Three-dimensional Stress State Research Articles

Strength characteristics and energy evolution of cement stone under true-triaxial loading conditions

Oil well cement is a commonly used cementing material in oil and gas extraction engineering. Due to the change of stress conditions, it is prone to microcracks or micro annular gaps, leading safety concerns. To study the strength characteristics and energy evolution of oil well cement stone under three-dimensional stress states, a series of true-triaxial compression tests were carried out. Based on the experimental results, the mechanical properties and energy evolution mechanisms during the true-triaxial compression process of cement stone under different stress states were explored. The research results indicate that there is an intermediate principal stress effect on the true-triaxial compressive strength of cement stone specimens. When the minimum principal stress remains constant, the strength of cement stone first increases and then decreases with the increase of intermediate principal stress. When the minimum principal stress is 5MPa, the specimen exhibits brittle failure, with ε1 at peak stress approximately 0.015. As the minimum principal stress increases to 20MPa, the specimen gradually transits to plastic failure, with ε1 at failure exceeding 0.04. The type of failure is determined by the energy ratio at the time of failure; when there is more elastic energy, it results in brittle failure, otherwise it leads to ductile failure. Due to the high porosity of cement stone samples, most of the input energy is converted into dissipated energy. At minimum principal stresses of 5MPa, 10MPa and 20MPa, the ratio of peak dissipated energy to total input energy fluctuates around 0.55, 0.73 and 0.85, respectively. There is a strong linear relationship between the peak elastic energy, peak dissipated energy, and total input energy, with fitting correlation coefficients of 0.847 and 0.996, indicating that the cement stone sample also follows the linear energy storage law. A strength criterion for cement stone was established based on the modified Lade criterion, which is more suitable for describing the strength of high porosity materials compared to traditional strength criteria. Finally, based on the interaction mechanism between energy accumulation and energy dissipation, a nonlinear model of energy evolution in cement stone was established. The results of the theoretical model are in good agreement with the experimental data.

Dynamic stress analysis of rock joints under railway loading

The proper estimation of stresses generated by the passage of a train is fundamentally important to the serviceability and longevity of railways, and yet very limited knowledge is available where the track substructure is built on a jointed rock mass. The present study introduces an analytical solution for estimating the ground stresses arising from moving wheel loads, causing a change in the three-dimensional stress state in the track formation, in relation to the stress variation with depth and along the longitudinal track section – that is, the direction of train passage. Based on 21 case histories, an array of field measurements and numerical simulations covering a wide range of freight tonnage, train speeds and different formation conditions were considered to validate the proposed analytical solution. The proposed methodology (analytical solution) was then applied to a jointed rock subgrade to determine the normal and shear stresses acting along a specific discontinuity plane. The main analytical outcome demonstrates that the orthogonal vertical and shear stresses present different and phase-shifted history plots for homogeneous ground conditions with principal stresses rotation. However, conversely for a jointed subgrade, the normal and shear stresses along the discontinuity have the same history plot pattern and are in phase. As a practical guide, the results from this study would help to define which cyclic loads should be applied in laboratory tests to simulate realistic traffic patterns of trains travelling over a jointed rock subgrade.

Brittle rock mass failure in deep tunnels: The role of infilled structural plane with varying dip angles

Filled rock discontinuities are commonly encountered in underground engineering excavations, and their mechanical behavior has a significant impact on the stability of surrounding rocks. However, their performance under three-dimensional stress conditions has rarely been explored in the past. In this study, we summarized the typical failure characteristics of filled structural planes in deep-buried tunnels and conducted laboratory true triaxial tests on granite specimens to investigate the effect of dip angle and infill thickness on the failure characteristics of high-stress brittle surrounding rocks. The study monitored the characteristics of vertical stress-displacement, acoustic emission, acceleration, and failure process of granite specimens during the loading process and revealed the failure mechanism of brittle rock masses under the influence of infill thickness and structural plane dip angle. The results showed that the dip angle of structural planes has a significant influence on the failure mode of rock masses in deep tunnels. There exists a critical range of dip angles, within which the rock mass undergoes shear slip failure along the pre-existing structural plane, manifested by rock spalling near the structural plane. Beyond this critical range, the rock mass experiences strain-type brittle failure, characterized by violent rock ejection. Furthermore, the filling of structural planes can change the failure mode of rock masses and lead to more severe damage when the friction coefficient of the structural plane falls below a specific threshold. Additionally, it was revealed that a perpendicular ejection force exists on the free surface of the rock mass under loading, which is approximately vertical to the normal direction and is the primary factor responsible for the formation of vertical parallel cracks within the rock mass. The findings of this work are of paramount importance for comprehending the failure characteristics of deep-buried tunnels containing infilling joints.

Adsorption–desorption characteristics of coal-bearing shale gas under three-dimensional stress state studied by low field nuclear magnetic resonance spectrum experiments

The micro-scale gas adsorption–desorption characteristics determine the macro-scale gas transport and production behavior. To reveal the three-dimensional stress state-induced gas adsorption–desorption characteristics in coal-bearing shale reservoirs from a micro-scale perspective, the coal-bearing shale samples from the Dongbaowei Coal Mine in the Shuangyashan Basin were chosen as the research subject. Isothermal adsorption–desorption experiments under three-dimensional stress state were conducted using the low field nuclear magnetic resonance (L-NMR) T2 spectrum method to simulate the in-situ coal-bearing shale gas adsorption–desorption process. The average effective stress was used as the equivalent stress indicator for coal-bearing shale, and the integral of nuclear magnetic resonance T2 spectrum amplitude was employed as the gas characterization indicator for coal-bearing shale. A quantitative analysis was performed to examine the relationship between gas adsorption in coal-bearing shale and the average effective stress. And a quantitative analysis was performed to examine the relationship between the macroscopic and microscopic gas quantities of coal-bearing shale. Experimental findings: (1) The adsorption–desorption process of coal-bearing shale gas follows the L-F function model and the D-A-d function model respectively with respect to the amount of gas and the average effective stress. (2) There is a logarithmic relationship between the macroscopic and microscopic gas quantities of coal-bearing shale during the adsorption–desorption process. This quantitatively characterizes the differences in the curves, which may be related to the elastic–plastic deformation, damage and fracture of the micropores in coal-bearing shale, as well as the hysteresis of gas desorption and the stress field of the gas occurrence state.

A Multiscale Method to Develop Three-Dimensional Anisotropic Constitutive Model for Soils

A multiscale method is presented to develop a constitutive model for anisotropic soils in a three-dimensional (3D) stress state. A fabric tensor and its evolution, which quantify the particle arrangement at the microscale, are adopted to describe the effects of the inherent and induced anisotropy on the mechanical behaviors at the macroscale. Using two steps of stress mapping, the deformation and failure of anisotropic soil under the 3D stress state are equivalent to those of isotropic soil under the triaxial compression stress state. A series of discrete element method (DEM) simulations are conducted to preliminarily verify this equivalence. Based on the above method, the obtained anisotropic yield surface is continuous and smooth. Then, a fabric evolution law is established according to the DEM simulation results. Compared with the rotational hardening law, the fabric evolution law can also make the yield surface rotate during the loading process, and it can grasp the microscopic mechanism of soil deformation. As an example, an anisotropic modified Cam-clay model is developed, and its performance validates the ability of the proposed method to account for the effect of soil anisotropy.

Finite Element for the Analysis of Massive Reinforced Concrete Structures with Cracking

The solid finite element has been developed for the calculation of massive reinforced concrete structures with cracks. When constructing a finite element in the compression-compression-compression mode, the Willam Warnke failure criterion was used. The tensile behavior of concrete was assumed to be linear before the crack initiation. Modern building codes require non-linear calculations of concrete and reinforced concrete structures, taking into account the real properties of concrete and reinforcement. In this regard, a technique has been developed and a solid finite element has been built, adapted to the PRINS software, which makes it possible to perform calculations of massive reinforced concrete structures, taking into account their actual work. Development of a method for calculating reinforced concrete structures under conditions of a three-dimensional stress state, taking into account the brittle fracture of compressed concrete and cracking in tensile concrete. Based on this technique, the implementation of a solid finite element in the PRINS software. To verify the developed finite element, a series of test calculations of a beam under three-point bending conditions was carried out. Comparison of the calculation results with the data of experiments by the authors confirmed the high accuracy and reliability of the results obtained. The developed solid finite element included in the PRINS software can be effectively used by engineers of design and scientific organizations to solve a wide class of engineering problems related to the calculations of massive reinforced concrete structures.

Establishment and engineering application of viscoelastic-plastic constitutive laws for creep modeling in interbedded rock masses

In order to study the creep behavior of the surrounding rock of the interbedded rock mass tunnel considering the time-dependent deformation, this paper proposes a viscoelastic-plastic seven-element model considering the stress threshold, and derives and establishes its creep equation under three-dimensional stress state. At the same time, the UMAT (User-defined Material) subroutine of the model is developed based on the ABAQUS software. The rationality of the seven-element model and the effectiveness of the subprogram are verified by rheological test results. Finally, the UMAT subroutine is applied to the numerical simulation of the creep behavior of soft and hard interbedded rock tunnels with different rock inclinations (α). The results show that the different rock inclination angles have different effects on the horizontal displacement of the ground above the tunnel, settlement deformation, and the convergence of the tunnel section. With the increase of the rock inclination (0 ≤ α ≤ 90°), the horizontal displacement of the surface on both sides is antisymmetric. When α is 0°, 45° and 90°, the horizontal displacement on both sides is equivalent. Surface subsidence decreases and then increases slowly. When α is 0° and 45°, the surface subsidence is the largest (12.4 mm) and the smallest (11.1 mm), respectively. The convergence values of the tunnel section change according to different parts of the tunnel. The convergence values of the arch top and arch bottom decrease continuously, and their maximum convergence values are 23.4 mm and 17.3 mm, respectively. The change trend of the arch waist and arch shoulder convergence values is the opposite. When α is 0°, the convergence value of the arch waist is maximum (3.5 mm). When α is 15°, the convergence value of the arch shoulder is the maximum (2.0 mm).

Failure characterization of fully grouted rock bolts under triaxial testing

Confining stresses serve as a pivotal determinant in shaping the behavior of grouted rock bolts. Nonetheless, prior investigations have oversimplified the three-dimensional stress state, primarily assuming hydrostatic stress conditions. Under these conditions, it is assumed that the intermediate principal stress (σ2) equals the minimum principal stress (σ3). This assumption overlooks the potential variations in magnitudes of in situ stress conditions along all three directions near an underground opening where a rock bolt is installed. In this study, a series of push tests was meticulously conducted under triaxial conditions. These tests involved applying non-uniform confining stresses (σ2 ≠ σ3) to cubic specimens, aiming to unveil the previously overlooked influence of intermediate principal stresses on the strength properties of rock bolts. The results show that as the confining stresses increase from zero to higher levels, the pre-failure behavior changes from linear to nonlinear forms, resulting in an increase in initial stiffness from 2.08 kN/mm to 32.51 kN/mm. The load-displacement curves further illuminate distinct post-failure behavior at elevated levels of confining stresses, characterized by enhanced stiffness. Notably, the peak load capacity ranged from 27.9 kN to 46.5 kN as confining stresses advanced from σ2 = σ3 = 0 to σ2 = 20 MPa and σ3 = 10 MPa. Additionally, the outcomes highlight an influence of confining stress on the lateral deformation of samples. Lower levels of confinement prompt overall dilation in lateral deformation, while higher confinements maintain a state of shrinkage. Furthermore, diverse failure modes have been identified, intricately tied to the arrangement of confining stresses. Lower confinements tend to induce a splitting mode of failure, whereas higher loads bring about a shift towards a pure interfacial shear-off and shear-crushed failure mechanism.

True-triaxial simulation of sandstone with full range of σ2 based on the Rigid-Body-Spring method

Accurately simulating the mechanical properties of rock under true-triaxial stress conditions (σ1 > σ2 > σ3) is crucial for understanding deep underground rock masses. Although discrete approaches have advantages in describing rock fracturing behavior, further development seems necessary for quantitatively simulating the intermediate principal stress σ2 effects on rock strength. In this paper, we study the failure mechanisms of microscopic contacts of a discrete approach, the modified Rigid-Body-Spring Method (mRBSM), in response to σ2. Numerical results indicate that the model calibrated by axisymmetric-triaxial tests underestimates the strength of Yunnan sandstone when the stress state approaches σ2 = σ1. To address this discrepancy, we propose a new microscopic contact failure criterion that takes into consideration not only the normal and shear stresses on the contact but also the average Cauchy stress tensor around the contact. Results show that the mRBSM with the new failure criterion is able to reproduce the true-triaxial strength of Yunnan sandstone more accurately than the original model, especially under high σ2 and σ3 conditions. In addition, the corresponding stress–strain curves and failure modes under the influence of σ2 still show reasonable trends under true-triaxial stress conditions. This study indicates that when simulating rock failure under three-dimensional stress conditions using the rigid-body-spring system, the contact failure criterion might be influenced by the local stress state.

Failure characteristics and energy distributions of surrounding rock in hard rock tunnels subjected to different three-dimensional stress conditions

To study the failure characteristics of surrounding rock near an excavated tunnel under different three-dimensional stress conditions, a TRW3000 rock true triaxial electro-hydraulic servo mutagenesis testing system was used to conduct triaxial loading experiments on granite specimens with a circular hole. A high-definition micro-camera system was used to record the failure process near the circular hole. The results show that spalling failure occurs near the holed specimen under low-stress conditions, and rockburst appears around the holed specimen under high-stress conditions. Increasing the horizontal stress can prevent the occurrence of rockburst. The evolution of strain energy density (SED) in the surrounding rock of the holed specimen under different stress conditions was simulated. The findings show that the distribution of SED around the circular hole decreases with an increase in horizontal stress. The SED distributions around the circular hole are basically consistent with the damage distributions in the experiments, namely, butterfly-shaped, ear-shaped, and ring-shaped.

Evaluation of coal mines’ rock mass gas permeability in the equivalent stress zone

Purpose. Based on a comparative analysis of the internal mechanism of shape change of rock samples, which were loaded in specified deformations mode, and geomechanical and gas-dynamic processes in coal mass, to establish a causal link between these phenomena. To qualitatively characterise their gas permeability as a function of the rock’s volume expansion. To justify the possibility of using a full “stress-strain” diagram as a technogenic methane deposit formation model and its spatial localisation. Methodology. Theoretical research is based on using the solid mechanic constitutive principles and results of studying the rock samples failure in the mode of specified strains. Findings. The ability to use a full “stress-strain” diagram for detecting and localising methane reservoirs during the coal seams excavation was proved during the research. It was found that the compaction threshold coincides with the bearing pressure maximum in front of the longwall face. This area corresponds to the rock mass with minimal porosity and minimal filtration, which allows considering it as an envelope of an artificial gas deposit. Regularities that connect the three-dimensional equivalent stress state with the final gas permeability of the gas-saturated coal mass were obtained. These data allow creating a predictive numerical geomechanical model of methane migration paths. Originality. The ability to use a full “stress-strain” diagram in the controlled strain mode for numerical modelling of gas permeability of a methane-saturated coal mass during the mining of coal seams and the determination of technogenic gas deposit boundaries are justified. Dependences of the current and final gas permeability on the rock’s mechanical characteristics in a post-peak strain state are obtained. Practical value. Functional dependencies based on geomechanical models are obtained that allow the identification and localisation of technogenic methane reservoirs in mines during coal seam excavation, with subsequent utilisation of the extracted gas. Furthermore, methane removal enhances mining safety by reducing the risk of gas dynamic phenomena while decreasing gas emissions into the atmosphere contributes to reducing the greenhouse effect.

A novel meso-damage constitutive model of rock under true triaxial stress with three-dimensional cracking strength, threshold and closure effect

Deep underground engineering is in a true three-dimensional stress state, and the adjustment of the three-dimensional stress state caused by engineering excavation will induce the fracture or even instability of the surrounding rock. However, three-dimensional mechanical model research suitable for the stability analysis of deep surrounding rock is very scarce. Therefore, a series of tests under different true triaxial stresses on two rocks (rhyodacite and marble) were conducted, and the characteristic strength (crack stable propagation initiation stress, crack unstable propagation initiation stress and peak strength) and deformation characteristics were further analyzed. After that, using the Lemaitre strain equivalence hypothesis and rock statistical damage theory, a new statistical damage constitutive model at true triaxial stress states was proposed, which introduced the three-dimensional strength criterion Modified Wiebols Cook to characterize the three-dimensional strength of the rock microelement. Therefore, the intermediate principal stress can be reasonably considered. The damage threshold, initial compaction effect and residual strength of the rock microelement at different true triaxial stress conditions were also considered. Then the relationships between the proposed model parameters and σ2 and σ3 were analyzed. Furthermore, sensitivity analysis of the influence of parameters m and F0 in proposed model on the shape of rock stress–strain curve and peak strength was also investigated. The comparison between the results predicted by proposed model and the experimental data shows that the new model established in this study can well simulate the prepeak and postpeak deformation characteristics of rock and the intermediate principal stress effect under true triaxial stress conditions.

Three-dimensional spatial stress state of highway subgrade under vehicle load: experimental evidence and evaluation model

ABSTRACT In this study, field tests of highway subgrade dynamic response were conducted using a self-developed regular dodecahedron soil pressure sensor (RDSPS). The RDSPS can be used to quantify the total stress of the subgrade soil element. The effects of different axle loads, speeds, and vehicle types on the dynamic response of the subgrade were investigated. The spatial distribution law of the vertical dynamic stress of the highway subgrade was examined. An estimation model for the vertical dynamic stress and the influence depth of vehicle load was developed. The results suggest that, under the action of a moving traffic load, the principal stress axis continues to rotate. Both vehicle speed and axle load have an impact on the dynamic response of the subgrade; however, for newly built highways, the impact of the vehicle axle load on the dynamic stress is more significant. The vertical dynamic stress increases linearly with the speed or axle load of the vehicle. The attenuation law of the vertical dynamic stress caused by different types of vehicles on the subgrade along the depth direction is basically consistent, and the attenuation speed changes from fast to slow, with the fastest attenuation rate in the base layer.

Research on the Modulus Decay Model under a Three-Dimensional Stress State of Asphalt Mixture during Fatigue Damage

Fatigue damage can develop within asphalt pavement due to the continuous impact of driving loads and natural elements. Understanding the process of asphalt mixtures’ fatigue damage is crucial for guiding the design, maintenance, and repair of asphalt pavement. This research aims to establish a model that characterizes the mixtures’ modulus decay under a three-dimensional (3-D) stress state. Firstly, asphalt mixes were subjected to direct tensile (DT), indirect tensile (IDT), unconfined compressive (UC) strength and fatigue tests, and the resulting data were analyzed. Then, modulus decay models under DT, IDT, and UC conditions were established, and the modulus decay patterns under the three loading modes were compared and analyzed. Finally, using the fatigue stress strength ratio Δ (a fatigue resistance index for asphalt mixtures that takes into account the impacts of stress state and loading rate), a unified characterization model for asphalt mixes’ modulus decay under a 3-D stress state was created. According to the study’s findings, asphalt mixes’ modulus decay during fatigue damage exhibits obvious nonlinear characteristics. While the asphalt mixes’ modulus decay law with various loading modalities is similar under the same conditions, the decay rate may differ. Essentially, the speed of the modulus decay of a certain asphalt mixture primarily depends on the value of Δ during service. A larger Δ indicates a faster modulus decay. This study offers a theoretical foundation for the conversion from material fatigue damage to structural fatigue damage, which is vital for enhancing the asphalt pavements’ construction quality and longevity.

Models of nonlinear deformation of concrete in a triaxial stress state and their implementation in the PRINS computational complex

Modern construction standards and regulations prescribe to carry out calculations of concrete and reinforced concrete structures in a nonlinear formulation with account of the real properties of concrete and reinforcement. However, the most of finite-element program complexes cannot perform such calculations in a nonlinear formulation with account of plastic deformations of concrete and reinforcement. To solve this problem, a methodology has been developed and a solid finite element adapted to the PRINS computing complex has been created, which made it possible to perform calculations of reinforced concrete structures considering their actual work. The aim of the study - development and implementation of a method for calculating reinforced concrete structures under conditions of a three-dimensional stress state, considering both brittle fracture and elastic-plastic deformation of concrete. A finite-element methodology, algorithm, and program for calculation of massive reinforced concrete structures with account of plastic deformations of concrete have been presented. The methodology is based on the modified Willam and Warnke strength criterion supplemented with the flow criterion. Two models of volumetric deformation of concrete have been regarded: the elastic model at brittle failure and the ideal elastoplastic model. An eight-node finite element with linear approximating functions of displacements implementing the mentioned deformation models is created. Verification calculations of a massive concrete structure in three-axial compression testify to the accuracy and convergence of the developed finite elements. The PRINS can be effectively used by engineers of designing and scientific organizations to solve a wide class of engineering problems related to calculations of building structures.

Investigation on fatigue characteristics of polyphosphoric acid modified high dosage bio-asphalt mixture under three-dimensional stress state

Polyphosphoric acid modified high dosage bio-asphalt (PMBA) could provide a new direction for saving asphalt resources. The fatigue performance of asphalt mixture is one of the critical factors that should pay more attention in achieving a life cycle cost-effective. This study aims to investigate the strength and fatigue performances of the PMBA mixture. The stress states of mixture tests include indirect tension, direct tension, and unconfined compressive, respectively. The S-N fatigue equations, Desai yield surface, and normalized fatigue equation were determined. At the same loading rate, the strength values of indirect tension and direct tension are close, while the strength of unconfined compressive is much larger than theirs. The S-N fatigue equations parameter obtained based on stress levels vary significantly under different stress states, with unconfined compressive being much more significant than indirect tension and direct tension. The Desai strength yield surface gradually expands outward with the increase of loading velocities. The diffusion rate decreases as the loading velocities increase. According to the Desai yield surface, the new fatigue stress path and stress ratio were defined. The normalized fatigue equation for PMBA mixture under various stress states is Nf=Δ-4.58R2=0.99. The normalized n value of PMBA asphalt mixtures is 4.58, 2.7% lower than that of SBS modified asphalt mixtures. This study is meaningful for applying high dosage bio-asphalt. The application of PMBA also provides new ideas for low carbon and environmental protection in road engineering.

A simple 3D elastoplastic constitutive model for overconsolidated soils based on the improved Hvorslev envelope

Soils encountered in geotechnical engineering exhibit a certain degree of overconsolidation and are in a three-dimensional (3D) stress state. The overconsolidated ratio (OCR) and the intermediate principal stress play a significant role in the strength and deformation of overconsolidated soils. By considering the variation of the peak stress ratio with OCR, and taking the zero-tension line and the critical state line as asymptotes, a continuous and smooth Hvorslev envelope is deduced to describe the peak strength of overconsolidated soils. Based on the proposed Hvorslev envelope, a simple 3D elastoplastic constitutive model for overconsolidated soils is directly established through the characteristic stress method. The constitutive relation of the proposed model is presented by the calculation of the total strain increment. All material parameters have clear physical meanings and can be determined by conventional laboratory tests. The isotropic compression path, the triaxial drained shearing path with constant p, the triaxial undrained shearing path, and the true triaxial drained shearing path with constant p and θ are adopted to demonstrate the essential features of the proposed model. The effectiveness of the proposed model is confirmed by comparing the experimental data from Lower Cromer till, Kaolin clay, and Fujinomori clay with corresponding model predictions.

A novel anisotropic deterioration mechanical model for rock ductile–brittle failure under 3D high-stress and its application

After the excavation of deep underground engineering, the three-dimensional (3D) stress state redistributes, leading to a significant change in the mechanical properties of the surrounding rock; however, the mechanical properties and parameter evolutions of rock under true triaxial stress are not clear, and theoretical mechanical models characterizing the above rock properties for surrounding rock stability analysis are very deficient. Therefore, in this research, a series of true triaxial compression tests and cyclic loading and unloading tests were carried out to investigate 3D stress-induced anisotropic behaviours, ductile–brittle characteristics and mechanical parameter evolutions. The determination methods of the mechanical parameters of Young’s modulus Ei, cohesion c, internal friction angle φ and dilation angle ψ in the rock failure process under true triaxial stresses were established, and the evolution functions of the mechanical parameters were proposed. A 3D degradation mechanical model for rock ductile–brittle and anisotropic behaviours under true triaxial stress was established. A numerical calculation program of the proposed model was implemented, and the simulation results agreed with the experimental results. The engineering application of the proposed model was further investigated with an improved index of rock fracture degree for the localized asymmetric failure of deep surrounding rock after excavation.

Study of the Tensile and Bonding Properties between Cement-Based Grout Materials and High-Strength Bolts

This study investigated the tensile and bonding properties between cement-based grouting materials (CBGM) and high-strength bolts. The associated failure mechanism, load-slip curve, ultimate pull-out load and bond stress were also studied. The effects of anchorage length and square steel tube restraint on the bonding properties were explored on the basis of 24 specimens used in central pull-out testing, and a bond stress–slip constitutive relationship model between high-strength bolts and CBGM was proposed. The results indicate that with the increase in the anchorage length of high-strength bolts, the failure modes of specimens can be divided into three types: the fracture failure of high-strength bolt that occurred when the anchorage lengths ranged from 18 d to 31 d, the specimens that experienced splitting failure with the constraint of square steel tube when the anchorage length was less than 15 d and the high-strength bolt that experienced pull-out failure without the constraint of square steel tubes. When the high-strength bolt experiences tensile failure, the ultimate pull-out load remains constant and the bond stress decreases as the anchorage length of high-strength bolts increases. Due to the lateral constrained effect of the square steel tube, the CBGM embodies a three-dimensional stress state, which can delay the generation and development of internal cracks and enhance the bond strength. A calculation formula was proposed to determine the bond strength between high-strength bolt and CBGM, and a constitutive model of the bond stress–slip constitutive relationship was modeled.

A novel three-dimensional M-K model by integrating GTN model for accurately identifying limit strains of sheet metal

To accurately identify the limit strains and improve the prediction accuracy of forming limits for sheet metal, a novel three-dimensional M-K model by integrating GTN model is developed and named 3D M-K-GTN model. By introducing the through-thickness normal stress and friction stress, the original M-K model is extended for adapting to the three-dimensional stress state. Furthermore, the empirical failure criterion is replaced by integrating the GTN model, and thus the limit strain is identified when the void volume fraction within the groove accumulates to a critical value. Besides, the voids nucleation and growth expressions are also re-derived and modified by embedding the friction stress and normal stress. Finally, the comparison between predicted forming limit curves and experimental limit strains of AA6016-T4 sheet indicates that the 3D M-K-GTN model solves the significant overestimation of prediction results under large strain paths, and always provides a high prediction accuracy for each strain path given.