Mixed-mode Crack Propagation Research Articles

Microstructural Controls on Mixed Mode Dynamic Fracture Propagation in Crystalline and Porous Granular Rocks

AbstractBrittle fracture propagation in rocks is a complex process due to significant grain‐scale heterogeneity and evolving stress states under dynamic loading conditions. In this work, we use digital image correlation and linear elastic fracture mechanics to make instantaneous measurements of the opening (mode I) and in plane shear (mode II) components of the stress intensity field during dynamic mixed mode crack initiation and propagation in crystalline and granular rocks. Both rock types display some similar fracture behaviors as observed in engineered materials, including rate dependent fracture initiation toughness and a direct relationship between propagation toughness and crack velocity; however, measured propagation toughness is higher than quasi‐static values at crack velocities well below the branching velocity in both rocks. Additionally, due to grain scale controls on the fracture process, mixed mode crack propagation is fundamentally different between these two rock types. Mixed mode propagation is energetically more favorable than pure opening mode propagation in sandstone, while the opposite is true in granite. Furthermore, following initiation, propagation in granite occurs so as to minimize the mode II contribution, irrespective of the initiation conditions, while fractures in sandstone maintain a non‐negligible mode II contribution during propagation across the sample.

Mixed-mode fatigue crack growth analysis of I-shaped steel girders with trapezoidal corrugated-webs

In bridge industry, the composite box-girder with corrugated steel webs and concrete flanges is quite a common practice over recent years. However, this innovation encounters fatigue-sensitive details not fully investigated, especially in the research area of mixed-mode fatigue crack growth. This study addresses the mixed-mode fatigue crack growth analysis of an I-shaped corrugated web girder associated with the represented web-to-flange detail. A linear elastic fracture mechanics (LEFM) based numerical framework was established to simulate the mixed-mode crack propagation. For comparison, a semi-analytical weight function framework was also introduced to predict the pure Mode-I crack propagation life of the same detail. Experimental results of a full-scale inclined corrugated web girder were employed to evaluate these two proposed frameworks. A parametric study was presented based on the proposed frameworks to investigate the effect of initial crack depth and crack shape on fatigue crack propagation life. By comparing the simulation results with test results, the applicability of the proposed LEFM-based numerical framework to the simulation of mixed-mode fatigue crack propagation behaviour in corrugated web girders was discussed.

Mixed mode crack paths in terms of plastic stress intensity factors based on conventional and strain gradient plasticity

The effect of an initial mode II loading on subsequent mixed-mode crack path was investigated through experiments and computations for 34X and P2M steels, 7050 aluminum, and Ti-6Al-4V alloys in a compact tension shear (CTS) specimen. An optical microscope and the drop potential method were used to determine the actual position of the crack tip on the curvilinear crack path monitoring and continuous measurements of the crack size were made along the path in the CTS specimen. In accordance with these experimental observations, the main feature of the initial mode II fracture is the crack growth on a curvilinear path. For subsequent mixed-mode crack propagation, the crack front continuously changes the shape and direction with each loading cycle. For a mode II loaded initial crack, the scenario becomes more complex because of the change in the propagation direction when a kinked crack is formed. The experimental data on crack growth for all tested materials are represented in terms of the elastic and new plastic stress intensity factors (SIFs). To this end, both the classical and the conventional mechanism-based strain-gradient plasticity (CMSGP) theories were employed. The material constitutive equation is implemented in a finite element code and the elastic and plastic fracture resistance parameters are calculated as a function of the position along the curvilinear crack path in all tested CTS specimens produced from 34X and P2M steels, 7050 aluminum and Ti-6Al-4V alloys. As a result of the finite element calculations, nonlinear amplitude factor solutions are determined across a wide range of material work hardening conditions, and the role of the Taylor’s intrinsic material length and both the coupled effect of these parameters are established. The potential of fundamental and practical applications of introduced fracture resistance parameters are discussed.

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.

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.

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.

Simulating quasi-static crack propagation by coupled peridynamics least square minimization with finite element method

In this work, we present an innovative simulation method for quasi-static crack propagation in mixed-mode, within a new framework of coupled peridynamics least square minimization and finite element method (PDLSM–FEM). As the inertia effect is neglected for quasi-static problems, the governing equations for PDLSM–FEM can be solved implicitly without iterations. Displacements and stresses from PDLSM–FEM are used to compute the interaction integral, an extended version of the J-integral. The stress intensity factors (SIFs) are derived from the interaction integral and used in the maximum circumferential tensile stress failure criterion to predict the onset of crack propagation and the direction of propagation. New contributions in this work include the following: (1) developing a new framework for coupling PDLSM and FEM; (2) pioneering a quasi-static method for simulating crack propagation in peridynamics; (3) developing an element-based approach for selecting the interaction integral contour; and (4) proposing and implementing an innovative method for handling peridynamic bond breakage and crack growth. Quasi-static crack propagation simulation in this work provides a computationally efficient way to find the shape of the crack, and the current PDLSM–FEM model is straightforward and can be easily introduced into commercial finite element codes. Three numerical examples are carried out, and the results show that the proposed method evaluates SIFs accurately and captures crack growth as expected.

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.

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.

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.

Experimental study on mode-I and mixed-mode crack propagation under tangentially incident P waves, S waves and reflected waves in blasts

A dynamic crack subjected to tangentially incident P and S waves, as well as reflected waves, was studied by caustics method in a PMMA plate under two explosions of 20 mg and 60 mg lead azide. Periodic P and S waves are visualized as fringes with different densities, which is verified by calculations of arriving time and different caustics patterns in the crack tip. Distorted and classical caustics patterns occur under P and S waves, respectively, indicating that crack-tip K-dominated stress field is destroyed by P waves, while it is less disrupted by S waves. Hence, modified caustics interpretation previously proposed by authors and classical one are employed to reproduce experimental caustics patterns under P and S waves, respectively, to obtain crack-tip positions, crack velocity, KI and KII. Results show that P-wave compression loading plays a dominant role to arrest the crack, despite temporary crack re-initiation by P-wave tension loading, and that S waves temporarily accelerate the crack with alternate positive and negative KII, indicating periodic shear stress is not a sustainable driving force for crack propagation. Reflected waves cause mixed-mode crack propagation with periodic crack deflections, in which peaks of KI correspond to peaks of crack velocity and flat crack path, however peaks of KII correspond to crack slowdowns and crack deflections. Finally, this paper emphasizes a comprehensive analysis including crack path, cack-tip positions, crack velocity, caustics shape, KI and KII is required for evaluating mixed-mode crack propagation in blasts.

A strain energy-based parameter for predicting the crack initiation angle of mixed mode fracture

In this paper, a new parameter is developed for I–II mixed mode fracture analysis based on the concepts of the volume strain energy and deviator strain energy. For most fracture conditions, it has been demonstrated that the final fracture state is not dominated by the plastic deformation during the fracture process. For instance, mixed mode crack propagation occurs not along the direction of the maximum plastic deformation near the crack-tip. Based on this phenomenon, it is considered that the crack growth behavior is promoted by the volume strain energy, but not the deviator strain energy. Therefore, a new fracture parameter defined by the ratio (H-parameter) of the volume strain energy density to the deviator strain energy density is proposed for the investigation of crack growth behavior. It is assumed that the crack initiation angle can be determined by the direction of the maximum value of the H-parameter near the crack tip. This newly proposed H-parameter method is validated by comparison with classical criteria and experiments with two materials (PMMA and polysilicon) and shows a good accuracy.

Fracture evaluation of ultra-high-performance fiber reinforced concrete (UHPFRC)

The development of numerical simulation for Ultra-high-performance concrete (UHPC) and Ultra-high-performance fiber-reinforced concretes (UHPFRC) is fundamental for the design and construction of related structures. The simplified engineering stress-strain relationship and the input values are necessary in the finite element modeling. Four-linear curves and modified Kent–Park model were proposed to describe the engineering tensile and compressive stress-strain relationship, respectively. An attempt was made to simulate the fracture of UHPC and UHPFRC using concrete damaged plasticity model and element deletion strategies. The predicted tensile and compressive behaviors of UHPC and UHPFRC were successfully validated by the test results in the literature. For a better understanding of the mechanical behavior of UHPC and UHPFRC exposed to biaxial loadings, mixed-mode crack propagation simulation on the double-notched specimens exposed to combined shear-tensile and shear-compressive forces was discussed.

Investigation of Mixed-Mode I/II Fracture under Impact Loading Using Split-Hopkinson Pressure Bar

Mixed-mode fracture of construction building materials under impact loading is quite common in civil engineering. The investigation of mixed-mode crack propagation behavior is an essential work for fundamental research and engineering application. A variable angle single cleavage semi-circle (VASCSC) specimen was proposed with which the dynamic fracture test was conducted by using a Split-Hopkinson pressure bar (SHPB). Notably, the mixed-mode crack propagation velocity could be detected by the synchronized crack velocity measuring system. With experimental results, the dynamic initiation stress intensity factors KI and KII were calculated by the experimental-numerical method. Additionally, the crack path of mixed-mode I/II fracture can be predicated precisely by using numerical method. Thus, a comprehensive approach of investigation on mixed-mode I/II fracture under impact loading was illustrated in this paper. The study demonstrates that the mixed-mode I/II crack would transform from complicated mode I/II to pure mode I during crack propagation, and several velocity decelerations induced crack deflection. The dynamic initiation fracture toughness of mixed-mode crack was determined by the experimental-numerical method. The VASCSC specimen has a great potential in investigating mixed-mode fracture problems with the SHPB device.

A generalized bond-based peridynamic model for quasi-brittle materials enriched with bond tension–rotation–shear coupling effects

A generalized bond-based micropolar peridynamic model is proposed to simulate the nonlinear deformation and mixed-mode crack propagation of quasi-brittle materials under arbitrary dynamic loads. The mechanical behaviors of the material points are reformulated by incorporating the bond tension–rotation–shear coupling effects, which makes the model suitable for complex discontinuous problems in both three-dimensional spatial and two-dimensional plane conditions. The governing equations of the bond are established by employing the Timoshenko beam theory to simulate the interaction between material points as well as the bond coupling effect. Three kinds of peridynamic parameters, corresponding to the compressive, shear and bending stiffness of the bond, are introduced to keep the consistence of the strain energy obtained from the proposed peridynamic model and from the continuum mechanics under arbitrary deformation fields. Moreover, a novel energy-based failure criterion, involving the maximum stretch, shear strain and rotation angle limits of the bond, is proposed to describe the nonlinear behaviors and progressive failure for general quasi-brittle materials. The proposed model is verified by providing comparisons between its results and those from known analytical solutions and experimental observations. The influences of the bond tension–rotation–shear coupling effect as well as the applicability of the proposed model for the wave propagation, complex deformation and mix-mode fracture problems are also investigated. Results show that the proposed model with the bond coupling effect will greatly improve the simulation accuracy for dynamic problems of quasi-brittle materials under complex loading conditions. Results also indicate that the proposed model can well capture the nonlinear deformation, crack propagation, as well as progressive failure of materials with variable Poisson’s ratios under complex combined loads

Effect of polypropylene fibers on the mode I, mode II, and mixed-mode fracture toughness and crack propagation in fiber-reinforced concrete

Concrete is the most commonly used material in civil engineering and also the most conventional and cheap material available in the market. Concrete cracking and fracturing may cause irreparable damages to concrete structures. Hence, fiber-reinforced concretes have been introduced in recent years as a strategy to eliminate these drawbacks to a large extent. In this study, crack propagation and the fracture toughness of concrete specimens without polypropylene (PP) fibers and those containing 0.2, 0.35, and 0.5 vol% of PP fibers were investigated through testing the straight notched Brazilian disc specimens. Furthermore, crack propagation from the pre-existing cracks in the specimens was investigated, and the mode I, mode II, and mixed-mode I/II fracture toughness were also calculated. The Brazilian disc (BD) test was performed on the specimens at different inclination angles of 0, 15, 28.83, 60, 75, and 90° relative to the direction of pre-existing cracks. According to the experimental results, at angles less than 75° (0 < α less than 75), the wing cracks initiated from the tip of pre-existing cracks, and as the loading continues, the path of crack propagation and growth approaches the loading direction. At angles equal to or greater than 75°, the crack initiated with a distance d from the crack tip. This distance was larger in the fiber-free specimens than in the fiber-reinforced specimens. The results showed that the mode I, mode II, and mixed-mode I/II fracture toughness of the specimen containing 0.35 vol% of PP fibers were higher than the fracture toughness of the fiber-free specimens or those containing other PP fiber levels.

A computational study of the mixed–mode crack behavior by molecular dynamics method and the multi – Parameter crack field description of classical fracture mechanics

The overarching objective of the paper is to analyze the mixed-mode crack propagation direction angles by molecular dynamics method and to investigate the validity of continuum-based linear elastic fracture mechanics crack growth criteria. For this purpose, an embedded atom potential (EAM) available in LAMMPS (Large-scale Atomic/Molecular Massively Parallel Simulator) molecular dynamics (MD) software is utilized to accurately pinpoint mixed-mode crack growth. The study is focused on the application of the different approaches for determination of the initial crack propagation angle. Copper and aluminum plates with the central crack under complex mechanical stresses (Mode I and Mode II loading) are studied by extensive MD simulations. Williams’ expansion for the crack tip fields containing the higher-order terms is used. The crack propagation direction angles for combinations of Mode I and Mode II loadings are obtained by 1) multi-parameter fracture mechanics approach based on three fracture mechanics criteria: maximum tangential stress (MTS), maximum tangential strain and strain energy density (SED); 2) atomistic modeling for the mixed-mode loading of the plane medium with the central crack. The temperature effects during fracture processes in MD simulations are considered and temperature field distributions for mixed-mode crack propagation are obtained.

Multi-partner benchmark experiment of fatigue crack growth measurements

The design of reliable structures and the estimation of the residual fatigue life of industrial parts containing flaws or cracks rely on our ability to predict the propagation of fatigue cracks. Whereas in industrial component cracks might have a complex path due to geometry and loading, lab experiments used for identifying crack propagation law are often in pure mode I. The paper presents a synthesis of an experimental benchmark performed in the context of a French national research network. A sample has been designed to produce mixed-mode crack propagation and variation of small scale yielding conditions. Two geometries and two maximum load levels are defined for the two tested materials: a stainless steel and an aluminum alloy. Around ten participants performed experiments using their usual instrumentation. Among the eight possible parameter sets, three are selected for which detailed results are presented. A satisfying overall agreement is obtained. But, some discrepancies are evidenced due either to limitations of the instrumentation or simply because from one lab to the other the applied load is not exactly the same. It is thus concluded that one of the most important issue is boundary conditions, which is confirmed by numerical simulations.