Limit analysis and homogenization of porous materials with Mohr–Coulomb matrix. Part I: Theoretical formulation

The present two-part study aims at investigating the specific effects of Mohr–Coulomb matrix on the strength of ductile porous materials by using a kinematic limit analysis approach. While in the Part II, static and kinematic bounds are numerically derived and used for validation purpose, the present Part I focuses on the theoretical formulation of a macroscopic strength criterion for porous Mohr–Coulomb materials. To this end, we consider a hollow sphere model with a rigid perfectly plastic Mohr–Coulomb matrix, subjected to axisymmetric uniform strain rate boundary conditions. Taking advantage of an appropriate family of three-parameter trial velocity fields accounting for the specific plastic deformation mechanisms of the Mohr–Coulomb matrix, we then provide a solution of the constrained minimization problem required for the determination of the macroscopic dissipation function. The macroscopic strength criterion is then obtained by means of the Lagrangian method combined with Karush–Kuhn–Tucker conditions. After a careful analysis and discussion of the plastic admissibility condition associated to the Mohr–Coulomb criterion, the above procedure leads to a parametric closed-form expression of the macroscopic strength criterion. The latter explicitly shows a dependence on the three stress invariants. In the special case of a friction angle equal to zero, the established criterion reduced to recently available results for porous Tresca materials. Finally, both effects of matrix friction angle and porosity are briefly illustrated and, for completeness, the macroscopic plastic flow rule and the voids evolution law are fully furnished.     

Material instabilities of anisotropic saturated multiphase porous media

This paper analyses the material instability of fully saturated multiphase porous media. On account of the fact that anisotropic mechanical behaviours are widely observed in saturated and partially saturated geomaterials, the anisotropic constitutive model developed by Rudnicki for geomaterials is used to model the anisotropic mechanical behaviour of the solid skeleton of saturated porous geomaterials in axisymmetric compression test. The inertial coupling effect between solid skeleton and pore fluid is also taken into account in dynamic cases. Conditions for static instability (strain localisation) and dynamic instability (stationary discontinuity and flutter instability) of fully saturated porous media are derived. The critical modulus, shear band angle for strain localisation, and the bound within which flutter instability may occur are given in explicit forms. The effects of material parameters on material instability are investigated in detail by numerical computations.     

A thermodynamical gradient theory for deformation and strain localization of porous media

In this work, a thermodynamically consistent gradient formulation for partially saturated cohesive-frictional porous media is proposed. The constitutive model includes a classical or local hardening law and a softening formulation with state parameters of non-local character based on gradient theory. Internal characteristic length in softening regime accounts for the strong shear band width sensitivity of partially saturated porous media regarding both governing stress state and hydraulic conditions. In this way the variation of the transition point (TP) of brittle-ductile failure mode can be realistically described depending on current confinement condition and saturation level. After describing the thermodynamically consistent gradient theory the paper focuses on its extension to the case of partially saturated porous media and, moreover, on the formulation of the gradient-based characteristic length in terms of stress and hydraulic conditions. Then the localization indicator for discontinuous bifurcation is formulated for both drained and undrained conditions.     

A micro–macro method for predicting the shear strength of brittle rock under compressive loading

Microcracks have great significance for shear strength of brittle rock in compression. A major challenge of this area is to establish the correlation of microcracks and macroscopic shear strength. A new micro–macro method is presented to predict the shear strength of brittle rock in compression. This method incorporates the microcrack model suggested by Ashby, Mohr–Coulomb failure criterion and a crack-strain relation. This crack–strain relation is presented to link the crack growth and axial strain by combining the micro and macro definitions from rock damage. The shear strength and stress–strain relationship of Jinping marble are theoretically investigated in detail. The rationality of this suggested method is verified by using the experimental results founded on Jinping marble. Effects of the initial microcrack size, friction coefficient and confining pressure on internal friction angle, cohesion, and shear strength are also discussed.     

Size effects on tensile and compressive strengths in metallic glass nanowires

Shear localization induced brittleness is the main drawback of metallic glasses which restricts their practical applications. Previous experiments have provided insights on how to suppress shear localization by reducing the sample size of metallic glasses to the order of 100 nm. In order to reveal the size effects and associated deformation mechanisms of metallic glasses in an even finer scale, we perform large-scale atomistic simulations for the uniaxial compression and tension of metallic glass nanowires. The simulation results show that, as the diameter of metallic glass samples decreases from 45 nm to 8 nm, the tensile yield strength increases while the compressive yield strength decreases. Homogeneous flow is observed as the governing deformation mechanism in all simulated metallic glass samples, where plastic shearing tends to initiate on the sample surface and propagate into the interior. To rationalize the size dependence of yield strengths, we propose a theoretical model based on the concept of surface stress and Mohr–Coulomb criterion. The theoretical predictions agree well with the simulation results, implying the important role of surface stress on the yielding of MGs below 100 nm. Finally, a discussion about the size effects of strength in metallic glasses at different length scales is provided. Our results suggest that the shear band energy and surface stress might be the two crucial parameters in determining the critical size required for the transition from shear localization to homogeneous deformation in MGs.     

A C2 continuous approximation to the Mohr–Coulomb yield surface

In spite of the development of more sophisticated constitutive models for soil, the Mohr–Coulomb yield criterion remains a popular choice for geotechnical analysis due to its simplicity and ease of use by practising engineers. The implementation of the criterion in finite element programs, however, presents some numerical difficulties due to the gradient discontinuities which occur at both the edges and the tip of the hexagonal yield surface pyramid. Furthermore, some implicit techniques utilising consistent tangent stiffness formulations are unable to achieve full quadratic convergence as the yield criteria is not C2 continuous. This paper extends the previous work of Abbo and Sloan (1995) through the introduction of C2 continuous rounding of the Mohr–Coulomb yield surface in the octahedral plane. This approximation, when combined with the hyperbolic approximation in the meridional plane (Abbo and Sloan, 1995), describes a yield surface that is C2 continuous at all stress states. The new smooth yield surface can be made to approximate the Mohr–Coulomb yield function as closely as required by adjusting only two parameters, and is suitable for consistent tangent stiffness formulations.     

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.     

Elliptical and circular FRP-confined concrete sections: A Mohr–Coulomb analytical model

An analytical stress–strain model is developed for predicting the compressive behavior of elliptical and circular fiber reinforced polymer (FRP)-confined concrete members. The model is based on a diagonal Poisson’s ratio formulation expressed as a function of the mechanical properties of the unconfined concrete and confining FRP jacket, the geometry of the concrete section, and the extent of internal damage in the confined concrete core. A Mohr–Coulomb yield criterion is introduced for analysis of the compressive behavior of confined concrete. Equilibrium and strain compatibility are used to obtain the ultimate compressive strength and strain of elliptical and circular FRP-confined concrete sections as a function of the effective confining stiffness of the FRP jacket. A simplified expression is derived for the FRP reinforcement ratio which precludes strain softening in elliptical and circular FRP-confined concrete sections.     

On the rheology of dilative granular media: Bridging solid- and fluid-like behavior

A new rate-dependent plasticity model for dilative granular media is presented, aiming to bridge the seemingly disparate solid- and fluid-like behavioral regimes. Up to date, solid-like behavior is typically tackled with rate-independent plasticity models emanating from Mohr–Coulomb and Critical State plasticity theory. On the other hand, the fluid-like behavior of granular media is typically treated using constitutive theories amenable to viscous flow, e.g., Bingham fluid. In our proposed model, the material strength is composed of a dilation part and a rate-dependent residual strength. The dilatancy strength plays a key role during solid-like behavior but vanishes in the fluid-like regime. The residual strength, which in a classical plasticity model is considered constant and rate-independent, is postulated to evolve with strain rate. The main appeal of the model is its simplicity and its ability to reconcile the classic plasticity and rheology camps. The applicability and capability of the model are demonstrated by numerical simulation of granular flow problems, as well as a classical shear banding problem, where the performance of the continuum model is compared to discrete particle simulations and physical experiment. These results shed much-needed light onto the mechanics and physics of granular media at various shear rates.     

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。     

An analytical homogenization method for heterogeneous porous materials

The objective of this work is to develop an analytical homogenization method to estimate the effective mechanical properties of fluid-filled porous media with periodic microstructure. The method is based on the equivalent inclusion concept of homogenization applied earlier for solid–solid mixture. It is assumed that porous media are described by the poroelastic constitutive law developed by Biot where porosity is a material parameter. By solving the governing equations of poroelasticity in Fourier transformed domain, the relation between periodic strain and eigenstrain in porous media is established. This relation is subsequently used in an average consistency condition involving both solid and fluid phase stresses and strains. The geometry of the porous microstructure is captured in the g-integral. This homogenization method can also be applied to estimate the equivalent properties of solid–fluid mixture where a pure solid and fluid can be modeled by assuming very low and high porosity, respectively. Several examples are considered to establish this new method by comparing with other existing analytical and numerical methods of homogenization. As an application, poroelastic properties of cortical bone fibril are estimated and compared with previously computed values.     

Loading rate and confining pressure effect on dilatancy,acoustic emission,and failure characteristics of fissured rock with two pre-existing flaws

Investigating the dilatancy, acoustic emission and failure characteristics of fissured rock are significant to ensure their geotechnical stability. In this paper, the uniaxial and triaxial compression experiments with AE monitoring under different loading rates were carried out on fissured rock specimens with the same geometrical distribution of two pre-existing flaws. The dilatancy and AE activity of these specimens were discussed, and the effects of the confining pressure and loading rate on the mechanical parameters and failure characteristics were analyzed. The results show that the exponential strength criterion is more suitable than the Mohr–Coulomb strength criterion to characterize the strength characteristics of fissured rock. The crack evolution and failure characteristics of fissured rock specimens are more complicated than those of intact rock specimens. The failure characteristics of the fissured rock follow the tensile shear coalescence model, crack branching occurs with increasing the loading rate, and the multi-section coalescence model is verified with increasing the confining pressure. The phenomena of stress drop and yield platform usually occur after the dilatancy onset, the specimen does not fail instantaneously, and the propagation and coalescence of cracks cause a sharp increase in the AE signals, circumferential strain, and volumetric strain.     

饱和多孔介质分析解的唯一性与应变局部化分岔

基于不连续性分岔基本理论导出了静态非渗流状态下弹塑性饱和多孔介质应变局部化发生时的临界硬/软化模量,利用二阶功正定性原理研究了两相问题分析解的唯一性问题,并给出了基于主轴空间下解的显式表达式。研究工作表明,在静态非渗流状态下,弹塑性饱和多孔介质分析的唯一性与应变局部化发生的临界条件除了在量值上与单相介质有着明显的不同外两者之间还有许多一致的特性,这些一致的特性对问题的分析是十分重要的。     

Predicting ductile fracture of low carbon steel sheets: Stress-based versus mixed stress/strain-based Mohr–Coulomb model

Two distinct implementations of the Mohr–Coulomb failure model are used in conjunction with a non-associated quadratic plasticity model to describe the onset of fracture in low carbon steel sheets. The stress-based version corresponds to the original Mohr–Coulomb model in stress space. For the mixed stress/strain-based version, the Mohr–Coulomb failure criterion is first transformed into the space of stress triaxiality, Lode angle parameter and equivalent plastic strain and then used as stress-state dependent weighting function in a damage indicator model. Basic fracture experiments including tensile specimens of different notch radii and a punch test are performed to calibrate the material parameters of the respective models. Subsequently, the models are used to predict the crack initiation in a Hasek test and during the stamping of an anticlastic structure. Unlike for the calibration experiments, the loading history during stamping is highly non-linear. Both models can be calibrated with similar accuracy, but the strain-based model predicts the instant of onset of fracture with greater accuracy in the stamping experiment which is an advantage of the empirical damage accumulation rule.     

Grain-scale processes governing shear band initiation and evolution in sands

A variety of experimental techniques have been used to advance understanding of strain localization phenomena in sands. However, all of these methods have fallen short in characterizing the evolution of the grain-scale processes that necessarily control shear band formation and growth in sands. This paper presents results of application of the non-destructive displacement measurement technique of digital image correlation (DIC) to measure two- and three-dimensional surface displacements on plane strain and axisymmetric sand specimens over short time steps. The abundance of local displacement data, high level of accuracy, and nearly continuous (spatially and temporally) record of displacement evolution afforded by the DIC technique has finally enabled a means to quantify local displacements to particulate-scale intensity. The data have been used to evaluate the local displacement mechanisms leading to the triggering of the formation of persistent shear bands, the timing of shear band formation with regard to the achievement of peak stress, and the character of displacements within fully formed shear bands. Insights are offered regarding the relation between strain localization and global stress-strain behavior, and the ensuing interpretations of shear banding as a hardening or softening phenomenon. Comparison of behavior between plane strain and triaxial tests offer additional perspective on the influences of three-dimensional stresses and boundary conditions on shear banding. The results further shed light on the micro-deformation mechanisms (i.e. buckling columns) responsible for the observed local strain non-uniformities that characterize “steady-state” shear band evolution.     

A microstructural elastoplastic model for unsaturated granular materials

The homogenization technique is used to obtain an elastoplastic stress–strain relationship for dry, saturated and unsaturated granular materials. Deformation of a representative volume of material is generated by mobilizing particle contacts in all orientations. In this way, the stress–strain relationship can be derived as an average of the mobilization behavior of these local contact planes. The local behavior is assumed to follow a Hertz–Mindlin’s elastic law and a Mohr–Coulomb’s plastic law. For the non-saturated state, capillary forces at the grain contacts are added to the contact forces created by an external load. They are calculated as a function of the degree of saturation, depending on the grain size distribution and on the void ratio of the granular assembly. Numerical simulations show that the model is capable of reproducing the major trends of a partially saturated granular assembly under various stress and water content conditions. The model predictions are compared to experimental results on saturated and unsaturated samples of silty sands under undrained triaxial loading condition. This comparison shows that the model is able to account for the influence of capillary forces on the stress–strain response of the granular materials and therefore, to reproduce the overall mechanical behavior of unsaturated granular materials.     

基于破坏概率的岩土试件剪切破坏角分析

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岩土体剪切破坏时剪切破坏角是一个一直在探讨的问题,传统理论解为Mohr-Coulomb解和Roscoe解,基于试件的宏观破坏现象受微观破坏概率的分布制约这一前提,从岩土体试件内部破坏概率分布函数的角度重新研究了这一问题。通过对试件内部微观破坏概率由微观到宏观的统计分析与坐标变换,发现在形式上破坏概率主轴即相当于主应力,并基此给出了破坏概率破坏莫尔圆。还对Mohr-Coulomb破坏准则与微观破坏概率统计之间的关系开展了探索分析,结果表明 Mohr-Coulomb准则是描述试件剪切破坏时大量微破坏行为共同作用的统计结果,Weibull分布仅是岩土试件内部破坏概率分布形式的一种描述,基于数学概率推导的岩土试件破坏模型能更好的与试件的破坏准则联系起来。

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On localization in saturated porous continua

Localization in elastic-plastic saturated porous media is investigated here using a linear perturbation approach. The adopted localization criterion corresponds to unbounded growth of perturbations. The critical conditions are compared with those obtained by a classical band analysis. While for one phase materials these conditions coincide, in the present context the linear perturbation approach leads in the limit of unbounded growth to the singularity of the undrained acoustic tensor, while the band analysis leads to the singularity of the drained acoustic tensor. Some general results clarifying the hierarchy of these two conditions are established for a quite general set of constitutive equations.     

Atomistic potentials based energy flux integral criterion for dynamic adiabatic shear banding

The energy flux integral criterion based on atomistic potentials within the framework of hyperelasticity–plasticity is proposed for dynamic adiabatic shear banding (ASB). System Helmholtz energy decomposition reveals that the dynamic influence on the integral path dependence is originated from the volumetric strain energy and partial deviatoric strain energy, and the plastic influence only from the rest part of deviatoric strain energy. The concept of critical shear banding energy is suggested for describing the initiation of ASB, which consists of the dynamic recrystallization (DRX) threshold energy and the thermal softening energy. The criterion directly relates energy flux to the basic physical processes that induce shear instability such as dislocation nucleations and multiplications, without introducing ad-hoc parameters in empirical constitutive models. It reduces to the classical path independent J-integral for quasi-static loading and elastic solids. The atomistic-to-continuum multiscale coupling method is used to simulate the initiation of ASB. Atomic configurations indicate that DRX induced microstructural softening may be essential to the dynamic shear localization and hence the initiation of ASB.     

Roles of plastic spin in shear banding

In this paper, the effects of plastic spin on shear banding and simple shear are examined systematically. Three types of plastic constitutive model with plastic spin are considered: (i) a non-coaxial model in which the direction of the plastic strain rate depends on that of the stress rate; (ii) a strain-softening model based on the J₂ flow theory; and (iii) the pressure-sensitive porous plasticity model. All the constitutive models are formulated in viscoplastic forms and in conjunction with non-local concepts that have been recently focused and discussed. First, behavior in simple shear is examined by numerical analysee with the aforementioned constitutive models. Moreover, some experimental evidences for stress response to simple shear are shown; that is, several large torsion tests of metal tubes and bars are carried out. Next, finite element simulations of shear banding in plane strain tension are performed. A critical effect of plastic spin on shear banding is observed for the noncoaxial model, while an almost negligible effect is observed for the porous model. The identical effects of plastic spin are observed, whether nonlocality exists or not. Finally, we discuss the relationship between the behavior in simple shear and the shear band formation. It is emphasized that this is a critical issue in predicting shear banding in macroscopic grounds.