Abstract

This work investigates systematically strain localization and failure mechanics for elastoplastic damage solids. Two complementary methodologies, i.e., traction-based discontinuities localized in an elastic solid and strain localization of a stress-based inelastic softening solid, are addressed. In the former it is assumed a priori that the discontinuity (band) forms with a continuous stress field and along the known orientation. A traction-based failure criterion is introduced to characterize the discontinuity (band) and the orientation is determined from Mohr’s maximization postulate. If the (apparent) displacement jumps are retained as independent variables, the strong/regularized discontinuity approaches follow, requiring constitutive models for both the bulk and discontinuity (band). Elimination of the displacement jumps at the material point level results in the embedded/smeared discontinuity approaches in which an overall inelastic constitutive model fulfilling the static constraint suffices. The second methodology is then adopted to check whether the assumed strain localization can occur and identify its consequences on the resulting approaches. The kinematic constraint guaranteeing stress boundedness/continuity upon strain localization is established for general inelastic softening solids. Application to a unified elastoplastic damage model naturally yields all the ingredients of a localized model for the discontinuity (band), justifying the first methodology. Two dual but not necessarily equivalent approaches, i.e., the traction-based elastoplastic damage model and the stress-based projected discontinuity model, are identified. The former is equivalent to the embedded/smeared discontinuity approaches, whereas in the later the discontinuity orientation and associated failure criterion, not given a priori, are determined consistently from the kinematic constraint. The bi-directional connections and equivalence conditions between the traction- and stress-based approaches are classified. Closed-form 2D results under plane stress condition are also given, with the classical Rankine, Mohr-Coulomb, von Mises and Drucker-Prager criteria analyzed as the illustrative examples. A generic failure criterion of either elliptic, parabolic or hyperbolic type, is then considered in a unified manner, resulting in many failure criteria frequently employed in practice.

Abstract

A new numerical model for the structural assessment of gravity dams by means of a semi-discrete approach is proposed. Gravity dams are massive structures, which their stability depends on the gravity loads applied into the structure. Mainly, its the structural assessment is performed by means of a gravity approach. However, this approach is too conservative and mostly does not reflect the real structural behaviour of the dam. In this context, there is the need of models that are simplified enough to allow a simple and fast parametric analyses. The proposed model idealizes the dam as a set of rigid elements, where the damage and the deformation are concentrated in the contact sides between adjacent elements. Thus, the elements are rigid, but the material is considered as deformable. As the proposed model is semi-discrete, it can detect separation or sliding between elements. However, initial contacts do not change during the analyisis and a relative continuity among elements exists, in order to simplify the computational effort. The effective performance of the proposed model is demonstrated by numerical validation and by comparisons with some numerical models presented in the literature.

Abstract

A new numerical model for the structural assessment of gravity dams by means of a semi-discrete approach is proposed. Gravity dams are massive structures, which their stability depends on the gravity loads applied into the structure. Mainly, its structural assessment is performed by means of a gravity approach. However, this approach is too conservative and mostly does not reflect the real structural behaviour of the dam. In this context, there is the need of models that are simplified enough to allow a simple and fast parametric analyses. The proposed model idealizes the dam as a set of rigid elements, where the damage and the deformation are concentrated in the contact sides between adjacent elements. Thus, the elements are rigid, but the material is considered as deformable. As the proposed model is semi-discrete, it can detect separation or sliding between elements. However, initial contacts do not change during the analyisis and a relative continuity among elements exists, in order to simplify the computational effort. The effective performance of the proposed model is demonstrated by numerical validation and by comparisons with some numerical models presented in the literature.

Abstract

This paper briefly reviews the formulations used over the last 40 years for the solution of problems involving tensile cracking, with both the discrete and the smeared crack approaches. The paper focuses on the smeared approach, identifying as its main drawbacks the observed mesh‐size and mesh‐bias spurious dependence when the method is applied ‘straightly’. A simple isotropic local damage constitutive model is considered, and the (exponential) softening modulus is regularized according to the material fracture energy and the element size. The continuum and discrete mechanical problems corresponding to both the weak discontinuity (smeared cracks) and the strong discontinuity (discrete cracks) approaches are analysed and the question of propagation of the strain localization band (crack) is identified as the main difficulty to be overcome in the numerical procedure. A tracking technique is used to ensure stability of the solution, attaining the necessary convergence properties of the corresponding discrete finite element formulation. Numerical examples show that the formulation derived is stable and remarkably robust. As a consequence, the results obtained do not suffer from spurious mesh‐size or mesh‐bias dependence, comparing very favourably with those obtained with other fracture and continuum mechanics approaches.

Abstract

This article presents the analysis of the structure of Mallorca Cathedral taking into account the influence on the structural behavior of auxiliary iron ties used during the construction process. Recent studies (Roca et al. 2012, 2013) presented some hypotheses about the construction process of the cathedral. This article complements the previous results by considering the use of auxiliary ties as temporary stabilizing device during the construction. Evidence of the use of ties during the construction has been recognized after a comprehensive survey. The study of the role of such ties and the effect of their later removal are studied by a FE analysis carried out on a representative bay of the structure. The study includes a time-dependent FE analysis after the removal of the ties to assess the long-term structural behavior. The results of the numerical analysis are compared with the deformation trends identified by means of a recent monitoring campaign.

Abstract

Damage-induced strain softening is of vital importance for the modeling of localized failure in frictional-cohesive materials. This paper addresses strain localization of damaging solids and the resulting consistent frictional-cohesive crack models. As a supplement to the framework recently established for stress-based continuum material models in rate form (Wu and Cervera 2015, 2016), several classical strain-based damage models, expressed usually in total and secant format, are considered. Upon strain localization of such damaging solids, Maxwell’s kinematics of a strong (or regularized) discontinuity has to be reproduced by the inelastic damage strains, which are defined by a bounded characteristic tensor and an unbounded scalar related to the damage variable. This kinematic constraint yields a set of nonlinear equations from which the discontinuity orientation and damage-type localized cohesive relations can be derived. It is found that for the “Simó and Ju 1987” isotropic damage model, the localization angles and the resulting cohesive model heavily depend on lateral deformations usually ignored in classical crack models for quasi-brittle solids. To remedy this inconsistency, a modified damage model is proposed. Its strain localization analysis naturally results in a consistent frictional-cohesive crack model of damage type, which can be regularized as a classical smeared crack model. The analytical results are numerically verified by the recently-proposed mixed stabilized finite element method, regarding a singly-perforated plate under uniaxial tension. Remarkably, for all of the damage models discussed in this work, the numerically-obtained localization angles agree almost exactly with the closed-form results. This agreement, on the one hand, consolidates the strain localization analysis based on Maxwell’s kinematics and, on the other hand, illustrates versatility of the mixed stabilized finite element method.

Abstract

This paper describes a procedure for the solution of problems involving tensile cracking using the so-called smeared crack approach, that is, standard finite elements with continuous displacement fields and a standard local constitutive model with strain-softening. An isotropic Rankine damage model is considered. The softening modulus is adjusted according to the material fracture energy and the element size. The resulting continuum and discrete mechanical problems are analyzed and the question of predicting correctly the direction of crack propagation is deemed as the main difficulty to be overcome in the discrete problem. It is proposed to use a crack tracking technique to attain the desired stability and convergence properties of the corresponding formulation. Numerical examples show that the resulting procedure is well-posed, stable and remarkably robust; the results obtained do not seem to suffer from spurious mesh-size or mesh-bias dependence.

Abstract

The aim of this contribution is to present a series of organised benchmarks that helps at validating the robustness of the FE implementation of a constitutive relation, its pertinence with respect to experiments, and quantitative and qualitative comparisons of structural elements. It is applied to an isotropic damage model used for concrete, coupled with a non local gradient formulation to avoid a spurious description of strain localisation. After the elementary (uniaxial monotonic, cyclic or triaxial loading) and structural (three point bending tests) simulations, an industrial application is presented in the form of a representative structural volume of a containment building for French nuclear power plants.

Abstract

Pure elastic damage models or pure elastic plastic constitutive laws are not totally satisfactory to describe the behaviour of concrete. They indeed fail to reproduce the unloading slopes during cyclic loading which define experimentally the value of the damage in the material. When coupled effects are considered, in particular in hydro-mechanical problems, the capability of numerical models to reproduce the unloading behaviour is essential, because an accurate value of the damage, which controls the material permeability, is needed. In the context of very large size calculations that are needed for 3D massive structures heavily reinforced and pre-stressed (such as containment vessels), constitutive relations ought also to be as simple as possible. Here an elastic plastic damage formulation is proposed to circumvent the disadvantages of pure plastic and pure damage approaches. It is based on an isotropic damage model combined with a hardening yield plastic surface in order to reach a compromise as far as simplicity is concerned. Three elementary tests are first considered for validation. A tension test, a cyclic compression test and triaxial tests illustrate the improvements achieved by the coupled law compared to a simple damage model (plastic strains, change of volumetric behaviour, decrease in the elastic slope under hydrostatic pressures). Finally, one structural application is also considered: a concrete column wrapped in a steel tube.

Abstract