Article

Density-driven damage mechanics (D3-M) model for concrete I: mechanical damage

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Abstract

Damage in concrete has been modelled using various approaches such as fracture mechanics, continuum damage mechanics and failure envelope theories. This study proposes a new approach to model the initiation of damage in concrete that addresses some limitations associated with the existing approaches. The proposed approach defines damage in terms of changes in the density of the material at the microscopic level, where such changes are induced by mechanical loading. The suggested approach is used to simulate the response of 2D concrete bodies to uni-axial tension and uni-axial compression. The simulation results indicate that the proposed model, by means of a single constitutive function, is able to correctly predict failure patterns and aptly capture the damage mechanisms under both uni-axial tension and uni-axial compression loadings using only the information related to the microstructure, the density field and the stiffness field. As a continuation, in Part II, the ability of the D3-M approach to model fully coupled chemo-mechanical damage in concrete using a single constitutive equation will be demonstrated.

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... Popular theories like CDM (Continuum Damage Mechanics) (Mazars et al. 1981), fracture mechanics (Bažant 1987) and Failure envelope theories (Wastiels 1982) do not capture the heterogeneity and pre-stressed state of concrete. In this study, the D3-M model (Murru et al. 2020) is used to model damage using the image obtained after processing using the algorithm described in this paper, as it models damage implicitly and in terms of density, to overcome the shortcomings of other models that were previously mentioned. ...
... where, p is the porosity of the ITZ, which is usually taken to be about 50% (Murru et al. 2020), and p ITZ and p C−S−H are the densities of the ITZ and the C-S-H paste respectively. From this equation, in our study, we take the density of the ITZ to be 1100 kg/m 3 . ...
... The processed images are used to analyse the mechanical response of a concrete cube to uniaxial tensile loads. The constitutive relationship used for this analysis is the Density 314 A. Udupa K. and P. Alagappan Driven damage Mechanics D3-M model (Murru et al. 2020). The resulting stress-strain curves are compared with those obtained previously to validate the accuracy of the algorithm. ...
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In the present world scenario, construction of tall building is being preferred due to rapid increase in the cost of land, lack of land availability and also to preserve land in rural areas for agricultural use (Smith and Coull 1991). The design of tall buildings is primarily governed by wind and seismic loads (Chaudhary et. al. 2019). The performance of the buildings depends on the structural configuration and the present study aims at comparing the performance of tall building with different configurations. In the present study, tall buildings have been investigated under the effect of seismic loads with different structural, namely, (i) special moment resisting frames, (ii) frame-shear wall buildings, (iii) outrigger systems, (iv) braced model and (v) hybrid model. The building models are assumed to be located in Guwahati, Assam which is one of the most earthquake-prone zones in India.Adetailed analysis of the building system is done by modeling the geometry, material section properties and boundary conditions. The sections are discretized using FE mesh using software “ETABS 2016”. Appropriate finite elements are assigned to beams and columns; slab is modeled as a rigid diaphragm to simulate the dynamic behavior of tall building structures. Initially, a modal analysis has been performed to get the natural frequency/period of the buildings. Further, seismic analysis has been carried out to get the performance levels and other responses. Firstly, pushover analysis is performed to obtain the collapse states of the buildings. Then, time-history analyses are carried out to get the dynamic behavior of the buildings. Five spectrum compatible ground motions compatible to Indian design spectrum at maximum earthquake level are considered for the time-history analysis. With the results obtained, a comparison is made for the performances of different structural systems. A detailed procedure of analysis of tall building systems has been discussed. The braced building has performed better as compared to the rigid frame structure. However, the frame-shear wall buildings give the best performance out of all building types. The outrigger and hybrid stiffening buildings over-estimate the performance levels. The conclusions drawn from this study are expected to be useful for academicians and structural designers/practicing engineers.
... The criterion used in this paper to determine when the solid starts softening or loses cohesion (beginning of yielding) was based on the use of some quantities that can be considered as generalization of the concept of principal stresses. Recently Rajagopal and co-workers [21] have proposed models, where one of the important variables is the mass density. Such models can be used to predict the initiation of failure, and that will be a central topic to be addressed in future works. ...
... However, in general we do not know whether the reference state is indeed stress free, and if it is not stress free, the idea of using a failure or switching criterion based on stress is not tenable. We have to use measures whose values are completely determined by the current state, see[21] for a discussion of the relevant issues. ...
Article
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An implicit constitutive relation is proposed to describe stress softening exhibited by solids, using as a basis the second law of thermodynamics, and appropriate choices for the specific Helmholtz potential and the rate of entropy production function. The implicit constitutive relation that is developed is a generalization of the earlier one-dimensional model developed by Rajagopal and Srinivasa (Int J Plast 71:1–9, 2015). One subclass is studied, wherein the stress is the controlling variable, which permits the study of a body that softens with the stress.
... where I is the identity transformation, ε is the linearized strain, the β i , i = 1, 2, 3 are scalar valued functions that can at most depend linearly on ε, but arbitrarily on the invariants of T, while β i , i = 0, 4, 5 depends on the invariants of T. Since by virtue of the balance of mass: ρ R = ρ(detF), which when linearized leads to ρ R = ρ(1 + trε), trε can be replaced by ρ, which makes the constitutive relation useful in describing the response of porous materials as the porosity determines the density of the material (see [23] for discussion of the development and relevance of such constitutive relations). Gainfully exploiting the fact that such constitutive relations can accommodate the material moduli can be functions of the density, [18,19] studied the problem of initiation of damage in concrete. A subclass wherein the constitutive relation is also linear in T is given by ...
... part ε(v) * are defined according to formulas (16) and (10). Here and in what follows, the dot implies the scalar product of tensors T · S = 3 i,j =1 T ij S ij and vectors, respectively, (21) is obtained in a standard way after multiplication of the equilibrium equation (17) with v i , summing it over i = 1, 2, 3 and integrating by parts over with the help of boundary conditions (19) and (20). The variational equations in (22) are derived from the constitutive equations in (18) after taking the scalar product with the test functions S * and q. ...
Article
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We study some mathematical properties of a novel implicit constitutive relation wherein the stress and the linearized strain appear linearly that has been recently put into place to describe elastic response of porous metals as well as materials such as rocks and concrete. In the corresponding mixed variational formulation the displacement, the deviatoric and spherical stress are three independent fields. To treat well-posedness of the quasi-linear elliptic problem, we rely on the one-parameter dependence, regularization of the linear-fractional singularity by thresholding, and applying the Browder–Minty existence theorem for the regularized problem. An analytical solution to the nonlinear problem under constant compression/extension is presented.
... We will see that this change in the value of the material moduli can lead to the geography of the stress and strain to alter, and more importantly to pronounced changes in the stress concentration factors, for the problems under consideration. Using such a model, Murru et al. have studied the onset of damage in concrete (see [22]). ...
Article
Many rocks, metals and concrete, are porous. This would then imply that their properties depend on the density. In this report we develop a constitutive relation to describe the response of elastic bodies that are linear in both the stress and the linearized strain with the material moduli depending on the density. Such a model is not possible within the context of the classical theory of linearized elasticity but is possible within the context of the implicit theory for elastic bodies that has been developed. The constitutive relations discussed in this report can be useful to describe the response of porous elastic bodies in the small displacement gradient regime. Using these constitutive relations, we study the stress concentration due to the presence of a circular hole in a plate due to uniaxial extension. We find that the stress concentration factor can be significantly different from that in the case of the classical linearized elastic solid.
... Of course, depending on whether λ 2 and λ 3 are positive or negative and the deformation in question and thus the density, locally material properties may increase or decrease in value. Using a similar model, Murru et al. (2020a), (2020b) studied damage that takes place in cement concrete. Their results are in qualitative agreement with experimental observation. ...
Preprint
In this short note we develop a constitutive relation that is linear in both the Cauchy stress and the linearized strain, by linearizing implicit constitutive relations between the stress and the deformation gradient that have been put into place to describe the response of elastic bodies (see \cite{rajagopal2003implicit}), by assuming that the displacement gradient is small. These implicit equations include the classical linearized elastic constitutive approximation as well as constitutive relations that imply limiting strain, as special subclasses.
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This chapter discusses damage models for concretes. It explains three types of models: isotropic damage model, nonlocal damage, and anisotropic damage model. The constitutive relation is valid for standard concrete with a compression strength of 30-40 MPa. Its aim is to capture the response of the material subjected to loading paths in which extension of the material exists. It should not be employed (1) when the material is confined because the damage loading function relies on extension of the material only, (2) when the loading path is severely nonradial, and (3) when the material is subjected to alternated loading. Further, the model provides a mathematically consistent prediction of the response of structures up to the inception of failure due to strain localization. After this point is reached, the nonlocal enhancement of the model is required. The purpose of nonlocal damage is to describe the nonlocal enhancement of the isotropic damage model. This modification of the model is necessary in order to achieve consistent computations in the presence of strain localization due to the softening response of the material. Further, anisotropic damage model describes a constitutive relation based on elastoplastic damage. This anisotropic damage model is compared to experimental data in tension, compression, compression-shear, and nonradial tension shear.
Article
The term ‘elasticity’ seems to conjure different images in different minds. After a discussion of the various interpretations of elasticity espoused by the pioneers, we discuss the notions of Cauchy elastic and Green elastic bodies, and whether Cauchy elastic bodies that are not Green elastic are reasonable from a physical standpoint. We then discuss a class of models, more general than classical Cauchy elastic bodies, and we find that such bodies need not be Green elastic. While a stored energy can be associated with these materials, the stress is not derivable from the stored energy. One can delineate conditions under which these models are thermodynamically consistent in that they meet the second law of thermodynamics; more precisely, the general class of bodies that is being described is incapable of dissipation in any process whatsoever. These models not only add to the repertoire of the elasticians in modeling solids that are incapable of dissipation, but also they seem to provide an opportunity for a genuinely new approach to the study of problems that result in singularities within the classical theory of linearized elasticity, such as that encountered in the rupturing and fracturing of solids. The generalized framework also provides a rational basis for developing linearized theories within which the linearized strain bears a nonlinear relationship to the stress.
Article
Due to its higher porosity, the interfacial transition zone (ITZ) in cementitious composite is often considered as a weak phase, compared to aggregate and bulk paste. In this paper, we present some results of a study on the ITZ including microstructure, thickness and porosity for the case of a ternary blended cementitious system, i.e., Portland cement, blast furnace slag and limestone filler. In particular, based on the backscattered electron image analysis and the HYMOSTRUC model, the ITZ microstructure, thickness and porosity are investigated in an elaborative way. The effects of casting factors such as curing age, water to binder (w/b) ratio and aggregate content are discussed, and two new formulas are proposed to fit the ITZ thickness and porosity. Results indicate that curing age influences both the ITZ thickness and porosity, while w/b ratio and aggregate content only influence the ITZ porosity.
Article
The microcracking in the fracture process zone ahead of a major crack is assumed to consist, in the initial stage, of a two-dimensional array of small circular(penny-shaped) cracks and, in the terminal stage, of a two-dimensional array of small circular ligaments, all located on the main crack plane. Both cases are solved in three-dimensions according to linear elastic fracture mechanics. The solution is approximate but asymptotically exact both for very small circular cracks and very small circular ligaments. The spacing of the cracks as well as the ligaments is governed by the spacing of the large aggregate pieces. The curve of the transverse displacement v due to cracks versus the remote applied normal stress is calculated and is found to exhibit snapback instability at which a negative slope changes to a positive slope and v reaches its maximum possible value. Since several other influencing physical mechanisms were neglected in the analysis, it still remains to be verified whether the snapback instability does actually occur in the concrete fracture process. The asymptotic behavior at ligament tearing is further analyzed, based on St.-Venant's principle, for arbitrary general three- and two-dimensional situations and it is shown that when the ligament transmits a force (mode I, II or III), its final tearing is always characterized by snapback instability, which determines maximum possible displacement due to crack. When, however, the ligaments transmit only a moment (bending or torsional), there is no snapback instability.
Article
This paper presents a concrete model that is capable of describing the response of concrete under bi-axial loading, with the features of simplicity and avoidance of convergence problems, often seen in plasticity based models. The proposed model incorporates the failure of concrete into a conventional continuum damage mechanics framework, where particular emphasises are placed on highlighting the different responses of concrete under tension and compression, as well as the different contributions of hydrostatic and deviatoric stress components on concrete damage. A weighted damage parameter and a damage multiplier are introduced to eliminate potential convergence problems and to reduce the effect of hydrostatic pressure on damage, respectively. Finally, several examples are provided and compared with experimental data.
Article
In the present paper a composite model is proposed to study the non-linear fracture response observed in a multi-phase material such as concrete. Mechanisms involved during crack propagation are discussed first. Effects of these mechanisms and their implications on the fracture behavior of multi-phase materials, are then examined. The proposed model shows that spurious energy induced by the ‘forced compliance’ composite action is the main contributor to the observed size-effect on fracture energy. Due to this spurious energy, higher values of fracture energy will be observed for larger specimens. Effects of specimen size on other fracture parameters are presented; comparisons of model predictions with available experimental results are also discussed.
Article
The behaviour of micro-concrete under tensile conditions has been studied in terms of the strain gradients. Dealing with the problem within the mechanics of simple body media, then it appears that the cracking limit state criterion is a function of the strain gradients. There is an increase of the threshold in stress and in strain. A stable microcracking development is assumed and such an assumption is confirmed by the analysis of the residual strains.
Article
The octahedral stress space is often used for representing the failure criterion of a material. After giving the necessary definitions, several drawbacks of this representation are considered for different combinations of tensile and compressive stresses. The emphasis is put on concrete, but most conclusions are valid for materials with a different compressive and tensile strength, and some of them are valid for all types of materials. It is concluded that a complementary representation in another space is necessary.RésuméL'espace des contraintes octahédriques est souvent utilisée pour représenter le critère de rupture d'un matériau. Après avoir énoncé les définitions nécessaires, plusieurs désavantages de cette représentation sont traités pour différentes combinaisons de contraintes de traction et de compression. Bien que l'accent est mis sur le béton, la plupart des conclusions est valable pour d'autres matériaux ayant une différente résistance en compression et en traction, et quelques conclusions sont valables pour tous les matériaux. Il est concludé qu'une représentation complémentaire dans un autre espace est nécessaire.
Article
This paper deals with the behavior of roller compacted concrete (RCC) cores in uniaxial tension cored from a practical RCC dam. Properties included are tensile strength, peak strain, modulus of elasticity, fracture energy, brittleness, complete stress-deformation curve, and stress-crack width curve both for the RCC matrix and the interface. Two categories of compressive strengths of RCC, 15 and 20 MPa, respectively, were tested in order to investigate the effect of RCC compressive strength on their behaviors. Two categories of specimen sizes of RCC, 150 X 300 mm and 250 X 500 mm cylindrical specimens, were also tested in order to investigate the effect of specimen size on their behaviors. The complete stress-deformation curves for RCC specimens were achieved by using a servohydraulic closed-loop testing system and four extensometers, of which the extensometer indicating maximum deformation is employed as feedback signal. Based on the stress-deformation curves, their stress-crack width curves were also obtained and theoretically modeled. The present results support the following conclusions. The uniaxial tensile strength of the RCC matrix is related to both the square root of its nominal compressive strength and the ratio of the maximum size of aggregate to the characteristic dimension of the specimen, but the uniaxial tensile strength of the interface is related only to the later. The modulus of elasticity both for the RCC matrix and for the interface are related to the square root of their nominal compressive strength and the ratio of the maximum size of the aggregate to the characteristic dimension of the specimen. The ultimate crack width of the RCC matrix is governed by the maximum size of the aggregate, but the ultimate crack width of the RCC interface is governed by the ratio of the maximum size of aggregate to the characteristic dimension of the specimen. The fracture energy of the RCC matrix is related to the square root of its nominal compressive strength and the ratio of the maximum size of the aggregate to the characteristic dimension of the specimen, whereas the fracture energy of the RCC interface is related to the square root of its nominal compressive strength and the maximum size of aggregate. The characteristic length of the RCC matrix and interface is related to both the square root of their nominal compressive strength and the ratio of the maximum size of the aggregate to the characteristic dimension of the specimen.
Article
The classical smeared cracking model widely used in finite-element analysis of concrete and rock cannot describe the size effect experimentally observed in brittle failures and exhibits spurious mesh sensitivity with incorrect convergence to zero energy dissipation at failure. The crack band model circumvents these deficiencies but has limitations with respect to mesh refinement, shear locking on zig-zag crack bands, and directional bias of the mesh. It is shown that all of these problems can be avoided by a nonlocal generalization, in which the damage that characterizes strain softening is considered to be a function of the spatial average of the positive part of the maximum principal strain. Two alternatives are presented: (1) Smeared cracking whose direction is fixed when cracks start to form; and (2) smeared cracking whose orientation rotates with the maximum principal strain. Furthermore, fracture tests on specimens of various sizes are analyzed by finite elements. It is shown that the model correctly reproduces the experimentally observed size effect and agrees with Bažant's size effect law. Orthogonal and slanted meshes are shown to yield approximately the same cracking zones and propagation directions. The model is easily programmed and computationally more efficient than the corresponding local version.
Article
A nonlinear model for the fracture process zone (FPZ) that exists at the tip of a crack in concrete is developed through the interactive use of experimental data and finite element analysis of crack-line-wedge-loaded double-cantilever- beam (CLWL-DCB) specimens. That model relates the clamping stress to the crack opening displacement (COD) and is characterized by three straight-line stress variations segmented by three critical CODs. In the first segment the clamping stress equals the concrete's tensile strength. In the second segment the crack closure decreases sharply with increasing CODs from the tensile strength at about 0.4 X 10-3 in. (0.01 mm) to 0.3 times that stress at about 2.0 X 10-3 in. (0.05 mm). The validity of that model is demonstrated by showing that it can predict the results of tests on five different groups of CLWL-DCB specimens subjected to mode I loadings.
Article
Attempts to apply linear elastic fracture mechanics (LEFM) to concrete have been made for several years. Several investigators have reported that when fracture toughness, Kk, is evaluated from notched specimens using conventional LEFM (measured peak load and initial notch length) a significant size effect is observed. This size effect has been attributed to nonlinear slow crack growth occurring prior to the peak load. A two parameter fracture model is proposed to include this nonlinear slow crack growth. Critical stress intensity factor, Kic, is calculated at the tip of the effective crack. The critical effective crack extension is dictated by the elastic critical crack tip opening displacement, CTODc. Tests on notched beam specimens showed that the proposed fracture criteria to be size independent. The proposed model can be used to calculate the maximum load (for Mode I failure) of a structure of an arbitrary geometry. The validity of the model is demonstrated by an accurate simulation of the experimentally observed results of tension and beam tests.
Article
On the basis of continuum constitutive models (stress vs. strain), the introduction of strong discontinuity kinematics (considering jumps in the displacement fields across a discontinuity interface) induces projected discrete constitutive models (traction-displacement jumps) in a consistent manner. Therefore, this projection provides possible links between the classical continuum strain-localization analysis and the non-linear (decohesive) fracture mechanics techniques. The strong discontinuity analysis shows that (bandwidth based) regularization of the hardening/softening parameter is the crucial modification to be done on the continuum model to achieve such a projection, and it also provides the strong discontinuity conditions that set restrictions on the stress state compatible with bifurcations in a strong discontinuity format. The methodology is illustrated on the basis of two classical families of non-linear constitutive models (scalar continuum damage and elasto-plasticity) for which the corresponding discrete constitutive models and the strong discontinuity conditions are derived.
Article
Smeared crack analysis models based on a nonlinear fracture mechanics (NLFM) crack propagation criterion are considered to study the two-dimensional static fracture behavior of plain concrete structures. A coaxial rotating crack model (CRCM) and a fixed crack model with a variable shear resistance factor (FCM-VSRF), both using a secant stiffness formulation, are considered in studying the behaviors of a notched shear beam, a model concrete gravity dam, and a full-scale concrete gravity dam: all have been experimentally or numerically investigated in the past. The responses obtained from smeared crack analyses are compared with those reported in the literature by other investigators. The CRCM appears to perform better than the FCM-VSRF in alleviating the stress-locking phenomenon generally observed in smeared crack analyses. However, the crack profiles predicted by the rotating crack model are prone to the directional bias caused by finite element meshes of the elementary beam problem and the model concrete dam. The two smeared crack propagation models (CRCM and FCM-VSRF) provide reasonable responses when a full-scale concrete gravity dam is analyzed.
Article
Concrete is modeled as a linear-elastic softening material and introduced into fracture mechanics. A discrete crack is considered with softening zones at the crack tips. Following the approach of Dugdale and Barenblatt, closing stresses are applied to the crack faces in the softening zone. The stresses are described by a power function. Relations are worked out between the remote stress on a cracked plate, the tensile strength of the material and the size of the softening zone. The finite width of a plate is considered and so are various stress distributions of the softening zone. Experiments were performed to establish the stress-strain behavior of concrete in deformation-controlled uniaxial tensile loading. The results show that nonlinear fracture mechanics can be applied to concrete in order to predict the load-bearing capacity of a cracked structure.
Article
A two-dimensional damage model is developed for concrete in uniaxial and biaxial tension up to the ultimate load. The model is based on the concept of the equivalence of complementary strain energies. Existing uniaxial and biaxial tension test data are used to establish the evolution of the damage parameters. It is shown that the latter predict accurately not only uniaxial and biaxial tension data but also flexural data from unrelated tests. It is further shown that, whereas the stresses/strains at first cracking and at or near ultimate load in tension and flexure are very different, the accumulated damage according to this model is not.
Article
This paper shows that, the size effect in flexure and shear strength for different concrete and reinforced concrete beam sizes subjected to concentrated and uniformly distributed loads can be successfully predicted on the basis of nonlinear fracture mechanics. The analysis was carried out by a computer simulation using the program ANACS (advanced nonlinear analysis of concrete structures), which was originally developed by the authors. In this investigation, the fictitious crack model was adopted with two orthogonal rod elements which involves the nonlinear fracture mechanics through their constitutive model to simulate the discrete crack path and represent the localized crack zone. By combining the arc-length calculation technique with the fictitious crack model, the postpeak behaviour can be predicted well even for snapback instability.
Article
The mechanical properties of cementitious composites have been observed to be sensitive to the rate of loading and this rate sensitivity has been attributed to the strain rate effects on cracking. A nonlinear fracture mechanics model is proposed to predict the strain rate effect on mode I fracture of concrete. This model requires three material properties (Critical Stress Intensity Factor, KIcs Critical Crack Tip Opening Displacement, CTODc and Young's Modulus, E) which can be determined from static tests. The analytical procedure is based on the observation that the pre-peak nonlinearity is due to the pre-peak (or stable or pre-critical) crack growth and that this pre-critical crack growth decreases with increase in rate of loading. KIcs and E are assumed to be rate independent while CTODc is assumed to decrease exponentially with the logarithm of the relative strain rate. The model predicted values correlated well with the experimentally observed trends in the strain rate effects on mode I fracture of concrete.
Article
Stress analysis of a compact compression specimen used for the determination of the fracture toughness of cementitious materials is carried out by the finite element method. The specimen, which is based on 100 mm cubes, contains two notches on opposite faces. It is found that the geometry and loading result in large tensile stresses at the root of the notch remote from the load. These stresses are sufficient to propagate the crack in the opening mode of fracture thus enabling the fracture toughness of the material to be determined. The fracture toughness is evaluated from the failure load and the stress intensity factor computed from the finite element results. Several finite element mesh refinements were employed and accurate estimates of the stress intensity factor were obtained by modelling the specimen by a relatively small number of elements. The accuracy of the results was largely independent of the evaluation method which included displacement extrapolation, conic section simulation, strain energy release rate and the J-integral. Whereas the stress intensity factor varied with the notch size, the tests conducted in the present work did not show significant variation in the fracture toughness.
Article
The evolution of permeability and elastic modulus for Type III portland cement pastes with water/cement ratios varying from 0.4 to 0.6 were measured using a beam-bending method. Young's modulus was independently verified by measuring the ultrasonic pulse velocity. The permeability ranged over 2 orders of magnitude, depending on the water/cement ratio and the age of the samples. The advantage of the beam-bending method is that the permeability results are obtained in a few minutes to a few hours, whereas conventional techniques take hours or days to measure permeability of this order of magnitude. More importantly, there is no need to maintain high pressure during the measurement period, so leaks are not a problem.
Article
The growth and development of the fracture process zone in plain concrete has been investigated. A fictitious crack model based noniterative numerical scheme is developed to study the fracture characteristics of specimens of different sizes and geometries. Results from numerical studies on four different geometrically similar specimen sizes and two different specimen geometries are reported and discussed. The finite element program developed accommodates linear as well as nonlinear softening laws for the fracture process zone in concrete. It is observed that the process zone reaches a steady state length which is specimen size as well as specimen geometry dependent. As long as the process zone is allowed to develop to its steady state length, the energy absorbed in the process zone appears to be size and geometry independent. Results from tests on three-point bending specimens and compact tension specimens reported in the literature have been compared with the numerical solutions obtained in this investigation. Specimen size and geometry dependence generally observed in these fracture experiments have been duplicated. The numerical model also successfully reproduces many of the other experimentally observed characteristics in the fracture of plain concrete.
Article
In classical constitutive models such as the Navier-Stokes fluid model, and the Hookean or neo-Hookean solid models, the stress is given explicitly in terms of kinematical quantities. Models for viscoelastic and inelastic responses on the other hand are usually implicit relationships between the stress and the kinematical quantities. Another class of problems wherein it would be natural to develop implicit constitutive theories, though seldom resorted to, are models for bodies that are constrained. In general, for such materials the material moduli that characterize the extra stress could depend on the constraint reaction. (E.g., in an incompressible fluid, the viscosity could depend on the constraint reaction associated with the constraint of incompressibility. In the linear case, this would be the pressure.) Here we discuss such implicit constitutive theories. We also discuss a class of bodies described by an implicit constitutive relation for the specific Helmholtz potential that depends on both the stress and strain, and which does not dissipate in any admissible process. The stress in such a material is not derivable from a potential, i.e., the body is not hyperelastic (Green elastic).
Article
Numerical analyses of large engineering structures undergoing highly localized deformations induced by material failure such as cracking in concrete or shear bands in soils still represent a challenge to the scientific community. In this paper, an efficient concept suitable for the analysis of those problems is presented. More precisely, an overview of the Strong Discontinuity Approach (SDA) is given. This specific approach is characterized by the incorporation of strong discontinuities, i.e. discontinuous displacement fields, into standard displacement-based finite elements by means of the Enhanced Assumed Strain (EAS) concept. The fundamentals of the SDA are illustrated and compared to those of other models based on discontinuous deformation mappings. The main part of this contribution deals with the numerical implementation of the SDA. Besides the original finite element formulation of the SDA, two more recently proposed algorithmic frameworks which avoid the use of the static condensation technique are presented. Both models result in a set of equations formally identical to that known from classical plasticity theory and, consequently, it can be solved by applying the return-mapping algorithm. Several recently suggested extensions of the SDA such as rotating surfaces of discontinuous displacements and intersecting discontinuities are discussed and investigated by means of finite element analyses. The applicability of the SDA as well as its numerical performance is illustrated by means of fully three-dimensional ultimate load analyses.
Article
This paper deals with the effect of the leaching process of cement-based materials on their mechanical properties. This process mainly induces a total leaching of Ca(OH)2 and a progressive decalcification of the C-S-H leading in turn to a gradient of C/S ratio in the leaching zone. Modeling of the deterioration of cement paste exposed to leaching consists of a decrease in the local elastic modulus with both a damage function d and an aging function L:E=E0 (1-d)(1-L). The main parameter fo the aging function is the residual calcium content in the material. This calcium content depends directly on the thickness of the degraded zone and on the different types of hydrates in the cement paste. The non-linearity of the mechanical behavior of the cement-based material is described by a damage function whose main state parameter is the equivalent strain. The characteristic parameters of the material are identified by a compressive loading test. The model of aging damage behavior proposed in this paper corresponds perfectly with the experimental results obtained in the case of a uniaxial compressive load.
Article
This paper describes the so called interfacial transition zone—ITZ—in concrete. This is the region of the cement paste around the aggregate particles, which is perturbed by the presence of the aggregate. Its origin lies in the packing of the cement grains against the much larger aggregate, which leads to a local increase in porosity and predominance of smaller cement particles in this region. The ITZ is region of gradual transition and is highly heterogeneous, nevertheless the average microstructural features may be measured by analysis of a large numbers of backscattered electron images of polished concrete samples. Such measurements show that the higher porosity present initially is significantly diminished by the migration of ions during hydration.
Article
This paper is the first part of an extended program to develop a theory of fracture in the context of strain-limiting theories of elasticity. This program exploits a novel approach to modeling the mechanical response of elastic, that is non-dissipative, materials through implicit constitutive relations. The particular class of models studied here can also be viewed as arising from an explicit theory in which the displacement gradient is specified to be a nonlinear function of stress. This modeling construct generalizes the classical Cauchy and Green theories of elasticity which are included as special cases. It was conjectured that special forms of these implicit theories that limit strains to physically realistic maximum levels even for arbitrarily large stresses would be ideal for modeling fracture by offering a modeling paradigm that avoids the crack-tip strain singularities characteristic of classical fracture theories. The simplest fracture setting in which to explore this conjecture is anti-plane shear. It is demonstrated herein that for a specific choice of strain-limiting elasticity theory, crack-tip strains do indeed remain bounded. Moreover, the theory predicts a bounded stress field in the neighborhood of a crack-tip and a cusp-shaped opening displacement. The results confirm the conjecture that use of a strain limiting explicit theory in which the displacement gradient is given as a function of stress for modeling the bulk constitutive behavior obviates the necessity of introducing ad hoc modeling constructs such as crack-tip cohesive or process zones in order to correct the unphysical stress and strain singularities predicted by classical linear elastic fracture mechanics. KeywordsElasticity–Implicit theories–Mode III fracture