Presents a computational algorithm for the numerical integration of triaxial concrete plasticity formulations. The specific material formulation at hand is the so‐called extended…
Abstract
Presents a computational algorithm for the numerical integration of triaxial concrete plasticity formulations. The specific material formulation at hand is the so‐called extended leon model for concrete. It is based on the flow theory of plasticity which entails isotropic hardening as well as fracture energy‐based softening in addition to non‐associated plastic flow. The numerical algorithm resorts to implicit integration according to the backward Euler strategy that enforces plastic consistency according to the closes‐point‐projection method (generalized radial‐return strategy). Numerical simulations illustrate the overall performance of the proposed algorithm and the significant increase of the convergence rate when the algorithmic tangent is used in place of the continuum operator.
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Eddy Pramono and Kaspar Willam
Numerical solutions in computational plasticity are severely challenged when concrete and geomaterials are considered with non‐regular yield surfaces, strain‐softening and…
Abstract
Numerical solutions in computational plasticity are severely challenged when concrete and geomaterials are considered with non‐regular yield surfaces, strain‐softening and non‐associated flow. There are two aspects that are of immediate concern within load steps which are truly finite: first, the iterative corrector must assure that the equilibrium stress state and the plastic process variables do satisfy multiple yield conditions with corners, Fi(σ, q) = 0, at discrete stages of the solution process. To this end, a reliable return mapping algorithm is required which minimizes the error of the plastic return step. Second, the solution of non‐linear equations of motion on the global structural level must account for limit points and premature bifurcation of the equilibrium path. The current paper is mainly concerned with the implicit integration of elasto‐plastic hardening/softening relations considering non‐associated flow and the presence of composite yield conditions with corners.
Lothar Haefner and Kaspar J. Willam
A simple beam element is developed for the solution of large deflection problems. The total Lagrangian formulation is based on the kinematic relations proposed by Reissner for…
Abstract
A simple beam element is developed for the solution of large deflection problems. The total Lagrangian formulation is based on the kinematic relations proposed by Reissner for finite rotations and stretching as well as shearing of plane beams. The motion is discretized by linear expansions of the global displacement components and the cross‐sectional rotation in two‐dimensional Euclidean space yielding a simple beam element with three degrees of freedom at the two nodes. The shear locking is reduced by selective integration in order to eliminate the spurious shear constraint similar to interdependent variable interpolation. The large rotation formulation is compared with two forms of moderate rotation theories which have been used in the past to develop the geometric stiffness properties for linear stability analysis of the so‐called Mindlin plate elements. The predictive value of different geometric stiffness approximations is assessed with several examples which range from the static and kinetic stability analysis of the classical Euler‐column to the large deflection problem of a clamped beam.
Gilles Pijaudier‐Cabot, Zdeněk P. Bažant and Mazen Tabbara
This paper presents a comparison of various models for strain‐softening due to damage such as cracking or void growth, as proposed recently in the literature. Continuum‐based…
Abstract
This paper presents a comparison of various models for strain‐softening due to damage such as cracking or void growth, as proposed recently in the literature. Continuum‐based models expressed in terms of softening stress—strain relations, and fracture‐type models expressed in terms of softening stress—displacement relations are distinguished. From one‐dimensional wave propagation calculations, it is shown that strain‐localization into regions of finite size cannot be achieved. The previously well‐documented spurious convergence is obtained with continuum models, while stress—displacement relations cannot model well smeared‐crack situations. Continuum models may, however, be used in general if a localization limiter is implemented. Gradient‐type localization limiters appear to be rather complicated; they require solving higher‐order differential equations of equilibrium with additional bourdary conditions. Non‐local localization limiters, especially the non‐local continuum with local strain, in which only the energy dissipating variables are non‐local, is found to be very effective, and also seems to be physically realistic. This formulation can correctly model the transition between homogeneous damage states and situations in which damage localizes into small regions that can be viewed as cracks. The size effect observed in the experimental and numerical response of specimens in tension or compression is shown to be a consequence of this progressive transition from continuum‐type to fracture‐type formulations.
Eduardo N. Dvorkin, Alberto M. Cuitiño and Gustavo Gioia
A concrete material model is presented. The model is based on non‐associated plasticity for the pre‐failure and ductile post‐failure regimes and fracture (smeared crack approach…
Abstract
A concrete material model is presented. The model is based on non‐associated plasticity for the pre‐failure and ductile post‐failure regimes and fracture (smeared crack approach) for the brittle post‐failure regime. The implementation of the constitutive model in the 2‐D elements of a general purpose non‐linear incremental finite element code is discussed. Some important numerical features of the implementation are the implicit integration of the stress/strain relation and the use of an efficient symmetric stiffness formulation for the equilibrium iterations.
In this study we examine the spectral properties of stiffness degradation at the constitutive level and at the levels of finite elements and their assemblies. The principal…
Abstract
In this study we examine the spectral properties of stiffness degradation at the constitutive level and at the levels of finite elements and their assemblies. The principal objective is to assess the effects of defects on the elastic stiffness properties at different levels of observation. In particular, we are interested in quantitative damage measures, which characterize the fundamental mode of degradation in the form of elastic damage at the level of constitutive relations and at the level of finite elements and structures.
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Peter Pivonka and Kaspar Willam
In this paper, we examine the influence of the third invariant in computational plasticity. For this purpose we consider the extended Leon model, an elasto‐plastic model for…
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In this paper, we examine the influence of the third invariant in computational plasticity. For this purpose we consider the extended Leon model, an elasto‐plastic model for concrete materials which accounts for the difference of shear strength in triaxial compression and triaxial extension. Consequently, the deviatoric trace of the loading surface is no longer circular like in von Mises and Drucker‐Prager plasticity. In the limit it approaches the triangular shape of the Rankine condition of maximum direct stress. Thereby, elliptic functions describe the out‐of‐roundness of the circular trace in terms of C1‐continuous functions of the Lode angle. The algorithmic aspects of the third invariant considerably complicate the computational implementation since the radial return method of J2‐plasticity does no longer maintain normality leading to loss of deviatoric associativity. The paper will focus on the computational issues near the three regions with high curvature at the compressive meridians with special attention on the lack of convergence of the plastic return algorithm and its slow rate of convergence in these regions. The algorithmic discussion at the constitutive level will be augmented by the axial plane‐strain compression test in order to illustrate the effect of the third invariant at the structural level of finite element analysis.
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Joshua J. Turner, Olena Kopystynska, Kay Bradford, Brian J. Higginbotham and David G. Schramm
High divorce rates have coincided with higher rates of remarriage. Although remarriages are more susceptible to dissolution than first-order marriages, less research has focused…
Abstract
High divorce rates have coincided with higher rates of remarriage. Although remarriages are more susceptible to dissolution than first-order marriages, less research has focused on factors that promote vulnerabilities among remarried couples. In the current study, the authors focused on whether predictors of divorce differ by the number of times someone has been married. The authors examined some of the most common reasons for divorce, as identified by parents who completed a state-mandated divorce education course (n = 8,364), while also controlling for participant sociodemographic characteristics. Participants going through their first divorce were more likely to identify growing apart and infidelity as reasons for seeking a divorce. Conversely, those going through a subsequent divorce were more likely to list problems with alcohol/drug abuse, childrearing differences, emotional/psychological/verbal mistreatment, money problems, physical violence, and arguing. Multivariate analyses indicated that sociodemographic factors were stronger predictors of divorce number than commonly listed reasons for divorce for both male and female participants. Implications for remarital and stepfamily stability and directions for future research are discussed.
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Rabah Hammoud, Rachid Boukhili and Ammar Yahia
A numerical model to simulate the impact of high temperature on the behavior of conventional concrete under chemoplastic framework is developed and validated. The model is based…
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A numerical model to simulate the impact of high temperature on the behavior of conventional concrete under chemoplastic framework is developed and validated. The model is based on new formulation of a constitutive law with new chemoplastic potential. By overlaying the chemoplastic potential on the modified Etse and Willam yielding surface, both defined on the Haigh-Westergaard coordinates, it was found that the two curves do not undergo similar stress state at the same strength parameter. For an adequate evaluation of normal vectors, each surface is forced to pass through the current stress state. Keeping the loading surface unchanged, the calculation of the plastic potential need to be modified. The proposed constitutive model is validated by comparing predicted and experimental data. The model is shown to be accurate to predict different stress states of concrete under different temperature levels.
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Various stress return algorithms in elastoplastic analyses using the finite element method require the evaluation of the contact (or penetration) stress state (Figure 1), defining…
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Various stress return algorithms in elastoplastic analyses using the finite element method require the evaluation of the contact (or penetration) stress state (Figure 1), defining the transition from elastic to elastoplastic behaviour. Various iterative schemes are commonly used to evaluate contact stress state with a great degree of precision, as subsequent analysis process (forward Euler, mid‐point rule stress return scheme) is greatly affected by the evaluation of the contact stress state, as has been stressed by several authors.