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...

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Theoretical stress strain model for confined concrete

http://mech.spbstu.ru/images/b/bd/Potyondy_Cundall_2004_A_bonded-particle_model_for_rock.pdf

A bonded-particle model for rock

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.

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Crack band theory for fracture of concrete

A plastic-damage model for concrete

National Institute of Standards and Technology における超伝導研究及び生活

Fracture of quasibrittle materials with a large zone of distributed cracking is simulated by the particle model (discrete element method). The particles at the microlevel interact only by central forces with a prescribed force-displacement or stress-strain relation, which exhibits postpeak softening and is characterized by microstrength and microfracture energy. It is shown that a regular lattice, even though capable of closely approximating isotropic elastic properties, exhibits strong directional bias favoring propagation along a few preferred directions. A randomly generated particle model has no such bias. With a proper choice of the microlevel constitutive law, it can realistically simulate fracture of an ice floe during impact on a rigid obstacle. Explicit integration of the equations of motion is used to simulate the impact process and to explore the effect of the floe size and its initial velocity on the failure pattern and the history of the contact force.

Particle Model for Quasibrittle Fracture and Application to Sea Ice

Considerable progress has been achieved in fractal characterization of the properties of crack surfaces in quasibrittle materials such as concrete, rock, ice, ceramics and composites. Recently, fractality of cracks or microcracks was proposed as the explanation of the observed size effect on the nominal strength of structures. This explanation, though, has rested merely on intuitive analogy and geometric reasoning, and did not take into account the mechanics of crack propagation. In this paper, the energy-based asymptotic analysis of scaling presented in the preceding companion paper in this issue [1] is extended to the effect of fractality on scaling. First, attention is focused on the propagation of fractal crack curves (invasive fractals). The modifications of the scaling law caused by crack fractality are derived, both for quasibrittle failures after large stable crack growth and for failures at the initiation of a fractal crack in the boundary layer near the surface. Second, attention is focused on discrete fractal distribution of microcracks (lacunar fractals), which is shown to lead to an analogy with Weibull's statistical theory of size effect due to material strength randomness. The predictions ensuing from the fractal hypothesis, either invasive or lacunar, disagree with the experimentally confirmed asymptotic characteristics of the size effect in quasibrittle structures. It is also pointed out that considering the crack curve as a self-similar fractal conflicts with kinematics. This can be remedied by considering the crack to be an affine fractal. It is concluded that the fractal characteristics of either the fracture surface or the microcracking at the fracture front cannot have a significant influence on the law of scaling of failure loads, although they can affect the fracture characteristics.

/pdf/scaling-of-quasibrittle-fracture-hypotheses-of-invasive-and-3lzbpgb296.pdf

Scaling of quasibrittle fracture : hypotheses of invasive and lacunar fractality, their critique and weibull connection

The universal size-shape effect law is a law that describes the dependence of nominal strength of specimen or structure on both its size and the crack (or notch) length, over the entire range of interest, and exhibits the correct small-size and large-size asymptotic properties as required by the cohesive crack model (or crack band model). The main difficulty has been the transition of crack length from 0, in which case the size effect is Type 1, to deep cracks (or notches), in which case the size effect is Type 2 and is fundamentally different from Type 1, with different asymptotes. In this transition, the problem is not linearizable because the notch is not much larger than the fracture process zone. The previously proposed universal law could not be verified experimentally for the Type 1-Type 2 transition because sufficient test data were lacking. The current study is based on recently obtained comprehensive fracture test data for three-point bend beams cast from one batch of the same concrete and cured and tested under identical conditions. The test data reveal that the Type 1-Type 2 transition in the previous universal law has insufficient accuracy and cannot be captured by Taylor series expansion of the energy release rate function of linear elastic fracture mechanics. Instead, the size effect for a zero notch and for the transitional range is now characterized in terms of the strain gradient at the specimen surface, which is the main variable determining the degree of stress redistribution by the boundary layer of cracking. The new universal law is shown to fit the comprehensive data quite well, with a coefficient of variation of only 2.3%.

/pdf/universal-size-shape-effect-law-based-on-comprehensive-4y340srcwv.pdf

Universal Size-Shape Effect Law Based on Comprehensive Concrete Fracture Tests

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.

Effect of Cracking on Drying Permeability and Diffusivity of Concrete

This paper deals with determination of macroscopic fracture characteristics of random particle systems, which represents a fundamental but little explored problem of micromechanics of quasibrittle materials. The particle locations are randomly generated and the mechanical properties are characterized by a triangular softening force-displacement diagram for the interparticle links. An efficient algorithm, which is used to repetitively solve large systems, is developed. This algorithm is based on the replacement of stiffness changes by inelastic forces applied as external loads. It makes it possible to calculate the exact displacement increments in each step without iterations and using only the elastic stiffness matrix. The size effect method is used to determine the dependence of the mean macroscopic fracture energy and the mean effective process zone size of two-dimensional particle systems on the basic microscopic characteristics such as the microscopic fracture energy, the dominant inhomogeneity spacing (particle size) and the coefficients of variation of the microstrength and the microductility. Some general trends are revealed and discussed.

Zdenek P. Bazant

Papers

Particle Model for Quasibrittle Fracture and Application to Sea Ice

Scaling of quasibrittle fracture : hypotheses of invasive and lacunar fractality, their critique and weibull connection

Universal Size-Shape Effect Law Based on Comprehensive Concrete Fracture Tests

Effect of Cracking on Drying Permeability and Diffusivity of Concrete

Macroscopic fracture characteristics of random particle systems