Experimental investigation of fracture damage of notched granite beams under cyclic loading using DIC and AE techniques

Fracture experiments of three‐point bending notched granite beams were performed under cyclic loading using digital image correlation (DIC) and acoustic emission (AE) techniques. The damage evolution process of the specimen under cyclic loading was analysed on the basis of AE ring count and b value. The strain and displacement fields and the fracture process zone (FPZ) ahead of the crack tip were revealed by DIC. The results showed that the AE characteristics of rock fracture indicated a noticeable Kaiser effect in the stage of cyclic loading and unloading. Moreover, when the loading force reached 70% of its peak value, the AE characteristics showed the Felicity effect. The damage produced during the loading‐unloading process contributed to the development of the cracks leading to the catastrophic fracture. Besides, a relatively high loading rate was found to help to suppress the development of the FPZ at the crack tip.     

用能量方法研究混凝土断裂过程区的力学性能

准脆性混凝土自由裂缝前缘断裂过程区的发展与其非线性断裂特征及尺寸效应现象密切相关。它的物理力学行为的量化分析对理解混凝土断裂破坏机理和建立适用于混凝土结构裂缝稳定分析和安全评估断裂准则尤为重要,一直是混凝土断裂力学研究的核心问题。该文依据Hillerborg给出的断裂能定义,给出了计算单位长度断裂过程区发展能量耗散的通用表达式。以三点弯曲梁为例,采用非线性软化本构关系,进一步给出了计算此平均能量耗散的具体步骤及对应的公式。在根据实测的三点弯曲梁的断裂能回归拟合了特征裂缝张开位移w0后,计算了每个试件整个断裂全过程中不同荷载时刻断裂过程区耗能的平均值。结果表明:随着裂缝扩展,断裂过程区能量耗散的变化和试件尺寸无关,可描述断裂过程区混凝土材料的力学性能。     

Experimental and numerical investigations on fracture process zone of rock–concrete interface

A crack propagation criterion for a rock–concrete interface is employed to investigate the evolution of the fracture process zone (FPZ) in rock–concrete composite beams under three‐point bending (TPB). According to the criterion, cracking initiates along the interface when the difference between the mode I stress intensity factor at the crack tip caused by external loading and the one caused by the cohesive stress acting on the fictitious crack surfaces reaches the initial fracture toughness of a rock–concrete interface. From the experimental results of the composite beams with various initial crack lengths but equal depths under TPB, the interface fracture parameters are determined. In addition, the FPZ evolution in a TPB specimen is investigated by using a digital image correlation technique. Thus, the fracture processes of the rock–concrete composite beams can be simulated by introducing the initial fracture criterion to determine the crack propagation. By comparing the load versus crack mouth opening displacement curves and FPZ evolution, the numerical and experimental results show a reasonable agreement, which verifies the numerical method developed in this study for analysing the crack propagation along the rock–concrete interface. Finally, based on the numerical results, the effect of ligament length on the FPZ evolution and the variations of the fracture model during crack propagation are discussed for the rock–concrete interface fracture under TPB. The results indicate that ligament length significantly affects the FPZ evolution at the rock–concrete interface under TPB and the stress intensity factor ratio of modes II to I is influenced by the specimen size during the propagation of the interfacial crack.     

The fracture process zone in asphalt mixture at low temperature

The fracture process zone (FPZ) is a key factor to mechanistically characterize material fracture. This study investigates the FPZ of asphalt mixture at low temperature. The fracture process under a semi-circular bend (SCB) test of seven asphalt mixtures that represent a combination of different factors was monitored using an acoustic (AE) system with eight piezoelectric sensors. The size of FPZ was estimated by locating micro-cracks that correspond to 95% AE energy before peak load in the vicinity of the initial crack tip. The experimental data illustrates the significant influence of test temperature on the behavior of the asphalt mixture. Comparison results showed that the size of the FPZ significantly depends on air voids and aggregate type, but is less depend on the asphalt content. It was found that at a very low temperature, different loading rates produced very close FPZ, both for the width and length. No obvious difference was observed on the width of the FPZ for the three different initial notch lengths, whereas the length of the FPZ was found significantly increases with the decrease of the notch length. The size of FPZ was also numerically estimated for one case with the cohesive zone model (CZM) calibrated by experimental data from the same SCB test. The FPZ size obtained with both methods agrees reasonably with each other.     

An experimental investigation on the FPZ properties in concrete using digital image correlation technique

This paper presents an experimental investigation on the properties of the fracture process zone (FPZ) in concrete using the digital image correlation (DIC) technique. Based on the experimental results, it is found that the FPZ length increases during crack propagation but decreases after the FPZ is fully developed. The FPZ length at the peak load and the maximum FPZ length increase with an increase in specimen height, but decrease by increasing the notch depth to specimen height ratio. It is also found that the crack extension length at the peak load is about 0.25 times the ligament length.     

Fracture Resistance of a Cracked Concrete Beam Post-strengthened with FRP Sheets

Fibre-reinforced plastic (FRP) composites have been increasingly used in rehabilitation and strengthening of concrete structures. Significant increases in stiffness and strength have been achieved by applying this technique. However, there is concern about the ductility or toughness performance of FRP/concrete hybrid structures, which is critical in the application of this technology. This paper presents a new theoretical method to predict the fracture resistance behaviour of FRP post-strengthened concrete flexural beams. No slip between the FRP and plain concrete matrix is assumed and Mode I fracture propagation is considered. The model is valid for a wide range of span-to-depth ratios and any crack length. The influence of the bridging stresses provided by the fracture process zone (FPZ) at the tip of a fictitious fracture is examined. The effect of various material and geometric parameters on the resistance curve and toughness of the hybrid structure is discussed, based on the numerical results from the developed theoretical formulae. The results provide a useful insight into the strengthening/toughening and the design of FRP sheet/concrete beam structures.     

Creep and fracture in concrete: a fractional order rate approach

The paper analyses the interaction between strain-softening and time-dependent behaviour in the case of quasi-static fracture of concrete. A viscous element based on a fractional order rate law is coupled with a micromechanical model for the fracture process zone. This approach makes it possible to include a whole range of dissipative mechanisms in a single rheological element. Creep fracture in mode I conditions is analysed through the finite element method, the cohesive (or fictitious) crack model and a new space and time integration scheme. The comparison with creep tests executed on three-point bending conditions shows a good agreement.     

Creep and fracture in concrete: a fractional order rate approach

The paper analyses the interaction between strain-softening and time-dependent behaviour in the case of quasi-static fracture of concrete. A viscous element based on a fractional order rate law is coupled with a micromechanical model for the fracture process zone. This approach makes it possible to include a whole range of dissipative mechanisms in a single rheological element. Creep fracture in mode I conditions is analysed through the finite element method, the cohesive (or fictitious) crack model and a new space and time integration scheme. The comparison with creep tests executed on three-point bending conditions shows a good agreement.     

Measurements and Calculations of the Width of the Fracture Process Zones on the Surface of Notched Concrete Beams

Abstract: The paper presents the results of experimental and theoretical investigations of the width of the fracture process zone (FPZ) on the surface of notched concrete beams during quasi‐static three‐point bending. To measure two‐dimensional deformations on the surface of beams, a Digital Image Correlation (DIC) technique was used. Laboratory tests were performed with different notched concrete beams. The experiments were simulated with two different isotropic continuum crack models under two‐dimensional conditions: an elasto‐plastic and a damage one with non‐local softening.     

Loading‐rate dependence of mode I crack growth in concrete

Mode I crack propagation process of concrete under relatively low loading rates which cover four orders of magnitude (0.2 μm/s to 2.0 mm/s) is investigated with three‐point bending (TPB) beams. All measured material properties exhibit rate sensitivity and follow a log‐linear relationship with the loading rate. A rate‐sensitive softening curve is established. The complete load‐crack mouth opening displacement (P‐CMOD) curve, crack propagation length, and fracture process zone (FPZ) length are simulated based on crack growth criterion with the fitted material parameters under those loading rates. Results show that the simulated P‐CMOD curves agree well with those of experimental measurements. It is clear that the peak load increases with the loading rate and so is the critical crack mouth opening displacement. Moreover, under the same load level, the length of the FPZ and the cohesive stress at the initial crack tip also increase with the increasing loading rate.     

Experimental Investigations of Fracture Process Using DIC in Plain and Reinforced Concrete Beams under Bending

The fracture behaviour of concrete and reinforced concrete beams under quasi‐static three‐point bending was comprehensively investigated with experiments at laboratory scale. The eight various concrete mixes were tested. The influence of the shape, volume and size of aggregate particles and reinforcement on concrete fracture under bending was studied. Displacements on the surface of concrete beams were measured by means of the digital image correlation (DIC) technique. Attention was paid to the formation of a localized zone and its characteristics. In order to avoid the effect of the search patch size and the cut‐off value at displacement and strain profiles, a consistent method was proposed to determine uniformly and accurately the width of a localized zone. Measured surface displacements from DIC were fitted by the error function ERF, whereas surface strains calculated from displacements were fitted by the usual normal distribution (Gauss) function. The width of a localized zone preceding a macro‐crack grew strongly with increasing maximum aggregate size and slightly with diminishing aggregate volume. It did not depend on the aggregate roughness and reinforcement presence.     

Efficient prediction of deterministic size effects using the scaled boundary finite element method

This paper develops an efficient numerical approach to predict deterministic size effects in structures made of quasi-brittle materials using the scaled boundary finite element method (SBFEM). Depending on the structure’s size, two different SBFEM-based crack propagation modelling methodologies are used for fracture analyses. When the length of the fracture process zone (FPZ) in a structure is of the order of its characteristic dimension, nonlinear fracture analyses are carried out using the finite element-SBFEM coupled method. In large-sized structures, a linear elastic fracture mechanics (LEFM)-based SBFEM is used to reduce computing time due to small crack propagation length required to represent the FPZ in an equivalent nonlinear analysis. Remeshing is used in both methods to model crack propagation with crack paths unknown a priori. The resulting peak loads are used to establish the size effect laws. Three concrete structures were modelled to validate the approach. The predicted size effect is in good agreement with experimental data. The developed approach was found more efficient than the finite element method, at least in modelling LEFM problems and is thus an attractive tool for predicting size effect.     

Fracture mechanic modeling of fiber reinforced polymer shear-strengthened reinforced concrete beam

A numerical method is developed to model shear-strengthening of reinforced concrete beam by using fiber reinforced polymer (FRP) composites. Tensile crack is simulated by a non-linear spring element with softening behavior ahead of the crack tip to model the cohesive zone in concrete. A truss element is used, parallel to the spring element, to simulate the energy dissipation rate by the FRP. The strain energy release rate is calculated directly by using a virtual crack closure technique. It is observed that the length of the fracture process zone (FPZ) increases with the application of FRP shear-strengthening. The present model shows that the main diagonal crack is formed at the support in the control beam while it appears through the shear span in the shear-strengthened beam. Another important observation is that the load capacity increases with the number of CFRP sheets in the shear span.     

II型III型动态裂纹尖端断裂过程区近似评估方法

动态裂纹尖端断裂过程区轮廓的确定问题仍然是一个没有得到完全解决的问题。基于弹性动力学的理论和复应力函数方法,提出一种伪应力函数方法,用于近似评估动态裂纹尖端应力场。通过与已知应力场计算结果对比,验证了伪应力函数的正确性。利用此近似方法通过Von Mises强度准则和Tresca强度准则,分别确定了不同强度准则条件下、不同裂纹扩展速度下断裂过程区的轮廓。计算结果表明:II型和III型动态裂纹尖端断裂过程区关于裂纹面对称分布,随着裂纹扩展速度增大而增大。当裂纹传播速度接近瑞利波速时,断裂过程区变化加剧。利用Tresca强度准则计算得到的动态裂纹尖端断裂过程区面积比利用Von Mises强度准则计算得到的断裂过程区的面积大。     

Extended structural criterion for numerical simulation of crack propagation and coalescence under compressive loads

An extension of the Neuber-Novozhilov structural fracture propagation criterion is presented for mode I (tensile) and mode II (shear) propagation under compressive loads. In addition to allowing numerical simulation of crack growth, the criterion can be used to model change of propagation mode, crack branching, and coalescence. The criterion can be applied effectively when the SIF is calculated accurately (at least three significant digits). A numerical method is suggested for this purpose that consists of complementing the complex variable hypersingular boundary element method (CVH-BEM) with special procedures for automatically tracing crack propagation and coalescence. The CVH-BEM code with the structural criterion has been used to investigate crack propagation in compression for both small and non-small fracture process zone (FPZ). The results of numerical experiments are in agreement with the analytical conclusions available for the case of small FPZ that indicates the possibility of three distinct patterns of crack propagation under external compressive loads. These are: (i) smooth curvilinear tensile (wing) cracks, (ii) stair-step propagation pattern with changing modes, and (iii) in plane shear propagation. The numerical study also indicates that when the critical size of the FPZ is large enough, the non-singular terms in the expansion of the stress functions strongly influence the crack trajectories. Specifically, this occurs when the size of the FPZ approaches a quarter of the half-length of the initial crack. Calculations for a closed initial crack in a half-space under compression illustrate the general features of crack propagation. Although the dominant direction of crack growth is that of the applied compressive stress, the pattern of propagation strongly depends on the particular geometry, critical size of the FPZ, and the ratio of shear-to-tensile microscopic strength.     

Size effect and quasi-brittle fracture: the role of FPZ

Fracture process zone (FPZ), or the crack-tip damage zone created by crack-bridging and micro-cracking activities, in a specimen of a concrete-like material is comparable to the crack size and un-cracked ligament, so fracture is typically quasi-brittle. Increasing or decreasing the specimen size, quasi-brittle fracture transition occurs towards the toughness-controlled or strength-controlled fracture, which is known as size effect (SE). In this study it is shown that the “size-dependent” quasi-brittle fracture transition is actually due to the interaction of FPZ with the nearest structure boundary rather than the size variation, and the widely-accepted SE for geometrically-similar specimens of different sizes is only a special case of quasi-brittle fracture controlled by the FPZ/boundary interaction. Relevant SE relations are critically reviewed and explained by emphasizing the key SE mechanism, FPZ/boundary interaction.     

Characterization of the nonlinear fracture resistance parameters for an aviation GTE turbine disc

The effects of crack growth rate model formulation, based on the elastic‐plastic and undamaged/damaged creep crack tip fields on the behaviour of low‐cycle fatigue and creep fracture resistance parameter behaviour, are represented by numerical calculations. The crack growth rate models include the fracture process zone size and damage parameters. An aviation gas turbine engine (GTE) rotating turbine disc is the focus of this innovative application of basic analytical and numerical solutions. For the GTE turbine disc, the constraint parameters, local fracture process zone sizes, and nonlinear plastic (K_p) and creep (K_cr) stress intensity factors are calculated by finite element analysis to characterize the fracture resistance along the semielliptical crack front as a function of the flaw aspect ratio, operation temperature, and disc rotation speed. Predictions of the creep‐fatigue crack growth rate and residual lifetime are given for different combinations of operation loading conditions and damage of the GTE turbine disc.     

A review of the J and I integrals and their implications for crack growth resistance and toughness in ductile fracture

The application of the J and the I-integrals to ductile fracture are discussed. It is shown that, because of the finite size of the fracture process zone (FPZ), the initiation value of the J-integral is specimen dependent even if the plastic constraint conditions are constant. The paradox that the I-integral for steady state elasto-plastic crack growth is apparently zero is examined. It is shown that, if the FPZ at the crack tip is modelled, the I-integral is equal to the work performed in its fracture. Thus it is essential to model the fracture process zone in ductile fracture. The I-integral is then used to demonstrate that the breakdown in applicability of the J-integral to crack growth in ductile fracture is as much due to the inclusion in the J-integral of progressively more work performed in the plastic zone as it is to non-proportional deformation during unloading behind the crack tip. Thus J _R -curves combine the essential work of fracture performed in the FPZ with the plastic work performed outside of the FPZ. These two work terms scale differently and produce size and geometry dependence. It is suggested that the future direction of modelling in ductile fracture should be to include the FPZ. Strides have already been made in this direction.     

Numerical and analytical investigation of the size-dependency of the FPZ length in concrete

The main objective of this paper is to study the size effect on the fracture characteristics in concrete structures. The numerical investigation is based on a mesoscale modeling approach. Analytically, two size effect laws are investigated: the classical Ba?ant SEL and a new size effect law based on the enrichment of the stress field on the crack tip. The mesoscopic approach is used to study the evolution of the tangential stress along the crack path in order to investigate the fracture process zone variation during the cracking process. In addition, different analytical governing equations are used to evaluate the size-dependency of the FPZ length.     

An experimental and theoretical comparison of CCNBD and CCNSCB specimens for determining mode I fracture toughness of rocks

The cracked chevron notched Brazilian disc (CCNBD) specimen, suggested by the International Society for Rock Mechanics for testing mode I fracture toughness of rocks, usually yields rather conservative toughness measurements, and the reasons have not been fully explored. In this study, the CCNBD method is compared with the cracked chevron notched semicircular bending (CCNSCB) method in the fracture process zone (FPZ) and its influence on the fracture toughness measurement. Theoretical analysis reveals that the FPZ is longer in the CCNBD specimen than in the CCNSCB specimen using a relatively large support span, the toughness measurement using the former is affected more seriously by the presence of FPZ, and thus the CCNBD method is usually, more or less, conservative compared with the CCNSCB method. These inferences are further validated by experimental results, which indicate that the CCNBD test indeed produces much lower fracture toughness values and even the results of 75‐mm radius CCNBD specimens are still lower than those of 25‐mm radius CCNSCB specimens. Consequently, due to smaller FPZ, the CCNSCB specimen with a relatively large span is more likely to produce comparably accurate or representative toughness value, and it may be more suitable than the CCNBD specimen for the engineering applications that require more representative or less conservative fracture toughness.