Showing papers by "Zdenek P. Bazant published in 2013"

Microplane model M7 for plain concrete. I: Formulation

Abstract: Mathematical modeling of the nonlinear triaxial behavior and damage of such a complex material as concrete has been a long-standing challenge in which progress has been made only in gradual increments. The goal of this study is a realistic and robust material model for explicit finite-element programs for concrete structures that computes the stress tensor from the given strain tensor and some history variables. The microplane models, which use a constitutive equation in a vectorial rather than tensorial form and are semimultiscale by virtue of capturing interactions among phenomena of different orientation, can serve this goal effectively. This paper presents a new concrete microplane model, M7, which achieves this goal much better than the previous versions M1–M6 developed at Northwestern University since 1985. The basic mathematical structure of M7 is logically correlated to thermodynamic potentials for the elastic regime, the tensile and compressive damage regimes, and the frictional slip regi...

Microplane Model M7 for Plain Concrete. II: Calibration and Verification

Abstract: The microplane material model for concrete, formulated mathematically in the companion paper, is calibrated by material test data from all the typical laboratory tests taken from the literature. Then, the model is verified by finite-element simulations of data for some characteristic tests with highly nonuniform strain fields. The scaling properties of model M7 are determined. With the volumetric stress effect taken from the previous load step, the M7 numerical algorithm is explicit, delivering in each load step the stress tensor from the strain tensor with no iterative loop. This makes the model robust and suitable for large-scale finite-element computations. There are five free, easily adjustable material parameters, which make it possible to match the given compressive strength, the corresponding strain, the given hydrostatic compression curve, and certain triaxial aspects. In addition, there are many fixed, hard-to-adjust parameters, which can be taken to be the same for all concretes. The opt...

On the Importance of Work-Conjugacy and Objective Stress Rates in Finite Deformation Incremental Finite Element Analysis

Abstract: This paper is concerned with two issues that arise in the finite element analysis of 3D solids. The first issue examines the objectivity of various stress rates that are adopted in incremental analysis of solids. In doing so, it is revealed that large errors are incurred by an improper choice of stress rate. An example problem is presented to show the implications of the choice of stress rate. The second issue addresses the need to maintain work-conjugacy in formulating and solving bifurcation buckling problems of 3D elastic solids. Four popular commercial codes are used to obtain buckling loads of an axially compressed thick sandwich panel, and it is shown that large errors in buckling load predictions are incurred as a result of violating the requirement of work-conjugacy. Remedies to fix the errors in the numerical solution strategy are given.

Relaxation of Prestressing Steel at Varying Strain and Temperature: Viscoplastic Constitutive Relation

Abstract: Recent studiesof excessivemultidecade deflectionsof prestressed segmentally erectedboxgirders revealedthatmore accurate pre- dictionsoftheprestresslossduetosteelrelaxationareneededforthedesignoflarge-spancreep-sensitivestructures.Inparticular,thelossneeds to be calculated as part of creep structural analysis, during which the strain of concrete to which the prestressing steel is bonded varies in each timestep.TheexistingempiricalformulasusedintheEuropeanModelCodeandAmericanpractice,whicharevalidonlyforconstantstrainand constant temperature, are here generalized to arbitrarily variable strain and temperature, heeding obvious asymptotic restrictions and the fact that steel is a viscoplastic material whose constitutive principles are well known. The resulting formula is a memoryless nonlinear equation for theviscoplasticstrainrateofsteelasafunctionofthecurrentstress,strain,andtemperature.Close fitsofallthemaintestdatafromtheliterature, including the available data on the effects of strain and temperature changes, are achieved. The effect of temperature is found to be quite im- portant and is formulated on the basis of the activation energy of viscoplastic flow of metals. Finally, the need for further tests at variable strain and variable temperature is emphasized. DOI: 10.1061/(ASCE)EM.1943-7889.0000533. © 2013 American Society of Civil Engineers. CE Database subject headings: Prestressing; Relaxation (mechanical); Viscoplasticity; Steel; Temperature effects. Author keywords: Prestress loss; Relaxation; Viscoplasticity.

Size Effect on Strength and Lifetime Distributions of Quasibrittle Structures Implied by Interatomic Bond Break Activation

Abstract: Engineering structures such as aircrafts, bridges, dams, nuclear containments and ships, as well as computer circuits, chips and MEMS, should be designed for failure probability < 10 to 10 per lifetime. However, the safety factors required to ensure it are still determined empirically, even though they represent much larger and much more uncertain corrections to deterministic calculations than do the typical errors of modern computer analysis of structures. Bažant and Pang recently developed (and presented at previous ECF) a new theory for the cumulative distribution function (cdf) for the strength of quasibrittle structures failing at macro-fracture initiation, and offered its simplified justification in terms of thermally activated inter-atomic bond breaks. Presented here is a refined justification of this theory based on fracture mechanics of atomic lattice cracks advancing through the lattice by small jumps over numerous activation energy barriers on the surface of the free energy potential of the lattice. For the strength of the representative volume element (RVE) of material, simple statistical models based on chains and bundles are inadequate and a model consisting of a hierarchy of series and parallel couplings is adopted. The theory implies that the strength of one RVE must have a Gaussian cdf, onto which a Weibullian (or power-law) tail is grafted on the left at the failure probability of about 10 to 10. A positive-geometry structure of any size can be statistically modeled as a chain of RVEs. With increasing structure size, the Weibullian part of the cdf of structural strength expands from the left tail and the grafting point moves into the Gaussian core, until eventually, for a structure size exceeding about 10 equivalent RVEs, the entire cdf becomes Weibullian. Relative to the standard deviation, this transition nearly doubles the distance from the mean to the point of failure probability 10. Contrary to recent empirical models, it is found that the strength threshold must be zero. This finding and the size effect on cdf has a major effect on the required safety factor. The theory is further extended to model the lifetime distribution of quasibrittle structures under constant load (creep rupture). It is shown that, for quasibrittle materials, there exists a strong size effect on not only the structural strength but also the lifetime, and that the latter is stronger. Like the cdf of strength, the cdf of lifetime, too, is found to change from Gaussian with a remote power-law tail for small sizes, to Weibullian for large sizes. Furthermore, the theory provides an atomistic justification for the powerlaw form of Evans’ law for crack growth rate under constant load and of Paris’ law for crack growth under cyclic load. For various quasibrittle materials, such as industrial and dental ceramics, concrete and fibrous composites, it is finally demonstrated that the proposed theory correctly predicts the experimentally observed deviations of strength and lifetime histograms from the classical Weibull theory, as well as the deviations of the mean size effect curves from a power law.

Constitutive Model for Material Comminuting at High Shear Rate

Abstract: The modeling of high velocity impact into brittle or quasibrittle solids is hampered by the unavailability of a constitutive model capturing the effects of material comminution into very fine particles. The present objective is to develop such a model, usable in finite element programs. The comminution at very high strain rates can dissipate a large portion of the kinetic energy of an impacting missile. The spatial derivative of the energy dissipated by comminution gives a force resisting the penetration, which is superposed on the nodal forces obtained from the static constitutive model in a finite element program. The present theory is inspired partly by Grady's model for comminution due to explosion inside a hollow sphere, and partly by analogy with turbulence. In high velocity turbulent flow, the energy dissipation rate is enhanced by the formation of micro-vortices (eddies) which dissipate energy by viscous shear stress. Similarly, here it is assumed that the energy dissipation at fast deformation of a confined solid gets enhanced by the release of kinetic energy of the motion associated with a high-rate shear strain of forming particles. For simplicity, the shape of these particles in the plane of maximum shear rate is considered to be regular hexagons. The rate of release of free energy density consisting of the sum of this energy and the fracture energy of the interface between the forming particle is minimized. The particle sizes are assumed to be distributed according to Schuhmann's power law. It is concluded that the minimum particle size is inversely proportional to the (2/3)-power of the shear strain rate, that the kinetic energy release is to proportional to the (2/3)-power, and that the dynamic comminution creates an apparent material viscosity inversely proportional to the (1/3)-power of the shear strain rate.