Modelling Of Strain Localization Research Articles

Discrete-Element Modeling of Strain Localization in Granular Medium at Passive Pressure Application to a Retaining Wall

Discrete-Element Modeling of Strain Localization in Granular Medium at Passive Pressure Application to a Retaining Wall

Cell division in deep material networks applied to multiscale strain localization modeling

Despite the increasing importance of strain localization modeling (e.g., failure analysis) in computer-aided engineering, there is a lack of effective approaches to capturing relevant material behaviors consistently across multiple length scales. We aim to address this gap within the framework of deep material networks (DMN) — a machine learning model with embedded mechanics in the building blocks. A new cell-division scheme is proposed to track the scale transition through the network, and its consistency is ensured by the physics of fitting parameters. Essentially, each microscale node in the bottom layer is described by an ellipsoidal cell with its dimensions back-propagated from the macroscale material point. New crack surfaces in the cell are modeled by enriching cohesive layers, and failure algorithms are developed for crack initiation and evolution in the implicit DMN analysis. Besides studies on a single material point, we apply the multiscale model to concurrent multiscale simulations for the dynamic crush of a particle-reinforced composite tube and various tests on carbon fiber reinforced polymer composites. For the latter, experimental validations on an off-axis tensile test specimen are also provided.

Controls on Asymmetric Rift Dynamics: Numerical Modeling of Strain Localization and Fault Evolution in the Kenya Rift

AbstractComplex, time‐dependent, and asymmetric rift geometries are observed throughout the East African Rift System (EARS) and are well documented, for instance, in the Kenya Rift. To unravel asymmetric rifting processes in this region, we conduct 2D geodynamic models. We use the finite element software ASPECT employing visco‐plastic rheologies, mesh‐refinement, distributed random noise seeding, and a free surface. In contrast to many previous numerical modeling studies that aimed at understanding final rifted margin symmetry, we explicitly focus on initial rifting stages to assess geodynamic controls on strain localization and fault evolution. We thereby link to geological and geophysical observations from the Southern and Central Kenya Rift. Our models suggest a three‐stage early rift evolution that dynamically bridges previously inferred fault‐configuration phases of the eastern EARS branch: (1) accommodation of initial strain localization by a single border fault and flexure of the hanging‐wall crust, (2) faulting in the hanging‐wall and increasing upper‐crustal faulting in the rift‐basin center, and (3) loss of pronounced early stage asymmetry prior to basinward localization of deformation. This evolution may provide a template for understanding early extensional faulting in other branches of the East African Rift and in asymmetric rifts worldwide. By modifying the initial random noise distribution that approximates small‐scale tectonic inheritance, we show that a spectrum of first‐order fault configurations with variable symmetry can be produced in models with an otherwise identical setup. This approach sheds new light on along‐strike rift variability controls in active asymmetric rifts and proximal rifted margins.

Variational Modelling of Strain Localization in Solids: A Computational Mechanics Point of View

Strain localization is one of the most challenging phenomena in solid mechanics. It occurs when strains concentrate in very narrow bands within a solid, typically referred to as localization bands. This behaviour is related to different types of failure mechanisms: fracture, shear bands and slip lines. Being a dissipative process, the modelling of strain localization could seem to preclude the use of variational methods. However, a sound framework to deal with this issue has been developed in the last few decades. In this contribution, we review the modelling of strain localization by means of variational methods systematically, presenting the main underlying theoretical concepts. The issue of irreversibility is approached by means of the theory of generalized standard materials, which constitutes the basis for the variational approach. Then, typical problems occurring in the modelling of strain localization are analyzed: the tendency to localize in a band of zero thickness with no dissipation, the determination of the geometry of the localization band, and the orientation bias of such band with respect to mesh alignment in finite element discretizations. We discuss solutions for these problems, focusing on the approach that tackles the description of the localization band using phase fields.

Implicit gradient softening plasticity for the modeling of strain localization in soils

A non-associative implicit gradient pressure-dependent softening plasticity model and its numerical implementation are presented to capture strain localization in soils. The thermodynamically consistent framework of the generalized micro-morphic approach by Forest (2009) is adopted. A micro-variable is introduced to serve as an additional kinematic variable; the micro-force balance containing higher-order generalized stresses is derived using the virtual power method, and it is then converted into a coupled Helmholtz equation by applying the second law of thermodynamics. The pressure dependency for applications in soils is further considered using a hyperbolic yield function, and a regularized non-associative constitutive model is formulated. Numerically, the Jacobian matrix is obtained by linearizing the residual vector and a finite element procedure coupling both displacements and micro-variable is implemented. The applicability of the proposed methodology to capture strain softening response accompanied by localized shear bands is verified using two numerical examples, and the computed numerical solutions exhibit excellent mesh-independent results. The plastic dissipation analysis further verifies the ability of the proposed model to regularize ill-posed boundary value problems induced by material softening. The influence of key parameters on the simulations and the effects of strain softening on the bearing behavior of soils are illustrated.

Numerical modeling of plastic deformation and failure around a wellbore in compaction and dilation modes

SummaryIn many wellbore stability analyses, the ability to forecast both the occurrence and extent of plastic deformation and failure hinges upon a fundamental understanding of deformation mode and failure mechanism in the reservoir rock. This study focuses on analyzing plastic zones, localized deformations, and failures around a borehole drilled overbalanced or underbalanced through a highly porous rock formation. Based on several laboratory experiments, porous rocks are prone to deform under both shear‐induced dilation and shear‐enhanced compaction mechanisms depending on the stress state. The shapes of the deformation and failure patterns around the borehole are shown, depending on the initial stress state and the local stress paths. The inquiry of the local stress paths in the near‐wellbore zone facilitates the understanding of the reasons for different types of failure mechanisms, including the mixed‐mode and the plastic deformation structures. The modification of the 2D plane strain condition by imitating third stress in the numerical scheme helps us bring the stress paths closer to the real state of loading conditions. Our modeling reveals that the transition from isotropic to anisotropic stress state is accompanied by an increase in the deviatoric part of effective shear tensor that leads to the development of inelastic deformation, degradation, and subsequent rock failure. Particular interest is devoted to the modeling of strain localization especially in compaction mode around a wellbore and computing the amount of stress concentration at the tips of dog‐eared breakouts. Stress concentration can result in a change in irreversible deformation mode from dilatancy to compaction, elucidating the formation of the shear‐enhanced compaction phenomenon at the failure tips in the direction of the minimum horizontal stress.

Material stability and instability in non-local continuum models for dense granular materials

A class of common and successful continuum models for steady, dense granular flows is based on the$\unicode[STIX]{x1D707}(I)$model for viscoplastic grain-inertial rheology. Recent work has shown that under certain conditions,$\unicode[STIX]{x1D707}(I)$-based models display a linear instability in which short-wavelength perturbations grow at an unbounded rate – i.e. a Hadamard instability. This observation indicates that$\unicode[STIX]{x1D707}(I)$models will predict strain localization arising due to material instability in dense granular materials; however, it also raises concerns regarding the robustness of numerical solutions obtained using these models. Several approaches to regularizing this instability have been suggested in the literature. Among these, it has been shown that the inclusion of higher-order velocity gradients into the constitutive equations can suppress the Hadamard instability, while not precluding the modelling of strain localization into diffuse shear bands. In our recent work (Henann & Kamrin,Proc. Natl Acad. Sci. USA, vol. 110, 2013, pp. 6730–6735), we have proposed a non-local model – called the non-local granular fluidity (NGF) model – which also involves higher-order flow gradients and has been shown to quantitatively describe a wide variety of steady, dense flows. In this work, we show that the NGF model also successfully regularizes the Hadamard instability of the$\unicode[STIX]{x1D707}(I)$model. We further apply the NGF model to the problem of strain localization in quasi-static plane-strain compression using nonlinear finite-element simulations in order to demonstrate that the model is capable of describing diffuse strain localization in a mesh-independent manner. Finally, we consider the linear stability of an alternative gradient–viscoplastic model (Bouzidet al.,Phys. Rev. Lett., vol. 111, 2013, 238301) and show that the inclusion of higher-order gradients does not guarantee the suppression of the Hadamard instability.

A phase-field model for strain localization analysis in softening elastoplastic materials

The present paper deals with the localization of strains in those structures consisting of materials exhibiting plastic softening response. It is assumed that strain localization develops in a finite thickness band separated from the remaining part of the structure by weak discontinuity surfaces. In view of the small thickness of the band with respect to the dimensions of the structure, the interphase concept is used for the mechanical modeling of the localization phenomenon. We propose a formulation for the quasi-static modeling of strain localization based on a phase-field approach. In this sense, the localization band is smeared over the volume of the structure and a smooth transition between the fully broken and the sound material phases is introduced. The mechanical performance of the model is illustrated for the case of uniaxial tensile test, discussing the instability of the force-displacement response, and for the case of three-point bending test, comparing the analytical results with the experimental ones.

Cyclic behavior of soils and numerical analyses in cold regions and seismic zones

The solutions of boundary value problems involving strain-softening material property contain serious difficulties from both modeling of strain-localization and a viewpoint of numerical procedure. Mesh size-dependent hardening modulus is considered to alleviate the mesh size-dependency of the solution. The elasto-plastic soil model with kinematic hardening model considering the cumulative deformation by cyclic loading is developed. In finite element analyses, the dynamic relaxation method combined with the generalized return-mapping algorithm is applied to the static drained and un-drained tri-axial tests and plane strain tests. The cyclic behavior of retaining wall problems by freeze and thaw in cold regions is also analyzed. Finally the dynamic progressive failure analysis of rockfill dam is carried out.

Elastoplastic axisymmetric Cosserat continua and modelling of strain localization

The theory and finite-element formula of the axisymmetric Cosserat continuum are presented by introducing an independent rotational degree of freedom into the axisymmetric continuum. A pressure-dependent elastic-plastic model and the corresponding consistent algorithm for the constitution integration of the model are presented in the framework of the axisymmetric Cosserat continuum theory. The numerical examples illustrate that the developed model can maintain the well-posedness of the boundary value problems; overcome the problems of mesh-dependency, which commonly occur in the classical continuum when the strain softening behaviour is incorporated; and capture the entire progressive failure process.

Experimental and numerical investigation of the fracture in steel welded joints

The work is devoted to the experimental and numerical investigation of fracture of the laser-welded specimen under quasi-static tension. In present study, two different steels stainless 316LN and P91 were welded with Laser Welding Process. The experimental study was carried out in Indira Gandhi Centre for Atomic Research (India). The evolution of the temperature field during each test was record by infrared camera CEDIP 420. Experimental results have shown quasi-wave character of the temperature evolution accompanying the specimen fracture process. During the experiments authors shown that accumulation of plastic strain irregular between joined materials, plastic flow in the P91 side overtakes plastic flow in the SS316LN side due to the various strain hardening rates of those materials. Numerical simulation of a tensile test was performed in the finite-element package Comsol Multiphysics under plane stress conditions. Simulation was carried out for two cases: small strain condition and large plastic strains, large plastic strains allow illustrate the necking effect of the specimen in the localization zone. The temperature evolution on the surface of a specimen has quantitative correspondences with the experimental results. Results of numerical simulation provided an explanation of the observed phenomena and let us to propose a model of strain localization.

Numerical modeling of strain localization in engineering ductile materials combining cohesive models and X-FEM

The present work aims at numerically predicting the current residual strength of large engineering structures made of ductile metals against accidental failure. With this aim in view, the challenge consists in reproducing within a unified finite element-based methodology the successive steps of micro-voiding-induced damage, strain localization and crack propagation, if any. A key ingredient for a predictive ductile fracture model is the proper numerical treatment of the critical transition phase of damage-induced strain localization inside a narrow band. For this purpose, the strong discontinuity cohesive model and the eXtended Finite Element Method are combined. A propagation algorithm is proposed and studied in the context of ductile materials. Physics-motivated criteria to pass from the phase of more or less diffuse damage to strain localization and from strain localization to crack propoagation are proposed. Finally, a 2D numerical example is shown to study the performance of the failure analysis model when implemented into an engineering finite element computation code, namely Abaqus.

Specific features of the transformation of spall cracks to localized shear bands

Investigations of specific features of the microstructure of the region where a spall crack transforms to an adiabatic shear band are based on a spall model of strain localization, which implies that adiabatic shear bands are induced by interference of unloading waves, and the value of the negative stress in the expansion region of these waves does not exceed the dynamic strength of the material. It is shown that the transformation region contains a tremendous number of dislocation ensembles, which is much greater than the number of dislocation ensembles generated by a shock wave. Detection of micrometer-sized fracture sites in the region of interference of unloading waves implies that small fracture sites are formed in a polycrystalline material on dislocations arising in the course of dynamic tension.

3D multiscale modeling of strain localization in granular media

A hierarchical multiscale modeling approach is used to investigate three-dimensional (3D) strain localization in granular media. Central to the multiscale approach is a hierarchical coupling of finite element method (FEM) and discrete element method (DEM), wherein the FEM is employed to treat a boundary value problem of a granular material and the required constitutive relation for FEM is derived directly from the DEM solution of a granular assembly embedded at each of the FEM Gauss integration points as the representative volume element (RVE). While being effective in reproducing the complex mechanical responses of granular media, the hierarchical approach helps to bypass the necessity of phenomenological constitutive models commonly needed by conventional FEM studies and meanwhile offers a viable way to link the macroscopic observations with their underlying microscopic mechanisms. To model the phenomenon of strain localization, key issues pertaining to the selection of proper RVE packings are first discussed. The multiscale approach is then applied to simulate the strain localization problem in a cubical specimen and a cylindrical specimen subjected to either conventional triaxial compression (CTC) or conventional triaxial extension (CTE) loading, which is further compared to a case under plane-strain biaxial compression (PBC) loading condition. Different failure patterns, including localized, bulging and diffuse failure modes, are observed and analyzed. Amongst all testing conditions, the PBC condition is found most favorable for the formation of localized failure. The CTC test on the cubical specimen leads to a 3D octopus-shaped localization zone, whereas the cylindrical specimen under CTC shows seemingly bulging failure from the outlook but rather more complex failure patterns within the specimen. The CTE test on a uniform specimen normally ends in a diffuse failure. Different micro mechanisms and controlling factors underlying the various interesting observations are examined and discussed.

On numerical aspects of different updating schedules for tracking fracture path in strain localization modeling

A comparative study is performed that investigates numerical features of different schedules, end-of-step vs. within iterations, for updating fracture path by employing the local and global tracking strategies. Embedded strong discontinuities within an enhanced finite element framework are used to model propagating discontinuities and fracture behavior of quasi-brittle materials. It is shown that end-of-step updating, which is a standard, can cause inaccuracies in peak strength and fracture energy for large time steps. Updating within iterations rectifies the accuracy issues, but at the expense of an increased computational cost. Both schedules yield comparable performance as the step size is refined.

Strain localization under dynamic loading

A spallation model of strain localization is suggested according to which localization bands under pulsed loading result from unloading wave interference such that negative stresses in the extension zone are lower than the ultimate strength of the material. The temperature in the localization band is estimated to be close to the environmental temperature.

A model of strain localization in porous sandstone as a function of tectonic setting, burial and material properties; new insight from Provence (southern France)

The analysis of three cataclastic band sets from Provence (France) reveals that the band density, their conjugate angles, their ratio of shear displacement to compaction, and the amount of cataclasis within the bands differ and can be expressed as functions of tectonic setting and petrophysical properties. We identify (1) a dense and closely spaced network of shear-enhanced (reverse) compaction bands; (2) a regularly spaced less dense network of reverse compactional shear bands; and (3) a localized network of normal shear bands. The field data show that strain localization is favored in an extensional regime and is characterized by shear bands with a large shear to compaction ratio and a small conjugate band angle. In contrast, distributed strain is favored in a contractional regime and is characterized by compactional bands with a low ratio of shear to compaction and a large conjugate band angle. To explain the mechanical origin of this strain localization, we quantify the yield strength and the stress evolution in extensional and contractional regimes in a frictional porous granular material. We propose a model of strain localization in porous sands as a function of tectonic stresses, burial depth, material properties, strain hardening and fluid pressure. Our model suggests that stress reduction, inherent to extensional regime, favors strain localization as shear bands, whereas stress increase during contraction favors development of compactional bands.

Hollow Cylinder Tests on Boom Clay: Modelling of Strain Localization in the Anisotropic Excavation Damaged Zone

Boom Clay is extensively studied as a potential candidate to host underground nuclear waste disposal in Belgium. To guarantee the safety of such a disposal, the mechanical behaviour of the clay during gallery excavation must be properly predicted. In that purpose, a hollow cylinder experiment on Boom Clay has been designed to reproduce, in a small-scale test, the Excavation Damaged Zone (EDZ) as experienced during the excavation of a disposal gallery in the underground. In this article, the focus is made on the hydro-mechanical constitutive interpretation of the displacement (experimentally obtained by medium resolution X-ray tomography scanning). The coupled hydro-mechanical response of Boom Clay in this experiment is addressed through finite element computations with a constitutive model including strain hardening/softening, elastic and plastic cross-anisotropy and a regularization method for the modelling of strain localization processes. The obtained results evidence the directional dependency of the mechanical response of the clay. The softening behaviour induces transient strain localization processes, addressed through a hydro-mechanical second grade model. The shape of the obtained damaged zone is clearly affected by the anisotropy of the materials, evidencing an eye-shaped EDZ. The modelling results agree with experiments not only qualitatively (in terms of the shape of the induced damaged zone), but also quantitatively (for the obtained displacement in three particular radial directions).

A polycrystal plasticity model of strain localization in irradiated iron

At low to intermediate homologous temperatures, the degradation of structural materials performance in nuclear environments is associated with high number densities of nanometric defects produced in irradiation cascades. In polycrystalline ferritic materials, self-interstitial dislocations loops are a principal signature of irradiation damage, leading to a mechanical response characterized by increased yield strengths, decreased total strain to failure, and decreased work hardening as compared to the unirradiated behavior. Above a critical defect concentration, the material deforms by plastic flow localization, giving rise to strain softening in terms of the engineering stress–strain response. Flow localization manifests itself in the form of defect-depleted crystallographic channels, through which all dislocation activity is concentrated. In this paper, we describe the formulation of a crystal plasticity model for pure Fe embedded in a finite element polycrystal simulator and present results of uniaxial tensile deformation tests up to 10% strain. We use a tensorial damage descriptor variable to capture the evolution of the irradiation damage loop subpopulation during deformation. The model is parameterized with detailed dislocation dynamics simulations of tensile tests up to 1.5% deformation of systems containing various initial densities of irradiation defects. The coarse-grained simulations are shown to capture the essential details of the experimental stress response observed in ferritic alloys and steels. Our methodology provides an effective linkage between the defect scale, of the order of one nanometer, and the continuum scale involving multiple grain orientations.

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.