Nonlinear Strength Criterion Research Articles

Assessing the Reliability of Rock Slopes Based on the Nonlinear Strength Criterion

Assessing the Reliability of Rock Slopes Based on the Nonlinear Strength Criterion

Multiscale modeling of diffuse damage and localized cracking in quasi-brittle materials under compression with a quadratic friction law

The diffuse damage and localized cracking of quasi-brittle materials (i.e., rocks and concretes) under compression can be delineated by a matrix-microcrack system, wherein a solid matrix phase is weakened by a large number of randomly oriented and distributed microcracks, and the macroscopic cracking is formed by a progressive evolution of microcracks. Several homogenization-based multiscale models have been proposed to describe this matrix-microcrack system, but most of them are based on a linear friction law on the microcrack surface, rendering a linear strength criterion. In this paper, we propose a new quadratic friction law within the local multiscale friction-damage (LMFD) model to capture the plastic distortion due to frictional sliding along the rough microcrack surface. Following that, a macroscopic Ottosen-type nonlinear strength criterion is rationally derived with up-scaling friction-damage coupling analysis. An enhanced semi-implicit return mapping (ESRM) algorithm with a substepping scheme is then developed to integrate the complex nonlinear constitutive model. The performance of LMFD model is evaluated compared to a wide range of experimental data on plain concretes, and the robustness of ESRM algorithm is assessed through a series of numerical tests. Subsequently, to effectively describe the localized cracking process, a regularization scheme is proposed by combining the phase-field model with the established LMFD model, and the discretization independent crack localization is numerically verified.

A Three-Dimensional Nonlinear Strength Criterion for Rocks Considering Both Brittle and Ductile Domains

A Three-Dimensional Nonlinear Strength Criterion for Rocks Considering Both Brittle and Ductile Domains

Damage creep model and application for sandy mudstone considering the effect of immersion deterioration

Abstract Aiming at the problem of performance degradation and large deformation of sandy mudstone after immersion in water, the compression test is carried out. The damage and deterioration law of strength and elastic modulus of immersed rock mass is analyzed, and the function relationship between rock damage variable and soaking time is obtained by fitting. Combined with the Hoek-Brown criterion and Mohr-Coulomb criterion, the calculation method of instantaneous cohesion and instantaneous internal friction angle of damaged rock mass is derived based on the tangent method, and the equivalent Mohr-Coulomb criterion which can effectively characterize the nonlinear strength characteristics of damaged rock mass is given. Based on the built-in creep model Cvisc of Flac3D, a damage creep model is given by introducing the damage factor and nonlinear strength criterion of immersed rock mass, which is more suitable for describing the creep characteristics of immersed sandy mudstone. The validity of the model is verified by comparing with the measured results.

Seismic active earth pressure for 3D earth retaining structure following a nonlinear failure criterion

Stability analysis for three-dimensional (3D) earth retaining structures (ERSs) is a pivotal problem in geotechnical engineering, particularly for seismically active areas. In this work, the 3D ERSs in soil are assessed for seismic stability following a nonlinear strength criterion. The power-law strength criterion and seismic forces are introduced into the 3D ERS stability analysis via a multi-cone failure mechanism. The coefficient of active soil pressure Ka is derived by exploiting the energy balance equation. The upper bound solutions, that is the maximum results of the Ka are captured with the help of a genetic algorithm. The validity of the current study is verified by a comparative analysis, and the effects of soil strength nonlinearity, the seismic forces, the soil-wall friction angle, the height, the inclined angle and the 3D geometric traits of ERSs on the Ka solutions and the stability of ERSs are investigated. It is indicated that the height and the 3D geometric characteristics of the ERSs will not only determine the Ka solutions directly, but also influence the impact of soil strength nonlinearity on the stability of ERSs. Strength criteria should be chosen combining the geometric characteristics of ERSs to derive more critical estimates on the stability of 3D ERSs. Additionally, a range of seismic and static stability charts used for preliminary design are put forward as well.

Pseudodynamic Estimation of 3D Active Earth Pressures with a Nonlinear Strength Criterion

This paper presents an analytical framework for predicting three-dimensional (3D) seismic active earth pressures for soils governed by a nonlinear strength criterion. The kinematic limit analysis method that incorporates a pseudodynamic approach is used to carry out the analysis. A generalized tangential technique is adopted to realize transition of the nonlinear criterion into the commonly used linear form. A 3D failure mechanism of rigid rotation is employed. Because the use of the modified pseudodynamic approach leads to a variation of seismic forces along the depth, the work rate due to earthquake effects is calculated by using a horizontal layer–wise summation method. By virtue of the balance equation of work and energy, the thrust of active earth pressures is determined, and the final solutions are expressed as dimensionless active earth pressure coefficients. A comparison between the solutions obtained in this paper and those of the existing studies shows the effectiveness of the developed method. A parametric analysis is carried out to investigate the influences of several parameters. The findings of this work can help understand the service behavior of retaining structures subjected to seismic loadings, which could provide references for designs.

Probability analysis of rock slope using FORM based on a nonlinear strength criterion

Various uncertainties are frequently encountered in the assessment of landslide stability. However, the uncertainty of parameters in the Mohr–Coulomb strength criterion is a major concern in the studies, neglecting the uncertainty related to the strength criterion itself that is used to describe the strength of geotechnical materials. To assess the reliability of landslides, a new probabilistic analysis method is proposed in this study. According to the traditional limit equilibrium framework, an improved method based on a nonlinear strength criterion is deduced to determine the safety factor of landslides. The local mean value of the random variables on the failure surface is estimated by the analytical method, avoiding random field simulation. The first-order reliability method (FORM) is applied to estimate the failure probability of landslides. An example is analyzed using the proposed probability analysis method, and its effectiveness is primarily verified. The results show that the traditional limit equilibrium method based on the linear Mohr–Coulomb strength criterion estimates the stability and failure probability of landslides inaccurately.

A micro-macro constitutive model for compressive failure of rock by using 4D lattice spring model

In this work, a micro-macro mechanics-based constitutive model is developed to describe the compressive failure process of rock. A micromechanics model is first developed based on the conceptual model of a micro cracked spring bond, where two micro parameters with clearly geometric meanings are introduced. By using the micromechanics model, it is possible to reproduce the whole nonlinear failure process of the rock under compressive loading, however, it fails to predict the reasonable quantitative relation between the crack initiation stress (σCI) and the direct tensile strength (DTS) that is commonly observed in laboratory tests. Therefore, a macroscopic failure criterion is introduced to determine the failure state of the spring bond instead of the spring deformation used in the micromechanics model. The numerical results indicate that the 4D-LSM with the micro-macro constitutive model can not only fully reproduce the whole nonlinear behavior of rock under different confining pressures, but also reproduce the quantitative relation between σCI and the DTS and the overall macroscopic nonlinear failure strength criterion defined by Hoek and Brown. The micro-macro constitutive model not only provides a feasible solution for better describing the complex nonlinear compressive failure process of rock but helps to better understand the internal failure mechanism of rock.

Mechanical Properties and Constitutive Model of the Cement-Improved Loess under Freeze-Thaw Conditions.

Cement-improved loess (CIL) is used as a common filler for subgrade construction projects in loess areas. The freeze-thaw (F-T) conditions have a significant effect on the stability of cement-improved loess subgrades in seasonally frozen regions. In this paper, the CIL samples, experiencing different numbers of F-T cycles at varying freezing temperatures, were used in consolidated undrained triaxial compression tests to investigate the effect of F-T conditions on the mechanical properties of CIL. The results show the stress-strain curves of CIL are of a strain-softening type with strong elastic brittleness. The initial tangent modulus of CIL increases with the growing confining pressure and gradually decreases with the increase in the F-T cycle number and the decreasing freezing temperature. It loses 46.4% of its original value after the twelfth F-T cycle with the confining pressure of 150 kPa and at the freezing temperature of −15 °C. The strength of CIL decreases with the increasing F-T cycle number, but it gradually tends to keep stable after the sixth F-T cycle. The strength also decreases with the reduction in the freezing temperature. It loses 37.7% of its original value after the twelfth F-T cycle with the confining pressure of 150 kPa and the freezing temperature of −15 °C. To express the nonlinearity correlation between the strength and confining pressure under F-T conditions, the Weibull function was applied and a nonlinear Mohr-Coulomb strength criterion was proposed. Through introducing a breakage rate function and a local strain coefficient, a binary-medium constitutive model consisting of bonded elements (soil-particle cohesion) and frictional elements (soil particles or soil aggregations) was established to describe the stress-strain relationships of CIL under F-T conditions. The test results indicated that the model can well describe the strain-softening phenomenon of the stress-strain curve of CIL and reflect the breakage mechanism of CIL.

Modified Statistical Damage Constitutive Model for Frozen Sand Based on Nonlinear Strength Criterion

The strength of frozen soil is a crucial criterion to judge the excavation of cross-passages or tunnels during artificial ground freezing. To better understand the combined influence of the temperature and confining pressure on the strength of frozen soil during freezing, typically triaxial compression tests were carried out on Harbin frozen sand under various temperatures and confining pressures. A nonlinear strength criterion considering the coupled experiment conditions was established according to the results obtained from the triaxial compression tests. Based on continuum damage mechanics, a modified statistical damage constitutive model was derived, in which the Weibull distribution was employed to describe the development of microcracks and defects, and the element strength was considered a random variable. The undetermined parameters in the model were solved using the nonlinear programming method. The effectiveness of the proposed model was verified by comparing the calculated and experimental results. The results indicate that the proposed model can reflect the nonlinear characteristics of the soil strength and accurately predict the stress—strain relationship of frozen sand, especially the phenomenon of pressure melting caused by high confining pressure.

A Nonlinear Strength Criterion of Frozen Saline Sandy Soil at High Pressures

The strength of frozen saline soil is influenced by various factors, such as confining pressure, salt content, and temperature. To investigate the influences of these factors on strength parameters, a series of triaxial tests were performed for frozen saline samples. According to the test results, as the mean principal stress increases, the failure strength increases firstly and then reaches a peak value, followed by a decreasing tendency. The pressure corresponding to ice melting is higher for frozen samples at a lower temperature. A strength function, which consists of cohesion and friction components, is developed in p ‐ q plane. Then, the relationships between temperature and strength parameters are given by fitting the test data. The envelope between those for the SMP and von Mises criteria is employed to describe the strength properties in π plane. By comparing the predicted data with test results, the developed strength criterion can well reproduce the strength variation of frozen sandy samples considering the effects of confining pressure, salt content, and temperature.

Numerical Study of Spherical Particle Breakage Based on Nonlinear Discrete Interface Bonded Model

To study the contact mechanical characteristics and breakage mechanism of single particle, the particle contact experiment is simulated using the discrete element method. In this paper, to solve the problem that the envelope of the shear strength of the classical model is linear, a nonlinear discrete surface bonded model is introduced by applying nonlinear tangential strength criterion. The new model is compared with the classical model to verify its validity. Then, based on the new model, the particle contact experiment is numerically simulated, and the ball-plane contact breakage mechanism is analysed and discussed from the mesoscopic perspective. The results reveal that the nonlinear discrete interface bonded model can provide shear properties similar to those of actual materials. The numerical result reflects the shear failure of the edge of the conical region and the tensile failure of the whole specimen. Combined with the analysis of the results of stress distribution, velocity field, and force chain distribution, the formation mechanism of the cone core can be obtained.

Developing a Three Dimensional (3D) Elastoplastic Constitutive Model for Soils Based on Unified Nonlinear Strength (UNS) Criterion

To achieve versatility, a unified nonlinear strength (UNS) criterion is put forward for capturing the complicated strength behaviors exhibited by geomaterials under three dimensional (3D) stress paths. The UNS criterion, widely covering meridian planes and octahedral planes, can serve for describing the nonlinear strength behaviors exhibited by soils, as well as confirm how the intermediate principal stress affects the strength of different materials. Based on UNS strength criterion, an elastoplastic constitutive model is presented, with the purpose of predicting the strength as well as deformation behavior exhibited by soils under 3D stress conditions. Besides, although the proposed model is extremely simple, it is fit for predicting the results of true triaxial tests in related literature with the help of the UNS criterion, and meanwhile can confirm how intermediate principal stress affects material strength and material deformation when the stresses are different.

Strength criterion for frozen silty clay considering the effect of initial water content

The initial water content and confining pressure significantly affect the deformation and strength behaviors of frozen soils. To investigate the influence of the initial state of water on the mechanical properties of frozen silty clay (FSC), triaxial compression tests are conducted on artificial FSC with five initial water contents (12.5%, 14.0%, 16.0%, 18.0%, and 20.0%) at the temperature of −6 °C. The results show that when the initial water content is 12.5%, the frozen soil samples present strain softening and strain hardening successively with increasing confining pressure. As the initial water content increases, the degree of strain hardening under the same confining pressure gradually weakens. The frozen soil samples mainly present strain softening when the initial water content is more than or equal to 18%. According to the test results, when the initial water contents ranging from 12.5% to 20%, the relationship between the strength and mean stress has three forms: increases nonlinearly, increases first and then decreases, and maintains an approximately constant value. Linear and nonlinear strength envelope equations based on different types of strength envelopes are proposed. Additionally, the relationship between the initial water content and internal friction angle is analyzed. In the p-q plane, based on the nonlinear strength characteristics of FSC with initial water contents of 12.5%, 14.0%, and 16.0%, a modified mean stress expression p* is derived to represent the nonlinear strength criterion equation according to the modified Cam-Clay model. Furthermore, a shape function in the deviatoric plane is proposed by modifying the Lade–Duncan model, and a strength criterion is established by integrating the strength functions in the p-q and deviatoric planes. The proposed strength criterion can effectively represent the nonlinear strength behaviors of FSC, including the effects of the initial water content, ice melting, and crushing.

Implementation of the Barton–Bandis Nonlinear Strength Criterion into Mohr–Coulomb Sliding Failure Model

The stability of rocky slope is determined mainly by discontinuities. The discrete element calculation method can be used to analyze the geometric features of rock structures and to deal with the nonlinear deformation and destruction of the rock mass that may be affected by the discontinuities. In this paper, the Barton–Bandis (B-B) nonlinear strength criterion was introduced to improve the Mohr–Coulomb (M-C) slip model of joint slope. The modified model could reflect the real-time shear strength in a changing stress state. Using the numerical calculation of the shear test of the structural plane, we found that the corrected slip curve fits well with the process before the failure of the shear test. Furthermore, the modified model can track the disintegration between blocks while sliding failure of joint slope. With an increase in the number of structural planes and the complexity of the relative location of blocks, the interaction force between blocks and the sliding failure mode of the joint slope would be more complex. Changing the nonlinear parameters with the stress state of the structure plane could effectively solve this problem.

Limit equilibrium analysis on the stability of rock wedges with linear and nonlinear strength criteria

The stability of a wedge is an important research topic for rock slopes. A rock wedge is formed by two sets of intersecting discontinuous structural surfaces distributed in a rock mass and cutting the rock slope. Hence, its stability is related to the occurrence of structural surfaces. To evaluate the stability of rock wedges, the limit equilibrium (LE) methods, which do not fully consider the shear forces on the slip surface of both sides of the rock wedge, are usually adopted. As a result, the calculated rock wedge factor of safety (FOS) may be overestimated. Here, a spatial model of a rock wedge is established from the occurrence parameters of the structural planes and slope surfaces. Moreover, reasonable assumptions of internal forces on both sides of a micro-unit body are adopted with the micro-unit body in the rock wedge as the analysis object. Subsequently, the analytical formulas for the normal and shear forces of the rock wedge with full consideration of the shear forces on the slip surface of both sides are derived by applying mechanical equilibrium conditions . Furthermore, the solution of the normal stress on the slip surface of a rock wedge under the assumption of a linear distribution is given. Afterwards, the formulas for calculating the FOS of a rock wedge under the linear Mohr–Coulomb strength criterion and nonlinear generalised Hoek–Brown strength criterion are derived. By comparing and analysing some rock slope examples, the feasibility of the proposed method is verified, and it is shown that the conventional LE methods overestimate the stability of rock wedges. In addition, stability charts of rock wedges are proposed to illustrate the influence trend of dips and dip directions of discontinuous structural planes on the rock wedge FOS when the structural planes have certain discreteness and distribution ranges. • A rigorous LE model is established for analysing the stability of rock wedges. • The FOS of a rock wedge under the nonlinear GHB strength criterion is calculated. • Stability charts of rock wedges are proposed and plotted.

Anisotropic deformability and strength of slate from NW-Spain

Slates are metamorphic rocks characterized by the pervasive occurrence of cleavage or foliation producing a highly anisotropic mechanical behavior characterized by fissility. Deformability and strength of these rocks are therefore dependent on the cleavage plane orientations relative to the principal stresses. In this study, the failure and deformability of these rocks are experimentally investigated by means of a set of standard uniaxial and triaxial compression tests on samples cut with the cleavage forming different angles. Propagation velocity measurements have also been taken in a good number of specimens. Compression tests show that deformability and strength are clearly anisotropic for this rock and that failure through the cleavage plane is observed in the range of dip angles from 15 to 75°. Transversely isotropic elastic parameters are fit based on improved existing approaches. Moreover, the strength of tested samples cut normal and parallel to foliation relevantly differ, something noted qualitatively in the past by some authors in some metamorphic anisotropic rocks. However, this difference has neither been explicitly reported for the case of other foliated sedimentary rocks such as shales, nor formalized in theoretical strength approaches. The triaxial compression experimental data on slates were fit with the Jaeger's plane of weakness (JPW) model. Strength criteria differentiating the strength in directions normal and parallel to foliation are proposed to adapt the JPW model to the observations in slate strength behavior. Other model improvement is proposed, the use of a non-linear strength criterion for the intact rock (Hoek-Brown), which shows to better represent observed strength laboratory results.

Friction-damage coupled models and macroscopic strength criteria for ice-saturated frozen silt with crack asperity variation by a micromechanical approach

This paper aims to present physical friction-damage coupled models and derive macroscopic nonlinear strength criteria for ice-saturated frozen silt. Two material scales are considered here. The material at the mesoscale is composed of a frozen silt matrix and embedded mesocracks. At the microscale, the frozen silt matrix is composed of elastic mineral grains and elastic ice crystals. Considering this microstructure, for the ice-saturated frozen silt, the plastic deformation is related to frictional sliding along the mesocracks, and damage is due to the propagation of mesocracks. We obtain the effective elastic properties with a two-step linear Mori-Tanaka homogenization procedure. For the second homogenization step, we deduce the macroscopic stress-strain relations and local thermodynamic forces associated with damage and plasticity. Then, an energy release-based damage criterion is developed to capture the evolution of damage, and a friction sliding criterion with an associated/nonassociated local plastic flow rule is introduced to describe the rate of plasticity. The friction sliding criterion considers the variation of surface asperities of mesocracks as a result of pressure melting and damage evolution. In this context, associated/nonassociated multiscale friction-damage models are established. Furthermore, associated/nonassociated macroscopic nonlinear strength criteria as inherent parts of the corresponding multiscale models are derived under a large range of compressive stresses. For application, a semi-implicit local numerical algorithm of the multiscale models is developed. The accuracy of associated/nonassociated multiscale models is assessed by comparing their numerical simulations with conventional triaxial compression (CTC) tests. Although the associated multiscale model correctly predicts the nonlinear strength behavior and axial deformation of the Lanzhou frozen silt under CTC tests with different temperatures, it fails to quantitatively reproduce volumetric deformation. Nonassociated multiscale model predictions with experimental data of volumetric deformation are significantly improved compared to the associated multiscale model.

Novel Model for Limit-Equilibrium Analysis of Slope Stability with a Nonlinear Strength Criterion

Slope stability has always been a concern of the geotechnical engineering community, and geotechnical scholars have continued to explore the reasonable failure mechanism of slopes and make the theoretical models of slope stability more reliable. This work established a novel model for limit-equilibrium (LE) analysis of slope stability with a nonlinear strength criterion. In this theoretical model, the thrust line of interslice force and corresponding increment are reasonably simplified. Then, the formula of normal stress on slip surface is derived by assuming the increment of normal interslice force. Meanwhile, strength reduction technology is applied to reduce the shear strength of the geotechnical body on the slip surface with slope factor of safety (FOS) to obtain shear stress on the slip surface. Thereafter, mechanical equilibrium conditions satisfied by the whole slope sliding body and newly introduced equation about increments of interslice force are used to solve LE stability of the slope. For stability analysis of a slope with a nonlinear strength criterion via a simple and feasible calculation process, an iterative loop solution strategy is adopted in the analysis. After comparison and analysis of several slope examples, the feasibility and rationality of the present method are verified. Moreover, the present method has high efficiency in terms of calculation speed.

Mechanical properties of thermal aged HTPB composite solid propellant under confining pressure

With the purpose of investigating the effects of confining pressure and aging on the mechanical properties of Hydroxyl-terminated polybutadiene (HTPB) based composite solid propellant, tensile tests of thermal accelerated aged propellant samples under room temperature and different confining pressure conditions were performed through the use of a self-made confining pressure device and conventional testing machine. Afterwards, the maximum tensile stress σm and the corresponding strain εm for the propellant under different test conditions were obtained and analyzed. The results indicate that confining pressure and aging can significantly affect the mechanical properties of HTPB propellant, and the coupled effects are very complex. On the one hand, the stress σm increases as a whole when confining pressure becomes higher or thermal aging time rises. Besides, this stress is more sensitive to aging with increasing confining pressure. There are almost three regions in the stress increments σmP−σm0/σm0 and thermal aging time curves for HTPB propellant. The maximum value of the stress increment σmP−σm0/σm0 for the propellant is about 98% at 7.0 MPa and 170 d. On the other hand, the strain εm decreases with increasing thermal aging time under the whole confining pressure conditions. However, the variation of this strain with confining pressure is more complex at various thermal aging time, which is different from that of unaged solid propellant in previous researches. In addition, this strain is slightly less sensitive to aging as the confining pressure increases. Furthermore, there is also a critical confining pressure in this investigation, whose value is between 0.15 MPa and 4.0 MPa. Beyond this critical pressure, the trends of the stress σm and the corresponding strain εm all change. Moreover, there are some critical thermal aging time for the stress increment (σmP−σm0)/σm0 and strain increment (εmP−εm0)/εm0 of HTPB propellant in this investigation, which are about at 35, 50 and 170 d. Finally, based on the twin-shear strength theory, a new modified nonlinear strength criterion of thermal aged HTPB propellant under confining pressure was proposed. And the whole errors of fitted results are lower than 6%. Therefore, the proposed strength criterion can be selected as a failure criterion for the analysis the failure properties of aged HTPB propellant under different confining pressures, the structural integrity of solid propellant grain and the safety of solid rocket motor during ignition operation after long periods of storage.