Strain injection techniques in numerical modeling of propagating material failure

The methodology proposed in this research work explores the use of the strain injection concept in a combination of classical strain localization methods and embedded strong discontinuities, to remove the flaws (stress locking and mesh bias dependence) of the former, and simultaneously abdicate of the global tracking algorithms usually required by the later. The basic idea is to use, after the bifurcation instant, i.e. after the time that elements are amenable to develop discontinuities, a mixed continuous displacements - discontinuous constant strains condensable finite element formulation (Q1/e0 ) for quadrilaterals in 2D. This formulation provides improved behavior results, specially, in avoiding mesh bias dependence. In a first, very short, stage after the bifurcation the concept of strong discontinuity is then left aside, and the apparent displacement jump is captured across the finite element length (smeared) like in classical strain localization settings. Immediately after, in a second stage, the kinematics of those finite elements that have developed deep enough strain localization is enriched with the injection of a weak/strong discontinuity mode that minimizes the stress locking defects. The necessary data to inject the discontinuity (the discontinuity direction and its position inside the finite element) is obtained by a post process of the strain-like internal variable field obtained in the first stage, this giving rise to a local (elemental based) tracking algorithm (the crack propagation problem) that can be locally and straightforwardly implemented in a finite element code in a non invasive manner. The obtained approach enjoys the benefits of embedded strong discontinuity methods (stress locking free, mesh bias independence and low computational cost), at a complexity similar to the classical, and simpler, though less accurate, strain localization methods. Moreover, the methodology is applicable to any constitutive model (damage, elasto-plasticity, etc.) without apparent limitations. Representative numerical simulations validate the proposed approach

Multified-based modeling of material failure in high performance reinforced cementitious composites

Cementitious materials such as mortar or concrete are brittle and have an inherent weakness in resisting tensile stresses. The addition of discontinuous fibers to such matrices leads to a dramatic improvement in their toughness and remedies their deficiencies. It is generally agreed that the fibers contribute primarily to the post-cracking response of the composite by bridging the cracks and providing resistance to crack opening (Suwaka & Fukuyama 2006). On the other hand, the multifield theory is a mathematical tool able to describe materials which contain a complex substructure (Mariano & Stazi 2005). This substructure is endowed with its own properties and it interacts with the macrostructure and influences drastically its behavior. Under this mathematical framework, materials such as cement composites can be seen as a continuum with a microstructure. Therefore, the whole continuum damage mechanics theory, incorporating a new microstructure, is still applicable. A formulation, initially based on the theory of continua with microstructure Capriz (Capriz 1989), has been developed to model the mechanical behavior of the high performance fiber cement composites with arbitrarily oriented fibers. This formulation approaches a continuum with microstructure, in which the microstructure takes into account the fibermatrix interface bond/slip processes, which have been recognized for several authors (Li 2003, Naaman 2007b) as the principal mechanism increasing the ductility of the quasi-brittle cement response. In fact, the interfaces between the fiber and the matrix become a limiting factor in improving mechanical properties such as the tensile strength. Particularly, in short fiber composites is desired to have a strong interface to transfer effectively load from the matrix to the fiber. However, a strong interface will make difficult to relieve fiber stress concentration in front of the approaching crack. According to Naaman (Naaman 2003), in order to develop a better mechanical bond between the fiber and the matrix, the fiber should be modified along its length by roughening its surface or by inducing mechanical deformations. Thus, the premise of the model is to take into account this process considering a microfield that represents the slipping fiber-cement displacement. The conjugate generalized stress to the gradient of this micro-field verifies a balance equation and has a physical meaning. This contribution includes the computational modeling aspects of the high fiber reinforced cement composites (HFRCC) model. To simulate the composite material, a finite element discretization is used to solve the set of equations given by the multifield approach for this particular case. A two field discretization: the standard macroscopic and the microscopic displacements, is proposed through a mixed finite element methodology. Furthermore, a splitting procedure for uncoupling both fields is proposed, which provides a more convenient numerical treatment of the discrete equation system. The initiation of failure in HPFRCC at the constitutive level identified as the onset of strain localization depends on the mechanical properties of the all compounds and not only on the matrix ones. As localization criteria is considered the bifurcation analysis in combination with the localized strain injection technique presented by Oliver et al. (Oliver et al. 2010a). It consists of injecting a specific localization mode during the localization stage, via mixed finite element formulations, to the path of elements that are going to capture the cracks, and, in this way, the spurious mesh orientation dependence is removed. Model validation was performed using a selected set of experiments that proves the viability of this approach. The numerical examples of the proposed formulation illustrated two relevant aspects, namely: 1) the role of the bonding mechanism in the strain hardening behavior after cracking in the HPFRCC and 2) the role that plays the finite element formulation in capturing the displacement localization in the localization stage

Stability and robustness issues in numerical modeling of material failure in the strong discontinuity approach

Robustness and stability of the Continuum Strong Discontinuity Approach (CSDA) to material failure are addressed. After identification of lack of symmetry of the finite element  formulation  and  material  softening  in  the  constitutive  model  as  possible  causes  of  loss  of  robustness,  two  remedies  are  proposed:  1) a  symmetric  version  of  the  elementary  enriched  finite  element  with  embedded  discontinuities,  and 2) an implicit explicit integration of the  internal variable, in the constitutive model, that renders the tangent constitutive algorithmic operator positive definite and constant. The combination of both developments leads to finite element formulations with constant and non-singular tangent structural stiffness, these allowing dramatic improvements in terms of robustness and computational costs. After assessing the convergence properties of the new strategies, three-dimensional numerical simulations of failure problems illustrate the performance of the proposed procedures

High-performance model reduction procedures in multiscale simulations

Technological progress and discovery and mastery of increasingly sophisticated structural materials have been inexorably tied together since the dawn of history. In the present era — the so-called Space Age —-, the prevailing trend is to design and create new materials, or improved existing ones, by meticulously altering and controlling structural features that span across all types of length scales: the ultimate aim is to achieve macroscopic proper- ties (yield strength, ductility, toughness, fatigue limit . . . ) tailored to given practical applications. Research efforts in this aspect range in complexity from the creation of structures at the scale of single atoms and molecules — the realm of nanotechnology —, to the more mundane, to the average civil and mechanical engineers, development of structural materials by changing the composition, distribution, size and topology of their constituents at the microscopic/mesoscopic level (composite materials and porous metals, for instance)

Multiscale formulation for material failure accounting for cohesive cracks at the macro and micro scales

This contribution presents a two-scale formulation devised to simulate failure in materials with het- erogeneous micro-structure. The mechanical model accounts for the activation of cohesive cracks in the micro-scale domain. The evolution/propagation of cohesive micro-cracks can induce material instability at the macro-scale level. Then, a cohesive crack is activated in the macro-scale model which considers, in a homogenized sense, the constitutive response of the intricate failure mode taking place in the smaller length scale.The two-scale model is based on the concept of Representative Volume Element (RVE). It is designed following an axiomatic variational structure. Two hypotheses are introduced in order to build the foundations of the entire two-scale theory, namely: (i) a mechanism for transferring kinematical information from macro- to-micro scale along with the concept of “Kinematical Admissibility”, relating both primal descriptions, and (ii) a Multiscale Variational Principle of internal virtual power equivalence between the involved scales of analysis. The homogenization formulae for the generalized stresses, as well as the equilibrium equations at the micro-scale, are consequences of the variational statement of the problem.The present multiscale technique is a generalization of a previous model proposed by the authors and could be viewed as an application of a general framework recently proposed by the authors. The main novelty in this article lies on the fact that failure modes in the micro-structure now involve a set of multiple cohesive cracks, connected or disconnected, with arbitrary orientation, conforming a complex tortuous failure path. Tortuosity is a topic of decisive importance in the modelling of material degradation due to crack propagation. Following the present multiscale modelling approach, the tortuosity effect is introduced in order to satisfy the “Kinematical Admissibility” concept, when the macro-scale kinematics is transferred into the micro-scale domain. There- fore, it has a direct consequence in the homogenized mechanical response, in the sense that the proposed scale transition method (including the tortuosity effect) retrieves the correct post-critical response.Coupled (macro-micro) numerical examples are presented showing the potentialities of the model to sim- ulate complex and realistic fracture problems in heterogeneous materials. In order to validate the multiscale technique in a rigorous manner, comparisons with the so-called DNS (Direct Numerical Solution) approach are also presented

Computational multiscale modeling of fracture problems and its model order reduction

This work focuses on the numerical modeling of fracture and its propagation in heterogeneous materials by means of hierarchical multiscale models based on the FE2 method, addressing at the same time, the problem of the excessive computational cost through the development, implementation and validation of a set of computational tools based on reduced order modeling techniques. For fracture problems, a novel multiscale model for propagating fracture has been developed, implemented and validated. This multiscale model is characterized by the following features: - At the macroscale level, were adapted the last advances of the Continuum Strong Discontinuity Approach (CSDA), developed for monoscale models, devising a new finite element exhibiting  good ability to capture and model strain localization in bands which can be intersect the finite element in random directions; for failure propagation purposes, the adapted Crack-path  field technique, was used. - At the microscale level, for the sake of simplicity, and thinking on the development of the reduced order model, the use of cohesive-band elements, endowed with a regularized isotropic   continuum damage model aiming at representing the material decohesion, is proposed. These cohesive-band elements are distributed within the microscale components, and their boundaries. The objectivity of the solution with respect to the failure cell size at the microscale, and the finite element size at the macroscale, was checked. In the same way, its consistency with respect to Direct Numerical Simulations (DNS), was also tested and verified. &nbsp

Contributions to the continuum modeling of strong discontinuities in two-dimensional solids

The objectives of this monograph are oriented to getting and efficient and robust computational tool that allows the simulation of complex problems in which strain localization aappears. All this relying on a mathematical model consistent from the classical continuum mechanics point of view

Continuum approach to computational multiscale modeling of propagating fracture

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Computational Homogenization of Inelastic Materials using Model Order Reduction

The present work is concerned with the application of projection-based, model reduction techniques to the ecient solution of the cell equilibrium equation appearing in (otherwise prohibitively costly) two-scale, computational homogenization problems. The main original elements of the proposed Reduced-Order Model (ROM) are fundamentally three. Firstly, the reduced set of empirical, globally-supported shape functions are constructed from pre-computed Finite Element (FE) snapshots by applying, rather than the standard Proper Orthogonal Decomposition (POD), a partitioned version of the POD that accounts for the elastic/inelastic character of the solution. Secondly, we show that, for purposes of fast evaluation of the nonane term (in this case, the stresses), the widely adopted approach of replacing such a term by a low-dimensional interpolant constructed from POD modes, obtained, in turn, from FE snapshots, leads invariably to ill-posed formulations. To safely avoid this ill-posedness, we propose a method that consists in expanding the approximation space for the interpolant so that it embraces also the gradient of the global shape functions. A direct consequence of such an expansion is that the spectral properties of the Jacobian matrix of the governing equation becomes a ected by the number and particular placement of sampling points used in the interpolation. The third innovative ingredient of the present work is a points selection algorithm that does acknowledge this peculiarity and chooses the sampling points guided, not only by accuracy requirements, but also by stability considerations. The eciency of the proposed approach is critically assessed in the solution of the cell problem corresponding to a highly complex porous metal material under plane strain conditions. Results obtained convincingly show that the computational complexity of the proposed ROM is virtually independent of the size and geometrical complexity of the considered representative volume, and this a ords gains in performance with respect to nite element analyses of above three orders of magnitude without signi cantly sacri cing accuracy |hence the appellation High-Performance ROM