The importance of accurate time-integration in the numerical modelling of P-wave propagation

The numerical dissipation characteristics of the Newmark and generalised-α time-integration schemes are investigated for P-wave propagation in a fully saturated level-ground sand deposit, where higher frequencies than those for S-waves are of concern. The study focuses on resonance, which has been shown to be of utmost importance for triggering liquefaction due to P-waves alone. The generalised-α scheme performs well, provided that the time-step has been carefully selected. Conversely, the dissipative Newmark method can excessively damp the response, changing radically the computed results. This implies that a computationally prohibiting small time-step would be required for Newmark to provide an accurate solution.     

Numerical Analysis of Blast-Induced Stress Waves in a Rock Mass with Anisotropic Continuum Damage Models Part 1: Equivalent Material Property Approach

Summary This paper uses the concept of anisotropic damage mechanics to analyze dynamic responses of a granite site under blasting loads. An anisotropic continuum damage model is suggested to model rock mass behavior under blasting loads. The effects of existing cracks and joints in the rock mass are considered by using equivalent rock material properties obtained from both field and laboratory test data. The anisotropic damage accumulations are simulated by continuous degradation of equivalent material stiffness and strength during loading process and are calculated using the exponential function with respect to the principal tensile strain in three directions. The suggested models are programmed and linked to an available computer program Autodyn3D through its user's subroutine capability. Stress wave propagation and damage zone in the rock mass induced by underground explosions are simulated. Numerical results of damaged area, peak particle velocity and acceleration attenuation as well as acceleration time histories and Fourier spectra are compared with those from independent field tests.     

Head-based isogeometric analysis of transient flow in unsaturated soils

An isogeometric analysis (IGA) is introduced to obtain a head-based solution to Richards equation for unsaturated flow in porous media. IGA uses Non-Uniform Rational B-Spline (NURBS) as shape functions, which provide a higher level of inter-element continuity in comparison with Lagrange shape functions. The semi-discrete nonlinear algebraic equations are solved using a combination of implicit backward-Euler time-integration and Newton-Raphson scheme. The time-step size is adaptively controlled based on the rate of changes in the pore pressure. The results from the proposed formulation are compared and verified against an analytical solution for one-dimensional transient unsaturated flow in a homogenous soil column. The proposed method is then applied to four more complex problems including two-dimensional unsaturated flow in a two-layered soil and a semi-circular furrow. The test cases in two-layered soil system involve sharp variations in the pressure gradient at the intersection of the two media, where the pore water pressure abruptly changes. It is shown that the proposed head-based IGA is able to properly simulate changes in pore pressure at the soils interface using fewer degrees of freedom and higher orders of approximation in comparison with the conventional finite element method.     

A robust numerical framework for simulating localized failure and fracture propagation in frictional materials

A computationally robust framework for simulating geomaterial failure patterns is presented in this paper. Finite element simulations which feature the use of embedded discontinuities to track material failure are known to suffer from convergence issues due to a lack of robustness. Oftentimes, complex time step-cutting schemes or arc-length methods are required in order to achieve convergence. This may invariably limit the complexity of constitutive models available for use in tracking nonlinear material behavior. To this end, we use an implicit–explicit integration scheme [Impl–Ex (Oliver et al. in Comput Methods Appl Mech Eng 195(52):7093–7114, 2006)] coupled with a novel constitutive model which allows for combined opening and shearing displacement in tension, as well as frictional sliding in compression. We show that this framework is suitable for capturing complex fracture patterns in geomaterial structures without the need for elaborate continuance schemes.     

Numerical integration during Fourier integral analysis

Numerical integration required during Fourier integral analysis is discussed. For the case of a long and prismatic elastic medium subject to three-dimensional loads applied at the surface (e.g. live load response of buried structures), the complexity of inverse integrals depends on the relative magnitude of the load width and the distance from the load in the longitudinal direction, as well as the longitudinal spacing of the loads. The inverse integrand of the applied surface loading is more difficult to evaluate compared to those for stresses and displacements. Selection of integration schemes based on successful inversion of the applied load provides accurate solutions of stress and displacement throughout the elastic body. The use of superposition when considering complex loading cases is beneficial. Copyright © 1999 John Wiley & Sons, Ltd.     

Explicit stress integration with error control for the Barcelona Basic Model: Part I: Algorithms formulations

The numerical integration of the stress–strain relationship is an important part of many finite element code used in geotechnical engineering. The integration of elasto-plastic models for unsaturated soils poses additional challenges associated to the presence of suction as an extra constitutive variable with respect to traditional saturated soil models. In this contribution, a range of explicit stress integration schemes are derived with specific reference to the Barcelona Basic Model (BBM), which is one of the best known elasto-plastic constitutive models for unsaturated soils. These schemes, however, do not address possible non-convexity of the loading collapse (LC) curve and neglect yielding on the suction increase (SI) line. The paper describes eight Runge–Kutta methods of various orders with adaptive substepping as well as a novel integration scheme based on Richardson extrapolation. The algorithms presented also incorporate two alternative error control methods to ensure accuracy of the numerical integration. Extensive validation and comparison of different schemes are presented in a companion paper. Although the algorithms presented were coded for the Barcelona Basic Model, they can be easily adapted to other unsaturated elasto-plastic models formulated in terms of two independent stress variables such as net stress and suction.     

双时间步法在二维浅水方程求解中的应用

针对传统算法效率低的问题,将隐式双时间步法应用于求解二维浅水方程,建立了非结构网格下高效的有限体积模型。在应用双时间步法时,虚拟时间层中的定常问题采用高效的隐式LU-SGS（Lower-Upper Symmetric Gauss-Seidel）方法进行迭代求解。通过模拟计算4个典型算例以及与传统显式算法进行比较,对模型精度、效率及处理实际问题能力进行检验,分析了时间步长、内迭代次数对模型性能的影响。结果表明,双时间步法放宽了稳定性对时间步长的限制,时间步长可取到显式格式10倍以上,计算耗时减少了50%以上,模型具有良好精度与适应性,具有较好的推广应用价值。     

低频循环荷载下盐岩轴向蠕变的Burgers模型分析

为了研究盐岩在低频循环荷载作用下的蠕变特性,采用恒轴压、循环围压的应力加载方式对盐岩开展了低频循环荷载蠕变试验。试验结果显示,当蠕变时间较短时,盐岩轴向蠕变曲线近似为平滑曲线,波动性不大;当时间增加到一定程度以后,盐岩变形随围压变化而发生明显的波动。总体来看,循环荷载下盐岩轴向蠕变曲线与Burgers模型蠕变曲线较为相似。将Burgers模型分解为Maxwell模型和Kelvin模型,忽略材料的疲劳效应,分别推导了恒轴压、循环围压应力加载方式下Maxwell模型和Kelvin模型的轴向蠕变方程,分析了荷载循环周期对模型蠕变性的影响规律;将循环荷载下Maxwell模型和Kelvin模型的轴向蠕变方程进行叠加,得到了循环荷载下Burgers模型的轴向蠕变方程,并利用盐岩蠕变试验结果对其合理性进行了验证。结果表明:拟合曲线和试验曲线吻合良好,该模型能够较好地反映盐岩在循环荷载作用下的蠕变特性和荷载变化对盐岩蠕变性的重要影响,特别是能够反映出不同荷载循环周期下盐岩变形随荷载变化而发生的明显的波动现象。     

Fuzzy-random probabilistic analysis of rock mass responses to explosive loads

This paper addresses the effects of randomness of initial damage in a rock mass and the critical tensile strain of the rock material on its dynamic responses and damage under explosive loads. A fuzzy definition is proposed to describe the fuzzy nature of failure phenomenon in a rock mass. The initial damage of the rock mass is estimated using the longitudinal and transverse elastic wave velocities. By using statistical analysis, the initial damage of the rock mass is found having the Beta distribution. The statistical estimation of a damage state and properties of randomly damaged rock mass are evaluated by the Rosenbluth's point estimate method. In numerical calculation, an isotropic continuum damage model with the initial damage and the cumulative damage dependent on an equivalent tensile strain is suggested to model the rock mass behavior under blast loads. A Beta distribution is proposed to represent the probabilistic distribution of the damage variable of the rock mass under explosive loads. Several types of membership functions are suggested to represent the fuzziness of material failure. Based on the fuzzy–random probabilistic theory, a model including both the effects of randomness and fuzziness is proposed for the failure analysis of rock mass under explosive loads. The suggested models are coded and linked with an available computer program AUTODYN2D through its user's subroutine capacity. The fuzzy failure probability and dynamic responses of the rock mass are calculated. Numerical results are compared with those obtained from independent field tests.     

Design of Raft Foundations for High-Rise Buildings on Jointed Rock

This paper presents calculations of displacements and bending moments in a 2-m-thick reinforced-concrete foundation slab using three-dimensional finite-element software. A preliminary paper was presented by Justo et al. (Rock Mech Rock Eng 43:287–304, 2010). The slab is the base of a tower of 137 m height above foundation, supported on jointed and partly weathered basalt and scoria. Installation of rod extensometers at different depths below foundation allowed comparison between measured displacements and displacements calculated using moduli obtained from rock classification systems and three material models: elastic, Mohr–Coulomb and hardening (H). Although all three material models can provide acceptable results, the H model is preferable when there are unloading processes. Acceptable values of settlement may be achieved with medium meshing and an approximate distribution of loads. The absolute values of negative bending moments (tensions below) increase as the rock mass modulus decreases or when the mesh is refined. The paper stresses the importance of adequately representing the details of the distribution of loads and the necessity for fine meshing to obtain acceptable values of bending moments.     

Integration schemes with substepping algorithms for creep analysis in geomaterials

This paper describes a robust and efficient methodology for predicting displacements, deformations, and stresses in geomaterials that are susceptible to creep. The methodology is based on two integration schemes, which consider substepping algorithms. The first scheme is used for integrating space-time relations in a global sense, whereas the second scheme is used for integrating stress-stain relations in a local sense. Different from previous studies, both integration schemes are easy to implement and general in the sense that they can be applied to any type of creep law. Through an in-house finite element simulator, several numerical tests are performed. They include triaxial and wellbore closure analyses considering soft soils and salt rocks. The results show that the combination of both schemes leads to stable and accurate solutions with reduced computational time.     

有侧向压力的岩石材料动态直接拉伸试验装置

岩石材料的动态拉伸力学特性是评价爆炸以及地震等动荷载作用下岩体工程响应以及安全的基本参数。介绍了一种有侧向压力的岩石动态直接拉伸试验研究装置,同时采用石膏和花岗岩材料对试验装置进行了尝试性试验。研究结果表明,岩石类脆性材料的抗拉强度随应变速率以及侧向压力的变化规律,与前人进行的岩石间接动态拉伸试验和及岩石类脆性材料（混凝土）双轴试验的变化规律具有一定相似性,因此所介绍的试验装置可以应用于岩石类脆性材料动态直接拉伸试验研究。     

Development of an implicit material point method for geotechnical applications

An implicit material point method (MPM), a variant of the finite element method (FEM), is presented in this paper. The key feature of MPM is that the spatial discretisation uses a set of material points, which are allowed to move freely through the background mesh. All history-dependent variables are tracked on the material points and these material points are used as integration points similar to the Gaussian points. A mapping and re-mapping algorithm is employed, to allow the state variables and other information to be mapped back and forth between the material points and background mesh nodes during an analysis. In contrast to an explicit time integration scheme utilised by most researchers, an implicit time integration scheme has been utilised here. The advantages of such an approach are twofold: firstly, it addresses the limitation of the time step size inherent in explicit integration schemes, thereby potentially saving significant computational costs for certain types of problems; secondly, it enables an improved algorithm accuracy, which is important for some constitutive behaviours, such as elasto-plasticity. The main purpose of this paper is to provide a unified MPM framework, in which both quasi-static and dynamic analyses can be solved, and to demonstrate the model behaviour. The implementation closely follows standard FEM approaches, where possible, to allow easy conversion of other FEM codes. Newton’s method is used to solve the equation of motion for both cases, while the formation of the mass matrix and the required updating of the kinematic variables are unique to the dynamic analysis. Comparisons with an Updated Lagrangian FEM and an explicit MPM code are made with respect to the algorithmic accuracy and time step size in a couple of representative examples, which helps to illustrate the relative performance and advantages of the implicit MPM. A geotechnical application is then considered, illustrating the capabilities of the proposed method when applied in the geotechnical field.     

盐岩三轴间隔疲劳特性试验研究

为研究地下盐穴储气库受间歇性周期注、采循环载荷作用下围岩的非连续疲劳破坏演化过程,对取自巴基斯坦的深层盐岩进行了室内三轴间隔疲劳试验研究,并分析了三轴状态下不同围压、不同应力等级对盐岩间隔疲劳的影响。试验结果表明：（1）与单轴间隔疲劳相比,围压的存在不仅提高了盐岩的抗压强度,而且增加了盐岩的疲劳寿命,围压越高增加的幅度越明显。（2）三轴间隔疲劳试验中,盐岩间隔后循环中的残余应变大于间隔前的残余应变。这与单轴间隔疲劳得到的结果一致,但围压的上升会导致盐岩的残余变形积累速度以及间隔前后循环残余的变形差值都减小。（3）随着应力等级的升高,残余应变以及时间间隔前后的残余应变差值都呈现增大的趋势。     

An assessment of the creep behaviour of Brazilian salt rocks using the multi-mechanism deformation model

Salt rocks are geomaterials that exhibit several peculiarities, which require a particular approach in rock mechanics. In the field, those rocks are usually found in layered/bedded deposits and in domes or similar structures. Creep is one of the main deformation mechanisms associated with salt rocks, and this phenomenon is highly dependent on the stress state, temperature and mineralogy. Salt rock mechanics for engineering applications requires the definition of a powerful constitutive model and this is an ongoing challenge. Among the many available models, one of the most sophisticated physical constitutive models for salt rocks is the multi-mechanism deformation creep model (MD model). The main contribution of this work is to present a first effort in the use of the MD model for Brazilian salt rocks. Material-sensitive parameters have been calibrated for the Brazilian halite through two methodologies. Salt is modelled as an elasto-viscoplastic material. Numerical simulations using the finite element method have been carried out for triaxial creep tests, Pre-salt wellbore closure and mining gallery convergence in order to validate the parameter set and the methodologies. Excellent results have been observed in most of the applications for validation. Even so, validation efforts should continue to consolidate the parameters and identify possible limitations.     

Behaviour of Track Ballast Under Repeated Loading

Najaf-sea quarry is located in Najaf city about 160 km south west of Baghdad the capital of Iraq. It is the main source that supplies track ballast for maintenance of existing railway network and construction of new railway lines in the middle and southern parts of Iraq. Track ballast experience a complex combination of stresses during its service lifetime, primarily from repeated axial loads of the trains in addition to stresses generated from the environmental conditions. The ideal evaluation of suitability of track ballast must be carried out under real field loading conditions, however such field tests are usually costly and time consuming. On the other hand laboratory model tests simulating field loads under limited boundary conditions can provide satisfactory indication about the suitability of the material. The present paper investigates the deformation characteristics of Najaf-sea track ballast, under repeated loading using model tests simulating ballast conditions under a selected track section. A test setup was designed and manufactured capable of applying both monotonic as well as repeated loading on the track section under different conditions. The repeated model tests which simulate as close as possible the field conditions shed the light on the generated settlement, modulus of deformation and degradation of the ballast particles under different repeated loading levels. Statistical analysis in terms of breakage index and repeated applied load revealed satisfactory correlations that help in understanding the overall performance of the ballast material. The results also demonstrate that 4–5 tamping are capable of controlling both the settlement and modulus of deformation of the ballast material.     

黏度时变性灌浆材料加固碎裂岩体应用研究

针对普通水泥浆流动时间长、早期强度低、可灌性差,在复杂岩体注浆中质量难以保证的问题,提出以普通水泥浆为基浆,对普通水泥浆适当掺加外掺剂进行改良,以调节水泥水化及硬化进程,形成SJP黏度时变性灌浆材料。其特征主要为浆液初始流动性好,浆液黏度增长缓慢,浆液过可泵时间后其黏度将迅速增大,浆液在可泵时间内保持良好的可灌性;浆液流动时间可控,可泵时间到初凝时间间隔短,浆液不易被冲蚀;同时浆液固结体具有前期强度增长快,后期强度高的特点;可以根据不同岩层特点,改变外掺剂的加入量,可以形成与地层良好适应性的灌浆材料。应用结果表明:对于陡倾、宽缝、碎裂岩体,SJP黏度时变性灌浆材料具有良好适宜性,可以作为锚杆灌浆材料,在减少材料用量的同时,可缩短工时,灌注质量满足设计要求。SJP灌浆材料已应用到坝基加固、房屋地基处理、地质灾害治理工程中。     

Stress‐driven integration strategies and m‐AGC tangent operator for Perzyna viscoplasticity and viscoplastic relaxation: application to geomechanical interfaces

The paper proposes a stress‐driven integration strategy for Perzyna‐type viscoplastic constitutive models, which leads also to a convenient algorithm for viscoplastic relaxation schemes. A generalized trapezoidal rule for the strain increment, combined with a linearized form of the yield function and flow rules, leads to a plasticity‐like compliance operator that can be explicitly inverted to give an algorithmic tangent stiffness tensor also denoted as the m‐AGC tangent operator. This operator is combined with the stress‐prescribed integration scheme, to obtain a natural error indicator that can be used as a convergence criterion of the intra‐step iterations (in physical viscoplasticity), or to a variable time‐step size in viscoplastic relaxation schemes based on a single linear calculation per time step. The proposed schemes have been implemented for an existing zero‐thickness interface constitutive model. Some numerical application examples are presented to illustrate the advantages of the new schemes proposed. Copyright © 2016 John Wiley & Sons, Ltd.     

Rheological analysis of tunnel excavations by means of coupled finite element (FEM)–boundary element (BEM) analysis

To facilitate the practical numerical analysis of tunnel structures by means of the finite element method in the case of viscoplastic properties of the rock mass and viscoelastic properties of the shotcrete, this method is coupled to boundary elements. In this way, the unchanged properties of the boundary element region provide enormous savings in computing time. In order to improve the numerical stability of such calculations, a variable time-step analysis was employed for each time step with an iterative correction method. Characteristic values are obtained from measured values by back-analysis.     

基于细观试验和离散元法的盐岩力学特性

当前盐岩的宏观力学模型通常是唯象模型,不能很好地解释盐岩受力变形破坏的真正物理基础。盐岩是由于化学沉积而形成的矿物集合体,是一种主要由NaCl和少量杂质组成的多晶体,其变形机制主要由晶粒与晶界的力学特性控制。通过扫描电镜（SEM）,获得盐岩晶粒的微细观结构特征,采用分子动力学方法和纳米压痕技术,确定盐岩晶粒和晶界的微细观力学参数;将盐岩晶粒作为块体,基于Voronoi多边形技术,建立盐岩的微细观数值模型;利用离散元方法,对盐岩试件在单轴压缩和直剪条件下的宏观力学行为进行了数值模拟。数值模拟结果与宏观力学试验结果吻合度高,表明基于盐岩微细观晶粒结构特征并结合离散元数值模拟的方法能够较好地研究盐岩的宏观力学性能及其材料物理基础。