基于物质点方法饱和多孔介质动力学模拟
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摘要
在实际工程中,如边坡失稳相联的土壤面破坏、石油开采中油层破裂诱发的地质沉降以及地下结构与土体间的相互作用等,均大量涉及了多孔介质力学问题,关于多孔多相材料(如土壤)力学行为的研究已经成为近年来科学与工程界的热点研究问题,相关研究工作具有广阔的工程应用背景。饱和多孔介质作为一种典型的多孔多相介质,本文将其作为研究对象。由于饱和多孔介质由固体骨架与孔隙流体组成,相对单相介质其固—液耦合响应更为复杂。有限元方法的成功提出使得饱和多孔介质的数值模拟成为可能;但是有限元方法存在着许多弱点,如:在处理大变形等问题时出现网格畸变,计算精度随之下降等。为了弥补这些不足,随着计算力学研究工作的深入,一些新的计算方法被提出,如离散元方法、流形元方法以及无网格方法等。本文所研究的物质点方法(Material Point Method,MPM)是具有代表性的一种无网格方法。
本文发展了用于饱和多孔介质动力学模拟的新的物质点方法,并将其应用到饱和多孔介质同固体间动态接触分析以及饱和多孔介质应变局部化分析中。本文各章节内容安排如下:
第一章,首先阐述了课题的研究背景与工程意义;并简要回顾了土力学的发展历史,介绍了几种具有代表性的无网格方法;最后概述了本文的研究工作。
第二章介绍了饱和多孔介质的经典控制方程。在细观上,饱和多孔介质由固、液两相组成,具有非均匀的结构。在宏观数值分析中假定饱和多孔介质是均匀、连续的;并且假定各相同时充满饱和多孔介质全域。在这些假设的基础上,应用连续介质力学理论框架,可以建立饱和多孔介质耦合数学模型,其由一组偏微分方程组成:饱和多孔介质混合物动量守恒方程、饱和多孔介质中孔隙流体传输方程——Darcy方程与孔隙流体质量守恒方程。
第三章详细介绍了传统物质点方法。物质点方法作为一种无网格方法,已经在固体力学领域成功地解决了很多问题,特别是在处理接触/碰撞、穿透/穿孔等问题时具有先天的优势,其可在不调用任何主从关系的基础上,模拟时间相关问题。本章显式地给出传统物质点方法的空间离散方程,详细描述了其算法实施过程;并介绍了物质点方法的优势与弱点。
第四章,基于饱和多孔介质u-p形式控制方程,提出了饱和多孔介质耦合物质点方法(coupling MPM),并显式地给出了其空间离散方程与相应的算法实施过程。耦合物质点方法通过一套物质点代表饱和多孔介质混合体。在耦合物质点方法中,本文首次提出了基于物质点方法的孔隙流体Darcy方程求解办法;并详细阐述了压强场边界条件的处理方式,通过引入边界压强层近似描述指定压强边界。最后通过数值算例,表明了所发展的耦合物质点方法的正确与有效性。
第五章,应用饱和多孔介质u-U形式控制方程,提出了饱和多孔介质两相物质点方法(two-phase MPM),并详细阐述了其算法流程。两相物质点方法通过引入两套物质点分别描述固体骨架变形与孔隙流体流动;并且针对固相和液相位移分别求解。应用显式时间积分算法,通过数值算例验证了两相物质点方法的正确与有效性。
第六章,在物质点方法算法框架下,提出了饱和多孔介质与固体介质间动力接触问题的数值模拟方案,其中饱和多孔介质与固体介质的力学响应分别通过耦合物质点方法与传统物质点方法模拟,二者通过本章提出的接触算法相结合,在保证饱和多孔介质与固体间不存在相互穿透(特别是孔隙流体不能通过接触面流动)的前提下,允许二者间的相互滑动。通过数值算例验证了所提出的接触数值模拟方案的有效性。
第七章,饱和多孔介质作为多相多孔材料,其应变局部化问题不仅与固体骨架变形有关,而且受到孔隙流体的影响。为了考虑饱和多孔介质中的流—固耦合特性,本章在破坏本构描述中引入了孔隙压强的影响,提出了改进的strong decohesion模型,并发展了相应的返回映射算法。在两相物质点方法算法框架下,通过应力应变场分岔分析,改进的strong decohesion模型与弹塑性本构关系相连接,模拟了饱和多孔介质的应变局部化现象,并对多孔介质混合物的承载力分析获得了客观的、非网格依赖的计算结果。
第八章总结全文内容,并展望下一步的研究工作。
在附录A中,具体介绍了二维饱和多孔介质物质点方法动力学分析程序Geo-MPM。Geo-MPM具有一定的通用性,在同一程序框架下实现了传统单相物质点方法、两相物质点方法与耦合物质点方法;并且Geo-MPM能够完成单相介质与饱和多孔介质的弹塑性与应变局部化分析、单相介质间以及饱和多孔介质与固体间的动力接触分析等。
In the recent years,more and more attention has been paid on the research ofmultiphase porous media,like geomaterials,which play important role in landslide related to the failure of soil foundation,land subsidence in petroleum extraction and the interaction between soil and underground structure in the geotechnical engineering.Saturated porous medium,as a kind of soils,is considered particularly in the dissertation.Because saturated porous medium is composed of solid skeleton and pore fluid,its simulation is more difficult than that of single-phase materials.A variety of the Finite Element Methods(FEMs) have been developed for such problem in the past decades;but they may have some disadvantages in some special cases,like mesh distortion in large strain or strain localization.To overcome the disadvantages of the FEM,some new numerical techniques,like meshless method,have been developed.The Material Point Method(MPM) is the representative one of meshless methods, which is proposed to predict the time-dependent problems of single-phase material in the previous works.
The MPM is extended in the dissertation to predict the coupling responses of saturated porous media.The new MPMs are applied to simulate the strain localization of saturated porous media and the dynamic contact between saturated porous media and solid bodies.The chapters of the dissertation are organized as follows.
The first chapter introduces the background of the research,and reviews the development of geomechanics.Several meshless methods,like SPH and EFG,are also surveyed briefly.At the end of the chapter,the contents of the dissertation are outlined.
Chapter 2 provides the physical description of saturated porous media,which is modeled as the solid-fluid mixture.Saturated porous medium has heterogeneous structures in space in the mireo-scale.To model the heterogeneous two-phase material as a continuum in the macro-scale,a fundamental assumption of every phase filling the entire domain is made, which means that all the phases are superposed at any material point.Under the assumption, the mathematical model of saturated porous media is obtained,which consists of a set of coupled partial differential equations:the momentum balance of the whole system,the transportation equation of pore fluid,Darcy's equation,and the mass balance of pore fluid.
In Chapter 3,the original MPM is reviewed,which has been used to solve some history-dependent problems,like contact/impact and penetration/perforation,in the solid mechanics without invoking any master/slave relationship as required in the conventional FEM.The discrete formula and the numerical algorithm of the MPM are provided explicitly.
Based on the u-p form governing equations of saturated porous media,the coupling MPM is proposed to predict dynamic responses of saturated porous media in Chapter 4.The material point refers to the mixture of saturated porous media in the coupling MPM.In the coupling MPM,the solving scheme is proposed for the Darcy's equation for the first time, and the treatments on the boundary conditions in the pore pressure field are described.The prescribed pressure boundary is imposed approximately with the use of the pressure boundary layer.The validity of the approach developed here is demonstrated in the comparison with the results of the FEM.It can be found that the interaction between solid skeleton and pore fluid in saturated porous media is successfully simulated.
In chapter 5,the two-phase MPM is developed to simulate the solid-fluid coupling in saturated porous media with the use of the u-U form governing eqatuions of saturated porous media.In the two-phase MPM,two sets of material points are invoked to represent the deformation of solid skeleton and the pore fluid flow.In this chapter,the numerical implementation of the two-hase MPM is provided explicitly,and the vadility of the two-phase is shown in the numerical examples.
The strategy for the dynamic contact between saturated porous media and solid bodies is proposed in the frame work of the MPM in chapter 6,in which the responses of saturated porous media and solids are predicted by the coupling MPM and the original MPM respectively.The link between these two kinds of MPMs is the contact algorithm proposed in the current chapter,in which the slip between saturated porous media and solid bodies is allowed following Coulomb friction law,and the inter-penetration between saturated porous media and solid bodies is forbidden.The results of numerical calculations demonstrate the validity of the proposed strategy which shines a light of modeling the problems of soil-structure interaction in a new and effective way.
Stain localization is one of the most frequently found mechanisms leading to progressive failure of materials.To simulate the strain localization of saturated porous media,a modified strong decohesion model is developed in Chapter 7,in which the solid-fluid coupling in saturated porous media is considered with bringing the effect of pore pressure into the traction on the decohesion surface.In the frame scheme of the two-phase MPM,the modified strong decohesion model is coupled with the bifurcation analysis of the stress-strain field,whose stress state is updated with the corresponding return mapping algorithm developed here. Objective and mesh-independent results are obtained in the representative examples.
The main contributions of the dissertation are summarized and the further works are suggested in chapter 8.
Appendix A introduces the code of Geo-MPM which is developed to predict the dynamic responses of saturated porous media.In Geo-MPM,three kinds of MPMs,the original MPM,the two-phase MPM and the coupling MPM,are implemented.Geo-MPM can simulate the ealstoplastic behaviour and strain localization of both saturated porous media and solids,and contact among solid bodies and between saturated porous media and solid bodies.
本文发展了用于饱和多孔介质动力学模拟的新的物质点方法,并将其应用到饱和多孔介质同固体间动态接触分析以及饱和多孔介质应变局部化分析中。本文各章节内容安排如下:
第一章,首先阐述了课题的研究背景与工程意义;并简要回顾了土力学的发展历史,介绍了几种具有代表性的无网格方法;最后概述了本文的研究工作。
第二章介绍了饱和多孔介质的经典控制方程。在细观上,饱和多孔介质由固、液两相组成,具有非均匀的结构。在宏观数值分析中假定饱和多孔介质是均匀、连续的;并且假定各相同时充满饱和多孔介质全域。在这些假设的基础上,应用连续介质力学理论框架,可以建立饱和多孔介质耦合数学模型,其由一组偏微分方程组成:饱和多孔介质混合物动量守恒方程、饱和多孔介质中孔隙流体传输方程——Darcy方程与孔隙流体质量守恒方程。
第三章详细介绍了传统物质点方法。物质点方法作为一种无网格方法,已经在固体力学领域成功地解决了很多问题,特别是在处理接触/碰撞、穿透/穿孔等问题时具有先天的优势,其可在不调用任何主从关系的基础上,模拟时间相关问题。本章显式地给出传统物质点方法的空间离散方程,详细描述了其算法实施过程;并介绍了物质点方法的优势与弱点。
第四章,基于饱和多孔介质u-p形式控制方程,提出了饱和多孔介质耦合物质点方法(coupling MPM),并显式地给出了其空间离散方程与相应的算法实施过程。耦合物质点方法通过一套物质点代表饱和多孔介质混合体。在耦合物质点方法中,本文首次提出了基于物质点方法的孔隙流体Darcy方程求解办法;并详细阐述了压强场边界条件的处理方式,通过引入边界压强层近似描述指定压强边界。最后通过数值算例,表明了所发展的耦合物质点方法的正确与有效性。
第五章,应用饱和多孔介质u-U形式控制方程,提出了饱和多孔介质两相物质点方法(two-phase MPM),并详细阐述了其算法流程。两相物质点方法通过引入两套物质点分别描述固体骨架变形与孔隙流体流动;并且针对固相和液相位移分别求解。应用显式时间积分算法,通过数值算例验证了两相物质点方法的正确与有效性。
第六章,在物质点方法算法框架下,提出了饱和多孔介质与固体介质间动力接触问题的数值模拟方案,其中饱和多孔介质与固体介质的力学响应分别通过耦合物质点方法与传统物质点方法模拟,二者通过本章提出的接触算法相结合,在保证饱和多孔介质与固体间不存在相互穿透(特别是孔隙流体不能通过接触面流动)的前提下,允许二者间的相互滑动。通过数值算例验证了所提出的接触数值模拟方案的有效性。
第七章,饱和多孔介质作为多相多孔材料,其应变局部化问题不仅与固体骨架变形有关,而且受到孔隙流体的影响。为了考虑饱和多孔介质中的流—固耦合特性,本章在破坏本构描述中引入了孔隙压强的影响,提出了改进的strong decohesion模型,并发展了相应的返回映射算法。在两相物质点方法算法框架下,通过应力应变场分岔分析,改进的strong decohesion模型与弹塑性本构关系相连接,模拟了饱和多孔介质的应变局部化现象,并对多孔介质混合物的承载力分析获得了客观的、非网格依赖的计算结果。
第八章总结全文内容,并展望下一步的研究工作。
在附录A中,具体介绍了二维饱和多孔介质物质点方法动力学分析程序Geo-MPM。Geo-MPM具有一定的通用性,在同一程序框架下实现了传统单相物质点方法、两相物质点方法与耦合物质点方法;并且Geo-MPM能够完成单相介质与饱和多孔介质的弹塑性与应变局部化分析、单相介质间以及饱和多孔介质与固体间的动力接触分析等。
In the recent years,more and more attention has been paid on the research ofmultiphase porous media,like geomaterials,which play important role in landslide related to the failure of soil foundation,land subsidence in petroleum extraction and the interaction between soil and underground structure in the geotechnical engineering.Saturated porous medium,as a kind of soils,is considered particularly in the dissertation.Because saturated porous medium is composed of solid skeleton and pore fluid,its simulation is more difficult than that of single-phase materials.A variety of the Finite Element Methods(FEMs) have been developed for such problem in the past decades;but they may have some disadvantages in some special cases,like mesh distortion in large strain or strain localization.To overcome the disadvantages of the FEM,some new numerical techniques,like meshless method,have been developed.The Material Point Method(MPM) is the representative one of meshless methods, which is proposed to predict the time-dependent problems of single-phase material in the previous works.
The MPM is extended in the dissertation to predict the coupling responses of saturated porous media.The new MPMs are applied to simulate the strain localization of saturated porous media and the dynamic contact between saturated porous media and solid bodies.The chapters of the dissertation are organized as follows.
The first chapter introduces the background of the research,and reviews the development of geomechanics.Several meshless methods,like SPH and EFG,are also surveyed briefly.At the end of the chapter,the contents of the dissertation are outlined.
Chapter 2 provides the physical description of saturated porous media,which is modeled as the solid-fluid mixture.Saturated porous medium has heterogeneous structures in space in the mireo-scale.To model the heterogeneous two-phase material as a continuum in the macro-scale,a fundamental assumption of every phase filling the entire domain is made, which means that all the phases are superposed at any material point.Under the assumption, the mathematical model of saturated porous media is obtained,which consists of a set of coupled partial differential equations:the momentum balance of the whole system,the transportation equation of pore fluid,Darcy's equation,and the mass balance of pore fluid.
In Chapter 3,the original MPM is reviewed,which has been used to solve some history-dependent problems,like contact/impact and penetration/perforation,in the solid mechanics without invoking any master/slave relationship as required in the conventional FEM.The discrete formula and the numerical algorithm of the MPM are provided explicitly.
Based on the u-p form governing equations of saturated porous media,the coupling MPM is proposed to predict dynamic responses of saturated porous media in Chapter 4.The material point refers to the mixture of saturated porous media in the coupling MPM.In the coupling MPM,the solving scheme is proposed for the Darcy's equation for the first time, and the treatments on the boundary conditions in the pore pressure field are described.The prescribed pressure boundary is imposed approximately with the use of the pressure boundary layer.The validity of the approach developed here is demonstrated in the comparison with the results of the FEM.It can be found that the interaction between solid skeleton and pore fluid in saturated porous media is successfully simulated.
In chapter 5,the two-phase MPM is developed to simulate the solid-fluid coupling in saturated porous media with the use of the u-U form governing eqatuions of saturated porous media.In the two-phase MPM,two sets of material points are invoked to represent the deformation of solid skeleton and the pore fluid flow.In this chapter,the numerical implementation of the two-hase MPM is provided explicitly,and the vadility of the two-phase is shown in the numerical examples.
The strategy for the dynamic contact between saturated porous media and solid bodies is proposed in the frame work of the MPM in chapter 6,in which the responses of saturated porous media and solids are predicted by the coupling MPM and the original MPM respectively.The link between these two kinds of MPMs is the contact algorithm proposed in the current chapter,in which the slip between saturated porous media and solid bodies is allowed following Coulomb friction law,and the inter-penetration between saturated porous media and solid bodies is forbidden.The results of numerical calculations demonstrate the validity of the proposed strategy which shines a light of modeling the problems of soil-structure interaction in a new and effective way.
Stain localization is one of the most frequently found mechanisms leading to progressive failure of materials.To simulate the strain localization of saturated porous media,a modified strong decohesion model is developed in Chapter 7,in which the solid-fluid coupling in saturated porous media is considered with bringing the effect of pore pressure into the traction on the decohesion surface.In the frame scheme of the two-phase MPM,the modified strong decohesion model is coupled with the bifurcation analysis of the stress-strain field,whose stress state is updated with the corresponding return mapping algorithm developed here. Objective and mesh-independent results are obtained in the representative examples.
The main contributions of the dissertation are summarized and the further works are suggested in chapter 8.
Appendix A introduces the code of Geo-MPM which is developed to predict the dynamic responses of saturated porous media.In Geo-MPM,three kinds of MPMs,the original MPM,the two-phase MPM and the coupling MPM,are implemented.Geo-MPM can simulate the ealstoplastic behaviour and strain localization of both saturated porous media and solids,and contact among solid bodies and between saturated porous media and solid bodies.
引文
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