基于内变量和张量函数表示定理的本构方程

针对各向同性材料,基于张量函数表示定理,建立了本构关系的张量不变性表示,其中,3个不可约基张量取决于应力的0～2次幂,且相互正交,3个系数由塑性应变增量和应力的不变量表示。基于塑性应变增量的不变量定义内变量,本构关系归结为确定内变量的演化。使用张量函数表示定理,给出了内变量演化方程的一般表达式,它取决于应力不变量的增量,因而与主轴旋转无关。讨论了如何根据试验资料和引入适当的假定,确定具体的演化方程。通过与塑性势理论和多重屈服面理论进行比较,表明所建模型是这些理论的最一般表示,且简捷直观、使用方便。     

An improved SPH method for saturated soils and its application to investigate the mechanisms of embankment failure: Case of hydrostatic pore‐water pressure

The method of smoothed particle hydrodynamics (SPH) has recently been applied to computational geomechanics and has been shown to be a powerful alternative to the standard numerical method, that is, the finite element method, for handling large deformation and post‐failure of geomaterials. However, very few studies apply the SPH method to model saturated or submerged soil problems. Our recent studies of this matter revealed that significant errors may be made if the gradient of the pore‐water pressure is handled using the standard SPH formulation. To overcome this problem and to enhance the SPH applications to computational geomechanics, this article proposes a general SPH formulation, which can be applied straightforwardly to dry and saturated soils. For simplicity, the current work assumes hydrostatic pore‐water pressure. It is shown that the proposed formulation can remove the numerical error mentioned earlier. Moreover, this formulation automatically satisfies the dynamic boundary conditions at a submerged ground surface, thereby saving computational cost. Discussions on the applications of the standard and new SPH formulations are also given through some numerical tests. Furthermore, techniques to obtain the correct SPH solution are also proposed and discussed throughout. As an application of the proposed method, the effect of the dilatancy angle on the failure mechanism of a two‐sided embankment subjected to a high groundwater table is presented and compared with that of other solutions. Finally, the proposed formulation can be considered a basic formulation for further developments of SPH for saturated soils. Copyright © 2011 John Wiley & Sons, Ltd.     

A new SPH-based approach to simulation of granular flows using viscous damping and stress regularisation

The smoothed particle hydrodynamics (SPH) method was recently extended to simulate granular materials by the authors and demonstrated to be a powerful continuum numerical method to deal with the post-flow behaviour of granular materials. However, most existing SPH simulations of granular flows suffer from significant stress oscillation during the post-failure process, despite the use of an artificial viscosity to damp out stress fluctuation. In this paper, a new SPH approach combining viscous damping with stress/strain regularisation is proposed for simulations of granular flows. It is shown that the proposed SPH algorithm can improve the overall accuracy of the SPH performance by accurately predicting the smooth stress distribution during the post-failure process. It can also effectively remove the stress oscillation issue in the standard SPH model without having to use the standard SPH artificial viscosity that requires unphysical parameters. The predictions by the proposed SPH approach show very good agreement with experimental and numerical results reported in the literature. This suggests that the proposed method could be considered as a promising continuum alternative for simulations of granular flows.     

一种适用于孔隙体积应变的有效应力方程

土力学奠基石Terzaghi有效应力原理被广泛应用于油藏孔隙和渗透率应力敏感研究中,然而其对于岩石孔隙体积应变的适用性存在争议。对颗粒不可压缩和颗粒可压缩的多孔介质分别进行了受力分析,推导了总体积、颗粒骨架、孔隙体积的有效应力表达式,与Biot、Skepmton有效应力方程对比,建立了适用于孔隙体积应变的新型有效应力方程,并进行了试验论证和应用举例。研究表明:在颗粒不可压缩多孔介质中,有效应力为超出平衡孔隙流压之外的颗粒间宏观等效应力;在颗粒可压缩变形多孔介质中,有效应力为其相同应变下的等效应力,有3种有效应力分别适用于总体积应变、颗粒体积应变、孔隙体积应变;新提出的孔隙体积有效应力方程与孔隙度、岩石总体积压缩系数、颗粒压缩系数、总应力和流压相关,4个理论计算式计算结果在3种多孔介质试验测试结果中的偏差均在5%以内;孔隙体积有效应力系数解决了如何定量增总应力来等效模拟储层降流压生产过程这一关键问题,3个压缩系数关系式理论计算准确方便。     

Frictional contact algorithms in SPH for the simulation of soil–structure interaction

Simulation of frictional contact between soils and rigid or deformable structure in the framework of smoothed particle hydrodynamics (SPH) is presented in this study. Two algorithms are implemented into the SPH code to describe contact behavior, where the contact forces are calculated using the law of conservation of momentum based on ideal plastic collision or using the criteria of partial penetrating. In both algorithms, the problem of boundary deficiency inherited from SPH is properly handled so that the particles located at contact boundary can have precise acceleration, which is critical for contact detection. And the movement and rotation of the rigid structure are taken into account so that it is easy to simulate the process of pile driving or movement of a retaining wall in geotechnical engineering analysis. Furthermore, the capability of modeling deformability of a structure during frictional contact simulations broadens the fields of SPH application. In contrast to previous work dealing with contact in SPH, which usually use particle‐to‐particle contact or ignoring sliding between particles and solid structure, the method proposed here is more efficient and accurate, and it is suitable to simulate interaction between soft materials and rigid or deformable structures, which are very common in geotechnical engineering. A number of numerical tests are carried out to verify the accuracy and stability of the proposed algorithms, and their results are compared with analytical solutions or results from finite element method analysis. Good agreement obtained from these comparisons suggests that the proposed algorithms are robust and can be applied to extend the capability of SPH in solving geotechnical problems. Copyright © 2013 John Wiley & Sons, Ltd.     

A generic approach to modelling flexible confined boundary conditions in SPH and its application

In this paper, a new approach to applying confining stress to flexible boundaries in the smoothed particle hydrodynamics (SPH) method is developed to facilitate its applications in geomechanics. Unlike the conventional SPH methods that impose confining boundary conditions by creating extra boundary particles, the proposed approach makes use of kernel truncation properties of SPH approximations that occur naturally at free-surface boundaries. Therefore, it does not require extra boundary particles and, as a consequence, can be utilised to apply confining stresses onto any boundary with arbitrary geometry without the need for tracking the curvature change during the computation. This enables more complicated problems that involve moving confining boundaries, such as confining triaxial tests, to be simulated in SPH without difficulties. To further enhance SPH applications in elasto-plastic computations of geomaterials, a robust numerical procedure to implement Mohr-Coulomb plasticity model in SPH is presented for the first time to avoid difficulties associated with corner singularities in Mohr-Coulomb model. The proposed approach was first validated against two-dimensional finite element (FE) solutions for confining biaxial compression tests to demonstrate its predictive capability at small deformation range when FE solutions are still valid. It is then further extended to three-dimensional conditions and utilised to simulate triaxial compression experiments. Simulation results predicted by SPH show good agreement with experiments, FE solutions, and other numerical results available in the literature. This suggests that the proposed approach of imposing confining stress boundaries is promising and can handle complex problems that involve moving confining boundary conditions.     

Static deformation of an orthotropic multilayered elastic half-space by two-dimensional surface loads

The transfer matrix approach is used to solve the problem of static deformation of an orthotropic multilayered elastic half-space by two-dimensional surface loads. The general problem is decoupled into two independent problems. The antiplane strain problem and the plane strain problem are considered in detail. Integral expressions for displacements and stresses at any point of the medium due to a normal line load and a shear line load, acting parallel to a symmetry axis, are obtained. In the case of a uniform half-space, closed form analytic expressions for displacements and stresses are derived. The procedure developed is quite easy and convenient for numerical computations.     

A comprehensive study on the parameters setting in smoothed particle hydrodynamics (SPH) method applied to hydrodynamics problems

Smoothed particle hydrodynamics (SPH) is a meshfree, Lagrangian particle method which has advantages in handling solids with extremely large deformation. Like any other numerical methods, cares must be taken to ensure its desirable accuracy and stability through considering several correction techniques in calculation. The selection of values for parameters in those correction approaches is a key step in SPH simulation, which is always difficult for new beginners to deal well with effectively. This paper examines the common inconsistency and instability problems in SPH method and studies its computational efficiency when applied to hydrodynamics problems with material strength like soil column collapse. We analyzed in detail how the correction techniques mitigate these inconsistency and instability problems. Also, the numerical testing results associate with different values for the parameters used in the correction techniques are provided for better understanding the influence of these parameters and for finding out the desirable values. It is found that (1) the SPH method is easily subjected to an inconsistency problem in the boundary area due to the boundary deficiency, and it can be treated well by adopting “virtual particles” contributing to the particle summations. (2) The numerical oscillation in SPH simulation can be mitigated effectively by artificial viscosity with the suggested parameter values. (3) The tension cracking treatment, artificial viscosity and artificial stress work well in removing the tensile instability problem in SPH method. In addition, the nearest neighboring particle searching (NNPS) algorithm, spacing ratio, smoothing length and time step influence the efficiency and accuracy of SPH method significantly. It is shown that SPH method with suggested parameters values can produce a very good result compared with the experimental result.     

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.     

Multiphase SPH modeling of free surface flow in porous media with variable porosity

This paper presents a numerical model for simulating free surface flow in porous media with spatially varying porosity. The governing equations are based on the mixture theory. The resistance forces between solid and fluid is assumed to be nonlinear. A multiphase SPH approach is presented to solve the governing equations. In the multiphase SPH, water is modeled as a weakly compressible fluid, and solid phase is discretized by fixed solid particles carrying information of porosity. The model is validated by several numerical examples including seepage through specimen, fast flow through rockfill dam and wave interaction with porous structure. Good agreements between numerical results and experimental data are obtained in terms of flow rate and evolution of free surface. Parameter study shows that (1) the nonlinear resistance law provides more accurate results; (2) particle size and porosity have significant influence on the porous flow.     

Finite-element analysis of time-dependent large-deformation problems

An updated Lagrangian finite-element formulation has been developed for time-dependent problems of soil consolidation involving finite deformations. Large plastic strains as well as rotations occur in such problems and nominal stress measures are introduced in the formulation to redefine stresses. This leads to corrective terms for equilibrium and yield violations in addition to geometric stiffening terms in the governing integral equations. The soil is considered to be either a linear elastic or an elastoplastic, critical-state material. Some simple numerical examples are studied to validate the formulation, followed by a detailed analysis of the problem of penetration of a pile into soil. The results of this problem are viewed with emphasis on the physical interpretation and practical significance.     

Simulation of mixed-mode fracture using SPH particles with an embedded fracture process zone

This paper focuses on the modelling of mixed-mode fracture using the conventional smoothed particle hydrodynamics (SPH) method and a mixed-mode cohesive fracture law embedded in the particles. The combination of conventional SPH and a mixed-mode cohesive model allows capturing fracture and separation under various loading conditions efficiently. The key advantage of this framework is its capability to represent complex fracture geometries by a set of cracked SPH particles, each of which can possess its own mixed-mode cohesive fracture with arbitrary orientations. Therefore, this can naturally capture complex fracture patterns without any predefined fracture topologies. Because a characteristic length scale related to the size of the fracture process zone is incorporated in the constitutive formulation, the proposed approach is independent from the spatial discretisation of the computational domain (or mesh independent). Furthermore, the anisotropic fracture responses of materials can be naturally captured thanks to the orientation of the fracture process zone embedded at the particle level. The performance of the proposed approach demonstrates its potentials in modelling mixed-mode fracture of rocks and similar quasi-brittle materials.     

Rocking vibration of a rigid circular disc in a transversely isotropic full‐space

In this paper, forced rocking vibration of a rigid circular disc placed in a transversely isotropic full‐space, where the axis of material symmetry of the full‐space is normal to the surface of the plate, is analytically investigated. Because of using the Fourier series and Hankel integral transforms, the mixed boundary‐value problem is transformed into two separate pairs of integral equations called dual integral equations. The dual integral equations involved in this paper are reduced to Fredholm integral equations of the second kind. With the aid of contour integration, the governing integral equation is numerically evaluated in the general dynamic case. The reduced static case of the dual integral equations is solved analytically and the vertical displacement, the contact pressure and the static impedance/compliance function are explicitly determined, and it is shown that the pressure in between the plate and the full‐space and the compliance function reduced for isotropic half‐space are identical to the previously published solutions. The dynamic contact pressure in between the disc and the space and also the related impedance function are numerically evaluated in general dynamic case and illustrated. It is shown that the singularity exists in the contact pressure at the edge of the disc is the same as the static case. To show the effect of material anisotropy, the numerical evaluations are given for some different transversely isotropic materials and compared. Copyright © 2010 John Wiley & Sons, Ltd.     

Variational principles and finite element method for stress analysis of porous media

A number of variational principles are established in this paper for the stress analysis of porous media with compressible constituents. A three-dimensional finite element method is proposed based on the variational principle. The finite element method thus established is applied to the study of the temperature, deformation and flow field associated with the water-flood technique in secondary recovery projects for oil exploration. In the study model, the layout of injection wells and production wells is considered to have a regular pattern where symmetry conditions exist. The injection fluid diffuses slowly into the formation through a vertical crack which is initially generated by an explosion. The problem is analysed by a plane strain formulation with the effects of heat conduction and convection included.     

Modelling the flow of self‐compacting concrete

A Lagrangian particle‐based method, smooth particle hydrodynamics (SPH), is used in this paper to model the flow of self‐compacting concretes (SCC) with or without short steel fibres. An incompressible SPH method is presented to simulate the flow of such non‐Newtonian fluids whose behaviour is described by a Bingham‐type model, in which the kink in the shear stress vs shear strain rate diagram is first appropriately smoothed out. The viscosity of the SCC is predicted from the measured viscosity of the paste using micromechanical models in which the second phase aggregates are treated as rigid spheres and the short steel fibres as slender rigid bodies. The basic equations solved in the SPH are the incompressible mass conservation and Navier–Stokes equations. The solution procedure uses prediction–correction fractional steps with the temporal velocity field integrated forward in time without enforcing incompressibility in the prediction step. The resulting temporal velocity field is then implicitly projected on to a divergence‐free space to satisfy incompressibility through a pressure Poisson equation derived from an approximate pressure projection. The results of the numerical simulation are benchmarked against actual slump tests carried out in the laboratory. The numerical results are in excellent agreement with test results, thus demonstrating the capability of SPH and a proper rheological model to predict SCC flow and mould‐filling behaviour. Copyright © 2010 John Wiley & Sons, Ltd.     

Finite element formulation and algorithms for unsaturated soils. Part I: Theory

This paper presents a complete finite‐element treatment for unsaturated soil problems. A new formulation of general constitutive equations for unsaturated soils is first presented. In the incremental stress–strain equations, the suction or the pore water pressure is treated as a strain variable instead of a stress variable. The global governing equations are derived in terms of displacement and pore water pressure. The discretized governing equations are then solved using an adaptive time‐stepping scheme which automatically adjusts the time‐step size so that the integration error in the displacements and pore pressures lies close to a specified tolerance. The non‐linearity caused by suction‐dependent plastic yielding, suction‐dependent degree of saturation, and saturation‐dependent permeability is treated in a similar way to the elastoplasticity. An explicit stress integration scheme is used to solve the constitutive stress–strain equations at the Gauss point level. The elastoplastic stiffness matrix in the Euler solution is evaluated using the suction as well as the stresses and hardening parameters at the start of the subincrement, while the elastoplastic matrix in the modified Euler solution is evaluated using the suction at the end of the subincrement. In addition, when applying subincrementation, the same rate is applied to all strain components including the suction. Copyright © 2003 John Wiley & Sons, Ltd.     

Breaking wave‐induced response and instability of seabed around caisson breakwater

Breaking‐wave‐induced dynamic response and instability of seabed around a caisson breakwater are investigated. A seabed‐rubble‐breakwater system is modeled using finite elements. The impact response of the porous seabed and rubble foundation is assumed to be governed by the coupled Biot equations, and three possible formulations are considered with respect to the inclusion of inertial terms. The response is presented in terms of shear stress and pore pressure distributions at three locations underneath the breakwater. The effect of seabed and wave parameters and the inertial terms on the impact response is investigated through parametric studies. Analyses show that usually partly dynamic formulation yields the largest response amplitudes as compared to the fully dynamic formulation, which is the most complete form. The instability of seabed and rubble mound as a result of instantaneous liquefaction is also studied. Breaking wave‐induced pressures in some cases are found to cause liquefaction in the rubble and the seabed. The effect of some parameters on the instability is found to be significant. Copyright © 2011 John Wiley & Sons, Ltd.     

Numerical analysis on seepage failures of dike due to water level-up and rainfall using a water–soil-coupled smoothed particle hydrodynamics model

For seepage failures of dike due to water level-up and rainfall, surface infiltration and strength change induced by suction reduction are important factors; thus, numerical analysis should consider the coupling of water and soil, as well as the effect of saturation to obtain more precise failure mechanism. Based on the advanced smoothed particle hydrodynamics (SPH) method, this work proposed a two-phase-coupled SPH model in coordination with a novel constitutive model for unsaturated soils. Then, a triaxial compression test is simulated to check the applicability of the SPH method on the soil phase. After that, the failure test of a dike due to water level-up is discretized and simulated, from which the seepage process, the distribution of maximum shear strain, the slip surface, and pore water pressure are obtained. The two-phase-coupled SPH model is also applied to a slope failure test of heavy rainfall, and the results are compared to the model test. Finally, a dike failure test due to rainfall is analyzed using the proposed SPH model to reproduce the surface infiltration and suction reduction. The proposed SPH model provides several insights of seepage failures and can be a helpful tool for the analysis of dike failures induced by water level-up and rainfall.     

等应力增量比路径下接触面应变软化模型

等应力增量比路径单剪试验条件下,部分土体与结构接触面表现出明显的应变软化和剪胀特性。基于广义位势理论,将土与结构的接触面问题看作应力空间上的二维问题,势函数取法向应力和切向应力,用塑性状态方程取代传统的屈服面,建立等应力增量比路径条件下的土与结构接触面应力-应变软化模型。采用指数函数,拟合单向压缩试验中法向应力与法向应变的关系,采用复合指数函数,拟合应力比与切向应变的关系,采用另一复合指数函数,拟合法向剪胀分量与切向应变的关系。通过对拟合函数进行微分,确定模型中待定系数的求解方法。结合试验结果对模型进行验证,模型拟合效果良好,具有一定的合理性。