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.     

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.     

Age-dependent size effect and fracture characteristics of ultra-high performance concrete

This paper presents an investigation of the age-dependent size effect and fracture characteristics of ultra-high performance concrete (UHPC). The study is based on a unique set of experimental data connecting aging tests for two curing protocols of one size and size effect tests of one age. Both aging and size effect studies are performed on notched three-point bending tests. Experimental data are augmented by state-of-the-art simulations employing a recently developed discrete early-age computational framework. The framework is constructed by coupling a hygro-thermo-chemical (HTC) model and the Lattice Discrete Particle Model (LDPM) through a set of aging functions. The HTC component allows taking into account variable curing conditions and predicts the maturity of concrete. The mechanical component, LDPM, simulates the failure behavior of concrete at the length scale of major heterogeneities. After careful calibration and validation, the mesoscale HTC-LDPM model is uniquely posed to perform predictive simulations. The ultimate flexural strengths from experiments and simulations are analyzed by the cohesive size effect curves (CSEC) method, and the classical size effect law (SEL). The fracture energies obtained by LDPM, CSEC, SEL, and cohesive crack analyses are compared, and an aging formulation for fracture properties is proposed. Based on experiments, simulations, and size-effect analyses, the age-dependence of size effect and the robustness of analytical-size effect methods are evaluated.     

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.     

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。     

Fracture process zone in cementitious materials

The current work is directed to the measurement and prediction of the fracture process zone (FPZ) with the compliance method. The method is renovated with a multi-cutting technique. A new compliance curve C _p is established while FPZ in a damaged specimen is removed stepwise by saw-cutting. The length of FPZ can be well determined from C _p in comparison with the compliance calibration curve. Results of two different mortars evidence that the multi-cutting method is applicable to cementitious materials.A general theory is presented in conjunction with the multi-cutting experiment. The bridging stress transferred within FPZ is evaluated from C _p by the theory. It is proven with a numerical simulation that the strain-softening relation derived from C _p predicts well the global load and displacement relationship. The extension of FPZ at various stages of fracture can also be predicted with the theory if parameters of the strain-softening are available. Both experimental and analytical results affirm the intrinsic connection among FPZ, fracture toughness and fracture energy.     

Pore pressure cohesive zone modeling of hydraulic fracture in quasi-brittle rocks

Hydraulic fracturing technology has been widely applied in the petroleum industry for both waste injection and unconventional gas production wells. The prevailing analytical solutions for hydraulic fracture mainly depend on linear elastic fracture mechanics. These methods can give reasonable prediction for hard rock, but are ineffective in predicting hydraulic fractures in quasi-brittle materials, such as ductile shale and sandstone. One of the reasons is that the fracture process zone ahead of the crack tip and the softening effect should not be neglected for quasi-brittle materials. In the current work, a set of chevron-notch three point bending tests were performed on sandstone samples from an oil field in Ordos Basin, Shaanxi province, China, and the results were compared with the cohesive zone method based on finite element analysis. The numerical results fit the experimental data well and it shows that the cohesive zone model and the Traction-Separation law used in the model are effective in modeling fracture nucleation and propagation in sandstone without considering the porous effect. A 3D pore pressure cohesive zone model was developed to predict nucleation and propagation of a penny-shaped fluid-driven fracture. The predictions were compared with the analytical asymptotic solutions and a field minifrac test from the literature; it shows that the proposed method can not only predict the length and aperture of hydraulic fracture well, but also predict the bottomhole pressure with reasonable accuracy. Based on analytical asymptotic and computational solutions, parametric studies were conducted to investigate the effects of different parameters on the fracture aperture and fracture length, fracture process zone and bottomhole pressure.     

Determination of cohesive laws of composite bonded joints under mode II loading

In this work, mode II cohesive laws of carbon–epoxy composite bonded joints were obtained using the direct method applied to the end notched flexure (ENF) test. The direct method is based on the differentiation of the relation between the evolution of the fracture energy (J_II) and the crack tip opening displacement in mode II (CTOD_II) during the test. A data reduction scheme based on equivalent crack length concept was used to obtain the evolution of the fracture energy during the test. The method allows overcoming problems related to identification of crack tip in mode II tests and the presence of a non-negligible fracture process zone (FPZ), which both difficult the right estimate of J_II. The digital image correlation technique (DIC) was used to monitor the CTOD_II, which was synchronized with the load–displacement data. A trapezoidal cohesive law was fitted to the experimental one in order to perform numerical simulations using finite element analysis. The main goal was to validate all the procedure used to get the cohesive laws. The good agreement obtained between the numerical and experimental load-CTOD_II curves and between the cohesive laws demonstrates the adequacy of the proposed procedure concerning the evaluation of the composite bonded joints cohesive laws under mode II loading.     

Measuring toughness and the cohesive stress-displacement relationship by the essential work of fracture concept

The complete fracture behaviour of ductile double edge notched tension (DENT) specimen is analysed with an approximate model, which is then used to discuss the essential work of fracture (EWF) concept. The model results are compared with the experimental results for an aluminium alloy 6082-O. The restrictions on the ligament size for valid application of the EWF method are discussed with the aid of the model. The model is used to suggest an improved method of obtaining the cohesive stress-displacement relationship for the fracture process zone (FPZ).     

Weight function approach for determining crack extension resistance based on the cohesive stress distribution in concrete

The use of universal form of weight functions for determining the K_R-curves associated with the cohesive stress distribution for complete fracture process of three-point bending notched concrete beam is reported in the paper. Closed form expressions for the cohesion toughness with linear and bilinear distribution of cohesive stress in the fictitious fracture zone during monotonic loading of structures are obtained. Comparison with existing analytical method shows that the weight function method yields results without any appreciable error with improved computational efficiency. The stability analysis and the size-effect study using K_R-curves of crack propagation are also described.     

Continuum coupled cohesive zone elements for analysis of fracture in solid bodies

Embedding cohesive surfaces into finite element models is a widely used technique for the numerical simulation of material separation (i.e. crack propagation). Typically, a traction-separation law is specified that relates the magnitude of the cohesive traction to the distance between the separating surfaces. Thus the characterization of fracture in such models is not directly coupled to the bulk constitutive response, in the sense that the cohesive traction does not explicitly depend on material stretching in the plane of the fracture surface. In this work, an initially-rigid cohesive-traction formulation that is coupled to the surrounding continuum is introduced as a further development of the cohesive zone idea. In this model, the traction-separation law - and therefore the fracture phenomenology - derives directly from the bulk constitutive law. The immediate goal is an improved cohesive zone framework that naturally and logically initiates cohesive separation behavior, and couples its evolution to the material state in the region of the crack tip. A cohesive element based on this model is implemented in an explicit three-dimensional finite element code. Proof-of-concept analyses using both linear elastic and Gurson void growth constitutive relations are presented. A three-point bend simulation is found to give good agreement with experimental results.     

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.     

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.     

Cohesive Crack with Rate-Dependent Opening and Viscoelasticity: I. Mathematical Model and Scaling

The time dependence of fracture has two sources: (1) the viscoelasticity of material behavior in the bulk of the structure, and (2) the rate process of the breakage of bonds in the fracture process zone which causes the softening law for the crack opening to be rate-dependent. The objective of this study is to clarify the differences between these two influences and their role in the size effect on the nominal strength of stucture. Previously developed theories of time-dependent cohesive crack growth in a viscoelastic material with or without aging are extended to a general compliance formulation of the cohesive crack model applicable to structures such as concrete structures, in which the fracture process zone (cohesive zone) is large, i.e., cannot be neglected in comparison to the structure dimensions. To deal with a large process zone interacting with the structure boundaries, a boundary integral formulation of the cohesive crack model in terms of the compliance functions for loads applied anywhere on the crack surfaces is introduced. Since an unopened cohesive crack (crack of zero width) transmits stresses and is equivalent to no crack at all, it is assumed that at the outset there exists such a crack, extending along the entire future crack path (which must be known). Thus it is unnecessary to deal mathematically with a moving crack tip, which keeps the formulation simple because the compliance functions for the surface points of such an imagined preexisting unopened crack do not change as the actual front of the opened part of the cohesive crack advances. First the compliance formulation of the cohesive crack model is generalized for aging viscoelastic material behavior, using the elastic-viscoelastic analog (correspondence principle). The formulation is then enriched by a rate-dependent softening law based on the activation energy theory for the rate process of bond ruptures on the atomic level, which was recently proposed and validated for concrete but is also applicable to polymers, rocks and ceramics, and can be applied to ice if the nonlinear creep of ice is approximated by linear viscoelasticity. Some implications for the characteristic length, scaling and size effect are also discussed. The problems of numerical algorithm, size effect, roles of the different sources of time dependence and rate effect, and experimental verification are left for a subsequent companion paper. This revised version was published online in July 2006 with corrections to the Cover Date.     

Ductile Fracture Study of Stainless Steel AISI 304L Thin Sheets Using the EWF Method and Cohesive Zone Modeling

The purpose of this study is to model the ductile fracture phenomenon using experimental and numerical methods. The stainless steel AISI 304L thin sheets are studied including two thicknesses (0.8 and 1.5 mm). A mechanical characterization is firstly done in order to obtain the main mechanical properties useful for the numerical modeling. In order to determine the essential work of fracture (EWF), DENT specimens are used involving the two thicknesses. The results obtained in terms of EWF for the two thicknesses are close; we find that the essential work of fracture can be considered as an intrinsic criterion for thin sheets. A cohesive zone modeling (CZM) is used in the present study; the model is represented by a traction–separation law (TS). The cohesive elements are implemented in the finite element model, and the material parameters of the model are determined by the mechanical and fracture characterizations. A satisfactory reproduction of the experimental tests is obtained. A good correlation is also obtained between the essential work of fracture determined experimentally and the work of separation used as cohesive zone model parameter.     

An extended finite element method with analytical enrichment for cohesive crack modeling

A recent approach to fracture modeling has combined the extended finite element method (XFEM) with cohesive zone models. Most studies have used simplified enrichment functions to represent the strong discontinuity but have lacked an analytical basis to represent the displacement gradients in the vicinity of the cohesive crack. In this study enrichment functions based upon an existing analytical investigation of the cohesive crack problem are proposed. These functions have the potential of representing displacement gradients in the vicinity of the cohesive crack and allow the crack to incrementally advance across each element. Key aspects of the corresponding numerical formulation and enrichment functions are discussed. A parameter study for a simple mode I model problem is presented to evaluate if quasi‐static crack propagation can be accurately followed with the proposed formulation. The effects of mesh refinement and mesh orientation are considered. Propagation of the cohesive zone tip and crack tip, time variation of the cohesive zone length, and crack profiles are examined. The analysis results indicate that the analytically based enrichment functions can accurately track the cohesive crack propagation of a mode I crack independent of mesh orientation. A mixed mode example further demonstrates the potential of the formulation. Copyright © 2008 John Wiley & Sons, Ltd.     

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.     

A two-way coupled multiscale model for predicting damage-associated performance of asphaltic roadways

This paper presents a quasi-static multiscale computational model with its verification and rational applications to mechanical behavior predictions of asphaltic roadways that are subject to viscoelastic deformation and fracture damage. The multiscale model is based on continuum thermo-mechanics and is implemented using a finite element formulation. Two length scales (global and local) are two-way coupled in the model framework by linking a homogenized global scale to a heterogeneous local scale representative volume element. With the unique multiscaling and the use of the finite element technique, it is possible to take into account the effect of material heterogeneity, viscoelasticity, and anisotropic damage accumulation in the small scale on the overall performance of larger scale structures. Along with the theoretical model formulation, two example problems are shown: one to verify the model and its computational benefits through comparisons with analytical solutions and single-scale simulation results, and the other to demonstrate the applicability of the approach to model general roadway structures where material viscoelasticity and cohesive zone fracture are involved.     

Bone fracture characterization using the end notched flexure test

A miniaturized version of the end notch flexure test was used in the context of pure mode II fracture characterization of bovine cortical bone. To overcome the difficulties intrinsic to crack length monitoring during its propagation an equivalent crack method was employed as data reduction scheme. The proposed method was validated numerically by means of a finite element analysis including a cohesive zone modeling and subsequently applied to experimental results to determine the fracture energy of bone under pure mode II loading. Finally, a cohesive law representative of fracture behavior of each specimen was determined employing an inverse method, considering a trapezoidal shape for the softening law. The consistency of the obtained results leads to the conclusion that the trapezoidal law is adequate to simulate fracture behavior of bone under mode II loading. The proposed testing setup and the employed data reduction scheme constitute powerful tools in which concerns fracture characterization of bone under pure mode II loading and can be viewed as the main outcomes of this work.     

A level set model for simulating fatigue-driven delamination in composites

This paper proposes a level set model for simulating delamination propagation in composites under high-cycle fatigue loading. For quasi-static loading conditions, interface elements with a cohesive law are widely used for the simulation of delamination. However, basic concepts from fatigue analysis such as the notion that the crack growth rate is a function of energy release rate cannot be embedded in existing cohesive laws. Therefore, we propose a model in which the cohesive zone is eliminated from the computation while maintaining the flexibility that the crack shape is not bound to element edges. The model is able to predict the delamination growth rate and its front shape accurately. To demonstrate the validity of the model, several tests under different fracture modes are conducted and the results are compared with experimental data, analytical solutions and results from cohesive zone analysis.