Strength Envelope Research Articles

Grain-based modelling of dynamic shear rupture of heterogeneous rock using a coupled continuum-discrete model

Dynamic shear rupture of rocks plays a significant role in the formation of earthquake faults and the stability of underground engineering structures at depth. This paper aims to explore shear strength, progressive fracturing and seismicity, and underlying shear failure mechanisms of heterogeneous rock under dynamic loads. A lab-scale direct shear test is conducted on a cubic granite sample by using a Triaxial Hopkinson bar (Tri-HB) system, and a coupled continuum-discrete element method is applied to simulate the functionality of the full-scale Tri-HB system for dynamic direct shear tests. The evolution of shear stress and strain distribution shows obvious concentration around the shear rupture band, which verifies the feasibility of the designed testing and modelling configuration for characterising dynamic shear stress and deformation. A grain-based discrete element method (GB-DEM) is adopted to represent the mineralogical heterogeneity of granite and to reveal multi-scale fracturing and seismic activities under dynamic shear loads. The microcracking process shows a transition of the dominant failure mode from intergranular crack to transgranular crack during the shear process. The shear rupture zone is typically initiated from the discrete fractures inclined at certain angles, which are gradually coalesced to form a main shear fracture that breaks the rock into two main blocks. The seismicity of the shear rupture process shows a drop in the b-value before the peak shear stress, followed by a certain degree of recovery at the post-failure stage. We found that shear mechanical response and rupture behaviour show a strong dependency on both strain rate and initial normal stress. The number of transgranular cracks is increased with increasing strain rate, which consequently results in intensively crushed mineral grains. Inclined fractures become steeper under higher initial normal stresses which further widen the shear rupture band. The normal-shear stress curves show a common trend of two linear increase stages followed by nonlinear damage and failure processes. The shear strength can be approximated by the Mohr-Coulomb strength envelope under various initial normal stresses and strain rates.

A Novel Radial Artery P-S Curve Model Based on Radial Vibration of Vascular Wall

In pulse wave analysis, the changing curve of pulse wave strength with continuous increasing pressure, that is, the P-S (pressure-strength) curve, contains abundant human physiological information, but there is no accurate model to describe the formation mechanism of the curve. Therefore, this paper proposes a modeling method of the radial artery P-S curve based on the radial vibration of the vascular wall. The modeling method includes three parts. Firstly, based on hemodynamics, we proposed the blood motion equation in the pulsation process of healthy people. Secondly, the motion equation of the vascular wall based on the fluid–structure interaction between blood motion and vascular wall was established. Finally, according to the elastic theory of the vascular wall, the relationship between pulse strength and extravascular pressure of blood vessels was found. To verify the accuracy and applicability of the model, this paper simulated the changes in the vascular wall stress and the intravascular pressure with the extravascular pressure during the process of vascular deformation. In addition, 69 healthy volunteers were selected to participate in this study. Based on the gradient compression, the pulse strength envelope under the continuous pressure sequence of the radial artery, namely the pulse P-S curve, was extracted. We also analyzed the relationship between the individual P-S curve difference and BMI. The results show that the actual human body data collection and analysis results are consistent with the theoretical model established in this paper, which indicates that the model can provide a novel idea for the evaluation of the state of the human body.

Representative sampling method for laboratory testing on shear strength of rock joints

The direct shear test is commonly used to determine the shear strength of rock joints. The measured shear strength, however, varies greatly with specimen positions due to the heterogeneity of rock joints. As a result, selecting specimens that represent the overall properties of rock joints is usually difficult. This paper aims to investigate the heterogeneity of rock joints and propose a new sampling method for selecting representative specimens. The roughness and shear strength variations of specimens taken from different positions of a natural rock joint were analyzed, and it was discovered that the heterogeneity of rock joint roughness is responsible for the heterogeneity of shear strength. The limitations of the traditional sampling method based on visual judgment were extensively investigated, revealing that shear strength parameters acquired by the traditional method contain large coefficient of variation (COV) values. To acquire trustworthy shear strength parameters, we proposed a representative sampling method based on the maximum likelihood estimation of the overall properties of rock joints. The number of determined representative specimens increases with the increase of normal stress. Representative specimens determined under low normal stresses can likewise exhibit the overall properties of the rock joint when subjected to high normal stresses. The Mohr-Coulomb and a nonlinear criterion were used to validate the derived representative specimens, demonstrating that the proposed method can produce reliable shear strength parameters and shear strength envelopes. Particularly, the determined representative specimens could derive shear strength parameters with relative errors less than 10% and COV values less than 0.1. The proposed method provides a quantitative and reliable tool for determining representative specimens to obtain reliable shear strength of rock joints.

Lithospheric strength and stress revisited: Pruning the Christmas tree

Whether lithospheric stress can reach the maximum level predicted by the Christmas-tree strength envelope is a fundamental question but with controversial answers. There is little controversy that a deforming lithosphere in high heat flow regions is likely critically stressed, i.e., at full yield at all depths, as described by the envelope. But different conceptual frameworks offer opposite views for very cold lithosphere, either at full yield or far below yield. Here, we use simple numerical models to investigate stresses in end-member cold cratonic lithosphere (e.g., Canadian Craton) in comparison with end-member warm plate-boundary lithosphere (e.g., Canadian Cordillera). The two key elements of our modelling are (1) that lithospheric stress builds up elastically with horizontal tectonic loading not only in the elastic–frictional brittle regime but also in the viscoelastic ductile regime, and (2) that the stress level is limited by the available tectonic force. In a cratonic lithosphere, the limiting tectonic force is sustained by competent rock material over a large depth range, represented by the competent thickness Tc that exceeds 90 km. The lithosphere undergoes mostly elastic deformation at a stress level of a few tens of MPa. While weakly stressed strong lithosphere can still produce limited earthquakes at shallow depths due to structural and stress heterogeneity, the lithospheric stress under horizontal tectonic loading is theoretically predicted to be orders of magnitude lower than predicted by the Christmas-tree envelope. Stresses in a real lithosphere may substantially deviate from this theoretical level because of spatiotemporal variations in rheology and structure. For example, the stress memory of past loading history in cold lithosphere may or may not be erased by more recent tectonic stresses. Because much of previous scientific debates on lithospheric stress levels and comparison with seismicity were focused on topographically induced flexural stress, we also investigate the effect of vertical loading. We show that the effective elastic thickness Te derived from the flexural response is a reasonable proxy for Tc derived from horizontal tectonic loading; a very large Te such as > 80 km is generally associated with very low tectonic stress far below yield. However, the flexure-induced bending stress is not directly comparable with seismicity because it may either enhance or suppress seismogenic stress in the crust.

Towards validation of crack nucleation criteria from V-notches in quasi-brittle metallic alloys: Energetics or strength?

We discuss crack nucleation observations from a series of experiments that have proven to be particularly challenging for model validation. In particular, we focus attention on crack nucleation from a series of V-notched specimens of an AISI 4340 steel alloy subjected to four-point bending. Details of the specimen geometry, loading, and experimentally-measured forces are provided for V-notches with opening angles of 30, 60, and 90 degrees. Despite the relatively simple geometry and loading, and the fact that the specimens exhibit small scale yielding, model-based simulations that can accurately reproduce the critical forces at crack nucleation across the full set of specimens have proven to be elusive. Along these lines, we present model-based simulation results from finite-element discretizations of phase-field models for fracture. We provide results using both a well-established phase-field model for brittle materials as well as a recently-developed phase-field model for ductile materials. Importantly, both models tie crack nucleation to critical values of energy, and both are shown to be lacking in their accuracy to varying degrees. We also use standard finite-element calculations of a calibrated plasticity model to demonstrate that much greater accuracy in the calculated critical forces can be obtained with the use of a particular strength envelope.

Experimental study on the influence of hydrostatic stress on the Lode angle effect of porous rock

To investigate the deformation mechanisms of rock under hydrostatic stress, destructive experiments were conducted on sandstone under different levels of hydrostatic stress and stress Lode angles. The results reveal that the shape of the strength envelope on the π plane gradually changes from the shape of the Lade criterion to the shape of the Drucker-Prage criterion with an increase in hydrostatic stress. Normally, there exists a deviation between the strain and stress paths for porous rocks on the π plane, and the deviation decreases with an increase in stress Lode angle and hydrostatic stress. A rock failure hypothesis based on the rock porous structure was proposed to investigate the reasons for the abovementioned phenomena. It was found that the shear expansion in the minimum principal stress direction is the dominant factor affecting the Lode angle effect (LAE); the magnitude of the hydrostatic stress induces the variation of the porous structure and influences the shear expansion. Therefore, the hydrostatic stress state affects the LAE. The failure hypothesis proposed in this paper can clarify the hydrostatic stress effect, LAE, and the variation of the rock strength envelope shape.

Nonlinear criterion for strength mobilization in brittle failure of rock and its extension to the tunnel scale

As underground excavations are getting deeper and field stresses increase, the behavior of intact rock blocks plays an increasingly important role in understanding and estimating the overall rock mass strength. To model the brittle behavior of intact rock blocks, the stress–strain curve is usually idealized considering a linear strength mobilization approach (cohesion-weakening-friction-strengthening, CWFS), however, it is well recognized that rock presents a nonlinear behavior in terms of the confining stress. This study extends the strength mobilization in brittle failure of rock using nonlinear criteria. To determine the model parameters, a standard statistical method that uses the complete laboratory stress–strain curves of the intact rock is employed. Several hypotheses of linear and nonlinear models are statistically compared for different types of rock and confining stress levels. Results demonstrate that the best approach to model the brittle failure of rock is to consider a nonlinear strength envelope, such as the Hoek-Brown criterion assuming a residual uniaxial compressive strength different from zero and a m i parameter that increases, both with simultaneous mobilization. This model helps to recreate high-confining conditions and a more realistic transition between peak and post-peak strength. The obtained parameters are discussed and compared with literature values to verify the validity and to develop guidelines for the estimation of parameters, providing an objective mobilization criterion. Finally, the nonlinear model was applied to a finite element code and extended to a tunnel scale in the brittle rock under high-stress conditions. A reasonable fit between the simulations and the in-situ overbreak measurements was found.

Compressive stress-strain relationships of concrete exposed to elevated temperatures based on mesoscopic damage method

Damage evolution in concrete after high temperature is a complicated procedure, in which the pre-peak strain hardening behaviour, the post-peak strain softening behaviour and the impact of high temperature play key roles. Uniaxial and biaxial compression damage models of concrete considering high temperature degradation effect are proposed based on damage theory and experimental phenomena. They consider that the destruction of concrete is actually the cumulative evolution course of the two meso-damage modes, rupture and yield. High temperature changes the mechanical performance in microstructure of concrete and the generation and propagation of microcracks. It could be described by adjusting the probability distributions which characterize the mesoscopic damage evolution. The damage constitutive model is employed to determine the stress-strain behaviour of concrete under uniaxial compression, and the calculated results are compared with the experimental results under different high temperature levels. Results indicate that the proposed model can not only predict the stress-strain behaviour with acceptable accuracy in macroscopic scale, but also reveal the damage evolution mechanism in mesoscopic scale. Finally, the constitutive behaviour under biaxial compression is also simulated to investigate the influence of high temperature on biaxial stress-strain behavior and strength envelope.

Bayesian Approach To Improve The Confidence Of The Estimation Of The Shear Strength Of Coarse Mine Waste Using Barton’s Empirical Criterion

The evaluation of the shear strength of waste rock is required for the verification of the stability of high waste dumps, especially those that reach hundreds of meters in height. Mine waste rock material in open pit mining contains particles of metric scale which precludes the utilisation of commercial laboratory testing equipment. To overcome testing limitations, the shear strength of waste rock is frequently estimated using the empirical criterion of Barton-Kjærnsli. This criterion takes into consideration the nonlinearity of the shear strength envelope, characterising the behaviour of very coarse granular materials submitted to high loads. In the criterion, a stress-dependent structural component of the shear strength is parametrised with the equivalent roughness (R) and equivalent strength (S) and the structural component is added to the basic friction angle (φb) of the parental rock to determine the shear strength of the waste rock material. This paper demonstrates the use of Bayesian inference to determine the best set of parameters φb, R and S that satisfied both: large-scale laboratory testing results characterising a waste rock material, and reconciliation data from observations of stability of the waste dumps. The methodology allows the estimation of project-specific model parameters that honour both, laboratory data and site performance information. This objective is achieved through the estimation of correction factors to downgrade the strength from laboratory to field scale.

Strength characteristics of brittle rocks under multi-stage intermediate principal stress loading

A series of single-stage true triaxial compression (TTC) tests and multi-stage intermediate principal stress (σ2) loading tests were carried out on three types of brittle rock specimens in this study to examine the influence of σ2 and the validity of obtaining strength parameters by using a multi-stage loading method under true triaxial stress conditions. There exists a critical σ2 at which the rock strength attains its ultimate value under both loading conditions. Under the same σ3 level, the critical σ2 is much smaller under the multi-stage σ2 loading compared with under the single-stage TTC. The test results also show that the strength determined under multi-stage σ2 loading is much smaller than that determined under single-stage TTC. The strength difference can attain 39%. A lower strength is measured under multi-stage σ2 loading because the local fracture plane induced by true triaxial stress is parallel to the σ2, and the strength of the rock specimen containing this type of local fracture plane is not affected by σ2. On the contrary, rock damage accumulates during multiple loading and unloading cycles. The current testing results indicate that the multi-stage loading test is not suitable for determining the strength envelope of brittle rock under true triaxial conditions.

Interference of areas of subcycle light pulses

Progress in reducing the duration of light pulses (down to one cycle of field oscillations or less), achieved in the last decades, has made it possible to experimentally observe and study a number of atomic-scale phenomena that occur at ultrafast times. Their study and observation are unavailable with conventional sources of monochromatic laser radiation. When using few- or subcycle pulses, a number of well-known phenomena in optics, which are well studied when using long monochromatic radiation sources, either lose their meaning, or their underlying physical mechanisms require revision. For example, the direct interference of subcycle pulses with their direct overlap is not possible due to their short duration. This note addresses the interference phenomenon, when a pair of subcycle pulses act on a medium, while do not simultaneously overlap inside it. In this case, it is not the pulses themselves that interfere in the medium, but the probability amplitudes of the bound states of the medium. Therefore the result of their impact on the medium can be interpreted as the interference of the areas of pulses—electric pulse areas (integral of the electric field strength over time) and envelope areas (integral of the slowly varying envelope of the electric field strength over time).

Bi-linear strength envelope of coarse- and fine-grained unsaturated soils with bimodal water-retention curve

Many engineering practices are carried out within the unsaturated soil. Soil-water retention curves (SWRCs) of many soils exhibit bimodal characteristics that significantly affect strength behavior, this issue still remains unresolved. This work employs suction stress concept to examine the uniqueness of strength parameters for coarse- and fine-grained unsaturated soils with bimodal SWRCs. Capillary degree of saturation is used to upscale pore-scale suction to macroscopic suction stress. In the suction stress-shear strength plane, both coarse- and fine-grained soils display the bi-linear strength envelope, differing from unsaturated soils with unimodal SWRC.

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.

Mechanical characteristics and permeability evolution of salt rock under thermal-hydro-mechanical (THM) coupling condition

The mechanical characteristics and gas tightness of salt rock play a decisive role in the safety evaluation of salt caverns. Triaxial compression coupled permeability tests were performed on Chinese impure salt rocks under thermal-hydro-mechanical (THM) coupling condition. The results revealed that the peak strength decreased with the decrease of confining pressure and the increase of temperature. The permeability of salt rock decreased with the increase of confining pressure, while the increase of temperature led to an increase in permeability under lower effective confining pressure, and a decrease in permeability under higher effective confining pressure. Moreover, there is a critical effective confining pressure, which controls the evolution trend of salt rock permeability with temperature. The dilatancy boundary and failure strength envelope of impure salt rock at different temperatures were given, which can be expressed by exponential and hyperbolic functions respectively. An empirical formula for the evolution of the final permeability of the salt rock with the effective confining pressure at different temperatures was established. Considering the effects of effective confining pressure and inelastic volume strain, a new model was proposed to predict the permeability evolution of salt rock under THM coupling condition at different temperatures. The results of this paper can help understand the mechanical characteristics and permeability evolution of salt rock under THM coupling condition, and provide the necessary basis for the safety evaluation of salt caverns.

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.

Influence of Block form on the Shear Behaviour of Soft Soil–Rock Mixtures by 3D Block Modelling Approaches

The influence of block forms on the shear behaviour of soil–rock mixtures with soft blocks (soft S–RMs) can be efficiently investigated by the discrete element method (DEM) on the basis of accurate 3D models accounting for the block breakage. This paper proposes a novel modelling approach, based on the spherical harmonics series, for the generation of 3D block geometries with different forms but same convexity and angularity. An already existing non-overlapping modelling approach was improved, characterized by a reduced computational cost, for the set-up of 3D block DEM models accounting for the block breakage. A number of soft S–RM DEM samples, subjected to numerical direct shear tests, were generated to analyze the influence of block forms and volumetric block proportion VBP on the mesoscopic and macroscopic behaviours. The results showed that the breakage degree is maximum for the spheroidal blocks, followed by the oblate, prolate and blade ones, due to the combined influence of the block frictional sliding and rotation. The shear strength of soft S–RMs is mainly controlled by the block interlocking and breakage, being maximum in the case of spheroidal block samples when the applied normal stress is low and in the case of prolate and blade ones for a high normal stress. It was found that a nonlinear Mohr–Coulomb criterion can provide a good description of the shear strength envelope of soft S–RMs. Soft S–RMs are characterized by a higher friction angle if composed by spheroidal and prolate blocks when the VBP is 40%, due to their elevated block interlocking, and in the case of prolate and blade blocks when the VBP is 60% at the higher normal stress, due to their lower block breakage degree.

Shear strength of bentonite saturated with saline solutions exhibiting variety of cations

Investigation of saline solutions on the shear strength (SS) of compacted bentonite is desideratum to evaluate the safety of engineered barriers in order to safely dispose of the high-level radioactive waste. Undrained shear experiments were carried out on Na-bentonite saturated with CaCl2, KCl, and NaCl solutions. The stress-strain relationship of all the bentonite samples exhibits strain softening behavior, and the peak stress increases with the increase of the saline solution concentration. The SS of samples in saline solutions at 0.1 mol/L (M) follows the order of NaCl< KCl < CaCl2, whereas, in 0.5 M salt solutions it follows the order of NaCl<CaCl2 < KCl, indicating that the cation exchange process has a major impact on the SS of bentonite. The modified effective stress concept p’ was used to express the SS in various salt solutions based on Mohr-Coulomb criterion. The results further revealed that the solution concentration had minimal effect on the angle of internal friction φ’. In addition, the strength envelope remains the same when samples are saturated with different concentrations of NaCl solution, unlike the samples saturated with KCl and CaCl2. At lower salinity concentration (0.1 M), the SS in CaCl2 solution approaches maximum value because of higher osmotic suction. On the contrary, the SS of bentonite saturated with KCl attains the highest value at high concentration (0.5 M), which is attributed with strong K-linkage and low hydration energy of potassium ion.

A constitutive model for geosynthetic interfaces considering nonlinear softening behavior

Geosynthetic interfaces within composite liner systems usually play an important role in failure events of landfills and should be well considered in stability assessment. A constitutive model based on the disturbed state concept is proposed to describe the shear behavior of geosynthetic interfaces under both static and dynamic conditions. The interface behavior is assumed to be governed by two states, i.e. the intact state and the fully disturbed state. The intact state is described with a bounding surface model with vanishing elastic range which has the advantages of a concise mapping rule and fewer parameters, and the fully disturbed state is characterized by a nonlinear strength envelope. The two states share the same normal stress, but the macro shear stress of the geosynthetic interface is the coupling result of the two states through a disturbance function related to accumulative plastic shear displacement. The model performs well in describing the monotonic shear behavior of typical geosynthetic interfaces for different drainage conditions and stress paths. Furthermore, a series of cyclic shear tests and shaking table tests on typical geosynthetic interfaces are also modeled, and the results indicate that the dynamic response and softening behavior can be well described with this model.

Artificial neural network–based constitutive relation modelling for the laminated fabric used in stratospheric airship

There have been gradually increasing interests in the stratospheric airship (SSA) as a cost-effective alternative to earth orbit satellites for telecommunication and high-resolution earth observation. Lightweight and high strength envelopes are the keys to the design of SSAs as it directly determines the endurance flight performance and loading deformation characteristics of the airship. Typical SSA envelope material is a laminated fabric, which is composed of fabric layer and other functional layers. Compared with conventional composite structures, the laminated fabric has complex nonlinear mechanical characteristics. Artificial neural network (ANN) has good processing ability to nonlinear information so that it is suitable to model the constitutive relation of laminated fabrics. In this work, an ANN based on the Scaled Conjugate Gradient (SCG) algorithm is proposed firstly to model the constitutive relation of fabric Uretek3216LV. Considering significant errors of the SCG ANN results, the network model is optimized through methods of selecting the number of hidden-layer nodes and training algorithms. Results show that the improved network model based on Bayesian Regularization (BR) algorithm and eight nodes of single hidden layer can better describe the constitutive relation of the laminated fabric than other conventional training algorithms. The proposed constitutive modelling method with ANN is expected to gain a deeper understanding of the mechanical mechanism and guide structural design of envelope material in further work.

Homogenization based kinematic limit analysis for finding the stability of an embankment over soft clay reinforced with granular columns

The present study examines the stability of an embankment of cohesive-frictional material placed over a purely cohesive stratum which is reinforced by means of uniformly spaced vertical granular cylindrical columns. By using the kinematic yield homogenization concept, the macroscopic yield strength envelope associated with the reinforced clay layer has been obtained numerically for different volumetric fractions of the granular column to the composite unit cell by using the semidefinite programming (SDP) technique. An upper bound approximation of the macroscopic yield criterion, using the least square method, is expressed as a summation of a number of convex ellipsoidal envelopes so that the solution for computing the stability factors for the embankment on granular column (in the presence of surcharge pressure) can be obtained later with the usage of the second order cone programming (SOCP) while performing the upper bound finite elements limit analysis. The methodology avoids the usage of any stress variable and the complete procedure has been clearly described and the associated code has been written in MATLAB. The study will be not only useful for designing embankments constructed over soft clay reinforced with granular columns but also for solving different geomechanics stability problems by using the homogenization based kinematic limit analysis approach.