岩体结构非平衡演化稳定与控制理论基础研究
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摘要
天然岩石受不同尺度节理、裂隙切割，构成裂隙岩体复合介质。各种裂隙岩体进一步构成大规模岩体工程结构。我国岩石工程规模与复杂度迅速增长，岩体工程结构稳定性问题突出。本文采用动态演化观点研究岩体结构稳定与控制问题，提出相对完整的岩体结构非平衡演化研究体系，重点研究该体系数学与力学相关理论基础。主要工作与创新性成果如下：
(1)非平衡态热力学基础研究：证明齐次Onsager流必然全等次，从而证明基于齐次Onsager流的Rice正则结构等价于Ziegler最大耗散率原理；该结论表明岩土类材料本构关系热力学基础需要建立在有旋热力学流范围。
(2)方向分布函数与组构张量：提出最一般形式的方向分布函数——张量型方向分布函数，建立其相对完备的组构张量代数，包括非对称与全对称组构张量解析解、收敛性与拟合精度、不同阶组构张量关系等。
(3) Kachanov-Rabotnov (K-R)损伤有效应力：基于矢量型方向分布函数建立K-R损伤有效应力三维各向异性微平面模型，实现严格宏—细观几何等效原理；该模型澄清连续体损伤力学若干基本问题，包括常用各向异性与各向同性有效应力模型适用条件、损伤效应张量正定性与Voigt对称性等。
(4)广义Hamilton原理：建立内变量热力学耗散材料体系广义Hamilton原理，相比经典弹性体Hamilton原理，只需将结构弹性应变能替换为特定内能与非弹性内耗散功之和，适用于非保守外力与内力以及有限变形情况；广义Hamilton原理是以下Lyapunov稳定与控制理论的连续介质力学基础。
(5) Rice非弹性体Lyapunov稳定与控制理论：基于Lyapunov第二类稳定性方法，证明理想Rice非弹性体在恒定外部作用条件与拟静力条件下的最小流动势原理，结合材料损伤效应阐明结构非平衡演化的一般规律；在此基础上提出结构稳定评价与失稳控制的有效方法，同时阐明新奥法原理、潘家铮最大最小原理等岩土工程实用稳定性原理的理论依据。
(6)变形加固理论：基于Rice非弹性体Lyapunov稳定与控制理论证明变形加固理论，并依托变形加固理论这一特例将前者应用于工程结构稳定评价与加固设计。结合有限元算例初步研究结构整体稳定性、加固措施效果与效率、结构长期稳定性监测与预测等岩土工程重要问题。
Natural rocks cut by joints and fissures form rock masses, which further constructlarge-scale rock engineering structures. Due to the rapid growth of the scale andcomplexity of rock engineering in China, the stability of rock engineering structures ispotentially threatened. In this paper, the stability and control of rock mass structures arestudied from the perspective of non-equilibrium evolution. An integrated researchframework is proposed, whose mathematical and mechanical fundamentals are focusedin this paper. The major achievements are listed as below:
(1) Non-equilibrium thermodynamics: It is proved that the degrees of a set ofhomogeneous Onsager fluxes must be identical and thus the Rice's normalitystructure based on homogeneous Onsager fluxes is equivalent to Ziegler'sprinciple of maximum rate of dissipation. Such conclusion means that thethermodynamic basis for the constitutive relations of geomaterials must beestablished in the range of rotational thermodynamic fluxes.
(2) Orientation distribution function (ODF) and fabric tensor: The mostgeneralized type of ODF is proposed as the tensor-valued ODFs. Integrated fabrictensor algebra for tensor-valued ODFs is established, including the analytical solutionsfor asymmetric and symmetric fabric tensors of any orders, the convergence andaccuracy analysis for the fabric tensor expansion and the relationship between fabrictensors of different orders.
(3) The Kachanov-Rabotnov (K-R) damage effective stress: Amicroplane-based model for3D anisotropic K-R damage effective stress isproposed based on vector-valued ODFs, which fulfills rigorous geometryequivalence between the micro and macro scopes. The model clarifies some basic issuesin continuum damage mechanics such as the applicability of some frequently usedanisotropic and isotropic effective stress models and the positive definiteness and Voigtsymmetry of the damage effect tensor.
(4) Generalized Hamilton's principle: A generalized Hamilton's principlefor dissipative material bodies is established in thermodynamics with internalvariables. Compared with the classical Hamilton's principle for elastic bodies, the generalized form has the elastic strain energy replaced by the summation ofthe specific internal energy and the inelastic dissipation work. The generalizedHamilton's principle is the continuum mechanical basis for the followingLyapunov theory for stability and control.
(5) The Lyapunov theory for stability and control of Rice inelastic bodies:Based on the Lyapunov's second method for stability, the principle of minimumflow potential for perfect Rice inelastic bodies is proved, which is thenincorporated with damage effects to reveal the non-equilibrium evolution lawsof continuum structures. Effective methods for stability evaluation and controlare proposed based on the evolution laws. The theoretical bases are clarified forsome important practical principles in geotechnical engineering such as the NewAustrian Tunneling Method and the Pan's principles of soil and rock stability.
(6) The deformation reinforcement theory: Based on the Lyapunov theoryestablished above, the deformation reinforcement theory is proved as a specialcase, through which the Lyapunov theory is applied to stability evaluation andreinforcement design for engineering structures. Combined with numericalexamples using finite element method, some fatal issues in geotechnicalengineering is preliminarily studied, such as global structural stability, effectand efficiency of reinforcement measures, monitoring and forecasting forlong-term stability of engineering structures.
(1)非平衡态热力学基础研究：证明齐次Onsager流必然全等次，从而证明基于齐次Onsager流的Rice正则结构等价于Ziegler最大耗散率原理；该结论表明岩土类材料本构关系热力学基础需要建立在有旋热力学流范围。
(2)方向分布函数与组构张量：提出最一般形式的方向分布函数——张量型方向分布函数，建立其相对完备的组构张量代数，包括非对称与全对称组构张量解析解、收敛性与拟合精度、不同阶组构张量关系等。
(3) Kachanov-Rabotnov (K-R)损伤有效应力：基于矢量型方向分布函数建立K-R损伤有效应力三维各向异性微平面模型，实现严格宏—细观几何等效原理；该模型澄清连续体损伤力学若干基本问题，包括常用各向异性与各向同性有效应力模型适用条件、损伤效应张量正定性与Voigt对称性等。
(4)广义Hamilton原理：建立内变量热力学耗散材料体系广义Hamilton原理，相比经典弹性体Hamilton原理，只需将结构弹性应变能替换为特定内能与非弹性内耗散功之和，适用于非保守外力与内力以及有限变形情况；广义Hamilton原理是以下Lyapunov稳定与控制理论的连续介质力学基础。
(5) Rice非弹性体Lyapunov稳定与控制理论：基于Lyapunov第二类稳定性方法，证明理想Rice非弹性体在恒定外部作用条件与拟静力条件下的最小流动势原理，结合材料损伤效应阐明结构非平衡演化的一般规律；在此基础上提出结构稳定评价与失稳控制的有效方法，同时阐明新奥法原理、潘家铮最大最小原理等岩土工程实用稳定性原理的理论依据。
(6)变形加固理论：基于Rice非弹性体Lyapunov稳定与控制理论证明变形加固理论，并依托变形加固理论这一特例将前者应用于工程结构稳定评价与加固设计。结合有限元算例初步研究结构整体稳定性、加固措施效果与效率、结构长期稳定性监测与预测等岩土工程重要问题。
Natural rocks cut by joints and fissures form rock masses, which further constructlarge-scale rock engineering structures. Due to the rapid growth of the scale andcomplexity of rock engineering in China, the stability of rock engineering structures ispotentially threatened. In this paper, the stability and control of rock mass structures arestudied from the perspective of non-equilibrium evolution. An integrated researchframework is proposed, whose mathematical and mechanical fundamentals are focusedin this paper. The major achievements are listed as below:
(1) Non-equilibrium thermodynamics: It is proved that the degrees of a set ofhomogeneous Onsager fluxes must be identical and thus the Rice's normalitystructure based on homogeneous Onsager fluxes is equivalent to Ziegler'sprinciple of maximum rate of dissipation. Such conclusion means that thethermodynamic basis for the constitutive relations of geomaterials must beestablished in the range of rotational thermodynamic fluxes.
(2) Orientation distribution function (ODF) and fabric tensor: The mostgeneralized type of ODF is proposed as the tensor-valued ODFs. Integrated fabrictensor algebra for tensor-valued ODFs is established, including the analytical solutionsfor asymmetric and symmetric fabric tensors of any orders, the convergence andaccuracy analysis for the fabric tensor expansion and the relationship between fabrictensors of different orders.
(3) The Kachanov-Rabotnov (K-R) damage effective stress: Amicroplane-based model for3D anisotropic K-R damage effective stress isproposed based on vector-valued ODFs, which fulfills rigorous geometryequivalence between the micro and macro scopes. The model clarifies some basic issuesin continuum damage mechanics such as the applicability of some frequently usedanisotropic and isotropic effective stress models and the positive definiteness and Voigtsymmetry of the damage effect tensor.
(4) Generalized Hamilton's principle: A generalized Hamilton's principlefor dissipative material bodies is established in thermodynamics with internalvariables. Compared with the classical Hamilton's principle for elastic bodies, the generalized form has the elastic strain energy replaced by the summation ofthe specific internal energy and the inelastic dissipation work. The generalizedHamilton's principle is the continuum mechanical basis for the followingLyapunov theory for stability and control.
(5) The Lyapunov theory for stability and control of Rice inelastic bodies:Based on the Lyapunov's second method for stability, the principle of minimumflow potential for perfect Rice inelastic bodies is proved, which is thenincorporated with damage effects to reveal the non-equilibrium evolution lawsof continuum structures. Effective methods for stability evaluation and controlare proposed based on the evolution laws. The theoretical bases are clarified forsome important practical principles in geotechnical engineering such as the NewAustrian Tunneling Method and the Pan's principles of soil and rock stability.
(6) The deformation reinforcement theory: Based on the Lyapunov theoryestablished above, the deformation reinforcement theory is proved as a specialcase, through which the Lyapunov theory is applied to stability evaluation andreinforcement design for engineering structures. Combined with numericalexamples using finite element method, some fatal issues in geotechnicalengineering is preliminarily studied, such as global structural stability, effectand efficiency of reinforcement measures, monitoring and forecasting forlong-term stability of engineering structures.
引文
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