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

Modeling of the evolution of distributed damage such as microcracking, void formation, and softening frictional slip necessitates strain-softening constitutive models. The nonlocal continuum concept has emerged as an effective means for regularizing the boundary value problems with strain softening, capturing the size effects and avoiding spurious localization that gives rise to pathological mesh sensitivity in numerical computations. A great variety of nonlocal models have appeared during the last two decades. This paper reviews the progress in the nonlocal models of integral type, and discusses their physical justifications, advantages, and numerical applications.

Nonlocal integral formulations of plasticity and damage: Survey of progress

A recent nonlocal damage formulation, in which the spatially averaged quantity was the energy dissipated due to strain-softening, is extended to a more general form in which the strain remains local while any variable that controls strain-softening is nonlocal. In contrast to the original imbricate nonlocal model for strain-softening, the stresses which figure in the constitutive relation satisfy the differential equations of equilibrium and boundary conditions of the usual classical form, and no zero-energy spurious modes of instability are encountered. However, the field operator for the present formulation is in general nonsymmetric, although not for the elastic part of response. It is shown that the energy dissipation and damage cannot localize into regions of vanishing volume. The static strain-localization instability, whose solution is reduced to an integral equation, is found to be controlled by the characteristic length of the material introduced in the averaging rule. The calculated static stability limits are close to those obtained in the previous nonlocal studies, as well as to those obtained by the crack band model in which the continuum is treated as local but the minimum size of the strain-softening region (localization region) is prescribed as a localization limiter. Furthermore, the rate of convergence of static finite-element solutions with nonlocal damage is studied and is found to be of a power type, almost quadratric. A smooth weighting function in the averaging operator is found to lead to a much better convergence than unsmooth functions.

Nonlocal Continuum Damage, Localization Instability and Convergence

A model for the characterization of concrete creep and shrinkage in the design of concrete structures is recommended. It is simplier, agrees better with the experimental data and is justified better theoretically than the previous models. The model complies with the general guidelines recently formulated by RILEM TC 107. Justification of the model and various refinements are to be published shortly in two parts.

Creep and shrinkage prediction model for analysis and design of concrete structures - model B3

The dependence of the fracture energy and the effective process zone length on the specimen size as well as the craek extension from the notch is analyzed on the basis of Ba<zant's approximate size effect law. The fracture energy and the effective length of the fracture process zone is defined on the basis of the extrapolation to an infinite specimen size, for which the definitions are independent of the shape of the specimen or structure. Both of these material properties are expressed in terms of the size effect law parameters and the function describing the nondimensional energy release rate. Ba<zant's size effect law for the nominal stress σN at failure is reformulated in a manner in which the parameters are the fracture energy and the effective (elastically equivalent) process zone length. A method to determine these material properties from σN-data by linear and nonlinear regressions is shown. This method permits these properties to be evaluated solely on the basis of the measured maximum loads of specimens of various sizes and possibly also of different shapes. Variation of both the fracture energy and the effective process zone length as a function of the specimen size is determined. The theoretical results agree with previous fracture tests of rock as well as concrete and describe them adequately in relation to the inevitable random scatter of the tests.

/pdf/determination-of-fracture-energy-process-zone-longth-and-uiu4zh3w3r.pdf

Determination of fracture energy, process zone longth and brittleness number from size effect, with application to rock and conerete

A particle model for brittle aggregate composite materials such as concretes, rocks, or ceramics is presented. The model is also applicable to the behavior of unidirectionally reinforced fiber composites in the transverse plane. A method of random computer generation of the particle system meeting the prescribed particle size distribution is developed. The particles are assumed to be elastic and have only axial interactions, as in a truss. The interparticle contact layers of the matrix are described by a softening stress‐strain relation corresponding to a prescribed microscopic interparticle fracture energy. Both two‐ and three‐dimensional versions of the model are easy to program, but the latter poses, at present, forbidding demands for computer time. The model is shown to simulate realistically the spread of cracking and its localization. Furthermore, the model exhibits a size effect on: (1) The nominal strength, agreeing with the previously proposed size effect law; and (2) the slope of the post‐peak l...

Zdenek P. Bazant

Papers

Nonlocal integral formulations of plasticity and damage: Survey of progress

Nonlocal Continuum Damage, Localization Instability and Convergence

Creep and shrinkage prediction model for analysis and design of concrete structures - model B3

Determination of fracture energy, process zone longth and brittleness number from size effect, with application to rock and conerete

Random particle model for fracture of aggregate or fiber composites