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).     

Boundary effect on concrete fracture and non-constant fracture energy distribution

This paper extends the local fracture energy concept of Hu and Wittmann [29] and [30], and proposes a bilinear model for boundary or size effect on the fracture properties of cementitious materials. The bilinear function used to approximate the non-constant local fracture energy distribution along a ligament is based on the assumption of the proportionality of the local fracture energy to the fracture process zone (FPZ) height and characterises the FPZ height reduction when approaching a specimen back boundary. The bilinear function consists of a horizontal straight line of the intrinsic fracture energy G_F and a declining straight line that reduces to zero at the back boundary. It is demonstrated that using the bilinear model, the size-independent fracture energy G_F can be estimated from the fracture energy data measured on laboratory-size specimens, and the intersection of these two linear functions, defined as the transition ligament, represents the influence of the back boundary on the fracture properties. It is also demonstrated that the specimen size alone is not sufficient to characterise the size effect in the fracture properties observed on laboratory-size specimens.     

An asymptotic approach to size effect on fracture toughness and fracture energy of composites

Size effect on fracture toughness and fracture energy of composites is investigated by a simple asymptotic approach. This asymptotic analysis based on the elastic/plastic fracture transition of a large plate with a small edge crack is extended to study fracture of composite. A reference crack length, a^*, is used in the model, which indicates an ideal elastic/plastic fracture transition defined by the yield strength and plane strain fracture toughness criteria. Experimental results of cementitious materials available in literature are analyzed and compared. It is shown that the common K_R-curves can also be obtained by the current asymptotic model. Furthermore, a local fracture energy distribution concept is also discussed and compared with the present asymptotic approach.     

Scaling properties of mortar fracture surfaces

Scaling properties of mortar crack surfaces are studied from mode I fracture specimens of six different sizes. Fracture surfaces initiated from a straight notch exhibit an anomalous dynamic scaling which involves two independent roughness indices: the universal local roughness exponent ζ_loc ≈ 0.8 and the global roughness exponent, estimated to ζ ≃ 1.35. We show that there exists a linear relationship between the specimen size and the maximum self-affine correlation length inducing a size effect on the roughness magnitude at saturation and this especially for the smallest length scales. Finally, we argue that anomalous roughening could be an inheritance of the changes in long range elastic interactions which take place in the fracture process zone of quasibrittle materials.     

Estimation of fracture toughness, driving force, and fracture energy for fractal cracks using the method of imaginary smooth crack

Theoretical expressions of the fracture toughness, driving force and fracture energy are derived for a perfect infinite plate with a solitary self-similar fractal crack of mode I considering the fractal effects of cracks in this work. The specific surface energy per unit fractal measure for concrete is fit based on test data. Thus, behaviors of the above four fracture parameters for concrete are studied as varying the initial length of the fractal crack.     

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。     

Influence of local fracture energy distribution on maximum fracture load of three-point-bending notched concrete beams

The maximum fracture load of a notched concrete beam has been related to the local fracture energy at the cohesive crack tip region analytically in this paper, and then the correlation between the size effects on the maximum fracture loads and the RILEM specific fracture energy is established. Two extreme conditions have been established, namely zero crack-tip bridging with zero local fracture energy and maximum crack-tip bridging with the maximum size-independent fracture energy. It is concluded that the local fracture energy at the crack tip region indeed varies with the initial crack length and the size of specimen. The tri-linear model for the local fracture energy distribution is confirmed by using the proposed simple analytical solution.     

Fractal aspects of ductile and cleavage fracture surfaces

The aim of this paper is to evaluate and interpret the three-dimensional variational fractal dimension of a ductile and a cleavage fracture surface. The fracture surface is acquired by fracturing Charpy impact and static loaded specimens of a low alloy steel in ductile-to-brittle transition temperature range, and reconstructed by a stereoscopic technique. The three-dimensional variational method for measuring fractal dimension is improved by shifting algorithm and tested on the Takagi surface using the local fractal dimension. We find very good fractal behaviour in the ductile area, however, fractal characteristics in the cleavage area are less noticeable. The results are discussed in thermodynamical terms and promote the idea that fractal behaviour reflects the quasi-static process and that the fracture mechanisms in the ductile fracture are independent of strain rate (at least up to 10³ s⁻¹).     

Crack morphology models for fracture toughness and fatigue strength analysis

Planar cracks represent an approximation, largely adopted in fracture mechanics and fatigue problems, of the physical reality, where cracks feature complex geometric morphologies related to material microstructure, residual stresses, material properties dispersions and so on. In the present paper, firstly a model to describe the influence of roughness and friction of the crack surfaces is reviewed in relation to the resulting near‐tip stress field and the fracture resistance under monotonic loading. Such a model is based on the Distributed Dislocation Technique, and considers a periodic profile of the crack. Then, some approximate theoretical models describing periodically kinked cracks are reviewed in their application to the estimation of fatigue strength of materials. In particular, the influence of crack path meandering on fatigue propagation is analysed by modelling the crack profile as a piecewise linear periodic curve in two dimensions. The same type of model is discussed within the framework of self‐similar fractal geometries. In the paper, emphasis is given to the effect of crack size on the fracture resistance and fatigue strength, where such an effect depends on the ratio between the characteristic length of the crack morphology and the nominal length of the crack itself.     

Estimation of fracture energy from the work of fracture and fracture surface area: I. Stable crack growth

Past attempts to determine fracture energy by the work of fracture (γ _WOF) technique, in most cases, have resulted in greater estimates due to the use of the cross-sectional area rather than the actual area of the fracture surface in calculations. The actual fracture surface area A _F of soda-lime-silica glass chevron-notch flexure specimens was estimated using atomic force microscopy. An equation for A _F was developed using the data from these tests. The use of A _F in the equation for γ _WOF resulted in γ _WOF values less than values reported from traditional fracture mechanics tests and from those obtained using the cross-sectional area. The implication is that the tortuosity of the fracture surface contributes to the energy expended during fracture and should be accounted for in the calculation of the fracture energy. These calculations provide an estimate for the minimum energy required to break bonds in the fracture process.     

Identification of concrete fracture parameters through size effect experiments

This paper deals with the identification of concrete fracture parameters through indirect methods based on size effect experiments. These methods utilize the size effect curve (structural strength versus structural size), associated with a certain specimen geometry, to identify the tensile strength and the initial fracture energy. These two parameters, in turn, are typically used to characterize the peak and the initial post-peak slope of the cohesive crack law. In the literature, two different approaches can be found for the calculation of the size effect curve: (a) an approach based on the polynomial interpolation of numerically calculated structural strengths of geometrically similar specimens of different sizes, and (b) the classical approach based on equivalent elastic fracture mechanics, which gives rise to the well-known Bažant’s size effect law (SEL). In this paper, the two approaches are first reviewed, the relationship between them is investigated, and a new procedure to identify the tensile strength using the SEL is proposed. Then several sets of experimental results, recently performed at the Politecnico di Milano, are analyzed with both approaches in order to assess their range of applicability and accuracy in the identification of the two fracture parameters specified above.     

Size effect and inverse analysis in concrete fracture

This paper summarizes the basic experimental and numerical results supporting an easy procedure to determine up to two fracture parameters based on numerically computed size effect curves. Furthermore, it supplies closed-form expressions to determine the initial linear segment approximation of the (stress vs. crack opening) softening curve of cohesive crack models for concrete, based only on the peak loads determined in splitting-tension (Brazilian) tests and in three-point-bending test on notched specimens. Knowledge of the initial segment, although not enough to describe all the fracture process of concrete structures, is enough to predict the fracture behavior of unnotched concrete structures prior and around the peak load. The same is true for notched structures provided their size is less than a limiting size, approximately defined in the paper. This revised version was published online in July 2006 with corrections to the Cover Date.     

Multifractal nature of concrete fracture surfaces and size effects on nominal fracture energy

Experimental evidence of the fractality of fracture surfaces has been widely recognized in the case of concrete, ceramics and other disordered materials. An investigationpost mortem on concrete fracture surfaces of specimens broken in direct tension has been carried out, yielding non-integer (fractal) dimensions of profiles, which are then related to the ‘renormalized fracture energy’ of the material. No unique value for the fractal dimension can be defined: the assumption of multifractality for the damaged, material microstructure produces a dimensional increment of the dissipation space with respect to the number 2, and represents the basis for the so-called multifractal scaling law. A transition from extreme Brownian disorder (slope 1/2) to extreme order (zero slope) may be evidenced in the bilogarithmic diagram: the nominal fracture energyG _F increases with specimen size by following a nonlinear trend. Two extreme scaling regimes can be identified, namely the fractal (disordered) regime, corresponding to the smallest sizes, and the homogeneous (ordered) regime, corresponding to the largest sizes, for which an asymptotic constant value ofG _F is reached.

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Resume On a largement établi la preuve expérimentale du caractère fractal des surfaces de rupture dans le cas du béton, des céramiques et d'autres matériaux ‘désordonnés’. Une étudepost mortem menée sur des surfaces de rupture d'échantillons cassés par traction directe révèle des dimensions non intégrales (fractales) des profils dont on a établi la relation avec l'énergie de rupture ‘renormalisée’ du matériau. Il n'est pas possible d'établir une valeur unique de la dimension fractale: en présumant la multifractalité de la microstructure du matériau endommagé, on obtient une augmentation dimensionnelle par rapport au numéro 2 et on établit la base de la loi dite d'échelle multifractale. Dans le diagramme à deux logarithmes on peut voir, une transition du désordre de Brown extrême (inclinaison 1/2) à l'ordre extrême (inclinaison zéro); l'énergie de fracture nominaleG _F augmente avec les dimensions de l'échantillon suivant une tendance non linéaire. On peut voir deux régimes extrêmes d'échelle, c'est-à-dire le régime fractal désordonné) . qui correspond aux dimensions minimales, et le régime homogène (ordonné), qui correspond aux dimensions maximales pour lesquelles on atteint une valeur constante asymptotique deG _F.

     

On a thermo-mechanical effect and criterion of crack propagation

A thermo-mechanical effect from partial conversion of fracture work into heat energy during crack propagation is considered with a simple mathematical model. It is assumed that the heat production zone in the vicinity of the crack tip is very small. Thus, the crack propagation process can be viewed as propagation of the crack in elastic material with a point thermal heat source fixed at the tip of the crack. This thermal heat source generates its own temperature and stress fields around the crack tip. As shown in this paper it also generates a negative stress intensity factor that specifies fracture mode I and has to be accounted for in the energetic fracture criterion. The model developed may help to explain many experimental observations such as the increase in the specific surface energy that accompanies an increase in the crack speed and why fracture mode I has a special role in crack propagation phenomena.     

Impact fracture of a ferritic steel in the lower shelf regime

This paper reports the results of a systematic investigation on the fracture of Charpy-V notch A508 steel specimens, tested in the lower shelf regime. The fracture energy has been determined for quasi-static, standard Charpy and one-point-bend impact. The results show a general trend for the fracture energy to increase with the loading rate, at the lower temperature (–160 °C). At this temperature, the roughness of the fracture surface increases markedly with the loading rate. The fractographic analysis shows the presence of 3–4 cleavage initiation sites situated at 100–800 m from the crack front, irrespective of the loading rate. Numerous cleavage microcracks are observed underneath the main fracture plane. The statistical analysis shows that the length distribution of the microcracks is adequately described by Weibull statistics. It is also found that the number of microcracks increases with the loading rate. It is suggested that the larger number of microcracks is responsible for the observed increased roughness and energy dissipation.     

Cohesive crack analysis of size effect

This paper deals with the analysis of size effect in concrete. An extensive campaign of accurate numerical simulations, based on the cohesive crack model, is performed to compute the size effect curves (CSEC) for typical test configurations. The results are analyzed with reference to the classical Ba?ant’s size effect law (SEL) to investigate the relationship between CSEC and SEL. This analysis shows that as specimen size tends to infinity, the SEL represents the asymptote of the CSEC, and that the SEL parameter known as the effective fracture process zone length is a material property which can be expressed as a function of the cohesive crack law (CCL) parameters. Finally, the practical implications of this study are discussed in relation to the use of the CSEC or the SEL for the identification of the CCL parameters through the size effect method.     

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.     

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.     

An approach to size effect in fatigue of metals using fractal theories

ABSTRACT As was experimentally observed by several authors, the fatigue strength of metallic materials decreases with increasing the specimen size. Such a decrease can be remarkable for very large structures like, for example, big cargo ships (some hundred meters long) transporting oil or other goods. Size effect in fatigue is herein explained by considering the fractal nature of the reacting cross sections of structures, that is, the renormalized fatigue strength is represented by a force amplitude acting on a surface with a fractal dimension lower than 2, where such a dimensional decrement depends on a self‐similar weakening of the material ligament, owing to the presence of cracks, defects, voids and so forth (microscopic level). However, this decrement tends to progressively disappear with increasing the structure size (macroscopic level), i.e. the effect of the material microstructure on the macroscopic fatigue behaviour gradually vanishes for structures large enough with respect to a characteristic microstructural size, this phenomenon being defined as multifractality. A multifractal scaling law for fatigue limit of metals is proposed, and some experimental results are examined in order to show how to apply the theoretical approach presented.     

Electrophysical method for evaluating the fracture surface energy of dielectric materials in contact with metal

A physical model is formulated and the method is substantiated for measuring the fracture surface energy of dieletric materials. An analogy between the distributions of electric field intensity and mechanical stresses close to the metal-dieletric surface contact is used.Translated from Izmeritelnaya Tekhnika, No. 10, pp. 52–56, October, 2004.