Elastic fields in two joined transversely isotropic media of infinite extent as a result of rectangular loading

This paper presents the closed‐form solutions for the elastic fields in two bonded rocks induced by rectangular loadings. Each of the two bonded rocks behaves as a transversely isotropic linear elastic solid of semi‐infinite extent. They are completely bonded together at a horizontal surface. The rectangular loadings are body forces along either vertical or horizontal directions and are uniformly applied on a rectangular area. The rectangular area is embedded in the two bonded rocks and is parallel to the horizontal interface. The classical integral transforms are used in the solution formulation, and the elastic solutions are expressed in the forms of elementary harmonic functions for the rectangular loadings. The stresses and displacements in the rocks induced by both the horizontal and vertical body forces are also presented. The numerical results illustrate the important effect of the anisotropic bimaterial properties on the stress and displacement fields. The solutions can be easily implemented for numerical calculations and applied to problems encountered in rock mechanics and engineering. Copyright © 2011 John Wiley & Sons, Ltd.     

A micro-mechanics-based elastic–plastic model for porous rocks: applications to sandstone and chalk

A micro-mechanics-based elastic–plastic model is proposed to describe mechanical behaviors of porous rock-like materials. The porous rock is considered as a composite material composed of a solid matrix and spherical pores. The effective elastic properties are determined from the classical Mori–Tanaka linear homogenization scheme. The solid matrix verifies a pressure-dependent Mises–Schleicher-type yield criterion. Based on the analytical macroscopic yield criterion previously determined with a nonlinear homogenization procedure (Shen et al. in Eur J Mech A/Solids 49:531–538, 2015), a complete elastic–plastic model is formulated with the determination of a specific plastic hardening law and plastic potential. The micro-mechanics-based elastic–plastic model is then implemented for a material point in view of simulations of homogeneous laboratory tests. The proposed model is applied to describe mechanical behaviors of two representative porous rocks, sandstone and chalk. Comparisons between numerical results and experimental data are presented for triaxial compression tests with different confining pressures, and they show that the micro-mechanical model is able to capture main features of mechanical behaviors of porous rock-like rocks.     

Fast multipole displacement discontinuity method (FM‐DDM) for geomechanics reservoir simulations

The displacement discontinuity method (DDM) is frequently used in geothermal and petroleum applications for modeling the behavior of fractures in linear‐elastic rocks. The DDM requires O(N²) memory and O(N³) floating point operations (where N is the number of unknowns) to construct the coefficient matrix and solve the linear system of equations by direct methods. Therefore, the conventional implementation of the DDM is not computationally efficient for very large systems of cracks, often limiting its application to small‐scale problems. This work presents an approach for solving large‐scale fracture problems using the fast multipole method (FMM). The approach uses both the DDM and a kernel‐independent version of the FMM along with a preconditioned generalized minimal residual algorithm to accelerate the solution of linear systems of equations using desktop computers. Using the fundamental solutions for constant displacement discontinuity in a two‐dimensional elastic medium, several numerical examples involving fracture networks representing fractured reservoirs are treated. Numerical results show good agreement with analytical solutions and demonstrate the efficiency of the FMM implementation of the DDM for large‐scale simulations. Copyright © 2015 John Wiley & Sons, Ltd.     

Mechanical behavior of porous solid considering mesoscopic elastic buckling

In this work, the elastic buckling of porous solids was investigated using a lattice spring model (LSM). The capability of the LSM to solve elastic buckling problems was comprehensively verified by comparing well-established numerical and analytical solutions. Following this, the buckling of a porous solid was studied, in which two porous structures were considered, ie, the random porous model and the Voronoi porous model. The results reveal that both the porosity and the shape of the pores influence the elastic buckling bearing capacity of the porous solid. Finally, the mechanical responses of a porous solid with an extra high porosity (0.85) were numerically investigated. Our numerical results demonstrated that the nonlinear elastic response of the porous solid might come from its mesoscale elastic buckling. This work shows the ability and promise of using the LSM as a fundamental numerical tool for the deep investigation of the buckling mechanical behavior of porous solids.     

岩石弹塑性损伤本构模型建立及在隧道工程中的应用

在实际隧道施工过程中,隧道开挖引起地下岩体应力重分布使得围岩的微裂纹扩展损伤,并伴随有塑性流动变形。在地下水环境中对于孔隙和微裂隙围岩介质受到应力作用时,在内部将产生高孔隙水压力影响岩石的力学性质,也改变了围岩的破坏模式。为了研究损伤引起的刚度退化和塑性导致的流动两种破坏机制的耦合作用,从弹塑性力学和损伤理论的角度出发,同时引入修正有效应力原理来考虑孔隙水压力的作用,建立基于Drucker-Prager屈服准则的弹塑性损伤本构模型;针对该本构模型推导了孔隙水压力作用下弹塑性损伤本构模型的数值积分算法-隐式返回映射算法,分别对预测应力返回到屈服面的光滑圆锥面或尖点奇异处两种可能的情况给出了详细的描述,隐式返回映射算法具有稳定性和准确性的特点;大多数弹塑性损伤模型中涉及参数多且不易确定的问题,采用反分析方法获得损伤参数,解决了损伤参数不易确定的难题;采用面向对象的编程方法,使用C++语言编制了弹塑性损伤本构求解程序,并对所建立的弹塑性损伤模型和所编程序进行了试验和数值两个方面的验证;最后将其在吉林抚松隧道工程中进行应用,模拟了塑性区和损伤区的发展变化。研究结果表明：所建立的弹塑性损伤本构模型能够较好地描述岩石的力学性能、塑性和损伤变化趋势,所编程序能够进行实际工程问题的模拟,对现场施工给予一定的指导。     

Vertical and horizontal deformation of an inhomogeneous elastic half-space

Vertical and horizontal deformations of surface footings have been studied for an inhomogeneous elastic half-space in which the shear modulus increases with an arbitrary power of depth, n, and Poisson's ratio is constant. A general solution for displacements has been obtained first for point loads applied in vertical and horizontal directions. These are then used in obtaining closed-form solutions for displacements of uniformly loaded circular and rectangular footings. Finally, a numerical method is described that can be used to analyse a rigid footing of an arbitrary shape, and results for rigid rectangular footings are given.     

Lagrangian meshfree particles method (SPH) for large deformation and failure flows of geomaterial using elastic–plastic soil constitutive model

Simulation of large deformation and post‐failure of geomaterial in the framework of smoothed particle hydrodynamics (SPH) are presented in this study. The Drucker–Prager model with associated and non‐associated plastic flow rules is implemented into the SPH code to describe elastic–plastic soil behavior. In contrast to previous work on SPH for solids, where the hydrostatic pressure is often estimated from density by an equation of state, this study proposes to calculate the hydrostatic pressure of soil directly from constitutive models. Results obtained in this paper show that the original SPH method, which has been successfully applied to a vast range of problems, is unable to directly solve elastic–plastic flows of soil because of the so‐called SPH tensile instability. This numerical instability may result in unrealistic fracture and particles clustering in SPH simulation. For non‐cohesive soil, the instability is not serious and can be completely removed by using a tension cracking treatment from soil constitutive model and thereby give realistic soil behavior. However, the serious tensile instability that is found in SPH application for cohesive soil requires a special treatment to overcome this problem. In this paper, an artificial stress method is applied to remove the SPH numerical instability in cohesive soil. A number of numerical tests are carried out to check the capability of SPH in the current application. Numerical results are then compared with experimental and finite element method solutions. The good agreement obtained from these comparisons suggests that SPH can be extended to general geotechnical problems. Copyright © 2008 John Wiley & Sons, Ltd.     

Expansion of cavities in layered elastic system

Theoretical formulations to the problem of expansion of spherical and cylindrical cavities in a layered elastic system are presented. The medium surrounding the cavity is assumed to consist of several layers, and each is idealized as a linear elastic solid. The solutions are expressed in matrix forms and can be obtained using a programmable calculator. The developed solutions are of some importance in geomechanics. The spherical cavity solution can be used to compute the settlement of single axially loaded piles. The predicted settlement using the analysis outlined herein agrees reasonably well with one case of field measurements. The cylindrical cavity solution can be used to manifest the effect of disturbed mass at the cavity wall on the stress–strain and deformation characteristics of the intact mass.     

Analytical modeling of glacier dynamics

Flow velocities and stresses within a glacier are determined by inverting known surface velocities with a specified glacier geometry. The surface velocities depend only weakly on the unknown velocities at the bed of a glacier, so the inversion is illposed and unstable. This instability causes both numerical computation errors and data errors to grow dramatically with depth, usually masking the actual velocity and stress solutions. To control the numerical errors, an analytical modeling scheme is presented which modifies the method of mean weighted residuals (used in finite element techniques). The resulting scheme impairs convergence by producing powerseries solutions, but in an advantageous tradeoff, the coefficients to the power series can be determined analytically rather than numerically. This leads to arbitrary order analytical powerseries solutions to the internal stress state of glaciers. The symbolic powerseries solutions can be evaluated at any point in the glacier with negligible roundoff and discretization errors. Analytical model accuracy is confirmed with known stress solutions for several widely used constitutive relations for ice.     

Development of hyperplasticity models for soil mechanics

Hyperplasticity theory was developed by Collins and Houlsby (Proc. Roy. Soc. Lon. A 1997; 453 :1975–2001) and Houlsby and Puzrin (Int. J. Plasticity 2000; 16 (9):1017–1047). Further research has extended the method to continuous hyperplasticity, in which smooth transitions between elastic and plastic behaviour can be modelled. This paper illustrates a development of a new constitutive model for soils using hyperplasticity theory. The research begins with a simple one‐dimensional elasticity model. This is extended in stages to an elasto‐plastic model with a continuous internal function. The research aims to develop a soil model, which addresses some of the shortcomings of the modified cam‐clay model, specifically the fact that it cannot model small strain stiffness, or the effects of immediate stress history. All expressions used are consistent with critical state soil mechanics terminology. Finally, a numerical implementation of the model using a rate‐dependent algorithm is described. Copyright © 2005 John Wiley & Sons, Ltd.     

Determination of Shear Strength and Three-dimensional Yield Strength for the Hoek-Brown Criterion

Summary. Most currently used techniques for analysing the stability of near surface structures, such as rock slopes, are based on the application of the effective Coulomb shear strength parameters cohesion c′, and the angle of friction φ′ on some known or anticipated shear surface subjected to an effective normal stress σ′_n. The most widely used of these techniques are the variants of the method of slices and related upper bound techniques. If the Hoek-Brown criterion is to be used to model the strength of near surface fractured rocks, it is necessary to determine equivalent Coulomb shear strength parameters for the specified level of effective normal stress. Calculation of the equivalent Coulomb parameters for the Hoek-Brown criterion for cases when a ≠ 0.5 is not a straightforward matter. A simple procedure for calculating instantaneous values of c′_i and φ′_i has been developed based on spreadsheet calculations and the application of a numerical optimisation routine. This procedure can also be applied to calculating the Hoek-Brown envelope plotted in shear stress/normal stress space. A simple closed form solution for c′_i and tan φ′_i has also been developed for the special case when a = 1. A three-dimensional version of the Hoek-Brown criterion has been developed by combining it with the Drucker-Prager criterion. This new yield criterion has been implemented by numerical solution of the governing equations. A simplification of this three-dimensional yield criterion has been developed by introducing an intermediate principal stress weighting factor. Comparison with published results demonstrates that this simplified criterion has the capacity to model the results of true triaxial tests for a range of different rock types over a wide range of stress levels. The new three-dimensional yield criterion has the advantage that its input parameters can be determined from routine uniaxial compression tests and mineralogical examination.     

Consolidation and deformation around stone columns: Numerical evaluation of analytical solutions

Coupled finite element analyses of the consolidation and deformation around stone columns have been performed to assess the accuracy of different analytical solutions. The numerical model reproduces the hypotheses of the closed-form solutions. In the model, a rigid load is applied to a unit cell formed by a fully penetrating column and its surrounding soil, and simple elastic or elasto-plastic soil models are used. The surface settlement, the dissipation of the pore pressure and the vertical stress concentration on the column are studied. These soil responses are accurately estimated with closed-form solutions that properly include the radial and plastic strains in the column. However, the surrounding soil does not yield for usual conditions, which reasonably justifies the elastic soil behavior assumed in the analytical solutions. The differences between drained and consolidation analyses are also evaluated. Comparing the numerical results with the closed-form solutions illustrates the implications of the assumptions of each approach.     

Discrete element method modeling of inherently anisotropic rocks under uniaxial compression loading

A new numerical approach is proposed in this study to model the mechanical behaviors of inherently anisotropic rocks in which the rock matrix is represented as bonded particle model, and the intrinsic anisotropy is imposed by replacing any parallel bonds dipping within a certain angle range with smooth‐joint contacts. A series of numerical models with β = 0°, 15°, 30°, 45°, 60°, 75°, and 90° are constructed and tested (β is defined as the angle between the normal of weak layers and the maximum principal stress direction). The effect of smooth‐joint parameters on the uniaxial compression strength and Young's modulus is investigated systematically. The simulation results reveal that the normal strength of smooth‐joint mainly affects the behaviors at high anisotropy angles (β > 45°), while the shear strength plays an important role at medium anisotropy angles (30°–75°). The normal stiffness controls the mechanical behaviors at low anisotropy angles. The angle range of parallel bonds being replaced plays an important role on defining the degree of anisotropy. Step‐by‐step procedures for the calibration of micro parameters are recommended. The numerical model is calibrated to reproduce the behaviors of different anisotropic rocks. Detailed analyses are conducted to investigate the brittle failure process by looking at stress‐strain behaviors, increment of micro cracks, initiation and propagation of fractures. Most of these responses agree well with previous experimental findings and can provide new insights into the micro mechanisms related to the anisotropic deformation and failure behaviors. The numerical approach is then applied to simulate the stress‐induced borehole breakouts in anisotropic rock formations at reduced scale. The effect of rock anisotropy and stress anisotropy can be captured. Copyright © 2015 John Wiley & Sons, Ltd.     

Life-time prediction for rocks under static compressive and tensile loads: a new simulation approach

The life time or time to failure of rocks under load is governed by microstructural defects, like microcracks, voids etc. The life time can be predicted either by empirical exponential laws or physical laws based on damage and fracture mechanics. The proposed numerical model is based on subcritical crack growth using the linear elastic fracture mechanical approach and is implemented as a numerical cellular automate. The algorithm considers both tensile and shear fracturing. Each cell contains a microcrack of random length according to a given probability function. Fracture growth is controlled by the Charles equation. Macroscopic cracks are the results of the coalescence of growing microcracks. Within the numerical approach elasto-plastic stress redistributions take place. If the stress intensity factors have reached the critical values or the microcrack has reached the zone dimension, the zone is considered as fractured and residual strength values are assigned. The proposed approach was applied to rock samples under uniaxial compressive and tensile loads (creep tests). Successful results were obtained in respect to the predicted life time, damage evolution and the fracture pattern. Conclusions for further improvements and extensions of this methodology were drawn.     

On the mechanical response of saturated cemented materials—part I: Theoretical considerations

Terzaghi's ‘effective stress’ principle, though originally developed for soils, has been extensively applied to water saturated cemented materials (such as concrete, rocks, masonry units, etc.). There are, however, serious reservations regarding its validity for this class of materials. In this paper an alternative stress decomposition is proposed. It is based on average stress measures in constitutents, i.e. the solid matrix and water. Constitutive relations governing the undrained response of a cemented aggregate mixture, treated as a brittle-plastic material, are derived. The framework is illustrated by some numerical examples pertaining to the behaviour of saturated concrete.     

Analytical 3D transient elastodynamic fundamental solution of unsaturated soils

Unsaturated soils are considered as porous continua, composed of porous skeleton with its pores filled by water and air. The governing partial differential equations (PDE) are derived based on the mechanics for isothermal and infinitesimal evolution of unsaturated porous media in terms of skeleton displacement vector, liquid, and gas scalar pressures. Meanwhile, isotropic linear elastic behavior and liquid retention curve are presented in terms of net stress and capillary pressure as constitutive relations. Later, an explicit 3D Laplace transform domain fundamental solution is obtained for governing PDE and then closed‐form analytical transient 3D fundamental solution is presented by means of analytical inverse Laplace transform technique. Finally, a numerical example is presented to validate the assumptions used to derive the analytical solution by comparing them with the numerically inverted ones. The transient fundamental solutions represent important features of the elastic wave propagation theory in the unsaturated soils. Copyright © 2010 John Wiley & Sons, Ltd.     

Nodal-based three-dimensional discontinuous deformation analysis (3-D DDA)

This paper presents a new numerical model that can add a finite element mesh into each block of the three-dimensional discontinuous deformation analysis (3-D DDA), originally developed by Gen-hua Shi. The main objectives of this research are to enhance DDA block’s deformability. Formulations of stiffness and force matrices in 3-D DDA with conventional Trilinear (8-node) and Serendipity (20-node) hexahedral isoparametric finite elements meshed block system due to elastic stress, initial stress, point load, body force, displacement constraints, inertia force, normal and shear contact forces are derived in detail for program coding. The program code for the Trilinear and Serendipity hexahedron elements have been developed, and it has been applied to some examples to show the advantages achieved when finite element is associated with 3-D DDA to handle problems under large displacements and deformations. Results calculated for the same models by use of the original 3-D DDA are far from the theoretical solutions while the results of new numerical model are quite good in agreement with theoretical solutions; however, for the Trilinear elements, more number of elements are needed.     

软硬相间岩层底板变形研究

引入损伤变量ω理论,分析水平状软硬相间岩层在竖向应力下的应力-应变关系：再引用弹性模量E和岩层厚度L两个变形参数建立软硬相间岩层底板变形参数方程,得出竖向应力随着底板深度的增加而减小的规律：运用数值模拟软件,建立煤层开采模型,模拟软硬相间岩层底板在采动作用下的变形过程。通过数值分析底板在采动作用下的应力-应变关系和位移特征,发现竖向应力作用下软岩受到的水平应力比硬岩大,竖向应变也比硬岩大,而竖向应力和水平应变变化不大,认为在采动影响下软硬相间岩层底板中的软岩具有吸收更多的应变能来保护软岩下硬岩的变形特点。     

On Garg's solution of Biot's equations for wave propagation in a one-dimensional fluid-saturated elastic porous solid

Garg's approximate analytical solutions of Biot's equations for wave propagation in a fluid-saturated elastic porous solid of infinite extent subjected to a velocity boundary condition of a Heaviside function at one end are examined for small and large drag. Garg's approximations were apparently introduced to facilitate exact inversion of Laplace transforms of certain quantities. The approximate solutions are compared with carefully evaluated numerical inverses of the Laplace transform solutions for different soils with widely varying properties. It is seen that for most soils (clays, silts and, sands) the error in Garg's approximate solutions in insignificant, and the solutions can be used as benchmarks for verifying numerical analysis procedures such as finite element codes.     

A Generalized Backward Euler algorithm for the numerical integration of an isotropic hardening elastoplastic model for mechanical and chemical degradation of bonded geomaterials

An extended version of the classical Generalized Backward Euler (GBE) algorithm is proposed for the numerical integration of a three‐invariant isotropic‐hardening elastoplastic model for cemented soils or weak rocks undergoing mechanical and non‐mechanical degradation processes. The restriction to isotropy allows to formulate the return mapping algorithm in the space of principal elastic strains. In this way, an efficient and robust integration scheme is developed which can be applied to relatively complex yield surface and plastic potential functions. Moreover, the proposed algorithm can be linearized in closed form, thus allowing for quadratic convergence in the global Newton iteration. A series of numerical experiments are performed to illustrate the accuracy and convergence properties of the algorithm. Selected results from a finite element analysis of a circular footing on a soft rock layer undergoing chemical weathering are then presented to illustrate the algorithm performance at the boundary value problem level. Copyright © 2002 John Wiley & Sons, Ltd.