduction of the non-coaxial characteristics. When 45° < α < 90°, the non-coaxial behavior initially diminishing and then strengthening with the q increasing.

The specimen is subjected to the compression-torsion shear state when α < 45°, with non-coaxial angle fluctuations limited to within 10°. Conversely, the soft rock is in tension–torsion shear state when α > 45°, where the non-coaxial angle fluctuations often exceed 10° and can reach up to 20°. Notably, the non-coaxial fluctuations in the tension–torsion state are significantly more severe than those observed in the compression-torsion state.

A non-coaxial stress rate expression method is developed, which considers the rotation of axial and circumferential stress on the radial stress action surface in principal stress space. It can effectively describe both the magnitude and direction of the principal stresses.

The research investigates the non-coaxial strain rate through loading mechanisms of principal stress magnitudes and orientations. In the spherical difference function method, a non-coaxial parameter (Δ) is introduced to quantify the non-coaxial characteristics of weakly cemented soft rock. The validation of the non-coaxial model is performed using experimental data and comparisons with other models, confirming its accuracy in depicting the flow direction of principal strain increments in weakly cemented soft rock.

The research results will provide insights into the non-coaxial plastic mechanical behavior of soft rock, which has significant implications for mining rock mechanics.

Acknowledgements

This work was financially supported by the National Natural Science Foundation of China (Grant Nos. 52104088, and 52374091), the Open Research Fund of State Key The indoor of Geomechanics and Geotechnical Engineering (No. SKLGME022021), and the China Postdoctoral Science Foundation (Grant No. 2022M713383). The authors gratefully acknowledge the helpful comments made by the reviewers.

Author contributions

J.S. Liu: Conceptualization, Funding acquisition, Methodology, Writing -review & editing. Z.Y. Zheng: Writing-original draft, Conceptualization. H. Zhou: Methodology, Resources. K.X. Zhu and Y. Wang: Investigation, Supervision. N. Zhou: Validation. S.Y. Wang: Investigation. M. Y. Sun: Supervision.

Data availability

All data, models, and code generated or used during the study appear in the submitted article.

Competing interests

The authors declare no competing interests.

Publisher's note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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