Modified Mohr-Coulomb Criterion Research Articles

Modified Mohr–Coulomb criterion for nonlinear strength characteristics of rocks

AbstractThe classical Mohr–Coulomb criterion has an extremely wide range of applications in the field of engineering geology, while it cannot accurately characterize rock failure under high confining stresses. Therefore, a nonlinear modification is implemented based on the Mohr–Coulomb criterion for achieving wider usage in all stress environments. The cohesion and internal friction angle are firstly calculated by regression based on the triaxial experiments of diorite and other typical rocks in different confining stress ranges, and the variation modes of cohesion and internal friction angle are then formalized and integrated into the classical Mohr–Coulomb criterion. Using the experimental results of 435 triaxial compression tests in existing publications, the predictions of the modified criterion are compared with those of the previous nonlinear criteria, which show that the predicted values of the modified criterion are in good agreement with the measured strength. Compared with the previous nonlinear criteria, the modified criterion has a wider application range and always maintains a higher accuracy.

A novel joining technology for metal and polymer sheets

This paper introduces a novel joining process for producing hybrid metal-polymer joints. The process, called hole hemming, involves deforming the metal sheet to establish a mechanical interlock with the polymer sheet, requiring neither heating nor auxiliary elements. The applicability of this process is tested for joining aluminum and polycarbonate (PC) sheets. Initially, an analytical design method is presented to achieve a connection without failure and with a mechanical interlock. Subsequently, the accuracy of the predictions is assessed through experiments and finite element simulations, employing the modified Mohr-Coulomb criterion for the prediction of ductile damage. Additionally, a new design for hole-hemmed joints, involving the incorporation of branches in the hole of the outer sheet, is introduced to expand the process window of this novel joining technology. Finally, the mechanical behavior of four different types of hybrid hole-hemmed joints (HHH joints) are evaluated through single-lap shear tests. The results show that the hole hemming process can successfully join AA6082-T4 and PC sheets, validating the proposed designs and ideas. The new hybrid joints demonstrate a maximum force and displacement of 1.6 kN and 12.9 mm, respectively, in the shear test, indicating significant potential of the proposed technology for joining metal and polymer sheets.

Static performance of multi-planar CFST chord-CHS brace KK joints

This paper investigates the static performance of jacket joints to promote the application of CFST jacket foundations for offshore wind power. An experimental and numerical study of the influence of loading patterns, concrete filling in the chord, geometric parameters, and material strength is examined. In total, five 1:6 scale tests and twenty-eight FEAs are described. The novelty is that a limit device is introduced to consider the interaction among the braces by connecting four braces. The experimental results indicate that, compared to symmetrical loading joints, the bearing capacity of CHS-KK joints decreased by 14% under anti-symmetric loading. Concrete filling in the chord alters the failure mode, enhancing the ultimate bearing capacity of the KK joint. However, its effect on the punching shear failure strength of the joint is limited. An FEA model incorporating the modified Mohr-Coulomb criterion (MMC) was established to explore the ultimate strength of CFST-KK joints, following validation through experiments. Through parameter analysis, it was found that the punching shear capacity of CFST-KK joints is significantly influenced by the diameter ratio of the brace to chord (β), the chord diameter to thickness ratio (2γ), the angle between the brace and chord axis (θ), and the chord steel strength. In addition, the recommended formula in GB50017–2017 for CHS-KK joint punching shear capacity conservatively predicted CFST-KK joint performance.

Development of a strain-based fracture assessment approach for X80 steel pipe welded girth by modified Mohr-Coulomb model

In the study, a strain-based fracture assessment scheme with the Modified Mohr-Coulomb (MMC) model is established to numerically characterize the tensile strain capacity (TSC) of X80 steel pipe welded girth. The TSC at crack initiation and local instability are predicted based on a lode-dependent MMC model accounting for crack onset and propagation phases. The combined experimental-numerical calibration of the MMC criterion for crack onset is undertaken for X80 pipeline steel, where seven geometrical specimens are designated to represent various stress states from low to high triaxiality, and the parameter determination of the MMC criterion for crack growth is carried on by fitting the single-edge notched tension tests. Finally, the comparison between the present work and PRCI and ExxonMobil in terms of TSC is performed. A relatively good agreement can be achieved, whereas PRCI gives non-conservative predictions of TSC at crack initiation when crack depth a/t is longer than 0.4 and ExxonMobil gives conservative predictions of TSC at local instability when internal pressure is small.

Ductile Fracture Prediction in Hole Hemming of Aluminum and Magnesium Sheets

The present work proposes a suitable approach for predicting ductile fracture in a new joining process by plastic deformation called hole hemming. This process creates a combined form- and force-fit joint and enables the joining of lightweight materials with varying formability without requiring heating or auxiliary elements. In this process, the joinability of materials is limited by the occurrence of fracture in the outer sheet, highlighting the crucial need to accurately predict its damage during the process design phase. In this study, five different fracture criteria, including the McClintock, Rice–Tracey, Normalized Cockroft–Latham, and Brozzo and Modified Mohr–Coulomb (MMC) criteria, are examined during the joining of a challenging combination of lightweight materials (aluminum AA6082-T4 and magnesium AZ31). These criteria are calibrated by a hybrid experimental–numerical method using three tests with distinct stress states. These criteria are then implemented into the finite element model of the hole hemming process, utilizing an appropriate user subroutine. The results show that the flange edge of the outer sheet is the most prone region to fracture during the joining process, and a criterion must be able to model the fracture behavior of the material from uniaxial tension to shear to accurately predict fracture in this area. Among the examined criteria, only the MMC criterion was capable of such modeling and accurately predicted the critical displacement of the punch in the hemming stage with a negligible error (about 1%). On the other hand, the prediction accuracy of the other criteria varied significantly depending on the calibration test, resulting in errors ranging from 8.6% to 75.5%. The error of 8.6% was achieved with the Normalized Cockroft–Latham criterion calibrated by a uniaxial tension test. Thus, based on the results, the MMC criterion is recommended for ductile fracture prediction in the hole hemming process, offering valuable insights to assist in process design.

Strength-based material layout optimization of solid reinforced concrete

Whether monetary, environmental, or otherwise, the cost of the structure is always an important aspect when designing reinforced concrete. A structure with a low cost is desirable, but other considerations must be made. First and foremost, the structures should be safe. The two conditions define the structural optimization problem that this paper seeks to solve as finding the structure with the lowest cost, which is also safe. Validation of reinforced concrete in the ultimate limit state is often performed using plasticity and strength-based methods. This paper combines Finite Element Limit Analysis (FELA) with a yield criterion that includes material layout variables to create a strength-based structural optimization framework for solid reinforced concrete structures. The yield criterion combines the Modified Mohr–Coulomb criterion, where variables scale the compressive strength, with the inclusion of smeared reinforcement, where variables scale the reinforcement ratios. The framework is presented and shown in examples using free optimization and examples where material layout variables in parts of the structure are coupled and optimized as a group.

Physical properties and tensile strength evolution of gypsum materials under different water content conditions

To investigate the physical properties and tensile strength evolution of gypsum materials under different water content conditions, three groups of gypsum standard samples were prepared, including natural, saturated and dehydrated gypsum samples. Electron microscopy, composition analysis, Brazilian splitting tests and numerical simulation analysis were carried out. The results are as follows. The larger the water content of the samples is, the higher the content of CaSO4·2H2O, the more flocculation structures on the surface, and the higher the longitudinal wave propagation speed. Saturation and dehydration change the composition of the sample and cause a change in its tensile strength and failure mode. The fitting effect of the modified Hoek-Brown criterion is obviously better than that of the modified Mohr-Coulomb criterion because it considers the influence of uniaxial compressive strength. The Balmer criterion introduces the index b and greatly improves its regression accuracy. The vertical compressive stress exhibits a concentrated distribution. At x = 0, the peak value of the vertical load appears at the two ends of contact between the specimen and the indenter and gradually decreases from both ends to the centre. The horizontal stress is mainly tensile stress, and the value at x = 0 is the theoretical tensile strength. It is uniformly distributed on the line directly affected by the vertical load and decreases to both ends of the specimen along the line. It is reduced to 0 at the left and right ends. The horizontal deformation arises from both sides of the disk tip and then spreads downwards in an arc during loading. The horizontal strain in the upper part of the disk is always larger than that in the lower part. The upper part reaches its tensile strength and then fails due to the continuous loading.

Chart-Based Granular Slope Stability Assessment Using the Modified Mohr–Coulomb Criterion

There are usually two types of slopes in nature: soil and rock. However, the granular slope has gradually come into view and has caused concern. In this paper, a modified Mohr–Coulomb (MC) criterion has been proposed to estimate the strength parameters of medium materials using a new parameter (tau_{0}) instead of the traditional MC parameter (c). Meanwhile, there is a new stability chart of a granular slope whose core is the dimensionless stability number Ns and dimensionless coefficient DR based on a modified MC criterion to estimate the safety factor of a granular slope preliminarily when the geometric and strength parameters of the slope are determined. A practical case is introduced to test the results, showing that the stability chart proposed in this paper is applicable to granular slopes. The research content and the proposed new viewpoints are intended to offer preliminary judgments on the stability analysis of granular slopes and provide guidance for the stability analysis and construction design of granular slope engineering.

Experimental study on the moisture migration and triaxial mechanical damage mechanisms of water-bearing coal samples

To investigate the internal moisture migration patterns and mechanical damage characteristics of water-bearing coal, X-ray diffraction (XRD), scanning electron microscopy (SEM), nuclear magnetic resonance (NMR) spectroscopy, acoustic emission (AE) tests, and conventional triaxial tests were carried out on coal samples saturated at different water pressures. The results showed that the saturated water content of the samples increased exponentially with the water pressure and reached a threshold value, from which the limiting water pressure and limiting water height were approximated. Under non-destructive immersion, the water absorption process of the coal samples was a combination of layer-by-layer saturation and downward water transport. As the water pressure increased, the number of pores indicated by the NMR analysis increased, and the degree of internal damage defined by the integrated area of the T2 curve also increased. These results were used to quantitatively characterize the damage degree of the samples. The triaxial test results showed that the peak strength, residual strength, modulus of elasticity, and Poisson's ratio of the saturated coal samples all decreased linearly or exponentially upon increasing the water pressure. The breaking angle and number of shear cracks increased, and localized deformation damage eventually developed in the form of narrow shear zones. A modified Mohr-Coulomb criterion based on water pressure and water level was proposed after the experimental results were verified by theoretical derivation. This work provides a scientific basis for studying rock weakening mechanisms in water environments.

Systematic calibration procedure for CFRP sheets and rods strengthened RC T-beams by using RBSM

In this study, a nonlinear 2D rigid body spring model (RBSM) was developed to verify the experimental works of Carbon Fiber Reinforced Polymer (CFRP) sheets and rods strengthened reinforced concrete (RC) T-shape beam (RC T-beam) section under a combination of bending and shearing loads. Modified Mohr–Coulomb criteria were adopted for modeling the plastic damage of concrete material. The orthotropic smeared layers representing the steel and CFRP materials were laminated on the concrete surface to model the reinforcing steel bars, stirrup steel, and CFRP sheets and rods. A mechanical behavior-based approach to fit the experimental results was discussed using the numerical results at each calibration stage. This study shows how the gradient of compressive and ultimate tensile strength of concrete affects the initial flexure behavior of the load-displacement curve of an RC T-beam. In contrast with finite element modeling, the RBSM can exhibit the crack propagation processes of element separation during the simulation. The calibrations showed the agreement of the models used in predicting the flexural behavior, ultimate load, and strengthening effects of the CFRP sheets and rods to the RC T-beam under bending loads. A systematic calibration procedure combined with the recommended use of energy-based criteria to evaluate the results of calibrated load-displacement curve was proposed.

Model for the Simulation of the Time-Dependent State in RC Elements

The paper presents an upgrade of the previously developed model for nonlinear 3D analysis of concrete structures extended with the possibility of simulation of the long-term effects (shrinkage and creep) under long-term static load. The origin model is based on the so-called multi-surface principle with modified Rankin criterion for dominant tensile influences (appearance and development of cracks) and the modified Mohr-Coulomb criterion for dominant compressive states (yielding and cracking of concrete). The material behaviour is described with elementary material parameters (modulus of elasticity, Poisson’s coefficient and uniaxial compressive and tensile strength of concrete) by standard tests. Sufficient accuracy along with a simple and effective description of the very complex behaviour of reinforced concrete structures, make this model advantageous. Creep and shrinkage are based on the procedure given by the fib Model Code 2010 and extended with a special extension for non-linear creeping. Two simple examples show the capabilities of the model, while a good agreement between numerical and experimental results indicates that the developed model can well describe long-term effects in reinforced concrete structures, and that the model is appropriate for standard engineering practice.

Effect of incorporating Lode angle parameter into a fracture criterion in predicting ballistic impact behavior of double-layered 2024-T351 aluminum alloy plates against blunt projectiles

Recent studies showed that the ductility of some metals was affected by not only stress triaxiality but also Lode angle parameter. However, the majority of impact-related simulations used Lode independent fracture criteria. The necessity of incorporating Lode angle parameter into a fracture criterion in predicting ballistic impact behavior of monolithic 2024-T351 aluminum alloy (AA2024-T351) plates has been previously addressed in the work of Xiao et al. [1]. This paper serves as an extension by considering plates with double-layered configuration and the same total thickness. Ballistic impact tests were performed on 4.82 + 4.82 mm thick double-layered plates. Blunt projectiles with dimensions of Φ 12.66×50.56 mm3 and a nominal mass of 50 g were used, and the impact velocities ranged from 133.4 to 363.8 m/s. According to the experimentally obtained data, ballistic limit velocities (BLVs) were calculated and fracture patterns of post-test plates were analyzed. It was found that for double-layered configurations, the first plate failed by shear plugging and the second plate failed by tensile stretching and global deformation. By replacing monolithic plates with double-layered ones, the BLV increased from 138.4 to 180.1 m/s, a 30% improvement in ballistic resistance. Subsequently, numerical simulations of ballistic impact tests were performed by software ABAQUS/Explicit. Mechanical property of AA2024-T351 was described by Johnson-Cook (JC) plasticity model accompanied with either the Lode independent JC fracture criterion or the Lode dependent modified Mohr-Coulomb (MMC) fracture criterion. The numerical results were compared with those obtained from experiments. It was found that MMC criterion predicted BLVs more accurately than JC criterion. The prediction errors for MMC and JC criteria were respectively 3.3% and 11.4%, with the former being merely one-third of the latter. The overall failure modes of double-layered plates were correctly predicted by both JC and MMC fracture criteria. However, details of the fracture patterns, such as the formation of fragments, the shape and number of ejected plugs, and the initiation and propagation of cracks, were better captured by MMC criterion than by JC criterion. Detailed analysis shows that the stress state governing ballistic impact tests locates in the range where Lode angle parameter is strong enough to induce a significant difference between JC and MMC criteria. Under such stress state, the ductility of AA2024-T351 is correctly predicted by the Lode dependent MMC criterion but is overestimated by the Lode independent JC criterion.

Extension of a shear-controlled ductile fracture criterion by considering the necking coalescence of voids

An isotropic ductile fracture criterion considering nucleation, growth, shear coalescence and necking coalescence of voids during plastic deformation is developed to model fracture behavior of ductile metals. The innovation of this fracture criterion is that it considers the mechanism of necking coalescence of voids on the basis of a shear-controlled ductile fracture criterion developed by the micro-mechanism (Lou et al., 2014). The damage from coalescence of voids is expressed as the sum of the damages from the shear coalescence of voids and necking coalescence of voids. We assume that the damage from necking coalescence of voids is controlled by the normalized normal stress at the maximum shear plane, the larger positive normalized normal stress at the maximum shear plane will promote the necking coalescence of voids, while the negative gives no effect on the necking coalescence of voids. After the establishment of the proposed fracture criterion, a detailed parametric study is performed to demonstrate the flexibility of the proposed criterion. Then, the proposed ductile criterion is employed to construct fracture loci for three different materials (AA 2024-T351, AA 6060-T6 and A710 steel). And fracture loci constructed are compared with those experimental data points to validate the performance of the proposed fracture criterion. For the purpose of comparison, the predicted results are compared with those predicted by the micro-mechanism-motivated criterion (Lou et al., 2014), the modified Mohr-Coulomb criterion (Bai and Wierzbicki, 2010). The better agreement of the predictions with the experimental data demonstrates that the proposed fracture criterion is able to improve the prediction accuracy for different metals under various stress states and the mechanism of necking coalescence exhibits a noticeable role in predicting the ductile fracture. Moreover, the proposed criterion is expected to be employed in finite element simulation to calculate the ductile fracture for different metals due to the high accuracy.

Creep-damage constitutive model based on fractional derivatives and its application in salt cavern gas storage

The long-term stability of oil / gas salt cavern storage is still considered as a puzzle, which leads to enormous challenges in its construction and operation. To describe the creep behavior of salt rock more accurately, a new fractional derivative creep-damage (FDCD) constitutive model, based on a modified Mohr-Coulomb criterion, is established by coupling the Hooke body, Abel dashpot and viscoplastic damage body. Previous creep models, especially those coupled with creep and damage, are rarely verified by very long-term creep experimental data. By fitting the creep experimental data of salt rock lasting 21,000 and 15,876 h (875 and 661.5 days), it is found that the proposed model can well characterize the experimental data, especially in the accelerated creep stage. Compared with the classic Nishihara and Burgers models, the FDCD constitutive model has significant advantages, with fewer parameters and higher accuracy. Then the FDCD constitutive model is implemented in FLAC3D with C++ language and applied to the deformation analysis of salt cavern gas storage. The deformation and volume shrinkage of salt caverns obtained by the numerical simulation are in good agreement with those measured by sonar. The FDCD constitutive model can be successfully applied to a numerical program for evaluating the long-term deformation of salt cavern gas storage.

Determination of critical criterion of tensile-shear failure in Brazilian disc based on theoretical analysis and meso-macro numerical simulation

To define the critical criterion between tensile failure and shear failure of Brazilian disc is crucial for determination of tensile strength under combination of tensile and compressive stresses. A critical criterion for tensile-shear failure including tensile truncation was proposed firstly based on Mohr-Coulomb criterion. Then, a new calculation method of shear strength parameters was deduced based on the critical state stress circle. Finally, effective diameter and range of compression-tension ratio of Brazilian disc under tensile failure were proposed, and failure process of Brazilian disc from meso to macro was carried out based on numerical simulation. The results show that the modified Mohr-Coulomb criterion can be determined by two indexes of compressive strength and tensile strength. The deduced compression-tension ratio is in good agreement with experiment results. The accuracy of tensile strength test results is closely related to this ratio. When the ratio is greater than 6, failure of rock can be regarded as ideal tensile failure. If the ratio is less than 6, wedge-shape failure in loading area is the main failure type. On this condition, even if radial crack occurs, it is still not considered to be caused by pure tension.

Statistical thermal damage constitutive model of rocks based on Weibull distribution

On the basis of Lemaitre’s strain equivalence theory, the effects of high temperatures on the mechanical parameters of rock are taken into account, and a thermal damage variable DT is introduced. A statistical thermal damage constitutive model is developed by adopting the modified Mohr-Coulomb criterion with the hypothesis that the mesoscopic unit strength of rock subjected to thermal treatment obeys a Weibull random distribution. The values of the model parameters are determined by the extreme value method according to the concept of yielding. The rationality of the model is verified by comparing the calculated results with experimental results from uniaxial and conventional triaxial experiments of granite, marble, and sandstone specimens that were subjected to different high-temperature treatments. The results indicate that the model can simulate the whole process of stress-strain relations of different lithological rocks under different confined pressure and temperature conditions; the model can also reflect the characteristics of strain softening. The model does not contain any unconventional rock mechanics parameters and is lithology independent, which is highly convenient for engineering applications. This research can provide a theoretical basis for the development and protection of high-temperature rock projects.

A new ductile fracture criterion considering both shear and tension mechanisms on void coalescence

A new ductile fracture criterion is proposed based on three stages of ductile fracture: void nucleation, growth and coalescence from the microscopic viewpoint. Based on the observation of SEM fracture surfaces of AA2024-T351 aluminum alloy sheet and bar samples under different stress states, it is assumed that the void aggregation is controlled by shear or shear-tension fracture mechanism according to the stress state. And the stress triaxiality is deemed as the only influence factor for controlling the void growth. The new ductile fracture criterion applied to a wide range of stress triaxiality is built. By fitting the available testing data of AA2024-T351 aluminum alloy and AISI 1045 steel, the fracture loci in the stress triaxiality, Lode parameter and equivalent fracture strain space ([Formula: see text]) built by the new criterion are compared with those by DF2014, Hu criterion, modified Mohr-Coulomb criterion (MMC) and Hosford- Coulomb (H-C) criterion. The fitting results prove better prediction accuracy for the new criterion. To further compare the stability of these criteria, the fracture loci in the space of ([Formula: see text]) for AA2024-T351 alloy are established by only fitting five tests. The new criterion can still well predict the equivalent fracture strain ([Formula: see text]). Compared with DF2014, Hu criterion, MMC and H-C criterion, the average errors of the new criterion are reduced by 26.72%, 20.07%, 31.78% and 34.62%, respectively. Furthermore, the maximum errors are reduced by 49.62%, 27.31%, 33.76% and 29.91%, separately. Therefore, the new fracture criterion has higher prediction accuracy and better prediction stability. Last but certainly not least, the new criterion can predict more accurately under high stress triaxiality conditions.

A Modified Mohr Coulomb Criterion for Rocks with Smooth Tension Cut-off

In this study, in order to solve the limitation of the modified Mohr-Coulomb criterion in the numerical implementation induced by the corners or apexes, a modified Mohr-Coulomb criterion with smooth tension cut-off has been proposed. Both the nonlinear meridian shape function and the deviatoric shape function have been studied, and the failure surface of the proposed criterion is smooth and derivable at any stress states. The criterion has been verified through numerical simulations and comparison with the experimental results. Furthermore, it will be easier to adopt this smooth criterion in the numerical algorithm without losing good qualities of modified Mohr-Coulomb criterion.

Fracture criteria applied to numerical simulation of blowout preventer ram shearing

The interest in predicting the force required for a subsea blowout preventer (BOP) to cut and isolate the drilling pipe has been growing since the Macondo accident. In previous research, numerical simulation using the nonlinear finite element method combined with the blowout preventer (BOP) shear test has been a common approach. As various fracture criteria exist, the selection and application of a suitable fracture criterion are of great interest to achieve a successful ram shearing simulation. In this paper, the CrashFEM criterion, previously employed in crashworthiness simulation, is first adopted in BOP ram shearing simulation to capture the onset of damage to the pipe. The Johnson-Cook (J-C) and Modified Mohr-Coulomb (MMC) criteria are utilized to carry out a comparative study. The TRIP 690 experimental data are used to derive the fracture parameters. These parameters enable the building of numerical models based on boundary and loading conditions for the BOP shear test. Based on the numerical model, a mesh sensitivity study is carried out. The simulation results using refined meshes show that the shearing forces predicted by the CrashFEM and MMC criteria are generally consistent with the experimental data. The results from the three criteria indicate differences originating from the shape of the fracture limit curve, the damage initiation concerning the equivalent plastic strain accumulation, and the damage distribution in the shearing area. These findings are explained in detail in the paper. A comprehensive application of the CrashFEM criteria for BOP shearing simulation is presented and recommendations are provided about the selection of suitable fracture criteria.

Failure Criteria for Isotropic Rocks Using a Smooth Approximation of Modified Mohr–Coulomb Failure Function

It is reported that strength of rocks is a function of three-dimensional (3D) stress state as observed from the large sets of experimental results under the true-triaxial condition. Triaxial compression test data in general is readily available for rocks, but in practice true-triaxial facilities are seldom used. This creates a requirement for 3D failure criterion, which can be used to predict the true-triaxial strength using simple triaxial compression test data. We have proposed three criteria by altering the deviatoric function of modified Mohr–Coulomb criterion (Singh et al. in Int J Rock Mech Min Sci 48(4):546–555, 2011). The proposed criteria represent the behaviour of rock from lower to very high confining pressures. The criteria have unique yield surfaces. The yield surface changes its shape with the change in the mean stress. It is observed beyond certain value of the mean stress; the yield surfaces converge to von-Mises yield surface which means the shape of the yield surface is independent of pressure beyond the certain value of mean normal stress. The criteria require three parameters that can be easily calculated from the triaxial compression test results. The proposed criteria also satisfy the Drucker stability postulate, hence could be adopted as a plastic potential function. For the assessment of the criteria, true-triaxial data from twelve rocks are considered. Results of the criteria are compared with the modified Mohr–Coulomb criteria both 2D and 3D proposed by Singh et al. (Int J Rock Mech Min Sci 48(4):546–555, 2011). Performance of the new failure criteria for the considered rocks are observed to be better than the modified Mohr–Coulomb criteria (2D and 3D).