Post-peak Stress Strain Curve Research Articles

Rock fractures developed in the Class II post-peak unloading

In this study, post-peak tests were conducted on sandstone specimens subjected to uniaxial compression under two different loading conditions: axial strain control and lateral strain control. The fractures developed in both loading regimes were investigated. Under axial strain control, Class I post-peak stress–strain curves (with negative slope) were obtained until the complete failure of the specimens. The observed macroscopic fractures have mixed orientations: sub-parallel and inclined to the axial loading direction. Under lateral strain control, the specimens were tested until different residual post-peak stresses; the tests yielded consistent Class II post-peak stress–strain curves (with positive slope). Interestingly, the tested specimens were found to be relatively intact, with only non-penetrating fractures observed on the lateral surfaces. X-ray CT scanning revealed that the subvertical macroscopic fractures were developed close to the lateral surfaces of the specimens in the post-peak stages, while the internal central parts remained visually undamaged. Analysis of longitudinal and shear wave velocities measured in the axial direction of sandstone specimens before and after post-peak tests under lateral strain control provided evidence of internal fracture (crack) development parallel to the loading axis in the internal central parts of the tested specimens. Concentration of these small fractures (cracks) is high, but it only marginally increases as the loading proceeds in post-peak region from 90% to 70% of the peak stress. This suggests that the Class II post-peak regime obtained under lateral strain control cannot be explained by pure unloading; also, the Class II post-peak unloading is mainly produced by the development of the large subvertical tensile fractures identified in the CT scans, rather than by the growth of these small fractures (cracks).

A damage constitutive model for concrete under uniaxial compression capturing strain localization

This paper presents a damage constitutive equation capturing the feature of strain localization and, on this basis, proposes a novel damage constitutive model for concrete under uniaxial compression. It is found that the proposed model can better simulate the stiffness degradation behavior of concrete under cyclic loading, well reproduce the dependency of the post-peak stress-strain curve on specimen or measuring length under monotonic loading, and provide FE simulation results that are insensitive to mesh size. Influence of the maximum aggregate size on the strain localization zone length is discussed.

Macro- and micromechanical characteristics and energy evolution law of micro/nano carbon fibre grouting samples

This study comprehensively explores the macro- and micromechanical attributes and energy evolution dynamics of micro/nano carbon fiber (CF) specimens. The investigation centers on a graded sand gravel grouting specimen, aiming to determine the optimal CF content through uniaxial compression tests conducted on samples containing 0.0 %, 0.5 %, 1.0 %, and 1.5 % CF. Utilizing CT and PFC2D simulation analyses, we establish the interrelation between micro- and macromechanics. The findings reveal similar failure modes across CF samples with varying contents, predominantly marked by shear failure in the principal crack of the inclined section, accompanied by proximal secondary cracks. The post-peak stress–strain curves exhibit notable divergences. Specifically, with increasing CF content, the total strain energy, elastic strain energy, and dissipated energy of the CF samples initially decline, subsequently rise, and then diminish once more. Furthermore, the 1.0%CF sample demonstrates heightened dissipated energy, uniaxial compressive strength, and elastic modulus in contrast to the 0%CF sample, showcasing superior macroscopic mechanical properties and enhanced toughness. A staged microcrack evolution unfolds, delineating defect initiation, pore evolution, crack development, and crack penetration stages. The non-uniform distribution of CF and sand gravel profoundly influences the propagation trajectory of minute cracks. Additionally, the failure modes and energy evolution trends from both models closely mirror experimental outcomes. The reinforcing characteristics and crack-resistant attributes of CF usher the transition from brittle failure (0%CF) to ductile failure (1.0%CF). PFC simulations highlight hysteresis in the model's damage and failure, underscoring CF's role in restraining grouting sample failure and shifting it from brittle (0%CF) to ductile (1.0%CF) failure, thus accentuating CF's pivotal role in impeding crack propagation. These findings furnish invaluable theoretical insights for bolstering grouting reinforcement in fractured rock masses.

Numerical study of unstable failure behavior in heterogeneous rock pillar

Strain bursting in deep-buried tunnels is a common engineering disaster closely related to the failure mechanism and loading system stiffness (LSS). The rock failure mechanism cannot be explicitly captured using the homogeneous model. In this study, the failure characteristics of heterogeneous rock pillars under different LSS were simulated using the combined finite-discrete element method (FDEM), and the grain-based model in FDEM was constructed using a C++ subroutine. The simulation results show that with the decrease of the LSS, the post-peak stress-strain curve of the specimen slows down, the specimen undergoes unstable failure, and the fragmentation degree of the specimen, released kinetic energy, and post-peak crack increase rate grow. Under soft loading conditions, rock heterogeneity is an important factor affecting the degree of rock failure. The ratio of shear cracks to tensile cracks, strain energy of the loading system (LSSE), and post-peak crack increase rate can be regarded as indicators for distinguishing between stable and unstable failure of rocks. Furthermore, by comprehensively considering the LSS, rock heterogeneity and failure phenomena, an empirical index for evaluating the degree of rock failure was proposed.

Prediction of post-peak stress-strain behavior for sensitive clays

Unlike typical soils, sensitive clays undergo extensive post-peak strength degradation with increasing strain and finally disintegrate into a remolded liquid state. Realization of post-peak stress-strain behavior of sensitive clay up to large strains is vital in assessing large deformation problems such as landslides and mud flows. The conventional experimental approaches are uncertain about accurately determining the post-peak stress curve up to large strains (>100%) owing to rapidly increasing testing problems at increasing strains. This necessitates the exploration of an alternative scientific approach to predict the complete stress-strain curve for sensitive clays, which is addressed in this paper. Post-peak stress-strain curves of sensitive clays for different sites are obtained by converting remolding index vs. strain energy curves. Using site-specific data from eastern Canada sites, a mathematical expression is proposed to predict the complete stress-strain curve. Subsequently, an equation is developed for predicting remolding energy based on the stress-strain curve. Finally, it is observed that the post-peak stress-strain behavior is highly site-specific and can be mathematically expressed with a combination of exponential and linear strain-softening curves. Overall, the knowledge of the complete stress-strain behavior contributes greatly to the prediction of post-failure movements closer to reality.

Post-Peak Stress–Strain Curves of Brittle Hard Rocks Under Different Loading Environment System Stiffness

Post-Peak Stress–Strain Curves of Brittle Hard Rocks Under Different Loading Environment System Stiffness

Post-peak Stress–Strain Curves of Brittle Rocks Under Axial- and Lateral-Strain-Controlled Loadings

Post-peak Stress–Strain Curves of Brittle Rocks Under Axial- and Lateral-Strain-Controlled Loadings

Post-peak Stress–Strain curves of brittle hard rocks under axial-strain-controlled loading

A super stiff rock mechanics testing machine called Stiffman is introduced in this paper. This test facility uses composite loading frames and relay loading by two loading hydraulic actuators and overcomes the technical difficulties of conventional rock mechanics testing machines such as insufficient loading system stiffness, inability to use axial-strain-controlled loading in the post-peak deformation stage, limited straining amount to reach the residual deformation stage. To validate the reliability of the newly developed test machine, a benchmark study was carried out first. Then, systematic uniaxial and triaxial compression tests under axial-strain-controlled loading for three brittle hard rocks were conducted and the results are compared with those obtained using a conventional rock mechanics testing system called Rockman. The test results of Rockman show that under uniaxial compression and triaxial compression under axial-strain-controlled loading at low confining pressures, the rock specimens failed violently in the post-peak deformation stage and it was not possible to obtain complete stress–strain curves of the rocks. On the other hand, it was possible to load the rocks under axial-strain-controlled loading in a stable fashion using Stiffman and complete stress–strain curves of the rocks under uniaxial compression and triaxial compression at low confining pressures could be obtained. Stiffman-tested rock specimens showed signs of spalling but stayed mostly in one piece. Finally, peak and residual strength envelopes of the three brittle rocks are given based on the test results of Stiffman. The experiment results show that Stiffman can be used to capture complete post-peak stress–strain curves of brittle rocks reliably under axial-strain-controlled loading. It provides a new platform for studying post-peak deformation behaviors of brittle hard rocks reliably in a loading control mode that other conventional test machines cannot provide.

Study on Shear Strength of Cement Face Between Rock and Concrete

The samples of slightly weathered granite drilled before the construction of Runyang suspended bridge's north anchorage are used as the research object in this paper. Through a lot of shear strength tests of concrete-bedrock rough cement face, the influence of the cement face roughness on the shear strength, shear deformation characteristics and shear failure mechanism were systematically studied. The results show that: The post-peak stress–strain curves of smooth and rough cement faces are obviously different. The shear stress–strain curve of cement face can be generalized into 3 types. The shear stress–strain model of rough cement face is the most general form, and the smooth cement face and no cementation are its special cases. The empirical relation between cement face roughness JRC and shear strength parameters is established by the fitting of experimental data. When the cement face is climbing, the internal friction angle increases with the climbing angle. When the climbing angle is constant, the increase of cohesion is positively correlated with the ratio of fluctuation difference and wavelength. When the climbing thrust is equal to the gnawing thrust, the specimen is transformed from climbing to gnawing.

Rock brittleness indices and their applications to different fields of rock engineering: A review

Brittleness is an important parameter controlling the mechanical behavior and failure characteristics of rocks under loading and unloading conditions, such as fracability, cutability, drillability and rockburst proneness. As such, it is of high practical value to correctly evaluate rock brittleness. However, the definition and measurement method of rock brittleness have been very diverse and not yet been standardized. In this paper, the definitions of rock brittleness are firstly reviewed, and several representative definitions of rock brittleness are identified and briefly discussed. The development and role of rock brittleness in different fields of rock engineering are also studied. Eighty brittleness indices publicly available in rock mechanics literature are compiled, and the measurement method, applicability and limitations of some indices are discussed. The results show that (1) the large number of brittleness indices and brittleness definitions is attributed to the different foci on the rock behavior when it breaks; (2) indices developed in one field usually are not directly applicable to other fields; and (3) the term “brittleness” is sometimes misused, and many empirically-obtained brittleness indices, which lack theoretical basis, fail to truly reflect rock brittleness. On the basis of this review, three measurement methods are identified, i.e. (1) elastic deformation before fracture, (2) shape of post-peak stress–strain curves, and (3) methods based on fracture mechanics theory, which have the potential to be further refined and unified to become the standard measurement methods of rock brittleness. It is highly beneficial for the rock mechanics community to develop a robust definition of rock brittleness. This study will undoubtedly provide a comprehensive timely reference for selecting an appropriate brittleness index for their applications, and will also pave the way for the development of a standard definition and measurement method of rock brittleness in the long term.

An Improved Statistical Damage Constitutive Model for Granite under Impact Loading

To study the impact properties of granite, the parameters (including the stress-strain curve, elasticity modulus, peak strength, and peak strain) of the test pieces in each group were determined via standard split-Hopkinson pressure bar tests. The results revealed that the prepeak stress-strain curves are approximately linear; the postpeak stress-strain curve declined sharply and exhibited the characteristics of brittle material failure after the stress exceeded the peak strength. In terms of the specimen form following failure, for increasing strain rate, the granite specimen became increasingly fragmented after failure. In addition, the single-parameter statistical damage constitutive model was improved, and a double-parameter statistical damage constitutive model for describing the total stress-strain curve of granite under the action of impact loading was proposed. The parameters of the statistical damage model, m and a, were obtained via fitting. The results revealed that the parameter m decreases with increasing elasticity modulus, whereas the parameter a increases. Similarly, the peak strength and the peak strain increased (in general) with increasing strain rate.

Numerical Investigation of Radial Strain-Controlled Uniaxial Compression Test of Äspö Diorite in Grain-Based Model

A complete stress–strain curve is difficult to obtain for Class II rocks because of the abrupt failure after peak stress. Although the servo-controlled loading system has been used successfully in laboratory tests to analyze the failure of Class II rocks, improved simulation techniques are still needed to control the failure of Class II rocks. In this study, the Class II behavior of Aspo diorite was reproduced explicitly using the GBM-UDEC model. The relative differentiations in all mechanical properties between the simulation and laboratory tests were within ± 5%, such as Young’s modulus, and slope of the post-peak stress–strain curve. Based on the stress and crack generation in UCS test simulation, the failure behavior of the Aspo diorite in the post-peak stress region could be classified into three stages: (V) unstable crack growth with decreasing stress, (VI) stable crack growth with increasing stress, (VII) unstable crack growth, and failure of the rock specimen with decreasing stress. Throughout the simulation, the microcrack investigation was conducted in the aspect of number, angle, relative strength of contacts, and location. In the pre-peak stress region, the cracks between 0° and 20° with respect to axial stress was dominant with over 62% of population among total cracks. Cracks were generated preferentially in relatively weak contacts. The ratio of brakeage in the weakest contact was 22%, while the average breakage value of the other stronger contacts was 16%. In the post-peak stress region, the number of cracks generated between 20° and 40° was approximately 1.4 times higher than those between 0° and 20° at Stages V and VII. There was a significant increase in the number of cracks in the strongest contacts when the model was at the beginning of Stage V and Stage VII. In Stage VII, microcracks were generated intensively in central column of the rock model. Moreover, crack coalescence was monitored in the inner column and rock spalling, complete detachment of block in rock specimen model, was observed at the lateral side of the rock model in Stage VII. In the energy calculation, 23% of the strain energy stored in the model was extracted immediately after peak stress to prevent the violent failure of the rock and most of the energy extracted resulted from grain blocks compared to contacts.

Mechanical properties of coral concrete subjected to uniaxial dynamic compression

The uniaxial compressive behavior of coral aggregate concrete was experimentally investigated under quasi-static to dynamic loading rates. Quasi-static compression tests of coral concrete at different curing ages were performed at a constant strain rate of 10−5 s−1 using an electro-hydraulic servo-controlled test machine. Dynamic impact loading tests were conducted at stain rates from 101.48 s−1 to 102.16 s−1 utilizing a 100-mm-diameter split Hopkinson pressure bar (SHPB) system. The strain rate effects on the mechanical properties of coral concrete were assessed in terms of the uniaxial compressive strength, energy dissipation, fractal dimension and failure pattern. The coral concrete exhibits high-early strength and the post-peak stress-strain curves behave in a more brittle manner than conventional concrete. A more remarkable rate-dependence in the compressive strength of coral concrete than other cement-based composites was observed as the dynamic increase factor (DIF) increased from 1.73 to 2.56 with the strain rate increasing from 30.12 s−1 to 143.32 s−1. Different from the failure pattern of conventional concrete, the fracture plane of coral concrete directly penetrated through the coral shingles rather than cracking in the interface between the cement mortar and the coarse aggregate under both quasi-static and dynamic loadings, ascribing to the low strength of coral aggregates and the high intensity of bonding interface. The ratio of the absorbed energy to incident energy is between 0.3–0.5 and tends to decrease with an increase in strain rate. A higher loading rate may lead to more energy absorption consumed by generating more fracture planes and smaller fragments, which is in consistent with a higher value of fractal dimension. The fractal dimension in the range of 2.027–2.302 for coral concrete was also found to be proportional to the logarithm of the loading strain rate.

An Experimental Study on Dynamic Mechanical Properties of Fiber-Reinforced Concrete under Different Strain Rates

Fiber-reinforced concrete (FRC) has a great advantage in earthquake-resistant structures, as compared with regular concrete. However, there are many difficulties in the construction and maintenance of concrete structures due to the high density and easy corrosion of the steel fiber in commonly used steel FRC. With the development of polymer material science, polyvinyl alcohol (PVA) fiber has been rapidly promoted for use in FRC because of its low density, high strength, and large elongation at break value. Dynamic uniaxial compression and splitting tensile experiments of FRC with PVA fiber were carried out with two matrix strengths (i.e., C30 and C40), which were blended with PVA fibers with a length of 12 mm in different volume contents (0, 0.2, 0.4, and 0.6%), at the age of 28 days, under different strain rates (i.e., 10−5, 10−4, 10−3, and 10−2 s−1). The results show that PVA has an obvious enhancing and toughening effect on concrete, which can improve its brittle properties and residual strength. With increasing strain rate, the compressive strength, split tensile strength, and elastic modulus increase to a certain extent, while the toughness index and the peak strain decrease to a certain degree. The post-peak deformation characteristic changes from a brittle failure of sudden caving to a ductile failure with dense cracking. The effect of PVA is different when enhancing the concrete with two different matrix strengths. The lower the matrix strength, the more obvious the enhancement effect of the fiber, showing characteristics of a higher compressive strength and low split tensile strength in FRC with low strength and a smoother post-peak stress–strain curve.

Influence of bedding and cleats on the mechanical properties of a hard coal

The extensive bedding and cleats have a great influence on the mechanical properties of coal. The previous research mainly focused on the bedding effect, and few studies took the influence of both bedding and cleats into consideration. The scanning electron microscopy (SEM), uniaxial compression experiments, and acoustic emission (AE) monitoring were conducted on one hard coal specimen with different bedding and cleat angles. The experimental results show that for the bedding dip of 0° and face and butt cleat dip of 90°, the properties of coal specimens were characterized by the highest peak strength, the drastic failure process, the post-peak brittle behavior, the long initial quiet stage, and the short violent booming stage of AE hits number. For the bedding and face cleat dip of 45°, and the butt cleat dip of 90°, the properties of coal specimens were characterized by the lowest peak strength, the slip failure mode, the step-like drops of the post-peak stress-strain curve, and the several peaks of AE hits number. On the other hand, for the bedding and butt cleat dip of 90°, and the face cleat dip of 0°, the properties of coal specimens were presented as tensile failure mode with plate cracks, the short non-linear stage and the slight step-like drops of stress-strain curve, the short initial quiet stage of AE hits number, and the large value of elastic modulus. Then, a theoretical model of coal containing bedding and cleats was established in this paper. The simulation results of peak strength and failure mode verified the proposed model. The simulation of coal strength shows that bedding had the decisive influence on the uniaxial compressive strength of coal, while the weakening effect of cleats was non-negligible. When bedding and cleats were taken into consideration together, the minimum peak strength was reduced and the bedding dip related to the minimum peak strength changed.

Dynamic tensile test of fly ash concrete under alternating tensile–compressive loading

In order to obtain the post-peak stress–strain curve of plain concrete, the strain-controlled uniaxial tensile tests are carried out on cylindrical specimens. The tensile cyclic tests are performed between a lower compressive load and an upper load to the envelope curve. Test results have shown that residual strain accumulates slowly when the concrete is subjected to alternating tensile–compressive loading, during which the microcracks increase and aggregate. Meanwhile, the rates of stiffness degradation with cycles are similar in all cases. In the study, an analytical model of the post-peak cyclic behavior of concrete is proposed. The strain is decomposed into classical linear elastic strain and non-classical strain described in the Presach–Mayergoyz model, which describes the hysteresis behavior of concrete well from a microscopic point of view. A good agreement is obtained between the test data and calculated results.

Study of Post-Peak Strain Softening Mechanical Behaviour of Rock Material Based on Hoek–Brown Criterion

In order to build the post-peak strain softening model of rock, the evolution laws of rock parameters m,s were obtained by using the evolutionary mode of piecewise linear function regarding the maximum principle stress. Based on the nonlinear Hoek–Brown criterion, the analytical relationship of the rock strength parameters m,s, cohesion c, and friction angle φ has been developed by theoretical derivation. According to the analysis on the four different types of rock, it is found that, within the range from 0 to σ3min, the peak hardness of the rock becomes smaller as the confining pressure increases and the degree of rock fragmentation decreases as well. The post-peak stress-strain curves obtained from the developed softening model are in good agreement with the laboratory test results under different confining pressures. In conclusion, the analytical method is reasonable, and it can predict the post-peak mechanical behaviour of rock well, which provides a new thought for the rock-softening simulation.

Influence of Loading System Stiffness on Post-peak Stress–Strain Curve of Stable Rock Failures

It is well known from laboratory testing that the rock failure process becomes unstable in a soft test machine due to excessive energy released from the machine. Great efforts had been devoted to increasing the loading system stiffness (LSS) of laboratory test machines to ensure that the post-peak stress–strain curve of rock can be obtained for underground rock engineering design. A comprehensive literature review on the development of stiff test machines reveals that because of the differences in the manufacturing arrangement of the test machines, LSS values of the test machines used for rock property testing are always finite and vary in a large range, and the influence of LSS on stable rock failure is less understood. A FEM-based numerical experiment is carried out to study the influence of LSS on the stress–strain curves of stable rock failure in uniaxial compression, with a focus on the post-peak deformation stage. Three test machine loadings including idealized rigid loading, platen loading, and frame–platen loading with finite LSS are considered, and the simulation results are analyzed and compared. The modeling results obtained from the simulations indicate that even if the LSS value is large enough to inhibit unstable rock failure, as long as LSS is finite, it has an influence on the post-peak stress–strain curve of rock. It is revealed that because the input energy supplied by the external energy source to drive the stable rock failure process is affected by the finite LSS of a test machine, the post-peak descending slopes of the stress–strain curves are all steeper than the post-peak descending slope obtained under an ideal loading condition of infinite LSS. An insight from this numerical experiment is that it might be more feasible to develop laboratory test machines with variable LSS that can match the local mine stiffness in the field for rock property testing.

A New Method to Evaluate Rock Mass Brittleness Based on Stress–Strain Curves of Class I

Brittleness is a key controlling parameter for rock engineering projects such as hydrocarbon production and other applications. In this paper, commonly used methods based on stress–strain curves of Class I for the calculation of rock brittleness are reviewed. In order to describe the rock brittleness more reasonable, the new index B i was proposed based on the stress drop rate obtained from post-peak stress–strain curve and the ratio of elastic energy released during failure to the total energy stored before the peak strength. Then the validity of B i is verified with experimental tests conducted on rock specimens drilled from the interlayer and oil layer through a well of Shengli Oilfield. Moreover, numerical simulation is performed to analyze the effects of primary mechanical parameters on the brittleness of rock masses. Based on experimental tests and numerical simulation results, the acoustic emission modes influenced by brittleness index B i are summarized. At last, correlation between acoustic emission mechanism and index B i is verified by comparing the acoustic emission modes of limestone under different levels of confining pressure and various types of coal.

Investigation on Mechanical Behaviors of Sandstone with Two Preexisting Flaws under Triaxial Compression

Triaxial compression experiments on sandstone samples with two preexisting closed non-overlapping flaws were performed to investigate the deformation and strength behaviors. Three types of preexisting closed flaw pair in sandstone samples, i.e., parallel low-dip (type B), parallel high-dip (type C), and composite high- and low-dip (type D), were considered as the typical arrangements of the non-overlapping crack pair. A general rule has been found that the arrangement of the flaw pair has greater impact on the rock deformation, strength, and crack coalescence pattern than the confining pressure (5–20 MPa). Experimental results showed that, compared with intact sandstone samples, the postpeak stress–strain curves of flawed samples distinctly demonstrate stress fluctuation. In particular, the unique prepeak stress–strain curves of the specimens with a low-dip flaw pair (type B) present oblique Z-shape with a double-peak stress. The stress for crack initiation σ ci, the critical stress of dilation σ cd, and the peak strength σ c of precracked sandstone samples are significantly lower than those of intact rock. The present numerical study, which is an extension of the test analysis, focuses on identifying the crack nature (tensile or shear) and coalescence process. These simulated crack coalescence patterns are in good agreement with the laboratory test results. The cracks of the precracked samples that contained flaws with small inclination angle (associated with either type B or type D) generally initiate at the inner flaw tips and eventually lead to simple direct shear coalescence. However, complex indirect shear coalescence appears in the model containing a steep preexisting flaw pair (associated with type B specimen), even though no coalescence occurs when σ 3 = 5 MPa.