Crack Opening Displacement Research Articles

Mixed FEM implementation of three-point bending of the beam with an edge crack within strain gradient elasticity theory

AbstractThis paper considers the problem of symmetrical three-point bending of a prismatic beam with an edge crack. The solution is obtained by the mixed finite element method within the simplified Toupin–Mindlin strain gradient elasticity theory. A mixed variational formulation of the boundary value problem for displacements–strains–stresses and their gradients is applied, simplifying the choice of approximating functions. The concept of energy balance is adopted to calculate the energy release rate with a virtual increase in crack length. The increment of the potential energy of an elastic body is determined by accounting for the strain and stress gradient contribution. Numerical calculations were performed using a quasi-uniform triangular mesh of the cross-type. The mesh refinement was applied in the vicinity of the crack tip, at the concentrated support, and the point of application of the transverse force, and uniform mesh partitioning was utilized in the rest of the beam. The fine-mesh analysis was carried out on the successively condensed meshes in the stress concentration domain for different values of the length scale parameter. The crack opening displacements and the distribution of strains and Cauchy stresses for various values of the length scale parameter are presented. An increase in this parameter increases the stiffness of the crack, which leads to a decrease in the crack opening displacements and a smooth closure of its faces at the crack tip. In addition, accounting for the scale parameter reduces the calculated values of strains and stresses near the crack tip. Based on the energy balance criterion, local fracture parameters such as the release rate of elastic energy at the crack tip and the stress intensity factor are determined for different values of the mesh step. The numerical calculations indicate the convergence of the obtained approximations. The main feature of solutions, which includes the strain gradient contribution, is the decrease in the values of the calculated parameters associated with the fracture energy compared to the classical elasticity theory.

RC dapped-end beams with various reinforcement layouts: An experimental investigation

Reinforced Concrete (RC) discontinuity regions (D-regions), such as dapped ends, can represent vulnerable zones in existing RC structures since they can exhibit a brittle failure mechanism. In common design practice, the ultimate strength of D-regions is assessed through Strut and Tie (S&T) models. Current design codes recommend two resistant mechanisms for dapped ends but do not provide information about the identification of steel bars involved in the resistant mechanism when the reinforcement is formed by distributed bars. An experimental campaign on dapped-end beams with various reinforcement layouts, including configurations without diagonal bars or with one or multiple layers of diagonal bars, has been carried out at the Structures and Material Testing Laboratory of the Department of Civil and Environmental Engineering in Florence. Results allow for identifying the influence of different reinforcement layouts on the crack pattern at the Service Limit State (SLS) and on the post-peak response. Both the strain on bars and the crack opening displacements are investigated until failure. The work highlights that layouts with distributed diagonal bars reduce the crack widths at the SLS compared to other configurations, increase the ductility, and allow for the timely detection of damaged dapped ends during in situ inspections.

Study on fracture behaviour of pipeline girth welds based on curved wide plate specimens: Experimental and numerical analysis

Understanding the fracture characteristics and strain capacity of pipelines with girth welds under external loads is crucial for evaluating pipeline safety. The use of curved wide plate (CWP) specimens closely resembling full-scale (FS) pipelines in geometry provides a more realistic representation of crack tip constraint states compared to small-size fracture toughness test specimens in laboratory settings. This study aims to investigate the ductile fracture characteristics of center cracks on the outer surface of girth welds in high-grade steel pipelines under external loads through large-scale CWP tensile tests and advanced numerical simulations. Tensile tests were conducted on CWP specimens using a kiloton tensile testing machine, with double crack opening displacement (COD) gauge monitoring crack mouth opening displacement (CMOD), and high-precision displacement sensors and strain gauges tracking deformation and strains at various positions. The study yielded significant experimental results, and a finite element (FE) model of the CWP was developed using the nonlinear finite element method (FEM) to validate the experimental data against simulation results. The FE model was then used to investigate the influence of material properties on crack propagation resistance and driving force curves. A method for determining the ultimate tensile strain capacity (UTSC) of pipelines based on actual material fracture toughness was proposed. This research contributes valuable experimental data for further exploration of fracture behaviour in large-scale pipeline girth welds and offers insights for safety evaluations of such welds.

Determination of the plastic J integral of ductile material using the XFEM with only Heaviside function and variable-node elements

In this paper, the extended finite element method (XFEM) with only Heaviside function is proposed for the elastic–plastic fracture mechanics (EPFM) modeling. The proposed method removes tip enrichment functions depending on the polar coordinates at the crack front, and a step function only determined by the level set function is utilized to track the crack front. To alleviate the volumetric locking phenomena caused by the plastic incompressibility, the B-bar method is incorporated into the three dimensional (3D) XFEM program. Therefore, the fully integration scheme can be chosen to ensure the accuracy when addressing large plastic deformation in EPFM analysis. Additionally, the material tangent stiffness matrix of Ramberg-Osgood constitutive is given, and the local refinement technique using variable-node elements is adopted to reduce the number of elements and nodes for efficient analysis. A Newton-Raphson iterative algorithm is developed to solve the nonlinear algebraic equations caused by material nonlinearity. Several numerical examples including the determination of crack opening displacement, and the fully plastic J integral in the ductile materials are presented to test the performance of the proposed method. Comparisons with the results from the existing methodologies show that the new enrichment scheme can save computational cost and obtain sufficient accuracy even in the case of 3D curved crack.

Displacement Interval Prediction Method for Arch Dam with Cracks: Integrated STL, MF-DFA and Bootstrap

Displacement prediction models based on measured data have been widely applied in structural health monitoring. However, most models neglect the particularity of displacement monitoring for arch dams with cracks, nor do they thoroughly analyze the non-stationarity and uncertainty of displacement. To address this issue, the influencing factors of displacement were first considered, with crack opening displacement being incorporated into them, leading to the construction of the HSCT model that accounts for the effects of cracks. Feature selection was performed on the factors of the HSCT model utilizing the max-relevance and min-redundancy (mRMR) algorithm, resulting in the screened subset of displacement influence factors. Next, displacement was decomposed into trend, seasonal, and remainder components applying the seasonal-trend decomposition using loess (STL) algorithm. The multifractal characteristics of these displacement components were then analyzed by multifractal detrended fluctuation analysis (MF-DFA). Subsequently, displacement components were predicted employing the convolutional neural network-long short-term memory (CNN-LSTM) model. Finally, the impact of uncertainty factors was quantified using prediction intervals based on the bootstrap method. The results indicate that the proposed methods and models are effective, yielding satisfactory prediction accuracy and providing scientific basis and technical support for the health diagnosis of hydraulic structures.

A detailed study on void growth and crack-void interaction in plastically compressible bilinear hardening and trilinear hardening-softening-hardening solids

This paper provides detailed investigations on void growth and crack-void interaction ahead of a moving crack-tip for a mode I crack in plastically compressible solid. A finite strain elastic-viscoplastic material model with hardening-hardening and hardening-softening-hardening hardness functions is used in the finite element simulation. The plastic compressibility leads to an increased crack opening displacement. The plastic compressibility together with material softening considerably increase stress triaxiality and decrease the void and crack growth. For plastically compressible solids, void growth is more in normal direction to the crack plane and increase in initial void size is accompanied by decrease in the crack and void growth.

Mechanical properties and cracking strength estimation of high-strength engineered cementitious composites considering the fracture process zone

In the present study, the effect of flaws on the mechanical properties of high-strength engineered cementitious composites (HS-ECC) was investigated. Compressive tests, tensile tests, and electron microscope scanning microscopic tests were designed to explore the influence mechanism of flaws on mechanical properties. The results show that the compressive failure of HS-ECC was primarily characterized by a high density of vertical micro-cracks, and the specimen maintained exceptional structural integrity. All tested mix ratios demonstrated tensile strain hardening. Notably, the test groups with an 18 mm fiber length exhibited superior crack width control, with an average crack width of less than 30 μm. A new fiber bridging model was proposed, informed by microstructural analysis and crack opening displacement observations. A theoretical formula for the cracking strength of HS-ECC was developed based on the fracture process zone concept. The calculated initial crack strength closely matches the experimental values, with a discrepancy range of [0, 17 %]. This close agreement provides robust validation for the accuracy and reliability of the enhanced cracking strength calculation theory, thereby supporting its application in the design of ECC.

Smart prediction of rock crack opening displacement from noisy data recorded by distributed fiber optic sensing

The commonly used method for estimating crack opening displacement (COD) is based on analytical models derived from strain transferring. However, when large background noise exists in distributed fiber optic sensing (DFOS) data, estimating COD through an analytical model is very difficult even if the DFOS data have been denoised. To address this challenge, this study proposes a machine learning (ML)-based methodology to complete rock’s COD estimation from establishment of a dataset with one-to-one correspondence between strain sequence and COD to the optimization of ML models. The Bayesian optimization is used via the Hyperopt Python library to determine the appropriate hyper-parameters of four ML models. To ensure that the best hyper-parameters will not be missing, the configuration space in Hyperopt is specified by probability distribution. The four models are trained using DFOS data with minimal noise while being examined on datasets with different noise levels to test their anti-noise robustness. The proposed models are compared each other in terms of goodness of fit and mean squared error. The results show that the Bayesian optimization-based random forest is promising to estimate the COD of rock using noisy DFOS data.

Spatial-temporal response and precursor characteristics of tensile failure on disc coal samples of different sizes combining AE, EMR and DIC techniques

Tensile fractures in deep coal and rock mass can readily trigger dynamic calamities like rock bursts, significantly impacting the safety of coal mining. To enhance our understanding of coal’s tensile failure mechanism, Brazilian splitting failure experiments were conducted on coal samples with prefabricated cracks of different sizes. Acoustic emission (AE), electromagnetic radiation (EMR) and digital image correlation (DIC) techniques were used to analyze the instability failure process and precursor characteristics of coal samples. The results showed that, for coal samples of the same thickness, the tensile strength and peak strain gradually decreased with the increase of coal sample size, whereas the elastic modulus increased gradually, highlighting more pronounced brittleness characteristics and revealing a noticeable size effect. The time-varying evolution of AE energy, EMR energy and VE (virtual extensometer) elongation during the tensile failure process of coal samples is closely correlated to the stress evolution trend, exhibiting distinct characteristics at various loading stages. The failure modes of coal samples of varying sizes exhibited a strong correlation with the spatial distribution of AE events, as well as the strain and displacement field measured by DIC. With the coal sample’s growing size, the percentage of AE high energy events rose while the total number of AE positioning events fell, the proportion of DIC strain concentration area decreased, displacement zoning characteristics became more obvious, and the critical opening displacement of cracks increased. The variance of AE, EMR and DIC related parameters in coal samples showed significant critical slowing down characteristics. Based on their response timescales, EMR energy was identified as an “Early warning”, AE energy as a “Medium warning”, and VE elongation as a “Short-imminent warning”. A comprehensive analysis of multi-parameter monitoring information proved beneficial in accurately pinpointing precursory response points of coal mine dynamic disasters, thereby enhancing monitoring precision.

Study on the dynamic fracture mechanical properties of ultra high-performance concrete under the influence of steel fiber content

Ultra-high performance concrete (UHPC) is characterized by its ultra-high strength and durability, making it suitable for use in large-span bridges, super high-rise buildings, and other special structures. These types of structures often have high requirements for crack control, making the exploration of UHPC's fracture mechanics properties of significant importance. Therefore, this paper sets up four steel fiber contents (0 %, 0.6 %, 1.2 % and 1.8 %) and four strain rates (10−5/s, 10−4/s, 10−3/s and 10−2/s), totaling 16 working conditions. The hydraulic servo system combined with digital image correlation (DIC) technology was used to conduct fracture mechanics tests on UHPC three-point bending beams. The test results show that the unstable fracture load of UHPC increases with the increase of strain rate and steel fiber content, and increasing the steel fiber content can enhance the strain rate effect of UHPC to a certain extent. Additionally, combined with the J-criterion model, this study proposed a dynamic criterion for the unstable fracture load of UHPC considering the influence of steel fiber content. The initial fracture toughness, unstable fracture toughness, and fracture energy of UHPC are significantly more affected by steel fiber content than by strain rate. Through DIC technology, the analysis of UHPC crack propagation shows that UHPC without steel fibers experiences rapid crack expansion in the Pre-90 % to Peak stage, with less influence from the strain rate. However, UHPC with steel fibers exhibits a more uniform crack growth rate, with the crack emergence phase gradually lagging as the strain rate increases. For UHPC without steel fibers, the crack opening displacement (COD) at the unstable fracture load point surges suddenly, while the COD development in UHPC with steel fibers is slower. At lower strain rates, an increase in steel fiber content leads to an earlier sudden increase phase in UHPC's COD; at higher strain rates, the influence of steel fiber content on the sudden increase phase of UHPC's COD is more scattered. The research results of this paper provide a theoretical basis for the calculation and analysis of the dynamic response of UHPC structures under seismic loads.

3D analysis of crack propagation in nuclear graphite using DVC coupled with finite element analysis

The initiation and propagation of cracks within nuclear graphite components compromise the lifespan of advanced gas-cooled reactors and even lead to safety incidents. A thorough understanding of fracture properties in nuclear graphite is crucial for the effective management and design of graphite structural components. In this work, an in-situ test of a double cleavage drilled compression (DCDC) specimen was conducted using X-ray tomographic imaging to fully analyze the 3D crack propagation behavior in IG-11 nuclear graphite. “Subvolume Splitting” digital volume correlation (SS-DVC) was employed to accurately quantify the displacement fields surrounding the crack. The entire 3D crack propagation and crack surface around the crack front were precisely captured through 3D imaging processing of the gray level residual (GLR) field. Additionally, crack opening displacements (CODs) and stress intensity factors (SIFs) were determined using finite element (FE) analysis with extended finite element method (XFEM). Experimental results indicate that although mode I crack remains predominant throughout the whole process, complex crack propagation exists around the crack front. The utilization of irregular crack surfaces in FE analysis helps to obtain refined displacement fields and accurate SIFs of the crack front. This work provides comprehensive insights into crack propagation and reveals the influence of complex crack morphology on SIF calculations, facilitating accurate prediction of structural integrity and rational design of graphite components.

Semi-analytical solutions of kinked edge cracks

A surface crack may advance either by further penetrating inward or by kinking out in response to possible variations in driving force or material properties. This paper provides a general approach to solve the planar elastic boundary value problem (BVP) of a kinked edge crack in a half-space. Semi-analytical solutions to BVPs with far-field uniform tension and concentrated loads on the free surface are given. The solution enable us to derive related information for reliability analysis, including stress fields, stress intensity factors (SIFs), and crack opening displacement (COD). An extension to the Cotterell–Rice formula (Cotterell and Rice, 1980) on SIFs for infinitesimal kinks is supplied, which is applicable to cracks with kinked segments of arbitrary length. Zooming in on a broadly concerned BVP of a cross-coating crack in a coating–substrate with a kink in the interface, we show the SIFs of the kink tip following an empirical formula K∗=αexp(−βah)+K∞∗ as a/h>0.2, where a is the length of the interface segment and h the coating thickness, and K∞∗ is the asymptotic solution by Thouless et al. (1987, 1989). The formula is in accord with existing solutions on the two asymptotic limits of the kinked part.

Analytical solution of two non‐identical edge cracks in an infinite strip under anti‐plane shear wave

AbstractThis article presents an extensive analytical solution addressing the interaction between two non‐identical edge cracks in an infinite orthotropic strip under anti‐plane shear waves. Most studies assume identical cracks or single edge crack in a strip, but this research breaks new ground by considering cracks of different sizes. By incorporating mixed‐type boundary conditions, the study derives dual integral equations. These equations are then transformed into a singular integral equation of Cauchy type with the aid of a trial solution and contour integration technique. The singular integral equation is further converted into a system of integral equations, which are solved numerically utilizing Jacobi polynomials. The obtained solutions are utilized to derive expressions for the stress intensity factor (SIF) and crack opening displacement (COD) at the crack tip using Krenk's interpolation formulae. The derived results are presented graphically and compared against existing solutions for single edge crack and symmetric edge cracks in static scenario.

Fracture energy of concrete after sustained loading

The fracture energy of concrete after sustained loading is a necessary parameter to calculate the residual carrying capacity of concrete structures in service. This study investigated the fracture energy of concrete after sustained loading by combining experimental observations and theoretical analyses. Firstly, a series of creep fracture tests were conducted on 100 three-point bending beams under various sustained loads, i.e., 75 %, 80 %, 85 %, and 90 % of peak loads. After different loading time, i.e., 10 %, 30 %, 50 %, 70 %, and 90 % of creep fracture lifetimes, the static fracture tests were performed until failure. Then, based on the relationship between the work done by external forces and the energy consumed by the crack propagation of specimens, an analytical model for the fracture energy of concrete after sustained loading was developed, describing the quantitative relationship between the fracture energy of concrete and the crack opening displacement, and the loading time. Finally, the impact of the loading time and sustained load levels on the fracture energy was investigated. The results indicated that the fracture energy of concrete after sustained loading decayed rapidly, with the decay rate gradually decreasing until stability over the loading time. Additionally, specimens subjected to lower sustained load levels experienced a longer loading time for the same crack opening displacement, indicating a greater decay degree of the fracture energy. The analytical model proposed in this study can be employed for predicting and evaluating the crack propagation process and residual carrying capacity of concrete structures during their service life.

Theoretical and experimental investigation into detection of early-stage propagation of double cracks with distributed fiber optic sensor

A strain transfer model is developed to describe the transfer of crack opening displacement (COD) of double cracks in a structural substrate to the distributed fiber optic sensor (DFOS). Global and local coordinate systems are set up to aid in the deduction of the transfer of CODs of the double cracks to the DFOS. The analytical model of strain transfer is first derived in the local coordinate system and then is transformed into the global coordinate system. Analytical results show that the DFOS fiber strain induced by CODs of the double cracks in the structural substrate is affected by both the spacing and the CODs of the double cracks, and the strain peak is suppressed as the crack spacing is less than the characteristic length which is defined as the range of influence of the COD of the individual crack. The influences of the material properties of the DFOS, the fiber coating, the bonding adhesive, and the DFOS-fiber coating interfacial stiffness are discussed. The developed model is also experimentally validated by monitoring CODs of double cracks in aluminum alloy plates with optical frequency domain reflectometer (OFDR) DFOSs. Experimental results show that the strain distribution measured by the OFDR DFOSs agree well with the analytical results of the developed model. The limitations of the developed model and future work anticipated are then discussed.

SINGULAR INTEGRAL EQUATIONS FOR A CRACK SUBJECTED NORMAL STRESS IN A HEATED PLATE

In this paper, a crack in a heated plate is investigated, subjected to normal stress. Employing the relationship between the uniform and perturbation fields, as well as complex potential functions and stresses, the problems of heat conduction and heat stress are modeled as singular integral equations. The derivatives of the crack opening displacement function and the temperature jump function serve as the unknown functions. Gauss integration rules are applied to solve the obtained equations numerically. Analysis of the stress intensity factors(SIFs) for some particular crack configurations is presented.

Reconstruction and prediction of Mode-I cohesive law using artificial neural network

Cohesive zone model (CZM) is a widely used tool for simulating both static and fatigue damage propagation for bonded area in composite materials. Under static loading, the damage propagation behavior is governed by a cohesive law (CL), the shape of which is an important property especially when non-linear damage mechanisms such as fiber bridging is involved. In this work, a CZM driven by artificially neural network (ANN) for simulating mode-I damage propagation is proposed, where the traction-separation relationship is calculated by a multi-layer perceptron (MLP). More importantly, a reconstruction method is proposed to extract the CL through the training of the neural network using a simple load-displacement (P–U) relation of a double cantilever beam (DCB) as the only inputs, without the need to measure the crack opening displacement (COD). The proposed method is first validated using finite element generated virtual experimental data with various CL shapes before being applied to experimental data, where good correlations are obtained, proving the effectiveness of the proposed method.

A deep learning informed-mesoscale cohesive numerical model for investigating the mechanical behavior of shield tunnels with crack damage

The present deep learning (DL)-based detection of crack damage of shield tunnels through images can only provide part of geometric information of the crack damage, which prevents the comprehensive understanding of the mechanical behavior of the cracked tunnel linings. This study bridges the gap between the geometric information identified by DL models and the transformation law of mechanical behavior of tunnel linings revealed by numerical methods. A mask-region-based hybrid attention convolutional neural network (Mask R-HACNN) and mesoscale cohesive numerical model are developed to obtain the geometric information of crack damage and investigate the structural response of segments and joints with crack damage, respectively. The numerical simulation results demonstrate that the relationship between in-depth crack length and crack opening displacement and the relationship between the degradation stiffness and in-depth crack length are independent of the loading path. Based on these findings, prediction formulas are established to interpret the relationships between the stiffness degradation coefficients, flexural stiffness, in-depth crack length, joint openings, and crack opening displacement. Then, the geometric information of crack damage obtained by the Mask R-HACNN is provided to the mesoscale cohesive numerical model to estimate the in-depth crack length, flexural stiffness, and stiffness degradation coefficient of cracked segments. Results show that the deep learning informed-mesoscale cohesive numerical model is satisfying in understanding the mechanical behavior of segments and joints during crack damage evolution.

Mechanical properties of polymer fibre reinforced concrete in the light of various standards

The article presents the results of a detailed experimental campaign including a compressive strength test, three- and four-point bending test (3PBT and 4PBT, respectively) of polymer fiber reinforced concrete with the addition of metakaolin. The comprehensive analysis included three Types of concrete mixture differing in amount and used polymer fibers. It was concluded that polymer fibers did not influence the maximum compressive and flexural tensile strength of concrete. On the other hand, they had a beneficial effect on the ductility, residual and equivalent flexural tensile strengths, and fracture energy of samples. The mixtures of Type 1 and 2 were characterized by softening behaviour but the mixture of Type 3 by soft-hardening behaviour. In the 3PBT, the residual flexural tensile strengths obtained according to EN 14651 did not correspond clearly with equivalent flexural tensile strengths calculated in compliance with RILEM TC 162-TDF. It is noteworthy that the effectiveness and correctness of equations presented in other work of the authors referring to dependencies between deflection, crack and tip mouth opening displacements for the 3PBT were confirmed on samples with different composition and fibers. Based on the 4PBT, the equivalent flexural tensile strengths according to JCI-SF4 standard were calculated and the correlations with the results from 3PBT were defined.

Acoustic emission and fracture morphology characteristics of thermal-damage granite under mixed mode I/III loading

To explore the fracture morphology and acoustic emission characteristics of thermally damaged granite under mixed mode I/III loading, a series of heat treatment and mixed mode I/III loading tests were conducted on edge-notched disc bend (ENDB) granite samples. First, the thermal damage samples’ strength and crack opening displacement were analyzed. Subsequently, based on 3D laser scanning technology, the fracture morphology of the sample was studied, and the fractal dimension was calculated. Then, the samples’ acoustic emission characteristics were discussed, including the ringing count, b value, spectrum signature, and AF-RA. Finally, an acoustic early warning index was proposed based on the critical slowing down theory, and the influence of temperature on the early warning effect was analyzed. The results show that under mixed mode I/III loading, the samples formed an antisymmetric curved surface with out-of-plane torsion. As the thermal damage effect is enhanced, the peak strength decreases linearly, and the fractal dimension increases linearly. The failure mode changes from sudden instability to progressive instability. Shear cracks gradually dominate the sample failure. The peak frequency signal of the sample is mainly distributed between 80 kHz and 400 kHz. The appearance of the sudden drop in the b value and a zero-frequency signal has an early warning effect. The quadratic function can describe the relationship between warning time and temperature. As the temperature increases, the deterioration effect of historical loading on the sample is enhanced, the development time of sample failure is prolonged, and the early warning effect is better.