Homogenized constitutive laws for rocks with elastoplastic fractures

This work deals with the development of a constitutive law for fractured rocks. Fractures are considered as penny‐shaped inclusions, whose constitutive law is deduced from an interface law and a regularization procedure. Such a method is applied to linear and non‐linear interface behaviours in order to reproduce effects such as an increase in stiffness during fracture closing, dilatancy or asperity surface degradation. Then, considering the fractured rock as a composite material, we use a Mori–Tanaka method to estimate the homogenized properties of the rock. Numerical experiments illustrate the interest of the proposed homogenization procedure. Copyright © 2007 John Wiley & Sons, Ltd.     

Effect of faceting on pore geometry in texturally equilibrated rocks: implications for low permeability at low porosity

The pore geometry of texturally equilibrated rocks is controlled by the interfacial energy ratio between grain boundaries and solid–liquid boundaries. Faceting at pore walls, which is a common feature of pore networks in rocks, strongly affects the liquid distribution. We investigated the effects of faceting on the equilibrium pore geometries based on image analysis of several systems with various degrees of faceting and dihedral angles. The degree of faceting was assessed by the F value, which is the ratio of the flat interface length at the pore wall to the length of total interfacial boundary between solid and liquid. The F values tend to increase with increasing liquid volume fraction. Little-faceted systems show relatively homogeneous liquid distribution. Moderately-faceted systems with a higher dihedral angle (∼55°) are characterized by development of large pores surrounded by faceted walls and complementary shrinkage of triple junction tubes, whereas strongly faceted systems with a low dihedral angle show no evidence of shrinking triple junction tubes, although most pores are surrounded by faceted pore walls. The faceted systems tend to produce more facet boundaries at the pore walls due to the difference of interfacial energies between the flat and curved surfaces. In the systems with the same degree of faceting, heterogeneity of liquid distribution tends to decrease with dihedral angle. For faceting systems, the permeability of texturally equilibrated rocks with low liquid fraction would be significantly decreased by the relative reduction of triple junction volumes or by closure of channels along grain edge due to the truncation of facet walls.     

Computer-aided design of constitutive laws

In this paper a method is presented which helps to develop and to check constitutive laws by means of computer experiments. The exposition has been carried out along the formalism of hypoplasticity, which has been introduced as a generalization of hypoelasticity. Several results of computer experiments are graphically represented.     

一种盐岩流变损伤模型

在Carter流变模型基础上,引进损伤,提出了“损伤增速界限”的概念,从而,建立了一种盐岩流变损伤模型,并根据实验数据验证了模型的正确性。研究表明,这种盐岩流变损伤模型,不但能很好地反映高应力水平下盐岩的流变损伤特性,而且,也能真实地描述低应力水平下盐岩的初始蠕变损伤和稳态蠕变损伤。 损伤增速界限对损伤演化和蠕变发展影响很大,关系着加速蠕变的发生与否。研究还表明,不同产地、不同实验条件下表述盐岩损伤的参数离散性不大,该模型的应用范围较广。     

Microplane constitutive model for porous isotropic rocks

The paper deals with constitutive modelling of contiguous rock located between rock joints. A fully explicit kinematically constrained microplane‐type constitutive model for hardening and softening non‐linear triaxial behaviour of isotropic porous rock is developed. The microplane framework, in which the constitutive relation is expressed in terms of stress and strain vectors rather than tensors, makes it possible to model various microstructural physical mechanisms associated with oriented internal surfaces, such as cracking, slip, friction and splitting of a particular orientation. Formulation of the constitutive relation is facilitated by the fact that it is decoupled from the tensorial invariance restrictions, which are satisfied automatically. In its basic features, the present model is similar to the recently developed microplane model M4 for concrete, but there are significant improvements and modifications. They include a realistic simulation of (1) the effects of pore collapse on the volume changes during triaxial loading and on the reduction of frictional strength, (2) recovery of frictional strength during shearing, and (3) the shear‐enhanced compaction in triaxial tests, manifested by a deviation from the hydrostatic stress–strain curve. The model is calibrated by optimal fitting of extensive triaxial test data for Salem limestone, and good fits are demonstrated. Although these data do not cover the entire range of behaviour, credence in broad capabilities of the model is lend by its similarity to model M4 for concrete—an artificial rock. The model is intended for large explicit finite‐element programs. Copyright © 2002 John Wiley & Sons, Ltd.     

Enhanced double-mechanism creep laws for salt rocks

The double-mechanism creep law (DM model) is an empirical creep constitutive model widely employed in Brazilian salt rock mechanics. This model often presents good performance in steady-state creep prediction. However, transient creep is not accounted for, and whenever early creep estimates are important, the contribution of this phase might be meaningful. This work adds value by presenting two alternatives to account for transient creep in the DM model. The first alternative couples the transient function from Sandia’s multi-mechanism deformation model to the DM model steady-state creep rate (EDMT model). The second alternative couples the DM model response to Norton’s power law when the strain rate given by the latter remains lower than the one from the former (EDMP model). These models can be implemented in numerical simulators at small code extensions of the DM model implementations. Applications from previous works by the authors are revisited to validate the formulations based on experimental data. EDMT and EDMP models differ in the formulation of transient creep and, consequently, in the time of transition between the transient and the steady-state phases. Both methods were successful in treating transient creep and in simulating experimental results.     

一种横观各向同性岩体蠕变模型

层状岩体由于层理面或者结构面的存在,在力学上具有横观各向同性的特点,现有的各向同性蠕变模型难以全面反映横观各向同性岩体的蠕变力学特性。为构建能够反映横观各向同性岩体的三维蠕变模型,以能反映瞬时应变、减速蠕变和稳态蠕变特征的黏弹性Burgers模型为基础,基于常泊松比假定,在三维各向同性蠕变本构方程的基础上,按照算子替换的方法,将横观各向同性柔度矩阵代替各向同性柔度矩阵,并考虑了平行和垂直层理方向岩体蠕变力学行为的差异性,推导了横观各向同性岩体的三维蠕变本构方程。根据本构方程的特点,提出了根据平行和垂直方向岩体蠕变试验结果进行三维蠕变本构模型中蠕变参数的辨识方法。将提出的模型应用于三轴蠕变试验参数辨识,从而获得了一套完整的三维蠕变参数,并与试验结果进行对比分析,从而验证了所提模型与方法的合理性和有效性。进一步,指出了传统蠕变试验设计方案的局限性,给出了横观各向同性材料蠕变试验设计建议。研究成果为研究岩体三维蠕变机制提供了新思路,可对岩体蠕变试验设计提供相应的科研支撑。     

一种岩石统计损伤本构模型的研究

基于岩石的非均质性,假设岩石微元粘聚力服从Weibull分布,微元破坏后其粘聚力降为0,结合统计理论和岩石剪切面上剪力平衡方程,建立了岩石发生剪切破坏时的损伤演化方程。该损伤演化方程考虑了破坏微元的抗压和抗剪能力,能够反映岩石峰后强度逐步丧失过程和残余强度。将该模型编入有限元程序,数值计算了不同围压下岩石应力-应变全过程曲线,与试验曲线比较取得了很好的一致性。     

A new Cosserat-like constitutive model for bedded salt rocks

Salt rocks are commonly used as geologic host rocks for storage of gas and crude oil, and are being considered for the disposal of radioactive waste. Different from the salt rock domes in many countries, the salt rock formations in China are usually laminar with many alternating layers, i.e. rock salt, anhydrite, and/or mudstone. Considering the unique stratigraphic characteristics of these salt rocks, a new Cosserat-like medium constitutive model is proposed in order to facilitate efficient modeling of the mechanical behavior of these formations. In this model, a new representative volume element, containing two different layers, is employed to simulate the compatibility of the meso-displacement between two different layers and also the bending effect. A new method for the deformation and failure analysis of bedded salt rocks is derived therefrom. Having the macro-average stresses, the conventional stresses in the different layers can be obtained in sequence. The conventional stresses can then be utilized in a routine way for the strength and failure analysis. For the initial numerical modeling, the new Cosserat-like medium is reduced to a transversely isotropic one. The simplified constitutive model for layered media is then implemented into FLAC3D codes. A test sample validates that the results by using the numerical model are in good agreement with that by using the built-in model, and the mesh size for the new model is reduced greatly. Finally, an application for the stability of oil storage caverns in deep thinly bedded salt rocks is carried out. The effects on convergence of storage caverns and on the failure of surrounding rock due to the presence of the mudstone interlayers (hard phase) are discussed in detail. Copyright © 2009 John Wiley & Sons, Ltd.     

不同地质成因岩石地应力分布规律的统计分析

通过收集我国大陆地区大量的实测地应力资料,筛选出了受局部影响小的实测数据,建立了岩浆岩、沉积岩和变质岩地应力随埋深分布的散点图,并根据各自的分布规律进行了相应的回归分析。结果发现,三大类岩石的地应力分布规律存在差异,其主要与岩石地质成因相关。三大类岩石的垂直应力总体上随埋深呈线性增大,量值接近各自上覆岩层自重;最大水平主应力随埋深均呈线性增大,相同埋深条件下,火成岩量值最大,变质岩次之,沉积岩最小;而沉积岩浅部最小水平应力大于岩浆岩;三大类岩石的水平差应力总体上随埋深也呈线性关系,相同埋深条件下,火成岩较大,而沉积岩较小;浅部岩浆岩平均水平主应力与垂直应力之比最大,而沉积岩较小。     

Flow laws of multiphase materials and rocks from end-member flow laws

Rheological properties of polyphase rocks play an important role in the dynamics of the lithosphere and asthenosphere. However, flow laws for large portions of the polyphase rocks in the Earth's crust and mantle have not been well determined. An analysis based on the theory of mixtures has been made to calculate the general flow laws of coarse (≥5 μm), nearly equant-grained (aspect-ratio ≤3), and massive polyphase rocks and materials for which only the flow laws and volume fractions of the constituents are taken into consideration in the modeling of the bulk rheology and effects of microstructure could be ignored. The theoretical analysis is based on three assumptions: (1) the polyphase composite and the monophase aggregates of its constituents obey the same kind of flow laws (linear, power or exponential), (2) there is no change in the operative deformation mechanism of each phase when it is in the composite as compared to when it is in a monophase aggregate, and (3) neither chemical (metamorphic) reactions take place among the constituent phases nor eutectic melting occurs due to the phase mixing. The proposed iterative process allows to predict, to the first approximation, the flow laws for a large number of polyphase rocks in terms of the experimentally determined flow laws of a relatively small number of monomineralic aggregates. Applications of this approach to typical polyphase rocks such as granite, diorite, diabase, aplite and websterite as well as to synthetic two-phase materials such as forsterite–enstatite mixtures and water ice–ammonia dehydrate aggregates yield quite accurate approximations to the experimental values.     

A phenomenological constitutive model for rocks with shear failure mode

In this paper, a phenomenological constitutive model is proposed to simulate the stress–strain behaviours of intact rocks with shear failure mode. The model captures a wide range of behaviours of rock material such as elastic, plastic, strain softening, ‘Class II’, strain localization, elastic modulus degradation, etc. The sensitivity of the stress–strain relation on the parameters is also investigated. Typical results obtained by testing a number of granite and marble specimens are used to validate the proposed model. Copyright © 2001 John Wiley & Sons, Ltd.     

Computational procedures for non-linear three-dimensional analysis with some advanced constitutive laws

Computational procedures for implementing some constitutive models are described and introduced in three-, and two-dimensional finite element procedures; here variable moduli, Drucker–Prager, critical state and cap models are considered. Consistent numerical schemes are presented with applications to a number of example problems. These procedures can provide successful results with advanced constitutive laws for three-dimensional analysis of a wide range of non-linear problems.     

冻融作用下岩石力-热-水耦合本构模型研究

在寒区隧道工程中,冻胀的水分迁移作用加剧了隧道围岩的冻融破坏。耦合水分迁移作用的岩石本构模型对防治冻融渗漏、崩塌等隧道围岩灾害有着重要意义。基于内状态变量理论,将冻胀过程中的水分迁移量作为本征变量引入Helmholtz自由能,在热力学框架下建立了一个岩石力-热-水耦合本构模型。模型描述了温度和水分迁移对冻融后岩石损伤阈值、等向强化饱和值、等向强化速率等力学参数的影响。模拟了岩石冻融后力学性质的劣化。区别于全量经验公式,本模型以增量形式给出,为复杂应力历史条件下的数值模拟提供了便利。通过将模型模拟曲线与冻融后岩石常规三轴压缩试验曲线进行对比,初步验证模型的可靠性,为实际寒区工程的冻胀破坏预测提供参考。     

Weathering potential index for rocks based on density and porosity measurements

Rock samples belonging to ten lithological types under different stages of weathering, were collected from different stratigraphical horizons at Bhagalpur. Their densities and porosities were determined experimentally and the data obtained were fitted empirically in a linear equation for each lithological type. The slopes of the curves, which were negative in each case, showed that the increase in porosity for the same decrease in density were in the order, white sandstone > ferruginous sandstone > white claystone > porphyritic gneiss > quartzite > pegmatite > amphibolite ≅ biotite gnejss > basalt ≅ dolerite. A new weathering potential index based on the density-porosity data was proposed and the values for a specific stage of weathering for all the lithological types studied fall within the same range.     

岩石流变损伤本构方程

根据岩石材料的微结构机理,基于简单机械模型,通过在不可逆应变和牛顿时间所构成的空间中合理的定义广义时间、引入四阶各向异性损伤张量,建立了岩石的流变损伤本构方程。该模型能够考虑复杂应力状态下材料的响应特性,各向异性损伤及其损伤的方向特征,静水压力的影响等。发展了相应的数值分析方法,并根据泥岩三轴蠕变试验结果进行了验证。     

An enhanced constitutive model for quasi-brittle rocks with localized damage

Acta Geotechnica - Localized deformation is the failure precursor of quasi-brittle rocks. The objective herein is to propose an enhanced constitutive model incorporating the evolution of localized...     

A constitutive model based on modified mixture theory for unsaturated rocks

Non-equilibrium thermodynamics and Biot poro-elasticity have been combined to give a coupled hydro-mechanical formulation for unsaturated rock. Darcy’s law for unsaturated flow has been derived from the dissipation process by using standard arguments of non-equilibrium thermodynamics, whereas Helmholtz free energy has been used to derive the relationship between stress and pore pressure changes. The resulting general framework accommodates both large and small deformation theories. When small deformations are assumed, the formulation is comparable with coupled equations derived using an alternative approach. For illustrative purposes, the formulation has been used to analyse a seasonally affected tunnel model. Numerical results for the desaturation in winter and resaturation in summer, of the zone near the tunnel wall, have been evaluated and compared with the findings of other researchers.     

A model for phlogopite development in low magnesium rocks

Phlogopites containing anomalously high values of Zn, Mn and Li have been identified in low-magnesium feldspathic schists and quartzites from the Precambrian of northern New Mexico. The phlogopite is volumetrically minor (<2%) and coexists with muscovite. The presence of hematite and piemontite in the phlogopite-bearing samples indicates high oxidizing conditions, but their presence in phlogopite-free mica schists nearby demonstrates that high fo₂ is insufficient to account for the phlogopite occurrences. A model for phlogopite development is presented which is based upon (1) high fo₂ conditions, (2) limited H⁺ during metamorphism, and (3) early mobilization of Mg (and Zn, Mn, Li) during incipient metamorphism. The absence of phlogopite in associated rocks which refect high fo₂ conditions is due to the larger amount of H⁺ available during metamorphism, resulting in the development of abundant muscovite rather than rare phlogopite.     

Scaling laws for hydraulic fractures driven by a power-law fluid in homogeneous anisotropic rocks

This study investigates parametric space of solutions for a planar hydraulic fracture propagating in a homogeneous anisotropic rock. It is assumed that the fracture has an elliptical shape and is driven by a power-law fluid. The purpose of this study is to investigate the influence of anisotropy and power-law fluid rheology on the parametric space of solutions. Rock anisotropy is represented by having two values of fracture toughness, one in the vertical direction and another one in the horizontal direction. Similarly, the effect of elastic anisotropy is approximated by using two different effective elastic moduli in the vertical and horizontal directions. In contrast to the isotropic case, for which there are four limiting solutions, the problem for anisotropic rocks features six different limiting cases. These cases represent competition between toughness and viscosity in the vertical and horizontal directions and competition between fluid storage inside the fracture and fluid leak-off into formation. Approximate expressions for the limiting solutions are obtained using global volume balance and tip asymptotic solutions. Despite the developed solutions rely on a series of approximations, they precisely capture all the scaling laws associated with the problem. Zones of applicability of these limiting solutions are calculated, and their dependence on the problem parameters is investigated.