Zdenek P. Bazant

Bio: Zdenek P. Bazant is an academic researcher from Northwestern University. The author has contributed to research in topics: Creep & Fracture mechanics. The author has an hindex of 82, co-authored 301 publications receiving 20908 citations. Previous affiliations of Zdenek P. Bazant include École Polytechnique Fédérale de Lausanne & Rensselaer Polytechnic Institute.

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.

Elastic and Fracture Behavior of Three-Dimensional Ply-to-Ply Angle Interlock Woven Composites: Through-Thickness, Size Effect, and Multiaxial Tests

Abstract: This work presents a comprehensive investigation of the elastic and fracture behavior of ply-to-ply angle interlock three-dimensional woven composites. The research investigated novel splitting and wedge-driven out-of-plane fracture tests to shed light on the tensile fracture behavior in the thickness direction and to provide estimates of the out-of-plane tensile strength and fracture energy. In addition, size effect tests on geometrically-scaled Single Edge Notch Tension (SENT) specimens were performed to fully characterize the intra-laminar fracture energy of the material and to study the scaling of structural strength in this type of three-dimensional composites. The results confirmed that size effect in the structural strength of these materials is significant. In fact, even if the range of sizes investigated was broader than in any previous size effect study on traditional laminated composites and two-dimensional textile composites, all the experimental data fell in the transition zone between quasi-ductile and brittle behavior. This implies strong damage tolerance of the investigated three-dimensional composites. The analysis of the data via Bazant's Type II Size Effect Law (SEL) enabled the objective characterization of the intra-laminar fracture energy of three-dimensional composites for the first time. Finally, Arcan rig tests combined with X-ray micro-computed tomography allowed unprecedented insights on the different damage mechanisms under multi-axial nominal loading conditions, particularly tension-dominated and shear-dominated conditions.

Inelastic Buckling of Concrete Column in Braced Frame

Abstract: The paper proposes an improved method of analysis of reinforced concrete columns in braced (no-sway) frames, which is suitable as a simple computer solution for design practice and is more realistic than the existing ACI and CBB methods. The elastic restraint provided by beams adjacent to columns is described by rotational springs. The inelastic behavior of concrete is defined by a uniaxial stress-strain curve with postpeak softening in compression and a zero strength in tension. Plasticity of reinforcement is also considered. The deflection curve is assumed to be a sine curve. The improvement consists in considering the wavelength as unknown and variable during loading. The problem is reduced to a system of seven nonlinear algebraic equations, which are easily solved for small increments of axial displacement by a standard library optimization algorithm. The convergence always occurs and is fast if the increments are small enough. The influence of various param­ eters on the load-deflection curve, the path in the diagram of axial load P versus moment M, and the failure envelope are studied. Various phenomena, such as the possibility of a concave P(M) path at constant load eccentricity, are explained. It is shown that the ACI approach is slightly conservative in most cases, although situations exist in which the ACI approach is either grossly overconservative or slightly unconservative.

Finite Strain Tube-Squash Test of Concrete at High Pressures and Shear Angles up to 70 Degrees

Abstract: In this study, a new type of concrete test, called the tube-squash test, that achieves, without fracturing under high pressures, enormous shear and deviatoric strains is developed. Tubes 76.2 mm and 63.5 mm in diameter, with wall thicknesses of 14.22 mm and 12.7 mm, made of highly ductile steel alloy, are filled with concrete. After curing, they are squashed in a high-capacity compression testing machine to half their original length, forcing the tubes to bulge. Normal and high-strength concretes at hydrostatic pressures of approximately 125 MPa are found to be remarkably ductile, capable of sustaining shear angles over 70 deg (1.225 rad) without visually detectable cracks or voids, though significant distributed mechanical damage does take place. Tests of cores drilled out from squashed tubes show that, after such enormous deformation, the concrete still retains approximately 25-35% of its initial uniaxial compression strength and initial stiffness and approximately 10-20% of its initial split-cylinder tensile strength. What makes the tube-squash test meaningful is the development of a relatively simple method of test results analysis that avoids solving the inverse nonlinear finite strain problem with finite elements despite high nonuniformity of the strain field. Approximate stress-strain diagrams of concrete at such large shear angles and strains are obtained by finite strain analysis of the middle cross section utilizing the measured lateral expansion, axial shortening, and bulge profile. The finite strain triaxial plastic constitutive law of the steel alloy needs to be determined first, and a method to do that is also formulated. Approximate stress-strain diagrams and internal friction angles of concrete are deduced from the test without making any hypotheses about its constitutive equation. Tests of tubes filled by hardened portland cement paste and cement mortar, as well as tubes with a snugly fitted limestone insert, show similar ductile shear strains and residual strengths.

Softening-Induced Dynamic Localization Instability: Seismic Damage in Frames

Abstract: This paper analyzes dynamic localization of damage in structures with softening inelastic hinges and studies implications for the seismic response of reinforced concrete or steel frames of buildings or bridges. First, the theory of limit points and bifurcation of the symmetric equilibrium path due to localization of softening damage is reviewed. It is proven that, near the state of static bifurcation or near the static limit point, the primary (symmetric) path of dynamic response or periodic response temporarily develops Liapunov-type dynamic instability such that imperfections representing deviations from the primary path grow exponentially or linearly while damage in the frame localizes into fewer softening hinges. The implication for seismic loading is that the kinetic energy of the structure must be absorbed by fewer hinges, which means faster collapse. The dynamic localizations are demonstrated by exact analytical solutions of torsional rotation of the floor of a symmetric and symmetrically excited frame, and of horizontal shear excitation of a building column. Static bifurcations with localization are also demonstrated for a portal frame, a multibay frame, and a multibay-multistory frame. The widely used simplification of a structure as a single-degree-of-freedom oscillator becomes invalid after the static bifurcation state is passed.

Theoretical stress strain model for confined concrete

Abstract: A stress‐strain model is developed for concrete subjected to uniaxial compressive loading and confined by transverse reinforcement. The concrete section may contain any general type of confining steel: either spiral or circular hoops; or rectangular hoops with or without supplementary cross ties. These cross ties can have either equal or unequal confining stresses along each of the transverse axes. A single equation is used for the stress‐strain equation. The model allows for cyclic loading and includes the effect of strain rate. The influence of various types of confinement is taken into account by defining an effective lateral confining stress, which is dependent on the configuration of the transverse and longitudinal reinforcement. An energy balance approach is used to predict the longitudinal compressive strain in the concrete corresponding to first fracture of the transverse reinforcement by equating the strain energy capacity of the transverse reinforcement to the strain energy stored in the concret...

Crack band theory for fracture of concrete

Abstract: A fracture theory for a heterogenous aggregate material which exhibits a gradual strain-softening due to microcracking and contains aggregate pieces that are not necessarily small compared to structural dimensions is developed. Only Mode I is considered. The fracture is modeled as a blunt smeard crack band, which is justified by the random nature of the microstructure. Simple triaxial stress-strain relations which model the strain-softening and describe the effect of gradual microcracking in the crack band are derived. It is shown that it is easier to use compliance rather than stiffness matrices and that it suffices to adjust a single diagonal term of the complicance matrix. The limiting case of this matrix for complete (continuous) cracking is shown to be identical to the inverse of the well-known stiffness matrix for a perfectly cracked material. The material fracture properties are characterized by only three parameters—fracture energy, uniaxial strength limit and width of the crack band (fracture process zone), while the strain-softening modulus is a function of these parameters. A method of determining the fracture energy from measured complete stres-strain relations is also given. Triaxial stress effects on fracture can be taken into account. The theory is verified by comparisons with numerous experimental data from the literature. Satisfactory fits of maximum load data as well as resistance curves are achieved and values of the three material parameters involved, namely the fracture energy, the strength, and the width of crack band front, are determined from test data. The optimum value of the latter width is found to be about 3 aggregate sizes, which is also justified as the minimum acceptable for a homogeneous continuum modeling. The method of implementing the theory in a finite element code is also indicated, and rules for achieving objectivity of results with regard to the analyst's choice of element size are given. Finally, a simple formula is derived to predict from the tensile strength and aggregate size the fracture energy, as well as the strain-softening modulus. A statistical analysis of the errors reveals a drastic improvement compared to the linear fracture theory as well as the strength theory. The applicability of fracture mechanics to concrete is thus solidly established.