Measurement of Characteristic Length of Nonlocal Continuum
01 Apr 1989-Journal of Engineering Mechanics-asce (American Society of Civil Engineers)-Vol. 115, Iss: 4, pp 755-767
TL;DR: The characteristic length of a heterogeneous brittle material such as concrete represents a material property that governs the minimum possible width of a zone of strain softening damage in nonlocal areas as mentioned in this paper.
Abstract: The characteristic length of a heterogeneous brittle material such as concrete represents a material property that governs the minimum possible width of a zone of strainsoftening damage in nonlocal...
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TL;DR: 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 as mentioned in this paper.
Abstract: 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.
1,171 citations
TL;DR: In this paper, the concept of representative volume element (RVE) is analyzed for elastic materials and the results were based on a statistical analysis of numerical experiments, where tests have been performed on a random heterogeneous material.
587 citations
TL;DR: In this article, a particle model for brittle aggregate composite materials such as concretes, rocks, or ceramics is presented, which is also applicable to the behavior of unidirectionally reinforced fiber composites in the transverse plane.
Abstract: 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...
487 citations
TL;DR: In this paper, a variational approach to brittle fracture approximates the crack evolution in an elastic solid through the use of gradient damage models, and a stability criterion in terms of the positivity of the second derivative of the total energy under the unilateral constraint induced by the irreversibility of damage is introduced.
Abstract: In its numerical implementation, the variational approach to brittle fracture approximates the crack evolution in an elastic solid through the use of gradient damage models. In this article, we first formulate the quasi-static evolution problem for a general class of such damage models. Then, we introduce a stability criterion in terms of the positivity of the second derivative of the total energy under the unilateral constraint induced by the irreversibility of damage. These concepts are applied in the particular setting of a one-dimensional traction test. We construct homogeneous as well as localized damage solutions in a closed form and illustrate the concepts of loss of stability, of scale effects, of damage localization, and of structural failure. Considering several specific constitutive models, stress
466 citations
Cites background from "Measurement of Characteristic Lengt..."
...For concrete, a brittle material, the typical values for the material parameters are (see, e.g., Bazant and Pijaudier-Cabot, 1989; Comi and Perego, 2001): E0 ¼ 29GPa, M ¼ 4:5MPa, Gc ¼ 70N/m: ð73Þ Using the relations (71), we find: 2D0 ¼ 106mm, ‘ ¼ 38mm, w1 ¼ 698N/m3: ð74Þ The value of the thickness…...
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Book•
01 Jan 1992
TL;DR: In this article, the authors present a state-of-the-art report on fracture mechanics of concrete: concepts, models and determination of material properties, including fracture models with softening zone.
Abstract: State-of-the-art report: Fracture mechanics of concrete: concepts, models and determination of material properties. Why fracture mechanics? Essential results from linear elastic fracture mechanics. Nonlinear fracture models with softening zone. Special nonlinear fracture models based on adaptions of LEFM. Size effect and brittleness of structures. Experimental or analytical determination of material fracture parameters. Factors influencing fracture parameters. Effect of reinforcement. Crack systems. Concluding remarks. References and bibliography. Appendix 1- derivations of some formulas. Conference papers. Material models for concrete fracture. Damage modelling. Numerical analysis of concrete fracture. Experimental methods and determination of fracture characteristics. Measurements of damage and size effect. Dynamic fracture. Fracture under shear. Fracture of reinforced concrete. Interaction between concrete and reinforcement. Fatigue and rate effects. Environment effects [temperature, shrinkage, corrosion]. Index.
343 citations
References
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01 Jan 2008
TL;DR: In this article, fracture mechanics is introduced into finite element analysis by means of a model where stresses are assumed to act across a crack as long as it is narrowly opened, which may be regarded as a way of expressing the energy adsorption in the energy balance approach.
Abstract: A method is presented in which fracture mechanics is introduced into finite element analysis by means of a model where stresses are assumed to act across a crack as long as it is narrowly opened. This assumption may be regarded as a way of expressing the energy adsorption GC in the energy balance approach, but it is also in agreement with results of tension tests. As a demonstration the method has been applied to the bending of an unreinforced beam, which has led to an explanation of the difference between bending strength and tensile strength, and of the variation in bending strength with beam depth.
5,564 citations
Additional excerpts
...The discrete crack models with a softening cohesive zone at the crack front, introduced for concrete by Hillerborg et al. (1976), can correctly describe localized cracking or damage, but may yield ambiguous results for situations where damage or cracking remains distributed (Bazant 1986)....
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TL;DR: In this article, fracture mechanics is introduced into finite element analysis by means of a model where stresses are assumed to act across a crack as long as it is narrowly opened, which may be regarded as a way of expressing the energy adsorption in the energy balance approach.
5,505 citations
01 May 1983
TL;DR: In this article, 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.
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.
3,102 citations
TL;DR: In this paper, a nonlocal damage formulation was extended to a more general form in which the strain remains local while any variable that controls strain-softening is nonlocal, and it was shown that the energy dissipation and damage cannot localize into regions of vanishing volume.
Abstract: 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.
815 citations
801 citations