Mixed-mode Crack Research Articles

Research on Failure Characteristics of Concrete Three-Point Bending Beams with Preexisting Cracks in Different Positions Based on Numerical Simulation

Concrete three-point bending beams with preexisting cracks are widely used to study the growth process of I-II mixed mode cracks. Studying the failure characteristics of preexisting cracks at different locations on concrete three-point bending beams not only has important scientific significance but also has a wide range of engineering application backgrounds in the safety assessment of engineering structures. In this paper, through several numerical experiments, the influence of preexisting cracks at different positions on the failure characteristics of concrete three-point bending beams is studied, and three typical failure modes are obtained. The failure process of the specimens with three typical failure modes is discussed in detail, and it is pointed out that the crack failure mode is tensile failure. The change trends of bearing capacity, acoustic emission quantity, and acoustic emission energy of three typical failure modes are analyzed. The maximum bearing capacity, the maximum acoustic emission quantity, and energy of three failure modes of concrete three-point bending beams generally show an increasing trend.

Investigation on in-plane and out-of-plane constraint effects for plates with I-II mixed mode cracks under biaxial compressive loading

Crack sizes, geometrical configurations of specimens and structures, as well as loading conditions have significant effect on in-plane and out-of-plane constraint effects. Safety assessment with constraint effects can effectively improve the accuracy. There are prominent discrepancies between compressive fracture and tensile fracture. Most of engineering structures are usually under biaxial loading conditions, so that the influence of biaxial compressive loading on fracture analysis with constraint effects is worth paying attention. The size effect factor (relative crack length a/W, thickness to width ratio t/W and crack inclination angle β) and load factor (lateral pressure coefficient λ) of central cracked plate (CCP) specimens with I-II mixed mode crack on constraint effects under biaxial compressive loading have been investigated through theoretical analysis and finite element analysis. In-plane non-singular T-stress (T11) and its components (Tx, Ty, and Txy), out-of-plane non-singular T-stress (T33), out-of-plane constraint parameter Tz and stress triaxiality factor h have been adopted to quantify constraint effects. It demonstrates that λ has remarkable influence on SIF, T11 and T33 under biaxial compressive loading. The effect of higher a/W (long cracks) on B11 is more noticeable when I-II mixed mode crack is closer to pure mode II crack or pure mode I crack. The proportion of shear fracture increases under biaxial compression with λ. With the increase of β, B33 decreases sharply and then increases rapidly with higher λ. Tz can directly reflect stress states and adequately describe the loss of out-of-plane constraint. The stress state of crack tip is more likely to asymptotic plane-strain condition ahead of crack tip under compression. There are crucial differences in variation range and tendency of Tz subjected to biaxial tensile and biaxial compressive loading. Moreover, Tz is insensitive to λ. Stress triaxiality factor h can illustrate in-plane and out-of-plane constraint effects including size effect and load factor (sensitive to a/W, t/W, β and λ), especially can be adopted as a unified constraint parameter for long cracks under biaxial compressive loading. The in-plane T-stress components can be directly calculated by T11 under biaxial compression. The empirical equations of KII and T11 containing size effect and load factor simultaneously are effective and convenient for applications.

Effect of mode mixity, temperature, binder content, and gradation on mixed mode (I/II) R-curve of asphalt concrete at low temperatures

R-curves provide the means for in-depth analysis of crack propagation in engineering materials. However, due to experimental costs and difficulties, the application of this approach has been very limited in hot mix asphalt (HMA) investigations to date. In the present research, single edge notched beam (SE(B)) tests are conducted on asphalt concrete (AC) mixtures in pure mode I and mixed mode (I/II) states at low temperatures. Mode mixity is induced in the test setup by fabricating an asymmetric notch with varying relative offset values of 0.3, 0.47, and 0.67 from the centerline of the beam where the loading is situated. Mode I and mixed mode R-curves were constructed by plotting cumulative fracture energy versus crack extensions obtained from digital images during each test. Based on the R-curves, the mixed mode (I/II) crack propagation regime of the HMA with regards to temperature variations (+5 °C, 0 °C, −5 °C, −10 °C, −15 °C, and −20 °C in this research), binder contents (4%, 4.5%, and 5%), and the nominal maximum aggregate size (NMAS) of 19 mm and 25 mm were determined and studied. A considerable stable crack growth zone was observed in the mixtures as the contribution of shearing mode is increased and the mode I contribution is decreased. The stable crack growth region was proceeded by unstable crack propagation in all the cases and the instability outweighed in lower temperatures, particularly in −15 °C and −20 °C. Nevertheless, the mixtures exhibited greater resistance to post-peak crack propagation in higher mode mixites which could be inferred from the progressively rising R-curve in the instability phase. It was also observed that the NMAS value of 25 mm could increase the cohesive energy and the blunting work required to initiate macrocracking while amplifying unstable crack growth at −20 °C and −15 °C in comparison with NMAS 19 mm.

Cracking feature and mechanical behavior of shield tunnel lining simulated by a phase-field modeling method based on spectral decomposition

Disturbed by the surrounding engineering construction, the shield tunnel lining may appear mixed-mode cracks such as segment cracking and joint crushing, which seriously damage its mechanical behaviors. To simulate the shield tunnel lining with mixed-mode cracks, a phase-field modeling method based on spectral decomposition is proposed, which unifies the critical energy release rate and crack-driving strain energy. The indoor full-scale tests are carried out to study the failure mechanism of the segment and joint. The phase-field modeling method is verified by the tests, and the influence of loading path on the cracking feature and mechanical behavior of the segment and joint are analyzed by the proposed numerical method. The test and numerical simulation indicate that the crack in the segment gradually develops from I-type crack (tensile crack) to Y-type crack (tensile-shear crack), and the joint is crushed due to local compression, and the crushed region expands gradually with the increase of the eccentricity. As the eccentricity decreases, the ultimate bearing capacity of the segment and joint is improved, while the ductility decreases. Then, the phase-field modeling method is extended to the stratum-structure model. As the surface load increases, the joints of the shield tunnel lining are crushed, and the top and bottom segments are cracked. Meanwhile, the bending moment is reduced due to the redistribution of internal forces caused by the segment cracking and joint crushing.

An empirical fracture equation of mixed mode cracks

On the basis of the multiaxial fatigue limit equation by Liu and Yan, an empirical fracture equation of mixed mode crack is proposed in this note. By means of the test data on the mixed crack fracture reported in literature, the empirical fracture equation has been verified to be not only simple in computation but also high in accuracy.

Mixed Mode Crack Propagation in Iliac Bone

Abstract Bone is a complex material that can be regarded as an anisotropic elastic composite material. The problem of crack propagation in human bone is analyzed by using a generalization of the maximum tensile stress criterion (MTS). The results concern the critical stress for crack propagation and the direction of the crack path in Iliac bone.

Investigation of Mixed-Mode Crack Propagation Behaviour under Impact Loading

The fractured rock masses are often subject to impact loading, giving rise to complicated mixed-mode cracking. The propagation of the cracks may result in structural instability and project disaster and often cannot be understood by studying only the mode I fracture. Discovering the mixed-mode crack behaviour is of great interest to engineering design. Here, we report on an investigation of mixed-mode crack propagation under impact loading. A Single Cleavage Simi-Ellipse configuration was proposed with different pre-crack dip angles. Split Hopkinson pressure bar impact tests were conducted and a high-speed camera was employed to capture the cracking behaviour. With the experimental-numerical method, the dynamic initial stress intensity factors KI and KII were calculated by the finite element method. The AUTODYN was applied to study mixed-mode crack propagation numerically. The investigation demonstrates that the mixed-mode crack would adjust fracture mode continuously in the whole propagation process companied with several extending decelerations. The crack initiation mode and the dynamic initiation stress intensity factor have a relationship with the impact loading orientation. The AUTODYN can be applied in predicting rock crack propagation, offering new insights into dynamic stability analyses in engineering design.

Characteristic-tensor method for efficient estimation of stress-intensity factors of three-dimensional cracks

A practically simple characteristic-tensor method (CTM) is proposed for evaluation of the stress state at crack tips subjected to arbitrary loading conditions. Stress-intensity factors (SIFs) in several representative types of three-dimensional cracks (such as the circular crack, elliptical crack, and inclined circular crack) were computed by the finite-element method. Even using a relatively coarse mesh, SIFs for mixed-mode crack problems could be estimated by the CTM with high accuracy. The results show that the CTM is a practical and reliable approach to the estimation of SIFs of 3D cracks that are important to the industrial application.

Experimental study on laminated glass responses of high-speed trains subject to windblown sand particles loading

China has built high-speed railways in desert areas. When high-speed trains run in wind-sand flows, wind pressure and sand impact increase the fracture probability of side-window laminated glass. This paper aims to explore the dynamic response and impact resistance performance of laminated glass in high-speed trains subjected to windborne sand particles loading using full-scale experimental tests. Dynamic pressure tests were carried out using an air cannon on the laminated glass to evaluate the glass response under wind dynamic pressure loading. Sandstone impact tests were conducted to study the laminated glass dynamic behavior during particle impact loading. The study results demonstrate that sand impact loading was the primary factor causing the laminated glass damage. The mixed-mode crack initiation and propagation were driven by the reflected tensile waves and Rayleigh surface waves under sandstone impacts. In addition, a statistical analysis of the scratch velocity of the laminated glass suggested that the size of sandstone and impact angle were sensitive to the scratch velocity in the sand particles impacts. This research provides practical help for designing crack-resistant laminated glass structures for high-speed trains.

T-stresses solution and out-of-plane constraint for central cracked plate (CCP) with I-II mixed mode crack under uniaxial compression

The crack tip constraint effect should be considered in the fracture assessment of structures with cracks, and the in-plane constraint and out-of-plane constraint should be illustrated simultaneously for the three-dimensional cracks in practical engineering structures. The constraint effect of central cracked plate (CCP) specimens with I-II mixed mode crack under uniaxial compression has been investigated quantitatively in present work. The relationship between T-stress components (Tx, Ty, Txy) and in-plane constraint parameter T11 has been established. The effects of a wide range of in-plane and out-of-plane geometrical configurations (including relative crack length a/W and thickness to width ratio t/W), material parameters (including elastic modulus E and Poisson's ratio ν), crack inclination angle β and friction coefficient μ corresponding to stress intensity factors (K), in-plane constraint parameter T11 and out-of-plane constraint parameters T33 and Tz at the crack tip have been calculated and discussed theoretically and numerically. It demonstrates that the constraint parameters can effectively express the stress-strain field at the crack tip. The relative crack length a/W has an influence on the in-plane and out-of-plane constraint parameters, and the thickness to width ratio t/W mainly affects the out-of-plane constraint parameters T33 and Tz. The out-of-plane constraint parameters should be considered for long cracks. There is a loss of out-of-plane constraint under compression when t/W less than or equal to 0.9, which can be much more directly and evidently investigated through the analysis of Tz than T33. The empirical equations for KII, T11 and Tz have been obtained according to theoretical analysis and simulation results, which are convenient and effective for application.

Effects of specimen type and thickness on elastic and elastic–plastic fracture parameters for I-II and I-III mixed mode cracks

This paper focuses on the effects of specimen type and thickness on the fracture parameters for I-II and I-III mixed mode cracks. Results show that mode III and mode II components emerge simultaneously near free surfaces of specimens under I-II and I-III mixed mode loadings, which is called coupling effect. For specimen under I-II mixed mode loading, the level of coupling effect is independent of specimen geometry. While for specimen under I-III mixed mode loading, the coupling effect is much affected by specimen geometry. Thicker middle cracked specimen has smaller coupling effect than thinner single edge notched specimen. Due to the non-negligible coupling effect on crack tip field and fracture behavior, a new mixity parameter taking the coupling effect into consideration is proposed to describe the mixed mode crack. Furthermore, mode I component is the main reason that constraint of I-II or I-III mixed mode crack exists, which is related with loading mode like tensile opening loading and bending opening loading. Specimen under tensile opening loading like middle cracked specimen is in a lower constraint than that under bending opening loading like single edge cracked specimen. Results also shows that the effect of specimen type on constraint is greater than that of thickness.

Quantitative fracture analysis of hard rock containing double infilling flaws with a novel DIC-based method

Natural or engineering rock joints often contains different kinds of infilling materials, such as clay, sand, or cement, which may significantly affect the crack initiation, propagation, and coalescence between joints. The cracking behaviours of the jointed rock masses have not been widely studied quantitatively, especially for the crack mechanism identification. Most previous studies identified the crack mechanisms, i.e. mode I, mode II, and mixed-mode, by visual inspection of the relative opening or sliding displacement across the cracks or the roughness of the newly formed fracture surfaces, while a quantitative method has not been developed to avoid being subjective. Thus, a novel DIC-based method is proposed for quantitatively identifying the crack path and fracture mechanism of the initiated cracks in rocks. The method involves three steps, including initial displacement measurement, displacement reconstruction around the discontinuity, and crack identification and displacement jump measurement. The key challenge of the proposed method lies in the displacement reconstruction at the discontinuity with the modified subset splitting technique, whose validity is verified by the computer experiments on synthetic images. With the novel DIC-based method, the cracking behaviors of flawed marble specimens considering different flaw geometries and infilling conditions are quantitatively studied. The experimental result indicates that the crack initiation, propagation, and coalescence of flawed marble specimens are closely related to infilling conditions. The mixed-mode cracks are more likely to initiate from cement-filled and resin-filled flaws, while mode I crack is prone to initiate from unfilled and gypsum-filled flaws. Besides, the infilling significantly alters the coalescence patterns at β=45°, while it plays a little role in crack coalescence at β=30° and 60°, mainly controlled by flaw geometry.

Evaluation of stress intensity factors under thermal effect employing domain integral method and ordinary state based peridynamic theory

In this article, several thermoelastic benchmark cases are studied within the framework of ordinary state based peridynamic theory (OSPD). By using OSPD, the limitations of geometrical discontinuity in fracture analysis can be overcome. Meanwhile, double nodes can also be avoided during crack definition. A domain integral method with thermal effect is applied in calculating the thermal stress intensity factors (TSIFs). Meanwhile, peridynamic differential operators are utilized to rewrite the spatial derivatives in the domain integral. Numerical investigations of TSIFs in the single and mixed-mode crack scenarios are provided respectively, and verified by the reference solutions. Good agreements between OSPD and the reference solutions show high performance and capability of the proposed method in thermoelastic fracture analysis.

The propagation speed of multiple dynamic cracks in fiber-reinforced cement-based composites measured using DIC

The primary purpose of this work is to explore the propagation speed and spreading direction of multiple dynamic cracks in steel fiber-reinforced cement-based composites using Digital Image Correlation (DIC). Three-point bending beams with an off-centered notch were employed to facilitate the initiation of mixed-mode crack. Specifically, beams with different fiber contents, 0% , 0.4% and 0.8% in volume fraction, were manufactured. Tests were carried out at low loading-line displacement rates (2.2 μm/s and 22 mm/s) using a servo-hydraulic machine, under impact loading (1.77 m/s and 3.55 m/s) using a drop-weight impact device. A high-speed video camera was used to record the process of (multiple) crack propagation. Besides the shear crack started from the notch tip, the flexural cracks initiated from the central bottom surface were also formed in all beams except those with a fiber content of 60 kg/m3 (0.8% in volume ratio) impacted at 1.77 m/s. The flexural cracks propagated with a maximum speed of 1297 m/s, whereas the shear cracks reached up to 1149 m/s. Such high speeds of multiple-crack propagation, more than 50% of the Rayleigh wave speed, have been rarely measured in the lab. In addition, the evolution of simultaneous crack formation was recorded; the influence of both fiber content and loading rate is examined. Furthermore, all three types of material were previously characterized through independent tests, the flexural strength, the toughness indices as well as the residual strength factors are reported.

Mixed-mode fracture in a soft elastomer

We present an experimental study on mixed-mode crack propagation in thin sheet specimens of a soft silicone elastomer. To introduce mixed-mode loading, we adopted the widely applied pure shear fracture testing geometry, but oriented the initial crack at an inclined angle relative to the direction perpendicular to the external tensile loading. Because of the inclined angle, locally the crack tip was subjected to both tensile and shear loading even though the global loading was tensile, thus resulting in a mixed Mode-I and Mode-II condition for the crack. Altering the inclined angle led to different degrees of mode-mixity and hence different crack propagation trajectories. Using a particle tracking method, we were able to measure the nonlinear deformation fields throughout the entire course of crack propagation. Based on the measured deformation fields, we used J-integral to calculate the energy release rate during crack propagation and found that the results were insensitive to the tangent direction of the crack trajectory, suggesting that the fracture toughness was approximately mode independent. Moreover, the tangent direction of the crack trajectory, when plotted in the reference configuration, was observed to be perpendicular to the direction of maximum principal stretch. The experimental results pave the way for establishing quasi-static mixed-mode fracture criteria for soft silicone elastomer, which has not yet been well understood in the literature.

Weight function calculation method for analyzing mixed-mode shear cracks in reinforced concrete beams

Based on a virtual crack extension (VCE) technique in finite element method (FEM), this paper proposes a rational method of calculating weight functions for inclined shear cracked RC beams with any geometry. As an inclined shear crack in an RC beam is geometrically determined by the absolute dimensions, shear span to beam depth ratio, crack inclined angle and crack initiation location, the relations between the weight functions and these parameters are investigated by either following theoretical equations or varying the parameters in a certain range. The range is determined to enclose the geometries of inclined critical shear cracks of a large set of RC beams which were tested under a bending load but failed due to the propagation of a critical inclined shear crack. Moreover, a method of calculating the weight functions of a shear cracked RC beam with any geometrical properties is developed numerically using the weight functions calculated and provided in this study. As a result, it is possible that the shear behaviors of RC beams can be even studied conveniently but also comprehensively following the theoretical fracture mechanic approach.

Effect of aggregate gradation on crack propagation in asphalt mixtures at low temperatures based on the Eshelby equivalent inclusion theory

This paper presents a study to evaluate the mixed-mode crack propagation behavior of several typical asphalt mixtures at low temperatures. Based on the Eshelby equivalent inclusion theory and Griffith micro-fracture theory, the basis for determining the I-II mix mode fracture types during crack propagation was established. A random particle-based growth algorithm was adopted to develop a heterogeneous DEM model for three typical asphalt mixtures: open-graded structure (OGFC-13), gap-graded structure (SMA-13), and dense-graded structure (AC-13) mixtures, and a virtual indirect tensile test was conducted. The characteristics of the development of I-II mix mode fractures for three asphalt mixtures at low temperatures were analyzed in four aspects: changes in the number of micro-cracks, proportions of crack types, energy release rates, and force chain diagrams of crack tip stress fields. The results indicate that: (1) From the perspective of the changes in the number of cracks, the dense-graded asphalt mixture has the largest number of micro-cracks, and the proportion of mode I cracks that are mainly due to tension is dominant, while the proportion of mode II cracks that are mainly due to shear is insignificant; (2) The crack energy release rate G(t) in the time-domain increases as the rate of internal energy consumption increases, along with a reduction in the energy storage rate, as the spatial distribution of aggregates change in the open-graded, gap-graded, and dense-graded asphalt mixtures. From an energy standpoint, it was found that more energy is required when the dense-graded asphalt mixture is fractured; (3) In terms of the stress field distributions in the crack tip regions, compared with the open-graded and gap-graded asphalt mixtures, the interior of a dense-graded asphalt mixture is less likely to develop stress concentrations, which is beneficial in suppressing the occurrence of a shear stress field, thereby reducing the occurrence of mode II low-temperature cracks. The results of this research provide theoretical support for the anti-cracking design of asphalt mixtures based on meso-characteristics.

Fracture Parameters and Cracking Propagation of Cold Recycled Mixture Considering Material Heterogeneity Based on Extended Finite Element Method.

Cold recycled mixture (CRM) has been widely used around the world mainly because of its good ability to resist reflection cracking. In this study, mixed-mode cracking tests were carried out by the designed rotary test device to evaluate the cracking resistance of CRM. Through the finite element method, the heterogeneous model of CRM based on its meso-structure was established. The cracking process of CRM was simulated using the extended finite element method, and the influence of different notch lengths on its anti-cracking performance was studied. The results show that the mixed-mode fracture test method can effectively evaluate the cracking resistance of CRM by the proposed fracture parameters. The virtual tests under three of five kinds of mixed-cracking modes have good simulation to capture the cracking behavior of CRM. The effect of notch length on the initial crack angle and the crack propagation process of the CRM is mainly related to the distribution characteristics of its meso-structure. With the increase of the proportion of Mode II cracking, the crack development path gradually deviates, and the failure elements gradually increase. At any mixed-mode level, there is an obvious linear relationship between the peak load, fracture energy, and the notch length.

Mixed Mode Crack Propagation in Polymers Using a Discrete Lattice Method

The fracture behavior of polymeric materials has been widely studied in recent years, both experimentally and numerically. Different numerical approaches have been considered in the study of crack propagation processes, from continuum-based numerical formulations to discrete models, many of the latter being limited in the selection of the Poisson’s coefficient of the considered material. In this work, we present a numerical and experimental analysis of the crack propagation process of polymethylmethacrylate beams with central and eccentric notches subjected to quasi-static three-point bending tests. The developed discrete numerical model consists of a regular triangular lattice model based on axial and normal interaction springs, accounting for nearest-neighbor interactions. The proposed model allows solving the above mentioned limitation in the selection of Poisson’s coefficient, incorporating a fracture criterion defined by a bilinear law with softening that includes the fracture energy in the formulation and allows considering a progressive damage. One of the main objectives of this work is to show the capacity of this lattice to simulate quasi-static fracture problems. The obtained results show that the proposed lattice model is capable of providing results close to the experimental ones in terms of crack pattern, peak load and initial stiffening.

Non-intrusive coupling of a 3-D Generalized Finite Element Method and Abaqus for the multiscale analysis of localized defects and structural features

This paper presents a multiscale computational framework able to resolve localized defects and features such as cracks and welds in large structures. It couples Abaqus models and 3-D Generalized Finite Element (GFEM) discretizations enriched with numerically-defined functions – the GFEM with global-local enrichments (GFEMgl). The structural-scale problem is modeled in Abaqus using a coarse, 3-D mesh, suitable for capturing the global response of the structure. The GFEMgl is used to accurately model localized features of interest that are otherwise ignored by Abaqus models. The GFEMgl utilizes enrichment functions provided by the solution of, potentially multiple, local problems solved in parallel. The coupling between Abaqus and GFEMgl models is handled by the Iterative Global-Local method (IGL). The proposed multiscale framework – denoted by the acronym IGL-GFEMgl in this work – is non-intrusive in the sense that the interactions between Abaqus and the GFEMgl solver require no modifications of Abaqus or knowledge about its implementation of the FEM. Numerical examples of a specimen undergoing localized plasticity, a hat-stiffened panel with a large number of spot welds, and a T-joint structure subjected to mixed-mode fatigue crack propagation are presented to demonstrate the accuracy and applicability of the proposed framework. The results show that the IGL-GFEMgl can achieve nearly the same accuracy as a Direct Finite Element Analysis (DFEA) while leveraging methodologies and algorithms implemented in commercial and research software. Moreover, it is also shown that the user time spent on model preparation can be greatly reduced when dealing with complex problems.