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

The paper presents two micromechanics arguments showing that continuum damage caused by microcracking ought to be nonlocal, defined by a spatial integral. Argument I is an analysis of a simplified model in which the microcracks are of such size, density, and arrangement that that they do not interact. The release of stored energy caused by the formation of one microcrack is calculated as a function of the associated relative displacement across the cell, which corresponds to the average strain of the macroscopic continuum. After imposing two homogenizing conditions, it is shown that damage is a nonlocal variable that is a function of the averaged (nonlocal) strain from a certain neighborhood of the given point. Argument II is an analysis of a body with arbitrary interacting cracks. The local damage is proportional to the forces applied on the cracks to replace the stresses before cracking. Crack formation changes the openings of the neighboring cracks, which represents an interaction described by crack in...

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Why Continuum Damage is Nonlocal: Micromechanics Arguments

This paper and its companion paper present the main results of an assessment of the fire in the Channel Tunel (the “Chunnel”), which destroyed a part of the concrete tunnel rings by thermal spalling. The study seeks (1) to evaluate the effect of thermal damage (loss of elastic stiffness) and thermal decohesion (loss of material strength) upon the stress state and cracking at a structural level; and (2) to check whether restrained thermal dilatation can explain the thermal spalling observed during the fire. In the present paper, a macroscopic material model for rapidly heated concrete is developed. It accounts explicitly for the dehydration of concrete and its cross-effects with deformation (chemomechanical couplings) and temperature (chemothermal couplings). The thermal decohesion is considered as chemoplastic softening within the theoretical framework of chemoplasticity. Furthermore, kinetics of dehydration, dimensional analysis, and thermodynamic equilibrium considerations show that a unique thermal dehydration function exists that relates the hydration degree to the temperature rise, provided that the characteristic time of dehydration is much inferior to the characteristic time of structural heat conduction. The experimental determination of the thermal dehydration function from in-situ measurements of the elastic modulus versus furnace temperature rise is shown from experimental data available from the chunnel concrete. Finally, by way of an example, the proposed constitutive model for rapidly heated concrete is combined with the three-parameter William-Warnke criterion extended to isotropic chemoplastic softening.

The chunnel fire. I: Chemoplastic softening in rapidly heated concrete

Crack growth caused by load repetitions in geometrically similar notched concrete specimens of various sizes is measured by means of the compliance method. It is found that the Paris law, which states that the crack length increment per cycle is a power function of the stress intensity factor amplitude, is valid only for one specimen size (the law parameters being adjusted for that size) or asymptotically, for very large specimens. To obtain a general law, the Paris law is combined with size-effect law for fracture under monotonic loading, proposed previously by Bazant. This leads to a size-adjusted Paris law, which gives the crack length increment per cycle as a power function of the amplitude of size-adjusted stress intensity factor. The crack growth is also characterized in terms of the nominal stress amplitude.

Size Effect in Fatigue Fracture of Concrete

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.

Zdenek P. Bazant

Papers

Why Continuum Damage is Nonlocal: Micromechanics Arguments

The chunnel fire. I: Chemoplastic softening in rapidly heated concrete

Size Effect in Fatigue Fracture of Concrete

Plastic-Fracturing Theory for Concrete

Shear Database for Reinforced Concrete Members without Shear Reinforcement Paper by Karl-Heinz Reineck, Daniel A. Kuchma, Kang Su Kim, and Sina Marx