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.     

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.     

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。     

Fully-developed FPZ length in quasi-brittle materials

This paper studies the development of fracture processes in quasi-brittle materials. We propose to use the length of the fracture process zone (FPZ) once it is fully developed as a material parameter. This assumption allows us to build an analytical formulation that reproduces the mechanical behavior of any specimen as a cohesive crack advances. Extensive comparisons with experimental results lead us to define a new characteristic length that commensurates with the fully-developed FPZ and that together with the new analytical model, is used to provide a complete and consistent study on the fracture process. In particular, the size-effect deriving from our formulation coincides with the statistical size-effect law of Ba?ant for small and medium sizes, whereas it smoothly converges to size-independent results as size increases. The analytical cohesive formulation developed here is validated against experimental results on various types of normal and high-strength concretes as well as construction ceramics for several experimental set-ups and test scales. Because of its simplicity as compared with numerical models for fracture, this analytical formulation constitutes a powerful tool for studying fracture processes in a wide variety of mechanical configurations. Meanwhile, analytical expression for a fully-developed FPZ length is given for a general type of cohesive law.     

基于定长裂缝试件的脆性材料尺寸效应实验方法

由于脆性或准脆性材料内各类微缺陷的影响,材料的力学性能,如名义破坏应力,　刚度以及断裂韧性等随试件的大小而改变,具有明显的尺寸效应。通常情况下,描述材料尺寸效应的Ｂａｚａｎｔ尺寸效应律是建立在一系列相似试件的基础上通过实验方法确定的。　本文提出了一种新的用含固定长度裂缝试件测定断裂韧性和有效断裂过程区大小的实验方法和计算公式。与相似试件测定方法相比,实验结果吻合很好。根据本文提出的定长裂缝试件实验方法,在保证与相似试件相同脆性指数范围的前提下,可以用小试件进行测量。     

A data reduction method based on the J-integral to obtain the interlaminar fracture toughness in a mode II end-loaded split (ELS) test

Various difficulties arise in the data reduction of the end-loaded split (ELS) test. On one hand, a small Fracture Process Zone (FPZ) at the crack front is assumed in the existing mode II end-loaded split test methodologies based on Linear Elastic Fracture Mechanics (LEFM). However, mode II fracture has been reported to involve large FPZ and a fuzzy crack tip. Furthermore, the ELS test, is usually affected by geometrical non-linearities.This work proposes a closed-form solution based on the J-integral to determine the interlaminar fracture toughness in an ELS test. This solution avoids the need to measure the crack length, and is applicable when a large FPZ is present, as occurs in adhesive bonded joints between CFRP. In addition, because the ELS test involves large vertical deflections, a correction of the formulation for large displacements has been implemented.This new methodology has been compared to other methods available in the literature based on LEFM by means of an experimental campaign of delamination tests using unidirectional CFRP specimens in order to make a first validation of the method.     

Size effect of concrete members applied with flexural compressive stresses

In this study, two types of special experiments are carried out to understand flexural compressive strength size effect of concrete members. The first type is an ordinary cylindrical specimen (CS) with a fully penetrated and vertically standing plate type notch at the mid-height of the specimen, which is loaded in compression at the top surface (e.g., in the parallel direction to the notch length). The second type is a general double cantilever beam (DCB), which is compression loaded in axial direction (e.g., in the parallel direction of the notch). For CS, an adequate notch length is taken from the experimental results obtained from the compressive strength experiment of various initial notch lengths. The trial tests to select the effective initial notch length show that CS with an initial notch length approximately greater than four times the maximum aggregate size fails without an additional increased load and in stable manner under Mode I failure mechanism. Therefore, the initial notch length to the maximum aggregate size ratio of 4.0 is used for all size specimens. For DCB, the eccentricity of loading points with respect to the axial axis of each cantilever and the initial notch length are varied. In both specimens, the compressive loads apply flexural compressive stresses on the crack tip region of the specimens. These two types of specimens fail by Mode I crack opening mechanism. By testing 3 geometrically proportional size specimens for CS and DCB, the experimental datum for flexural compression size effect of concrete are obtained. Using the obtained flexural compressive strength size effect datum, regression analyses are performed using Levenberg-Marquardt's least square method (LSM) to suggest new parameters for the modified size effect law (MSEL). The analysis results show that size effect is apparent for flexural compressive strength of specimens with an initial notch. For CS, the effect of initial notch length on flexural compressive strength size effect is apparent. For DCB, flexural compressive size effect is dependent on the eccentricity of loading points with respect to the axial axis of the cantilever beam. In other words, if DCB specimen is applied with greater tensile stress at the crack tip, the size effect of concrete becomes more distinct. The results show that the flexural compressive strength size effect of initial notch length variation of DCB exists but directly dependent on the loading location. This is due to the fact that the sizes of fracture process zone (FPZ) of all DCB specimens are similar regardless of the differences in the specimen slenderness ratio, but the flexural compressive and tensile stress combinations resulting in stress concentration at the crack tip region has direct effect on size effect of concrete members.     

考虑骨料尺寸的混凝土岩石边界效应断裂模型

该文比较了边界效应模型（BEM）和尺寸效应模型（SEM）在研究材料断裂性能方面的不同。提出了由处于准脆性断裂状态的三点弯曲试件的峰值荷载P_max,同时确定材料参数--断裂韧度K_IC与拉伸强度f_t的理论与方法。由于实验室条件下混凝土试件高度W与骨料最大粒径d_max的比例W/d_max约为5~20,试件的非均质性明显,破坏为准脆性断裂控制。因此,区别于以连续介质力学为基础的应用于准脆性断裂研究的力学模型,该文研究将骨料最大粒径d_max引入相应的断裂模型解析表达式中,由参数组合β d_max来计算结构峰值状态对应的裂缝扩展量,通过离散参数β的不同取值,实现了对材料参数--断裂韧度与拉伸强度的准确预测。基于不同学者的相同尺寸W而不同初始裂缝长度a₀,以及相同初始缝高比a₀/W而不同尺寸W的几何相似的砂浆、混凝土及岩石类材料试件的试验成果（骨料最大粒径d_max从1.2 mm~40 mm变化）,验证了所提理论与方法的合理性。     

Estimation of width of fracture process zone in spruce wood by radial tensile test

Fracture behavior of spruce wood under radial tension was analyzed using nonlinear fracture mechanics as wood is classified as a quasi-brittle material. Stress-strain relationship with a strain-softening branch was obtained by digital image analysis and stress redistribution process, and the energy release rate of serial end-matched specimens was measured by performing a single-edge-notched tension test.The width of the fracture process zone (FPZ) was estimated by comparing two kinds of fracture energies. One was the dispersion energy per unit area to model strain localizations using a discontinuum model of damage theory, by integrating the stress-strain function with the strain-softening branch. The other was the energy release rate to determine crack growth. From this analysis, we determined that the width of the FPZ ranged from 0.3 mm to 0.5 mm in the radial direction. However, for a few specimens, the approximate stress-strain function could not be fitted into the stress-strain relationship obtained by the image analysis; it was observed that the fracture planes of these specimens tended to be more or less inclined.     

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.     

Brittle fracture in rounded-tip V-shaped notches

Two failure criteria are proposed in this paper for brittle fracture in rounded-tip V-shaped notches under pure mode I loading. One of these criteria is developed based on the mean stress criterion and the other based on the point stress criterion which both are well known failure criteria for investigating brittle fracture in elements containing a sharp crack or a sharp V-notch. To verify the validity of the proposed criteria, first the experimental data reported by other authors from three-point bend (TPB) and four-point bend (FPB) tests on PMMA at −60 °C and Alumina–7% Zirconia ceramic are used. Additionally, some new fracture tests are also carried out on the rounded-tip V-notched semi-circular bend (RV-SCB) specimens made of PMMA for various notch opening angles and different notch tip radii. A very good agreement is shown to exist between the results of the mean stress criterion and the experimental data.     

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.     

Cohesive crack model description of ductile to brittle size-scale transition: dimensional analysis vs. renormalization group theory

The present paper is a review of the research works carried out on the cohesive crack model and its applications at the Politecnico di Torino during the last decade. The topic encompasses experimental, numerical and theoretical aspects of the cohesive crack model. The research work followed two main directions. The early work concerns the development and the implementation of the cohesive crack model, which has been shown to be able to simulate experiments on concrete specimens and structures. It is referred to as the dimensional analysis approach, since it succeeds in capturing the ductile-to-brittle transition by increasing the structural size owing to the different physical dimensions of two material parameters: the tensile strength and the fracture energy.On the other hand, the later research direction aims at extending the classical cohesive model to quasi-brittle materials showing (as they often do) fractal patterns in the failure process. This approach is referred to as the renormalization group (or fractal) approach and leads to a scale-invariant cohesive crack model. This model is able to predict the size effects even in tests where the classical approach fails, e.g. the direct tension test.The two research paths, therefore, complete each other, allowing a deeper insight into the ductile-to-brittle transition usually detected when testing quasi-brittle material specimens or structures at different size-scales.     

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.     

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.     

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。     

Probability distribution of energetic-statistical size effect in quasibrittle fracture

The physical sources of randomness in quasibrittle fracture described by the cohesive crack model are discussed and theoretical arguments for the basic form of the probability distribution are presented. The probability distribution of the size effect on the nominal strength of structures made of heterogeneous quasibrittle materials is derived, under certain simplifying assumptions, from the nonlocal generalization of Weibull theory. Attention is limited to structures of positive geometry failing at the initiation of macroscopic crack growth from a zone of distributed cracking. It is shown that, for small structures, which do not dwarf the fracture process zone (FPZ), the mean size effect is deterministic, agreeing with the energetic size effect theory, which describes the size effect due to stress redistribution and the associated energy release caused by finite size of the FPZ formed before failure. Material randomness governs the statistical distribution of the nominal strength of structure and, for very large structure sizes, also the mean. The large-size and small-size asymptotic properties of size effect are determined, and the reasons for the existence of intermediate asymptotics are pointed out. Asymptotic matching is then used to obtain an approximate closed-form analytical expression for the probability distribution of failure load for any structure size. For large sizes, the probability distribution converges to the Weibull distribution for the weakest link model, and for small sizes, it converges to the Gaussian distribution justified by Daniels' fiber bundle model. Comparisons with experimental data on the size-dependence of the modulus of rupture of concrete and laminates are shown. Monte Carlo simulations with finite elements are the subject of ongoing studies by Pang at Northwestern University to be reported later.     

Effects of finite notch width on the fracture of chevron – notched specimens

The chevron notched three-point bend test specimen is often used for measuring the fracture toughness of brittle materials such as ceramics. Specimen sizes are often very restricted when testing advanced materials due to limited volume of material available or high material costs. Since the minimum chevron notch width is limited by the size of the cutting wheels or wire saw used to produce it, as the sample size gets small enough, the notch width becomes large in relation to the sample size. It is shown via finite element analysis that the notch width has an important effect on the stress intensity factors of short cracks. The minimum in the normalized stress intensity factor versus crack length is lost, rendering the usual analysis of the experimental results invalid and contributing greatly to decreased fracture stability of such specimens. Previous analytical and numerical studies do not take into account the width of the chevron notch. Based on the calculations, a guideline to permissible notch widths is introduced. This revised version was published online in July 2006 with corrections to the Cover Date.     

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.     

Examination of the failure behaviour of wood with a short crack in the tangential-radial system by single-edge-notched bending test

Single-edge-notched tests of a tangential-radial system were conducted on agathis specimens to analyze the failure behaviour of wood with a short crack. The nominal bending strength and mode I critical stress intensity factors of the specimens with various crack lengths were measured, and the influence of the crack length on these properties was examined. The nominal bending strength of the cracked specimens was significantly lower than that of a crack-free specimen, even when the crack was extremely short. This finding suggests that the fracture mechanics theory is essential for analyzing the failure behaviour of wood with a very short crack. However, the mode I critical stress intensity factor still depended on the crack length. When considering the fracture process zone developing at the crack tip, the critical intensity factor could be predicted effectively.