A thermo-plastic constitutive law for brittle-plastic behavior of rocks at high temperatures

Mechanical properties of rocks change under the influence of, temperature. Stress at the onset of yielding, ultimate strength, dilatancy, strain hardening and softening, and the confining pressure at brittle-ductile transition are all reduced by the increasing temperature. This study presents a framework of constitutive modeling of thermo-brittle-plastic behavior of rocks which encompasses these changes. The constitutive law is based on a thermo-plasticity theory first proposed for metals byPrager (1958). Two phenomenological mechanisms have been identified as central for the modeling: temperature dependence of the yield locus (thermal softening), and temperature dependence of the strain-hardening function (thermally enhanced ductility). Material parameters for two rocks, Carrara marble and Westerly granite, were determined on the basis of additional hypotheses. These parameters are used in numerical simulations of triaxial tests at different temperatures. The obtained stress-strain curves compare well to the experimental results. The changes with temperature in the stress at the onset of yielding are more accurately reproduced that the evolution of hardening or softening. Suggestions for possible improvements and future research directions are indicated.     

华北地区四种岩石在高压下的破坏特征

利用我国第一台1GPa高压三轴容器进行了岩石破坏实验.所用的样品为济南辉长岩、昌平花岗岩、房山大理岩和周口店石灰岩.得到的主要结果是:(1)在相应于地壳的压力范围内辉长岩和花岗岩仍然发生脆性破坏, 其强度和韧度随围压增加而增加, 破坏角也随围压增加而增大, 破坏前发生体积膨胀现象并出现声发射活动增加的前兆;(2)大理岩和石灰岩样品在0.1——0.8GPa的围压范围内发生鼓状变形, 不形成明显的主断层面, 样品承载能力随变形增加而增加, 出现应变硬化现象.在变形过程中声发射活动水平极低。这些实验标志着我国岩石力学三轴实验的围压水平有所提高其结果对震源物理的研究具有参考价值.      

The Elastic Strain Energy of Damaged Solids with Applications to Non-Linear Deformation of Crystalline Rocks

Laboratory and field data indicate that rocks subjected to sufficiently high loads clearly deviate from linear behavior. Non-linear stress–strain relations can be approximated by including third and higher-order terms of the strain tensor in the elastic energy expression (e.g., the Murnaghan model). Such classical non-linear models are successful for calculating deformation of soft materials, for example graphite, but cannot explain with the same elastic moduli small and large non-linear deformation of stiff rocks, such as granite. The values of the third (higher-order) Murnaghan moduli estimated from acoustic experiments are one to two orders of magnitude above the values estimated from stress–strain relations in quasi-static rock-mechanics experiments. The Murnaghan model also fails to reproduce an abrupt change in the elastic moduli upon stress reversal from compression to tension, observed in laboratory experiments with rocks, concrete, and composite brittle material samples, and it predicts macroscopic failure at stress levels lower than observations associated with granite. An alternative energy function based on second-order dependency on the strain tensor, as in the Hookean framework, but with an additional non-analytical term, can account for the abrupt change in the effective elastic moduli upon stress reversal, and extended pre-yielding deformation regime with one set of elastic moduli. We show that the non-analytical second-order model is a generalization of other non-classical non-linear models, for example “bi-linear”, “clapping non-linearity”, and “unilateral damage” models. These models were designed to explain the abrupt changes of elastic moduli and non-linearity of stiff rocks under small strains. The present model produces dilation under shear loading and other non-linear deformation features of the stiff rocks mentioned above, and extends the results to account for gradual closure of an arbitrary distribution of initial cracks. The results provide a quantitative framework that can be used to model simultaneously, with a small number of coefficients, multiple observed aspects of non-linear deformation of stiff rocks. These include, in addition to the features mentioned above, stress-induced anisotropy and non-linear effects in resonance experiments with damaged materials.     

The Effect of Varying Damage History in Crystalline Rocks on the P- and S-Wave Velocity under Hydrostatic Confining Pressure

Cracks play a very important role in many geotechnical issues and in a number of processes in the Earth’s crust. Elastic waves can be used as a remote sensing tool for determining crack density. The effect of varying crack density in crystalline rock on the P- and S-wave velocity and dynamic elastic properties under confining pressure has been quantified. The evolution of P- and S-wave velocity were monitored as a suite of dry Westerly granite samples were taken to 60, 70, 80 and 90 % of the unconfined uniaxial strength of the sample. The damaged samples were then subjected to hydrostatic confining pressure from 2 MPa to 200 MPa to quantify the effect of varying crack density on the P- and S-wave velocity and elastic properties under confining pressure. The opening and propagation of microcracks predominantly parallel to the loading direction during uniaxial loading caused a 0.5 and 6.3 % decrease in the P- and S-wave velocity, respectively. During hydrostatic loading, microcracks are closed at 130 MPa confining pressure. At lower pressures the amount of crack damage in the samples has a small but measureable effect. We observed a systematic 6 and 4 % reduction in P- and S-wave velocity, respectively, due to an increase in the fracture density at 2 MPa confining pressure. The overall reduction in the P- and S-wave velocity decreased to 2 and 1 %, respectively, at 50 MPa. The elastic wave velocities of samples that have a greater amount of microcrack damage are more sensitive to pressure. Effective medium modelling was used to invert elastic wave velocities and infer crack density evolution. Comparing the crack density results with experimental data on Westerly granite samples shows that the effective medium modelling used gave interpretable and reasonable results. Changes in crack density can be interpreted as closure or opening of cracks and crack growth.     

Simulation of torsional shear test results with neuro-fuzzy control system

In the first part of this study, a series of stress-controlled hollow cylinder cyclic torsional triaxial shear tests were conducted on loose to medium dense saturated samples of clean Toyoura sand to investigate its liquefaction behavior. A uniform cyclic sinusoidal loading at a 0.1 Hz frequency was applied to air-pluviated samples where confining pressure and relative density was varied. Cyclic shear stress–strain changes, the number of cycles to reach liquefaction and pore pressure variations were recorded. Results indicate that the liquefaction resistances of uniform sands are significantly affected by the method of sample preparation and initial conditions.     

Modelling of Strong Ground Motions from 1991 Uttarkashi,India, Earthquake Using a Hybrid Technique

We present a simple and efficient hybrid technique for simulating earthquake strong ground motion. This procedure is the combination of the techniques of envelope function (Midorikawa et al. Tectonophysics 218:287–295, 1993) and composite source model (Zeng et al. Geophys Res Lett 21:725–728, 1994). The first step of the technique is based on the construction of the envelope function of the large earthquake by superposition of envelope functions for smaller earthquakes. The smaller earthquakes (sub-events) of varying sizes are distributed randomly, instead of uniform distribution of same size sub-events, on the fault plane. The accelerogram of large event is then obtained by combining the envelope function with a band-limited white noise. The low-cut frequency of the band-limited white noise is chosen to correspond to the corner frequency for the target earthquake magnitude and the high-cut to the Boore’s f _max or a desired frequency for the simulation. Below the low-cut frequency, the fall-off slope is 2 in accordance with the ω² earthquake source model. The technique requires the parameters such as fault area, orientation of the fault, hypocenter, size of the sub-events, stress drop, rupture velocity, duration, source–site distance and attenuation parameter. The fidelity of the technique has been demonstrated by successful modeling of the 1991 Uttarkashi, Himalaya earthquake (Ms 7). The acceptable locations of the sub-events on the fault plane have been determined using a genetic algorithm. The main characteristics of the simulated accelerograms, comprised of the duration of strong ground shaking, peak ground acceleration and Fourier and response spectra, are, in general, in good agreement with those observed at most of the sites. At some of the sites the simulated accelerograms differ from observed ones by a factor of 2–3. The local site geology and topography may cause such a difference, as these effects have not been considered in the present technique. The advantage of the technique lies in the fact that detailed parameters such as velocity-Q structures and empirical Green’s functions are not required or the records of the actual time history from the past earthquakes are not available. This method may find its application in preparing a wide range of scenarios based on simulation. This provides information that is complementary to the information available in probabilistic hazard maps.     

Ultrasonic Velocities, Acoustic Emission Characteristics and Crack Damage of Basalt and Granite

Acoustic emissions (AE), compressional (P), shear (S) wave velocities, and volumetric strain of Etna basalt and Aue granite were measured simultaneously during triaxial compression tests. Deformation-induced AE activity and velocity changes were monitored using twelve P-wave sensors and eight orthogonally polarized S-wave piezoelectric sensors; volumetric strain was measured using two pairs of orthogonal strain gages glued directly to the rock surface. P-wave velocity in basalt is about 3 km/s at atmospheric pressure, but increases by > 50% when the hydrostatic pressure is increased to 120 MPa. In granite samples initial P-wave velocity is 5 km/s and increases with pressure by < 20%. The pressure-induced changes of elastic wave speed indicate dominantly compliant low-aspect ratio pores in both materials, in addition Etna basalt also contains high-aspect ratio voids. In triaxial loading, stress-induced anisotropy of P-wave velocities was significantly higher for basalt than for granite, with vertical velocity components being faster than horizontal velocities. However, with increasing axial load, horizontal velocities show a small increase for basalt but a significant decrease for granite. Using first motion polarity we determined AE source types generated during triaxial loading of the samples. With increasing differential stress AE activity in granite and basalt increased with a significant contribution of tensile events. Close to failure the relative contribution of tensile events and horizontal wave velocities decreased significantly. A concomitant increase of double-couple events indicating shear, suggests shear cracks linking previously formed tensile cracks.     

不同围压条件下的圆形巷道岩爆过程模拟

利用FLAC模拟了不同围压条件下圆形巷道的岩爆过程。为了模拟巷道开挖,利用编写的FISH函数删除巷道内部的单元。岩石服从摩尔库仑剪破坏与拉破坏复合的破坏准则,破坏之后呈现应变软化-理想塑性行为。模拟结果表明：当围压较低时,剪切应变集中区域呈圆环状,围岩能保持稳定,不出现剪切带;当围压增加到一定程度时,围岩中出现了“狗耳”形的V形坑,发生岩爆,但围岩也还能保持稳定;当围压进一步增加时,围岩中出现了多条狭长的剪切带,巷道的整个断面均遭到了破坏,发生强烈的岩爆。随着围压的增加,V形岩爆坑变大、变深,剪切带花样的对称性变差;在高围压时,剪切带花样与塑性力学中的滑移线网有类似之处。     

Stiffness degradation of natural fine grained soils during cyclic loading

Cyclic behavior of natural fine grained soils under a broad range of strains were investigated considering the effects of plasticity index and changes in confining pressures based on cyclic triaxial tests. A total of 98 stress controlled cyclic triaxial tests were conducted on normally consolidated and slightly overconsolidated samples. The investigation was divided into two parts. The first part consists of stress controlled cyclic triaxial tests under different stress amplitudes that were conducted to estimate the modulus reduction and the thresholds between nonlinear elastic, elasto-plastic and viscoplastic behavior. The second part involves the investigation of the undrained stress–strain behavior of fine grained soils under irregular cyclic loadings. The results showed that the elastic threshold is approximately equal to 90% of G_max. Another transition point was defined as the flow threshold where the value of tangent of shear modulus ratio changes for the second time. Simple empirical relationships to estimate the dynamic shear modulus and damping ratio was formulated and compared with the similar empirical relationships proposed in the literature. The results provide useful guidelines for preliminary estimation of dynamic shear modulus and damping ratio values for fine grained soils based on laboratory tests.     

岩石损伤模量分析

从能量的角度出发定义损伤变量,通过应力-应变曲线卸载段有效应力与总应力之间的假定关系,确定损伤模量的计算方法。在单轴和三轴条件下,对岩石损伤相关参数进行统计研究,重点对损伤模量进行计算和对比分析。结果表明,单轴及低围压下损伤模量基本呈逐渐减小的趋势,而随着围压的增大,损伤模量呈先减小后保持动态稳定的趋势。损伤变量的计算充分考虑到围压的影响,可进一步增进对岩石损伤的认识。     

Fracture in Westerly Granite under AE Feedback and Constant Strain Rate Loading: Nucleation, Quasi-static Propagation, and the Transition to Unstable Fracture Propagation

New observations of fracture nucleation are presented from three triaxial compression experiments on intact samples of Westerly granite, using Acoustic Emission (AE) monitoring. By conducting the tests under different loading conditions, the fracture process is demonstrated for quasi-static fracture (under AE Feedback load), a slowly developing unstable fracture (loaded at a `slow' constant strain rate of 2.5 × 10<sup>−6</sup> /s) and an unstable fracture that develops near instantaneously (loaded at a `fast' constant strain rate of 5 × 10⁻⁵ /s). By recording a continuous ultrasonic waveform during the critical period of fracture, the entire AE catalogue can be captured and the exact time of fracture defined. Under constant strain loading, three stages are observed: (1) An initial nucleation or stable growth phase at a rate of ~ 1.3 mm/s, (2) a sudden increase to a constant or slowly accelerating propagation speed of ~ 18 mm/s, and (3) unstable, accelerating propagation. In the ~ 100 ms before rupture, the high level of AE activity (as seen on the continuous record) prevented the location of discrete AE events. A lower bound estimate of the average propagation velocity (using the time-to-rupture and the existing fracture length) suggests values of a few m/s. However from a low gain acoustic record, we infer that in the final few ms, the fracture propagation speed increased to 175 m/s. These results demonstrate similarities between fracture nucleation in intact rock and the nucleation of dynamic instabilities in stick slip experiments. It is suggested that the ability to constrain the size of an evolving fracture provides a crucial tool in further understanding the controls on fracture nucleation.     

Experimental Study on the Stick-slip Phenomenon of Intact Granite under Solid Confining Pressure

In this paper,compression tests of intact granite samples have been made in a triaxial testing machine with solid confining pressure.From the tests,the influences of confining pressure and loading rate(axial strain rate)on the deformation and fracture process of rock samples,on the stress drop and recurrence interval of stick-slip events,and on the geometric distribution of the main fracture have been studied.The experimental results show that the loading rate influences the stress drop and recurrence interval of stick-slip events greatly.At lower loading rates,the stress drop of stick-slip events is greater,their recurrence interval is longer and shows no regularity in distribution.When the loading rate goes higher,the stress drop will become smaller and the recurrence interval will tend to be constant and stick-slip events show a quasi-periodicity.At lower confining pressures and strain rates,the main fracture may evolve into 2 X-shaped conjugate shear faults; whereas at higher confining pressures and     

Influence of confining pressure and impact loading on mechanical properties of amphibolite and sericite-quartz schist

In order to investigate the dynamic mechanical properties of amphibolite and sericite-quartz schist under confi ning pressure, two rocks are subjected to impact loadings with different strain rates and confi ning pressures by using split Hopkinson pressure bar equipment with a confi ning pressure device. Based on the experimental results, the stress-strain curves are analyzed and the effects of confi ning pressure and strain rates on the dynamic compressive strength, peak strain and failure mode are summarized. The results show that:(1) The characteristics of two rocks in the ascent stage of the stressstrain curve are basically the same, but in the descent stage, the rocks gradually show plastic deformation characteristics as the confi ning pressure increases.(2) The dynamic compressive strength and peak strain of two rocks increase as the strain rate increases and the confi ning pressure effects are obvious.(3) Due to the effect of confi ning pressure, the normal stress on the damage surface of the rock increases correspondingly, the bearing capacity of the crack friction exceeds the material cohesion and the slippage of the fractured rock is controlled, which all lead to the compression and shear failure mode of rock. The theoretical analysis and experimental methods to study the dynamic failure mode and other related characteristics of rock are useful in developing standards for engineering practice.     

Velocity Measurements and Crack Density Determination During Wet Triaxial Experiments on Oshima and Toki Granites

— A set of experiments on four samples of Oshima Granite at 15, 40 and 60 MPa confining pressure have been performed in order to investigate the damage behavior of granite submitted to deviatoric stress. In addition an experiment on one sample of Toki Granite at 40 MPa confining pressure was performed, in order to compare and elucidate the structural effects. Using acoustic emission data, strain measurements and elastic wave velocities allow to define consistently a damage domain in the stress space. In this domain, microcracking develops. The microcracking process is, in a first stage, homogeneous and, close to failure, localized. Elastic wave velocities decrease in the damage domain and elastic anisotropy develops. Using Kachanov's model (1993), elastic wave velocities have been inverted to derive the full second-order crack density tensor and characterize the fluid saturation state from the fourth-order crack density tensor. Crack density is strongly anisotropic and the total crack density close to failure slightly above one. The results indicate that the rock is saturated in agreement with the experimental conditions. The model is thus shown to be very appropriate to infer from elastic wave velocities a complete quantitative characterization of the damaged rock.     

Constitutive theories on viscoelastoplasticity and damage of frozen soil

The constitutive theory on the viscoelastoplasticity and damage of frozen soil is based on the continuous mechanics and thermodynamics. The basic principles of the theory, dissipation potential function and damage model are presented. The constitutive theory explains the mechanical properties of frozen soils under complicated stresses, especially under high confining pressures which make frozen soil harden and soften. The agreement between the calculated results by the constitutive theory and the experimental results of triaxial creep of frozen soil is seen to be very good.     

Compaction and Failure in High Porosity Carbonates: Mechanical Data and Microstructural Observations

We investigated systematically the micromechanics of compaction in two carbonates of porosity above 30%, Majella grainstone and Saint Maximin limestone. The composition, grain size and pore surface area of these rocks were determined. Hydrostatic compression experiments were performed under dry and wet conditions beyond the onset of grain crushing. A significant water weakening effect was observed in both rocks. A set of conventional triaxial experiments was also performed on both rocks under dry conditions at confining pressures ranging from 3 to 31 MPa. Microstructural observations were carried out on the deformed samples. The mechanical behavior of these high porosity carbonates is dominated by shear-enhanced compaction associated in most cases with strain hardening. Stress-induced cracking and grain crushing are the dominant micromechanisms of deformation in both rocks. In Majella grainstone, compactive shear bands appeared at low confinement, in qualitative agreement with the deformation bands observed in the field. At higher confining pressures, compaction localization was inhibited and homogeneous cataclastic flow developed. In Saint Maximin limestone, compaction localization was observed at all confining pressures. An increasing number of compactive shear bands at various orientations appeared with increasing strain. These new data suggest that compaction localization is important in the mechanical compaction of high porosity carbonates.     

Nonlinear elastic behavior of fiber-reinforced soil under cyclic loading

Experimental investigations and modeling of nonlinear elasticity of fiber-reinforced soil under cyclic loading at small strain are conducted in this paper. The investigations include three aspects. First, cyclic shear tests are conducted using conventional triaxial apparatus. Twenty-seven specimens with three different fiber contents are employed to conduct triaxial cyclic shear tests under different confining pressure and loading repetition. Effects of geofiber, confining pressure and loading repetition on elastic shear modulus of reinforced soil are studied and analyzed. Second, a hyperbolic function is introduced to describe the nonlinear stress–strain skeletal curve under cyclic loading. Nonlinear elastic modulus is expressed as a function of shear strain and two variables A and B that are related to the initial tangential modulus and ultimate cyclic loading stress, respectively. In the present paper, variables A and B both are further assumed to be functions of geofiber content, confining pressure and loading repetition. Finally, eight constitutive coefficients of the nonlinear elastic model are calibrated using stress–strain curves from cyclic triaxial shear tests. The calibration of parameters is conducted using the technique of the linear regression for multiple variables. Impacts and effects of geofiber, confining pressure and loading repetitions on soil nonlinear elastic behavior are discussed.     

Examination of a Rock Failure Criterion Based on Circumferential Tensile Strain

—Uniaxial compression, triaxial compression and Brazialian tests were conducted on several kinds of rock, with particular attention directed to the principal tensile strain. In this paper we aim to clarify the effects of the experimental environment—such as confining pressure, loading rate, water content and anisotropy—on the critical tensile strain, i.e., the measured principal tensile strain at peak load.¶It was determined that the chain-type extensometer is a most suitable method for measuring the critical tensile strain in uniaxial compression tests. It is also shown that the paper-based strain gage, whose effective length is less than or equal to a tenth of the specimen’s diameter and glued on with a rubber-type adhesive, can be effectively used in the Brazilian tests.¶The effect of confining pressure P _C on the critical tensile strain ? _TC in the brittle failure region was between ?0.02 × 10^?10 Pa^?1 and 0.77 × 10^?10 Pa^?1. This pressure sensitivity is small compared to the critical tensile strain values of around ?0.5 × 10^?2. The strain rate sensitivities ?? _TC /?{log(d|?|/dt)} were observed in the same way as the strength constants in other failure criteria. They were found to be from ?0.10 × 10^?3 to ?0.52 × 10^?3 per order of magnitude in strain rate in the triaxial tests. The average magnitude of the critical tensile strain ? _TC increased due to the presence of water by 4% to 20% for some rocks, and decreased by 22% for sandstone. It can at least be said that the critical tensile strain is less sensitive to water content than the uniaxial compressive strength under the experimental conditions reported here. An obvious anisotropy was observed in the P-wave velocity and in the uniaxial compressive strength of Pombetsu sandstone. It was not observed, however, in the critical tensile strain, although the data do show some variation.¶A "tensile strain criterion" was proposed, based on the above experimental results. This criterion signifies that stress begins to drop when the principal tensile strain reaches the critical tensile strain. The criterion is limited to use within the brittle failure region. The critical tensile strain contains an inelastic strain component as well as an elastic one. It is affected by the strain rate, however, it is relatively insensitive to the confining pressure, the presence of water and anisotropy.