Crack Band Approach Research Articles

Progressive failure analysis of laminar composites under compression using smeared crack-band damage model and full layerwise theory

This paper presents an original finite element (FE) model that integrates the smeared crack band (SCB) approach and full layerwise plate theory (FLWT). The model enhances the computational efficiency of progressive failure analysis (PFA) of laminar composites in compression, by utilizing the layerwise approach which reduces a 3D model to a 2D one. The model distributes damage throughout the FE domain, with fracture mechanisms represented by material stiffness degradation controlled by damage variables (based on equivalent strains specifically defined for each failure mode). Mesh dependency issues are addressed by scaling fracture energy using a characteristic element length, and failure initiation and modes are determined using the 3D Hashin failure criterion.Accurately describing lamina response in fiber direction under compression requires linear-brittle softening with a stress plateau. The study showed that a model considering 30 % of residual stress accurately predicts maximum stress regardless of mesh refinement, demonstrating results’ minor dependence from the selected element size.The model accuracy has been confirmed by comparing the obtained results against experimental and benchmark data from the literature. The size effect study demonstrated a decrease in maximum stress of the open-hole laminates in compression with increasing specimen in-plane size. This trend is consistent with experimental and reference numerical observations, confirming the model accuracy and applicability even for relatively coarse meshes. Therefore, computational efficiency is improved, with preserved accuracy of conventional solid finite element models.

Mesoscale modelling of progressive damage and failure in single-lap and double-lap thin-ply laminated composite bolted joints

Composite bolted joints are being used extensively in primary load-bearing structures of modern aircrafts. However, simulating progressive damage and failure of the bolted composite structures under high bearing loading is still challenging. In this paper, numerical simulation and experimental verification on the bearing failure of single-lap and double-lap thin-ply laminated composite bolted joints were carried out for the entire loading process. 3D explicit finite element models have been developed using Abaqus/Explicit together with a 3D physically-based intralaminar damage model implemented in a VUMAT subroutine. An interface cohesive-zone model based on the Benzeggagh-Kenane (B-K) law was used to simulate delamination initiation and evolution. For the intralaminar damage, the initiation was determined based on Pinho’s failure criteria while the evolution was regularized with the crack-band approach to alleviate mesh-size dependence. Element deletion and in-situ effect as practical numerical strategy are discussed and analyzed in detail. Element deletion is used to prevent excessive element distortion and capture corresponding post-peak bearing failure behavior. In-situ effect has been proved to have a key influence on the damage initiation and progression of thin-ply composite matrix. The final simulation results are shown to agree well with experimental data and reproduce the mechanical response curves. More importantly, the model can accurately capture the local crushing of bolt hole in the later loading stage. This study shows that the numerical model can be helpful for the design and optimization of composite bolted structures.

Lattice modelling of substandard RC beam-column joints considering localization issues

The strain adjustment for quasi-brittle materials to minimize the mesh size effect and localization issues are investigated by the nonlinear lattice modelling approach for substandard reinforced concrete (RC) beam-column joints (BCJs). The lattice model is generated by implementing the truss analogy for the concrete and reinforcement steel members in RC BCJs. The relevant uniaxial material properties in OpenSees are assigned to the truss members. The reinforcement bond-slip relationship is modelled by zero-length springs attached to steel and concrete elements. The modelling approach is validated by comparing the simulated responses to the experimental behaviour of inadequately detailed RC test specimens. A 3-story 3-bay frame tested with a pseudo-dynamic testing procedure and two substandard RC BCJs tested under quasi-static cyclic loading, one of which contains a slab and an out-of-plane beam, are simulated. To avoid the mesh size effect and localization issues, the strain-based softening function in the uniaxial material property of concrete is adjusted by adopting the crack band approach. Analysis results of the rigid joint model assumption, which does not account for the joint deformation, are also compared with the outcomes of the lattice model. By minimizing the mesh size effect, more realistic results are obtained using the lattice modelling technique with high computational efficiency and less analysis efforts. The model accuracy in reproducing the experimental behaviour and response quantities of prime interest is at a reasonable level for the investigated test specimens.

Compressive damage modeling of fiber-reinforced composite laminates using 2D higher-order layer-wise models

A refined progressive damage analysis of fiber-reinforced laminated composites subjected to compressive loads is presented here. The numerical analysis exploits higher-order theories developed using the Carrera Unified Formulation, specifically 2D plate theories with Lagrange polynomials to enhance the kinematic approximation through each ply's thickness resulting in a layer-wise structural model. The CODAM2 material model, based on continuum damage mechanics, governs the intralaminar composite damage. The Hashin criteria and the crack-band approach provides failure initiation and propagation, respectively. Fiber micro-buckling and kinking is taken into account via the use of nonlinear post-peak softening models. It is shown that linear-brittle stress-strain softening is effective for accurate compressive strength predictions. A series of numerical assessments on coupon-level composite laminates is carried out to verify the proposed numerical framework while its validation is demonstrated by successfully applying the numerical tool to test cases for which experimental data is available from the literature. Various through-the-thickness structural models are evaluated to provide insights for proper modeling. Numerical assessments considered quasi-isotropic laminates, the compressive strength and size-effects under brittle fracture of notched laminates, and progressive damage characteristics due to stable crack growth in compact compression tests. The results show the possibility of using coarser meshes than those used in standard FEM approaches as the accuracy of predictions is preserved through the use of higher-order structural theories.

Structural size effect: Experimental, theoretical and accurate computational assessment

In this paper, experimental evidence, theoretical predictions and the finite element modelling of the structural size effect in cracking problems of quasi-brittle materials are discussed and assessed against each other. The fracture process is modelled through the crack band approach, using an isotropic damage constitutive law. The correct dissipation of the fracture energy, essential for modelling the phenomenon with precision, is introduced. An enhanced accuracy mixed finite element formulation is used to ensure mesh bias independent results.Several experimental campaigns where size effect is investigated are numerically reproduced in 2D and in 3D to assess the feasibility and the performance of the method. For this, mode I and mixed mode I and II fracture situations are considered in notched and unnotched beams.The correlation of the experimental results with the numerical simulations shows the capacity of the mixed FE formulation to reproduce crack paths, force-displacement curves and collapse mechanisms with precision for a wide range of structural sizes. The enhanced accuracy FE formulation eliminates the spurious mesh dependency that is characteristic of standard FE simulations.In addition, the model is able to follow Bazant’s size effect law with precision. Results confirm that the energy release rate in the progressing fracture is the fundamental cause of size effect in quasi-brittle materials. This is additionally verified in a study of the relative influence of statistical and energetic size effect. Computations show that the essential requirements to suitably simulate the phenomenon are (1) a fracture model ensuring the correct energy dissipation at the crack and (2) a method guaranteeing mesh objective results.

Tension‐Compression Damage Model with Consistent Crack Bandwidths for Concrete Materials

This paper proposes a tension‐compression damage model for concrete materials, formulated within the framework of thermodynamics of irreversible processes. The aim of this work is to solve the following problems: the premature divergence of numerical solutions under general loading conditions due to the conflict of tensile and compressive damage bounding surfaces, which is a result of the application of the spectral decomposition method to distinguish tension and compression, and the unsatisfactory reproduction of distinct tension‐compression behaviors of concrete by strain‐driven damage models. The former is solved by the sign of the volumetric deformation, while the latter is solved via two separated dissipation mechanisms. Moreover, of specific interest is an improved solution to the problem of mesh‐size dependency using consistent crack bandwidths, which takes into account situations with irregular meshes and arbitrary crack directions in the context of the crack band approach. The performance of the model is validated by the well‐documented experimental data. The simplicity and the explicit integration of the constitutive equations render the model well suitable for large‐scale computations.

Gradient‐Enhancement of a Rock Model Applied to Numerical Simulations of Tunnel Advance

AbstractRock is classified as a frictional cohesive material characterized by nonlinear stress‐strain relations including softening leading to structural failure. To describe the mechanical behavior of rock comprising the accumulation of irreversible strains, strain hardening, and the degradation of stiffness and strength, an isotropic damage plasticity model for intact rock and rock mass was proposed in [1]. In the context of numerical simulations of tunneling, considering strain softening of rock allows to predict the formation of shear bands emanating from the tunnel surface due to the excavation process, which is crucial for engineers to forecast potentially dangerous situations related to failure. To ensure objective results in the softening regime upon mesh refinement, in the aforementioned rock model the concept of the crack band approach by Bažant and Oh was adopted. As a remedy of known deficiencies related to this regularization approach, the rock model is extended by the over‐nonlocal implicit gradient‐enhancement proposed in [2]. Accordingly, a nonlocal field of the damage‐driving variable, implicitly defined as the solution of a Helmholtz‐like partial differential equation, is introduced yielding a fully coupled system of second‐order PDEs. In the present contribution, the over‐nonlocal gradient enhancement of the rock model is presented and applied together with a linear‐elastic material model for shotcrete to finite element simulations of a deep tunnel constructed by the New Austrian Tunneling Method. Based on this real example, objectivity of the results with respect to the finite element discretization is discussed.

Examination of two regularized damage-plasticity models for concrete with regard to crack closing

Continuum models of fracture should be equipped with a localization limiter to prevent pathological discretization sensitivity of finite element simulations. Moreover, stiffness recovery, called also crack closing effect, should be reproduced in the modelling of quasi-brittle materials when reversed loading is applied. In this paper the efficiency of two different established damage-plasticity models for concrete is assessed from the viewpoint of the two above-mentioned aspects. The first description is the so-called concrete damaged plasticity model, available in the ABAQUS package. In this model the plasticity theory is not only augmented with stiffness degradation and recovery, but also with the crack band approach and viscoplastic regularization. The gradient-enhanced damage description, which is the second one considered, can also be coupled to plasticity. In this model additional averaging equation prevents the pathological discretization sensitivity of fracture simulations. Two fields, displacements and averaged strain measure, are interpolated and suitable finite elements are programmed by the authors in the FEAP package. Basic concepts of the numerical analysis of the crack closing phenomenon are reviewed. Both models are capable of avoiding mesh-sensitivity, but it is achieved in a different manner and it is shown in the paper that the users of ABAQUS should employ the crack band options with particular care. Representative examples in the context of the examined issues are demonstrated: uniaxial tension of a bar and a cantilever beam. The results obtained using both the models are compared and the effects of regularization and crack closing are illustrated.

Discontinuous Cell Method (DCM) for the simulation of cohesive fracture and fragmentation of continuous media

In this paper, the Discontinuous Cell Method (DCM) is formulated with the objective of simulating cohesive fracture propagation and fragmentation in homogeneous solids without issues relevant to excessive mesh deformation typical of available Finite Element formulations. DCM discretizes solids by using the Delaunay triangulation and its associated Voronoi tessellation giving rise to a system of discrete cells interacting through shared facets. For each Voronoi cell, the displacement field is approximated on the basis of rigid body kinematics, which is used to compute a strain vector at the centroid of the Voronoi facets. Such strain vector is demonstrated to be the projection of the strain tensor at that location. At the same point stress tractions are computed through vectorial constitutive equations derived on the basis of classical continuum tensorial theories. Results of analysis of a cantilever beam are used to perform convergence studies and comparison with classical finite element formulations in the elastic regime. Furthermore, cohesive fracture and fragmentation of homogeneous solids are studied under quasi-static and dynamic loading conditions. The mesh dependency problem, typically encountered upon adopting softening constitutive equations, is tackled through the crack band approach. This study demonstrates the capabilities of DCM by solving multiple benchmark problems relevant to cohesive crack propagation. The simulations show that DCM can handle effectively a wide range of problems from the simulation of a single propagating fracture to crack branching and fragmentation.

Modelling the failure of reinforced concrete with nonlocal and crack band approaches using the damage-plasticity model CDPM2

A nonlocal extension of the damage-plasticity model CDPM2 is proposed. The performance of this extension is evaluated in comparison with a crack band version of the same model for describing the failure of reinforced concrete. In particular, the influence of mesh size on the structural response in the form of load–displacement curves and strain distributions is studied for a reinforced concrete beam subjected to four point bending. The nonlocal model provides mesh independent load–displacement curves and strain profiles, whereas the peak loads and strain profiles obtained with the crack band model depend on the element size.

The simulation of crack opening-closing and aggregate interlock behaviour in finite element concrete models

The ability to model crack closure behaviour and aggregate interlock in finite element concrete models is extremely important. Both of these phenomena arise from the same contact mechanisms and the advantages of modelling them in a unified manner are highlighted. An example illustrating the numerical difficulties that arise when abrupt crack closure is modelled is presented and the benefits of smoothing this behaviour are discussed. We present a new crack-plane model that uses an effective contact surface derived directly from experimental data and which is described by a signed-distance function in relative-displacement space. The introduction of a crack-closure transition function into the formulation improves its accuracy and enhances its robustness. The characteristic behaviour of the new smoothed crack-plane model is illustrated for a series of relative-displacement paths. We describe a method for incorporating the model into continuum elements using a crack band approach and address a previously overlooked issue associated with scaling the inelastic shear response of a crack-band. A consistent algorithmic tangent and associated stress recovery procedure are derived. Finally, a series of examples are presented demonstrating that the new model is able to represent a range of cracked concrete behaviour with good accuracy and robustness. This article is protected by copyright.

Numerical aspects of the crack band approach

The crack band approach has been used in many engineering computations as a simple technique eliminating or reducing the sensitivity of numerical results to the size of finite elements in simulations that involve strain localization due to softening. In the present paper, this traditional technique is carefully analyzed and potential sources of errors are identified. The role of various components of the modeling strategy is elucidated by simple examples dealing with a one-dimensional tensile test and a two-dimensional model of a notched beam subjected to bending. The influence of the specific choice of finite elements, integration scheme and formula for estimation of the crack band width is analyzed. Particular attention is devoted to the differences among band width estimation methods based on element area or volume, on element projection, and on the idea proposed by Oliver [32]. The results are summarized in the form of recommendations which can contribute to reduction of errors in practical simulations.

Simulation of corroded RC structures using a three-dimensional irregular lattice model

Deteriorative effects of steel corrosion on the structural response of reinforced concrete are simulated for varying degrees of corrosion. The simulation approach is based on a three-dimensional irregular lattice model of the bulk concrete, in which fracture is modeled using a crack band approach that conserves fracture energy. Frame elements and bond link elements represent the reinforcing steel and its interface with the concrete, respectively. Polylinear stress-slip properties of the link elements are determined, for several degrees of corrosion, through comparisons with direct pullout tests reported in the literature. The link properties are then used for the lattice modeling of reinforced concrete beams with similar degrees of corrosion of the main reinforcing steel. The model is successful in simulating several important effects of steel corrosion, including increased deflections, changes in flexural cracking behavior, and reduced yield load of the beam specimens.

A reinforced concrete frame element with shear effect

A novel flexibility-based 1D element that captures the material nonlinearity and second order P-<TEX>$\Delta$</TEX> effects within a reinforced concrete frame member is developed. The formulation is developed for 2D planar frames in the modified fiber element framework but can readily be extended to 3D cases. The nonlinear behavior of concrete including cracking and crushing is taken into account through a modified hypo-elastic model. A parabolic and a constant shear stress distribution are used at section level to couple the normal and tangential tractions at material level. The lack of objectivity due to softening of concrete is addressed and objectivity of the response at the material level is attained by using a technique derived from the crack band approach. Finally the efficiency and accuracy of the formulation is compared with experimental results and is demonstrated by some numerical examples.

Numerical Modeling of Concrete-FRP Debonding Using a Crack Band Approach

In this study, numerical procedures are proposed to predict the structural behavior of concrete members strengthened with fiber-reinforced polymeric (FRP) sheets or plates. The concept of damage band or crack band is introduced and used for predicting the debonding failure of the concrete-epoxy interface formed when FRP sheets or plates are externally bonded to a concrete substrate. In the crack band approach, all the processes taking place during the failure of a concrete-epoxy interface are smeared in a band of fixed width. This makes the approach attractive from a modeling point of view since continuum theories, along with softening relations, can be used to model the damage which causes debonding of the interface. In order to validate this approach, numerical predictions, using the concept of crack band, are compared against experimental results obtained from tests of concrete blocks and reinforced concrete beams strengthened with FRP. In particular, the capability of the proposed numerical approach t...

Fracture of fiber-reinforced cement composites: effects of fiber dispersion

Short-fiber reinforcement is commonly used to improve various properties of cement-based materials, including their resistance to shrinkage cracking, post-cracking strength, and toughness. The spatial distribution of fibers within a structural component exhibits local variations that depend on many factors, including the methods of production. Local measures of fiber volume fraction can be much lower than the global average and such regions can act as flaws within the composite material. In addition, fiber orientation is restricted by any near surfaces of the structural component. Such non-uniformities of the fiber distribution can significantly affect fracture behavior and complicate the interpretation of fracture test results. This paper concerns the computational modeling of fiber- reinforced cement composites (FRCC), with attention to the effects of non-uniform fiber distributions. The simulations utilize lattice models of the matrix phase of the composite, based on the Voronoi tessellation of irregularly positioned nodes within the structural domain. A crack band approach provides energy-conserving, mesh-insensitive descriptions of fracture through the irregular lattice. Each fiber is explicitly modeled within the lattice framework. The fiber distributions are synthetically constructed within the structural domain, although experimentally determined fiber positions could be used. Models of simple tensile tests demonstrate a dependence of post-cracking strength and toughness on the spatial distributions of the fibers.

Spectral Stiffness Microplane Model for Quasibrittle Composite Laminates—Part II: Calibration and Validation

The spectral stiffness microplane (SSM) model developed in the preceding Part I of this study is verified by comparisons with experimental data for uniaxial and biaxial tests of unidirectional and multidirectional laminates. The model is calibrated by simulating the experimental data on failure stress envelopes analyzed in the recent so-called “World Wide Failure Exercise,” in which various existing theories were compared. The present theory fits the experiments as well as the theories that were best in the exercise. In addition, it can simulate the post-peak softening behavior and fracture, which is important for evaluating the energy-dissipation capability of composite laminate structures. The post-peak softening behavior and fracture are simulated by means of the crack band approach which involves a material characteristic length.

Formulation and numerical implementation of a constitutive law for concrete with strain-based damage and plasticity

A triaxial constitutive law for concrete within the framework of isotropic damage combined with plasticity is proposed in this paper. It covers typical characteristics of concrete like non-linear uniaxial compression and tension, bi- and triaxial failure criteria and dilatancy with a unified strain-based approach. Thus, this model is quite simple and especially suitable for strain-driven methods like common finite elements. It is complemented with a regularization method based on the crack band approach. A further issue is discussed with procedures for the model parameter determination for a wide range of concrete grades. The application of the model is demonstrated with typical benchmark tests for plain concrete.

Three dimensional combined fracture–plastic material model for concrete

This paper describes a combined fracture–plastic model for concrete. Tension is handled by a fracture model, based on the classical orthotropic smeared crack formulation and the crack band approach. It employs the Rankine failure criterion, exponential softening, and it can be used as a rotated or a fixed crack model. The plasticity model for concrete in compression is based on the Menétrey–Willam failure surface, the plastic volumetric strain as a hardening/softening parameter and a non-associated flow rule based on a nonlinear plastic potential function. Both models use a return-mapping algorithm for the integration of constitutive equations. Special attention is given to the development of an algorithm for the combination of the two models. The suggested combination algorithm is based on a recursive substitution, and it allows for the two models to be developed and formulated separately. The algorithm can handle cases when failure surfaces of both models are active, but also when physical changes such as crack closure occur. The model can be used to simulate concrete cracking, crushing under high confinement and crack closure due to crushing in other material directions. The model is integrated in a general finite element package ATENA and its performance is evaluated by comparisons with various experimental results from the literature.

Crack band model of fracture in irregular lattices

An irregular lattice model is used to simulate mode I fracture in softening materials, such as concrete. Lattice geometry is based on a three-dimensional Voronoi discretization of the material domain. The Voronoi diagram provides scaling rules for the elemental stiffness relations, leading to an elastically uniform representation of the material for simple modes of straining. Fracture is represented using a crack band approach, in which the dimensions of the crack band are also scaled according to the Voronoi diagram. The material is viewed as homogeneous and the energy dissipation mechanisms active at finer scales are lumped into a cohesive crack relation. This energy conserving crack band approach is objective with respect to the irregular geometry of the lattice. Model accuracy and performance are demonstrated through simulated fracture testing of concrete specimens under uniaxial tension and flexural loadings. Basic qualities of the simulation approach, demonstrated here for homogeneous models of concrete, are applicable toward simulating fracture in multi-phase systems where material features are explicitly modeled.