Showing papers by "Zdenek P. Bazant published in 1979"

Physical model for steel corrosion in concrete sea structures­ theory

Abstract: A physical-mathematical model is developed for concrete exposed to sea water. The model describes: (1)Diffusion of oxygen, chloride ions, and pore water through the concrete cover of reinforcement; (2)ferrous hydroxide near steel surface; (3)the depassivation of steel due to critical chloride ion concentration; (4)the cathodic and anodic electric potentials depending on oxygen and ferrous hydroxide concentrations according to Nernst equation; (5)the polarization of electrodes due to changes in concentration of oxygen and ferrous hydroxide; (6)the flow of electric current through the electrolyte in pores of concrete; (7)the mass sinks or sources of oxygen, ferrous hydroxide, and hydrated red rust electrodes, based on Faraday law; and (8)the rust production rate, based on reaction kinetics. To enable calculations, numerical values of all coefficients are indicated. The theory is completed by formulating the problem as an initial-boundary value problem.

Blunt Crack Band Propagation in Finite Element Analysis

Abstract: A propagating smeared crack band of blunt front is much simpler to model by finite elements than a sharp interelement crack, especially when the propagation direction is unknown. For concrete or rock, a smeared crack band is also more realistic. A strength criterion is generally used for the propagation, but this is not objective because of a strong spurious dependence of results on the chosen element size, Three objective methods, which avoid the use of singularity elements, are proposed. Method A is based on the rate of energy release by the blunt crack band. In method B, the usual strength criterion is used but an adjustment of the strength value according to the element size is proposed. In method C, the propagation direction and crack advance are determined by fitting the Mode I asymptotic series to nodal displacements around the crack front, using an optimization subroutine. Special merits of each method are analyzed and solutions of example problems are compared with exact ones.

PHYSICAL MODEL FOR STEEL CORROSION IN CONCRETE SEA STRUCTURES­ ApPLICATION

Abstract: The theoretical physical model developed in a companion paper, is applied to a simplified calculation of corrosion rates and times to corrosion cracking of concrete cover. Setting up approximate estimates of effective resistance of the corrosion cell, and treating oxygen and chloride ion transport through concrete cover as quisi-stationary and one-dimensional, the corrosion problem is reduced to ordinary differential equations in time. For determining the extents of cathodic and anodic areas and the thickness of the rusting layer, a new principle stating that the actual corrosion current is maximum is postulated. Various steady-state corrosion processes are then analyzed, and after developing approximate formulas for the time of steel depassivation due to chloride ions and for cover cracking due to volume expansion of rust, a number of illustrative numerical examples are given. Diffusivities for chloride ions and oxygen are shown to be usually the controlling factors.

Plastic-Fracturing Theory for Concrete

Abstract: Incremental plasticity and fracturing (microcracking) material theory are combined to obtain a nonlinear triaxial constitutive relation that is incrementally linear. The theory combines the plastic stress decrements with the fracturing stress decrements, which reflect microcracking, and accounts for internal friction, pressure sensitivity, inelastic dilatancy due to microcracking, strain softening, degradation of elastic moduli due to microcracking, and the hydrostatic nonlinearity due to pore collapse. Failure envelopes are obtained from the constitutive law as a collection of the peak points of the stress strain response curves. Six scalar material functions are needed to fully define the monotonic response. One function, the dilatancy due to microcracking, is determined theoretically based on Budianski-O’Connell’s calculation of the effective elastic contents of a randomly microcracked elastic material by the self consistent method for composites.

Nonlinear creep of concrete adaptation and flow

Abstract: A nonlinear integral-type creep law is developed by generalizing the linear superposition integral for the creep rate rather than the total strain. At low service stress level there is a significant (though previously overlooked) nonlinearity that consists in gradual stiffening or adaptation to a sustained compressive stress. It is modeled by a stress-dependent acceleration of the age-dependence of stiffness, and by an adaptation parameter whose rate is a function of the stress and age. The high-stress nonlinearity that consists of a weakening of the stiffness, is essentially without memory and is described by an additive rate-type flow term. Its stress dependence and the flow rate decay is modeled by kinematic hardening. An extension to elevated temperatures, which agrees with recovery data, is indicated. Although uniaxial creep is of primary interest, a rational triaxial generalization involving proper stress invariants is derived.

Thermodynamics of Solidifying or Melting Viscoelastic Material

Abstract: Investigated are thermodynamic restrictions on rate-type creep laws for porous materials which slowly solidify while carrying load (aging of concrete, i.e., gradual filling of pores in concrete by hydration products) or slowly melt (gradual dehydration of concrete at high temperature). Thermodynamic potentials (Helmholtz free energy and complementary or Gibbs free energy) are determined. The chemical dissipation of elastic energy is calculated and the condition of its positiveness is proposed; this requires that elastic relations be introduced in terms of stress and strain rates for solidifying materials and in terms of stresses and strains for melting materials. Some creep laws used in practice are found to have thermodynamically inadmissible form. Creep laws of thermodynamically correct form are shown. The known forms of such laws often cannot, however, fit available long-time creep test data for concrete very well, unless some material parameters or rates are allowed to have thermodynamically inadmissible negative values for short periods of time.

Approximate Relaxation Function for Concrete

Abstract: Presented is an approximate algebraic formula for calculating the relaxation function for aging concrete. The formula is general; it applies to any form of the creep function. Compared to the previously used effective modulus method, the formula reduces the error from up to 37% to within 2% relative to the exact solution according to the superpositon principle. Calculation of long-time relaxation does not require knowledge of the elastic modulus and of the creep function for load duration below one day. The formula enables direct calculation of the age-adjusted effective modulus from the creep function, and thus it allows dispensing with the table of the aging coefficient. This is particularly useful in case of a more sophisticated creep model which reflects the fact that the creep functions for different humidities and sizes have different (nonproportional) shapes.

Concrete reinforcing net: optimum slip-free limit design

Abstract: The proposed slip-free design assures desired safety against large frictional shear slip of crack surfaces, which assures a reduced extent of cracking and damage to concrete. Orthogonal nets in shell or plate walls under in-plane forces are considered. A reinforcement that is up to about 34% heavier is obtained when large shear forces in the bar directions are present. The case when one principal force is compressive is also analyzed, and here the differences become still larger (up to about 67%). The optimum classical (frictionless) limit design is found to be equivalent to the optimum service stress design (except for a common scaling factor), which is an objectionable feature, and the difference between them is due solely to safety factors. In the slip-free design, which is almost as simple as the frictionless design, the critical crack direction leading to the lightest possible reinforcement is not at a 45° angle with the bars, as in the frictionless design, but deviates from it substantially. The safe domain in the plane whose coordinates are the reinforcement ratios is still a hyperbola, but with inclined asymptotes.

Viscoplasticity of Normally Consolidated Clays

Abstract: A constitutive law is developed to model the behavior of normally consolidated isotropic cohesive soils under multidimensional stress or strain paths. This law represents the endochronic form of viscoplasticity and involves a number of material variables that are defined in terms of semi-empirical expressions and model: (1)Strain softening and hardening; (2)densification and dilatancy; (3)frictional aspects; and (4)strain rate dependence of the response. Furthermore, by considering saturated soils as two-phase media, the model accounts for the development of pore pressures due to the volumetric strain in the solid skeleton. Data reported in the literature are used to demonstrate the applicability of the approach, and approximate correlations between material parameters are established to predict the stress-strain-pore response of normally consolidated clays.

Viscoplasticity of Transversely Isotropic Clays

Abstract: A viscoplastic constitutive relation of the endochronic type (i.e., the inelastic strain increments are characterized by an intrinsic time) is formulated to describe the behavior of transversely isotropic clays produced by one-dimensional consolidation. The formulation contains eight material parameters in addition to those needed for isotropic clays. The hardening and softening functions and the densification-dilatancy function are assumed to be given by the same expressions previously found for isotropic clays, but the invariants involved in these expressions are replaced by the proper transversely isotropic invariants. The pore pressure is determined from the volume change and the compressibility of the water, and the constitutive relation is written in terms of the effective stresses. The elastic moduli are assumed to be functions of hydrostatic stress and inelastic dilatancy, and they are correlated with the consolidation stress. Experimental curves of axial strain for various anisotropically consolidated clays have been fit by a time-independent version of the theory, and a satisfatory agreement has been achieved.