Elliptic yield cap constitutive modeling for high porosity sandstone

Field and laboratory investigators have observed thin, tabular zones of localized compressional deformation without shear in high porosity sandstone. These ‘compaction bands’ display greatly reduced porosity, and may affect the withdrawal of fluids from reservoirs. Studies addressing band formation as a type of strain localization predict the onset of the bands in a range of constitutive parameters roughly consistent with experiments, but are highly dependent on the constitutive relation used. In particular, the hardening modulus in shear and the slope of the yield surface in a plot of shear stress versus mean compressive stress are critical to localization predictions. Previous yield cap constitutive models employed a single deformation criterion, linking hydrostatic and shear response. In this work, we propose an elliptic yield cap model employing separate inelastic deformation parameters along each axis of the ellipse. The two deformation parameters allow the proposed surface to change in aspect ratio as it deforms, and allow a negative hardening modulus in shear without a negative hydrostatic modulus. Some cases with simplified modeling are shown for illustrative purposes, followed by a comparison with existing models. The proposed model displays similar strain behavior to the other models, but predicts localization under less restrictive conditions.     

Simulation and interpretation of diffuse mode bifurcation of elastoplastic solids

The diffuse mode bifurcation of elastoplastic solids at finite strain is investigated. The multiplicative decomposition of deformation gradient and the hyperelasto-plastic constitutive relationship are adapted to the numerical bifurcation analysis of the elastoplastic solids. First, bifurcation analyses of rectangular plane strain specimens subjected to uniaxial compression are conducted. The onset of the diffuse mode bifurcations from a homogeneous state is detected; moreover, the post-bifurcation states for these modes are traced to arrive at localization to narrow band zones, which look like shear bands. The occurrence of diffuse mode bifurcation, followed by localization, is advanced as a possible mechanism to create complex deformation and localization patterns, such as shear bands. These computational diffuse modes and localization zones are shown to be in good agreement with the associated experimental ones observed for sand specimens to ensure the validity of this mechanism. Next, the degradation of horizontal sway stiffness of a rectangular specimen due to plane strain uniaxial compression is pointed out as a cause of the bifurcation of the first antisymmetric diffuse mode, which triggers the tilting of the specimen. Last, circular and punching failures of a footing on a foundation are simulated.     

Hydrogen induced shear localization of the plastic flow in metals and alloys

Hydrogen enhanced localized plasticity (HELP) is a viable mechanism for hydrogen embrittlement supported by experimental observations. According to the HELP mechanism, hydrogen induced premature failures result from hydrogen induced plastic instability which leads to hydrogen assisted localized ductile processes. The objective of this work is to reveal the role of hydrogen in possibly localizing the macroscopic deformation into bands of intense shear using solid mechanics methodology. The hydrogen effect on material deformation is modeled through the hydrogen induced volume dilatation and the reduction in the local flow stress upon hydrogen dissolution into the lattice. Hydrogen in assumed to reside in both normal interstitial lattice sites (NILS) and reversible traps associated with the plastic deformation. The analysis of the plastic deformation and the conditions for plastic flow localization are carried out in plane strain uniaxial tension. For a given initial hydrogen concentration in the unstressed specimen, a critical macroscopic strain is identified at which shear localization commences.     

Micromechanical analysis of damage in saturated quasi brittle materials

In this paper, we propose a micromechanical analysis of damage and related inelastic deformation in saturated porous quasi brittle materials. The materials are weakened by randomly distributed microcracks and saturated by interstitial fluid with drained and undrained conditions. The emphasis is put on the closed cracks under compression-dominated stresses. The material damage is related to the frictional sliding on crack surface and described by a local scalar variable. The effective properties of the materials are determined using a linear homogenization approach, based on the extension of Eshelby’s inclusion solution to penny shaped cracks. The inelastic behavior induced by microcracks is described in the framework of the irreversible thermodynamics. As an original contribution, the potential energy of the saturated materials weakened by closed frictional microcracks is determined and formulated as a sum of an elastic part and a plastic part, the latter entirely induced by frictional sliding of microcracks. The influence of fluid pressure is accounted for in the friction criterion through the concept of local effective stress at microcracks. We show that the Biot’s effective stress controls the evolution of total strain while the local Terzaghi’s effective stress controls the evolution of plastic strain. Further, the frictional sliding between crack lips generates volumetric dilatancy and reduction in fluid pressure. Applications of the proposed model to typical brittle rocks are presented with comparisons between numerical results and experimental data in both drained and undrained triaxial tests.     

Influence of void nucleation on ductile shear fracture at a free surface

An approximate continuum model of a ductile, porous material is used to study the influence of the nucleation and growth of micro-voids on the formation of shear bands and the occurrence of surface shear fracture in a solid subject to plane strain tension. Bifurcation into diffuse modes is analysed for a plane strain tensile specimen described by these constitutive relations, which account for a considerable plastic dilatancy due to void growth and for the possibility of non-normality of the plastic flow law. In particular, bifurcation into surface wave modes and the possible influence of such modes triggering shear bands is investigated. For solids with initial imperfactions such as a surface undulation, a local material inhomogeneity on an inclusion colony, the inception and growth of plastic flow localization is analysed numerically. Both the formation of void-sheets and the final growth of cracks in the shear bands is described numerically. Some special features of shear band development in the solid obeying non-normality are studied by a simple model problem.     

Analysis of deformation localization based on proposed theory of ultrasonic wave velocity propagating in plastically deformed solids

The localization of plastic deformation is discussed as “stationary discontinuity” characterized by a vanishing velocity of an acceleration wave derived using the author’s proposed theory of ultrasonic wave velocities propagating in plastically deformed solids. To formulate the proposed theory, the elasto-plastic coupling effect was introduced to consider the elastic stiffness degradation due to the plastic deformation. The driving force of the deformation localization is caused by the yield vertex effect, which introduces a pronounced softening of the shear modulus, and geometrical softening due to double slip caused by lattice rotations. In the present paper, it is examined theoretically and experimentally that the diagonal terms of the introduced elasto-plastic coupling tensor represent a slight hardening followed by a pronounced softening of the elastic modulus induced by the point defect development caused by cross slides among dislocations at multiple slip stages similar to the yield vertex effects. The off-diagonal terms represent geometrical softening induced by lattice rotations such as texture evolution. Then, based on the coincidence of the onset strains between localization and acceleration waves of vanishing velocity, the diagrams of diffuse necking, localized necking and forming limit are analyzed by applying the proposed acoustic tensor, which is based on the generalized Christoffel tensor derived by the author, and solving cut off conditions of the quasi-longitudinal wave to determine the onset strains of deformation localization and localization modes. As a result, diagrams of diffuse necking, localized necking and forming limit were obtained. Moreover, the localization modes were determined and distinguished as the SH-mode, SV-mode, tearing mode and splitting mode.     

A numerical methodology for investigating the formation of adiabatic shear bands

In this paper, we present a numerical approach for analyzing thermo-visco-plastic deformation in one dimension. The method, which is accurate to second order, is based on integration along the characteristic lines. It is able to simulate fully localized plastic flow with high resolution and good efficiency. We apply this numerical scheme to the analysis of shear localization, emphasizing the interactions between a single shear band and its surroundings and among the members of a periodic array of shear bands. It is found that a shear band may grow intermittently due to interactions with other bands. The developed method is specifically adequate for analyzing the self-organized multiple adiabatic formation process, which will be discussed in the follow-up paper.     

Effect of porosity on the properties of strain localization in porous media under undrained conditions

Within the general framework of mixture theory and by introducing the fictitious “fluid phase” as a mixture of a liquid and a gas, the conditions for localization of deformation into a shear band in the incremental response of partially saturated and fully saturated elastic–plastic porous media under undrained conditions are derived. The effect of porosity is included in the derivation. The explicit analytical expressions of the direction of shear band initiation and the corresponding hardening modulus of the porous media for the plane strain case are deduced, and a parametric analysis is made of the influence of the porosity on the properties of strain localization based on Mohr–Coulomb yield criterion. It is found that the dependence of the shear banding properties of partially saturated porous media on the porosity is related to the stress states and Poisson's ratio. However, the properties of the strain localization for the fully saturated porous media are almost independent of Poisson's ratio. Finally, on the basis of Mohr–Coulomb yield criterion, some solutions of the shear banding orientation for water-saturated granular materials are obtained, which are proved to be in good agreement with the experimental results reported by other researchers.     

Analysis of anisotropy and plastic spin effects on localization phenomena

Summary The main objective of the paper is the investigation of the influence of the anisotrophy and plastic spin effects on criteria for adiabatic shear band localization of plastic deformation. A theory of thermoplasticity is formulated within a framework of the rate-type covariance material structure with a finite set of internal state variables. The theory takes into consideration such effects as plastic non-normality, plastic-induced anisotropy (kinematic hardening), micro-damage mechanism, thermomechanical coupling and plastic spin. The next objective of the paper is to focus attention on cooperative phenomena in presence of the plastic spin, and the discussion on the influence of synergetic effects on localization criteria. A particular constitutive law for the plastic spin is assumed. The necessary condition for a localized plastic deformation region to be formed is obtained. This condition is accomplished by the assumption that some eigenvalues of the instantaneous adiabatic acoustic tensor vanish. A procedure has been developed which allows us to discuss two separate groups of effects on the localization phenomenon along a shear band. Plastic spin, spatial covariance and kinematic hardening effects are investigated at an isothermal process in an undamaged solid. In the second case, an adiabatic process in a damaged solid is discussed when the spatial covariance terms and the plastic spin are neglected. Here the thermomechanical coupling, micro-damage mechanism and kinematic hardening effects are examined. For both cases, the criteria for adiabatic shear band localization are obtained in an exact analytical form. Particular attention is focused on the analysis of the following effects: (i) plastic non-normality; (ii) plastic spin; (iii) covariant terms; (iv) plastic strain-induced anisotropy; (v) micro-damage mechanism; (vi) thermomechanical couplings. Cooperative phenomena are considered, and synergetic effects are investigated. A discussion of the influence of the plastic spin, kinematic hardening and covariant terms on the shear band localization conditions is presented. A numerical estimation of the effects discussed is given. Received 24 April 1997; accepted for publication 23 December 1997     

On the localization and failure in aluminum shells due to crushing induced bending and tension

The paper is concerned with the accurate numerical simulation of localized deformation that can develop into necking and failure, induced by combined bending and tension in shell structures. The study is motivated by the need to establish the onset and evolution of such failures in imploding underwater structures. Such localized zones of deformation are shown to develop under controlled conditions in experiments on Al-6061-T6 cylindrical shells crushed laterally by rigid punches. The crushing induces gradually developing local depressions in the shells, at radially constrained locations. As the crushing progresses, the depressions with a width of the order of the shell wall thickness, deepen, increase their span, become neck-like and develop inclined failures. In the experimental set-up used, the crushing was terminated when the first of four such depressions that develop ruptured. The shell was sliced along the principal plane of crushing and the most deformed cross sections of the necks were measured. The crushing experiments were simulated numerically using solid FE models. The material was modeled as a finitely deforming elastic–plastic solid that hardens isotropically using the von Mises, the non-quadratic isotropic Hosford and anisotropic Yld04–3D yield functions suitably calibrated. While the overall structural response was reproduced well by all models, differences were observed in the evolution of localization in the depressions. For the von Mises yield function, the localized deformation was significantly milder than in the experiments. The isotropic Hosford yield function produced necks that were closer to the experimental ones, while Yld04–3D produced results that were very close to the measurements. Clearly, and in concert with other applications, the adoption of a non-quadratic yield function is necessary for reproduction of localized and other challenging deformation histories in Al alloys. The addition of anisotropy in such models improves further the predictions.     

Randomness and slip avalanches in gradient plasticity

While localization of deformation at macroscopic scales has been documented and carefully characterized long ago, it is only recently that systematic experimental investigations have demonstrated that plastic flow of crystalline solids on mesoscopic scales proceeds in a strongly heterogenous and intermittent manner. In fact, deformation is characterized by intermittent bursts (‘slip avalanches’) the sizes of which obey power-law statistics. In the spatial domain, these avalanches produce characteristic deformation patterns in the form of slip lines and slip bands. Unlike to the case of macroscopic localization where gradient plasticity can capture the width and spacing of shear bands in the softening regime of the stress–strain graph, this type of mesoscopically jerky like localized plastic flow is observed in spite of a globally convex stress–strain relationship and may not be captured by standard deterministic continuum modelling. We thus propose a generalized constitutive model which includes both second-order strain gradients and randomness in the local stress–strain relationship. These features are related to the internal stresses which govern dislocation motion on microscopic scales. It is shown that the model can successfully describe experimental observations on slip avalanches as well as the associated surface morphology characteristics.     

Adiabatic shear band localization of inelastic single crystals in symmetric double-slip process

Summary The main objective of the present paper is the development of a viscoplastic regularization procedure valid for an adiabatic dynamic process for multi-slips of single crystals. The next objective is to focus attention on the investigation of instability criteria, and particularly on shear band localization conditions.To achieve this aim, an analysis of acceleration waves is given, and advantage is taken of the notion of the instantaneous adiabatic acoustic tensor. If zero is an eigenvalue of the acoustic tensor, then the associated discontinuity does not propagate, and one speaks of a stationary discontinuity. This situation is referred to as the strain localization condition, and corresponds to a loss of hyperbolicity of the dynamical equations. It has been proved that for an, adiabatic process of rate-dependent (elastic-viscoplastic) crystal, the wave speed of discontinuity surface always remains real and different from zero. It means that for this case the initial-value problem is well-posed. However, for an adiabatic process of rate-independent(elastic-plastic) crystal, the wave speed of discontinuity surface can be equal zero. Then the necessary condition for a localized plastic deformation along the shear band to be formed is as follows: the determinant of the instantaneous adiabatic acoustic tensor is equal to zero. This condition for localization is equivalent to that obtained by using the standard bifurcation method. Based on this idea, the conditions for adiabatic shear band localization of plastic deformation have been investigated for single crystals. Particular attention has been focused on the discussion of the influence of thermal expansion, thermal plastic, softening and spatial covariance effects on shear band localization criteria for a planar model of an f.c.c. crystal undergoing symmetric primary-conjugate double slip. The results obtained have been compared with available experimental observations.Finally, it is noteworthy that the viscoplasticity regularization procedure can be used in the developing of an unconditionally stable numerical integration algorithm for simulation of adiabatic inelastic flow processes in ductile single crystals, cf. [21].The paper has been prepared within research programme sponsored by the Committee of Scientific Research under Grant 3 P404 031 07.     

岩土中的剪切带局部化问题研究:回顾与展望

回顾了圆弧滑动面理论的产生及其在土坡抗滑动稳定分析和极限承载力计算中的应用，并指出了圆弧滑动面理论和刚塑性理论及极限平衡条件的关系及其局限性。介绍了剪切带局部化问题的研究现状，包括一些热点研究领域和最新研究成果。着重介绍了用子负荷面模型模拟超固结黏性土剪切带局部化和用动态剪切带单元模拟摩尔一库仑材料剪切带局部化的最新研究成果。对剪切带局部化问题研究提出了几个主要发展方向。     

Heterogeneous deformation in ductile FCC single crystals in biaxial stretching: the influence of slip system interactions

The heterogeneity of deformation in ductile FCC single crystals is investigated by both numerical simulations and an analytic approach. The constitutive behaviour is based on a generalized storage recovery model and takes into account the interactions between slip systems previously obtained by dislocation dynamics simulations. In biaxial stretching, the simulations show the activation of a large number of slip systems and their localization in mutually excluding zones. As a result, a microstructure of lamellar type is formed in the early stages of the deformation. These numerical results are complemented by a linear stability analysis showing that the heterogeneous deformation pattern is triggered by instability modes of the single crystal. Furthermore, the interaction matrix is playing a key role as the partition is found to originate from slip system interactions. The partition is driven by the strongest interaction, which is in most cases the collinear interaction. A comparison with an experimental study in simple shear yields useful information about how to check the respective strength of some interactions. The collinear interaction is not involved in that case, but its effect can be verified by reproducing the experiment on a crystal with a different orientation.     

Evolving internal length scales in plastic strain localization for granular materials

A numerical model in the Cosserat continuum for strain localization phenomena in granular materials is developed and proposed in this paper. The model assumes a constant internal length scale that is used to describe the shear band thickness. However, it is observed that the internal length scales need to change to accommodate the possible change in the contact surface between the particles, damage of the particles or/and any change in the local void ratio within the domain, which will change the shear band thickness. The mathematical formulations used in the present numerical model were equipped with evolution equations for the length scales through the Micropolar theory, those formulations are proposed and discussed in this paper. The evolution equations of the internal length scales describe any possible change in the contact surface between the particles, damage of the particles if exists and/or any change in the local void ratio within the domain. Hence, the strain localization described by the enhanced model with evolving internal length scales is more accurate and closer to the real solution. The solution for the shear bands thickness shows more accurate correlation with the experimental results and less dependency on the mesh size when such evolution equations are used. Moreover, the shear band thickness and inclination evolve during the deformation process.     

Finite element analysis of localization in FCC polycrystalline sheets under plane stress tension

Localization phenomena in thin sheets subjected to plane stress tension are investigated. The sheet is modelled as a polycrystalline aggregate, and a finite element analysis based on rate-dependent crystal plasticity is developed to simulate large strain behaviour. Accordingly, each material point in the specimen is considered to be a polycrystalline aggregate consisting of a large number of FCC grains. The Taylor model of crystal plasticity theory is assumed. This analysis accounts for initial textures as well as texture evolution during large plastic deformations. The numerical analysis incorporates certain parallel computing features. Simulations have been carried out for an aluminum sheet alloy, and the effects of various parameters on the formation and prediction of localized deformation (in the form of necking and/or in-plane shear bands) are examined.     

Numerical simulation of plastic-flow localization for simple shear

An algorithm was developed to numerically simulate plastic-flow localization for simple shear of a thermally plastic and viscoplastic material. The algorithm is based on solving the partial differential equations describing continuum flow. The closing equation is the constitutive relation known in the literature as the power law linking the plastic-strain rate to the flow stress, temperature, and accumulated plastic strain. Calculated relations for the time evolution of the shear-band width and the temperature and plastic strains localized in it agree satisfactorily with experimental relations. Good agreement with experimental results is also obtained for the sample temperature distribution at the developed stage of the localization process.Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 1, pp. 173–180, January–February, 2005     

平头弹冲塞靶板的绝热剪切数值模拟

针对绝热剪切形成时由于变形高度局域化,塑性功产生的热导致局部高温,有时会伴随动态再结晶(DRX)的现象,采用一种考虑动态再结晶过程的绝热剪切破坏准则,利用有限元方法模拟了Arne工具钢平头弹冲塞Weldox 460 E钢靶板的实验.数值模拟揭示了剪切带产生、传播的过程,温度分布情况表明在绝热剪切带中具备动态再结晶形成的...