SECOND GRADE MODELING FOR THE STRAIN LOCALIZATION ANALYSIS OF LAYERED MATERIALS WITH DAMAGING INTERFACES (original) (raw)
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A multi-scale strain-localization analysis of a layered strip with debonding interfaces
The paper is focused on the multi-scale modeling of shear banding in a two-pha se linear elastic periodically layered strip with damaging interfaces. A two-dimensional layered strip is considered subjected to transverse shear and is assumed to have a finite thickness along the direction of the layers and an infinite extension along the direction perpendicular to layering. The strip is analyzed as a second-gradi ent continuum resulting from a second-order homogenization procedure developed by the Authors, here specialized to the case of layered materials. This analysis is also aimed to understand the influence on the strain localization and post-peak structural response of the displacement boun dary conditions prescribed at the strip edges. To this end, a first model representative of the strip with warping allowed at the edges is analyzed in which the strain localization process is obtained as a results of a bifurc ation in analogy to the approach by . A second model is analyzed in which the warping of the edge is inhibited and the damage propagates from the center of the specimen without exhibiting bifurcation phenomena. For this latter case the effects of a possible interactio n between the shear band and the boundary shear layer are considered, which are influenced mainly by the characteristic lengths of the model and the strip length. For realistic values of the relevan t parameters it is shown that the boundary conditions have a small effe cts on the elastic response and on the overall strength of the model. Conversely, the boundary conditions have a significant effe ct on the shear band location, the post-peak response and the structural brittleness. Since the model parameters directly depend on the material microstructure as a result of the homogenization process, both the extensio n of the shear band and the occurrence of snap-back in the post-peak phase may be controlled in terms of the constitutive parameters and of the geometry of the phases.
Boundary condition effects on multiscale analysis of damage localisation
The choice of boundary conditions used in multiscale analysis of heterogeneous materials affects the numerical results, including the macroscopic constitutive response, the type and extent of damage taking place at the microscale and the required size of the Representative Volume Element (RVE). We compare the performance of periodic boundary conditions and minimal kinematic boundary conditions applied to the unit cell of a particulate composite material, both in the absence and presence of damage at the particle–matrix interfaces. In particular, we investigate the response of the RVE under inherently non-periodic loading conditions, and the ability of both boundary conditions to capture localization events that are not aligned with the RVE boundaries. We observe that, although there are some variations in the evolution of the microscale damage between the two methods, there is no significant difference in homogenized responses even when localization is not aligned with the cell boundaries.
Damage localization and rupture with gradient damage models
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International Journal of Plasticity, 2009
Sheet metal forming processes generally involve large deformations together with complex loading sequences. In order to improve numerical simulation predictions of sheet part forming, physically-based constitutive models are often required. The main objective of this paper is to analyze the strain localization phenomenon during the plastic deformation of sheet metals in the context of such advanced constitutive models. Most often, an accurate prediction of localization requires damage to be considered in the finite element simulation. For this purpose, an advanced, anisotropic elastic-plastic model, formulated within the large strain framework and taking strain-path changes into account, has been coupled with an isotropic damage model. This coupling is carried out within the framework of continuum damage mechanics. In order to detect the strain localization during sheet metal forming, Rice's localization criterion has been considered, thus predicting the limit strains at the occurrence of shear bands as well as their orientation. The coupled elastic-plastic-damage model has been implemented in Abaqus/implicit. The application of the model to the prediction of Forming Limit Diagrams (FLDs) provided results that are consistent with the literature and emphasized the impact of the hardening model on the strain-path dependency of the FLD. The fully three-dimensional formulation adopted in the numerical development allowed for some new results -e.g. the out-of-plane orientation of the normal to the localization band, as well as more realistic values for its in-plane orientation.
Computer Methods in Applied Mechanics and Engineering
Integral Non-Local (INL) formulations are often used to regularize Continuum Damage computations, in the presence of stress softening for instance. The introduction of a characteristic/internal length allows for avoiding pathological mesh dependency. Some questions concerning the identification of the characteristic length, its possible evolution during damage process and the need for special treatments of non-locality operators near boundaries (e.g. edges, cracks) are however still open. A physical request is that material points separated by a crack (or an highly damaged zone) should not interact. Despite what is done in standard Integral Non-Local theories, this can be obtained by allowing non-local interactions to evolve depending on mechanical fields (e.g. damage, strain, stress). The Eikonal Non-Local (ENL) formulation provides a novel interpretation of damage dependent non-local interactions. Based on the Wentzel-Kramers-Brillouin (WKB) approximation for high-frequency wave propagation in a damaged medium, this formulation defines the interaction distances as the solution of a stationary damage dependent Eikonal equation. It allows for the modeling of non-local interactions which gradually vanish in damaged zones, thus ensuring a progressive transition from diffuse damage to fracture in a natural way. The numerical implementation and properties of this regularization technique are investigated and discussed. From a numerical viewpoint, a Fast Marching method is used to compute non-local interaction distances between Gauss integration points. Geodesic distances are then used to define the kernel of weighting function to be used in integral non-local averaging. Several numerical results of quasi-statics simulations of quasi-brittle fracture in isotropic media are presented.
A Multiscale Method for Damage Analysis of Quasi-Brittle Heterogeneous Materials
Computer Modeling in Engineering & Sciences, 2019
A novel multiscale algorithm based on the higher-order continuum at both micro-and macrostructural level is proposed for the consideration of the quasi-brittle damage response of heterogeneous materials. Herein, the microlevel damage is modelled by the degradation of the homogenized stress and tangent stiffness tensors, which are then upscaled to govern the localization at the macrolevel. The C 1 continuity finite element employing a modified case of Mindlin's form II strain energy density is derived for the softening analysis. To the authors' knowledge, the finite element discretization based on the strain gradient theory is applied for the modeling of damage evolution at the microstructural level for heterogeneous materials for the first time. The advantage of the novel C 1 finite element formulation in comparison with the standard finite element discretization in terms of the regularization efficiency as well as the objectivity has been shown. An isotropic damage law is used for the reduction of the constitutive and nonlocal material behaviour, which is necessary for the physically correct description of the localization formation in quasi-brittle materials. The capabilities of the derived finite element to capture the fully developed localization zones are tested on a random representative volume element (RVE) for several different loading cases. By employing the conventional second-order computational homogenization, the microstructural material constitutive response is averaged over the whole RVE area. In order to model the loss of structural integrity when sharp localization is formed across RVE, the specific conditions which detect a completely formed localization zone are developed. A new failure criterion at the microstructural level has been proposed. The derived finite element formulation, as well as the multiscale damage algorithm, are implemented into the finite element program ABAQUS. The capabilities of the presented multiscale scheme to capture the effects of the deformation localization are demonstrated by few benchmark numerical examples.
A Multiscale Micro-Continuum Model to Capture Strain Localization in Composite Materials
International Journal for Multiscale Computational Engineering, 2012
This paper presents a plasticity/damage formulation in the context of the physically based micro-continuum theory for multiphase materials described in a companion paper (see Vernerey, A physically-based micro-continuum theory, Mech. Adv. Mater. Struct., 2012). Based on a micro-structurally motivated decomposition of the deformation, the presented inelastic formulation is capable of characterizing the independent plastic/damage processes occurring in different phases (such as fiber or inclusions) and predicting the overall material behavior. The inelastic constitutive relation can thus be cast in a simple, physically motivated form, compared to conventional models. Such a formulation is thus very attractive for establishing a link between materials structure and properties. To illustrate the presented framework, we apply the micro-continuum model to the tensile failure of fiber-reinforced composite and compare it to a "brute force" approach in which the microstructure is explicitly modeled. We show that the model captures accurately the evolution of various features that cannot be calculated with conventional methods such as the independent stress, strain, and damage in the matrix and fibers and the fiber/matrix interface. Moreover, the existence of a size effect during failure is accounted for correctly.
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The macroscopic material properties of the cohesive cracks are obtained from the inelastic deformation manifested in a localization band (modelled with a continuum damage theory) at the microscopic scale. The macroscopic behavior of the adhesive crack is derived from the response of a microscale sample representing the microstructure inside the adhesive crack. In this manuscript, we extend the theory presented in Comput.
Shear band localization via local J2 continuum damage mechanics
Computer Methods in Applied Mechanics and Engineering, 2004
This paper describes a novel formulation for the solution of problems involving shear band localization using a local isotropic J 2 continuum damage model and mixed linear simplex (triangles and tetrahedra). Stabilization methods are used to ensure existence and uniqueness of the solution, attaining global and local stability of the corresponding discrete finite element formulation. Consistent residual viscosity is used to enhance robustness and convergence of the formulation. Implementation and computational aspects are also discussed. A simple isotropic local J 2 damage constitutive model is considered, either with linear or exponential softening. The softening modulus is regularized according to the material mode II fracture energy and the element size. Numerical examples show that the formulation derived is fully stable and remarkably robust, totally free of volumetric locking and spurious oscillations of the pressure. As a consequence, the results obtained do not suffer from spurious mesh-size or mesh-bias dependence, comparing very favourably with those obtained with the ill-posed standard approaches. , strategy, a variant of the first, the nonlinear constitutive laws, for plasticity or damage, are made dependent not only on the local inelastic strain, but also on its second gradient, which is computed according to some additional relation which couples it to the local strain, . In the third, micropolar, strategy, the usual non-polar description of Continuum Mechanics is substituted by other nonstandard theory, like the CosseratÕs continuum, see, for instance, [9] and [10]. In the fourth, viscous-regularized, strategy, the rate-independent format is substituted by a regularized version, also dependent on the strain-rate, see .
Continuum media from classical mechanics cannot appropriately reproduce the evolution of materials exhibiting strong heterogeneities in the strain field, e.g. strain localization. Models without a microscale representation cannot properly reproduce the microscale mechanisms that trigger the strain localization, in addition, first gradient relations don't present any length parameter in the formulation. This results in a model without a characteristic length that cannot exhibit any objective band width. In this paper, techniques to introduce an internal length will be enumerated. Microstuctured materials will be retained and in particular Second Gradient model will be exposed and used along with a FEMxDEM approach. Numerical results showing the abilities of the enriched model will conclude the text.