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.     

Modelling the fracture of concrete under mixed loading

A simple and efficient numerical procedure for mixed mode fracture of quasibrittle materials is shown: This technique predicts crack trajectories as well as load-displacement or load-CMOD responses. The model is based on the cohesive crack concept and uses the local mode I approach. Numerical results agree quite well with three experimental sets of mixed mode fracture of concrete beams; one from Arrea and Ingraffea, another from García, Gettu and Carol and from a nonproportional loading by the authors. In constrast to more sophisticated models, this method offers two major advantages: it requires only material properties measured by standardized methods and it can easily be implemented with general multipurpose finite element codes.     

Influence of shear parameters on mixed–mode fracture of concrete

The influence of the mode II fracture parameters on the mixed mode fracture experimental tests of quasibrittle materials is studied. The study is based on experimental results and numerical analyses. For the numerical study, a procedure for mixed mode fracture of quasibrittle materials is presented. The numerical procedure is based on the cohesive crack approach, and extends it to mixed mode fracture. Four experimental sets of mixed mode fracture were modelled, one from Arrea and Ingraffea and another from a nonproportional loading by the authors, both with bending concrete beams. Two other sets of experimental fracture were modelled, based on double-edge notched testing; in these tests an important mode II is beforehand expected. The numerical results agree quite well with experimental records. The influence of the main parameters for mode II fracture on the mixed mode fracture is studied for the four experimental set of tests and compared with these results. In all them, large changes in the mode II fracture energy hardly modify the numerical results. The tangential and normal stresses along the crack path during the loading proccess are obtained, also with different values of the mode II fracture energy. For the studied experimental tests it is concluded that the crack is initiated under mixed mode but propagated under predominant mode I. This allows a development of mixed mode fracture models, mainly based on standard properties of the material measured by standard methods, avoiding the problems associated with the measurement of mode II fracture parameters, such as mode II fracture energy and cohesion.     

An embedded cohesive crack model for finite element analysis of mixed mode fracture of concrete*

An embedded cohesive crack model is proposed for the analysis of the mixed mode fracture of concrete in the framework of the Finite Element Method. Different models, based on the strong discontinuity approach, have been proposed in the last decade to simulate the fracture of concrete and other quasi‐brittle materials. This paper presents a simple embedded crack model based on the cohesive crack approach. The predominant local mode I crack growth of the cohesive materials is utilized and the cohesive softening curve (stress vs. crack opening) is implemented by means of a central force traction vector. The model only requires the elastic constants and the mode I softening curve. The need for a tracking algorithm is avoided using a consistent procedure for the selection of the separated nodes. Numerical simulations of well‐known experiments are presented to show the ability of the proposed model to simulate the mixed mode fracture of concrete.     

Constitutive models for cohesive zones in mixed-mode fracture of plain concrete

Discrete mixed-mode fracture (modes I and II) of plain concrete is investigated using a coupled and an uncoupled cohesive zone constitutive model in a finite element context. Fracture surfaces are confined to inter-element boundaries that are not necessarily coincident with the actual fracture surfaces. For this reason, traction components on the cohesive zone do not correspond to actual values either. In this work is demonstrated that only the coupled model is able to cope with these spurious traction components, that must decrease with crack opening. It is shown also that, in this regard, the key variable is the plastic potential adopted in the integration of tractions. Three mixed-mode fracture examples were tested in this work: a three-point single-edge notched beam, double-edge notched plates under variable lateral and normal deformation and four-point double-edge notched beams. A good fitting with experiments was obtained only for the coupled model. Mode II parameters can change in a large range without noticeable change in results, at least in the tested examples.     

Choice of standard fracture test for concrete and its statistical evaluation

The main characteristics of the cohesive (or fictitious) crack model, which is now generally accepted as the best simple fracture model for concrete, are (aside from tensile strength) the fracture energies G _F and G _f corresponding to the areas under the complete softening stress-separation curve and under the initial tangent of this curve. Although these are two independent fracture characteristics which both should be measured, the basic (level I) standard test is supposed to measure only one. First, it is argued that the level I test should measure G _f, for statistical reasons and because of relevance to prediction of maximum loads of structures. Second, various methods for measuring G _f (or the corresponding fracture toughness), including the size effect method, the Jenq-Shah method (TPFM), and the Guinea et al. method, are discussed. The last is clearly the most robust and optimal because: (1) it is based on the exact solution of the bilinear cohesive crack model and (2) necessitates nothing more than measurement of the maximum loads of notched specimens of one size, supplemented by tensile strength measurements. Since the identification of material fracture parameters from test data involves two random variables, f_t (tensile strength) and G _f, statistical regression should be applied. But regression is not feasible in the original Guinea et al.'s method. The present study proposes an improved version of Guinea et al.'s method which reduces the statistical problem to linear regression thanks to exploiting the systematic trend of size effect. This is made possible by noting that, according to the cohesive (or fictitious) crack model, the zero-size limit _N0 of nominal strength _N of a notched specimen is independent of F _f and thus can be easily calculated from the measured f_t. Then, the values of _N0 obtained from the measured f_t values, together with the measured _N-values of notched specimens, are used in statistical regression based on the exact size effect curve calculated in advance from the cohesive crack model for the chosen specimen geometry. This has several advantages: (1) the linear regression is the most robust statistical approach; (2) the difficult question of statistical correlation between measured f_t and the nominal strength of notched specimens is bypassed, by virtue of knowing the size effect trend; (3) the resulting coefficient of variation of mean G _f is very different and much more realistic than in the original version; (4) the coefficient of variation of the deviations of individual data from the regression line is very different from the coefficient of variation of individual notched test data and represents a much more realistic measure of scatter; and (5) possible accuracy improvements through the testing of notched specimens with different notch lengths and the same size, or notched specimens of different sizes, are in the regression setting straightforward. For engineering purposes where high accuracy is not needed, the simplest approach is the previously proposed zero-brittleness method, which can be regarded as a simplification of Guinea et al.' method. Finally, the errors of TPFM due to random variability of unloading-reloading properties from one concrete to another are quantitatively estimated.     

Artificial neural network‐based analysis of effective crack model in concrete fracture

Many non‐linear fracture models have been proposed by design codes and investigators to determine fracture parameters of cement‐based materials. To characterise failure of concrete structures, the effective crack model (ECM) needs two fracture parameters: the effective crack length a_e and the critical stress intensity factor . Nevertheless, ECM requires a closed‐loop testing system and the calculation of a_e needs considerable computational effort. For this reason, ECM is simulated with an artificial neural network (ANN) in this study. The main benefit of using an ANN approach is that the network is built directly on experimental data by using the self‐organizing capabilities of the ANN. The presented fracture model was developed by utilising 464 noisy test data taken from the literature, which were obtained via different test methods in different laboratories. The results of an ANN‐based ECM look viable and very promising.     

Influence of specimen geometry on determination of double-K fracture parameters of concrete: a comparative study

Comparative study on analytical method, simplified method and weight function approach for determination the double-K fracture parameters using three-point bend and compact tension tests specimen geometries is presented in the paper. The input data required for numerical calculations are obtained using Fictitious Crack Model. The study reports that the double-K fracture parameters computed depends on factors such as initial-notch length/depth ratios, specimen geometry and size-effect. In addition, it is demonstrated that the use of weight function will further improve the computational efficiency without loss of accuracy.     

A multifractal analysis of fatigue crack growth and its application to concrete

As is well-known, strength of materials is influenced by the specimen or structure size. In particular, several experimental campaigns have shown a decrease of the material strength under static or fatigue loading with increasing structure size, and some theoretical arguments have been proposed to interpret such a phenomenon. As far as fatigue crack growth is concerned, limited information on size effect is available in the literature, particularly for so-called quasi-brittle materials like concrete. In the present paper, by exploiting concepts of fractal geometry, some definitions of fracture energy and stress intensity factor based on physical dimensions different from the classical ones are discussed. A multifractal size-dependent fatigue crack growth law (expressing crack growth rate against stress intensity factor range) is proposed and used to interpret relevant experimental data related to concrete.     

Some recent developments in computational modelling of concrete fracture

Some of the most important aspects of numerical modelling of cracking in concrete are reviewed. After a discussion of the three main lines in modelling cracking – discrete crack models, smeared representations and approaches using lattice models – a concise treatment including comparative studies is given of the various smeared crack approaches that exist to date. Next a discussion is presented of some issues pertaining to the sensitivity of numerical results on the fineness of the mesh and the direction of the mesh lines, and on size effects in concrete structures. This revised version was published online in July 2006 with corrections to the Cover Date.     

Proposal of a new concept on creep fracture remnant life for a precracked specimen

Abstract

The creep life time of a smooth specimen can be predicted using existing laws for creep deformation and steady state creep rate. When crack growth behaviour is involved, it is necessary to construct a law of creep crack growth rate to predict creep fracture life. Creep fracture life can be measured by integrating the law of creep crack growth rate. One example is the creep crack growth rate, represented by the parameter Q*. In this study, we investigated the applicability of this prediction method to creep fracture remnant life for a cracked specimen. The Ω criterion is proposed to predict creep fracture remnant life for a smooth specimen for creep ductile materials. In this study, the correlation between Q*_L derived from the paremeters Q* and Ω is investigated. The correlation between Q_L* and Ω provided a unified theoretical prediction law of creep fracture remnant life for high-temperature creep-ductile materials in the range from smooth to precracked specimens.     

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.     

Embedded representation of fracture in concrete with mixed finite elements

As an alternative to the smeared and discrete crack representations, an embedded representation of fracture for finite element analysis of concrete structures is presented. The three-field Hu–Washizu variational statement is extended to bodies with internal discontinuities. The extended variational statement is then utilized for formulating elements with a discontinuous displacement field. The new elements are capable of modelling different deformation modes of an internal discontinuity at the element level. The satisfactory performance of the embedded crack representation is verified by several case studies on concrete fracture.     

混凝土断裂过程区长度计算方法研究

该文基于粘聚裂缝概念,以起裂韧度作为裂缝起裂和扩展的准则,提出了混凝土断裂过程区长度的计算方法。以Ⅰ型裂缝为例,计算了不同初始缝长和起裂韧度情况下的断裂过程区长度值,结合以往大体积混凝土的试验数据对其进行了验证。进而分析了断裂过程区长度的影响因素,结果表明：断裂过程区长度随初始缝长的增大而逐渐增大,随起裂韧度的增大而逐渐减小。     

ATENA — A tool for engineering analysis of fracture in concrete

Advanced constitutive models implemented in the finite element system ATENA serve as rational tools to explain the behaviour of connection between steel and concrete. Three nonlinear material models available in ATENA are described: crack band model based on fracture energy, fracture-plastic model with non-associated plasticity and microplane material model. Nonlinear simulation using these advanced constitutive models can be efficiently used to support and extend experimental investigations and to predict behaviour of structures and structural details.     

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.     

Simplified equations for determining double‐K fracture parameters of concrete for 3‐point bending test

The paper presents simplified polynomial equations for determining the double‐K fracture parameters of concrete for 3‐point bending beams with variable strengths and material properties of concrete. The derived equations avoid complexities involved in computations of fracture parameters using existing analytical methods. The input data required for systematic computation in the study for deriving the nondimensional fracture parameters are obtained using a fictitious crack model. It is inferred that for a relative size of initial crack length, critical load and corresponding crack opening displacement maintain a linear relationship in their nondimensional forms. The value of critical mouth opening displacement can also be determined for known value of peak load using the derived nondimensional equation, thus avoiding the measurement of the crack mouth opening displacement in the experiment. Further, the derived polynomial equations predict the double‐K fracture parameters of concrete with negligible error as compared to those obtained based on experimental results.     

Numerical study of mixed-mode fracture in concrete

In the present paper, a finite element code based on the microplane model for concrete is used for the analysis of typical mixed-mode geometries: a Single-Edge-Notched beam, a Double-Edge-Notched specimen and a Dowel-Disk specimen. A local smeared fracture finite element analysis is carried out. As a regularization procedure, the crack band method is used. The principal objective of the study was to investigate whether the smeared fracture finite element code is able to predict mixed-mode fracture of concrete with no optimisation of the material model parameters. Comparison between experimental and numerical results shows that the used code predicts structural response and crack patterns realistically for all cases investigated. Moreover, it is shown that for most of the studied geometries a mixed-mode fracture mechanism dominates at crack initiation, however, with increase of the crack length mode-I fracture becomes dominant and the specimens finally failed in mode-I fracture.