Continuum Strong Discontinuity Approach Research Articles

Analysis of 3D quasi-brittle solids failures by crack growth using the strong discontinuity approach with the boundary element method

The continuum strong discontinuity approach (CSDA) is suitable for the analysis of material failures of quasi-brittle solids when constitutive damage models are applied to kinematic equations with displacement discontinuities. Its association with the implicit version of the boundary element method (BEM), using cells with embedded strong discontinuity, allows the analysis of material failures with crack growth. Such cells are mandatory for the domain discretization, which are restricted to the region where the crack is supposed to occur. They are introduced in the model as the analysis progresses, without the need to re-evaluate the pre-existing terms of the matrices, but only through their appropriate expansion. Discontinuity in the displacement field is quantified, after internal equilibrium verification, assuming a uniform displacement jump inside each cell. This work uses this methodology for the analysis of three-dimensional solids under external loads. An elastic-degrading isotropic constitutive model with an exponential softening law is considered, with the strong discontinuity regime imposed directly after the end of the elastic regime. This is the first time that such cells are applied for full three-dimensional mixed mode fracture problems, in the context of the boundary element method. The results presented in the numerical examples are in good agreement with the references.

Three dimensional cells with embedded strong discontinuity for material failure analysis by the boundary element method

The Boundary Element Method (BEM) associated with the Continuum Strong Discontinuity Approach (CSDA) can be used as a non-geometric alternative for the modelling of crack growth in physically non-linear three-dimensional problems of solid mechanics. This work makes use of the refereed approach through the implementation of volumetric hexahedral cells with embedded discontinuity that enables the evaluation of discontinuous jumps in the displacement field along a crack discontinuity surface. This is made with an internal equilibrium verification, using a regularized isotropic damage constitutive model. Although this formulation has been extensively studied for two-dimensional problems over the past few years, this is the first time that a three-dimensional extension is presented. The strong discontinuity regime in the cells were imposed directly after the end of the elastic regime, as a typical behaviour of the fracture process of brittle materials. These cells are needed only at the specific region of the solid domain where the cracks propagation is supposed to occur, while the remaining non-discretized regions of domains are considered to work in elastic regime. Some simple numerical examples shows the feasibility of such methodology.

A new class of cells with embedded discontinuity for fracture analysis by the boundary element method

SummaryMaterial failure analysis are addressed by the continuum strong discontinuity approach within the implicit formulation of the boundary element method in this work. An automatic cell generation algorithm is used to track cracks during the loading processes in the nonlinear analysis. As a major novelty, a new class of cells with embedded discontinuity is developed in which nonuniform displacement jump components are considered. It is shown that this new approach eliminates the stress locking effect verified in propagation analyses using cells with embedded uniform discontinuous displacement field, since the relative rotational motion between the two portions of a cell can now be properly captured.

On the use of high aspect ratio finite elements to model hydraulic fracturing in deformable porous media

Hydraulic fracturing is a technique based on the injection of a viscous fluid at high pressure into an engineered well with the intention of initiating and propagating multiple fractures in a rock formation containing hydrocarbons to increase well-reservoir connectivity. This work proposes a framework to simulate the initiation and propagation of hydraulically induced fracture based on the Continuum Strong Discontinuity Approach (CSDA) and solid finite elements with high aspect ratio (HAR) in the context of conventional continuum constitutive (stress–strain) relationships (based on damage theory). The porous media considered here are deformable and the hydro-mechanical problem is solved in a fully-coupled manner. Full details about the proposed continuous approach to model hydraulic fractures are presented, including the finite element equations and their approximations. The new approach is validated against analytical and numerical solutions. Moreover, the influence of the dimensions of the HAR interface elements on the results is also investigated.

Two-Dimensional Numerical Model of the Fracture Process in Steel Fibre Reinforced Concrete with the Continuum Strong Discontinuity Approach and Functional Data Analysis

Abstract This paper presents the formulation of a two-dimensional numeri-cal model able to describe the fracture process in structural mem-bers of steel fibre reinforced concrete (SFRC) from the volume ratio of the fibres and the mechanical properties of the compo-nents: a concrete matrix and a set of steel fibres with a random orientation. The relationship between the stress and the strain fields of the composite material is obtained using the mixture theory with a compatibility strain of its component materials. The concrete matrix is represented with a scalar damage constitutive model with a softening strain and a different strength in tension and compression. The mechanical strain of an insulated fibre and the slip between the fibre and the matrix are simultaneously de-scribed with a one-dimensional plasticity constitutive model. The cracking of the composite material indicates a jump in the dis-placement field and non-bounded values of the strain field, which are represented by the Continuum Strong Discontinuity Ap-proach. The model has been implemented in the framework of the nonlinear analysis with the Finite Element Method, using con-stant strain triangular elements. Moreover, the fibres distribution and orientation change randomly in each finite element and each simulation or observation. The structural responses of the simula-tions are treated as curves and analysed by tools from the Func-tional Data Analysis. Confidence intervals for the structural re-sponse are built using bootstrap methodology. Finally, experi-mental tests of SFRC members subjected to tension and bending are simulated. The structural response and the cracking patterns obtained from the numerical simulation are satisfactory.

Crack propagation using the continuum strong discontinuity approach by the BEM: some numerical remarks

Some numerical remarks regarding the crack evolution in failure analysis by the BEM using cells with embedded strong discontinuities are addressed in this work. A comparative study between the generation of these cells at any iteration or only after a step convergence is firstly performed. Moreover, an analysis is carried out related to the cells size growth throughout the iterative-incremental process. As reference, some classical problems whose experimental results are available in the literature are used for the numerical analysis which is performed considering the implicit formulation of the boundary element method together with the continuum strong discontinuity approach. It was verified that the results are coincident for different numbers of steps considered in the simulations when cells are generated during any iteration, showing step size independence in this case, while the same is not true for the case of cells generated only after step convergence, in which a large number of steps are required for a good accuracy. Finally, it is shown that a small increase in cell size throughout the analysis contributes to the reduction in numerical processing time without significantly affecting the results accuracy.

A boundary element method formulation for quasi-brittle material fracture analysis using the continuum strong discontinuity approach

The implicit formulation of the boundary element method is applied to bi-dimensional problems of material failure involving, sequentially, inelastic dissipation with softening in continuous media, bifurcation and transition between weak and strong discontinuities. The bifurcation condition is defined by the singularity of the localization tensor. Weak discontinuities are related to strain localization bands of finite width, which become increasingly narrow until to collapse in a surface with discontinuous displacement field, called strong discontinuity surface. To associate such steps to the fracture process in quasi-brittle materials, an isotropic damage constitutive model is used to represent the behaviour in all of them, considering the adaptations that come from the strong discontinuity analysis for the post-bifurcation steps. The crack propagation across the domain is done by an automatic cells generation algorithm and, in this context, the fracture process zone in the crack tip became totally represented.

Strain-injection and crack-path field techniques for 3D crack-propagation modelling in quasi-brittle materials

This paper presents a finite element approach for modelling three-dimensional crack propagation in quasi-brittle materials, based on the strain injection and the crack-path field techniques. These numerical techniques were already tested and validated by static and dynamic simulations in 2D classical benchmarks [Dias et al., in: Monograph CIMNE No-134. International Center for Numerical Methods in Engineering, Barcelona, (2012); Oliver et al. in Comput Methods Appl Mech Eng 274:289–348, (2014); Lloberas-Valls et al. in Comput Methods Appl Mech Eng 308:499–534, (2016)] and, also, for modelling tensile crack propagation in real concrete structures, like concrete gravity dams [Dias et al. in Eng Fract Mech 154:288–310, (2016)]. The main advantages of the methodology are the low computational cost and the independence of the results on the size and orientation of the finite element mesh. These advantages were highlighted in previous works by the authors and motivate the present extension to 3D cases. The proposed methodology is implemented in the finite element framework using continuum constitutive models equipped with strain softening and consists, essentially, in injecting the elements candidate to capture the cracks with some goal oriented strain modes for improving the performance of the injected elements for simulating propagating displacement discontinuities. The goal-oriented strain modes are introduced by resorting to mixed formulations and to the Continuum Strong Discontinuity Approach (CSDA), while the crack position inside the finite elements is retrieved by resorting to the crack-path field technique. Representative numerical simulations in 3D benchmarks show that the advantages of the methodology already pointed out in 2D are kept in 3D scenarios.

Experimental and numerical modeling of shear behavior of laminated Guadua bamboo for different fiber directions

The shear strength and the crack pattern of laminated Guadua bamboo (LGB) are studied in this work, by means of both experimental tests and numerical simulation, considering the three different orthogonal directions of the fibers: (i) when the fiber is parallel to the loading direction, (ii) when the fiber is perpendicular to the loading direction without crossing the reduced area plane, and (iii) when the fiber is perpendicular to the loading direction and crossing the reduced area plane. The slats orientation and disposition in LGB were experimentally studied, when the contact surface between slats is parallel to the reduced area plane and when it is perpendicular. Likewise, the cases when the contact surface between slats matches with the reduced area plane is tested and when it is not. The numerical model of finite elements is based on the continuum strong discontinuity approach for composite materials and the Weibull probability model for the failure progressive of fibers. The crack direction obtained in the numerical simulation is approximately equal to the one observed in the corresponding experimental test, regardless of the disposition slats. LGB with fibers parallel to the loading direction exhibits a crack that matches with the reduced area plane. In LGB with fibers perpendicular to the loading direction that do not cross the reduced area plane, the crack is inclined 45 degrees with respect to loading direction. In LGB with fibers perpendicular to the loading direction and crossing the reduced area plane, several cracks are made up perpendicularly to loading direction and parallel to the fiber orientation.

The Strong Discontinuity Approach as a limit case of strain localization in the implicit BEM formulation

The Implicit Formulation of the Boundary Element Method (BEM) is used to deal with physically non-linear 2D-problems in solids. An isotropic damage constitutive model, equipped with a strain softening rule, is applied to increasingly narrow bandwidths, determined by mesh refinement, showing the typical post-peak mesh-dependence behaviour. The Continuum Strong Discontinuity Approach (CSDA), characterized by the introduction of discontinuous jumps in the displacement field together with an equilibrium verification (using continuous constitutive models) on the discontinuity line, is also applied to the implicit formulation of the BEM. By a simple numerical example, it is shown that, beyond the mesh independence, the CSDA results represent the limit case of a zero localization bandwidth. Moreover, to demonstrate the accuracy of the CSDA, other three examples involving concrete fracture are presented and the obtained results are compared with experimental or analytical data available in the literature.

Strain injection techniques in dynamic fracture modeling

A computationally affordable modeling of dynamic fracture phenomena is performed in this study by using strain injection techniques and Finite Elements with Embedded strong discontinuities (E-FEM). In the present research, classical strain localization and strong discontinuity approaches are considered by injecting discontinuous strain and displacement modes in the finite element formulation without an increase of the total number of degrees of freedom. Following the Continuum Strong Discontinuity Approach (CSDA), stress–strain constitutive laws can be employed in the context of fracture phenomena and, therefore, the methodology remains applicable to a wide number of continuum mechanics models. The position and orientation of the displacement discontinuity is obtained through the solution of a crack propagation problem, i.e. the crack path field, based on the distribution of localized strains. The combination of the above mentioned approaches is envisaged to avoid stress-locking and directional mesh bias phenomena.Dynamic simulations are performed increasing the loading rate up to the appearance of crack branching, and the variation in terms of failure modes is investigated as well as the influence of the strain injection together with the crack path field algorithm. Objectivity of the presented methodology with respect to the spatial and temporal discretization is analyzed in terms of the dissipated energy during the fracture process. The dissipation at the onset of branching is studied for different loading rate conditions and is linked to the experimental maximum velocity observed before branching takes place.

2D mesoscale model for concrete based on the use of interface element with a high aspect ratio

The mesostructure of concrete plays a very important role in the process of initiation and propagation of cracks. Microcracks tend to start in the Interfacial Transition Zone (ITZ) and propagate toward the mortar matrix until a macrocrack formation. Seeking to better understand the influence of the concrete mesoscopic structure, translated macroscopically in the form of loss of stiffness and energy dissipation, this work proposes a 2D mesoscale model in which the concrete is modeled as a heterogeneous three-phase material composed of coarse aggregates, mortar matrix and ITZ. The coarse aggregates are generated from a grading curve and placed into the mortar matrix randomly. Interface solid finite elements with a high aspect ratio are used to represent the ITZ and the crack process based on a mesh fragmentation technique. These interface elements present the same kinematics as the Continuum Strong Discontinuity Approach (CSDA), which allows the use of a continuum constitutive relation to describe their behavior. Thus, an appropriate continuum tension damage model is adopted to describe the complex nonlinear behavior of concrete due to the crack phenomenon. Initially, the proposed mesoscale approach is applied in uniaxial tensile tests to study the influence of the size, volume and distribution of the coarse aggregates within the mortar matrix. Then, three-point bending beams are simulated in mesoscale and the results compared with the experimental ones. The results showed that the proposed 2D mesoscale model presents the same kinds of characteristics that real 3D concrete shows, considering the effects of the mesostructure constituents.

On the use of finite elements with a high aspect ratio for modeling cracks in quasi-brittle materials

A new technique for modeling cracks in quasi-brittle materials based on the use of interface solid finite elements is presented. This strategy named mesh fragmentation technique consists in introducing sets of standard low-order solid finite elements with a high aspect ratio in between regular (or bulk) elements of the mesh to fill the very thin gaps left by the mesh fragmentation procedure. The conception of this strategy is supported by the fact that, as the aspect ratio of a standard low-order solid finite element increases, the element strains also increase, approaching the same kinematics as the Continuum Strong Discontinuity Approach. As a consequence, the analyses can be performed integrally in the context of the continuum mechanics, and complex crack patterns can be simulated without the need of tracking algorithms. A tension damage constitutive relation between stresses and strains is proposed to describe crack formation and propagation. This constitutive model is integrated using an implicit–explicit integration scheme to avoid convergence drawbacks, commonly found in problems involving discontinuities. 2D and 3D numerical analyses are performed to show the applicability of the presented technique. Relevant aspects such as the influence of the thickness of the interface elements and mesh objectivity are investigated. The results show that the technique is able to predict satisfactorily the behavior of structural members involving different crack patterns, including multiple cracks, without significant mesh dependency provided that unstructured meshes are used.

Continuum approach to computational multiscale modeling of propagating fracture

A new approach to two-scale modeling of propagating fracture, based on computational homogenization (FE2), is presented. The specific features of the approach are: (a) a continuum setting for representation of the fracture at both scales based on the Continuum Strong Discontinuity Approach (CSDA), and (b) the use, for the considered non-smooth (discontinuous) problem, of the same computational homogenization framework than for classical smooth cases. As a key issue, the approach retrieves a characteristic length computed at the lower scale, which is exported to the upper one and used therein as a regularization parameter for a propagating strong discontinuity kinematics. This guarantees the correct transfer of fracture energy between scales and the proper dissipation at the upper scale. Representative simulations show that the resulting formulation provides consistent results, which are objective with respect to size and bias of the upper-scale mesh, and with respect to the size of the lower-scale RVE/failure cell, as well as the capability to model propagating cracks at the upper scale, in combination with crack-path-field and strain injection techniques. The continuum character of the approach confers to the formulation a minimal intrusive character, with respect to standard procedures for multi-scale computational homogenization.

Continuum Approach to Computational Multi-Scale Modeling of Fracture

This paper presents a FE2 multi-scale framework for numerical modeling of the structural failure of heterogeneous quasi-brittle materials. The model is assessed by application to cementitious materials. Using the Continuum Strong Discontinuity Approach (CSD), innovative numerical tools, such as strain injection and crack path field techniques, provide a robust, and mesh-size, mesh-bias and RVE-size objective, procedure to model crack onset and propagation at the macro-scale.

Strain localization, strong discontinuities and material fracture: Matches and mismatches

The work focuses on the connections of strain localization modeling of material failure and discrete fracture mechanics. It is an attempt to give an answer to the old question of whether the finite element solutions of material failure problems based on strain localization techniques, using standard continuum stress–strain constitutive models equipped with strain softening, have physical sense as solutions of de-cohesive fracture mechanics problems. Based on some well-established links of the Continuum Strong Discontinuity Approach (CSDA) to material failure and cohesive fracture mechanics, some objective indicators are proposed to assess the quality of strain localization results. These indicators are simply derived on the basis of the inelastic strain distribution provided by the strain localization solutions and the direction of propagation of the localization band. They can be computed without knowledge of the exact fracture mechanics solution of the problem, and used as error indicators in a large variety of material failure situations. The proposed indicators are assessed, by means of their application to evaluation of a number of strain localization solutions of benchmark problems. Issues as the influence of the mesh structure and alignment, type of constitutive model and considered finite element techniques are examined. On the light of the obtained results, classical flaws in strain localization solutions, as mesh bias dependence and stress-locking are reinterpreted.

A macroscopic damage‐plastic constitutive law for modeling quasi‐brittle fracture and ductile behavior of concrete

SUMMARYA new phenomenological macroscopic constitutive model for the numerical simulation of quasi‐brittle fracture and ductile concrete behavior, under general triaxial stress conditions, is presented. The model is particularly addressed to simulate a wide range of confinement stress states, as also, to capture the strong influence of the mean stress value in the concrete failure mechanisms.The model is based on a two‐surface damage‐plastic formulation. The mechanical behavior in different domains of the stress space is separately described by means of a quasi‐brittle or ductile material response: (i) For positive values of the mean stress (tensile states), an isotropic continuum damage model with strain softening is considered. In this context, and in order to avoid the Boundary Value Problem ill‐posedness induced by the softening law, a regularization technique based on the Continuum Strong Discontinuity Approach (CSDA) is adopted, which results equivalent to a damage model with embedded cohesive cracks providing anisotropic responses. (ii) A plastic model governs the material behavior when the mean stress is negative (confinement states). It is based on the classical plastic flow theory. In particular, a yield criterion similar to that of Willam and coauthors, which depends on the three stress invariants, is used. Additional features defining the plastic response are: an isotropic strain hardening law and a non‐associative flow rule.The paper presents the numerical implementation of the model using an efficient integration algorithm, namely, the Impl‐Ex scheme. Several widely known experimental tests (such as uniaxial, biaxial and triaxial tests) carried out on concrete specimens are used to calibrate and validate the performance of the proposed formulation. Finally, a classical 2D reinforced concrete beam example is analyzed in order to show the predictive capability of the model in structural analysis applications. Copyright © 2011 John Wiley & Sons, Ltd.

Numerical modelling of the fracture process in reinforced concrete by means of a continuum strong discontinuity approach. Part II: application to shear panels

The numerical simulation results of the fracture process in reinforced concrete shear panels are presented in this work. The simulation used a model based on the continuum strong discontinuity approach (CSDA) and mixing theory. CSDA describes strain localization and formation of discontinuity associated with the appearance of a crack. On the other hand, mixing theory represents composite material behaviour which is formed by a simple concrete matrix and one or two bundles of long reinforcement bars. The behaviour of simple concrete and steel is represented by a two dimensional damage model and one-dimensional plasticity model, respectively. The model has been implemented in the finite element method which considers plane stress, infinitesimal strain and static loads. Three panels are simulated, reinforced in one or two ways; they are mainly subjected to shear forces. The numerical simulation results as well as structural response and cracking patterns were satisfactory.

Numerical modeling of the fracture process in reinforced concrete by means of the continuum strong discontinuity approach. Part I: formulation

Reinforced concrete structures generally refers to beams, columns and walls which are constituted by complex lattices of steel bars embedded in a concrete matrix, exhibiting multiple cracks due to high external loads. This paper presents the formulation of a numerical model aimed at describing the fracture process in reinforced concrete, from the volumetric ratio of concrete and steel Crack formation and propagation in a composite material is described in the model by an enhanced strain field, such as that established in the continuum strong discontinuity approach and mixture theory. The composite material is constituted by a concrete matrix and one or two steel bar orthogonal packages. The steel and concrete are represented by a one-dimensional plasticity model and a scalar damage model having different tension and compression strength, respectively. The dowel action and the bond-slip effects between the bars and the matrix are described with additional models relating component material stress and strain. It is concluded that the proposed model can easily be implemented in the finite element method, due to several conventional nonlinear numerical process characteristics which remain. The model would also allow the problem to be analysed at macroscopic scale, thereby avoiding a finite element mesh having to be constructed for each component material and its interaction effects and reducing computational costs.

Mesoscopic model to simulate the mechanical behavior of reinforced concrete members affected by corrosion

In this contribution, a finite element methodology devised to simulate the structural deterioration of corroded reinforced concrete members is presented. The proposed numerical strategy has the ability to reproduce many of the well-known (undesirable) mechanical effects induced by corrosion processes in the embedded steel bars, as for example: expansion of the reinforcements due to the corrosion product accumulation, damage and cracking patterns distribution in the surrounding concrete, degradation of steel–concrete bond stress transfer, net area reduction in the reinforcements and, mainly, the influence of all these mentioned mechanisms on the structural load carrying capacity predictions. At the numerical level, each component of the RC structure is represented by means of a suitable FE formulation. For the concrete, a cohesive model based on the Continuum Strong Discontinuity Approach (CSDA) is used. Steel bars are modeled by means of an elasto-plastic constitutive relation. The interface is simulated using contact-friction elements, with the friction degradation as a function of the degree of corrosion attack. Two different (and coupled) mesoscopic analyzes are considered in order to describe the main physical phenomena that govern the problem: (i) an analysis at the cross section level and (ii) an analysis at the structural member level. The resultant mechanical model can be used to simulate generalized reinforcement corrosion. Experimental and previous numerical results, obtained from the available literature, are used to validate the proposed strategy.