R-curve behavior and roughness development of fracture surfaces

We investigate the idea that the fractal geometry of fracture surfaces in quasibrittle materials such as concrete, rock, wood and various composites can be linked to the toughening mechanisms. Recently, the complete scaling analysis of fracture surfaces in quasibrittle materials has shown the anisotropy of the crack developments in longitudinal and transverse directions. The anomalous scaling law needed to describe accurately these particular crack developments emphasizes the insufficiency of the fractal dimension, usually used to characterize the morphology of fracture surfaces. It is shown that a fracture surface initiating from a straight notch, exhibits a first region where the amplitude of roughness increases as a function of the distance to the notch, and a second one where the roughness saturates at a value depending on the specimen size. Such a morphology is shown to be related to an R-curve behavior in the zone where the roughness develops. The post R-curve regime, associated with the saturation of the roughness, is characterized by a propagation at constant fracture resistance. Moreover, we show that the main consequence of this connection between anomalous roughening at the microscale and fracture characteristics at the macroscale is a material-dependent scaling law relative to the critical energy release rate. These results are confirmed by fracture experiments in Wood (Spruce and Pine).     

Energetic–statistical size effect simulated by SFEM with stratified sampling and crack band model

The paper presents a model that extends the stochastic finite element method to the modelling of transitional energetic–statistical size effect in unnotched quasibrittle structures of positive geometry (i.e. failing at the start of macro‐crack growth), and to the low probability tail of structural strength distribution, important for safe design. For small structures, the model captures the energetic (deterministic) part of size effect and, for large structures, it converges to Weibull statistical size effect required by the weakest‐link model of extreme value statistics. Prediction of the tail of extremely low probability such as one in a million, which needs to be known for safe design, is made feasible by the fact that the form of the cumulative distribution function (cdf) of a quasibrittle structure of any size has been established analytically in previous work. Thus, it is not necessary to turn to sophisticated methods such as importance sampling and it suffices to calibrate only the mean and variance of this cdf. Two kinds of stratified sampling of strength in a finite element code are studied. One is the Latin hypercube sampling of the strength of each element considered as an independent random variable, and the other is the Latin square design in which the strength of each element is sampled from one overall cdf of random material strength. The former is found to give a closer estimate of variance, while the latter gives a cdf with smaller scatter and a better mean for the same number of simulations. For large structures, the number of simulations required to obtain the mean size effect is greatly reduced by adopting the previously proposed method of random property blocks. Each block is assumed to have a homogeneous random material strength, the mean and variance of which are scaled down according to the block size using the weakest‐link model for a finite number of links. To check whether the theoretical cdf is followed at least up to tail beginning at the failure probability of about 0.01, a hybrid of stratified sampling and Monte Carlo simulations in the lowest probability stratum is used. With the present method, the probability distribution of strength of quasibrittle structures of positive geometry can be easily estimated for any structure size. Copyright © 2007 John Wiley & Sons, Ltd.     

Scaling of quasibrittle fracture: asymptotic analysis

Fracture of quasibrittle materials such as concrete, rock, ice, tough ceramics and various fibrous or particulate composites, exhibits complex size effects. An asymptotic theory of scaling governing these size effects is presented, while its extension to fractal cracks is left to a companion paper [1] which follows. The energy release from the structure is assumed to depend on its size D, on the crack length, and on the material length c _f governing the fracture process zone size. Based on the condition of energy balance during fracture propagation and the condition of stability limit under load control, the large-size and small-size asymptotic expansions of the size effect on the nominal strength of structure containing large cracks or notches are derived. It is shown that the form of the approximate size effect law previously deduced [2] by other arguments can be obtained from these expansions by asymptotic matching. This law represents a smooth transition from the case of no size effect, corresponding to plasticity, to the power law size effect of linear elastic fracture mechanics. The analysis is further extended to deduce the asymptotic expansion of the size effect for crack initiation in the boundary layer from a smooth surface of structure. Finally, a universal size effect law which approximately describes both failures at large cracks (or notches) and failures at crack initiation from a smooth surface is derived by matching the aforementioned three asymptotic expansions. Walter P. Murphy Professor of This revised version was published online in July 2006 with corrections to the Cover Date.     

Influence of Stress Singularities on Scaling of Fracture of Metal-Composite Hybrid Structures

It has been recently shown that the nominal structural strength of metal-composite structures depends on the structure size, and such dependence is strongly influenced by the stress singularities. Nevertheless, previous studies only focused on structures that exhibit very strong stress singularities, which are close to the crack-like stress singularity. In the actual engineering designs, due to the mismatch of material properties and complex structural geometries, many metalcomposite structures may contain stress singularities that are much weaker than the crack-like stress singularity. This paper presents a numerical study on the size dependence of scaling of fracture of metal-composite hybrid structures for a wide range of stress singularities. The numerical examples include a series of metalcomposite hybrid beams with a V-notch under three-point bending with different notch angles, which lead to various magnitudes of stress singularities. By assuming that the bimaterial interface is weaker than both metal and composite, we use a mixed-mode cohesive element model to simulate the fracture behavior of these hybrid beams. It is shown that the resulting size effect curves strongly depend on the magnitude of stress singularities. The simulation results agree well with a recently developed energetic-statistical scaling model.     

Self-affinity of fracture surfaces and implications on a possible size effect on fracture energy

As demonstrated within the last 15 years by numerous experimental studies, tensile fracture surfaces exhibit a self-affine fractal geometry in many different materials and loading conditions. In the last few years, some authors proposed to explain an observed size effect on fracture energy by this fractality. However, because they did not consider a lower bound to this scale invariance (which necessarily exists, at least at the atomic scale), they had to introduce a new definition of fracture energy with unconventional physical dimensions. Moreover, they were unable to reproduce the observed asymptotic behavior of the apparent fracture energy at large specimen sizes. Here, we show that this is because they considered self-similar fracture surfaces (not observed in nature) instead of self-affine. It is demonstrated that the ignorance of the self-affine roughness of fracture surfaces when estimating the fracture energy from the work spent to crack a specimen necessarily leads, if the work of fracture is proportional to the fracture area created, to a size effect on this fracture energy. Because of the self-affine (instead of self-similar) character of fracture surfaces, this size effect follows an asymptotic behavior towards large scales. It is therefore rather limited and not likely detectable for relatively large sample sizes (10^–1 m). Consequently, significant and rapid increases of the apparent fracture energy are more likely to be explained mainly by other sources of size effect.     

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.     

Scaling of quasibrittle fracture: hypotheses of invasive and lacunar fractality, their critique and Weibull connection

Considerable progress has been achieved in fractal characterization of the properties of crack surfaces in quasibrittle materials such as concrete, rock, ice, ceramics and composites. Recently, fractality of cracks or microcracks was proposed as the explanation of the observed size effect on the nominal strength of structures. This explanation, though, has rested merely on intuitive analogy and geometric reasoning, and did not take into account the mechanics of crack propagation. In this paper, the energy-based asymptotic analysis of scaling presented in the preceding companion paper in this issue [1] is extended to the effect of fractality on scaling. First, attention is focused on the propagation of fractal crack curves (invasive fractals). The modifications of the scaling law caused by crack fractality are derived, both for quasibrittle failures after large stable crack growth and for failures at the initiation of a fractal crack in the boundary layer near the surface. Second, attention is focused on discrete fractal distribution of microcracks (lacunar fractals), which is shown to lead to an analogy with Weibull's statistical theory of size effect due to material strength randomness. The predictions ensuing from the fractal hypothesis, either invasive or lacunar, disagree with the experimentally confirmed asymptotic characteristics of the size effect in quasibrittle structures. It is also pointed out that considering the crack curve as a self-similar fractal conflicts with kinematics. This can be remedied by considering the crack to be an affine fractal. It is concluded that the fractal characteristics of either the fracture surface or the microcracking at the fracture front cannot have a significant influence on the law of scaling of failure loads, although they can affect the fracture characteristics. Walter P. Murphy, Professor| of This revised version was published online in July 2006 with corrections to the Cover Date.     

Asymptotic stress intensity factor density profiles for smeared-tip method for cohesive fracture

The paper presents a computational approach and numerical data which facilitate the use of the smeared-tip method for cohesive fracture in large enough structures. In the recently developed K-version of the smeared tip method, the large-size asymptotic profile of the stress intensity factor density along a cohesive crack is considered as a material characteristic, which is uniquely related to the softening stress-displacement law of the cohesive crack. After reviewing the K-version, an accurate and efficient numerical algorithm for the computation of this asymptotic profile is presented. The algorithm is based on solving a singular Abel's integral equation. The profiles corresponding to various typical softening stress-displacement laws of the cohesive crack model are computed, tabulated and plotted. The profiles for a certain range of other typical softening laws can be approximately obtained by interpolation from the tables. Knowing the profile, one can obtain with the smeared-tip method an analytical expression for the large-size solution to fracture problems, including the first two asymptotic terms of the size effect law. Consequently, numerical solutions of the integral equations of the cohesive crack model as well as finite element simulations of the cohesive crack are made superfluous. However, when the fracture process zone is attached to a notch or to the body surface and the cohesive zone ends with a stress jump, the solution is expected to be accurate only for large-enough structures.     

Experimental investigation of size effect in concrete fracture under multiaxial compression

Series of scaled hollow-cylinder experiments in a size range 1: 4 were performed to investigate the size effect on strength and fracture of concrete subject to multiaxial compression. A notable size effect was observed during the tests with strength decrease as specimen size increased. Fracture processes were examined using impregnation techniques and their results indicated splitting type mechanisms to take place, which were encircling the inner-holes in a rather uniform manner. Interpretation of the results showed that the observed size effect attributes to a combination of structural (e.g. geometry imposed stress gradients) and material (statistical) size effects.     

Linking fresh and durability properties of paste to SCC mortar

In the last years many approaches to design SCC have been developed, but it remains a very complex process since it is necessary to manipulate several variables and understand their effects on concrete behaviour (fresh and hardened state). The prediction of concrete or mortar behaviour based on paste properties will be a significant contribution to simplify SCC design. With this purpose, two statistical experimental designs were carried out, one at paste level and the other at mortar level, to mathematically model the influence of mixture parameters on fresh and durability properties. The derived numerical models were used to define an area, labelled by self-compacting zone at paste level (SCZ), where fresh properties of the paste enable the design of SCC mortar. Furthermore, in order to extend this link to durability properties, the effect of including aggregate in cement paste was evaluated by means of the electrical resistivity test.     

Fatigue and fracture properties of thin metallic foils

Metallic thin foils are essential structural parts in microsystems,which may be subjected to fatigue loading caused by thermal fluctuations and mechanical vibrations influencing their reliability in numerous engineering applications. It is well known that the fatigue properties of bulk material cannot be adopted for small scaled structures. For a better understanding of the `size-effect' in the present investigation fatigue crack growth near threshold in the high cycle fatigue regime and associated fracture processes were studied. Free- standing rolled and electrodeposited Cu-, Mo- and Al foils of thickness from 20 m to 250 m in different conditions have been tested in a special experimental set up operating at R=–1 and a testing frequency of 20 kHz. At a given constant strain value the fatigue crack growth behaviour has been recorded accompanied by intermittent observation of the change of the dislocation structure in the vicinity of the growing crack by use of the electron channeling contrast imaging (ECCI)-technique in a scanning electron microscope (SEM). In a load shedding technique fatigue threshold stress intensity factor values have been derived and compared with data of bulk material. Typical crack growth features were detected depending on thickness and grain sizes of the foils. Various criteria (compliance, extent of plastic zones and plastic strain gradients) were selected for the explanation of this anomalous behaviour. Additionally fractomicrographs of uniaxial strained and fatigued foils have been studied to obtain further insight of the effect of dimensional constraint.     

Fatigue and fracture properties of thin metallic foils

Metallic thin foils are essential structural parts in microsystems, which may be subjected to fatigue loading caused by thermal fluctuations and mechanical vibrations influencing their reliability in numerous engineering applications. It is well known that the fatigue properties of bulk material cannot be adopted for small scaled structures. For a better understanding of the `size-effect' in the present investigation fatigue crack growth near threshold in the high cycle fatigue regime and associated fracture processes were studied. Free-standing rolled and electrodeposited Cu-, Mo- and Al foils of thickness from 20 m to 250 m in different conditions have been tested in a special experimental set up operating at R=–1 and a testing frequency of 20 kHz. At a given constant strain value the fatigue crack growth behaviour has been recorded accompanied by intermittent observation of the change of the dislocation structure in the vicinity of the growing crack by use of the electron channeling contrast imaging (ECCI)-technique in a scanning electron microscope (SEM). In a load shedding technique fatigue threshold stress intensity factor values have been derived and compared with data of bulk material. Typical crack growth features were detected depending on thickness and grain sizes of the foils. Various criteria (compliance, extent of plastic zones and plastic strain gradients) were selected for the explanation of this anomalous behaviour. Additionally fractomicrographs of uniaxial strained and fatigued foils have been studied to obtain further insight of the effect of dimensional constraint.     

新拌砂浆的流变性

确认了漏斗测试砂浆流变性（相对塑性黏度和相对屈服应力）方法的有效性后,用它对几个影响砂浆流变性的因素进行了测定。     

Scaling global fracture behavior of structures-sized laminated composites

The propagation of damage in laminated fiber composites similar to a crack-like configuration is studied experimentally. Of particular interest is the behavior of a `macro-crack' or `global crack' that propagates through a large scale structure (e.g., airplane fuselage or wing) with the potential interaction of stringers or other reinforcements. The question is considered whether such damage propagation can be understood in terms of classical fracture mechanics. Because the damage zone (wake width) can be very extensive and may be measured in terms of inches, the question arises as to the scalability of associated `fracture' phenomena. An integral part of this investigation is thus an examination of the size of the test specimens to establish whether a minimum size is required to relate fracture at larger scales by laboratory specimens. Using (globally anisotropic) 32-lamina composite specimens, proportioned like compact tension specimens, it is found that test `coupons' on the order of 45 to 50 cm on a side (18×18 in.²) or larger are needed to begin simulating large scale structures for establishing a global equivalent of a fracture energy for `crack propagation' simulations. Displacement controlled tests on groups of three specimen sizes (15, 30 and 46 cm on a side) indicate that scaling can be accomplished through the square root of the linear specimen or crack dimension. In certain lay-ups `run-away delamination' severs the surface laminae so that the reinforcement action of stringers is jeopardized. Damage at the `global crack front' is quantified through an effective area relation, with a characteristic value for the mostly intra-laminar initiation of damage (intra-lamina cracks) at a sharp notch, and another value conceived of as governing the onset of unstable growth.     

基于强度尺寸效应的准脆性材料脆性指标研究

长期以来,如何评价准脆性材料的脆性一直是一重要问题。然而,现有的大多数脆性指标存在明显的局限性,即显著受试件尺度的影响,往往反映的是材料的脆性行为,而不是材料性质。基于Bazant尺寸效应律,强度与试件尺寸双对数曲线图中两条渐近线的交点或与之相关的有效断裂过程区长度可反映材料脆性。该脆性指标测定简单方便、真实可靠,能广泛应用于各种准脆性材料脆性评价。     

Strength and fracture properties of asbestos-cement mortar composites

The strength and fracture properties of random asbestos fibre-reinforced cement mortar composites are reported in this paper. The fibre content varies between 5% and 20% by weight. Both the ultimate tensile strength ( _t) and the modulus of rupture ( _b) increase with increasing fibre-volume fraction. These results are shown to agree satisfactorily with the law of mixtures modified for randomly oriented short fibre-reinforced composites. The critical stress intensity factor (K _c) and the specific work of fracture (R) have been determined using three-point bend edge-notched beams and grooved double-cantilever-beam (DCB) specimens. There is generally good agreement between these two physical quantities estimated from the two testpiece geometries. It is shown that the fibre pull-out mechanism is dominant in the fracture of asbestos cements and that the specific work of fracture can be reasonably well predicted by considering the energies absorbed in both the pull-out and the fibre/matrix interfacial debonding processes.     

Effect of crack opening on carbon dioxide penetration in cracked mortar samples

The paper addresses the effect of crack opening on the ability of carbon dioxide to diffuse along a crack. The experimental tests were carried out on mortar samples. A mechanical expansive core was used to generate cracks of constant width across the thickness of the sample. Cracked specimens with crack openings ranging from 9 to 400 μm were exposed to accelerated carbonation for 65 days. Then they were removed to determine the depth of carbonation perpendicular to the crack path. Theses depths were compared to the measured ones on the reference samples. The results show that crack opening significantly influences the ability of carbon dioxide to diffuse along the crack. Indeed, the carbonation depth perpendicular to the crack wall indicates a lower capacity to diffuse in cracks less than 41 μm in width. For crack openings ranging from 9 to 41 μm, there was still diffusion along the crack path. Moreover, carbonation of the interface between steel and mortar was observed inducing a depassivation of the reinforcement. For the duration of the experiments, there was no diffusion in crack openings of less than 9 μm. The effect of interlocking phenomena between the fracture surfaces on the ability of carbon dioxide to diffuse along the crack, was also studied. The results showed that interlocking phenomena in cracks is the main factor limiting the diffusion of carbon dioxide in fine cracks.     

Early age fracture properties of microstructurally-designed mortars

This paper compares the fracture properties as well as crack initiation and propagation of real and equivalent mortars. The development of the elastic modulus, tensile strength, and fracture energy at different hydration stages were determined by inverse analysis of load-displacement curves obtained by the compact tension test (CTT). Further, the impact of the moisture content on the aforementioned material properties was also tested on oven-dried equivalent mortars. Digital image correlation (DIC) was used to follow the crack initiation and propagation.The elastic modulus, tensile strength, and fracture energy support the validity of the equivalent mortars approach. The load-displacement curves obtained by the CTT were also compared to those simulated by finite element method showing excellent correlations. DIC revealed the formation of similar crack patterns at comparable load levels between the two mortars. At early age, the moisture content has a considerable influence on the tensile strength and the fracture energy.     

多维度评定计量法在面部识别中的运用

研究采用多维度评定计量法(MultidimensionalScaling),让46名被试对成对人脸作相似性判断及对单个人脸作主观评价,以此来探索面部识别中的心理过程。结果表明:①男女被试对人脸相似性判断的标准有差异,女性被试判断维度是人脸的性别、种族和异常性,而男性被试的判断依据是人脸的年龄、种族和异常性;②人脸的社会意义在相似性判断中并未起作用,但是男女被试对同一人脸的主观评价上有差异。     

Using torsional bar testing to determine fracture toughness

A new method to determine fracture toughness K _IC of materials is introduced. A round-rod specimen having a V-grooved spiral line with a 45° pitch is tested under pure torsion. An equibiaxial tensile/compressive stress state is effectively created to simulate conventional test methods using a compact-type specimen with a thickness equivalent to the full length of the spiral line. K _IC values are estimated from the fracture load and crack length with the aid of a three-dimensional finite element analysis. K _IC of 7475-T7351 aluminium is estimated to be 51.3 MPa √m, which is higher than the vendor's value in the TL orientation by ∼0.8% and higher than 0.5T compact tension (CT) value by 6%; A302B steel yields 54.9 MPa √m being higher than CT test value by ∼2%. Good agreement between the K _IC values obtained by different methods indicates the proposed method is sound and reliable.