Linear Elastic Fracture Mechanics Model Research Articles

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.

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.

Fracture and size effect analysis in concrete using 3-D G/XFEM and a CZM-LEFM correlation model: Validation with experiments

The existence of process zones at crack tips in quasi-brittle materials is responsible for the structural size-dependent fracture strength. Hence, accurate prediction tools should not only be capable of predicting the fracture propagation and its load–displacement attributes, but should render themselves capable of the size effect analysis as well. This paper applies a robust 3-D generalized/eXtended finite element method (G/XFEM) algorithm to fracture propagation in concrete and validates its capability to capture the size effect behavior based on the cohesive zone model (CZM). Equipped with the capability of mesh adaptivity, the framework utilized in discretizing the resulting system of equations allows the adoption of independent representation for the background solid mesh and the fracture surfaces without the need for conformity between the two. The fracture simulations on a series of geometrically similar concrete beams under three-point bending have shown very good agreement with available experimental results in terms of both load–displacement (peak load and post-peak) characteristics and size effect behavior. The study also models the experimental problems using Linear Elastic Fracture Mechanics (LEFM) to investigate the structure’s size threshold below which the applicability of LEFM diminishes. The effect of beam geometry and concrete material properties on the LEFM accuracy is assessed by calculating the deviations between LEFM and bilinear cohesive model solutions. The simulation results show that power functions best describe the effect of beam’s depth and tensile strength, and linear functions best fit the effect of elastic modulus and fracture toughness. Subsequently, the study presents and validates an empirical correlation model that modifies the LEFM solution to predict similar ultimate strength as the CZM in addition to capturing the concrete size effect law successfully.

Crack-parallel stress effect on fracture energy of plastic hardening polycrystalline metal identified from gap test scaling

The gap test is a new type of fracture test developed in 2020, in which the end supports of a notched beam are installed with gaps that close only after the elasto-plastic pads next to notch introduce a desired constant crack-parallel compression σxx (also called the T-stress). The test uses the size effect method to identify how such a compression alters the material fracture energy, Gf, and the characteristic size cf of the fracture process zone (FPZ). In 2020, experiments showed that a moderate σxx doubled the Gf of a quasibrittle material (concrete) and a high σxx reduced its Gf to almost zero. A preliminary study by Nguyen et al. (2021) showed that the gap test can be extended to plastic-hardening polycrystalline metals. A generalized scaling law with an intermediate asymptote for large-scale yielding in small structures was derived, and limited tests of aluminum alloy showed its applicability. In this study, geometrically scaled gap tests of notched three-point bend fracture specimens of aluminum are conducted at three different levels of σxx. An extended structural strength scaling law that captures the transition from the micrometer-scale FPZ through millimeter-scale yielding zone (YZ) to large-scale structures which follow linear elastic fracture mechanics (LEFM) is derived and then applied to analyze the effect of σxx. Presented here are the gap tests of aluminum alloy, in which three different levels of σxx are applied to scaled notched four-point-bend beams of depths D = 12, 24, 48 and 96 mm. Using an extended size effect law for plastic-hardening metals, it is found that, at crack-parallel stress σxx≈−40% of the yield strength, the critical J-integral value gets roughly quadrupled, not only because of the well-known enlargement of the hardening YZ whose width is of millimeter scale, but also because of the increase of the FPZ width of micrometer scale. These results can be reproduced neither by line crack models, including the LEFM, cohesive crack and phase-field models, nor by peridynamic and various nonlocal models that ignore the tensorial nature of the material stress at the crack tip. The crack band models, being able to represent an FPZ of finite width and a YZ whose size evolves depending on σxx, can capture the effect of crack-parallel stresses provided that a realistic 3D tensorial damage constitutive model is used. Here, Bai–Wierzbicki’s model is shown to capture the σxx effect on the Gf and Jcr qualitatively.

No Unique Scaling Law for Igneous Dikes

AbstractIn linear elastic fracture mechanics (LEFM), veins, dikes, and sills grow in length when the stress intensity factor at the tip reaches a critical value: the host rock fracture toughness . This criterion is applied broadly in LEFM models for crack growth and it is often assumed that the pressure inside the crack is uniform. When applied to intrusion length versus thickness scaling, a significant issue arises in that derived , which is about 100–1,000 times that of measured values for rocks at upper crustal depths. The same scaling relationships applied to comparatively short mineral vein data gives , approaching the expected range. Here we propose that intrusions preserve non‐equilibrated pressures as cracks controlled by kinetics, and therefore cannot be treated in continuum with fracture‐controlled constant pressure (equilibrium) structures such as veins, or many types of scaled analogue model. Early stages of dike growth (inflation) give rise to increasing length and thickness, but magma pressure gradients within intrusions may serve to drive late‐stage lengthening at the expense of maximum thickness (relaxation). For cracks in 2D, we find that intrusion scaling in non‐equilibrium growth is controlled by the magma injection rate and initial dike scaling, effective (2D) host rock modulus, magma viscosity and cooling rate, which are different for all individual intrusions and sets of intrusions. A solidified intrusion can therefore achieve its final dimensions via many routes, with relaxation acting as a potentially significant factor, hence there is no unique scaling law for dike intrusions.

Tensile and fracture behavior of silica fibers from the Venus flower basket (Euplectella aspergillum)

The extraordinary mechanical properties shown by many biological materials stem from their unique hierarchical structures. The spicules of the deep-sea glass sponge, Euplectella aspergillum, show layered architecture and are known to improve toughness in various loading conditions. In the present work, we conducted a systematic study on the tensile and fracture behavior of E. aspergillum’s spicules. Tensile tests were performed on three different gage lengths of spicules that compose anchoring structures of E. aspergillum. The cross-sectional area of each spicule was accurately measured using an x-ray microscope. The effect of gage length and varying cross-sectional area on the tensile behavior of spicules is discussed in detail. The interplay between two failure initiation sites, namely, weakest-link flaws and minimum cross-sectional area locations was observed. Weibull statistics were used to quantify the variability in the strengths of spicules as a function of their gage length. The Weibull modulus was observed to decrease with the increase in gage length. The possibility of a bimodal flaw population for a higher gage length of spicules due to the interplay between failure initiation sites is also discussed. The fractography study was utilized to understand the failure and toughening mechanisms of spicules in tensile loading. The fracture toughness of the central core was quantified from fracture surfaces using a linear elastic fracture mechanics model. The present study shows that the failure of spicules initiates in the outer layers and proceeds progressively to the center. The layers surrounding the central core of the spicule resist the crack propagation and increase its toughness.

Finite element simulation of quasi-static tensile fracture in nonlinear strain-limiting solids with the phase-field approach

We investigate a quasi-static tensile fracture in nonlinear strain-limiting solids by coupling with the phase-field approach. A classical model for the growth of fractures in an elastic material is formulated in the framework of linear elasticity for deformation systems. This linear elastic fracture mechanics (LEFM) model is derived based on the assumption of small strain. However, the boundary value problem formulated within the LEFM and under traction-free boundary conditions predicts large singular crack-tip strains. Fundamentally, this result is directly in contradiction with the underlying assumption of small strain. In this work, we study a theoretical framework of nonlinear strain-limiting models, which are algebraic nonlinear relations between stress and strain. These models are consistent with the basic assumption of small strain. The advantage of such framework over the LEFM is that the strain remains bounded even if the crack-tip stress tends to the infinity. Then, employing the phase-field approach, the distinct predictions for tensile crack growth can be governed by the model. Several numerical examples to evaluate the efficacy and the performance of the model and numerical algorithms structured on finite element method are presented. Detailed comparisons of the strain, fracture energy with corresponding discrete propagation speed between the nonlinear strain-limiting model and the LEFM for the quasi-static tensile fracture are discussed.

Mixed-mode cohesive laws and the use of linear-elastic fracture mechanics

Small-scale cohesive-zone models based on potential functions are expected to be consistent with the important features of linear-elastic fracture mechanics (LEFM). These include an inverse-square-root K-field ahead of a crack, with the normal and shear stresses being proportional to the mode-I and mode-II stress-intensity factors, KI and KII, the work done against crack-tip tractions being equal to KI2+KII2∕Ē, where Ē is the appropriate modulus, and failure being controlled by the toughness. The use of an LEFM model also implicitly implies that the partition of the crack-tip work into shear and normal components is given by a phase angle defined as ψK=tan−1KII∕KI. In this paper, we show that the partition of crack-tip work in a cohesive-zone model is consistent with LEFM if the normal and shear deformations across an interface are uncoupled. However, we also show that this is not the case for coupled cohesive laws, even if these are derived from a potential function. For coupled laws, LEFM cannot be used to predict the partition of work at the crack tip even when the small-scale requirements for LEFM conditions being met; furthermore, the partition of the work may depend on the loading path. This implies that LEFM cannot be used to predict mixed-mode fracture for interfaces that are described by coupled cohesive laws, and that have a phase-angle-dependent toughness.

Crack growth models for multiaxial fatigue in a ship’s propeller shaft

A premature fatigue failure of a large intermediate propeller shaft in a shuttle tanker is discussed and analyzed. The short fatigue life consists mainly of a crack growth phase. Life predictions are carried out by crack growth modelling based on engineering fracture mechanics. The purpose of the present investigation is to identify the most likely loading modes based on the evolution of the crack propagation. A Linear Elastic Fracture Mechanics Model (LEFM) is applied with the stress intensity factor range entering the Paris law as a key parameter. Existing formulas for the geometry functions are supplemented by more detailed stress intensity factor calculations pertaining to small semi-elliptical surface cracks subjected to stress mode I. Enhanced geometry functions are proposed as a function of the relative crack depth and the crack shape aspect ratio. The ability of the fracture mechanics model to reconstruct the observed crack path and crack shape development is emphasized. Various loading modes and multi-axial stress states are applied to pursue the observed crack behavior. The observed semi-elliptical crack shapes and the shift in crack planes are included in the analysis.

A study of DCT specimen in the verification of the influence of the geometric variation in the stress intensity factor

The objective of this work is to verify the influence of the stress intensity factor in the linear elastic fracture mechanics model. The model consists in a Disk-shaped Compact Tension specimen (DCT) of concrete material. The methodology considers a comparative study of an analytical approach from the literature and numerical simulations. These numerical simulations are performed in ANSYS Workbench program by the use of the Finite Element Method (FEM). The results show that the solutions obtained are satisfactory for the comparative study.
  

Sharp contrasts in observed and modeled crevasse patterns at Greenland's marine terminating glaciers

Abstract. Crevasses are affected by and affect both the stresses and the surface mass balance of glaciers. These effects are brought on through potentially important controls on meltwater routing, glacier viscosity, and iceberg calving, yet there are few direct observations of crevasse sizes and locations to inform our understanding of these interactions. Here we extract depth estimates for the visible portion of crevasses from high-resolution surface elevation observations for 52 644 crevasses from 19 Greenland glaciers. We then compare our observed depths with those calculated using two popular models that assume crevasse depths are functions of local stresses: the Nye and linear elastic fracture mechanics (LEFM) formulations. When informed by the observed crevasse depths, the LEFM formulation produces kilometer-scale variations in crevasse depth, in decent agreement with observations. However, neither formulation accurately captures smaller-scale variations in the observed crevasse depths. Critically, we find that along-flow patterns in crevasse depths are unrelated to along-flow patterns in strain rates (and therefore stresses). Cumulative strain rate is moderately more predictive of crevasse depths at the majority of glaciers. Our reliance on lidar limits the inference we can make regarding fracture depths. However, given the discordant patterns in observed and modeled crevasses, we recommend additional in situ and remote sensing analyses before Nye and LEFM models are considered predictive. Such analyses should span extensional and compressive regions to better understand the influence of advection on crevasse geometry. Ultimately, such additional study will enable more reliable projection of terminus position change and supraglacial meltwater routing that relies on accurate modeling of crevasse occurrence.

Validation of a 3-D adaptive stable generalized/eXtended finite element method for mixed-mode brittle fracture propagation

In this paper, a Stable Generalized/eXtended Finite Element Method (SGFEM) is combined with mesh adaptivity for the robust and computationally efficient simulation of mixed-mode brittle fracture propagation. Both h-refinement around the fracture front and p-enrichment of the analysis domain are used to control discretization errors. A Linear Elastic Fracture Mechanics (LEFM) model based on Griffith’s criterion is adopted. LEFM scaling relations are used at each fracture propagation step to back calculate SIFs that meet Griffith’s criterion. As a result, no iterations are necessary to find loading scaling parameters or fracture size that meets Griffith’s criterion. The method is validated against several experimental data sets for mode I and mode I+II fracture propagation problems. Very good agreement between SGFEM and experimental results is observed. These include fracture path, Crack Opening Displacement (COD), and load and fracture length versus COD curves. The computational efficiency of the method is also assessed.

A novel tearing residual strength model for architectural coated fabrics with central crack

This study concerns the tearing residual strength models of architectural woven coated fabrics. Firstly, central crack tearing tests were conducted, the evolution of tearing residual strength with the increase of initial crack length is investigated. Secondly, the commonly used tearing residual strength models, including linear elastic fracture mechanics (LEFM) models and stress field models, are summarized and analyzed. Due to their inherent defects in theoretical foundation, the prediction precision is merely determined by curve fitting, no physical meaning can be drawn from these models. Finally, an asymptotic model based on fracture mechanics theory is proposed. The relative size of fracture process zone (FPZ) is adopted to distinguish the transition from strength-controlled fracture to quasi-brittle fracture and finally to toughness-controlled fracture. The prediction results agree very well with the test data, and most of test data lies in nonlinear fracture range which has never been considered.

Energy release rate of hyperelastic solids with a nanocrack

ABSTRACT The purpose of this study is to explore the hyperelastic effect on the energy release rate of a crack extension at the nanoscale. A molecular statics computation has been carried out to characterise the atomistic nature of fracturing. The concept of the J-integral is employed to measure the energy release rate for Ni and Si single crystals having a hyperelastic nature. The obtained J-integral is compared to the energy release rates predicted using the two-specimen method and the finite-element method. The results show that the effect of the highly-localized nonlinear zone in front of the crack tip potentially induces a breakdown of the linear elastic fracture mechanics model.

A non-local continuum poro-damage mechanics model for hydrofracturing of surface crevasses in grounded glaciers

AbstractHydrofracturing can enhance the depth to which crevasses propagate and, in some cases, allow full depth crevasse penetration and iceberg detachment. However, many existing crevasse models either do not fully account for the stress field driving the hydrofracture process and/or treat glacier ice as elastic, neglecting the non-linear viscous rheology. Here, we present a non-local continuum poro-damage mechanics (CPDM) model for hydrofracturing and implement it within a full Stokes finite element formulation. We use the CPDM model to simulate the propagation of water-filled crevasses in idealized grounded glaciers, and compare crevasse depths predicted by this model with those from linear elastic fracture mechanics (LEFM) and zero stress models. We find that the CPDM model is in good agreement with the LEFM model for isolated crevasses and with the zero stress model for closely-spaced crevasses, until the glacier approaches buoyancy. When the glacier approaches buoyancy, we find that the CPDM model does not allow the propagation of water-filled crevasses due to the much smaller size of the tensile stress region concentrated near the crevasse tip. Our study suggests that the combination of non-linear viscous and damage processes in ice near the tip of a water-filled crevasse can alter calving outcomes.

Influence of the Test Configuration and Temperature on the Mechanical Behaviour of WC-Co

In this work, the effect of the test configuration and temperature on the mechanical behaviour of cemented carbides (WC-Co) with different carbide grain sizes (dWC) and cobalt volume fractions (VCo), implying different binder mean free paths (λCo), was studied. The mechanical strength was measured at 600 °C with bar-shaped specimens subjected to uniaxial four-point bending (4PB) tests and with disc specimens subjected to biaxial ball-on-three-balls (B3B) tests. The results were analysed within the frame of the Weibull theory and compared with strength measurements performed at room temperature under the same loading conditions. A mechanical degradation greater than 30% was observed when the samples were tested at 600 °C due to oxidation phenomena, but higher Weibull moduli were obtained as a result of narrower defect size distributions. A fractographic analysis was conducted with broken specimens from each test configuration. The number of fragments (Nf) and the macroscopic fracture surface were related to the flexural strength and fracture toughness of WC-Co. For a given number of fragments, higher mechanical strength values were always obtained for WC-Co grades with higher KIc. The observed differences were discussed based on a linear elastic fracture mechanics (LEFM) model, taking into account the effect of the temperature and microstructure of the cemented carbides on the mechanical strength.

Critical bending load of CFRP panel with shallow surface scratch determined by a tensile strength model

Carbon fibre reinforced polymer (CFRP) composite structures may experience light surface damage such as shallow surface scratch around 50 μm or less in applications. In this study, critical bending loads of 2D woven CFRP panels with and without shallow surface scratches were assessed under three-point-bending (3-p-b) conditions and then modelled by a simple strength theory for composite fracture. This closed-form tensile strength solution is a non- Linear Elastic Fracture Mechanics (LEFM) model derived specifically for heterogeneous composites. Direct tensile tests of the CFRP panels (without shallow surface scratch) were also performed, and measured tensile strength was compared with that from 3-p-b tests. The relative error between the two different tests was only around 6.7%. The ply thickness around 140 μm was selected as the controlling composite microstructure parameter or characteristic composite unit Cch in this non-LEFM model. This simple model shows that a suitable composite characteristic microstructure measurement such as Cch = ply thickness can significantly simplify composite fracture analysis. A simple statistical analysis using a 2D normal distribution methodology for strength measurements was also introduced so that the experimental scatters can be analysed. The 95% reliability band can be conveniently used in safe design of composite structures.

Electrochemically induced fractures in crystalline silicon anodes

This communication presents our recent discovery on electrochemically induced fracture behavior of a single-crystal silicon (Si) anode, with particular focus on the initialization and propagation of cracks. Intensive efforts have investigated mechanical properties and cracking behavior of Si electrodes overextended electrochemical cycles. However, it is still elusive how cracks nucleate and propagate in the crystalline Si anodes at the very early stage. Here, we examined the morphology evolution of the Si (100) wafer electrodes during the first electrochemical cycle, and we found that the formation of <110>-oriented buckles—induced by the anisotropic lithiation in crystalline Si during 1st lithiation—results in cracks propagated along the <110> direction in the buckled region during 1st delithiation. Surprisingly, as the delithiation proceeds, new cracks oriented along the <100> direction appear in the intact area where no macroscopic deformation occurred during the previous lithiation. The <100>-oriented cracks have not been studied previously. Herein, we introduce a linear elastic fracture mechanics model to help understand underlying mechanisms for such crack propagation. The findings in this work provide significant insights into the fracture behavior and formation mechanism, as well as possible strategies to inhibit crack propagations of Si electrodes at the beginning stage of cycling.

Fatigue resistance of the deck plate in steel orthotropic deck structures

Steel orthotropic decks in bridges and ferries have shown to be prone to fatigue cracking. A typical fatigue crack observed in these structures initiates at the root of the weld between the deck plate and stringer and grows into the deck plate. Due to limited fatigue test data available and the deviating characteristics of this type of crack as compared to other types of fatigue cracks, the fatigue strength is uncertain. In this study, a linear elastic fracture mechanics model has been developed for this type of crack, providing insight into its fatigue performance. The model predicts a relatively high fatigue resistance which is close to the results of fatigue tests. The model further predicts a relatively long residual fatigue life after crack detection, providing large inspection intervals.

A methodology for fretting fatigue life estimation using strain-based fracture mechanics

This work presents a methodology to estimate the lifetime of components under fretting fatigue using strain-based fracture mechanics (SBFM). The latter was originally designed to model small crack behavior in notches, where the behavior of materials is non-linear, i.e. outside of the linear elastic fracture mechanics (LEFM) domain. The SBFM model is basically a LEFM model modified to consider the effects of material nonlinearity and does not require two stages (initiation and propagation) to model fatigue life. To obtain these solutions, 2D finite element model simulations were performed and later adjusted by a 3D correction factor to consider a 3D elliptical crack propagation. The plasticity property of the material was considered by using the Ramberg-Osgood curve in association with Neuber’s rule. The simulations estimated the fatigue life of 7050-T7451 Al and 2024-T351 Al components for several stress levels and the results were then compared to experimental data. The methodology was able to predict short and long crack behaviors, as well as crack arrests, and the results show that the method can satisfactorily predict fretting fatigue lives.