Zdeněk P. Bažant

Concrete fracture models: Testing and practice

Abstract The existing fracture models for concrete and the testing methods for fracture energy and other fracture characteristics are reviewed and some new results on the relationship between fracture testing and size effect are presented. The limitations of the cohesive crack model are discussed. The discrepancy between the fracture energy values measured by Hillerborgs work-of-fracture method and the size effect method is explained and mathematically described by the recently proposed broad-range size effect law. The implications of the recently identified large statistical scatter of the fracture energy values measured by the work of fracture, compared to those measured by the size effect method or Jenq–Shah method, are discussed. Merits of various testing methods are analyzed. A testing procedure in which the maximum loads of notched beams of only two different sizes in the ratio 2:1 and two different notch depths are tested is proposed and a least-square procedure for calculating the fracture parameters is given. A simplified testing procedure with an empirical coefficient, in which only the maximum loads of identical notched and unnotched beams of one size are tested, is also proposed as an alternative. To improve the size effect description for small sizes, the small-size asymptotics of the cohesive crack model is determined and a formula matching this asymptotics, as well as the large-size linear elastic fracture mechanics asymptotics, is presented. Finally, various arguments for introducing fracture mechanics into concrete design practice are reviewed and put into the perspective of safety factors.

Surface singularity and crack propagation

Abstract The three-dimensional singular stress field near the terminal point 0 of the crack front edge at the surface of an elastic body is investigated, using spherical coordinates r , θ , φ and assuming all three displacements to be of the form r λ p p F ( θ , λ ) where p = distance from the singularity line (crack front edge or notch edge) and p = given constant . The variational principle governing the displacement distribution on a unit sphere about point 0, which has previously been obtained from the differential equations of equilibrium, is now derived more directly from potential energy. The previously developed finite element method on the unit sphere is used to reduce the problem to the form k ( λ ) X = 0 where X = column matrix of the nodal values of displacements on the unit sphere and k ( λ ) = square matrix , all coefficients of which are quadratic polynomials in λ. It is proven that the variational principle as well as matrix k must be nonsymmetric, which means that complex eigenvalues A are possible. The dependence of A upon Poissons ratio v for Mode I cracks whose front edge is normal to the surface is solved numerically and it closely agrees with the analytical solution of Benthem. Previously unavailable solutions for Modes II and HI and for cracks (of all modes) with inclined front edge and inclined crack plane are also obtained. By energy flux argument, it is found that the front edge of a propagating crack must terminate at the surface obliquely, at a certain angle whose dependence upon the inclination of the crack plane is also solved. The angle is the same for Modes II and III, but different for Mode I. For this mode, the surface point trails behind the interior of the propagating crack, while for Modes II and III it moves ahead. Consequently, a combination of Mode I with Modes II and III is impossible at the surface terminal point of a propagating crack whose plane is orthogonal. When the plane is inclined, the three stress intensity factors can combine only in certain fixed ratios. The angle of crack edge is a function of the angle of crack plane. Some results with complex λ for two-material interfaces are also given.

Statistical prediction of fracture parameters of concrete and implications for choice of testing standard

This article shows how the fracture energy of concrete, as well as other fracture parameters such as the effective length of the fracture process zone, critical crack-tip opening displacement and the fracture toughness, can be approximately predicted from the standard compression strength, maximum aggregate size, water‐cement ratio, and aggregate type (river or crushed). A database, consisting of 238 test data, is extracted from the literature and tabulated, and approximate mean prediction formulae calibrated by this very large data set are developed. A distinction is made between (a) the fracture energy, Gf, corresponding to the area under the initial tangent of the softening stress‐separation curve of cohesive crack model, which governs the maximum loads of structures and is obtained by the size effect method (SEM) or related methods (Jenq‐Shah two-parameter method and Karihaloo’s effective crack model, ECM) and (b) the fracture energy, GF, corresponding to the area under the complete stress‐separation curve, which governs large postpeak deflections of structures and is obtained by the work-of-fracture method (WFM) proposed for concrete by Hillerborg. The coefficients of variation of the errors in the prediction formulae compared to the test data are calculated; they are 17.8% for Gf and 29.9% for GF, the latter being 1.67 times higher than the former. Although the errors of the prediction formulae taking into account the differences among different concretes doubtless contribute significantly to the high values of these coefficients of variation, there is no reason for a bias of the statistics in favor of Gf or GF. Thus, the statistics indicate that the fracture energy based on the measurements in the maximum load region is much less uncertain than that based on the measurement of the tail of the postpeak load‐deflection curve. While both Gf and GF are needed for accurate structural analysis, it follows that if the testing standard should measure, for the sake of simplicity, only one of these two fracture energies, then Gf is preferable. D 2002 Elsevier Science Ltd. All rights reserved.

Moisture diffusion in cementitious materials Adsorption isotherms

Abstract This article describes an improvement on a previous model proposed by Bažant and Najjar, in which moisture diffusivity and moisture capacity are treated as separate parameters. These parameters are evaluated from independent test results, and are shown to depend on the water:cement ratio, curing time, temperature, and cement type. The moisture capacity is obtained as the slope of the adsorption isotherm. A mathematical model is developed and is shown to predict experimental adsorption isotherms of Portland cement paste very well. In the present form, the model is not applicable to high temperatures.

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 cf 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.

Prediction of concrete creep and shrinkage: past, present and future

The first part of the paper summarizes various aspects of the prediction of concrete creep and shrinkage to be discussed in the conference lecture. They include the theories of physical mechanism, prediction models, constitutive equations, computational approaches, probabilistic aspects, and research directions. The second part then presents two new prediction models. One of them deals with the approximate prediction formulae for pore relative humidity distributions, required for realistic creep and shrinkage analysis, and the other deals with the extrapolation of short time measurements of creep and shrinkage into long times.

Three-dimensional constitutive model for shape memory alloys based on microplane model

Abstract A new model for the behavior of polycrystalline shape memory alloys (SMA), based on a statically constrained microplane theory, is proposed. The new model can predict three-dimensional response by superposing the effects of inelastic deformations computed on several planes of different orientation, thus reproducing closely the actual physical behavior of the material. Due to the structure of the microplane algorithm, only a one-dimensional constitutive law is necessary on each plane. In this paper, a simple constitutive law and a robust kinetic expression are used as the local constitutive law on the microplane level. The results for SMA response on the macroscale are promising: simple one-dimensional response is easily reproduced, as are more complex features such as stress–strain subloops and tension–compression asymmetry. A key feature of the new model is its ability to accurately represent the deviation from normality exhibited by SMAs under nonproportional loading paths.

Shear fracture tests of concrete

Symmetrically notched beam specimens of concrete and mortar, loaded near the notches by concentrated forces that produce a concentrated shear force zone, are tested to failure. The cracks do not propagate from the notches in the direction normal to the maximum principal stress but in a direction in which shear stresses dominate. Thus, the failure is due essentially to shear fracture (Mode II). The crack propagation direction seems to be governed by maximum energy release rate. Tests of geometrically similar specimens yield maximum loads which agree with the recently established size effect law for blunt fracture, previously verified for tensile fracture (Mode I). This further implies that the energy required for crack growth increases with the crack extension from the notch. The R-curve that describes this increase is determined from the size effect. The size effect also yields the shear fracture energy, which is found to be about 25-times larger than that for Mode I and to agree with the value predicted by the crack band model. The fracture specimen is simple to use but not perfect for shear fracture because the deformation has a symmetric component with a non zero normal stress across the crack plane. Nevertheless, these disturbing effects appear to be unimportant. The results are of interest for certain types of structures subjected to blast, impact, earthquake, and concentrated loads.RésuméOn a procédé aux essais en rupture d’échantillons de béton et de mortier à entailles symétriques dans le voisinage desquelles s’exerçaient des efforts concentrés qui ont déterminé une zone de concentration de contraintes de cisaillement. Les fissures à partir des entailles ne se sont pas propagées dans la direction normale à la contrainte maximale principale mais dans une direction où dominaient les contraintes de cisaillement. L’endommagement est donc dû d’abord à une rupture de cisaillemnt (Mode II). La direction de propagation des fissures paraît déterminiée par le taux maximal de restitution d’énergie. Les essais de corps d’épreuve géométriquement similaires indiquent des charges maximales qui concordent avec la loi récemment établie de l’effet de dimension pour les ruptures qui étaient précédemment vérifiées pour les ruptures en tension (Mode I). Ceci par la suite implique que l’énergie requise pour la croissance de fissure augmente avec l’extension de fissure à partir de l’entaille. Plus la fissure est loin de l’entaille, plus il faut d’énergie. La courbe de résistance que décrit ce phénomène se détermine à partir de l’effet de dimension. Celui-ci détermine également l’énergie de rupture en cisaillement qui se trouve être environ 25 fois supérieure à celle de la rupture en tension et concorder avec la valeur annoncée par le modèle de bande de fissure. Le corps d’épreuve de rupture est simple d’emploi mais pas parfaitement adapté à la rupture par cisaillement du fait que la déformation présente une composante symétrique avec la contrainte normale à travers le plan de fissuration. Néanmoins, les effets perturbateurs semblent être négligeables. Les résultats présentent de l’intérêt pour certains types de structures exposées à des charges concentrées sismiques d’impact et explosives.

Drying of concrete as a nonlinear diffusion problem

Abstract Numerous experimental data on drying of concrete and cement paste are subjected to computer analysis. It is found that for a satisfactory fit of the data the diffusion coefficient must be considered to be a function of pore relative humidity (or specific water content), which makes the diffusion problem of drying nonlinear. The diffusion coefficient is shown to decrease sharply (about 20-times) when passing from 0.9 to 0.6 pore humidity, while below 0.6 it appears to be approximately constant. Improvement over the linear theory used in the past is very substantial and indicates that a realistic prediction of drying is possible.

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.