Post-peak Regime Research Articles

Micro- and macro-scale fracture behaviour of brittle rocks: Comparison between the conventional Brazilian test and the advanced universal snap-back indirect tensile test (AUSBIT)

Post-peak behaviour is crucial for the estimation of rock mass fracturing in cave mining operations where hard rocks can exhibit class-II or snap-back response when subjected to loading. Despite the rapid development of research into class-II rocks under compression, the corresponding behaviour in tensile tests has rarely been investigated, which is critical considering the complexity of rock mass fracturing under various stress states. The post-peak response of brittle rocks involves abrupt micro-fracturing, leading to brittle macro-scale behaviour. Controlling the fracture process using the Advanced Universal Snap-Back Indirect Tensile test (AUSBIT) allowed the acquisition of the complete macro-scale class-II behaviour in the post-peak regime, facilitating the use of advanced techniques for insights into both micro and macro-scale fracture. In this study, the AUSBIT tests with digital image correlation (DIC) and acoustic emission (AE) instrumentation were conducted to analyse the progressive failure in Calca granite and Gosford sandstone specimens. Post-test observations of the fracture surfaces were performed using a scanning electron microscope (SEM). From a macroscale viewpoint, the lateral strain control in AUSBIT enabled controlled cracking with significant lateral strain extension prior to failure accompanied by gradual energy dissipation and higher rates of AE activity as smaller magnitudes of energy are being released by each AE hit or microcrack compared to conventional Brazilian tests. The stable microcrack propagation was also identified from SEM observations with more uniform profiles of microcracks and less debris observed in AUSBIT specimens. These findings were more significant in Calca granite, which verified its extreme class-II behaviour while also demonstrating the efficiency of AUSBIT in controlling the violent failure of high-strength brittle rocks commonly encountered in deep mining projects, leading to the acquisition of more accurate material behaviour in terms of micro and macro-scale post-peak features which was unattainable from conventional indirect tensile tests.

Full-scale tests of two-storey precast reinforced concrete shear walls: Investigation of strength and deformation capacity

Precast concrete structures behave differently than equivalent in-situ cast solutions mainly because of the narrow connections between the precast elements. In this context, the keyed shear connections are of particular interest, as they have to ensure structural integrity in order to transfer shear- and normal stresses between adjacent elements. Extensive experimental studies (based on push-off tests) of this type of connection have been conducted in the past. These tests typically showed significant softening in the post-peak regime of the shear-slip relationship. It is for this reason important to study how effectively such connections can be utilised on the level of structural systems, where redistribution of internal forces is often required to reach the global load carrying capacity. This paper presents results from full-scale testing of two precast shear wall structures. The structures were two stories high and simulated a typical segment of precast buildings in practice. Each test structure was composed of 12 precast elements (decks and walls) and tested to failure under a combination of vertical and horizontal loads. The main varying parameter between the two tests is the design shear capacity of the keyed connections, which was doubled from test T2 to test T3 by increasing the diameter and the yield strength of the U-bar loops. Both structures displayed a ductile failure with significant deformation capacity. However, despite the higher strength in the keyed connections, the load carrying capacity of test T3 was not increased compared to test T2. This behaviour can be explained by a comprehensive analysis of the measured deformation field obtained from 2D Digital Image Correlation. By comparing the deformations with information obtained from independent component push-off tests with equivalent connections, it is possible to pinpoint the different stress regimes that the critical connection experienced, when the test structure reached its load carrying capacity. The results show that at that point in time, large parts of the critical connection were still in the pre-peak shear-slip regime, while another part located between two window openings was already at the end of the post-peak regime with practically no residual strength left. The findings indicate that the effective design strength of connections should be chosen with caution when modern numerical rigid-plastic methods are used to calculate the load carrying capacity of precast structural systems.

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).

Frequency dependence of acoustic emission with particle interaction and failure process in dry sands during triaxial compression

To investigate the behaviour of sands at the particle-scale during shear failure, a series of triaxial compression tests was conducted on dry sands with varying confining stresses and relative densities. These tests were complemented by acoustic emission (AE) monitoring to analyse the frequency-dependent AE signals and characterise particle interactions in terms of crushing (>100 kHz) and rearrangement (<100 kHz). The results reveal a linear relationship between the relative breakage index (Br) and the logarithm of cumulative high-frequency AE hits, regardless of the relative density. With this, particle crushing, as well as particle rearrangement, is demonstrated to occur throughout the entire shearing. Particle crushing increases more rapidly to prevail from the very beginning to the general yield, thereafter at a slower increasing rate to become less predominant. Its evolution in the post-peak regime is related to the failure pattern, which accumulates at a constant and most rapid rate when the specimen is ‘barrelling’; however, it decays rapidly in the presence of shear banding and eventually reaches an equilibrium state with particle rearrangement once the critical stress state is attained. Accordingly, two types of approximate evolution trends are suggested to estimate particle crushing upon shearing depending on the failure patterns.

Toughness of network materials: Structural parameters controlling damage accumulation

Many materials have a network of fibers as their main structural component and are referred to as network materials. Their strength and toughness are important in both engineering and biology. In this work we consider stochastic model fiber networks without pre-existing cracks and study their rupture mechanism. These materials soften as the crosslinks or fibers fail and exhibit either brittle failure immediately after the peak stress, or a more gradual, ductile rupture in the post peak regime. We observe that ductile failure takes place at constant energy release rate defined in the absence of pre-existing cracks as the strain derivative of the specific energy released. The network parameters controlling the energy release rate are identified and discussed in relation to the Lake-Thomas theory which applies to crack growth situations. We also observe a ductile to brittle failure transition as the network becomes more affine and relate the embrittlement to the reduction of mechanical heterogeneity of the network. Further, we confirm previous reports that the network strength scales linearly with the bond strength and with the crosslink density. The present results extend the Lake-Thomas theory to networks without pre-existing cracks which fail by the gradual accumulation of distributed damage and contribute to the development of a physical picture of failure in stochastic network materials.

A better understanding of the mechanics of borehole breakout utilizing a finite strain gradient-enhanced micropolar continuum model

Borehole breakout denotes the failure in rock mass subjected to drilling, caused by stress concentrations exceeding the material strength. Depending on the material properties, the preexisting in situ stress state, and the borehole dimensions, different types of borehole breakout, such as spiral-shaped breakout or v-shaped breakout, are distinguished in the literature. In the present work, we address the influence of the material properties on the predicted borehole breakout mode in a comprehensive finite element study. To this end, we employ a gradient-enhanced micropolar damage-plasticity model based on the Mohr–Coulomb strength criterion, formulated in the finite strain regime, which is calibrated based on experimental results from plane strain compression tests on Red Wildmoor sandstone. In the numerical study, the influence of the in situ stress state, the material friction angle, plastic dilation, post peak residual strength, and the inherent material length scale parameters are investigated. Thereby, we demonstrate that depending on the material parameters, substantially different failure modes, characterized by strongly localized shear bands or diffuse failure zones, are predicted. It is shown that in particular the brittleness of the material in the post peak regime has a major influence on the predicted breakout type. Moreover, a statistical validation of the results is obtained by considering different random field distributions of the initial material strength.

Machine-learning for damage assessment of rubble stone masonry piers based on crack patterns

Under seismic actions, stone masonry buildings are prone to damage. To assess the severity of damaged masonry buildings and their failure modes, engineers connect these problems to surface crack features, such as the crack width and the extent of cracking. We aim to further these assessments in this study, wherein we propose using simple machine learning models to predict: 1) three ratios encoding the degradation of stiffness, strength, and displacement capacity of damaged rubble stone masonry piers as a function of the observed crack features and the applied axial load and shear span ratio; and 2) the pre-peak vs. post-peak regime, based on the crack features.When predicting the stiffness, force, and drift ratios, the prediction error is significantly reduced when the axial load and shear span ratio are included in the feature vector. Furthermore, when predicting the pre-peak vs. post-peak regime, simple machine learning models such as the k-nearest neighbor and the logistic regression result in remarkable accuracy.The obtained results have significant implications on the automated post-earthquake assessment of masonry buildings using image data. It is shown based on documented laboratory test data, that, by selecting proper crack features and incorporating information about the kinematic and static boundary conditions, even simple machine learning models can predict accurately the damage level caused to a rubble masonry pier. The three crack features used in this study are the maximum crack width, length density, and complexity dimension. The pipeline developed in this paper is general enough and is applicable to other masonry typologies and elements upon new evaluation of crack features and image data.

Fracture mechanisms of intact rock-like materials under compression

The fracture mechanism in natural rock is closely related to the nucleation and interaction of microcracks. However, a comprehensive understanding of the crack evolution in intact rocks remains unclear. This paper presents a systematic investigation on the microcrack nucleation, coalescence and propagation in conjunction with detailed characterization of local stress distribution and bond strength information by means of the discrete element-based modelling. It was found that crack initiation dominates at the elastic stage and crack coalescence takes control at the unstable crack growth stage. Crack growth in both the stages is sensitive to the local heterogeneity of the sample and local stress distribution. In the post-peak regime, crack growth becomes intensified and localised, forming the shear zone and leading to a complete disintegration of a sample.

Experimental Study on Progressive Damage Evolution in Rocks Subjected to Post-peak Cyclic Loading History

Abstract Because of the release of in situ rock stress, rocks surrounding deep underground excavations are mostly driven into a post-peak damage state. Thus, investigations on cyclic loading-induced damage of rocks are of paramount significance in terms of the long-term stability of rock structures. In this study, using a damage-controlled testing method, the effect of post-peak cyclic loading on the failure behavior of different rock types was comprehensively evaluated. This investigation revealed that post-peak cyclic loading imposes no significant influence on the overall failure behavior of rocks. The evolution of damage variables also indicated that the large portion of axial strain damage (Da) occurs during the first cycle at the peak strength, whereas lateral strain damage (Dl) evolved more rapidly than Da did in the post-peak regime under further loading and unloading cycles. The cumulative permanent axial strain (∑εap) evolved nonlinearly with the axial stress ratio (σi/σa−peak) in the post-peak regime, which was manifested as sparse-dense-sparse hysteretic loops. A rapid reduction in the crack damage stress ratio was observed for rocks in the initial sparse hysteretic loops, which was followed by a lower decrease rate in the subsequent dense hysteretic loops, and an almost constant value was finally reached in the last sparse hysteretic loops. Furthermore, tangent Young’s modulus (Etan) and Poisson’s ratio (υ) evolved in three main stages: One, Etan first increased and then decreased in the initial sparse hysteretic loops, while υ continuously increased, representing significant deformation in the lateral direction. Two, Etan decreased in the dense hysteretic loops, while υ increased at a higher rate for stronger rocks because of the decrease in Da accumulation. Three, the increase rate of υ declined in the second set of sparse hysteretic loops, and Etan decreased until complete failure occurred.

Strain localisation in sensitive clays: can rate dependency provide mesh independent results?

Finite element modelling involving strain-softening materials becomes mesh dependent if not regularised by a mesh independent length scale. The objective of this paper is to investigate whether the inherent strain rate dependency of a sensitive clay can result in mesh independent results in the post-peak strain-softening regime. It describes first a constitutive model that accurately reproduces the rate-dependent behaviour obtained from laboratory tests on block samples of a natural sensitive clay. Then finite element analyses of soil columns under constant shear displacement rate are conducted. The boundaries are closed but internal flow of pore water is allowed. Strain localisation is triggered using a zone with lower undrained shear strength. The results show an interval of regularised strain-softening behaviour before the solution becomes mesh dependent. Pore water flow is found to delay the onset of mesh dependency. In conclusion, this paper achieves a regularised force displacement response for a strain-softening clay, without the need to introduce an artificial internal length scale.

A localizing gradient plasticity model for ductile fracture

The conventional gradient enhanced plasticity model uses a nonlocal gradient regularization to avoid mesh sensitivity issues present in the continuum-based local plasticity models. These conventional gradient plasticity models are found to manifest spurious spreading of localization zone during the post-peak regime, leading to non-physical structural and fracture response. Therefore, in the present work, a localizing gradient plasticity (LGP) model is proposed to avoid these non-physical responses. The LGP model avoids spurious responses by including the effects of the fracture processes (at micro and macro level) in the model description through the micromorphic framework. The fracture processes are typically found to initiate as a diffused region of plastic deformation (called necking region), which localizes to form cracks upon further loading. An interaction function is adopted in the LGP model to incorporate the localization of the fracture processes. The enhanced capability of the LGP model is demonstrated through one-dimensional and planar problems with tensile, compressive and mixed-mode loading. The numerical results show that the LGP model can arrest the spurious widening of the localization region resulting in a narrow region of high deformation representing failure.

Phase-field modeling of electromechanical fracture in piezoelectric solids: Analytical results and numerical simulations

The optimal design, reliable manufacture and durable utilization of electromechanical systems demand objective modeling of failure under coupled electric field and mechanical actions. This is still a challenging and largely an open issue despite the large volumes of contributions from the discontinuous approach and the phase-field community for brittle fracture. In this work we propose a length scale insensitive phase-field cohesive zone model (PF-CZM) for electromechanical fracture in linear piezoelectric solids, overcoming all the issues of mesh dependence, crack tracking complexity, length scale sensitivity and crack nucleation inconsistency. More specifically, in the electromechanically coupled phase-field constitutive relations, the evolution law for the crack phase-field is established from the geometric regularization of sharp cracks and the energetic criterion in global form. In order to capture the unsymmetric mechanical and electrical fracture behavior, the mechanical energy release rate contributed from the positive cone of the effective stress in energy norm is employed as the crack driving force, resulting in a hybrid formulation. With a set of phase-field characteristic functions optimal for cohesive fracture, the critical stress upon crack nucleation and the post-peak softening regime are derived in close-form for the first time, and the experimentally observed effects of the electric field on the fracture load are predicted qualitatively. The proposed electromechanical PF-CZM is numerically implemented in the multi-field finite element method and applied to several representative examples. In all cases, as those for purely mechanical problems, the predicted crack patterns and fracture loads are insensitive to the phase-field length scale and the experimental results are well reproduced. In particular, the effect of electric field on the deflection of crack paths can be captured. These merits make the proposed PF-CZM promising in the modeling of electromechanical fracture in piezoelectric solids.

Comparative 3D DEM simulations of sand\u2013structure interfaces with similarly shaped clumps versus spheres with contact moments

Three-dimensional simulations of a monotonic quasi-static interface behaviour between initially dense cohesionless sand and a rigid wall of different roughness during tests in a parallelly guided direct shear test under constant normal stress are presented. Numerical modelling was carried out by the discrete element method (DEM) using clumps in the form of convex non-symmetric irregularly shaped grains. The clumps had an aspect ratio of 1.5. A regular grid of triangular grooves (asperities) along the wall with a different height at the same distance was assumed. The numerical results with clumps were directly compared under the same conditions with our earlier DEM simulations using pure spheres with contact moments with respect to the peak and residual interface friction angle, width of the interface shear zone, ratio between grain slips and grain rotations, distribution of contact forces and stresses. The difference between the behaviour of clumps and pure spheres with contact moments proved to be noticeable in the post-peak regime due to a different particle shape. The rolling resistance model with pure spheres was proved to be limited for capturing particle shape effects. Three different boundary conditions along the interface were proposed for micropolar continua, considering grain rotations and grain slips, wall grain moments and wall grain forces, and normalized interface roughness. The numerical results in this paper offer a better understanding of the interface behaviour of granular bodies in DEM and FEM simulations.

Novel findings on the dilatancy and non-coaxiality of sand under generalised loading

The dilatancy and non-coaxiality of sand are investigated under generalised loading including rotation of the stress principal axes (PA), in drained and undrained conditions. Weak non-coaxiality is exhibited under radial loading being dependent on the stress direction relative to the deposition direction, as well as on state, and gradually diminishing beyond peak failure. On the contrary, non-coaxiality is stronger and persists in the post-peak regime when sand is subjected to rotation of the stress PA. In conditions involving undrained rotation of the stress PA at constant total stress principal values (PV), distinct non-coaxiality patterns and elastic–plastic coupling are observed under increasing stress ratio, while a steady state is ultimately attained. This study shows that sand exhibits strong non-coaxiality and contracts immediately upon initiating the rotation of the stress PA at constant effective stress PV very close to critical state (CS), while it was previously dilating on the failure surface in a coaxial deformation mode, under radial loading; the phenomenon becomes increasingly intense as CS is approached. The degree of non-coaxiality and associated contractancy becomes higher when the previous shearing becomes more persistent, in terms of shear strain accumulation, while the influence of pre-shearing is stronger than that of the degree of stress rotation and the level of η = q/p′, p′, e, and b = (σ′2 − σ′3)/(σ′1 − σ′3), but diminishes gradually during stress rotation.

Behavior of recycled steel fiber reinforced concrete under uniaxial cyclic compression and biaxial tests

In this study, the behavior of recycled steel fiber reinforced concrete (RSFRC) is investigated through uniaxial cyclic compression and biaxial tests conducted on 56 cubic specimens with four admixtures of RSF-content. The results obtained from cyclic compression tests show that the presence of RSFs reduces the damage of specimens in the post-peak regime which leads to an increase of residual strength. Furthermore, the biaxial failure-stress envelopes constructed based on the biaxial testing results indicate that an increase in RSF-content expands the failure envelope. Consistently, maximum biaxial strength for all specimens is found to occur at a stress confinement ratio α=~0.44-0.48. Moreover, measurements show that the failure biaxial strain significantly increases with increasing the RSF-content. The biaxial test results are found in good agreement with the literature on steel fiber reinforced concretes (SFRCs) indicating that industrial steel fibers can be substituted by RSFs which are environment-friendly. Employing these results, a model for the biaxial behavior of RSFRCs is proposed which can be employed in the constitutive relations required for the modeling of RSFRCs as well as SFRCs.

A 3D gradient-enhanced micropolar damage-plasticity approach for modeling quasi-brittle failure of cohesive-frictional materials

Continuum models based on the combination of the theories of plasticity and damage mechanics pose a powerful framework for representing the highly nonlinear material behavior of cohesive-frictional materials. However, non-associated plastic flow rules for representing the inelastic volumetric expansion of such materials may result in unstable material behavior and, accordingly, strongly mesh-dependent results in finite element simulations. Regularization techniques such as the gradient-enhanced continuum or similar nonlocal approaches, which work well for regularizing mode I failure, are often not sufficient as a remedy. In contrast to the latter, the theory of the micropolar continuum represents a suitable framework for regularizing non-associated plastic flow and shear band dominated failure properly, but it fails to do so for mode I failure. Hence, in the present contribution, a combination of the theories of the micropolar continuum and the gradient-enhanced continuum for regularizing both shear band dominated failure and mode I failure is presented. By incorporating a 3D damage-plasticity model for concrete into the proposed framework, it is demonstrated that the proposed model constitutes a physically sound and numerically stable approach for modeling the nonlinear material behavior of concrete in both the pre-peak and the post-peak regime for a broad variety of loading conditions.

Experimental investigation on fracturing process of marble under biaxial compression

Abstract In this study, servo-controlled biaxial compression tests were conducted on marble specimens to investigate their failure characteristics and fracturing process. The complete stress-strain curves were obtained, and the three-dimensional (3D) features of the failure surfaces were acquired by 3D laser scanning. Acoustic emission (AE) monitoring and moment tensor (MT) analysis were used in combination to better understand the fracturing mechanism of marble under biaxial compression. It was noted that a type of 3D stepwise cracking behaviour occurred on the fracturing surfaces of the examined specimens. The stress dropped multiple times, and a repeated fracturing mode corresponding to the repeated stress drops in the post-peak regime was observed. Three substages, i.e. stress stabilisation, stress decrease and stress increase, were identified for a single fracturing mode. Then quantitative and statistical analyses of the fracturing process at each substage were discussed. Based on the testing results, it was found that at the stress stabilisation substage, the proportion of mixed-mode fractures increased. At the stress decrease substage, the proportion of mixed-mode fractures decreased, and the tensile or shear fractures increased. At the stress increase substage, the proportion of mixed-mode or tensile fractures decreased, and the shear fractures increased. Finally, a conceptual model for the stepwise crack formation was proposed.

A rate-dependent model for sand to predict constitutive response and instability onset

A constitutive model has been proposed for predicting rate-dependent stress–strain response of sand and further implemented in finite element code to explore the influence of strain rate on the localization behavior of sand. The proposed model simulates various constitutive features of sand subjected to higher strain rates, e.g., enhanced shear strength, early peak followed by a softening response, reduced compression for loose sand, etc., which have been reported in the literature. Numerical simulations predict a delayed onset of strain localization and increase in the band angle with increasing strain rate. Strains are noticed to localize in the hardening regime for loose sand, whereas for denser state localization emerges in the post-peak regime.

A combined theoretical-experimental approach for modelling ductile fracture of cold-reduced G450 steel sheet

A thermodynamics-based coupled damage-plasticity model for high strength cold-reduced steel is developed on the basis of a generic form, allowing the use of any yield criterion and damage evolution rule. The model possesses essential features of ductile failure, including large plastic deformation prior to fracture and the effects of both triaxiality and Lode angle on the failure. In conjunction with theoretical development, experimental work for both furthering the understanding of ductile fracture in high strength cold-reduced steel and validation of the proposed model under different loading conditions is also performed. In particular, advanced non-contact measurement techniques based on Digital Image Correlation (DIC) are used not only for characterising the material failure/data in both pre- and post-peak phases but also for calibration of parameters and validation of the proposed model. While the pre-peak hardening responses with homogeneous deformation can be described by a nonlinear hardening rule, in the post-peak regime, localisation of deformation before complete fracture requires the use of energy in both characterisation of the material failure and calibration of the model. In this sense, the essential work of fracture obtained from the experiments is linked with parameters controlling the fracture of the material and also used for a simple regularisation technique based on the concept of smeared deformation. The model responses are validated against experimental data in terms of both macro responses and evolving failure patterns, demonstrating good potential for applications in failure analysis of structures made of high strength cold-reduced steel.

Fatigue fracture of fibre reinforced concrete in flexure

An experimental study on the behaviour of fibre reinforced concrete under flexure is presented, where pre-cracked notched beams were tested under cyclic loading for concrete with different dosages of hooked-ended steel fibres. The fatigue crack growth and stiffness degradation during the cycles are discussed in term of the rate of evolution of crack opening, and the critical crack opening is identified, beyond which the crack growth becomes unstable. S–N curves have been proposed with respect to the critical crack opening as these would be more relevant to design avoiding sudden failure. It is evident that higher dosages of fibres improve the fatigue life of cracked concrete only when the magnitudes of the cyclic loading are high, and that the effect of dosage is insignificant at lower magnitudes. For the fibre concretes considered here, the endurance limit seems to be about 50% of the pre-crack load in the post-peak regime.