Linear Elastic Fracture Mechanics Research Articles

Applicability investigation on failure assessment diagram of ASME code section XI - Appendix H for ferritic nuclear piping

The failure assessment diagram (FAD) is a widely used tool for evaluating the fracture acceptance of structural components. In addition to linear elastic fracture mechanics, elastic-plastic fracture mechanics, and limit load analysis, the ASME Code incorporated FAD as an alternative safety evaluation method for flaws in nuclear pipes starting from 2004 edition. This paper conducts a series of finite element elastic-plastic fracture mechanics analyses to establish the specific –based FAD curves considering various loading and geometry conditions for ferritic pipes containing various circumferential flaws. By comparing with the suggested FAD of ASME Code Section XI - Appendix H, it is found that the radius-to-thickness ratio of pipes and the flaw depth influence the applicability of FAD, but with minor effects of crack length. In addition, the effects of material properties between base metal and weld materials on FAD are also discussed. The result of this work could be a reference when applying the FAD to evaluate the flawed piping for nuclear power plants.

Micromechanical properties characterization of 4H–SiC single crystal by indentation and scratch methods

Growth and micromachining of single crystal silicon carbide (SiC) have drawn extensive attention in wide bandgap semiconductors and remained challenging. The micromechanical properties (i.e., hardness, elastic modulus, fracture toughness), which are the key to machinability and machining quality (and therefore the yield of the grown SiC), of 4H–SiC were assessed by nanoindentation, microhardness, and microscratch tests. Consistent values of true indentation hardness of 4H–SiC can be obtained by different approaches without the need for the area function of the indenter. Accurate determination of elastic modulus (i.e., 583.5 GPa) and indentation hardness (i.e., 38 GPa) of 4H–SiC by the instrumented indentation technique requires low loads with slight indentation-induced damage. Consistent values of fracture toughness Kc of 4H–SiC were obtained by different methods such as the Vickers/Berkovich indenter-induced cracking method (2.1 MPa·m1/2), energy-based nanoindentation approaches (the critical void volume fraction of 0.062 should be used for 4H–SiC when the deterioration of indentation hardness is used), and linear elastic fracture mechanics (LEFM)-based scratch approach with a spherical indenter (2.6 MPa·m1/2). This work provides the guidance for the mechanical machining of 4H–SiC wafers and the paradigm of micromechanical characterization of brittle solids by indentation and scratch technologies.

How Stress Barriers and Fracture Toughness Heterogeneities Arrest Buoyant Hydraulic Fractures

AbstractIn our study, we investigated the impact of changes in Mode I fracture toughness and stress barriers on fully developed planar, buoyant hydraulic fractures assuming linear elastic hydraulic fracture mechanics. We present scaling-based arguments to predict the interaction type and use numerical simulations to validate our findings. Through a two-dimensional simplification, we estimate the lower limit for the fracture to feel a change in fracture toughness (so-called immediate breakthrough). Our simulations show that this approach only captures the order of magnitude of the toughness jump necessary for immediate breakthrough compared to the actual value due to three-dimensional solid effects, emphasizing their importance in such systems. We show that we can estimate the occurrence of indefinite containment at depth by considering that lateral spreading occurs at an approximately constant height. However, timing predictions in the case of a transient containment suffer from our simplified approach, which cannot model the injection history of the spreading constant height fracture. The same observations regarding immediate breakthrough and indefinite containment hold when considering stress barriers using pressure-scale-based arguments. Our study shows that the required toughness changes for fracture arrest are more significant than the observed values in the field. In contrast, stress barriers with a magnitude of around 1 MPa are generally sufficient to contain buoyant hydraulic fractures indefinitely. Stress barriers, in combination with other arrest mechanisms, are thus the most prominent mitigation factor of buoyant growth in industrially created hydraulic fractures.

Tensile-Tear cracking resistance of RAP incorporated asphalt Mixtures, enhanced by Warm mix asphalt additive and nano calcium carbonate

Asphalt mixture is a commonly used material for road paving. However, its production is expensive and requires significant amounts of virgin materials. To address this issue, adopting a less harmful production process and incorporating recycled materials can be beneficial. In this study, the focus was on investigating the cracking resistance parameters of warm mix asphalt (WMA) mixtures prepared with high contents of reclaimed asphalt pavement (RAP), along with the use of nano calcium carbonate (NC) as a modifier. In this context, the cracking resistance of the mixtures was studied in two −12 and + 25 °C temperatures under pure mode I, two mixed modes I/III, and pure mode III loading. The results revealed that adding RAP increased the fracture toughness and fracture energy, with an average increase of 30 and 20 %, respectively. Incorporating NC also improved the mixture’s resistance to cracking, enhancing fracture energy by up to 30 % and fracture toughness by up to 20 %. The optimal content of NC was determined to be in the range of 2 to 4 %, depending on the variables considered. Considering the linear elastic fracture mechanics, several theoretical criteria were used to predict the fracture data, and accurate predictions were achieved. Statistical analyses were also conducted to evaluate the impact of variables on cracking resistance; as an example of outcomes, it is seen fracture energy of pure mode III is about twice the mode mode I.

Measurement of the interfacial energy release rate (ERR) and cohesive law parameters of polyvinyl butyral laminated glass using a novel beam-type test specimen

The polyvinyl butyral (PVB) laminated glass (LG) structures are widely used in architectural and automotive sectors. Determination of the interfacial fracture energy (i.e. G) between the PVB layer and glass is one of the most vital procedures for ensuring the structural safety and fracture resistance of these kinds of components. The present study investigates the adhesion characteristics between glass and PVB interface in a LG composite structure. A new simple test fixture is introduced to measure the mixed-mode fracture energy from pure mode I (i.e. GI) to pure mode II (i.e. GII). By making use of the concept of Equivalent material Concept (EMC), linear elastic fracture mechanics (LEFM) formulations together with finite element (FE) analysis can be used to compute the compliance parameter of the system and pure modes I/II fracture energies. Besides, the interaction of cohesive forces between the PVB layer and glass are simulated by using a FE method based on a bi-linear cohesive zone model (CZM) technique. In this regard, the cohesive law parameters are computed for mode I and mode II in an iterative process. Then, the obtained pure mode cohesive law parameters are utilized to predict the mixed-mode load–displacement diagrams. The obtained results demonstrate the satisfactory accuracy of the cohesive law parameters and fracture energies in predicting the mixed-mode loading results across four different loading rates (i.e. LR = 2, 5, 10, and 15 mm/min).

Mechanics of abrasion-induced particulate matter emission

Microplastic pollution constitutes a substantially detrimental type of environmental contamination and poses threats to human health. Among the sources of airborne and marine microplastics, evidence indicates that non-exhaust emissions resulting from tire abrasion and other organic materials have emerged as a notable contributor. However, the mechanistic understanding of abrasion emission of organic materials has remained elusive. To fill the gap, we here develop a multi-scale abrasion mechanics model using the principles of linear elastic fracture mechanics. Macroscopically, material wear and tear can be viewed as a process of macro-crack propagation associated with the fatigue fracture. Microscopically, we consider the effect of microcracks propagating under cyclic loading on the material modulus and energy release rate during fatigue fracture. This framework leads to an evaluation of the effective energy release rate for the abrasion-induced emission of particulate matter, thus leading to a calculation of the concentration of the emitted particulate matter with varied sizes. The theory is validated by corresponding experiments and high consistency is exhibited between the theoretical and experimental results. This research constructs a quantitative relationship between fracture mechanics and abrasion emissions. This research not only paves the way for a mechanistic understanding of particulate matter pollution from a solid mechanics perspective but also offers rational guidance for modern society to alleviate airborne particulate matter and marine microplastic abrasion emissions.

Estimating the remaining service life of in-situ grouted post-tension anchors using flaw tolerance limit plots

Brittle failure of grouted post-tensioned anchors used in trunnion girders at dams has given rise to concerns and questions surrounding their continued use and remaining service life. This paper reviews the brittle failure of five grouted post-tensioned anchor rods due to stress corrosion cracking at dams in California. To provide an early warning system to dam owners and expand on available resources for practicing engineers, this study derives plain strain fracture toughness from the failed grouted post-tension anchors using cross sectional photos. Using derived and available fracture toughness values, flaw tolerance limit plots, which can quickly illuminate the severity of a defect for various anchor diameters, were developed for semicircular and straight fronted flaw types using linear elastic fracture mechanics and stress intensity factors for both ASTM A322 and ASTM A722 materials. To help demonstrate the use of flaw tolerance limit plots and estimate the remaining service life of an in-situ anchor, an example problem was developed.

Optimising fatigue crack growth predictions for small cracks under variable amplitude loading

The fatigue cracks in a fighter aircraft that pose the greatest threat to structural integrity and availability are usually less than 1 mm for most of their lives. However, it has long been recognised that small cracks can grow significantly faster than long cracks at the same stress intensity range (ΔK). This means predictions for small cracks growing under realistic spectrum loading can be significantly non-conservative when using linear elastic fracture mechanics and long crack-based empirical rate data. Previous work showed empirical rate data based on small cracks grown under constant amplitude (CA) loading could significantly improve predictions. However, such data still have limitations related to an inability to predict the spectrum history effects that influence small cracks. This paper presents a method to optimise a model to predict fatigue crack growth rates for cracks loaded with realistic spectra. This method is demonstrated for small cracks in Aluminium Alloy 7050-T7451 grown under fighter wing root spectrum loading. Numerical optimisation is used to select model parameters that minimise fatigue crack growth rate prediction errors for a training dataset. This purpose-built model consistently outperformed a generic model based on the growth of small cracks under CA loading.

Universal slope-based J-integral methods for characterization of the mode I, mode II and mixed mode I/II fracture behaviour of adhesively bonded interfaces

Universal slope-based J-integral methods have been developed for the determination of the energy release rate for adhesively bonded joints under mode I, mode II and mixed-mode (I/II) loading conditions. The individual J components corresponding to the mode I and mode II loading were separated based on the J-integral decomposition theory. The proposed methods use the slopes of the substrates at various locations to characterize the energy release rate and thus avoid the measurement of crack lengths, which are especially suitable for characterizing the tough interfaces associated with large fracture process zones ahead of crack tips. Under linear elastic deformation, the slope-based J equations were found to be equivalent to classical G equations based on linear elastic fracture mechanics (LEFM). Both experimental and numerical testing of adhesively bonded joints were undertaken to validate the slope-based J equations. The universal slope-based J-integral methods provide a reliable alternative to the measurement of G for adhesive joints or laminated composites undergoing nonlinear or inelastic deformations where conventional LEFM is not valid. It is shown that LEFM, even when coupled with an effective crack length approach, can be inaccurate when damage occurs in a test specimen away from the fracture process zone, as was seen here in mode II. Slope-based J equations can avoid these inaccuracies with a careful selection of contour paths. Slope based methods are therefore strong candidates for selection in future test standards for mode II fracture characterisation of structural adhesive joints.

Mode I fracture propagation and post-peak behavior of sandstone: Insight from AE and DIC observation

Microcracks generated at the crack tip, particularly in quasi-brittle materials such as sandstone, give rise to a non-linear region known as the fracture process zone (FPZ), which cannot be analyzed using linear elastic fracture mechanics (LEFM). To gain insights into the nonlinear fracturing behavior of rock, this study conducted three-point bending experiments on sandstone beams with prefabricated notch. Stable mode-Ⅰ fracture propagation during the post-peak stage was achieved by the monotonic increment of crack mouth opening displacement (CMOD) measured by a clip gauge. The acoustic emission (AE) and digital image correlation (DIC) techniques were utilized to visually track the detailed crack propagation process from both micro and macro perspectives. Nonlinear fracture parameters, including FPZ length, were identified by AE and DIC measurements and compared with the theoretical calculation. A nonlinear fracture criterion relating applied load, equivalent crack length, and stress intensity factor were established, from which the theoretical CMOD-P curve at the post-peak stage were obtained and validated with experimental measurement from MTS machine. The results reveal that microcracks formulating the FPZ materialize in the pre-peak stage, with the majority emerging post-peak to facilitate the propagation of the traction-free crack and FPZ in the loading direction, as evidenced by AE locations and DIC displacement gradient. Throughout the crack propagation process, over 65 % of microcracks are estimated to be tensile mode based on both AF/RA and polarity analysis, aligning with anticipated outcomes from macroscopic three-point bending fracture tests. Energy dissipation primarily occurs within the FPZ to facilitate separation of crack faces, with the dissipated energy region advancing towards the beam boundary. By integrating AE source locations and DIC displacement gradient, the FPZ length of the 3-point-bending sandstone specimens is determined to be lp = 8.1 mm, slightly exceeding the theoretical FPZ lengths calculated by the Irwin model (lp = 7.0 mm) and the Dugdale-Barenblatt model (lp = 7.6 mm). Using a non-linear fracture criterion that incorporates the FPZ and employing the in-time equivalent crack length measured by AE and DIC as an intermediate variable, the theoretical applied load-crack length curve during the post-peak stage is calculated, showing close agreement with load cell measurements with a negligible error of approximately 2 %. Furthermore, the theoretical post-peak CMOD vs. load curve aligns with experimentally obtained results, demonstrating a non-linear crack propagation speed at a constant CMOD rate, wherein crack propagation speed decreases as crack length increases.

The demarcation problem in the applied sciences

The demarcation problem in the applied sciences refers to the challenge of discerning between progressive, stagnant, and improper model development programs. We examine the demarcation problem from the perspectives of two influential philosophers of science, Imre Lakatos and Thomas Kuhn, using the evolutionary development of linear elastic fracture mechanics (LEFM) for illustration. We propose objective criteria.

Fracture toughness testing of metallic materials based on scratch tests

As a crucial parameter for determining a material's resistance to crack propagation, the convenient characterization of fracture toughness has attracted much attention. The scratch test is an alternate method for determining fracture toughness because it does not require notches or pre-existing cracks and only requires a minimal sample volume. In this study, the product of strength and elongation of materials is incorporated into a modified model based on the energy size effect law to calculate the fracture toughness of metallic materials using scratch tests. To validate the proposed model, scratch tests of DP800, 18CrNiMo7-6, 4140, and 4340 are performed to characterize their fracture toughness. The fracture toughness of these four metallic materials determined from scratch tests utilizing the proposed model, Akono's linear elastic fracture mechanics (LEFM) model, Akono's microscopic energetic size effect law (MESEL) model, Hubler's MESEL model, and Liu's MESEL model are compared with the results from a compact tensile test or single edge notched beam test. Our results indicate that our modified approach is applicable to metallic materials. The proposed model could be reduced to the linear elastic fracture mechanics model. This study offers an efficient way of characterizing the fracture toughness of materials using the scratch method.

Structural design against brittle fracture: Optimizing energy release rate and experiment

Despite substantial advancements in optimizing structural designs for stiffness, the field of design against fracture is still in its early stages. This paper introduces a fundamental and standardized paradigm for structural design against brittle fracture, achieved through the minimization of the energy release rate G within the framework of linear elastic fracture mechanics (LEFM). By leveraging the Griffith criterion expression and the total potential energy in LEFM, we propose an efficient approach for structural optimization that enhances the toughness of architected materials. The energy release rate G is evaluated a synergistic application of finite element analysis and finite difference methods, yielding a precision of 0.6% compared to benchmarks in published handbooks. We demonstrate significantly reduced G values compared to standard stiffness and stress-oriented topology optimization, which directly contributes to increased toughness. To comprehensively analyze the fracture process encompassing damage, crack initiation, propagation, and failure, we employ post-processing phase field fracture modeling on the optimally designed architected materials. By performing multi-objective optimization of the stiffness and energy release rate G, the toughness and extreme load-bearing capacity can be simultaneously improved. The resulting perforated architecture effectively outperforms homogeneous structures in both toughness and peak load. Additionally, we extend our analysis to various configurations, such as non-center cracks, pure shear loading, and adherence to the standard 3D compact tension ASTM E-399-72. The designed structures are additively manufactured and experimentally validated, demonstrating a toughness increase of over 10 times compared to stress-based design. These contributions offer promising prospects for advancing the design of future engineering structures and materials with a favorable balance of critical fracture resistance characteristics.

Data-Driven Dynamic Inversion Method for Complex Fractures in Unconventional Reservoirs

Abstract Hydraulic fracturing is a crucial technology for enhancing the recovery of oil and gas from unconventional reservoirs. Accurately describing fracture morphology is essential for accurately predicting production dynamics. This article proposes a new fracture inversion model based on dynamic data-driven methods, which is different from the conventional linear elastic fracture mechanics model. This method eliminates the need to consider complex mechanical mechanisms, resulting in faster simulation speeds. In the model, the fracture morphology is constrained by combining microseismic data and fracturing construction data, and the fracture tip propagation domain is introduced to characterize the multi-directionality of fracture propagation. The simulated fracture exhibits a multi-branch fracture network morphology, aligning more closely with geological understanding. In addition, the influence of microseismic signal intensity on the direction of fracture propagation is considered in this study. The general stochastic approximation (GSA) algorithm is employed to optimize the direction of fracture propagation. The proposed method is applied to both the single-stage fracturing model and the whole well fracturing model. The research findings indicate that in the single-stage fracturing model, the inverted fracture morphology aligns closely with the microseismic data, with a fitting rate of the fracturing construction curve exceeding 95%, and a microseismic data fitting rate exceeding 93%. In the whole well fracturing model, a total of 18 sections were inverted. The fitting rate between the overall fracture morphology and the microseismic data reached 90%. The simulation only took 5 minutes, demonstrating high computational efficiency and meeting the needs of large-scale engineering fracture simulation. This method can effectively support geological modeling and production dynamic prediction.

Investigation on the size effect of compacted clay mode I fracture based on NDB specimens

The size effect is closely related to the cracking resistance of compacted clay, which is crucial in engineering applications, but little attention has been given to it in previous studies. This study investigated the size effect on mode I fracture behaviors of compacted clay using single-edge notched deep beam (NDB) specimens. Through the digital image correlation (DIC) method, the crack initiation mechanism of compacted clay was analyzed. The results showed that the specimen size and dimensionless crack length did not change the failure pattern, while peak load, peak displacement, and fracture energy gradually increased with the increase of specimen size. Based on the load–displacement curves and the horizontal strain evolution laws, the three horizontal strain stages at the crack tip crack initiation point were defined, i.e., the steady strain growth stage, the nonlinear strain growth stage, and the nonlinear strain intensification stage. Through the Bažant size effect model (SEM), the fracture process zone length Cf and mode I fracture toughness KIC of compacted clay without the size effect were inferred. The separated form of SEM was applied, and the contribution ratios of the strength criterion and linear elastic fracture mechanics were quantitatively analyzed. Combing the characteristics of the NDB specimens, a prediction model considering the support spacing, specimen structure parameters, and fracture toughness was established.

Methods and models for fracture mode partitioning: A review

In the problem of a crack lying in the interface between two dissimilar materials, the mathematical expressions for the stress and displacement fields, derived from linear elastic fracture mechanics, feature oscillatory singularities near the crack tip. This makes it impossible to define the mode mixity mathematically. To overcome this issue, several alternative methods for mode partitioning and analytical, semi-analytical, empirical, or combined models performing mode partitioning in linear elastic, mixed-mode fracture problems have been proposed over the last 35 years. This paper attempts the first review of the topic. It begins by recalling the fundamental scientific problem through reference on the fracture mechanics of bimaterial interfaces. Next, it collects over 36 models performing mode partitioning, proposes classifying them into five broad categories (models based on the global method, models based on the local method, models employing crack-tip force measures, models employing a cohesive interface, and special models), and describes and critically reviews them. A comprehensive discussion is then provided, and the paper closes with justification of concluding remarks and prospective directions for future research.

Complexity of crack front geometry enhances toughness of brittle solids

AbstractBrittle solids typically fail by growth and propagation of a crack from a surface flaw. This process is modelled using linear elastic fracture mechanics, which parameterizes the toughness of a material by the critical stress intensity factor, or the prefactor of the singular stress field. This widely used theory applies for cracks that are planar, but cracks typically are not planar, and instead are geometrically complex, violating core tenets of linear elastic fracture mechanics. Here we characterize the crack tip kinematics of complex crack fronts in three dimensions using optical microscopy of several transparent, brittle materials, including hydrogels of four different chemistries and an elastomer. We find that the critical strain energy required to drive the crack is directly proportional to the geodesic length of the crack, which makes the sample effectively tougher. The connection between crack front geometry and toughness has repercussions for the theoretical modelling of three-dimensional cracks, from engineering testing of materials to ab-initio development of novel materials, and highlights an important gap in the current theory for three-dimensional cracks.

Study of the mechanical behavior and multiscale simulation of the crack propagation in a bilinga wooden beam

The mechanical behavior of local wood species (Bilinga) in the south west region in Cameroon during rainy and dry seasons and the mechanical behavior of wooden beam under bend loading are studied. The three points flexural tests were used to determine the mechanical properties of the wood under study. ANSYS 2020 R1 finite element (FE) software is used for numerical simulations at a macroscopic level using one of the newer technologies called Smart crack growth, which was introduced in the 2019 version. The geometry was modeled in SolidWorks with an initial crack length of 4 and 8 mm introduced in each sample and then imported to ANSYS workbench for further analysis with ANSYS which has all the tools to perform linear fracture. The stress intensity factor (SIF) determines the fracture toughness of a material which is subjected to linear-elastic fracture mechanics (LEFM) where a variable of the critical stress intensify is denoted as KIc. The fatigue crack growth was modeled using Paris’ law. The crack growth was simulated based on Mode I crack specimen with an initial crack length of 4 and 8 mm, respectively. The stochastic multiscale modeling of crack growth on meso- and microscale is used to compare the crack growth rate in the approach of a heterogeneous material and taking into account the microstructure and fracture mechanism of the Bilinga wood. The results of stochastic modeling of the crack growth in the array of cracks and pores of a characteristic size shows that the simulation is close to FE-modeling results. Therefore, the stochastic simulation of the crack growth in wood at meso- and microscale shows the lower local stress intensity factors and slower crack growth due to the existence of the scale-time hierarchy. The crack growth rate vcr at a macroscale ranges within 0.845 – 0.9 × 10–3 m/sec which corresponds to the macroscopic value of the fracture toughness KIc.

Cavity growth – or lack thereof – in imperfectly roll bonded sheet metal

Roll bonding is a well-established solid-state joining method that has also found commercial use. In current industrial practice, a near-perfectly bonded interface is commonly aimed for, achieved by optimizing surface preparation and rolling parameters. On the other hand, an intriguing recent study has shown that roll bonding scrap metal has the potential to provide significant environmental benefits while not jeopardizing performance, despite the resulting imperfect interface with a distribution of cavities. This present work focuses on the mechanical aspects of this observation, aiming to provide a deeper understanding of the influence (or lack thereof) of the cavity content on the mechanical behavior of a roll bonded stainless steel, and a roll bonded mild steel. Imperfectly bonded samples of both steels are studied using mechanical tests and microstructure characterization, which both revealed that the interface cavities exhibit limited growth and no crack nucleation or propagation into the interface. A numerical finite element model of the imperfectly bonded sheet confirmed a tendency for low cavity growth by showing that opening stresses on the interface cavity remain low for various geometries and loading conditions, in line with linear elastic fracture mechanics calculations. Overall, this experimental-numerical study provides insights regarding the unexpected formability (and thus the wide applicability) of imperfectly roll bonded sheets.

Configurational force method enables fracture assessment in soft materials

Configurational mechanics offers a framework for quantifying the tendency of defects to alter the material configuration. When applied to fracture mechanics, configurational forces can be used to quantify the propensity of cracks to propagate. An alternative, well-established approach involves analytical solutions for crack tip displacement fields. However, these solutions typically apply to a limited range of constitutive behaviors and oftentimes to the linear small strain regime. The ease of calculating configurational forces in a numerical Finite Element implementation, along with their applicability to soft fracture at large strains, motivates the study of their performance as a standalone fracture framework. In contrast to the majority of works that remain theoretical and numerical, our study includes a robust experimental approach to configurational forces at finite strains. We report tensile experiments on a soft elastomer with pre-cuts ante fracture initiation. In a first attempt to approach the J-integral via configurational forces, we explore the performance of the linear elastic fracture mechanics solutions on Pacman-shaped domains that reproduce the crack tip vicinity. Then, we implement the entire boundary value problem with three-dimensional simulations that replicate the empirical tensile deformation of the soft elastomer samples. Subsequently, the results are benchmarked against estimations of the J-integral obtained through a bespoke finite strain analytical crack tip solution. With the successful validation of the configurational force method at finite strains, we aim to establish a pipeline for the calculation of configurational forces in a standalone manner and circumventing the need for close-form analytical solutions.