Zdenek P. Bazant

Nonlocal continuum damage, localization instability and convergence

A recent nonlocal damage formulation, in which the spatially averaged quantity was the energy dissipated due to strain-softening, is extended to a more general form in which the strain remains local while any variable that controls strain-softening is nonlocal. In contrast to the original imbricate nonlocal model for strain-softening, the stresses which figure in the constitutive relation satisfy the differential equations of equilibrium and boundary conditions of the usual classical form, and no zero-energy spurious modes of instability are encountered. However, the field operator for the present formulation is in general nonsymmetric, although not for the elastic part of response. It is shown that the energy dissipation and damage cannot localize into regions of vanishing volume. The static strain-localization instability, whose solution is reduced to an integral equation, is found to be controlled by the characteristic length of the material introduced in the averaging rule. The calculated static stability limits are close to those obtained in the previous nonlocal studies, as well as to those obtained by the crack band model in which the continuum is treated as local but the minimum size of the strain-softening region (localization region) is prescribed as a localization limiter. Furthermore, the rate of convergence of static finite-element solutions with nonlocal damage is studied and is found to be of a power type, almost quadratric. A smooth weighting function in the averaging operator is found to lead to a much better convergence than unsmooth functions.

Determination of Fracture Energy from Size Effect and Brittleness Number

A series of tests on the size effect due to blunt fracture is reported and analyzed. It is proposed to define the fracture energy as the specific energy required for crack growth in an infinitely large specimen. Theoretically, this definition eliminates the effects of specimen size, shape, and the type of loading on the fracture energy values. The problem is to identify the correct size-effect law to be used for extrapolation to infinite size. It is shown that Bazants recently proposed simple size-effect law is applicable for this purpose as an approximation. Indeed, very different types of specimens, including three-point bent, edge-notched tension, and eccentric compression specimens, are found to yield approximately the same fracture energy values. Furthermore, the R-curves calculated from the size effect measured for various types of specimens are found to have approximately the same final asymptotic values for very long crack lengths, although they differ very much for short crack lengths. The fracture energy values found from the size effect approximately agree with the values of fracture energy for the crack band model when the rest results are fitted by finite elements. Applicability of Bazants brittleness number, which indicates how close the behavior of specimen or structure of any geometry is to linear elastic fracture mechanics and to plastic limit analysis, is validated by test results. Comparisons with Mode II shear fracture tests are also reported.

Size Effect on Diagonal Shear Failure of Beams Without Stirrups

The paper presents the results of recent tests on diagonal shear failure of reinforced concrete beams without stirrups. The results indicate a significant size effect and show a good agreement with Bazants law for size effect. Scatter of the test results is much lower than that previously found by studying extensive test data from the literature, which have not been obtained on geometrically similar beams. The tests also show that preventing bond slip of the longituudinal bars causes an increase of the brittleness number of the beam. It is concluded that the current design approach, which is intended to provide safety against the diagonal crack initiation load, should be replaced or supplemented by a design approach based on the ultimate load, in which a size effect of the fracture mechanics type, due to release of stored energy must be taken into account.

Mathematical model for kinetics of alkali-silica reaction in concrete

Vast though the literature on the chemistry of the alkali ‐ silica reaction (ASR) in concrete has become, a comprehensive mathematical model allowing quantitative predictions seems lacking. The present study attempts a step toward this goal. While two distinct problems must be dealt with, namely, (1) the kinetics of the chemical reaction with the associated diffusion processes and (2) fracture mechanics of the damage process, only the former is addressed here. The analysis is focused on the recent attempts by C. Meyers and W. Jin to incorporate ground waste glass (mainly, bottle glass) into concrete. With minor adjustments, though, the model can be applied to ASR in natural aggregates as well. A characteristic unit cubic cell of concrete containing one spherical glass particle is analyzed. A spherical layer of basic ASR gel grows radially inward into the particle, controlled by diffusion of water toward the reaction front. Modification of the solution for the case of mineral aggregates with veins of silica is also indicated. Imbibition of additional water from the adjacent capillary pores, which causes swelling of the gel, is described as a second diffusion process, limited by the development of pressure due to resistance of concrete to expansion. The water used up to form the basic ASR gel and imbibed to cause its swelling appears as a sink term in the non-linear diffusion equation for the global water transport through a concrete structure. The differential equations are integrated numerically. The study of the effects of various parameters provides improved understanding of the ASR, and especially the effect of glass particle size. Full prediction will require measurements of some parameters of the reaction processes. D 2000 Elsevier Science Inc. All rights reserved.

Fracture properties and brittleness of high-strength concrete

The size effect method for determining material fracture characteristics, as previously proposed by Bazant is applied to typical high-strength concrete. Geometrically similar 3-point bending specimens are tested and the measured peak load values are used to obtain the fracture energy, the fracture toughness, the effective length of the fracture process zone, and the effective critical crack tip opening displacement. The brittleness of the material is shown to be objectively quantified through the size-effect method. Comparing the material fracture properties obtained with those of normal strength concrete shows that an increase of 16 % in compressive srength causes: (1) increase of fracture toughness; (2) decrease of effective fracture process zone length; (3) more than doubling of the brittleness number. The brittleness number, however, is still not high enough to permit the use of linear elastic fracture mechanics. The R-curves are demonstrated to derive according to the size effect law exclusively from the maximum loads of specimens of various sizes and yield remarkably good predictions of the load-deflection curves.

A thermodynamically consistent approach to microplane theory. Part I. Free energy and consistent microplane stresses

Microplane models are based on the assumption that the constitutive laws of the material may be established between normal and shear components of stress and strain on planes of generic orientation (so-called microplanes), rather than between tensor components or their invariants. In the kinematically constrained version of the model, it is assumed that the microplane strains are projections of the strain tensor, and the stress tensor is obtained by integrating stresses on microplanes of all orientations at a point. Traditionally, microplane variables were defined intuitively, and the integral relation for stresses was derived by application of the principle of virtual work. In this paper, a new thermodynamic framework is proposed. A free-energy potential is defined at the microplane level, such that its integral over all orientations gives the standard macroscopic free energy. From this simple assumption, it is possible to introduce consistent microplane stresses and their corresponding integral relation to the macroscopic stress tensor. Based on this, it is shown that, in spite of the excellent data fits achieved, many existing formulations of microplane model were not guaranteed to be fully thermodynamically compliant. A consequence is the lack of work conjugacy between some of the microplane stress and strain variables used, and the danger of spurious energy dissipation/generation under certain load cycles. The possibilities open by the new theoretical framework are developed further in Part II companion paper.

Elastodynamic Near-Tip Stress and Displacement Fields for Rapidly Propagating Cracks in Orthotropic Materials

The near-tip angular variations of elastodynamic stress and displacement fields are investigated for rapid transient crack propagation in isotropic and orthotropic materials. The 2-dimensional near-tip displacement fields are assumed in the general form r/sup P/ T(t, c) K(theta, c), where c is a time-varying velocity of crack propagation, and it is shown that p = 0.5. For isotropic materials, K(theta, c) is determined explicitly by analytical considerations. A numerical procedure is employed to determine K(theta, c) for orthotropic materials. The tendency of the maximum stresses to move out of the plane of crack propagation as the speed of crack propagation increases is more pronounced for orthotropic materials, for the case that the crack propagates in the direction of the larger elastic modulus. The angular variations of the near-tip fields are the same for steady-state and transient crack propagation, and for propagation along straight and curved paths, provided that the direction of crack propagation and the speed of the crack tip vary continuously.

EFFECT OF TEMPERATURE AND HUMIDITY ON FRACTURE ENERGY OF CONCRETE

Fracture experiments were conducted at temperatures from 20 to 200 C (68 to 392 F) to determine the dependence of the Mode I fracture energy of concrete on temperature as well as the specific water content. The fracture energy values were determined by testing geometrically similar specimens of sizes in the ratio 1:2:4:8 and then applying Bazants size effect law. Three-point bend specimens and eccentric compression specimmns are found to yield approximately the same fracture energies, regardless of temperature. To describe the temperature dependence of fracture energy, a recently derived simple formula based on the activation energy theory (rate process theory) is used and verified by test results. The temperature effect is determined both for concrete predried in an oven and for wet (saturated) concrete. By interpolation, an approximate formula for the effect of moisture content on fracture energy is also obtained. This effect is found to be small at room temperature but large at temperatures close to 100 C (212 F).

ENERGETIC-STATISTICAL SIZE EFFECT IN QUASIBRITTLE FAILURE AT CRACK INITIATION

The size effect on the nominal strength of quasibrittle structures failing at crack initiation, and particularly on the modulus of rupture of plain concrete beams, is analyzed. First, an improved deterministic formula is derived from the energy release caused by a boundary layer of cracking (initiating fracture process zone) whose thickness is not negligible compared with beam depth. To fit the test data, a rapidly converging iterative nonlinear optimization algorithm is developed. The formula is shown to give an excellent agreement with the existing test data on the size effect on the modulus of rupture of plain concrete beams. The data range, however, is much too limited; it does not cover the extreme sizes encountered in arch dams, foundations, and retaining walls. Therefore, it becomes necessary to extrapolate on the basis of a theory. For extreme sizes, the Weibull type statistical effect of random material strength must be incorporated into the theory. Based on structural analysis with the recently developed statistical nonlocal model, a generalized energetic-statistical size effect formula is developed. The formula represents asymptotic matching between the deterministic-energetic formula, which is approached for small sizes, and the power law size effect of the classical Weibull theory, which is approached for large sizes. In the limit of infinite Weibull modulus, the deterministic-energetic formula is recovered. Data fitting with the new formula reveals that, for concrete and mortar, the Weibull modulus is approximately equal to 24 rather than 12, the value widely accepted so far. This means that, for extreme sizes, the nominal strength (modulus of rupture) decreases, for two-dimensional (2D) similarity, as the -1/12 power of the structure size, and for 3D similarity, as the -1/8 power (whereas the -1/4 power has been assumed thus far). The coefficient of variation characterizing the scatter of many test results for one shape and one size is shown not to give the correct value of Weibull modulus because the energetic size effect inevitably intervenes. The results imply that the size effect at fracture initiation must have been a significant contributing factor in many disasters (for example, those of Malpasset Dam, Saint Francis Dam and Schoharie Creek Bridge.)

Effect of Cracking on Drying Permeability and Diffusivity of Concrete

The increase of overall drying permeability and diffusivity of concrete due to cracking is determined experimentally and formulated mathematically. The test specimens are C-shaped beams deformed by a tie rod and reinforced on the tensile face so that uniformly spaced cracks are produced. The difference in the loss of weight for various drying periods between cracked and uncracked specimens is measured and used to quantify the effect on permeability and diffusivity. The overall drying diffusivity and permeability in the cracking direction, which is theoretically proportional to the crack width cubed and inversely proportional to the crack spacing, is found to increase about 2.25 times for crack width 0.1 mm and crack spacing 70 mm. Although appreciable, this value is two orders of magnitude less than the theoretical upper bound predicted on the basis of viscous flow calulation if it is assumed that the cracks are of constant thickness, have planar walls, and are continuous. It is concluded that even though the major cracks are seen to be continuous on the specimen surface, they must be discontinuous in the specimen interior, perhaps being interconnected by much narrower necks with a width about 10 times smaller. This fact is of interest for deducing fracture process zone models from visual observations of cracks on the specimen surface. Although approximate, the presently derived formula for the increase of diffusivity and permeability is directly usable in finite element programs for drying or wetting of concrete.