Maximum Tangential Stress Research Articles

Fracture parameters and crack initiation Assessment Employing Mixed-mode I/II Fracture Criterion in Laminate Composites Using Digital Image Correlation Method

The present study aims to have fracture parameters, such as stress intensity factors (SIFs) and J-integral, as well as the crack initiation angle, extracted through the utilization of a novel mixed-mode (I/II) fracture criterion founded on the maximum energy release rate (MERR) and maximum tangential stress (MTS). The digital image correlation (DIC) method is employed in the analysis of crack behavior. Of note, several key findings are highlighted by our research. Results show a slight decrease in the mode I stress intensity factor (KI) with an increase in the crack inclination angle (φ). Interestingly, a bell-shaped curve characterizes the mode II stress intensity factor (KII), with its peak occurring at φ = 45. Comparison of experimental outcomes with finite element (FE) based modeling reveals consistent trends in both KI and KII from both sources. The comparison of experimentally determined crack initiation angles with predictions from the MERR and MTS criteria offers insightful observations. For φ ≤ 30, both criteria closely match experimental outcomes, as the crack initiation corresponds to φ. However, for φ > 30, these criteria underestimate the crack initiation angles, leading to a slower decrease in kink angles compared to observations. Further analysis involves a comparison between the two criteria and experimental results. Here, MERR emerges as being more accurate in prediction than MTS. Lastly, Simpson's rule is employed to calculate the J-integral, contributing to a comprehensive understanding of crack behavior. The reliability of the new mixed-mode fracture criteria and the DIC method in evaluating crack tip conditions and predicting minor kink angles is established by our study. Not only is understanding enhanced by our findings, but practical applications in predicting crack initiation and behavior in complex materials are also offered.

A strength-stress coupling criterion for rockburst: Inspirations from 1114 rockburst cases in 197 underground rock projects

Rockburst is a kind of engineering geological disaster induced by artificial engineering excavation. Qualitative and quantitative statistical analysis of rockburst cases is the primary work and research basis for rockburst prediction and discrimination. In this study, 1114 rockburst cases are collected and the qualitative and quantitative characteristics are analyzed. It is found that rockburst is prone to occur in rocks that are hard, complete, dry, and unweathered. Strong rockburst occurs more frequently in rocks with high uniaxial compressive strength (σc), elastic modulus, and elastic energy index, and in rock masses with high initial maximum principal stress (σ1)and tangential stress. Notably, rockburst may occur only when σc≥ 60 MPa and σ1≥ 10 MPa. On this basis, the “Human–Rock–Environment” three-element mechanism of rockburst is presented. Guided by this conception, the rationality and defects of the traditional strength-stress ratio (SSR) criterion are investigated. The SSR criterion does not adequately consider the absolute difference between the rock strength and geostress, as well as their combined effects. Subsequently, a strength-stress coupling (SSC) criterion is proposed from a two-dimensional scale.This criterion is characterized by considering both the absolute boundary conditions of σcand σ1, as well as their coupling interval. Five intervals of weak, weak–moderate, moderate, moderate–strong, and strong rockburst are divided in the two-dimensional chart and their distinguishing conditions are given. The existence of two transition zones for weak–moderate and moderate–strong reflects the coupling effect of rock strength and geostress when rockburst occurs, confirming the rationality and scientificity of the SSC criterion. The verification results show that the SSC criterion features higher accuracy as compared with the SSR criterion.

A new criterion based on the ratio of strain energy density for mixed-mode crack initiation prediction in ductile materials

In consideration of the different mechanisms of microscopic separation governed by dilation and microscopic shearing controlled by distortion on crack initiation, a new criterion based on the ratio of two strain energy densities (RSED) is proposed by differentiating the dilatational and distortional parts in strain energy density. The influence of the weights of two strain energy densities under different stress states on fracture resistance of crack initiation is also considered in the RSED criterion. The size and shape of the core region do not affect the direction and fracture resistance of crack initiation predicted by the RSED criterion. Moreover, the direction predicted by the RSED criterion is theoretically demonstrated to be identical to the orientations predicted by the maximum stress triaxiality (MST) criterion and the T criterion on crack initiation. The crack initiation directions which are obtained by the experiments for different materials and predicted by various criteria are found to be mostly coincident near the mode-I crack, and so are the fracture resistances. In the vicinity of the mode-II crack, the direction of crack initiation predicted by the RSED criterion shows a superior consistency for the experimental results of the ductile materials reported by Theocaris et al. (1982) and Kong et al. (1995) over the three classical criteria, such as the maximum tangential stress (MTS), the maximum energy release rate (MERR) and minimum strain energy density (MSED). The RSED criterion also has the potential to amend the issue of the underestimation for KIIC/KIC in some typical materials, e.g., the face-centered-cubic (f.c.c) metals. Therefore, the RSED criterion prefers to predict the direction and fracture resistance of crack initiation in ductile materials.

Implementation of the Displacement Discontinuity Method in Geotechnical Case Studies

This paper uses the displacement discontinuity method, one of the boundary element methods, to solve two major engineering problems. The first one addresses the safe design of underground excavations in fractured rock masses. The implemented method was used to control the slip of discontinuities passing through a circular opening at 45°. Special contact elements were used to simulate a possible slip on the cracks. At the same time, stress intensity factors (SIFs) were calculated using the gradient elasticity theory (special tip elements where numerical integrations are needed were excluded). The crack propagation due to shear slip occurrence was defined using the criterion of maximum tangential stress at an angle of 71° from the initial crack direction. The second one involved in the crack’s propagation was solved by applying pressure to the circular opening, while a part of it was transferred to the cracks, approximating the mechanism of hydraulic fracture. Finally, the implementation of higher elasticity elements in the cracks provided an accurate estimation of SIFs, showing an error of about 4%, as confirmed by comparisons with existing type I loading solutions.

On the decay of fracture toughness of sandstone subjected to freeze–thaw cycles

AbstractIn cold regions, progressive cracking, caused by freeze–thaw cycles, plays an important role in deteriorating engineering rock mass. Employing central cracked Brazilian disks (CCBDs) of sandstone, this paper aims at investigating the variations of falling rate among different modes of fracture toughness exposed to freeze–thaw cycles and the applicability of fracture criterion in predictions. Results revealed that different modes of fracture toughness exhibited comparable descending trends after varied freeze–thaw cycles. There was an agreement between the predictions via the generalized maximum tangential stress criterion (GMTS) and the test results. In specified conditions, the GMTS criterion incorporating fracture process zone of 0 freeze–thaw cycle can be applied to estimations for different freeze–thaw cycles. With increasing freeze–thaw cycles the fractal dimensions of mode I and mixed‐mode I–II fracture surfaces displayed a growing trend, indicating that the fracture surfaces became rougher and more complicated.

Loading rate effect on mixed-mode I + II fracture of V-notched Brazilian disc rock specimen under impact loads

Dynamic failure processes of rock masses containing numerous discontinuities including pores, cracks and notches have been the major issues in structural safeties of underground buildings subjected to impact loads, such as earthquake, blast and vehicle vibration. At present, most of the studies are focused on the dynamic fracture behaviors of rocks containing crack-like defects, while the studies on rocks containing notch-type defects are very rare, especially under mixed-mode loading conditions. This paper aims to study the effects of loading rate on mixed-mode I + II fracture characteristics of flattened V-notched Brazilian disc rock specimen under impact loads by means of an split Hopkinson pressure bar (SHPB) system combined with a high-speed camera. Dynamic fracture tests were performed on a total of 125 groups of V-notched rock specimens with different notch angles (0°, 30°, 45°, 60°, and 90°) and loading angles (0°, 30°, 45°, 60°, and 90°) at different air pressures (0.16 MPa, 0.18 MPa, 0.20 MPa, 0.22 MPa, and 0.24 MPa). Based on dimensional analysis, a numerical method was proposed to establish the relationship between mixed-mode notch stress intensity factor (NSIF) at the V-notch tip and external axial load. The dynamic notch fracture toughness values of flattened V-notched Brazilian disc rock specimens were calculated from the maximum values of dynamic force in the SHPB tests according to the proposed numerical method. The effects of loading rate, notch angle, and loading angle on the mixed-mode I + II notch fracture toughness value and crack initiation angle were quantitatively analyzed, which were verified using a modified maximum tangential stress (MTS) criterion by considering the loading rate effect. It is found that the loading rate has a significantly positive effect on the notch fracture toughness value, but has a negative effect on the crack initiation angle due to the inertial effect under dynamic impact loads. Generally, pure mode I fracture occurs for the loading angle of 0° or 90°, pure mode II fracture occurs for the loading angle varying from 35° to 45°, and mixed-mode fracture occurs for other loading angles. Good consistencies of the mixed-mode I + II notch fracture toughness values and crack initiation angles of the V-notched rock specimens are observed between the modified MTS criterion prediction and experimental results. This work may improve the understanding of the dynamic fracture behaviors of rock masses containing notch-type defects during tunnel excavation and underground construction process.

Thermally nonlinear analysis of propagating cracks under generalized thermal shock

In this paper, the thermally nonlinear response of stationary and moving cracks in isotropic media subjected to a hyperbolic thermal shock in the framework of the eXtended Finite Element Method (XFEM) is examined. For this purpose, the thermally nonlinear fully coupled Lord-Shulman (LS) model is considered as governing equations. In addition, two cases of material properties, temperature-dependent (TD) and temperature-independent (TID), are assumed. The geometry, including a predefined crack, is modeled using the XFEM for spatial discretization. For modeling the dynamic crack growth within the XFEM, the Maximum Tangential Stress (MTS) criterion is modified based on the stress field near the tip of a growing crack to calculate the dynamic crack growth angle. Besides, the nonlinear form of the Newmark integration scheme is implemented for the numerical integration in the time domain. Finally, a method based on one of the Barnet-Lothe tensors for isotropic materials is employed to extract the stress intensity factors (SIFs). In the provided examples, the effects of thermal nonlinearity, the temperature-dependency of material properties, and the LS relaxation time on the crack propagation path are explored.

Prediction of the effect of shot peening residual stress on fretting fatigue behaviour

Fretting occurs at two contact bodies, which sustain oscillation loading. The fretting phenomenon decreases the fatigue lifetime dramatically. Shot peening is a commonly used surface treatment to enhance the resistance to damage because the residual stress is introduced to the surface of the material. Some studies showed that residual stress was helpful to improve fatigue lifetime. Under fretting fatigue condition, there are two main stages: crack initiation stage and the crack propagation stage. In this paper, the authors apply the Critical Plane (CP) method combined with the Theory of Critical Distance (TCD) to study the fretting fatigue crack initiation behaviour under the influence of shot peening residual stress, including the crack initiation position, orientation and the crack initiation lifetime. Meanwhile, the Linear Elastic Fracture Mechanics (LEFM) theory and the Extension Maximum Tangential Stress (EMTS) criterion are adopted to study crack propagation behaviour considering shot peening residual stress effect. The results show that the methods applied in this paper can make good predictions of fretting fatigue behaviour, including both initiation and propagation stages. Meanwhile, the influence of residual stress in these two stages are studied as well.

A nontraditional method for reducing thermoelastic stresses of variable thickness rotating discs

Stresses reductions and/or raising the load-carrying capacity for a mechanical structure are always great dilemmas for researchers. In this article, a novel method is proposed, and its efficiency is examined for achieving these goals on functionally graded rotating nonuniform thickness discs. The originality of this method relies on comprising a geometrically well-defined area, into the whole structure, with certain homogeneous properties including density, thermal expansion coefficient, and elasticity matrix. This area acts as a reducer of the maximum values of various stress components. The solution of the magnetoelastic/magneto-thermoelastic problem is accomplished using the finite element method. The disc is subjected to partial uniform outer pressure, whereas, upon applying thermal loads; the thermal boundary conditions are considered symmetric. The proposed method is found to be beneficial as the obtained results demonstrated the ability to reduce the maximum stresses with different percentages depending on the location, angular width, and properties of the predefined area. This is reflected by an attainable decrease in the maximum compressive tangential stress and the von Mises stress by approximately 20.7% and 12.5%, respectively, under certain conditions.

Investigation of Constraint Effect and Fracture Mode for Mixed Mode Inclination Surface Crack in Infinite Plate Under Compression

Abstract The stress field, constraint effect, and fracture mode transition at crack tip of mixed mode I-II-III inclination surface crack under compression have been investigated. The effects of geometrical configurations (relative crack depth and aspect ratio), friction coefficient, and biaxial scale factor on stress intensity factor (KII and KIII) and in-plane constraint parameter T-stress are quantitatively studied, the stress field at different crack inclination angles under tension and compression are compared, the failure mode at special locations along crack front of inclination surface crack is analyzed according to the generalized maximum tangential stress criterion (GMTS). The relative crack depth has slight effect on stress intensity factor and T-stress, and aspect ratio has a significant effect on stress intensity factor and T-stress. The friction coefficient decreases the magnitude of stress intensity factor and increases the magnitude of T-stress, the greater the crack inclination angle is, the more pronounced the effect is when crack inclination angle greater than 30 deg. The stress distribution around crack tip under tension and compression is completely different. At free surface, the crack will failure in-plane shear mode II sliding crack, and at the deepest part of crack, the crack will start as out-plane shear mode III tearing crack under compression.

Analysis of Fracture Characteristics of Ore Rock Based on GMTS Criterion

In this paper, the Tongkaiyu ore-rock is taken as the research object, aiming to analyze the fracture characteristics of the mixed type I, II and I–II faults in ore-rock by using the appropriate fracture criterion, and analyze the fracture development mode. Considering the non-singular term (T stress) in the formula of stress field sequence at crack tip, the generalized maximum tangential stress (GMTS) criterion is established, which is consistent with the boundary conditions of the central straight Brazilian disk (CSTBD) specimen. Through CSTBD splitting test and three-dimensional particle flow code (PFC3D) numerical simulation, the Angle between the central direct joint and the loading direction was changed, and the fracture toughness and fracture development law of type I, type II and type I–II mixed fracture were obtained. The theoretical and simulation results show that, compared with the maximum tangential stress (MTS) criterion, the GMTS criterion has an advantage in judging the fracture characteristics of mixed types I, II and I–II in ore rock.

A logarithmic model for predicting fracture trajectory of pre-cracked rock specimen under compression

Researches on the fracture trajectory of pre-cracked specimen are mostly focused on experiments and numerical simulations, and theoretical studies are usually limited to the fracture criterion, crack initiation angle and load, with few theoretical studies on the prediction of the complete fracture trajectory. This work first derives stress intensity factor (SIF) and T-stress expressions related to crack geometry, friction and deformation parameters, and physical properties of the rock. Then, the classical maximum tangential stress (MTS) fracture criterion is modified, followed by the development and validation of a logarithmic model for predicting the fracture trajectory of pre-cracked specimen made of diverse brittle or quasi-brittle rock materials under compression. The results shown that the crack geometry, friction and deformation parameters have significant effects on SIF and T-stress under compression. The present study demonstrates a strong concurrence between the anticipated fracture trajectory of pre-cracked specimens subjected to compressive loading and diverse experimental outcomes. Furthermore, the fracture trajectory is predominantly influenced by the crack geometry, friction and deformation parameters, and the mechanical characteristics of the specimen.

A Theoretical and Experimental Investigation on the Fracture Mechanism of Center-Symmetric Closed Crack in Compacted Clay under Compression–Shear Loading

In this study, a modified maximum tangential stress criterion by considering T-stress and uniaxial compression tests have been utilized to theoretically and experimentally reveal the fracture initiation mechanism of a center-symmetric closed crack in compacted clay. The results show that wing cracks occur in the linear elastic phase of the stress-strain curve. In the plastic phase of the stress-strain curve, the wing cracks extend gradually and the shear cracks occur. The crack initiation stress and peak stress of compacted clay first decrease with the rise in pre-crack inclination angle (β = 0°–40°), and then increase with the rise in pre-crack inclination angle (β = 50°–90°). When the pre-crack inclination angle is relatively small or large (β ≤ 10° or β ≥ 70°), the crack type is mainly tension cracks. Secondary shear cracks occur when the pre-crack inclination angle is 10°–80°. When the dimensionless crack length is larger than 0.35, the crack types include wing-type tension cracks and secondary shear cracks. The experimental results were compared with the theoretical values. It was found that the critical size rc of compacted clay under compression-shear loading was 0.75 mm, smaller than the value calculated by the empirical formula (12 mm). The MTS criterion considering T-stress can be used to predict the compression-shear fracture behavior of compacted clay.

Fracture assessment of U-notched diagonally loaded square plates additively manufactured from ABS with different raster orientations

This paper investigates the fracture behavior of 3D-printed notched specimens under various combinations of mode I and mode II loadings, including pure mode I and pure mode II loadings. The study employs fused deposition modeling (FDM) to fabricate U-notched diagonally loaded square plate (UNDLSP) specimens using Acrylonitrile Butadiene Styrene (ABS) filaments. To experimentally measure the fracture load and fracture initiation angles, 120 UNDLSP specimens are 3D printed with different notch tip radii, notch orientation angles, and [0, 90], [45, −45] raster orientations. Tensile characteristics and translaminar fracture toughness of ABS plates with [0, 90] and [45, −45] raster orientations are obtained using dog-bone and cracked DLSP samples, respectively. Considering the layered nature and anisotropy of the material, the virtual isotropic material concept (VIMC) is utilized along with two well-known stress-based brittle fracture models: the maximum tangential stress (MTS) and the mean stress (MS) criteria. The test results are predicted using the fracture limit curves and fracture initiation angle curves developed for these fracture models in terms of the notch stress intensity factors (NSIFs). Furthermore, a general equation is established to estimate the fracture initiation angle of U-notched ABS specimens under mixed mode I/II loading, irrespective of geometry, raster orientation, and notch tip radius, based on the curves obtained from the MTS and MS criteria. The fracture surfaces of the UNDLSP specimens are examined using scanning electron microscopy (SEM) to study the micro-mechanisms involved in the fracture process. The findings indicate that both the VIMC-UMTS and VIMC-UMS criteria accurately predict the experimentally achieved fracture loads. Additionally, it is observed that the fracture initiation angle curves for both criteria are nearly identical and accurate. The significance of this research is not only providing numerous new experimental results on mixed mode I/II fracture of notched 3D-printed ABS plates, but also providing simple, fast, accurate, and robust criteria for predicting the critical loads of such plates, without the need for conducting layer-by-layer failure analysis using progressive damage models.

Crack deflection in shale-liked layered rocks under three-point bend loading

The bedding angle has a significant influence on the fracturing of anisotropic geomaterials, such as shale rock, and crack deflection usually takes place during crack propagation. To better understand and estimate the tensile crack deflection in shale, 35 large notched deep beam (NDB) specimens with different bedding angles—0° (Short-Transverse), 15°, 30°, 45°, 60°, 75°, and 90° (Arrester)—were tested under three-point bend (3PB) loading. For mode I dominated loadings, significant expected anisotropies in both fracture toughness and crack deflecting behavior were observed. The nominal mode I fracture toughness showed a progressive decrease from the arrester to the short-transverse orientation, with values from 1.371 MPa·m1/2 to 0.731 MPa·m1/2. When the bedding angle was less than or equal to 60°, the crack deflection induced the difference between the nominal and actual fracture toughness. By introducing the weak plane model to describe the fracture toughness of shale, an anisotropic K-type maximum tangential stress criterion (KMTS criterion) was presented. Using the KMTS criterion, the deflection angle and fracture load for an arbitrary bedding angle could be obtained conveniently with fracture toughness values in both arrester and short-transverse orientations. The comparison between theoretical predictions and experimental data demonstrated that the KMTS criterion could provide a dependable prediction, particularly for the slight deflection when the crack propagates within the matrix.

Thermal effects on prediction accuracy of dense granite mechanical behaviors using modified maximum tangential stress criterion

Thermal effects on prediction accuracy of dense granite mechanical behaviors using modified maximum tangential stress criterion

Fretting fatigue crack propagation under out-of-phase loading conditions using extended maximum tangential stress criterion

In this paper, a numerical crack growth analysis is conducted by using the Maximum Tangential Stress (MTS) criterion and its extension, which considers the contact stress between the crack surfaces. Moreover, fretting fatigue is a multiaxial non-proportional loading problem, and most components are subjected to asynchronous loads. Therefore, this paper mostly focuses on the influence of loading phase difference (Φ) on the crack propagation behavior. Meanwhile, for the fretting fatigue problem, as there is a relationship between the crack initiation and propagation, the effect of initiation characteristics, including crack initiation position and direction is studied under different loading phase differences using Linear Elastic Fracture Mechanics (LEFM) theory. It is observed that the predicted crack path and lifetime by using the extended MTS criterion and Paris’ law is in good agreement with the experimental results in the literature. It is found that the phase difference has a strong effect on the crack propagation behavior. Furthermore, we found that the crack initiation position has a greater influence on crack propagation behavior than the crack initiation direction.

Effects of Support Friction on Mixed-Mode I/II Fracture Behavior of Compacted Clay Using Notched Deep Beam Specimens under Symmetric Fixed Support

This paper investigates the effects of support friction on mixed-mode I/II fracture behavior of compacted clay using notched deep beam (NDB) specimens under symmetric fixed support. Numerical models of 330 NDB specimens were established considering the crack inclination angle, crack length, support span, and support friction coefficient, and the normalized fracture parameters (YI, YII, and T*) of NDB specimens were calibrated. The numerical results showed that the values of YI, YII, and T* decreased at different degrees after considering the support friction. Notably, the support friction coefficient could significantly change the loading pattern at the crack tip. To verify this phenomenon, 12 compacted clay NDB specimens were prepared, and a mixed-mode I/II fracture test was performed under fixed support conditions; the phenomenon of asymmetric crack propagation was studied. The test data were processed using the numerical calibration results of YI, YII, and T* with and without consideration of friction. Afterward, the test data were compared and analyzed by combining the generalized maximum tangential stress (GMTS) and the maximum tangential stress (MTS) criteria. The analysis indicated that the real fracture characteristics of compacted clay NDB specimens could not be reflected when conducting mixed-mode I/II fracture tests under symmetric fixed support conditions if the test results were analyzed by YI, YII, and T* without considering support friction, as in previous studies.

A general interaction integral for dynamically propagating cracks in anisotropic materials

In this paper, some relevant tools for numerical analysis of dynamically propagating cracks in anisotropic materials are discussed within the framework of the eXtended Finite Element Method (XFEM). A general interaction integral that is domain-independent is proposed to determine the Stress Intensity Factors (SIFs) for both stationary and dynamically propagating cracks in anisotropic materials. Besides, a set of functions for enriching the nodes of the tip element, which are applicable for modeling both quasi-stationary and dynamically propagating cracks in anisotropic materials, are also proposed. In addition, a modified form of the Maximum Tangential Stress (MTS) criterion is manipulated for estimating the direction of the propagating crack which takes into account the asymptotic stress field near a propagating crack tip in anisotropic materials under mixed mode plane loading. Finally, the accuracy of the suggested method is then confirmed using some numerical examples that involve both stationary and dynamically propagating cracks. Additionally, a thorough investigation is done into the impacts of material anisotropy and material orientation on the path of dynamically propagating cracks in orthotropic structures.

Mixed mode crack growth behaviour considering plasticity-induced and roughness-induced closure

A theoretical analytical solution of the Christopher-James-Patterson (CJP) model in mixed mode is presented in this paper, where the influence of roughness-induced crack closure (RICC) on fatigue crack growth is considered and a new effective stress intensity factor range (ΔKP-R) is proposed as the driving force for crack propagation. The solution showed that both the maximum tangential stress criterion (MTS) and the minimum strain energy density criterion (S) can reasonably well predict the crack deflection angle. Experimental testing on EA4T axle steel samples showed that both the 8 % compliance offset criterion in the conventional ASTM procedure and the parabolic-line method were suitable for the determination of the crack-opening force. The solution includes plasticity-induced crack closure (PICC) and RICC effects, therefore the crack propagation behaviour can be described in a more effective way. Compared with the traditional mixed model crack growth models with higher fitting correlation coefficient, the fitting effect of the CJP model has been improved by 12 % in this solution, and the residual stress in the pre-crack stage increased the crack-closure level. Due to the influence of residual stress, the rate of roughness value (RA) showed a fluctuation of 4.5 %, which indirectly affected the fatigue crack growth rate (FCGR) of the test.