基于参数变分原理的非均质Cosserat体弹塑性有限元分析
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摘要
基于经典连续介质力学理论的有限元法由于其高效性和灵活性，已在航空、航天、机械、土木与核工程等领域的问题求解中得到了广泛的应用。然而，对于如颗粒材料、多孔介质和金属基复合材料等某些特殊材料，如采用基于经典连续介质理论的有限元法进行分析，则会由于材料的非均匀性而遇到多方面的挑战。同样对于如应变局部化等一些特殊物理现象，采用传统的有限元法进行数值模拟时会遇到不同程度的问题，有时甚至不能得到正确的结果。因此有必要对经典的有限元法进行改进，发展适用于非均质材料计算与应变软化问题分析的新的计算方法。
为了更好地研究非均质材料中微观结构对其宏观整体力学性能的影响，Ghosh等提出了Voronoi单胞有限元法来描述非均质材料的弹塑性力学性能。通过在基体单元中引入一形状、尺寸和分布任意的夹杂相来模拟真实的非均质材料微观结构，由此简化了有限元网格剖分，减少了有限元网格数量，从而提高了有限元法计算效率。
为了模拟材料中应变局部化带以及尺度效应的存在，1909年法国学者Cosserat兄弟提出了Cosserat模型。除了具有一般的平动自由度外，Cosserat模型中还引入了一独立的旋转自由度。通过在本构关系中引入材料内尺度参数对问题实现了正则化，从而可以很好地模拟材料应变局部化问题，提供非网格依赖性的结果。
对于上述非线性问题求解通常采用的是增量迭代法。但传统的迭代方法在迭代过程中经常会遇到收敛速度慢或发散的问题，并且当结构刚度矩阵发生变化时需要修正刚度矩阵和对刚度矩阵重新求逆，这就使得计算时间变长，计算效率降低。为了克服这一缺陷，钟万勰等提出的参数变分原理和参数二次规划法将变分方法推广到一些经典变分原理不能适用的地方，同时避免了迭代过程。对于弹塑性分析，该方法避免了Drucker假定的限制，同时还可以应用到非关联塑性本构模型，非法向流动和应变软化问题求解中。
本文基于参数变分原理，构建了Cosserat体弹塑性有限元分析模型并用于求解应力集中和应变局部化问题；利用Voronoi有限元法提出了适用于非均质材料宏观弹塑性性能计算的参数二次规划算法；进一步将Cosserat模型应用到Voronoi有限元法中，提出了基于Cosserat理论的Voronoi杂交有限元法，并采用参数变分原理进行新模型的分析计算。各章节具体内容安排如下：
第一章综述了非均质材料等效力学性能研究概况。主要包括：与材料微结构无关的普适关系及其发展，非均质材料有效模量上下限的讨论，采用细观力学方法和有限元法进行非均质材料有效性质计算以及非均质微极介质等效方法等。同时还介绍了Cosserat连续体理论的产生、发展、相关的数值研究以及Cosserat模型在工程实际中的应用情况。
第二章介绍了参数变分原理及参数二次规划算法。首先阐述了经典变分原理的局限性，由此说明参数变分原理的基本思想，在此基础上论述了基于经典连续体理论的参数最小势能原理和参数最小余能原理。通过变换将系统弹塑性问题转化为线性互补问题，给出了弹塑性分析参数二次规划模型。
第三章在Cosserat连续体理论框架下构造了三个平面四边形等参单元，给出了对应于上述三个等参单元的分片试验和数值结果，并将单元应用到小孔应力集中问题上以验证Cosserat模型的特性。
第四章基于参数变分原理构建了适用于Cosserat体弹塑性计算的新算法，证明了基于Cosserat连续体理论的参数最小势能原理，建立了对应于Cosserat体弹塑性问题的参数二次规划模型，应用于平面应变软化问题的计算，得到了非网格依赖性的结果。
第五章提出了基于参数变分原理的Voronoi单元弹塑性新算法并应用于非均质材料等效力学性能预测计算中。给出了无夹杂和含夹杂Voronoi单元有限元列式推导，证明了相应的含夹杂Voronoi单元参数最小余能原理，建立了对应的参数二次规划求解模型，并以此为基础研究了非均质材料微结构夹杂对其整体力学性能的影响。
第六章采用Voronoi有限元法进行非均质Cosserat材料弹塑性分析，给出了基于Cosserat理论的参数余能原理，并构建了相应的参数二次规划模型。基于所提出的新方法，研究了非均质Cosserat材料微结构对其整体力学性能的影响。
结论部分对全文主要内容进行了总结，并对进一步的研究内容和工作进行了展望。
本文的研究工作是国家自然科学基金资助项目(50679013，10421202)、长江学者和创新团队发展计划、国家基础性发展规划项目(2005CB321704)资助计划的一部分。
Due to the high efficiency and flexibility, the classic continuum theory based finiteelement method (FEM) has been applied in a wide range of engineering areas such asaerospace, astronauics, mechanism, civil and nuclear engineerings. However, it will facemany challenges when the classical continuum theory based finite element method is used tosolve the mechanical problems of heterogenous materials such as granular materials, porousmedia and metal matrix composite materials, becasue of the strong heterogeneity of thesematerials. The same problem will exist in the numerical simulation of some special physicalphenomena such as strain localization in material deformation, even the incorrect results maybe obtained. Therefore, it is necessary to improve the classic FEM and develop some newFEMs for efficient computation of heterogeneous materials and strain localization problems.
In order to study the effects of micro structures on the macro equavilent mechanicsbehavior of heterogeneous materials, Ghosh and coworkers proposed a material basedVoronoi cell finite element method (VCFEM) for the numerical analysis of the elastic-plasticmechanics properties of heterogeneous materials. In this method, each cell which has asecond phase with arbitrariness in shape, size and distribution is treated as a finite element forthe simulation of the micro structure of actual materials. As a result, the finite element meshgeneration is simplified, the number of finite elements is reduced and the correspondingcomputational efficiency is increased.
In attempt to simulate the strain localization and size effect in materials, the Cosseratmodel was proposed by Cosserat brothers in 1909. The kinematics of a Cosserat model ischaracterized by an independant rotation degree of freedom, besides the displacement degreesof freedom in the element. By introducing the internal length scale in the constitutiveformulation, the regularization of the problem is realized. The strain localization problem canbe solved correctly by the Cosserat model and the results turn out to be mesh-independent.
Traditionally, the incremental iteration method is widely adopted in solution of thenon-linear problems. Nevertheless, it would result in the problems of low convergent speed ornon-convergence. When the stiffness matrix changes, modification and inversion of thestiffness matrix have to be done and this will prolong the computation time and lower theefficiency. To overcome the limitations of the classic methods, Zhong and coworkersproposed the parametric variational principle (PVP) and the parametric quadratic programming method (PQPM) in which the variational method is generalized to adapt to theproblems where the classical variational principle can not be used and the iteration procedurescan be avoided. For the elastic-plastic problem, this method can avoid the limitation of theDrucker hypothesis. It can also be applied to the non-associated plastic constitutive model,non-normal sliding and strain softening problems.
Based on the PVP, a new elastic-plastic FEM is developed in the area of the Cosserattheory for the solution of the stress concentration problem and the strain localization problem.A new PQPM based on the VCFEM is proposed to apply to the macro elastic-plastic analysisof heterogeneous materials. Moreover, this paper makes an effort in incorporating theCosserat theory in the VCFEM for the elastic-plastic analysis of heterogeneous materialsbased on the PVP. The chapters are divided as follows:
In Chapter 1, the author summarizes the study on the equivalent mechanics properties ofheterogeneous materials (including the development of the universal relationship which isirrelevant to the micro structure, the discussion of the limitations of the effective moduli, thecomputation of the effective properties of heterogeneous materials with the micromechanicsmethod and the FEM, and the equivalence of the heterogeneous micropolar materials) and thepresentation, the evolution, the corresponding numerical studies of the Cosserat model and itsapplications in engineering.
In Chapter 2, the PVP and PQPM are described. At first, the limitations of the classicvariational principle are discussed and the basic idea of the PVP is given. Then, theparametric minimum potential energy principle and the parametric minimum complementaryenergy principle are developed. By transferring the corresponding elastic-plastic problem tothe linear complementary problem, the parametric quadratic programming model isconstructed.
In Chapter 3, three quadrilateral plane isoparametric elements based on the linearCosserat theory are constructed. The patch tests for the verification of the elements aredescribed. The stress concentration problem around a circular hole is solved to validate thecharacteristics of the Cosserat model.
In Chapter 4, based on the PVP, a new algorithm is developed for the elastic-plasticanalysis of the Cosserat continuum. The parametric minimum potential energy principle andthe corresponding PQPM of the Cosserat continuum are verified and constructed, separately.Strain localization problems are computed numerically with the new method and meshindependent results can be obtained.
In Chapter 5, PVP based VCFEM is developed for the computation of the effectivemechanics properties of heterogeneous materials. The finite element formulations of theVoronoi elements with and without inclusion are deduced. The corresponding parametric minimum complementary energy principle of Voronoi element with inclusion is verified. ThePQPM for the Voronoi elements with and without inlusion are presented and the influence ofmicrostructure on the overall mechanics properties of heterogeneous materials is studied.
In Chapter 6, the elastic-plastic analysis of heterogeneous Cosserat materials is carriedout with the VCFEM. The parametric complementary energy principle of the Cosserat theoryis developed and the PQPM for the VCFEM are established. Based on the new method,influence of microscopic heterogeneities on the overall mechanical responses ofheterogeneous materials is studied.
Finally, the main contributions of the dissertation are concluded and some possiblefurther research work are suggested.
The research of the dissertation is supported by the National Natural Science Foundationof China (50679013, 10421202), the Program for Changjiang Scholars and InnovativeResearch Team in University of China (PCSIRT) and the National Key Basic ResearchSpecial Foundation of China (2005CB321704).
为了更好地研究非均质材料中微观结构对其宏观整体力学性能的影响，Ghosh等提出了Voronoi单胞有限元法来描述非均质材料的弹塑性力学性能。通过在基体单元中引入一形状、尺寸和分布任意的夹杂相来模拟真实的非均质材料微观结构，由此简化了有限元网格剖分，减少了有限元网格数量，从而提高了有限元法计算效率。
为了模拟材料中应变局部化带以及尺度效应的存在，1909年法国学者Cosserat兄弟提出了Cosserat模型。除了具有一般的平动自由度外，Cosserat模型中还引入了一独立的旋转自由度。通过在本构关系中引入材料内尺度参数对问题实现了正则化，从而可以很好地模拟材料应变局部化问题，提供非网格依赖性的结果。
对于上述非线性问题求解通常采用的是增量迭代法。但传统的迭代方法在迭代过程中经常会遇到收敛速度慢或发散的问题，并且当结构刚度矩阵发生变化时需要修正刚度矩阵和对刚度矩阵重新求逆，这就使得计算时间变长，计算效率降低。为了克服这一缺陷，钟万勰等提出的参数变分原理和参数二次规划法将变分方法推广到一些经典变分原理不能适用的地方，同时避免了迭代过程。对于弹塑性分析，该方法避免了Drucker假定的限制，同时还可以应用到非关联塑性本构模型，非法向流动和应变软化问题求解中。
本文基于参数变分原理，构建了Cosserat体弹塑性有限元分析模型并用于求解应力集中和应变局部化问题；利用Voronoi有限元法提出了适用于非均质材料宏观弹塑性性能计算的参数二次规划算法；进一步将Cosserat模型应用到Voronoi有限元法中，提出了基于Cosserat理论的Voronoi杂交有限元法，并采用参数变分原理进行新模型的分析计算。各章节具体内容安排如下：
第一章综述了非均质材料等效力学性能研究概况。主要包括：与材料微结构无关的普适关系及其发展，非均质材料有效模量上下限的讨论，采用细观力学方法和有限元法进行非均质材料有效性质计算以及非均质微极介质等效方法等。同时还介绍了Cosserat连续体理论的产生、发展、相关的数值研究以及Cosserat模型在工程实际中的应用情况。
第二章介绍了参数变分原理及参数二次规划算法。首先阐述了经典变分原理的局限性，由此说明参数变分原理的基本思想，在此基础上论述了基于经典连续体理论的参数最小势能原理和参数最小余能原理。通过变换将系统弹塑性问题转化为线性互补问题，给出了弹塑性分析参数二次规划模型。
第三章在Cosserat连续体理论框架下构造了三个平面四边形等参单元，给出了对应于上述三个等参单元的分片试验和数值结果，并将单元应用到小孔应力集中问题上以验证Cosserat模型的特性。
第四章基于参数变分原理构建了适用于Cosserat体弹塑性计算的新算法，证明了基于Cosserat连续体理论的参数最小势能原理，建立了对应于Cosserat体弹塑性问题的参数二次规划模型，应用于平面应变软化问题的计算，得到了非网格依赖性的结果。
第五章提出了基于参数变分原理的Voronoi单元弹塑性新算法并应用于非均质材料等效力学性能预测计算中。给出了无夹杂和含夹杂Voronoi单元有限元列式推导，证明了相应的含夹杂Voronoi单元参数最小余能原理，建立了对应的参数二次规划求解模型，并以此为基础研究了非均质材料微结构夹杂对其整体力学性能的影响。
第六章采用Voronoi有限元法进行非均质Cosserat材料弹塑性分析，给出了基于Cosserat理论的参数余能原理，并构建了相应的参数二次规划模型。基于所提出的新方法，研究了非均质Cosserat材料微结构对其整体力学性能的影响。
结论部分对全文主要内容进行了总结，并对进一步的研究内容和工作进行了展望。
本文的研究工作是国家自然科学基金资助项目(50679013，10421202)、长江学者和创新团队发展计划、国家基础性发展规划项目(2005CB321704)资助计划的一部分。
Due to the high efficiency and flexibility, the classic continuum theory based finiteelement method (FEM) has been applied in a wide range of engineering areas such asaerospace, astronauics, mechanism, civil and nuclear engineerings. However, it will facemany challenges when the classical continuum theory based finite element method is used tosolve the mechanical problems of heterogenous materials such as granular materials, porousmedia and metal matrix composite materials, becasue of the strong heterogeneity of thesematerials. The same problem will exist in the numerical simulation of some special physicalphenomena such as strain localization in material deformation, even the incorrect results maybe obtained. Therefore, it is necessary to improve the classic FEM and develop some newFEMs for efficient computation of heterogeneous materials and strain localization problems.
In order to study the effects of micro structures on the macro equavilent mechanicsbehavior of heterogeneous materials, Ghosh and coworkers proposed a material basedVoronoi cell finite element method (VCFEM) for the numerical analysis of the elastic-plasticmechanics properties of heterogeneous materials. In this method, each cell which has asecond phase with arbitrariness in shape, size and distribution is treated as a finite element forthe simulation of the micro structure of actual materials. As a result, the finite element meshgeneration is simplified, the number of finite elements is reduced and the correspondingcomputational efficiency is increased.
In attempt to simulate the strain localization and size effect in materials, the Cosseratmodel was proposed by Cosserat brothers in 1909. The kinematics of a Cosserat model ischaracterized by an independant rotation degree of freedom, besides the displacement degreesof freedom in the element. By introducing the internal length scale in the constitutiveformulation, the regularization of the problem is realized. The strain localization problem canbe solved correctly by the Cosserat model and the results turn out to be mesh-independent.
Traditionally, the incremental iteration method is widely adopted in solution of thenon-linear problems. Nevertheless, it would result in the problems of low convergent speed ornon-convergence. When the stiffness matrix changes, modification and inversion of thestiffness matrix have to be done and this will prolong the computation time and lower theefficiency. To overcome the limitations of the classic methods, Zhong and coworkersproposed the parametric variational principle (PVP) and the parametric quadratic programming method (PQPM) in which the variational method is generalized to adapt to theproblems where the classical variational principle can not be used and the iteration procedurescan be avoided. For the elastic-plastic problem, this method can avoid the limitation of theDrucker hypothesis. It can also be applied to the non-associated plastic constitutive model,non-normal sliding and strain softening problems.
Based on the PVP, a new elastic-plastic FEM is developed in the area of the Cosserattheory for the solution of the stress concentration problem and the strain localization problem.A new PQPM based on the VCFEM is proposed to apply to the macro elastic-plastic analysisof heterogeneous materials. Moreover, this paper makes an effort in incorporating theCosserat theory in the VCFEM for the elastic-plastic analysis of heterogeneous materialsbased on the PVP. The chapters are divided as follows:
In Chapter 1, the author summarizes the study on the equivalent mechanics properties ofheterogeneous materials (including the development of the universal relationship which isirrelevant to the micro structure, the discussion of the limitations of the effective moduli, thecomputation of the effective properties of heterogeneous materials with the micromechanicsmethod and the FEM, and the equivalence of the heterogeneous micropolar materials) and thepresentation, the evolution, the corresponding numerical studies of the Cosserat model and itsapplications in engineering.
In Chapter 2, the PVP and PQPM are described. At first, the limitations of the classicvariational principle are discussed and the basic idea of the PVP is given. Then, theparametric minimum potential energy principle and the parametric minimum complementaryenergy principle are developed. By transferring the corresponding elastic-plastic problem tothe linear complementary problem, the parametric quadratic programming model isconstructed.
In Chapter 3, three quadrilateral plane isoparametric elements based on the linearCosserat theory are constructed. The patch tests for the verification of the elements aredescribed. The stress concentration problem around a circular hole is solved to validate thecharacteristics of the Cosserat model.
In Chapter 4, based on the PVP, a new algorithm is developed for the elastic-plasticanalysis of the Cosserat continuum. The parametric minimum potential energy principle andthe corresponding PQPM of the Cosserat continuum are verified and constructed, separately.Strain localization problems are computed numerically with the new method and meshindependent results can be obtained.
In Chapter 5, PVP based VCFEM is developed for the computation of the effectivemechanics properties of heterogeneous materials. The finite element formulations of theVoronoi elements with and without inclusion are deduced. The corresponding parametric minimum complementary energy principle of Voronoi element with inclusion is verified. ThePQPM for the Voronoi elements with and without inlusion are presented and the influence ofmicrostructure on the overall mechanics properties of heterogeneous materials is studied.
In Chapter 6, the elastic-plastic analysis of heterogeneous Cosserat materials is carriedout with the VCFEM. The parametric complementary energy principle of the Cosserat theoryis developed and the PQPM for the VCFEM are established. Based on the new method,influence of microscopic heterogeneities on the overall mechanical responses ofheterogeneous materials is studied.
Finally, the main contributions of the dissertation are concluded and some possiblefurther research work are suggested.
The research of the dissertation is supported by the National Natural Science Foundationof China (50679013, 10421202), the Program for Changjiang Scholars and InnovativeResearch Team in University of China (PCSIRT) and the National Key Basic ResearchSpecial Foundation of China (2005CB321704).
引文
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