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

http://cee.northwestern.edu/people/bazant/PDFs/Papers/495.pdf

Identification of viscoelastic C-S-H behavior in mature cement paste by FFT-based homogenization method

Recent investigations prompted by a disaster in Palau revealed that worldwide there are 69 long-span segmental prestressed-concrete box-girder bridges that suffered excessive multi-decade deflections, while many more surely exist. Although the excessive deflections were shown to be caused mainly by obsolescence of design recommendations or codes for static creep, some engineers suspect that cyclic creep might have been a significant additional cause. Many investigators explored the cyclic creep of concrete experimentally, but a rational mathematical model that would be anchored in the microstructure and would allow extrapolation to a 100-year lifetime is lacking. Here it is assumed that the cause of cyclic creep is the fatigue growth of pre-existing microcracks in hydrated cement. The resulting macroscopic strain is calculated by applying fracture mechanics to the microcracks considered as either tensile or, in the form of a crushing band, as compressive. This leads to a mathematical model for cyclic creep in compression, which is verified and calibrated by laboratory test data from the literature. The cyclic creep is shown to be proportional to the time average of stress and to the 4th power of the ratio of the stress amplitude to material strength. The power of 4 is supported by the recent finding that, on the atomistic scale, the Paris law should have the exponent of 2 and that the exponent must increase due to scale bridging. Exponent 4 implies that cyclic creep deflections are enormously sensitive to the relative amplitude of the applied cyclic stress. Calculations of the effects of cyclic creep in six segmental prestressed concrete box girders indicate that, because of self-weight dominance, the effect on deflections absolutely negligible for large spans ( > 150 m ) . For small spans ( 40 m ) the cyclic creep deflections are not negligible but do not matter since the static creep causes in such bridges upward deflections. However, the cyclic creep is shown to cause in bridges with medium and small spans ( 80 m ) a significant residual tensile strain which can produce deleterious tensile cracking at top or bottom face of the girder.

http://www.civil.northwestern.edu/people/bazant/PDFs/Papers/537.pdf

Theory of cyclic creep of concrete based on Paris law for fatigue growth of subcritical microcracks

The spacing and width of cracks in a parallel crack system is approximately analyzed using the energy criterion of fracture mechanics as well as the strength criterion. The energy criterion indicates that the crack spacing is a function of the axial strain of the bars and also depends on bar spacing, bar diameter, fracture energy of concrete, and its elastic modulus. Both the energy and strength criteria yield a minimum strain necessary to produce any cracks. The energy criterion and the bond slip conditions further yield a lower bound on possible spacing of continuous cracks. The rules for the formation of shorter, partial length, cracks are also set up. Approximate expressions for the crack width at, and away, from the bars are derived. Numerical comparisons indicate satisfactory agreement with existing test data, and lend theoretical support to one aspect of the empirical Gergely-Lutz formula obtained by statistical regression analysis of test data. Finally, formation of skew cracks in a biaxially stressed and biaxially reinforced plate is analyzed and a crack spacing formula is derived for one typical case.

Spacing of Cracks in Reinforced Concrete

This article attempts to review the progress achieved in the understanding of scaling and size effect in the failure of structures. Particular emphasis is placed on quasibrittle materials for which the size effect is complicated. Attention is focused on three main types of size effects, namely the statistical size effect due to randomness of strength, the energy release size effect, and the possible size effect due to fractality of fracture or microcracks. Definitive conclusions on the applicability of these theories are drawn. Subsequently, the article discusses the application of the known size effect law for the measurement of material fracture properties, and the modeling of the size effect by the cohesive crack model, nonlocal finite element models and discrete element models. Extensions to compression failure and to the rate-dependent material behavior are also outlined. The damage constitutive law needed for describing a microcracked material in the fracture process zone is discussed. Various applications to quasibrittle materials, including concrete, sea ice, fiber composites, rocks and ceramics are presented.

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Scaling of structural failure

The multi scale approach was pioneered by Tadmor, Ortiz and Phillips [1] for atomistic-based quasi-continuum analysis of dislocations and hardening plasticity of polycrystalline metals. In that case, the structural failure is due to necking, which is caused by nonlinear geometric effects of finite strain, or to sharp fracture, which is modelled separately (see also [2]). There can be no dispute that the multi scale approach is realistic, delivering to the continuum macroscale an essential information on the physical behavior on the subscale. However, applying the multi scale approach to failure due to interacting crack systems, or to softening damage such as distributed cracking, is an entirely different matter. To clarify it, let us discuss a few typical multi-scale approaches representative of a flood of recent publications.

Zdenek P. Bazant

Papers

Identification of viscoelastic C-S-H behavior in mature cement paste by FFT-based homogenization method

Theory of cyclic creep of concrete based on Paris law for fatigue growth of subcritical microcracks

Spacing of Cracks in Reinforced Concrete

Scaling of structural failure

Can Multiscale-Multiphysics Methods Predict Softening Damage and Structural Failure?