Is there Link between the Type of the Volumetric Strain Curve and Elastic Constants, Porosity, Stress and Strain Characteristics ?

The stress [crack damage stress (σ _cd) and uniaxial compressive strength (σ _c)] and strain characteristics [maximum total volumetric strain (ε _cd), axial failure strain (ε _af)], porosity (n) and elastic constants [elastic modulus (E) and Poisson’s ratio (ν)] and their ratios were coordinated with the existence of two different types (type 1 and type 2) of volumetric strain curve. Type 1 volumetric strain curve has a reversal point and, therefore, σ _cd is less than the uniaxial compressive strength (σ _c). Type 2 has no reversal point, and the bulk volume of rock decreases until its failure occurs (i.e., σ _cd = σ _c). It is confirmed that the ratio between the elastic modulus (E) and the parameter λ = n/ε _cd strongly affects the crack damage stress (σ _cd) for both type 1 and type 2 volumetric strain curves. It is revealed that heterogeneous carbonate rock samples exhibit different types of the volumetric strain curve even within the same rock formation, and the range of σ _cd/σ _c = 0.54–1 for carbonate rocks is wider than the range (0.71 < σ _cd/σ _c < 0.84) obtained by other researchers for granites, sandstones and quartzite. It is established that there is no connection between the type of the volumetric strain curve and values of n, E, σ _cd, ν, E/(1 ? 2ν), M _R = E/σ _c and E/λ. On the other hand, the type of volumetric strain curve is connected with the values of λ and the ratio between the axial failure strain (ε _af) and the maximum total volumetric strain (ε _cd). It is argued that in case of small ε _af/ε _cd–small λ, volumetric strain curve follows the type 2.     

岩石脆性临界破坏的波速特征分析

根据岩石加载破坏过程中应力-应变和波速-应变曲线所反映的的波速变化特征,研究了岩石声波传播速度与其加载变形过程的相关性。研究表明:岩石破裂前波速及特征参数随着变形破坏阶段变化会有显著改变,特别是应力水平到达70%⁸0%之后,当加载破坏过程中岩石的裂隙大量增加后波速参数出现突变。这是由于在膨胀点附近岩石中微裂隙迅速丛集式增长,改变了岩石内部的微观结构,导致岩石波速降低。伴随着波速降低,波速走时急剧增大,离散度增大;S波和P波的振幅减小,但其振幅比急剧增大;S波、P的波Q值也会出现突变。通过对岩石加载过程中的波速变化与应力-应变曲线之间的关联性分析,利用波速信息变化可以实现对岩石脆性破裂前兆的识别。     

The relationship between the deformation mechanism and permeability on brittle rock

In order to study the relationship between the deformation mechanism and permeability on brittle rock, the curves of stress–strain and permeability–strain of brittle rock were analyzed, the results indicated that the permeability of brittle rock increased rapidly at the critical failure point due to the microstructure changed seriously during the compaction, elastic deformation, and progressive failure process. When the rock bridges or asperities on the internal structural plane were ruptured partly or completely after the peak stress, a sudden decrease in rock strength occurred and the peak permeability could be the maximum. The strain at the critical point and turning point on the stress–strain curve corresponded to that at the rapid increase point and the peak point on the permeability–strain curve according to the renormalization theory and curves of stress–strain and permeability–strain. In addition, the ratios could be described by mathematical expressions. The conclusions could be proved reasonably by the experimental studies and examples of statistical analysis.     

Study of nonlinear behaviors of geological materials

Two polynomial constitutive equations of nonlinear stress–strain relationship were used to construct two nonlinear 1-D wave equations with external pressure (source term) applied. Nonlinear model 1 was a concave downward curve, and nonlinear model 2 was a concave upward curve. The time-dependent stress and strain of a 300-m length were calculated. The computation for nonlinear model 1 terminated at time t?=?5.0 s. The evaluated stress versus position at different times was mainly at position x?=?0 m. The stress versus position had a concave point at position x?=?0.3 m. Between x?=?0 m and x?=?0.3 m, the graph of stress versus position showed a small convex upward curve. Moreover, at the two sides of x?=?0.3 m, the strain position had distinctively different slopes. The distinctive difference in the slope of strain at position x?=?0.3 m can therefore be used to estimate the rupture position of a rock. The terminal evaluation time for nonlinear model 2 was at t?=?1.55 s. The stress versus position and the strain versus position to time change were within the 0–3 m interval from the pressure end. Time increase produced the phenomenon of stress and strain solitons. These stress and strain solitons moved forward and increased in peak value with time. During the compression process, the stress and strain soliton resulted in instability which rendered the rock situation more easily broken. The position of the rock rupture might have occurred away from the pressure side.     

New models to predict rock failure by linking the volume dilatant point with the peak stress point

The rock mass failure process can be divided into several distinct deformation stages: the compaction stage, elastic stage, stable failure stage, accelerated failure stage, and post-peak stage. Although each stage has been well studied, the relationship among the stages has not been established. Here, we establish two models which are the Strain model Q and Energy density model S by using the renormalization group theory and investigate the mechanical relationship between the volume dilatant point and peak stress point on the rock stress-strain curve. Our models show that the strain ratio (ε _f /ε _c ) and energy ratio (E _f /E _c ) at the volume dilatant point and peak stress point are solely functions of the shape parameter m. To verify our models, we further studied the failure process of rock specimens through several uniaxial compression experiments and found that the relationship between ε _f /ε _c or E _f /E _c and m shares a notably similar pattern to that from our theoretical model. However, the ε _f /ε _c and E _f /E _c values in our experiments are slightly smaller than those predicted by the models. In brief, we demonstrate that our models can be used to predict the failure process of the laboratory-scale hard brittle rock samples.     

Experimental Study on Acoustoelastic Character of Rock Under Uniaxial Compression

This paper made a research about the change rule of elastic wave velocity with stress applied on rock from theoretical and experimental aspect. Firstly, a mathematical model of P-wave velocity and confining pressure of rock was set up from the point of acoustoelastic character. Effect of axial stress on P-wave velocity in granite and sandstone during uniaxial compression process was studied experimentally by using GAW-2000 rock mechanical testing system and RSM-SY5 ultrasonic wave testing system, and the relation curves of axial stress with P-wave velocity were obtained. Based on test data, acoustoelastic theoretical formulas of granite and sandstone were established and the best empirical formulas were fitted by using regression method. Meanwhile, a comparative analysis of the empirical and theoretical calculated values was carried out. Finally, the reliability of applying acoustoelastic theoretical formula in hard rock range was further verified based on the experimental data of granitic gneiss. The results show that the P-wave velocity experiences a rapid increase, gentle increase and then a sharp fall during the uniaxial compression process. The B-value in acoustoelastic theoretical formula (proportion coefficient determined by elastic modulus and third-order elastic constant) decreases exponentially with axial stress. The acoustoelastic theoretical formula can effectively reflect the relationship between rock acoustic velocity and stress within the allowable error, which can be the theoretical foundation of acoustoelastic geo-stress measurement of subsurface rock mass.     

Bonding degradation and stress–dilatancy response of weakly cemented sands

The progressive bond breakage of artificially cemented sands induced by shear straining was investigated through conventional isotropically consolidated drained triaxial compression tests. Sand specimens were prepared with a low degree of cementation by adopting a chemical grout. Test results were interpreted in terms of two stress–dilatancy theories for cohesive-frictional materials proposed in literature. The influence of debonding on the stress–dilatancy behaviour of cemented sands was analysed with particular emphasis on the ‘delayed dilatancy’ phenomenon. A bonding degradation curve was determined for each test relating the interparticle cohesion (c) to the magnitude of the total plastic strain vector (ε_d) and a bond degradation rate factor (D_c) was assessed from each curve. The maximum value of interparticle cohesion (c₀) before the onset of bond degradation under shearing was found to correspond with a sharp decrease in the soil stiffness of the specimens. The influence of the effective confining stress (p^′_c) on both c₀ and D_c parameters gathered from each test was also ascertained.     

Comparison Between Stress and Strain Quantities of the Failure–Deformation Process of Fennoscandian Hard Rocks Using Geological Information

The aim of this paper was to compare the stress and strain quantities that are related to the failure–deformation process of hard rock. The data used here was obtained from laboratory uniaxial compression tests performed on different types of Fennoscandian hard rocks. The failure–deformation process quantities were compared at each deformation stage and for each single specimen. Moreover, geological information such as the rock origin process and the rock characteristics of the specimens were studied and linked to the stress and strain quantities. The purpose was to investigate the influence of the rock origin process and rock characteristics on these quantities. The main results of this study showed that the normalized crack damage lateral strain (ε _3cd/ε _3p ) and the volumetric strain (ε _crv?ci and ε _v?cd) quantities were strongly affected by the grain size. The normalized and volumetric quantities are weakly dependent on the mineral composition.     

A Criterion for Brittle Failure of Rocks Using the Theory of Critical Distances

This paper presents a new analytical criterion for brittle failure of rocks and heavily over-consolidated soils. Griffith’s model of a randomly oriented defect under a biaxial stress state is used to keep the criterion simple. The Griffith’s criterion is improved because the maximum tensile strength is not evaluated at the boundary of the defect but at a certain distance from the boundary, known as half of the critical distance. This fracture criterion is known as the point method, and is part of the theory of critical distances, which is utilised in fracture mechanics. The proposed failure criterion has two parameters: the inherent tensile strength, σ ₀, and the ratio of the half-length of the initial crack/flaw to the critical distance, a/L. These parameters are difficult to measure but they may be correlated with the uniaxial compressive and tensile strengths, σ _c and σ _t. The proposed criterion is able to reproduce the common range of strength ratios for rocks and heavily overconsolidated soils (σ _c/σ _t = 3–50) and the influence of several microstructural rock properties, such as texture and porosity. Good agreement with laboratory tests reported in the literature is found for tensile and low-confining stresses.     

Memory Effects in Rock Salt Under Triaxial Stress State and Their Use for Stress Measurement in a Rock Mass

Summary Regularities of memory effects in rock salt specimens under triaxial stress state were investigated. Each specimen was subjected to two loading cycles. The first cycle was axisymmetric triaxial compression (σ₁ >σ₂=σ₃). The second cycle was uniaxial compression in the direction of σ₁ of the first cycle. Distinct acoustic emission (AE) and deformation memory effects were observed in the second cycle at the stress value equal to a linear combination of the first cycle principal stresses given by σ₁− (k + 1) σ₃, where k is about 0.5–0.6 for rock salt. Anomalies in deformation curves were found to be more reliable than the AE methods in distinguishing memory symptoms. The necessary pre-requisite for memory formation in the first cycle was that σ₁ exceeded the elastic limit, corresponding to the given confining stress σ₃. Inflections in uniaxial stress versus axial strain and lateral strain curves, in the second cycle, were observed at equal stress values if in the first cycle σ₁ exceeded the elastic limit and memory-forming damage was induced. If there was no memory-forming damage, those inflections were seen at different stress values. This characteristic was used to distinguish between true memory effects and natural characteristic points in deformation curves derived from rock salt testing. A new memory symptom was established, namely a turn point in curve “uniaxial stress versus differential coefficient of lateral strains”. The results form a basis for application of the memory effects for stress measurement in rock salt masses.     

Development of a Nonlinear Model for Soft Rock and its Applications

A phenomenological model has been developed for soft rock based on the results of a series of triaxial compression (TC) tests conducted on Kobe sandstone with a very high precision measurement. From the analysis and interpretation of the test results, it has been found that small strain Young’s modulus (E^e) was a function of the major principal stress. E^e for elastic strains of soft rock was assumed to be cross-anisotropic. A damage function has been used to derive the appropriate elastic Young’s modulus when subjected to shear loading. As the basic stress–strain relation, the relationship between the tangent modulus and the shear stress level was used. The differential form of which was subsequently integrated by a 4th order Runge–Kutta solver to obtain the stress–strain relation. The model of soft rock is based on an isotropic strain hardening elasto-plastic framework which takes into account the pressure sensitivity, cross-anisotropy, degradation of Young’s modulus with the degree of mobilized shear stress and the nonlinearity of the shear stress-shear strain relationship. Although the model was developed from the analysis of the TC tests results of Kobe sandstone, it was also applied to the other types of soft rock or stiff geomaterials. Plate loading tests were conducted at a level of 61 m below the ground level at the bottom of a large excavated shaft at four locations. Finally, the model was used to simulate the plate loading test results successfully. This model was successfully calibrated with Akashi sandstone and applied in the simulation for the settlement of Akashi-Kaikyo Bridge piers. The simulations were carried out for both drained and undrained condition by changing the Poisson’s ratio. The layering information beneath the foundations were used in the FEM simulation. The use of very accurate Young’s modulus from the field shear wave velocity test was the key to the successful simulation of the settlement under bridge pier foundations.     

泊松比对岩样破坏模式及全部变形特征的影响

利用编写的计算岩样全部变形特征的FISH函数, 采用FLAC模拟了泊松比不同时单缺陷岩石试样的破坏及全部变形特征。在峰前及峰后, 本构模型分别取为线弹性模型及莫尔库仑剪破坏与拉破坏复合的应变软化模型。高泊松比使岩样发生由单一剪切破坏向复杂破坏转变、破坏区域的面积增加、剪切带倾角降低, Coulomb、Roscoe及Arthur理论对此无法解释。不同泊松比时计算得到的峰前应力-轴向应变曲线、应力-侧向应变曲线、侧向应变-轴向应变曲线、体积应变-轴向应变曲线的线性阶段与平面应变压缩条件下的线弹性解吻合。若泊松比超过1/3, 通过计算得到的平面应变压缩泊松比可大于0.5, 这被数值模拟确认。泊松比的增加使峰后的侧向应变-轴向应变曲线、体积应变-轴向应变曲线、计算得到的泊松比-轴向应变曲线变得不陡峭, 使峰后的应力-侧向应变曲线变得陡峭, 使破坏的前兆变得不明显。      

Experimental and theoretical investigation on the compression behavior of sand-marine clay mixtures within homogenization framework

In this paper, the compression behavior of sand-marine clay mixtures was investigated, both experimentally and theoretically. The test data reveal that the Normal Compression Line of a sand-clay mixture depends on both the sand fraction and the initial water content of the clay matrix. The local stress in the clay matrix σ′_c is approximately close to the overall stress of the sand-clay mixture σ′ for a sand mass fraction of 20%. The stress ratio, σ′_c/σ′, falls significantly with increasing overall stress for a sand fraction of 60%, which may be attributed to the formation of clay bridges between adjacent sand particles. A compression model was formulated within the homogenization framework. First, a homogenization equation was proposed, which gives a relationship between the overall stiffness E and that of the clay matrix Ec. Then, a model parameter ξ was incorporated considering the sensitivity of the structure parameter on the volume fraction of the clay matrix. Finally, a simple compression model with three model parameters was formulated using the tangent stiffness. Comparisons between the experimental data and simulations reveal that the proposed model can well represent the compression curves of the sand-marine clay mixtures observed in the laboratory.     

Dynamic soil properties for microzonation of Delhi,India

Delhi, the capital of India, has experienced mild seismic shaking during several earthquakes in the past. The large variations of depth to bedrock and ground water table coupled with different soil types at different locations of Delhi necessitate a seismic microzonation study. Dynamic soil properties such as shear wave velocity, modulus reduction and damping characteristics of local soils are the basic and essential input parameters for conducting even a preliminary ground response analysis which is an essential input in microzonation studies. Shear wave velocity is not measured routinely due to its high cost and lack of the required expertise. Several researchers in the past developed correlations between shear wave velocity (V _s ) and routinely measured N values. In the present study, shear wave velocity profiles measured in the field at more than 80 borehole locations to a depth of about 20 to 32m using Spectral Analysis of Surface Waves (SASW) are presented and correlations between shear wave velocity and N values are also presented for use by engineers and designers. Results of strain and stress controlled cyclic triaxial tests on remoulded samples of sand-silt mixtures in the high strain range are used for generating the modulus reduction and damping curves and are compared with the well-known curves in the literature. The results presented in this article can be used for microzonation studies as well as site specific ground response analyses at Delhi.     

Laboratory Validation of Ultra-Soft Soil Deformation Model

Ultra-soft soil with high moisture content will experience large strain deformation under one-dimensional compression with little or no gain in effective stress. Such deformation behaviour does not comply with Terzaghi’s effective stress gain theory. The e-log s_v^￠ \sigma_{v}^{\prime } relationship of ultra-soft soil is non-linear with large compression index in the first order of log cycle. This paper proposes three compression indices (C_c1^* C_{c1}^{*} , C_c2^* C_{c2}^{*} and C_c3^* C_{c3}^{*} ) for stresses covering three log cycles. Good prediction of settlement magnitude is possible with these newly proposed compression parameters for ultra-soft soil. In addition, implicit finite difference model applying the large strain theory is also proposed and validated with results from laboratory measurements. The time factor curves for ultra-soft soil with large strain compression are also proposed and validated.     

Influence of different strain rates on hydro-mechanical behaviour of reconstituted unsaturated soil

The hydro-mechanical behaviour of a reconstituted unsaturated soil under different suctions and strain rates was studied through various rate-controlled unsaturated/undrained triaxial tests. The fully saturated reconstituted specimens were desaturated to four different initial suctions (s₀?=?0, 100 kPa, 200 kPa and 300 kPa) and then triaxially sheared (conventional triaxial compression) at three different strain rates in undrained conditions (\(\dot{\varepsilon }_{1} = 0.001\) h^?1, 0.01 h^?1, and 0.1 h^?1). The observed hydro-mechanical behaviour during shearing including the volumetric strain, deviatoric stress, degree of saturation and suction is presented and discussed in this paper. The results indicate that when the strain rate rises at the given initial suctions (or pore water pressures), the maximum deviatoric stress (q_max), critical net stress ratio (M) and critical state suction (s_c) increase but the degree of saturation (S_rc) and volumetric strain at the critical state (ε^c_v ) reduce. The critical effective stress ratio (M′) is not dependent on the strain rate for saturated and unsaturated samples. The critical state lines for unsaturated soils with the constant strain rates are parallel with each other in the e???lnp′ space.

     

Effects of salt solutions on the hydro-mechanical behavior of compacted GMZ01 Bentonite

In this study, the effects of salinity of infiltrating solutions on the swelling strain, compressibility, and hydraulic conductivity of compacted GMZ01 Bentonite were investigated. After swelling under vertical load using either distilled water or NaCl solutions with concentrations of 0.1, 0.5 M, and 1 M, laboratory oedometer tests were conducted on the compacted GMZ01 Bentonite. Based on the oedometer test results, hydraulic conductivity was determined using the Casagrande’s method. Results show that the swelling strain of highly compacted GMZ01 Bentonite decreases as the concentration of NaCl solution increases. The compression index C _c ^* increases and then turns to decrease with an increase in the vertical stress or a decrease in the void ratio for different solutions, and the C _c ^* decreases as the concentration of NaCl solution increases. The secondary consolidation coefficient C _α increases linearly with the increase of the compression index C _c ^* . Furthermore, a bi-linear relationship between the swelling index C _s ^* and the secondary consolidation coefficient C _α can be characterized clearly. The hydraulic conductivity increases as the concentration of NaCl solution increases, however, this increase can be prevented if a high confining stress is applied.     

An Experimental Study of the Fracture Coalescence Behaviour of Brittle Sandstone Specimens Containing Three Fissures

To analyse the fracture coalescence behaviour of rock, rectangular prismatic sandstone specimens (80?×?160?×?30?mm in size) containing three fissures were tested under uniaxial compression. The strength and deformation behaviours of the specimens are first analysed by investigating the effects of the ligament angle β₂ on the peak strength, peak strain and crack initiation stress of the specimens. To confirm the sequence of crack coalescence, a photographic monitoring technique is used throughout the entire period of deformation. Based on the results, the relationship between the real-time crack coalescence process and the axial stress–strain curve of brittle sandstone specimens is also developed, and this relationship can be used to evaluate the macroscopic deformation characteristics of pre-cracked rock. The equivalent strain evolution fields of the specimen, with α?=?β₁?=?45° and β₂?=?90°, are obtained using the digital image correlation technique and show good agreement with the experimental results of pre-cracked brittle sandstone. These experimental results are expected to improve the understanding of fracture mechanisms and be used in rock engineering with intermittent structures, such as deep underground excavated tunnels.     

Application of the strain energy to estimate the rock load in squeezing ground condition of Eamzade Hashem tunnel in Iran

Estimation of rock load is a very important issue because the selection of a support system is highly related to this parameter. Several methods are used to estimate this parameter such as experimental, empirical, and numerical methods. This study propose a new empirical method to estimate the rock load in squeezing ground condition using actual collapses data of Emamzade Hashem tunnel of Iran based on the ration of the post-failure residual strain energy to the pre-failure stored strain energy. Prediction of squeezing ground condition in this study is performed based on Jethwa, Singh, and Hoek criterions. Results show that some sections in shale and sandstone of the Shemshak formation are prone to squeezing. Finally, the relation between the rock load and the ratio of the post-failure residual strain energy to the pre-failure stored strain energy, Ψ, in squeezing ground condition is estimated. Based on the statistical analysis, the maximum correlation between both parameters is achieved when Alejano’s equations are used to estimate the drop modulus. As the rock mass behavior changes from elastic–plastic to elastic–brittle, the drop modulus changes from 0 to infinite. The reason is that by increasing the quality of rock mass and reducing the minimum principal stresses, the ratio of post-failure residual strain energy to pre-failure stored strain energy and rock load height (H _p) reduce. So, regression analysis is used to investigate the relation between the rock load height and the ratio of post-failure residual strain energy to pre-failure stored strain energy, and finally, a formulation is presented to determine rock load height based on power function.     

Stress–Strain Model for Carbonate Rocks Based on Haldane’s Distribution Function

Summary. This paper presents a model created by the author to predict stress–strain relationships for weak to strong carbonate rocks (σ_c < 100 MPa) exhibiting axial strains up to 1%. The stress–strain model based on Haldane’s distribution function (Haldane, 1919) relates the axial stress (or normalized axial stress) to the square of an exponential function where the exponent is axial strain. To obtain accurate stress–strain relationship over the whole pre-failure strain with the proposed stress–strain model, it is necessary to have only one datum point (peak axial stress and maximum axial strain at this peak stress). It is shown that the stress–strain relationships observed in laboratory compression tests on samples collected from six carbonate rock formations (chalk, dolomites and limestones) from different parts of Israel, agree well with the stress–strain prediction model proposed by the author.