Drucker-Prager Criterion Research Articles

Experimental Study on Damage Constitutive Model of Rock under Thermo-Confining Pressure Coupling

Underground engineering frequently encounters the challenges of high temperatures and high confining pressures. The combined effects of temperature and confining pressure can significantly alter the mechanical properties of rock. Evaluating the impact of these factors on rock is a crucial aspect of engineering. This study investigates the mechanical properties of rocks under various temperature and stress conditions, focusing on the deformation and failure characteristics of four typical rock types: granite, red sandstone, gray sandstone, and shale under temperature-confining pressure coupling. The results indicate that high temperatures cause internal structural damage and crack propagation in rocks, leading to a reduction in compressive strength and elastic modulus. Conversely, high confining pressures can inhibit crack propagation and enhance rock deformation capacity. Additionally, significant differences were observed in the mechanical responses of different rocks; red sandstone and shale predominantly exhibited shear failure under the coupled effects of temperature and confining pressure, whereas granite and gray sandstone exhibited bulging failure. Based on the experimental results, an elastic modulus fitting model considering the temperature-confining pressure coupling effect was proposed, and the parameters of the Drucker–Prager criterion were modified. A constitutive model was constructed to accurately reflect the stress–strain state of rocks under the coupled effects of temperature and confining pressure. The constitutive model results show good agreement with the experimental findings.

Dry granular column collapse: Numerical simulations using the partially regularized [formula omitted]-model via stabilized finite elements and phase field formulation

We revisit the gravitational collapse of a 2D column of dry granular material surrounded by air, using a continuum mechanics approximation. By employing the Cahn-Hilliard phase-field equation as an interface capturing technique and by coupling it with the Cauchy equation, we numerically simulate this multiphase system, eliminating the need for any ad-hoc numerical adjustment to prevent the finger formation of light fluid between the material and the solid boundary due to the no-slip boundary condition. We implement the μ(I)-rheology in our stabilized Finite Element method, highlighting the presence of instabilities when using this constitutive law. Our study is characterized by three main goals. First, we address the instability issue by implementing the partially regularized formulation of the μ(I)-rheology proposed by Barker and Gray (2017). An important outcome is that using shock-capturing terms in the momentum equation can significantly smooth these oscillations by adding dissipation in the direction of the gradients. Second, we systematically study the fluid dynamics under realistic conditions. Our results accurately replicate the material dynamics during collapse, confirming three distinct stages: free-fall, spreading, and cessation. We identify two regions in the material during the spreading phase: a quasi-static zone with negligible velocities and deformations, and a flowing layer exhibiting high shear rates. These observations closely align with experimental data. Additionally, we examine the evolution of the yielded/unyielded regions based on the Drucker-Prager criterion, and we also explore an empirical criterion, based on a critical value of the velocity norm, that satisfactorily separates these regions. Finally, we perform an extensive parametric study covering a wide range of rheological parameters.

The influence of mineral inclusion on the effective strength of rock-like geomaterials

The influences of microstructure on the macroscopic mechanical behavior of a composite with a porous matrix reinforced by mineral inclusions are investigated in the present work by both numerical and theoretical methods. The mineral inclusions are embedded at the mesoscopic scale and much bigger than the pores which are located at the microscopic scale. In order to consider the properties of the studied rock-like geomaterials, such as the dissymmetry between tension and compression, the solid phase at the microscopic scale is assumed to obey to a Drucker–Prager criterion. Based on the studied microstructure, the Fast Fourier Transform (FFT) based numerical method is firstly adopted to investigate the macroscopic plastic yield stress of the studied composite. Different microstructure having different volume fraction of inclusion, micro-porosity and frictional coefficient of the solid phase are considered. Based on these numerical results, the existing theoretical yield criterion is estimated. One finds that it should be improved for the case of a microstructure having a high inclusion content. Then, a new macroscopic yield criterion is constructed in the present work by using the modified secant method. This criterion ameliorates fundamentally the one proposed in Shen et al. (2013), specially for the case of a deviatoric loading. It is then estimated and validated by comparing with the FFT based numerical results which were carried out in this work with different volume fractions of heterogeneous phase.

A length-free phase field model for epoxy adhesive materials based on modified Drucker–Prager criterion

Characterizing the cracking behavior of epoxy adhesive material is of great importance for composite structures. Recently, the phase field model has emerged as the preferred method for predicting the crack propagation. This paper presents a length-free phase field model considering the characteristics of epoxy adhesive materials. The proposed model is formulated within a framework that decomposes the phase field fracture driving force, ensuring that it adequately considers the tension/compression asymmetry of the epoxy adhesive materials. The effects of hydrostatic pressure, deviational stress, and friction are incorporated into the model by employing a modified Drucker-Prager failure criterion. The failure criterion is included in the phase field model using the effective stress invariants. Furthermore, the internal length scale, previously considered to lack real physical meaning, is reformulated based on material strength. This model addresses the challenge of measuring the internal length of epoxy materials. It serves as a valuable reference for applying phase field models to epoxy materials. The accuracy of the model is validated through an L-shaped plate simulation, and an attempt using the model to predict the crack development in thick epoxy adhesive joints is undertaken. Several representative samples are simulated, including three double cantilever beams with different configurations. The numerical results demonstrate a good correlation with experimental data, highlighting the potential of the model in simulating the crack propagation in epoxy adhesive joints.

Hop-to-Hug algorithm: Novel strategy to stable cutting-plane algorithm based on convexification of yield functions

Numerical challenges, incorporating non-uniqueness, non-convexity, undefined gradients, and high curvature, of the positive level sets of yield function are encountered in stress integration when utilizing the return-mapping algorithm family. These phenomena are illustrated by an assessment of four typical yield functions: modified spatially mobilized plane criterion, Lade criterion, Bigoni-Piccolroaz criterion, and micromechanics-based upscaled Drucker-Prager criterion. One remedy to these issues, named the "Hop-to-Hug" (H2H) algorithm, is proposed via a convexification enhancement upon the classical cutting-plane algorithm (CPA). The improved robustness of the H2H algorithm is demonstrated through a series of integration tests in one single material point. Furthermore, a constitutive model is implemented with the H2H algorithm into the Abaqus/Standard finite-element platform. Element-level and structure-level analyses are carried out to validate the effectiveness of the H2H algorithm in convergence. All validation analyses manifest that the proposed H2H algorithm can offer enhanced stability over the classical CPA method while maintaining the ease of implementation, in which evaluations of the second-order derivatives of yield function and plastic potential function are circumvented.

Thermal-mechanical modelling on cumulative freeze-thaw deformation of porous rock in elastoplastic state based on Drucker-Prager criterion considering dilatancy

The comprehensive understanding of freeze-thaw (F-T) deterioration and deformation properties of rock is a critical basis for frost damage prevention on cold regional rock engineering. In freezing process, plastic strain generates in rock when the stress induced by pore ice pressure (PIP) becomes higher than yield strength, and unrecoverable deformation accumulates in rock under multiple F-T cycles. Therefore, this study presents a thermal-mechanical modelling on cumulative F-T deformation of rock under cyclic F-T action. In the modelling of mechanical action, the microporomechanics theory is applied with a spherical element model in the derivation of equilibrium equations of rock, and the Drucker-Prager criterion is utilized with dilatancy effect of rock considered to describe unrecoverable plastic deformation cumulated in rock under multiple F-T cycles. The thermal process is modelled with unfrozen water content and latent heat accounted. The proposed thermal-mechanical modelling method is validated based on experiments and numerical simulations of F-T deformation of sandstone, and it is able to simulate the temperature and F-T deformation of rock in acceptable precision. The numerical distributions of temperature, PIP, maximum principal strain and displacement are exhibited and show that the plastic region begins to develop in rock matrix with PIP exceeding the elastoplastic critical pressure when temperature drops to about −2 °C and grows rapidly when temperature drops from −2 to −6 °C. Several F-T cycles later, large unrecoverable plastic strain remains in sandstone though PIP dissipates after thawing. The cumulative F-T deformation, porosity increment and maximum of plastic volume ratio are much greater under condition with freezing temperature of −20 °C, compared with condition −10 °C. Besides, as cycle number increases, the plastic volume ratio becomes smaller and its decrease rate attenuates.

Numerical analysis by comparing the Drucker-Prager and Mohr-Coulomb behaviors to optimize the best sites for constructing a new tunnel adjacent to the old one

The tunnels are one of the most significant infrastructures for facilitating mobility in mountainous terrain. It has several benefits, including shortening distances, time, and ease of transportation. To make it easier, developed countries create several tunnels close to each other. Engineers face the necessity of building these new tunnels near existing ones; this infrastructure must be safe. In this paper, for this numerical study, we used the OptumG2 software to analyze the position of a new tunnel created in six locations: bottom, top, left (two places at 12 m and 32 m), and right (two places at 12 m and 32 m) by the finite element method. This study was analyzed using two criteria: the equivalent of Drucker Prager and Mohr Coulomb. We are interested in the total field of the displacement, the vertical displacement of the model, the displacement, bending moment, shear, and normal force of the concrete lining. The findings of this numerical analysis using the Drucker Prager and Mohr Coulomb behaviors to determine the ideal locations for building a new tunnel next to an existing one show that the new bottom tunnel might have a significant influence on the stability of the existing tunnel. It was also observed that the maximum displacement present when we add a new tunnel at the bottom meets the Drucker-Prager criteria.

Dynamic tensile mechanical properties of red sandstone under different pulse widths and amplitudes: Brazilian disk experiment and macroscopic and microscopic analysis

To investigate the dynamic tensile mechanical properties of red sandstone under varying impact pulse widths and amplitudes, we employed the response surface methodology to examine the dynamic response characteristics of sandstone under different bullet lengths and impact velocities. Subsequently, a dynamic damage constitutive model for sandstone was developed. Macroscopic and microscopic features of sandstone were analyzed through digital image correlation (DIC) and scanning electron microscopy (SEM) experiments. Additionally, ANSYS software was utilized to analyze stress wave characteristics under controlled shock waves and the impact of geological factors on rock fracturing effects. The findings revealed the following. First, the length and impact velocity of bullets exhibit an interactive effect on the tensile response characteristics of sandstone. Peak load, energy consumption rate, and energy density display a positive correlation with bullet velocity. Second, according to the DIC results, the fractal dimension of the crack is negatively correlated with the length of the bullet during equal energy impacts. Third, microscopic failure modes included fractures along particle cementation and through particles, with section roughness decreasing as bullet length increased under equivalent energy impacts. Fourth, a dynamic damage constitutive model (R2 ≥ 0.84) was established based on the Zhu-Wang-Tang (Z-W-T) model and Drucker-Prager (D-P) criterion, clarifying model parameters and influence rules. Fifth, under controlled shock waves, the pulse width facilitates crack propagation, while the pulse amplitude initiates crack formation. Optimal rock breaking efficiency is achieved when stress waves exhibit a large pulse width and low radiation values, meeting specific threshold conditions.

Elastic–plastic criterion solution of deep roadway surrounding rock based on intermediate principal stress and Drucker–Prager criterion

AbstractExcavation‐induced rock failure and deformation near an underground opening boundary is closely associated with the intermediate principal stress, strain softening and rock mass dilation. By combining theoretical analysis and numerical simulation to explore the mechanical evolution of the roadway surrounding rock during excavation. The elastic–plastic criterion solutions for surrounding rock stress, displacement, and the plastic zone of a circular roadway were deduced by including the intermediate principal stress, strain softening and rock mass dilation, based on the Drucker–Prager criterion and the nonassociated flow rule. Furthermore, ABAQUS finite element software was utilized for numerical simulations to scrutinize the influence of mechanical parameters on stress redistribution, surface displacement, and plastic range. The results of the numerical simulations verify the reliability of the theoretical analysis and affirm the high consistency of the results with the theoretical solution. The research findings indicate that the strength of surrounding rock is significantly influenced by the intermediate principal stress, and the deformation and failure of rock mass are primarily impacted by rock mass dilation, while strain softening mainly affects the thickness of the fractured zone. Moreover, the study reveals that enhancing the residual strength of the surrounding rock can improve its resistance to deformation and failure. Furthermore, the excavation of the roadway is observed to increase the original rock stress in the surrounding rock, but increasing the ground support strength effectively controls the deformation and failure of surrounding rock. Ultimately, the research outcomes offer valuable references for engineering calculations and ground support design of surrounding rock in deep roadways.

Influence of advance direction on surrounding rock stability of main ramp under deep high ground stress

Due to the widespread use of trackless equipment in metal mines, the main ramp has become a major development roadway in large mines. Under deep high ground stress condition, the stability of the surrounding rock of the main ramp is subject to significant variability due to the change in the advance direction, leading to structural failure in different areas of the roadway. Because the support design scheme of surrounding rock of main ramp will not pay attention to the advance direction of roadway, leading to the key support area of the surrounding rock is not clear, resulting failure of the local supporting form occur frequently. Therefore, this paper adopts the combination of on-site investigation and numerical simulation to carry out the stability analysis of the surrounding rock under different advance directions of the roadway. Through on-site investigation, various failure modes of surrounding rock induced by the change of the advance direction of the roadway are summarized, and the key supporting area of surrounding rock of main ramp is determined according to the angle change between the roadway advance direction and the intermediate principal stress. The extent of the risk zone of the roadway surrounding rock is defined based on the Drucker-Prager criterion, which can be used as the new basis for the design of roadway support parameters. The results show that: As the angle between the advance direction and the intermediate principal stress increases, the degree of deformation of the surrounding rock decreases linearly, and the volume of the plastic and risk zones of the surrounding rock decreases nonlinearly. Based on the distribution of the strain energy density of the roadway surrounding rock, when the angle between the advance direction and the intermediate principal stress is small, the roadway surrounding rock is more prone to the floor heave phenomenon. When the angle between the advance direction and the intermediate principal stress is larger, the roadway surrounding rock is prone to the full-section failure phenomenon. Based on the block theory, when the advance direction is changed, the location and production of key blocks have significant variability, leading to the need for reasonable adjustment of the support parameters for the surrounding rock. The research results of this paper are of guiding significance for the optimization of the surrounding rock support scheme of the main ramp.

Compaction elastoplastic damage constitutive model of rock under hydro-mechanical coupling: A heuristic modeling method based on internal state variables theory

Based on the internal state variables (ISV) theory, this paper employs a heuristic modeling approach to investigate the compaction elastoplastic damage constitutive model of rocks under hydro-mechanical coupling. The heuristic modeling method demonstrated herein can seamlessly integrate with existing constitutive theories, facilitating a broader spectrum of research on hydro-mechanical coupling constitutive models. Furthermore, a numerical solution was conducted in the principal axes via stress spectrum decomposition, which can well reflect the distinct characteristics of compaction deformation and the influence of water pressure on rock stress distribution. Utilizing the classic Drucker-Prager (D-P) criterion, the modeling method and numerical algorithm were implemented to derive a hydro-mechanical coupling compaction elastic–plastic damage constitutive model. This contribution expands the application domain of the classic D-P criterion within the context of hydro-mechanical coupling environments. The rationality of the model was confirmed through compression test data under various water pressures and confining pressures. Water pressure increases nonlinear compaction deformation by changing the opening degree of microcracks under low confining pressure, which has not been paid enough attention in existing research. This study thoroughly considers the influence of water pressure on compaction deformation. The developed process of “heuristic modeling method-numerical method-model application” is helpful for enriching the constitutive theory of rock under hydro-mechanical coupling.

Steel Ladle Slag Zone Lining Optimization Considering Irreversible Material Behavior

The optimization of refractory linings is a complex task due to the many factors that influence the thermomechanical behavior. This study applies the multiparameter optimization method technique for order preference by similarity to the ideal solution to a commercially used slag zone lining of a secondary metallurgical steel ladle, considering the influence of different material models in a finite element (FE) simulation. The dataset is acquired by varying the working lining brick shape, initial expansion allowance (IEA), and isolation layer thickness. The maximum irreversible strain, joint opening at the hot face, and steel shell temperature are determined from the FE simulation results and used as the input values for the optimization. A larger circumferential brick dimension combined with low IEA is favorable for the von Mises creep model. The main contribution is from the IEA (98.53%). The optimized lining for the Drucker–Prager criterion has a brick with a small circumferential dimension and larger IEA. The main influencing factor is the brick shape (85.06%). The differences in the optimization results and factor contributions can be explained by the contribution of the constitutive models to the simulation results before and after the thermal shock.

Simulation of Plastic Deformation Failure of Tillage Tools Based on the Smoothed Particle Hydrodynamics Method

The problems of large deformations, failures, and fractures that agricultural tillage tools may encounter during the cultivation process has long been a concern in the field of agricultural machinery design and manufacturing. It is important to establish a more accurate numerical model to effectively predict tools’ plastic deformation failures and ductile fracture failures. This research develops a numerical model for predicting the plastic deformation failure and ductile fracture failure of agricultural tillage tools using the smoothed particle hydrodynamics (SPH) method and the Johnson–Cook constitutive model. The model uses the Drucker–Prager criterion to describe the elastic–plastic constitutive behavior of the soil, the von Mises criterion to describe the Johnson–Cook constitutive model of the tool, and the coupling condition with the Lennard-Jones repulsive force to describe the interaction between the tool and soil. The numerical results show that the proposed model can effectively simulate the interaction between the tool and soil, as well as the tool’s plastic deformation failure and ductile fracture failure during the agricultural cultivation process. It can also predict the variation trend of the cutting force of the tool. This helps to provide a new approach for the numerical simulation of such problems.

Concrete strength criteria for biaxial and triaxial state of stress

Based on the experiments conducted by the authors, a concrete strength criteria has been developed, which allows us to take into account biaxial and triaxial states of stress in strength calculations of massive concrete and reinforced concrete structures. The developed strength criteria is adapted to a spatial eight-node finite element (solid type) and implemented in the PrinCe programme. In order to conduct the developed criteria operational check, both experimental data and calculation results of other widely used strength criteria for concrete were compared. Thus, to verify the proposed criteria under triaxial compression, the calculation results were compared with experimental data as well as with the Willam-Warnke criterion and the modified Drucker-Prager criterion. The comparison shows that in the low hydrostatic stresses mode results of the above criteria converge both with each other and with the experimental data. In the medium hydrostatic stress mode , the criteria proposed by the authors and the Willam-Warnke criterion show close results, while the modified Drucker-Prager criterion, on the contrary, gives 20 % overestimation of concrete strength. Comparison of the results of concrete biaxial tests shows good agreement with the criteria developed by the authors. Thus, for heavy coarse concrete the differences in strength values were not more than 3,5 %, for heavy fineconcrete and expanded clay concrete — not more than 4 %.

Study on mechanical characteristics and damage model of layered sandstone after high temperature action

Rock engineering, which includes slopes, tunnels, and mines, often encounters stratified rocks. These projects are also frequently exposed to special environments of high temperatures, such as deep underground or fire-related conditions. It is of significant importance to conduct research on the damage characteristics and constitutive models of stratified rocks under high-temperature conditions to accurately reflect the influences of rock structure characteristics, geological conditions, and load effects on the damage and deformation characteristics of rock engineering. Under five temperature conditions (20, 200, 400, 600, 800 ℃), the intact sandstone rock samples and the layered sandstone samples are subjected to high-temperature treatment, followed by triaxial compression tests. Based on existing research on statistical damage constitutive models for rocks, a high-temperature layered rock statistical damage constitutive model is established by introducing the Weibull distribution function and high-temperature, bedding, and load coupling damage variables, under the condition that the microelement strength follows the Drucker-Prager (D-P) criterion. The results indicate that the peak strength, damage threshold, elastic modulus, and longitudinal wave velocity show a "U"-shaped trend with an increasing bedding angle, with an opening upwards. As the temperature increases, the anisotropy of the rock initially increases and then decreases, with obvious ductile characteristics after the temperature reaches 600 ℃. The analysis of damage threshold, stress-strain curve, and macroscopic failure morphology shows that the 60° dipping angle sandstone is prone to undergo compressive-shear failure along the weak plane of bedding, exhibiting low toughness mechanical characteristics. Theoretical curves of the statistical damage constitutive model for high-temperature rock are in good agreement with the Triaxial shear test curve of sandstone, which indicates that the constitutive model can reflect the stress-strain process of layered sandstone after high-temperature action, and verifies the applicability of the model. This model does not include unconventional mechanical parameters and can reflect the ductility, brittleness, and strength characteristics with clear physical meanings. The findings of the study can offer theoretical support for computing and numerically modeling rock mechanics after high-temperature action.

A modeling study of elastoplastic rock failure regime based on finite discrete elements

Numerical simulation has become a pivotal method in geotechnical engineering for analyzing rock failure. However, traditional model lacks comprehensive elastoplastic rock characterization. Therefore, a new finite-discrete element model using the Drucker-Prager criterion and cohesive elements based on a traction-separation law to accurately simulate elastoplastic rock breaking has been built. A scratch test simulation is conducted to describe the mechanical response and failure characteristics of the rock during the cutting process. Rock failure characteristics derived from numerical simulations align closely with experimental observations. These simulations successfully replicate both the ductile and brittle regimes observed during scratch tests. Furthermore, they provide insights into the primary driving factors behind rock failure under different regimes. Variation of cutting force and specific energy obtained from the model are entirely consistent with experimental results, with the specific energy decreasing following a −0.5 power law, thus explaining the energy dissipation rule of the specific energy decrease. In brittle regime, crush zones and the resulting rough shearing surface are observed, plastic damage is also found, revealing that the cutter tip is the key structure influencing them. The model established in this paper holds significant importance for rock mechanical property evaluation and the optimization design of cutter structures.

Damage constitutive model of jointed rock mass considering structural features and load effect

Abstract Rock masses in underground engineering are usually damaged, which are caused by rock genesis and environmental stress. Studying the constitutive relationship between rock strength and deformation under loading is crucial for the design and evaluation of such scenarios. The new damage constitutive model considering the dynamic change of joint damage was developed to describe the behavior of rocks under loading in this work. First, considering the influence of jointed rock mass structural features in their entirety, the Drucker–Prager criterion and the Hoek–Brown criterion were combined. Second, based on the idea of macro–micro coupling, the calculation formulae of damage variables were derived. Finally, the damage constitutive model of the jointed rock mass was established, and the proposed model was fitted and compared with the test data. Results show that the variation rules for damage value and peak strength are opposite, and the stress–strain is highly sensitive to changes in the parameter s of the model. Moreover, the proposed model can accurately describe the effect of joint deterioration on the entire process of rock mass compression failure, which shows that the damage constitutive models are useful for evaluating the strength characteristics of jointed rock mass in engineering practice.

Research on the effects of heating and cooling processes on the mechanical properties of yellow rust granite

Geological hazards caused by high-temperature rocks cooling down after encountering water are closely related to underground mining and tunneling projects. To fully understand the impact of temperature changes on the mechanical properties of rocks, yellow rust granite samples were subjected to heating-natural cooling and heating-water cooling cycles to experimentally study the effects of these processes on the mechanical properties of the samples. The mechanism of the heating-cooling process on the macromechanical properties of the rock was discussed. Based on the Drucker-Prager criterion and Weibull distribution function, a damage variable correction factor was introduced to reflect the post-peak strain softening characteristics, and a thermo-mechanical coupled damage constitutive model of the granite was established. The results showed that in the natural cooling mode, the mechanical properties deteriorate significantly when the temperature exceeded 600 ​°C, and the failure mode changed from brittle failure to ductile failure. In the water cooling mode, the peak strength and deformation modulus increased at temperatures below 400 ​°C with an increase in the cycle number, while at 600 ​°C, the peak strength and elastic modulus notably decreased. The peak strain increased with the increase of the cycle number and temperature at all temperatures, and the failure mode of the granite tended to change from tensile failure mode to shear failure mode. The experimental results were used to validate the damage constitutive model. The shape parameter r and scale parameter S in the Weibull distribution function of the model were used as indicators to reflect the brittleness degree and peak strength. This study helps to understand the behavior of rocks in high-temperature environments, in order to prevent and mitigate potential geological hazards.

A rock creep model based on Atangana-Baleanu fractional derivative with coupled damage effects of pore water pressure, stress and time

Pore water pressure is an important factor affecting long-term rock failure and instability. On the basis of the assumptions that the rock microunit strength follows Weibull distribution and the rock failure compliances Drucker-Prager criterion, a pore water pressure-stress coupled damage (PSCD) variable is constructed in this paper. Considering the effect of pore water pressure, stress and time (PST), a pore water pressure-stress-time coupled damage (PSTCD) variable is put forward by introducing the damage variable considering time effect. Based on the characteristics and advantages of fractional differential theory in viscoelastic analysis of rock materials, one-dimensional and three-dimensional pore water pressure-stress-time coupled damage fractional (PSTCDF) creep models are proposed by Atangana-Baleanu (AB) fractional derivative. By fitting and analyzing the laboratory experimental data of rock creep under the combined action of PST, the relevant parameters for PSTCDF creep model can be deduced. The analysis of the goodness of fit and the generalization accuracy prove the rationality, validity and generalization of the proposed model. The sensitivity of the PSTCDF model is analyzed by different the stress level, pore water pressure, fractional derivative order and damage variable coefficient. The results show that the creep model changes flexibly when the model parameters change, and the specific creep state of rock can be reflected by reasonable selection of the model parameters.

Study on Rock Failure Criterion Based on Elastic Strain Energy Density

Uniaxial and five conventional triaxial compression tests were conducted on sandstone to obtain the evolution laws of the input energy density, elastic strain energy density, and dissipative energy density. The input and dissipative energy densities increased with increasing axial strain; the elastic strain energy density increased with increasing axial strain at the pre-peak stage and decreased after the peak. According to the linear change rule between the peak elastic strain energy density and confining pressure, the energy density failure criterion of sandstone was established, and the criterion has high precision and few parameters, and the parameters have clear physical meaning. Moreover, the expression of the energy density failure criterion was similar to the classical Hoek-Brown criterion, but its adaptability was more extensive. The strength calculation results for seven different rocks under different confining pressures calculated using the energy density failure criterion were consistent with the experimental values, and the calculation error was smaller than that of the Mohr–Coulomb criterion and Drucker–Prager criterion, verifying the accuracy and applicability of the criterion.