Inverse Estimation of Cohesive Fracture Properties of Asphalt Mixtures Using an Optimization Approach

Tensile cracking in asphalt pavements due to vehicular and thermal loads has become an experimental and numerical research focus in the asphalt materials community. Previous studies have used the discrete element method (DEM) to study asphalt concrete fracture. These studies used trial-and-error to obtain local fracture properties such that the DEM models approximate the experimental load-crack mouth opening displacement response. In the current study, we identify the cohesive fracture properties of asphalt mixtures via a nonlinear optimization method. The method encompasses a comparative investigation of displacement fields obtained using both digital image correlation (DIC) and heterogeneous DEM fracture simulations. The proposed method is applied to two standard fracture test geometries: the single-edge notched beam test, SE(B), under three-point bending, and the disk-shaped compact tension test, DC(T). For each test, the Subset Splitting DIC algorithm is used to determine the displacement field in a predefined region near the notch tip. Then, a given number of DEM simulations are performed on the same specimen. The DEM is used to simulate the fracture of asphalt concrete with a linear softening cohesive contact model, where fracture-related properties (e.g., maximum tensile force and maximum crack opening) are varied within a predefined range. The difference between DIC and DEM displacement fields for each set of fracture parameters is then computed and converted to a continuous function via multivariate Lagrange interpolation. Finally, we use a Newton-like optimization technique to minimize Lagrange multinomials, yielding a set of fracture parameters that minimizes the difference between the DEM and DIC displacement fields. The optimized set of fracture parameters from this nonlinear optimization procedure led to DEM results which are consistent with the experimental results for both SE(B) and DC(T) geometries.     

Using Semi Circular Bending Test to Evaluate Low Temperature Fracture Resistance for Asphalt Concrete

This work presents a repeatable semi circular bending (SCB) fracture test to evaluate the low temperature fracture resistance of asphalt mixture. The fracture resistance of six asphalt mixtures, which represent a combination of factors such as binder type, binder modifier, aggregate type, and air voids, and two testing conditions of loading rate and initial notch length, was evaluated by performing SCB fracture tests at three low temperatures. Fracture energy was calculated from the experimental data. Experimental results indicated strong dependence of the low temperature fracture resistance on the test temperature. Experimental plots and low coefficient of variation (COV) values from three replicates show a satisfactory repeatability from the test. The results of the analysis showed that fracture resistance of asphalt mixtures is significantly affected by type of aggregate and air void content. Experimental results also confirmed the significance of binder grade and modifier type with relation to cracking resistance of asphalt mixtures. Analysis of result also indicated that both the loading rate and initial notch length had significant effect on the fracture energy at the highest test temperature, whereas the effect was strongly diluted at the two lower temperatures. No clear trend was found with the fracture peak load from either the effect of loading rate or notch length.     

Analytical representation of the non-square-root singular stress field at a finite angle sharp notch

The stress field near the tip of a finite angle sharp notch is singular. However, unlike a crack, the order of the singularity at the notch tip is less than one-half. Under tensile loading, such a singularity is characterized by a generalized stress intensity factor which is analogous to the mode I stress intensity factor used in fracture mechanics, but which has order less than one-half. By using a cohesive zone model for a notional crack emanating from the notch tip, we relate the critical value of the generalized stress intensity factor to the fracture toughness. The results show that this relation depends not only on the notch angle, but also on the maximum stress of the cohesive zone model. As expected the dependence on that maximum stress vanishes as the notch angle approaches zero. The results of this analysis compare very well with a numerical (finite element) analysis in the literature. For mixed-mode loading the limits of applicability of using a mode I failure criterion are explored.     

Discrete fracture modeling of asphalt concrete

A heterogeneous fracture approach is presented for modeling asphalt concrete that is composed of solid inclusions and a viscous matrix, and is subjected to mode-I loading in the fracture test configuration. A heterogeneous fracture model, based on the discrete element method (DEM), is developed to investigate various fracture toughening mechanisms of asphalt materials using a high-resolution image processing technique. An energy-based bilinear cohesive zone model is used to model the crack initiation and propagation of materials, and is implemented as a user-defined model within the discrete element method. Experimental fracture tests are performed to investigate various fracture behavior of asphalt concrete and obtain material input parameters for numerical models. Also, bulk material properties are necessary for each material phase for heterogeneous numerical models; these properties are determined by uniaxial complex modulus tests and indirect tensile strength tests. The main objective of this study is to integrate the experimental tests and numerical models in order to better understand the fracture mechanisms of asphaltic heterogeneous materials. Experimental results and numerical simulations are compared at different test conditions with excellent agreement. The heterogeneous DEM fracture modeling approach has the potential capability to understand various crack mechanisms of quasi-brittle materials.     

Direct Extraction of Cohesive Fracture Properties from Digital Image Correlation: A Hybrid Inverse Technique

The accuracy of an adopted cohesive zone model (CZM) can affect the simulated fracture response significantly. The CZM has been usually obtained using global experimental response, e.g., load versus either crack opening displacement or load-line displacement. Apparently, deduction of a local material property from a global response does not provide full confidence of the adopted model. The difficulties are: (1) fundamentally, stress cannot be measured directly and the cohesive stress distribution is non-uniform; (2) accurate measurement of the full crack profile (crack opening displacement at every point) is experimentally difficult to obtain. An attractive feature of digital image correlation (DIC) is that it allows relatively accurate measurement of the whole displacement field on a flat surface. It has been utilized to measure the mode I traction-separation relation. A hybrid inverse method based on combined use of DIC and finite element method is used in this study to compute the cohesive properties of a ductile adhesive, Devcon Plastic Welder II, and a quasi-brittle plastic, G-10/FR4 Garolite. Fracture tests were conducted on single edge-notched beam specimens (SENB) under four-point bending. A full-field DIC algorithm was employed to compute the smooth and continuous displacement field, which is then used as input to a finite element model for inverse analysis through an optimization procedure. The unknown CZM is constructed using a flexible B-spline without any “a priori” assumption on the shape. The inversely computed CZMs for both materials yield consistent results. Finally, the computed CZMs are verified through fracture simulation, which shows good experimental agreement.     

内聚力模型裂纹问题分析的解析奇异单元

内聚力模型已经被广泛应用于需要考虑断裂过程区的裂纹问题当中,然而常用的数值方法应用于分析内聚力模型裂纹问题时还存在着一些不足,比如不能准确的给出断裂过程区的长度、需要网格加密等。为了克服这些缺点,论文构造了一个新型的解析奇异单元,并将之应用于基于内聚力模型的裂纹分析当中。首先将虚拟裂纹表面处的内聚力用拉格拉日插值的方法近似表示为多项式的形式,而多项式表示的内聚力所对应的特解可以被解析地给出。然后利用一个简单的迭代分析,基于内聚力模型的裂纹问题就可以被模拟出来了。最后,给出二个数值算例来证明本文方法的有效性。     

Determination of Creep Compliance of Asphalt Concrete from Notched Semi-Circular Bend (SCB) Test

The feasibility of determining the creep compliance of asphalt concrete from a notched Semi-Circular Bend (SCB) test specimen was investigated. The objective of this study was to propose a combined test methodology that can provide both viscoelastic and fracture properties of asphalt concrete mixtures tested at low temperatures. Finite element (FE) analyses were performed to understand the stress state in the SCB, and an optimal load range producing appreciable displacement measurements while preserving the linear viscoelastic conditions was identified. Expressions that relate displacement measurements, from particular regions of the SCB specimen, to creep function were derived. The validity of the proposed SCB creep method was tested both by numerical simulations and experimental testing. Good agreement was found between the creep function obtained from SCB and those obtained from the Three-Point Bending Beam (3PBB) and the Indirect Tensile (IDT) creep test.     

环形含内钝缺口试件在斜加载作用下弹塑性场的分布特征

运用有限元数值计算方法,对含内缺口的环形试件在受到与缺口对称面成不同角度的载荷作用下,缺口前端附近的弹塑性场分布特征进行了研究,得到了缺口前端应力和应变场以及塑性区的变化情况,此外,选取缺口弧度不同的模型,探讨了缺口顶端钝化程度对弹生场的影响,为了对计算结果的可靠性进行评价,运用高精度云纹干涉实验力学方法进行了验证性试验,试验得到的位移,应变分布情况与计算结果基本吻合。文中得到的结果为钝口试件的材     

Effects of adhesive thickness on global and local Mode-I interfacial fracture of bonded joints

The interfacial fracture of adhesively bonded structures is a critical issue for the extensive applications to a variety of modern industries. In the recent two decades, cohesive zone models (CZMs) have been receiving intensive attentions for fracture problems of adhesively bonded joints. Numerous global tests have been conducted to measure the interfacial toughness of adhesive joints. Limited local tests have also been conducted to determine the interface traction-separation laws in adhesive joints. However, very few studies focused on the local test of effects of adhesive thickness on the interfacial traction-separation laws. Interfacial toughness and interfacial strength, as two critical parameters in an interfacial traction-separation law, have important effect on the fracture behaviors of bonded joints. In this work, the global and local tests are employed to investigate the effect of adhesive thickness on interfacial energy release rate, interfacial strength, and shapes of the interfacial traction-separation laws. Basically, the measured laws in this work reflect the equivalent and lumped interfacial fracture behaviors which include the cohesive fracture, damage and plasticity. The experimentally determined interfacial traction-separation laws may provide valuable baseline data for the parameter calibrations in numerical models. The current experimental results may also facilitate the understanding of adhesive thickness-dependent interface fracture of bonded joints.     

On the determination of the cohesive zone properties of an adhesive layer from the analysis of the wedge-peel test

An extensive numerical study of the mechanics of the “wedge-peel test” is performed in order to analyze the mode I steady state debonding of a sandwich structure made of two thin plastically deforming metallic plates bonded with an adhesive. The constitutive response of the metallic plates is modeled by J₂ flow theory, and the behavior of the adhesive layer is represented with a cohesive zone model characterized by a maximum separation stress and the fracture energy. A steady-state finite element code accounting for finite rotation has been developed for the analysis of this problem. Calculations performed with the steady-state formulation are shown to be much faster than simulations involving both crack initiation and propagation within a standard, non-steady-state code. The goal of this study is to relate the measurable parameters of the test to the corresponding fracture process zone characteristics for a representative range of adherent properties and test conditions. An improved beam bending model for the energy release rate is assessed by comparison with the numerical results. Two procedures are proposed for identifying the cohesive zone parameters from experimental measurements.     

Theoretical development and experimental validation of a thermally dissipative cohesive zone model for dynamic fracture of amorphous polymers

A thermally dissipative cohesive zone model is developed for predicting the temperature increase at the tip of a crack propagating dynamically in a nominally brittle material exhibiting a cohesive-type failure such as crazing. The model assumes that fracture energy supplied to the crack tip region that is in excess of that needed for the creation of new free surfaces during crack advance is converted to heat within the cohesive zone. Bulk dissipation mechanisms, such as plasticity, are not accounted for. Several cohesive traction laws are examined, and the model is then used to make predictions of crack tip heating at various crack propagation speeds in the nominally brittle amorphous polymer PMMA, observed to fail by a crazing-type mechanism. The heating predictions are compared to experimental data where the temperature field surrounding a high speed crack in PMMA was measured. Measurements are made in real time using a multi-point high speed HgCdTe infrared radiation detector array. At the same time as temperature, simultaneous measurement of fracture energy is made by a strain gauge technique, and crack tip speed is monitored through a resistance ladder method. Material strength can be estimated through uniaxial tension tests, thus minimizing the need for parameter fitting in the stress-opening traction law. Excellent agreement between experiments and theory is found for two of the cohesive traction law temperature predictions, but only for the case where a single craze is active during the dynamic fracture of PMMA, i.e. crack tip speed up to approximately 0.2c_R. For higher speed fracture where subsurface damage becomes prominent, the line dissipation model of a cohesive zone is inadequate, and a distributed damage model is needed.     

An isolated cohesive crack in tension

The shape of the back-calculated stress-separation curves obtained from the in-situ fracture of first-year sea ice in the Arctic and Antarctic is convex, and radically different from those for concrete. In these tests, the process zone size changes with crack growth, but the nature of this change differs with the test conditions, load-control or displacement control. These results prompted a closer examination of the cohesive crack model using the simplest cracked configuration, a finite cohesive crack in an infinite elastic medium loaded in tension by a uniform stress at infinity. Different types of strain softening are examined: rectangular softening, linear softening, prescribed cohesive stresses, and prescribed cohesive crack-opening displacements. For each of these cases, crack nucleation is examined; close attention is paid to test control conditions, be they load-control or fixed-grip. The test control conditions alter the fracture, crack nucleation, crack growth, and process zone size behavior significantly. Accurate approximate solutions to linear softening are presented and examined.     

Damage Evolution and Crack Propagation in Semicircular Bending Asphalt Mixture Specimens

The damage and fracture behaviors of semicircular bending(SCB) asphalt mixture specimens with different orientation notches are experimentally and numerically investigated. In the numerical simulations, asphalt mixture is modeled as a two-phase material, namely a mix of coarse aggregates and asphalt mastic, and the mechanical behavior of asphalt mastic is characterized with the damage constitutive model and the damage-based fracture criterion. Some SCB experiments are performed on the asphalt mixture specimens with different orientation notches to validate the numerical method. Finally, the effects of notch orientation and aggregate distribution on crack path, damage distribution, and the load vs.displacement relation are numerically evaluated.     

金属材料层裂破坏的内聚力模型

本文把内聚力单元嵌入到连续介质有限元单元之间,构建了一个层裂破坏的内聚力模型,以计及层裂破坏过程中能量耗散行为。采用该模型对平板撞击条件下的20号钢层裂实验进行了数值模拟研究,重点讨论了内聚力模型特征参数对计算结果的影响规律。研究结果表明采用指数型损伤演化行为的内聚力模型可以较好地描述弹塑性材料层裂破坏过程中的非线性能量耗散行为。利用一发实测自由面速度波剖面对计算结果进行对比校准,可确定内聚力模型特征参数。该特征参数可同样成功地预示不同撞击速度下的层裂实验,获得的模拟曲线与实验曲线之间符合程度很好,特别是自由面速度“回跳”后波形振荡周期和幅值与实验结果非常接近。这表明了内聚力模型在描述层裂过程中能量耗散行为方面具有较好适用性,并且不难由简单实验标定相关的特征参数。     

Ductile–brittle transitions in the fracture of plastically deforming,adhesively bonded structures. Part II: Numerical studies

To enable the effective and reliable use of structural adhesive bonding in automotive applications, the cohesive properties of a joint need to be determined over a wide range of loading rates. In this paper, a strategy for determining these properties has been described and used to analyze a set of experimental results presented in a companion paper. In the particular system studied, a crack growing in a toughened quasi-static mode could make a catastrophic transition to a brittle mode of fracture. The cohesive parameters for both the toughened and brittle modes of crack growth were determined by comparing numerical predictions from cohesive-zone simulations to the results of experimental tests performed using double-cantilever beam specimens and tensile tests. The cohesive parameters were found to be essentially rate-independent for the toughened mode, but the toughness dropped by a factor of four upon a transition to the brittle mode. The results of wedge tests were used as an independent verification of the cohesive parameters, and to verify that the quasi-static properties remained rate-independent to very high crack velocities corresponding to conditions of low-velocity impact. The effects of friction, and the use of the wedge test to determine cohesive parameters, were also explored.     

Pure mode II fracture characterization of composite bonded joints

A new data reduction scheme is proposed for measuring the critical fracture energy of adhesive joints under pure mode II loading using the End Notched Flexure test. The method is based on the crack equivalent concept and does not require crack length monitoring during propagation, which is very difficult to perform accurately in these tests. The proposed methodology also accounts for the energy dissipated at the Fracture Process Zone which is not negligible when ductile adhesives are used. Experimental tests and numerical analyses using a trapezoidal cohesive mixed-mode damage model demonstrated the good performance of the new method, namely when compared to classical data reduction schemes. An inverse method was used to determine the cohesive properties, fitting the numerical and experimental load–displacement curves. Excellent agreement between the numerical and experimental R-curves was achieved demonstrating the effectiveness of the proposed method.     

Local Interface Shear Fracture of Bonded Steel Joints with Various Bondline Thicknesses

Interfacial fracture is a critical issue for extensive applications of adhesively bonded structures to a variety of modern industries. Extensive global experimental tests have been conducted to measure the global behavior of adhesively bonded joint, such as ultimate load capacity and toughness. Recently, several studies have also been employed to characterize the local interfacial traction–separation laws. However, very few tests have investigated the dependency of the local interfacial constitutive laws on the adhesive thickness, particularly, under Mode-II loading conditions. In this work, six typical adhesive thicknesses (from 0.1 mm to 1.0 mm) are prepared for the bonded joints with a configuration of end notched flexure (ENF) specimen to realize the Mode-II fracture loading (shear fracture). With a recently developed analytical model, the global energy release rates of the ENF specimens are experimentally measured. Meanwhile, with the image analysis technique, the local slips between the two adherends are obtained. Finally, based on the J-integral theory, the local interfacial constitutive laws at different bondline thicknesses are obtained. Several experimental findings are reported in this work. This work may provide valuable baseline experimental data for the input in cohesive zone model (CZM) based analytical and numerical simulations.     

Experimental and Numerical Analysis of an In-Plane Shear Specimen Designed for Ductile Fracture Studies

An in-plane shear specimen made of dual phase steel designed for ductile fracture studies is presented and then analyzed experimentally and numerically. In the experiment, digital image correlation (DIC) technique is utilized to measure the deformation of the specimen. Based on the implicit nonlinear FE solver Abaqus/Standard, numerical analysis of the specimen is performed by using plane stress and solid elements respectively. The elongation of the specimen’s gauge length and the shear strain distribution within the shear zone are compared between the experimental and numerical results and a general good agreement is obtained. Thereafter, based on calculated results, the stress state of the shear zone is investigated in detail. It is shown that the shear stress is dominant within the shear zone despite of the emergence of normal stresses. The deformation is concentrated in the shear zone, where the incipient fracture is most likely to occur. The stress triaxiality and the Lode parameter at the fracture initiation are found to be maintained at a relatively low level, which implies that the stress state achieved by the specimen is close to pure shear. The present study demonstrates that the proposed in-plane shear specimen is suitable for investigation of the fracture behavior of high strength materials under shear stress states.     

STOCHASTIC ELASTO-PLASTIC FRACTURE ANALYSIS OF ALUMINUM FOAMS

Model I quasi-static nonlinear fracture of aluminum foams is analyzed by considering the effect of microscopic heterogeneity. Firstly, a continuum constitutive model is adopted to account for the plastic compressibility of the metallic foams. The yield strain modeled by a two- parameter Weibull-type function is adopted in the constitutive model. Then, a modified cohesive zone model is established to characterize the fracture behavior of aluminum foams with a cohesive zone ahead of the initial crack. The tensile traction versus local crack opening displacement relation is employed to describe the softening characteristics of the material. And a Weibull statistical model for peak bridging stress within the fracture process zone is used for considering microscopic heterogeneity of aluminum foams. Lastly, the influence of stochastic parameters on the curve of stress-strain is given. Numerical examples are given to illustrate the numerical model presented in this paper and the effects of Weibull parameters and material properties on J-integral are discussed.     

Size effect in quasibrittle failure: Analytical model and numerical simulations using cohesive zone model

The recent rewriting of the Ba?ant’s size effect law (Morel, 2008) which has suggested the existence of an additional asymptotic regime for intermediate structure sizes is now compared to numerical simulations of fracture of geometrically similar notched structures of different sizes extending over 2.4 decades. The quasibrittle fracture behavior is simulated through cohesive zone model (bilinear softening) using a constant set of cohesive parameters whatever the specimen size D is. The R-curves resulting from the load–displacement responses are estimated and appear as size-independent. On this basis, the different asymptotic regimes expected for the size effect on fracture properties at peak load such as the relative crack length, the resistance to crack growth and the nominal strength are shown in fair agreement with the size effect observed on the results obtained from numerical simulations.