Maximum Energy Release Rate Criterion Research Articles

Formation of transverse cracks in cross ply laminates in an environment of nonuniformly distributed fibers

This work is concerned with extracting characteristic features of transverse crack formation in cross ply laminates under axial tension considering nonuniform distribution of fibers in the ply cross section. The fiber-matrix debond crack initiation sites are determined first by the dilatation induced cavitation criterion for an epoxy matrix. The growth of the debond cracks is analyzed by calculating the energy release rates using the virtual crack closure technique. The kink-out of the debond cracks into the matrix is determined in short incremental steps by the maximum energy release rate criterion. Different scenarios are considered for the linking up of the kinked-out cracks to form continuous transverse cracks. By studying two different degrees of fiber distribution nonuniformity, the interactive effects due to the fiber distribution on the transverse crack formation are clarified.

Experimental and numerical assessment of fracture toughness variations in steam boiler components after a long period of operation

This study presents an experimental and numerical assessment of the fracture toughness of steam boiler components, which have been in operation for more than 100,000 h. The materials tested are components of two different parts of a steam turbine unit: the pipeline section that connects the headers with turbines, which is made of 13HMF steel, and the valve chamber section, which is made of L17HMF steel grades. Full-size arc-shaped tension specimens (AT) are employed for the experimental evaluation. AT and single-edge notch tension (SENT) specimens are used for numerical assessment. The finite element method (FEM) formulations are developed and examined in Abaqus, which is set up as a shell model with the specimen thickness designated as the mid-surface for the analysis. The notch is modeled using a contour integral definition based on the maximum energy release rate criterion. The fracture toughness is quantified using the stress intensity factor (KI), and the results are compared. The valve chamber is found to have a higher fracture toughness than the pipeline section. The effect of specimen geometry is also noticeable, and the fracture toughness of SENT and AT specimens are compared. The results are useful in assessing the component’s durability after long operation.

Wmic-GMTS and Wmic-GMERR criteria for micron-scale crack propagation in red-bed soft rocks under hydraulic action

Micron-scale crack propagation in red-bed soft rocks under hydraulic action is a common cause of engineering disasters due to damage to the hard rock–soft rock–water interface. Previous studies have not provided a theoretical analysis of the length, inclination angle, and propagation angle of micron-scale cracks, nor have they established appropriate criteria to describe the crack propagation process. The propagation mechanism of micron-scale cracks in red-bed soft rocks under hydraulic action is not yet fully understood, which makes it challenging to prevent engineering disasters in these types of rocks. To address this issue, we have used the existing generalized maximum tangential stress (GMTS) and generalized maximum energy release rate (GMERR) criteria as the basis and introduced parameters related to micron-scale crack propagation and water action. The GMTS and GMERR criteria for micron-scale crack propagation in red-bed soft rocks under hydraulic action (abbreviated as the Wmic-GMTS and Wmic-GMERR criteria, respectively) were established to evaluate micron-scale crack propagation in red-bed soft rocks under hydraulic action. The influence of the parameters was also described. The process of micron-scale crack propagation under hydraulic action was monitored using uniaxial compression tests (UCTs) based on digital image correlation (DIC) technology. The study analyzed the length, propagation and inclination angles, and mechanical parameters of micron-scale crack propagation to confirm the reliability of the established criteria. The findings suggest that the Wmic-GMTS and Wmic-GMERR criteria are effective in describing the micron-scale crack propagation in red-bed soft rocks under hydraulic action. This study discusses the mechanism of micron-scale crack propagation and its effect on engineering disasters under hydraulic action. It covers topics such as the internal-external weakening of nano-scale particles, lateral propagation of micron-scale cracks, weakening of the mechanical properties of millimeter-scale soft rocks, and resulting interface damage at the engineering scale. The study provides a theoretical basis for the mechanism of disasters in red-bed soft-rock engineering under hydraulic action.

Predicting the Critical Grout Pressure in Anisotropic Rocks Based on the Maximum Energy Release Rate Criterion

Predicting the Critical Grout Pressure in Anisotropic Rocks Based on the Maximum Energy Release Rate Criterion

An Anisotropic Peridynamic Model for Simulating Crack Propagation in Isotropic and Anisotropic Rocks.

In this work, we present a peridynamic-based simulation method for modeling quasi-static fracture propagation in isotropic and anisotropic rock within the framework of peridynamic least square minimization (PDLSM). The original isotropic elastic PDLSM is further extended to investigate fracture propagation in anisotropic materials in this study. The proposed AN-PDLSM model integrates an anisotropic model in fracture mechanics to analyze the failure process of anisotropic rocks. An important advancement in this research lies in the incorporation of the maximum energy release rate criterion (MERR) into the PDLSM framework for the first time. This enhancement enables accurately determining crack propagation and the associated crack angles. The proposed model utilizes the energy release rate calculated through the J-integral method to assess bond breakage, and it employs a mesh-independent, piecewise linear fracture model to describe crack propagation. The proposed method fully combines the merits of traditional fracture mechanics with the unique capabilities of peridynamics. To demonstrate the effectiveness of the proposed model, a simulation of fracture evolution in isotropic plates subjected to semi-circular bending tests is presented and compared with experimental results. It is shown that the proposed model accurately replicates fracture trajectories in isotropic specimens. In the context of anisotropic rock, the effect of a weak coefficient on crack morphology is discussed in order to obtain a suitable value. Additionally, the impact of bedding angles on fracture paths through our proposed model is also explored, revealing excellent agreement with experimental results. The findings in this research demonstrate that the proposed AN-PDLSM model is exceptionally proficient at capturing the intricate, oscillating crack paths observed in anisotropic rock materials.

Investigation of crack propagation behavior on shales with different inclination based on ultra-fast time resolution method

The layer structures of shale greatly influence its crack propagation behavior. Nevertheless, the correlation between crack extension rates and mechanical parameters has not been well studied, especially, the relationship between the energy release rate and crack propagation rates. Consequently, there are unresolved issues regarding the effects of layer structure on the propagation of cracks in shale. In this paper, a series of three-point compressive loading tests were conducted on seven groups of shale semi-circular bend (SCB) samples with an inclination angle of 0°-90°. The entire fracture process was captured using the ultrafast time-resolution method with a resolution capability of up to 15 picoseconds. In addition, fracture surface information was obtained by a high-precision laser three-dimensional scanning system, and this information was utilized to categorize crack propagation behaviors, such as propagation along weak planes, intermittent penetration, and deflection penetration. Furthermore, a quantitative relationship was established between the energy-release rate and crack propagation rate, based on the first law of thermodynamics and the maximum energy release rate (MERR) criteria proposed by Griffith. The experimental results showed that there is a linear relationship between the energy-release rate and the propagation rate, which meets well with the established relationship. The experimental results indicated that symmetric loading does not result in significant angle localization during crack initiation. This suggests that crack propagation behavior in shale is a reflection of the interplay between energy-release rates and fracture resistance. Finally, this study evaluated the influence of the bedding angle on dissipated energy, and absorbed energy.

Kinking prohibition enhancement of interface crack in artificial periodic structures with local resonators

In this study, the propagation and kinking of an interface crack between two dissimilar artificial periodic structures with local resonators is investigated. By Fourier transform, the dynamic fracture problem is derived as an equation with Wiener–Hopf type. An additional band gap can be created by local resonators. During the dynamic failure, elastic waves will be excited continuously from the crack tip and result in the energy dissipation. Moreover, the energy release ratio which characterizes the fracture resistance is obtained. The meta-arrest property of the artificial periodic structure with local resonators is illustrated. Based on inverse Fourier transform, the displacement field is given to further understand the deformation and oscillation of crack faces. Based on the crack-tip field, the maximum energy release rate criterion is presented to discuss about whether the crack can propagate through kinking out of the interface. Comparing with the model without local resonators, we find that crack kinking can be prohibited in the proposed artificial periodic structures. Finally, finite element simulation and experiment are performed to show the good agreement with the theoretical predictions.

On the mechanistic driving force for short fatigue crack path

The growth of fatigue cracks at the microstructural level is significant for many engineering industries. The multifarious nature of crack propagation at microstructure length scales necessitates the use of advanced modelling techniques to predict fatigue life accurately. A key contributor to this is the path of the crack, which also remains challenging to predict; the maximum slip criterion has been used widely in the literature but is not fully adequate. This paper tests the hypothesis that for short crystallographic cracks, there is some other factor responsible for guiding their extension on a particular plane. With this, using a crystal plasticity finite element modelling framework, two new energy-based methods are developed: a microstructure-sensitive maximum energy release rate criterion, and a maximum normal stored energy density criterion. Results demonstrate, for the two cases studied, both methods offer a significant improvement over the maximum slip criterion when applied to real microstructures. In particular, the maximum normal stored energy method gives closest agreement with experiments, while maximum energy release rate offers new insights into a mechanism for crack bifurcation.

Cracking direction in graphene under mixed mode loading

Crack branching in graphene under complex stresses is investigated by molecular dynamics simulations and boundary-layer models. Using the maximum energy release rate (MERR) criterion, an analogy of Wulff curve is plotted considering the anisotropic fracture toughness of hexagonal graphene, which shows the direction of crack kinking (local bump on the curve) is consistent with that of the weak fracture toughness (along zigzag edge). The angle of crack kinking prefers 0°/30°/60°, rather than the angle predicted by maximum circumferential stress criterion. The T-stress along crack front, obtained by the over-deterministic method, can indicate the possibility of kink by its transformation between positive and negative, especially for zigzag cracks. This study incorporates Wulff-like curve and MERR criterion to provide an accurate prediction of nanocrack paths in graphene.

An extended multiphase hybrid-stress finite element method for modelling interface crack propagation between two dissimilar materials

In this work, an extended multiphase hybrid-stress finite element method (X-MHSFEM) is developed to investigate the stress intensity factor, kinking angle, and propagation path of an interface crack. Further, we have simulated the propagation of the interface crack by using the X-MHSFEM. Several types of element models containing the interface and matrix cracks are introduced in the propagation process of the interface crack. The derivation of the functional of the element containing the two materials, one interface, and all types of cracks, is performed. Stress functions are enriched by the asymptotic singular items that accounts for the oscillatory behavior to describe the stress concentration in the vicinity of the interface crack-tip, and the stress concentration near the matrix crack-tip are also captured by inserting certain singular stress terms of Williams expansion into the physical field of whole model. The stress intensity factors of the tips of the interface and matrix cracks are calculated by the least square method. The maximum and relative maximum circumferential stress criterion are employed as the fracture criteria for predicting the interface crack propagation along the interface or the deflection into layers. Further, the matrix crack propagation angle within the layers is determined by the maximum energy release rate criterion. A remeshing algorithm is employed to implement the propagation of the interface and matrix cracks. The effectiveness and accuracy of the present method are validated by comparing the results, obtained by the new method, with the experimental observation and other numerical results. Therefore, the extended multiphase hybrid-stress finite element method can provide an effective medium to deal with the interface crack fracture problems.

Prediction of Crack Propagation of Nano-Crystalline Coating Material Prepared from (SAM2X5): Experimentally and Numerically

The fracture and crack growth of materials can be practically and conveniently predicted through numerical analysis and linear elastics fracture mechanics. On this basis, the current study aims to present experimental work supported by a numerical technique for mimicking the crack propagation by Version 5.6 of COMSOL Multiphysics (version 5.6), used for the simulation of the coating made from Fe-based amorphous material with a thickness of 300 µm. The paper shows the effects of mixed-mode loading on cohesive zone parameters attained from load-crack mouth opening displacement (CMOD) curves. The microstructure dominates the fracture, which in mode I is altered from all-transgranular cleavage to nearly all-intergranular structure in mode II. Two common criteria for failure are linked to the mixed-mode results: Maximum energy release rate criterion (Maximum G) and maximum tensile stress criterion (Maximum S). However, distinguishing between the two criteria is made impossible by the large scatter in the data. The stress intensity factor is the basis for the. The stress intensity factor is the leading parameter facilitated by the singular element and should be estimated with accuracy. With the aim of comparing each criterion and illustrating the numerical schemes’ robustness, a number of examples are presented. It can be concluded that the Maximum G and Maximum S were successful and accurate in predicting the propagation of the Fe-based amorphous material prepared on mild steel.

Study on in-plane double-crack propagation in the rib-to-deck welded toe of steel-bridge

The fatigue problem of two in-plane cracks in rib-to-deck welded toe of orthotropic steel bridge decks is an important issue. Traditionally, related work focuses on single-crack propagation in these joints; however, many practical observations have shown that the initial weld cracks of structures are characterized by multiple cracks in a small range. Thus, the main goal of this paper is to investigate the mechanism of and guidelines for in-plane double-crack propagation in rib-to-deck welded toe of orthotropic steel bridge decks. In this paper, the rib-to-deck welded toes in the orthotropic steel deck of the Anqing Yangtze River Bridge are taken as the background. Based on extended finite element method theory, single-crack propagation theory and the multiple-crack coalescence law, the stress intensity factors at the crack tip are calculated by the interaction integral method, and the propagation directions are determined using the maximum energy release rate (Merr) criterion. The whole process of in-plane double-crack propagation, coalescence and failure in rib-to-deck welded toe of long-span steel bridge decks and their fatigue life are simulated according to the principle of an equivalent crack front. Meanwhile, the most unfavorable conditions of cracking parameters are determined by changing the spacing and shape ratio of the two cracks. Then, the fatigue loading experiment is carried out on equal proportion specimens under appropriate cracking parameters. The results reveal the whole process of in-plane double-crack propagation in rib-to-deck welded toe of orthotropic steel bridge decks, and the verification shows that experimental results agree well with the simulation results.

Virtual element method for mixed-mode cohesive fracture simulation with element split and domain integral

We present a computational framework for mixed-mode cohesive fracture simulation based on the virtual element method (VEM). To represent an arbitrary crack path, the element splitting scheme is developed on a polygonal mesh to capitalize its flexibility in element shape. For the accurate evaluation of a crack-tip stress field and crack propagation direction, the virtual grid-based stress recovery scheme is tailored for VEM in conjunction with the maximum strain energy release rate criterion. The mixed-mode fracture examples are illustrated to validate the accuracy and robustness of the proposed computational scheme. Numerical results demonstrate that the domain integral method with the stress recovery scheme captures an accurate crack path without oscillation under the biaxial tensile stress state. Furthermore, the computed cracks using the element splitting scheme show that smooth and curved patterns on polygonal elements are in good agreement with the experimental results.

Mixed-mode dynamic fracture behavior of a rubber toughened epoxy adhesive under stress wave loading

This work seeks to measure mixed-mode fracture parameters for a dynamically growing crack in a core shell rubber toughened epoxy adhesive (EPONTM 828 with PARALOID™ EXL-2691J particles). Digital image correlation technique coupled with ultrahigh-speed photography and a hybrid finite element-based post-processing approach are used to extract fracture parameters. The dynamic loading is accomplished by using a striker to induce stress waves in an incident bar in contact with a free-standing semi-circular specimen. A companion finite element analysis is performed for investigating suitable experimental parameters and configurations for implementing the approach and determining specific experimental parameters (crack length, crack angle, and impact velocity) necessary to achieve the desired experimental conditions (mode mixity at crack initiation and strain rate). The critical fracture parameters are used to generate mixed mode fracture envelope and fracture toughness-mode mixity relationship for the material. Further investigation is carried out to evaluate various fracture criteria for predicting initial crack kink angles. The measured dynamic fracture quantities are also compared with the quasi-static counterparts to assess strain rate-dependence of the adhesive. The results reveal a decrease in critical stress intensity factors of approximately 10 % and 30 % for the mode I and mode II states, respectively, with increasing strain rate. Post-initiation, the dynamic crack propagates at a near-constant mode I stress intensity factor of ∼1.4 MPa-√m. Under dynamic conditions, the critical, effective stress intensity factors are relatively constant with respect to mode mixity. This is in contrast to quasi-static results where the effective values increased with respect to mode mixity. Experimentally measured crack kink angles agree well with those predicted by the maximum energy release rate criteria and the maximum tangential stress criteria, particularly under mode I dominant conditions.

Experimentally measuring weak fracture toughness anisotropy in graphene

The two-dimensional nature of graphene offers a number of interesting mechanical properties. Amongst these, fracture toughness has received substantial interest, yet computational works have not reached a consensus regarding anisotropy in its fracture energy when graphene is loaded in armchair or zigzag directions. Here, we resolve the steps involved during fracture of graphene by carrying out in situ tensile tests. Embryo cracks nucleated from the graphene edges are observed to deflect into major cracks with local kinking features, as explained by an evolving stress intensity factor during crack advance. Extended finite element analysis with the maximum energy release rate criterion is used to model the fracture process. We determine a weak degree of anisotropy in the fracture toughness, Gc(armchair)/Gc(zigzag), of 0.94, which aligns with previous predictions from first-principles calculations and observed growth kinetics of graphene crystals in experiments.

How to obtain a more accurate maximum energy release rate for mixed mode fracture

Energy release rate (ERR) is considered one of the most widely employed fracture criteria. However, for mixed mode fracture situations, the exact computation of ERR for an infinitesimal branch crack is challenging. With some assumptions, Nuismer's theoretical expression (Int. J. Fracture. 11 (1975)) has been used to determine the ERR. In this work, we investigate its deviation. Based on St. Venant's theorem, it is first proved that the stress intensity factor (SIF) of a branch crack is almost independent of its length when the branch crack is small enough as compared to the main crack, which enables us to more accurately obtain the ERR and SIF of an infinitesimal branch crack by simulating a finitely short branch crack in finite element simulations. More accurate curves for the maximum ERR and the corresponding branch angle under any mixed mode are provided. From results of more accurate simulation, we find that Nuismer's model has the maximum difference of 11.2% in the maximum ERR and the 7.1% difference in the branch angle prediction. These are also the differences of predictions between the maximum circumferential stress criterion and the maximum energy release rate criterion.

On Reliable Prediction of Fracture Path in Anisotropic Rocks

We evaluate the accuracy of two well-known fracture growth theories in predicting crack growth path in anisotropic rocks, through comparison with new experimental data. In doing so, the results of fracture toughness tests on metamorphic Grimsel Granite under four different ratios of mixed-mode I/II loadings are reported, and the experimental kink angle and the effective fracture toughness are compared with the predictions of the maximum tangential stress (MTS), and the maximum energy release rate (MERR) criteria. Comparing theoretical predictions based on the classical forms of the two criteria with the experimental data reveals that the energy-based criterion is unable to accurately describe the fracture growth behavior. We suggest a reformulation of the MERR criterion and show that the modified form give significantly better predictions of fracture growth trajectories. The evaluation of these criteria also illustrates that the MTS and modified MERR criteria cannot provide accurate predictions if T-stress is absent.

Research on the anisotropic fracture behavior and the corresponding fracture surface roughness of shale

Understanding the anisotropic fracture mechanism of shale is of significance to assess the development of a stimulated reservoir volume. In this study, three-point bending tests were carried out using notched deep beam specimens of shale with 7 different bedding dip angles to investigate the anisotropic fracture behavior. The anisotropic characteristics of the fracture toughness, energy release rate, inelastic zone and crack propagation path were comprehensively analyzed and highlighted. The results showed that the fracture toughness decreases with increasing bedding dip angle. When the bedding dip angle is 0° and 90°, the fracture toughness reaches the maximum and minimum values, respectively, with a ratio of 1.60. The variation tendency of the energy release rate with the bedding inclination angle is consistent with that of the fracture toughness, but the ratio of the maximum value to the minimum value of the energy release rate is about 3.24. The size of the inelastic zone first decreases and then increases at different bedding inclination angles from 0° to 90° and reaches the maximum size of 1.51 mm when the bedding plane inclination angle is 45°. The crack propagates along the bedding plane rather than straight along the initial crack direction when the bedding dip angle is higher than 45°. As the bedding plane dip angle increases from 0° to 45°, the crack surface roughness of the tested shale first increases, then decreases and finally reaches the maximum. The discussion indicated that the presence of the Mode II fracture has a considerable influence on the crack surface roughness and crack propagation path, although it has little effect on the failure load. Compared with the maximum energy release rate (MERR) criterion, the maximum tangential stress (MTS) criterion seems more accurate in predicting the anisotropic Mode I fracture toughness. The applicability of these two classical crack initiation criteria of fracture mechanics has been analyzed based on our experimental results, which demonstrates the validity of the MTS criterion to assess the anisotropic fracture toughness of shales.

Effect of the distribution of lip-shaped matrix cracks in continuous fibre reinforced ceramic matrix composites on failure progress under transverse tension

ABSTRACT This paper develops a numerical model that contained random lip-shaped matrix cracks based on the maximum energy release rate criterion for the prediction of failure progress of Continuous Fibre-reinforced Ceramic Matrix Composites (CFCC) under transverse tension. Fibre distribution layout is created based on the actual microstructure and the Random Sequential Adsorption algorithm is used to generate the distribution of lip-shaped matrix cracks. The influence of the lip-shaped matrix crack distribution and crack density on the failure progress and stress–strain response was investigated. It was found that the onset of damage initiation commenced from the lip-shaped matrix crack when the interface fracture energy was smaller than the critical interface fracture energy. The failure progress and the predicted results were in good agreement with the experimental results. The different matrix cracks distribution affected the failure progress. Finally, the tensile strength and the elastic modulus both decreased with the crack density increasing.

An Experimental and Numerical Investigation on the Initiation and Interaction of Double Cracks in Rocks under Hydromechanical Coupling

The growth of double cracks is the main factor leading to progressive rock failure under hydromechanical coupling. The initiation modes and interaction behaviors of double cracks were investigated by using laboratory tests, and the influences of water pressure were analyzed. The maximum energy release rate criterion was modified to determine the crack growth characteristics. A numerical model was established and then verified by the test results. Based on the simulation, the distribution of stress fields and key fracture parameters of double cracks was investigated. Then, initiation characteristics and interaction behaviors of parallel and nonparallel cracks were quantitatively analyzed. The results indicate that the increase in water pressure leads to the crack initiation being inclined to the original surfaces and the growth length along the crack fronts tending to be uniform; the small tensile stress zones are formed close to the crack tips, and significant compressive stress zones are formed at both sides of the crack surfaces; stress superposition and interaction occur when crack spacing is less than 2.5a; the interactive weakening effect is mainly present in the inner side (rock bridge zone) of cracks, while a certain degree of interactive enhancement effect exhibits in the outer sides; the cracks are much easier to initiate at the outer wing cracks when the spacing is less than the critical length (0.5a); and cracks with a dip angle of 45° are much easier to initiate at the endpoints of long axis. The research results provide certain theoretical guidance for the safety assessment of underground engineering.