含液多孔介质中失稳现象理论研究及应变局部化的有限元—无网格耦合方法
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摘要
研究含液多孔介质的失稳现象,诸如基坑工程或隧洞的开挖过程中可能出现的地基沉降甚至坑壁垮塌;以及边坡或堤坝由地震或暴雨所引发的滑动破坏(通常称为滑坡现象或边坡失稳)具有很重要的工程和理论意义。本论文从理论分析和数值模拟两个方面致力于研究在静、动力荷载作用下含液(特别是饱和)多孔介质中驻波间断、颤振失稳和应变局部化等破坏现象和过程。
在多孔介质受到冲击或爆炸等高频模态占主导地位的脉动荷载作用时,人们需要研究介质中应力波的传播过程。工程中许多多孔材料为塑性应变软化材料。在应力波的传播过程中,介质内某处的受力状态将首先达到材料的极限承载能力,并伴随以在介质中局部狭窄区域内急剧发生非弹性应变为特征的应变局部化现象和承载能力的急剧下降。
本论文基于饱和与非饱和含液多孔介质的非线性动力.渗流耦合模型(广义Biot模型)。计及介质中流固两相的惯性耦合,具体考虑模拟介质压力相关弹塑性本构行为的非关联Drucker-Prager准则,忽略液相和固相颗粒的压缩性。详细分析了二维情况下含液饱和多孔介质在动力荷载作用下波传播过程的间断和失稳,导出了在波传播过程中产生驻波间断和动力颤振失稳的临界条件。驻波间断是由于应变软化导致材料失稳的结果,它并不一定意味着完全丧失波通过间断面在介质中继续传播的能力。颤振失稳则是因为模拟含液多孔介质固体骨架的非关联塑性本构行为所致,它可以先于驻波间断、即在塑性硬化阶段发生;但它仅可能在含液多孔介质中发生,对于固体材料即使为非关联塑性连续体也不可能发生颤振失稳现象。
波的逸散性意味着波的相速度随频率而变化。这一性质与正确模拟波在因应变软化引起的应变局部化区域中的传播密切相关。本论文基于上述耦合模型,对单轴应变一维情况讨论了非线性饱和—非饱和多孔介质中波传播过程的失稳现象和逸散性。分析了流固粘性耦合、流固惯性耦合、流固混合体的压缩性、孔隙饱和度及固体骨架材料在高应变速率下粘弹塑性本构行为等因素对失稳与逸散性的影响。由此所获得的结果和结论将为克服含液多孔介质在强动荷载作用下波传播过程数值模拟的困难提供理论基础和线索。
实验观察表明,在粘性土等多孔介质材料中因应变软化引起的在局部区域发生并急剧发展塑性变形的剪切带具有一定的宽度。此外,剪切带的萌发、发展直至最终形成是一个渐进破坏过程。为数值模拟和再现这一渐进破坏过程,本论文工作中作为正则化机制引入梯度塑性模型。对饱和多孔介质(也能作为退化情况用于固体材料)提出了一个归结为线性互补问题(Linear Complementary Problem)求解过程的梯度塑性连续体有限元—无网格耦合方法。
为模拟材料的弹塑性本构行为,对固体材料和饱和多孔介质分别采用von-Mises准则和非关联Drucker-Prager准则。利用在积分点上定义的离散塑性乘子值和采用基于移动最小二乘(Moving Least-Square)的无网格法插值近似假定塑性乘子场。而位移和压力场则利用定义在节点上的离散值采用有限元法插值近似。因而可充分发挥无网格法与有限元法的各自优势,而避免它们的各自缺点。
通过建立平衡方程的弱形式实现空间离散化,结合在积分点上逐点满足而不是积分意义下满足非局部本构方程和屈服条件,导出相应的线性互补问题标准型。并通过Lexico-Lemke算法求解。发展了一个基于向后欧拉返回映射积分方案和利用Newton-Raphson方法的全局迭代过程的一致性算法,使得空间离散的平衡方程和在每个积分点上的非局部本构方程和屈服准则在每次全局迭代中同时满足。值得强调指出,所提出方法在保证二阶收敛率的同时无需形成非局部一致性切线刚度矩阵;另外,对于非关联塑性模型,所导出的为线性互补问题求解的全局广义刚度阵仍保持对称。数值结果表明,所发展的模型和一致性算法能正确模拟由应变软化引起的以应变局部化为特征的渐进破坏过程。
The study of instability phenomena occurring in saturated porous media,such as the subsidence of soil foundation,the collapse of pit walls in the pit construction or the excavation process of tunnels,the landslide of soil slopes and dikes subjected to earthquake or rainstorm loads,is of great significance in sciences and engineering.The present dissertation devotes to the failure phenomena and processes in the saturated(particularly fully saturated) porous media subjected to static and dynamic loads,i.e.stationary discontinuity,flutter instability and strain localization.
The wave propagation problems have to be investigated while the high frequency modes of the loading pulse dominate the response of the porous media due to impact or explosive loading.A great number of engineering porous materials are classified as plastic strainsoftening materials.The stress and strain states at some local points where traveling waves through will first reach the limit load-carrying capability.Which follow to occur are strain localization characterized by intensely increasing inelastic deformation into narrow bands around the local points and a reduction of the load-carrying capability due to strain-softening.
The present work is carried out on the basis of the non-linear coupled hydro-dynamic model named after the generalized Biot model for saturated and unsaturated porous media. The inertial coupling between the solid skeleton and pore fluid is incorporated into the model to account for the response to the excitation with high frequency modes.In what follows the non-associated Drucker-Prager criterion is particularly considered to simulate the pressure dependent elasto-plastic constitutive behavior in the media.With no consideration of compressibility of solid grains and the pore fluid,the discontinuity and instability of the wave propagation in saturated poro-elastoplastic media are analyzed for the plane strain problems in detail.The critical conditions for the stationary discontinuity and flutter instability to occur in the wave propagation in poro-elastoplastic media are derived and formulated.It is found that stationary discontinuity can be regarded as a result of material instability due to strain softening and does not necessarily mean that the media will entirely lose the ability to wave propagation passing through the surface of discontinuity.Flutter instability stems from non-associated plasticity used to simulate the non-linear constitutive behavior of the solid skeleton of the porous medium and may occur prior to the stationary discontinuity,i.e.at the plastic strain hardening stage.This phenomenon can only occur for saturated porous medium. In the solids even when non-associated plasticity is considered,no flutter instability may occur.
The dispersivity of wave propagation implies that phase velocity of a single harmonic wave is a function of the angle frequency.This property of wave propagation is intrinsically related to a correct simulation of wave propagation in the zone where the strain localization due to strain softening occurs.Based on the coupled hydro-dynamic model mentioned above, the instability and dispersivity of wave propagation in inelastic saturated/unsaturated porous media in one dimensional problem are analyzed.The effects of the following factors on the instability and dispersivity are discussed.They are the viscous and inertial couplings between the solid and fluid phases,compressibility of the mixture composed of solid grains and pore fluid,degree of saturation,visco-plastic(rate dependent inelastic) constitutive behavior of the solid skeleton under high strain rate.The results and conclusions obtained by the present work will provide some bases and clues for overcoming the difficulties in numerical modeling of wave propagation in the media subjected to strong and shock loadings.
It has been experimentally observed in many engineering materials such as in the clay that a shear band of intense plastic deformation caused by strain softening possesses a finite width.In addition,the threshold and evolution of the shear band until its final formation is a progressive failure process.To numerically simulate and reproduce the progressive process the gradient plasticity model is introduced as a regularization mechanism in the present work. A coupled finite element and meshfree method attributed to a solution procedure of linear complementary problem(LCP) for gradient plasticity continuum for both saturated porous medium and the solid is presented.
The von-Mises criterion and non-associated Drucker-Prager criterion are respectively adopted to model elasto-plastic constitutive behaviors in the solid and the saturated porous medium.With the mesh-free(MF) method based on moving least-square approximation (MLS) procedure,the plastic multiplier field is assumed and approximately interpolated in terms of its discretized values defined at the integration points.Whereas the displacements and pore pressure fields are discretized in terms of their discretized values defined at the nodal points with finite element(FE) interpolation approximations.Hence,respective advantages of both FE and MF methods are exploited and their respective weak points are avoided.
The weak form of the equilibrium equation along with the non-local constitutive equation and the non-local yield criterion locally enforced at each integration point are combined to mathematically educe a normal form of LCP solved by means of Lexico-Lemke algorithm.A consistent algorithm based on backward-Euler return mapping integration scheme with a global iterative procedure based on the Newton-Raphson method is devised to simultaneously satisfy at each iteration the discretized momentum conservation equation as well as the non-local constitutive equation and non-local yield criterion at each of local integration points.It is remarked that there is no need to derive non-local consistent tangent elasto-plastic modulus matrix in the proposed method while the second convergence rate for the solution of the boundary problem of gradient plasticity continuum is still retained. Moreover,the global generalized stiffness matrix for the LCP solver derived by the proposed method remains symmetric even for the non-associated plasticity model.The numerical results demonstrate the validity of the proposed model and numerical method in the simulation of progressive failure process characterized with the strain localization problem due to strain softening.
在多孔介质受到冲击或爆炸等高频模态占主导地位的脉动荷载作用时,人们需要研究介质中应力波的传播过程。工程中许多多孔材料为塑性应变软化材料。在应力波的传播过程中,介质内某处的受力状态将首先达到材料的极限承载能力,并伴随以在介质中局部狭窄区域内急剧发生非弹性应变为特征的应变局部化现象和承载能力的急剧下降。
本论文基于饱和与非饱和含液多孔介质的非线性动力.渗流耦合模型(广义Biot模型)。计及介质中流固两相的惯性耦合,具体考虑模拟介质压力相关弹塑性本构行为的非关联Drucker-Prager准则,忽略液相和固相颗粒的压缩性。详细分析了二维情况下含液饱和多孔介质在动力荷载作用下波传播过程的间断和失稳,导出了在波传播过程中产生驻波间断和动力颤振失稳的临界条件。驻波间断是由于应变软化导致材料失稳的结果,它并不一定意味着完全丧失波通过间断面在介质中继续传播的能力。颤振失稳则是因为模拟含液多孔介质固体骨架的非关联塑性本构行为所致,它可以先于驻波间断、即在塑性硬化阶段发生;但它仅可能在含液多孔介质中发生,对于固体材料即使为非关联塑性连续体也不可能发生颤振失稳现象。
波的逸散性意味着波的相速度随频率而变化。这一性质与正确模拟波在因应变软化引起的应变局部化区域中的传播密切相关。本论文基于上述耦合模型,对单轴应变一维情况讨论了非线性饱和—非饱和多孔介质中波传播过程的失稳现象和逸散性。分析了流固粘性耦合、流固惯性耦合、流固混合体的压缩性、孔隙饱和度及固体骨架材料在高应变速率下粘弹塑性本构行为等因素对失稳与逸散性的影响。由此所获得的结果和结论将为克服含液多孔介质在强动荷载作用下波传播过程数值模拟的困难提供理论基础和线索。
实验观察表明,在粘性土等多孔介质材料中因应变软化引起的在局部区域发生并急剧发展塑性变形的剪切带具有一定的宽度。此外,剪切带的萌发、发展直至最终形成是一个渐进破坏过程。为数值模拟和再现这一渐进破坏过程,本论文工作中作为正则化机制引入梯度塑性模型。对饱和多孔介质(也能作为退化情况用于固体材料)提出了一个归结为线性互补问题(Linear Complementary Problem)求解过程的梯度塑性连续体有限元—无网格耦合方法。
为模拟材料的弹塑性本构行为,对固体材料和饱和多孔介质分别采用von-Mises准则和非关联Drucker-Prager准则。利用在积分点上定义的离散塑性乘子值和采用基于移动最小二乘(Moving Least-Square)的无网格法插值近似假定塑性乘子场。而位移和压力场则利用定义在节点上的离散值采用有限元法插值近似。因而可充分发挥无网格法与有限元法的各自优势,而避免它们的各自缺点。
通过建立平衡方程的弱形式实现空间离散化,结合在积分点上逐点满足而不是积分意义下满足非局部本构方程和屈服条件,导出相应的线性互补问题标准型。并通过Lexico-Lemke算法求解。发展了一个基于向后欧拉返回映射积分方案和利用Newton-Raphson方法的全局迭代过程的一致性算法,使得空间离散的平衡方程和在每个积分点上的非局部本构方程和屈服准则在每次全局迭代中同时满足。值得强调指出,所提出方法在保证二阶收敛率的同时无需形成非局部一致性切线刚度矩阵;另外,对于非关联塑性模型,所导出的为线性互补问题求解的全局广义刚度阵仍保持对称。数值结果表明,所发展的模型和一致性算法能正确模拟由应变软化引起的以应变局部化为特征的渐进破坏过程。
The study of instability phenomena occurring in saturated porous media,such as the subsidence of soil foundation,the collapse of pit walls in the pit construction or the excavation process of tunnels,the landslide of soil slopes and dikes subjected to earthquake or rainstorm loads,is of great significance in sciences and engineering.The present dissertation devotes to the failure phenomena and processes in the saturated(particularly fully saturated) porous media subjected to static and dynamic loads,i.e.stationary discontinuity,flutter instability and strain localization.
The wave propagation problems have to be investigated while the high frequency modes of the loading pulse dominate the response of the porous media due to impact or explosive loading.A great number of engineering porous materials are classified as plastic strainsoftening materials.The stress and strain states at some local points where traveling waves through will first reach the limit load-carrying capability.Which follow to occur are strain localization characterized by intensely increasing inelastic deformation into narrow bands around the local points and a reduction of the load-carrying capability due to strain-softening.
The present work is carried out on the basis of the non-linear coupled hydro-dynamic model named after the generalized Biot model for saturated and unsaturated porous media. The inertial coupling between the solid skeleton and pore fluid is incorporated into the model to account for the response to the excitation with high frequency modes.In what follows the non-associated Drucker-Prager criterion is particularly considered to simulate the pressure dependent elasto-plastic constitutive behavior in the media.With no consideration of compressibility of solid grains and the pore fluid,the discontinuity and instability of the wave propagation in saturated poro-elastoplastic media are analyzed for the plane strain problems in detail.The critical conditions for the stationary discontinuity and flutter instability to occur in the wave propagation in poro-elastoplastic media are derived and formulated.It is found that stationary discontinuity can be regarded as a result of material instability due to strain softening and does not necessarily mean that the media will entirely lose the ability to wave propagation passing through the surface of discontinuity.Flutter instability stems from non-associated plasticity used to simulate the non-linear constitutive behavior of the solid skeleton of the porous medium and may occur prior to the stationary discontinuity,i.e.at the plastic strain hardening stage.This phenomenon can only occur for saturated porous medium. In the solids even when non-associated plasticity is considered,no flutter instability may occur.
The dispersivity of wave propagation implies that phase velocity of a single harmonic wave is a function of the angle frequency.This property of wave propagation is intrinsically related to a correct simulation of wave propagation in the zone where the strain localization due to strain softening occurs.Based on the coupled hydro-dynamic model mentioned above, the instability and dispersivity of wave propagation in inelastic saturated/unsaturated porous media in one dimensional problem are analyzed.The effects of the following factors on the instability and dispersivity are discussed.They are the viscous and inertial couplings between the solid and fluid phases,compressibility of the mixture composed of solid grains and pore fluid,degree of saturation,visco-plastic(rate dependent inelastic) constitutive behavior of the solid skeleton under high strain rate.The results and conclusions obtained by the present work will provide some bases and clues for overcoming the difficulties in numerical modeling of wave propagation in the media subjected to strong and shock loadings.
It has been experimentally observed in many engineering materials such as in the clay that a shear band of intense plastic deformation caused by strain softening possesses a finite width.In addition,the threshold and evolution of the shear band until its final formation is a progressive failure process.To numerically simulate and reproduce the progressive process the gradient plasticity model is introduced as a regularization mechanism in the present work. A coupled finite element and meshfree method attributed to a solution procedure of linear complementary problem(LCP) for gradient plasticity continuum for both saturated porous medium and the solid is presented.
The von-Mises criterion and non-associated Drucker-Prager criterion are respectively adopted to model elasto-plastic constitutive behaviors in the solid and the saturated porous medium.With the mesh-free(MF) method based on moving least-square approximation (MLS) procedure,the plastic multiplier field is assumed and approximately interpolated in terms of its discretized values defined at the integration points.Whereas the displacements and pore pressure fields are discretized in terms of their discretized values defined at the nodal points with finite element(FE) interpolation approximations.Hence,respective advantages of both FE and MF methods are exploited and their respective weak points are avoided.
The weak form of the equilibrium equation along with the non-local constitutive equation and the non-local yield criterion locally enforced at each integration point are combined to mathematically educe a normal form of LCP solved by means of Lexico-Lemke algorithm.A consistent algorithm based on backward-Euler return mapping integration scheme with a global iterative procedure based on the Newton-Raphson method is devised to simultaneously satisfy at each iteration the discretized momentum conservation equation as well as the non-local constitutive equation and non-local yield criterion at each of local integration points.It is remarked that there is no need to derive non-local consistent tangent elasto-plastic modulus matrix in the proposed method while the second convergence rate for the solution of the boundary problem of gradient plasticity continuum is still retained. Moreover,the global generalized stiffness matrix for the LCP solver derived by the proposed method remains symmetric even for the non-associated plasticity model.The numerical results demonstrate the validity of the proposed model and numerical method in the simulation of progressive failure process characterized with the strain localization problem due to strain softening.
引文
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