Modes Of Strain Localization Research Articles

Experimental study of compaction localization in carbonate rock and constitutive modeling of mechanical anisotropy

AbstractSedimentary rocks are inherently anisotropic and prone to strain localization. While the influence of rock anisotropy on the brittle/dilative regime has been studied extensively, its influence on the ductile/compactive regime is much less explored. This paper discusses the anisotropic behavior of a high‐porosity carbonate rock from central Europe (the Maastricht Tuffeau). A set of triaxial tests with concurrent x‐ray tomography has been performed at different confining pressures. The anisotropic characteristics of this rock have been investigated by testing samples cored at different inclinations of the bedding, thus revealing non‐negligible effects of the coring direction on yielding and compaction behavior. Specifically, samples cored perpendicular to bedding display higher strength and longer stages of post‐yielding deformation before manifesting re‐hardening. Despite such alterations of the inelastic response, Digital Image Correlation has revealed that the strain localization mode is independent of the coring direction, thus being primarily affected by the confinement level. To capture the observed interaction between material anisotropy and compaction behavior at the continuum‐scale, an elastoplastic constitutive law has been proposed. For this purpose, a set of tensorial bases has been introduced to replicate how the oriented rock fabric modulates the yielding and plastic flow characteristics of the material. The analyses show that the impact of the coring direction on yield function and plastic flow rule is fundamentally different, thus requiring the use of distinct projection strategies (a strategy here defined heterotopic mapping). The performance of the model, studied through parametric analyses and by calibrating the experimental results, illustrates the improved capability of the proposed constitutive approach when applied to strongly anisotropic porous rocks.

Discrete Element Method Modeling for the Failure Analysis of Dry Mono-Size Coke Aggregates.

An in-depth study of the failure of granular materials, which is known as a mechanism to generate defects, can reveal the facts regarding the origin of the imperfections, such as cracks in the carbon anodes. The initiation and propagation of the cracks in the carbon anode, especially the horizontal cracks below the stub-holes, reduce the anode efficiency during the electrolysis process. The failure analysis of coke aggregates can be employed to determine the appropriate recipe and operating conditions in order to avoid the formation of cracks in the carbon anodes. In this paper, it will be shown that a particular failure mode can be responsible for the crack generation in the carbon anodes. The second-order work criterion is employed to analyze the failure of the coke aggregate specimens and the relationships between the second-order work, the kinetic energy, and the instability of the granular material are investigated. In addition, the coke aggregates are modeled by exploiting the discrete element method (DEM) to reveal the micro-mechanical behavior of the dry coke aggregates during the compaction process. The optimal number of particles required for the failure analysis in the DEM simulations is determined. The effects of the confining pressure and strain rate as two important compaction process parameters on the failure are studied. The results reveal that increasing the confining pressure enhances the probability of the diffusing mode of the failure in the specimen. On the other hand, the increase of strain rate augments the chance of the strain localization mode of the failure in the specimen.

Finite element simulation of ductile fracture in polycrystalline materials using a regularized porous crystal plasticity model

In the present study, a hypoelastic–plastic formulation of porous crystal plasticity with a regularized version of Schmid’s law is proposed. The equation describing the effect of the voids on plasticity is modified to allow for an explicit analytical solution for the effective resolved shear stress. The regularized porous crystal plasticity model is implemented as a material model in a finite element code using the cutting plane algorithm. Fracture is described by element erosion at a critical porosity. The proposed model is used for two test cases of two- and three-dimensional polycrystals deformed in tension until full fracture is achieved. The simulations demonstrate the capability of the proposed model to account for the interaction between different modes of strain localization, such as shear bands and necking, and the initiation and propagation of ductile fracture in large scale polycrystal models with detailed grain description and realistic boundary conditions.

Porous plasticity revisited: Macroscopic and multiscale modeling

Porous plasticity aims to model the growth and coalescence of voids leading to ductile failure. The GTN model (1984), resulting from heuristic modifications to Gurson's homogenized hollow sphere model (1977), is used in numerous publications. The Rousselier model (1981), developed in the framework of continuum thermodynamics, is apparently similar. Both models are effective in numerical calculations, but the reasons why they perform well were not investigated in details in the existing literature, as regards transition to uniaxial deformation, relations between various modes of strain localization, finite element discretization, regularization. In the present paper, we propose first to revisit both models and to compare their fundamentally different mechanical behaviors. For stress triaxiality larger than some critical value, it is shown that theoretically the GTN model cannot achieve strain localization in a plane but only pointwise localization for the ultimate mechanical state (stress tensor equal to zero). The larger the void volume fraction (void growth), the smaller the stress triaxiality critical value. Fortunately, discretization transforms the pointwise localization into volume localization and with an appropriate Cartesian finite element mesh a more or less planar sheet of integration points can be obtained. The Rousselier model can achieve strain localization in a plane at all stress triaxialities and discretization also transforms this localization into volume localization with a characteristic element size. Second, multiscale modeling of both plasticity and ductile damage (not limited to void damage) is an essential way of progress for laboratory specimen calculations. The Rousselier model can be incorporated into polycrystalline models based on crystal plasticity, with reasonable computation times provided a reduced texture with a small number of crystallographic orientations is used. It can be coupled with a new Coulomb ductile fracture model at the slip system scale and with secondary void nucleation and growth models at the grain and slip system scales, respectively. The multiscale model is applied to aluminum CT and KAHN specimens and to steel round notched specimens.

In Situ Observation on Rate-Dependent Strain Localization of Thermo-Induced Shape Memory Polyurethane.

In situ monotonic tensile experiments of thermo-induced shape memory polyurethane (SMPU) at different loading rates were carried out by the digital image correlation (DIC) method and infrared camera FLIR®-A655sc in natural convection (NC) and forced convection (FC) conditions, respectively. The multiform strain localization of SMPU was observed by the DIC method, and the influence of thermo–mechanical coupling on the strain localization was analyzed by using the FLIR to measure the temperature field caused by the internal heat generation. The experimental results show that the strain localization mode strongly depends on the strain rate and convection condition, and the strain localization mode can be transformed by changing the convection condition from NC to FC. The competition mechanism between the strain hardening induced by the increasing loading rate and strain softening induced by the internal heat generation is indicated, the transition modes of strain localization are clarified, and the influences of thermo–mechanical coupling on shape memory effect are discussed.

Strain rate dependence of tensile strength and ductility of nano and ultrafine grained coppers

An ultrafine grained Cu with a wide distribution of grain size and an average grain size of d= 110nm was prepared by electric brush-plating technique. Tensile properties, microstructure and deformation surface morphologies of this ultrafine grained Cu were compared with two results on an electric brush-plated nanocrystalline Cu (d= 59nm) and an electrodeposited ultrafine grained Cu (d= 200nm) previously prepared by our group. Based on these comparisons, the strain rate dependences of strength and ductility of these three materials and underlying mechanisms were studied. It was revealed that the pronounced strain rate sensitivity (larger m value) of these three materials is a result of the competition between the dislocation and grain boundary mediated deformations. The grain boundary mediated deformation plays a very important role in controlling the tensile ductility of these three materials, which affects the deformation accommodation by promoting different modes of strain localization and hence leads to the completely different strain rate dependences of tensile ductility (the elongations decrease, remains unchangeable and increase with increasing strain rate).

A lattice discrete particle model for pressure-dependent inelasticity in granular rocks

This paper details the formulation, calibration, and validation of a Lattice Discrete Particle Model (LDPM) for the simulation of the pressure-dependent inelastic response of granular rocks. LDPM is formulated in the framework of discrete mechanics and it simulates the heterogeneous deformation of cemented granular systems by means of discrete compatibility/equilibrium equations defined at the grain scale. A numerical strategy is proposed to generate a realistic microstructure based on the actual grain size distribution of a sandstone and the capabilities of the method are illustrated with reference to the particular case of Bleurswiller sandstone, i.e. a granular rock that has been extensively studied at the laboratory scale. LDPM micromechanical parameters are calibrated based on evidences from triaxial experiments, such as hydrostatic compression, brittle failure at low confinement and plastic behavior at high confinement. The results show that LDPM allows exploring the effect of fine-scale heterogeneity on the inelastic response of rock cores, achieving excellent quantitative performance across a wide range of stress conditions. In addition, LDPM simulations demonstrate its capability of capturing different modes of strain localization within a unified mechanical framework, which makes this approach applicable for a wide variety of geomechanical settings. Such promising performance suggests that LDPM may constitute a viable alternative to existing discrete numerical methods for granular rocks, as well as a versatile tool for the interpretation of their complex deformation/failure patterns and for the development of continuum models capturing the effect of micro-scale heterogeneity.

Influence of the intermediate principal stress on the strain localization mode in porous sandstone

Compaction bands and dilation bands are zones of either pure compaction or pure dilation, characterized by decreased or increased porosity, respectively. In contrast with commonly observed shear bands, there are relatively few reports of compaction bands or dilation bands, particularly in field settings. This work explores one possible explanation for these isolated observations: unique stress conditions, which occur infrequently, are required to form these bands. In particular, the influence of the intermediate principal stress on localization conditions is examined. Theoretical results, using a bifurcation approach to localization, reveal that conditions for compaction band formation are least restrictive when the intermediate principal stress is equal to the minimum compressive stress, while dilation band conditions are least restrictive when the intermediate principal stress is equal to the maximum compressive stress. However, the occurrence of one of these favorable stress states does not guarantee formation of one of these deformation modes, since certain conditions on the inelastic material parameters (i.e., friction factor and dilation coefficient) must also be satisfied. These material parameters vary with mean stress; thus satisfaction of compaction band or dilation band conditions requires not only a specific relationship between the principal stresses but also a loading path resulting in a favorable mean stress, such that inelastic material parameters fall in the required range. In a laboratory environment, where principal stresses are controllable, these conditions should be readily achievable. However, if these specialized stress conditions are uncommon in field settings, theory suggests that compaction bands and dilation bands may form infrequently.

On some topics for the numerical simulation of ductile fracture

In this work, we analyze some aspects of the macroscopic Gurson–Tvergaard–Needleman (GTN) constitutive model when it is addressed to solve ductile fracture problems by means of numerical simulations: (i) The analytical solutions of the material discontinuous bifurcation problem is performed. Closed and exact formulas are obtained. The so determined critical conditions and the developed strain localization mode are afterward studied and compared in crack growth problems. Even when this methodology of analysis is rather standard at the present, the conclusions drawn from this study differ significantly from that obtained with a similar analysis in quasi-brittle fracture problems. (ii) A new very robust numerical integration method for the GTN model, namely the Impl–Ex Method, is proposed. It is a low computational cost algorithm, equivalent to a linear problem per each integration step, with a reasonable precision for engineering purposes. Its accuracy and convergence rate is assessed by means of an error study applied to a ductile fracture test simulation. (iii) A detailed analysis of a plane strain ductile crack growth problem is performed in a material containing two size-scale of voids. In the analysis, particular attention is given to the mesh size dependence and to the coalescence of the larger void.

Shear bands in plane strain compression of loose sand

Experimental results are presented which characterize the behaviour of a loose, fine-grained, water-saturated sand tested under globally undrained and drained conditions in a plane strain apparatus. The objective of this investigation is to provide insight into the phenomenon of shear banding in loose sand. Together with local measurements of boundary forces and deformations, stereophotogrammetry is used to investigate the progression of strain localization in plane strain compression. Typical results and findings concerning the evolution of non-homogeneous deformation are presented in detail. Shear banding occurred in both undrained and drained experiments on loose masonry sand. In general, temporary modes of strain localization, observed during macroscopically ‘uniform’ deformations of a specimen, gave way to a single, persistent shear band. A clear pattern of onset of the formation of the persistent shear band, mobilization of the maximum effective friction and complete formation of the band was observed in all tests. The stress state when the localization begins is very close to, yet precedes that corresponding to the maximum mobilized friction. The persistent shear bands evolve with changing width and orientation. L'exposé présente les résultats d'essais de compression plane visant à caractériser le comportement de sable meuble, à grains fins et saturé d'ean dans des conditions générales non drainées et drainées. L'objectif de cette étude est de mieux faire comprendre la formation de bandes de cisaillement dans le sable meuble. En plus de mesurer les forces limites et les déformations, on a eu recours à la stéréophotogrammétrie pour étudier la progression de la localisation des tensions sous l'effect d'une compression plane. L'exposé présente en détail les résultats et constatations types concernant l'évolution de déformations non homogénes. Lors d'essais non drainés et drainés, des bandes de cisaillement se sont formées dans du sable de construction meuble. En règlegénérale, la formation d'une seule bande de cisaillement persistante s'est substituée aux modes temporaires de localisation des contraintes, observés pendant des déformations macroscopiquement «uniformes« d'une éprouvette. Dans tous les essais, le début de la formation d'une bande de cisaillement persistante, la mobillsation du frottement efficace maximal et la formation complète de la bande ont suivi un schéma bien défini. L'état de contrainte où commence la localisation est très proche de celui qui correspond au frottement mobilisé maximal, mais le précède. Les bandes de cisaillement persistantes évoluent en fonction de la largeur et de l'orientation

Strain hardening at high strain in aluminum alloys and its effect on strain localization

The modes of strain localization in the tensile testing of a sheet sample are diffuse necking, localized necking and, in some materials, localization in an unstable shear band. In a tensile test of a rate insensitive material, the normalized strain hardening parameter,H = (1/σ)(dσ/de) has the values ofH = 1 for diffuse necking andH = 0.5 for localized necking. Curves ofH vs strain were obtained up to large values of plastic strain using the hydraulic bulge test. The materials selected were commercially important sheet alloys in the condition normally used for forming. It is shown that the materials have similarH vs strain curves in the range of uniform tensile straining, but the curves diverge widely at higher strains whereH falls below 1. This has important consequences on strain localization behavior. The limit strains of the alloys in simple tension and punch stretching show reasonable correlation with their values ofH and those alloys which are susceptible to catastrophic shear failure have low values ofH at high strains. Strain rate sensitivity adds to or subtracts from theH values obtained in this study and has an additional influence on strain localization.