基于Cosserat连续体模型的应变局部化有限元模拟
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摘要
当在经典连续体计算模型中引入应变软化本构行为时，模型的初边值问题在数学上将成为不适定，并导致病态的有限元网格依赖解。为正确地数值模拟由应变软化引起以在局部狭窄区域急剧发生和发展的非弹性应变为特征的应变局部化现象，必须在经典连续体中引入某种类型的正则化机制以保持应变局部化问题的适定性。
采用引入了高阶连续结构的Cosserat连续体理论是引入正则化机制的主要途径之一。在Cosserat连续体中引入了旋转自由度和相应产生的微曲率，引入了与微曲率能量共轭的对偶应力、以及作为正则化机制在本构方程中具有“特征长度”意义的内尺度参数。本论文基于Cosserat连续体理论，在应变局部化有限元数值模拟与工程应用方面作了如下的工作：
提出了一个压力相关弹塑性Cosserat连续体模型。具体考虑非关联Drucker-Prager屈服准则。分离应力和应变速率向量的偏量和球量部分，在Cosserat连续体理论的框架下，推导出基于压力相关弹塑性Cosserat连续体本构模型的一致性算法：率本构方程积分的返回映射算法和压力相关弹塑性切线本构模量矩阵的闭合型显式表示。避免了计算切线本构模量矩阵时的矩阵求逆，保证了数值求解过程的收敛性与计算效率。利用所发展的模型和有限元法，数值模拟由应变软化引起的平面应变问题中的应变局部化现象。数值例题结果表明，所发展模型对保持应变局部化边值问题适定性、再现应变局部化问题特征：塑性应变局限于局部处急剧发生和发展以及随塑性变形发展的整体承载能力下降的有效性。
对Cosserat连续体模型中的本构参数Cosserat剪模、软化模量及内部长度参数对应变局部化数值模拟结果的影响进行了研究。指出在一定的取值范围内，Cosserat剪模对数值模拟结果几乎没有影响，并给出了具体数值计算时的取值范围；软化模量绝对值越大，后破坏段的荷载-位移曲线越陡，计算得到的剪切带宽度越窄；内部长度参数越大，后破坏段的荷载-位移曲线越平缓，计算得到的剪切带越宽。数值结果表明，u8ω8与u8ω4插值单元均具有模拟应变局部化的良好性能，但后者具有更好一点的模拟后破坏过程的能力。
由所发展的压力相关弹塑性Cosserat连续体模型，对一些典型的岩土工程如边坡、开挖、挡土墙及地基等结构中以应变局部化为特征的渐进破坏现象进行了有限元数值模拟。数值结果表明，对于岩土工程中广泛存在的非关联塑性流动或应变软化非弹性材料，Cosserat连续体模型具有保持应变局部化边值问题适定性、再现应变局部化问题特征的能力。
建立了适用于饱和多孔介质中应变局部化分析及动力渗流耦合分析的Biot-Cosserat连续体模型。基于饱和多孔介质动力渗流耦合分析的Biot理论，将固体骨架看作Cosserat连续体，并考虑与微极转角自由度相应的惯性，建立了饱和多孔介质动力渗流耦合分析的Biot-Cosserat连续体模型。基于Galerkin加权余量法，对所发展的模型推导了以固体骨架广义位移(包含旋转)及孔隙水压力为基本未知量的有限元公式。利用所发展的数值模型，对包含压力相关弹塑性固体骨架材料的饱和多孔介质进行了动力渗流耦合分析与应变局部化有限元模拟。结果表明，所发展的两相饱和多孔介质动力渗流耦合分析的Biot-Cosserat连续体模型能保持饱和两相介质应变局部化问题的适定性及模拟饱和多孔介质中由应变软化引起的应变局部化现象的有效性。
发展了基于Cosserat连续体的非光滑多重屈服面的CAP模型，考虑Cosserat连续体理论的特点，发展了非线性本构速率方程积分的一致性算法，且率本构方程积分的返回映射算法和一致性弹塑性切线模量矩阵均为显式表示，避免了计算切线本构模量矩阵时的矩阵求逆。利用所发展的模型和一致性算法，数值模拟了平面应变条件下的边坡稳定和隧道开挖问题，结果表明，所发展的模型和一致性算法能够保持应变局部化问题的适定性，保证数值求解过程的收敛性和计算效率。说明将Cosserat连续体理论推广应用于各种岩土体材料模型是可能的。
基于Cosserat连续体模型的有限元过程，对两个分别由填筑与开挖引起破坏的工程实例进行了数值模拟。填筑问题中的土体材料为服从非关联塑性流动的理想弹塑性材料，而开挖问题中的土体材料为应变软化弹塑性材料。数值结果表明，与基于经典连续体的有限元分析过程相比，所发展的基于Cosserat连续体模型的有限元过程具备保持应变局部化问题的适定性、模拟整个渐进破坏过程的能力。
As strain softening constitutive behavior is incorporated into a computational model inthe frame of classical plastic continuum theories, the initial and boundary value problem ofthe model will become ill-posed, resulting in pathologically mesh-dependent solutions.Furthermore, the energy dissipated at strain softening is incorrectly predicted to be zero, andthe finite element solutions converge to incorrect, physically meaningless ones as the elementmesh is refined. To correctly simulate strain localization phenomena characterized byoccurrence and severe development of the deformation localized into narrow bands of intenseirreversible straining caused by strain softening, it is required to introduce some type ofregularization mechanism into the classical continuum model to preserve the well-posednessof the localization problem.
One of the radical approaches to introduce the regularization mechanism into the modelis to utilize the Cosserat micro-polar continuum theory, in which high-order continuumstructures are introduced. As two dimensional problems are concerned, a rotational degree offreedom with the rotation axis orthogonal to the 2D plane, micro-curvatures as spatialderivatives of the rotational degree of freedom, coupled stresses energetically conjugate to themicro-curvatures and the material parameter defined as internal length scale are introduced inthe Cosserat continuum. In this paper, the Cosserat micro-polar continuum theory isintroduced into the FEM numerical model, which is used to simulate the strain localizationphenomena occuring in solid and saturated porous media.
First, a pressure-dependent elastoplastic Cosserat continuum model is presented. Thenon-associated Drucker-Prager yield criterion is particularly considered. Splitting the scalarproduct of the stress rate and the strain rate into their deviatoric and spherical parts, theconsistent algorithm of the pressure-dependent elastoplastic model is derived in theframework of Cosserat continuum theory, i.e. the return mapping algorithm for the integrationof the rate constitutive equation and the closed form of the consistent elastoplastic tangentmodulus matrix. The matrix inverse operation usually required in the calculation ofelastoplastic tangent constitutive modulus matrix is avoided, that ensures the second orderconvergence rate and the computational efficiency of the model in numerical solutionprocedure. The strain localization phenomena due to strain softening are numericallysimulated using the developed model with corresponding finite element method. Numerical results of the plane strain examples illustrate the capability and performance of the presentmodel in keeping the well-posedness of the boundary value problems with strain softeningbehavior incorporated and in reproducing the characteristics of strain localization problems, i.e. intense plastic straining development localized into the narrow band and a significantreduction of the load-carrying capacity of the structure in consideration.
Next, the effects of constitutive parameters, such as Cosserat shear module, the softeningmodule and the internal length scale for modelling the softening behaviour and adhering tothe Cosserat continuum, on the numerical results of the simulation of the strain localization, are studied. It is pointed out that the value of Cosserat shear module chosen within certainrange has no effect on the results; the greater the absolute value of the soften modules is, thesteeper the post-failure curve of load-displacement is and the narrower the width of shearband is; the greater the value of the internal scale is, the flatter the post-failure curve ofload-displacement is and the wider the width of shear band is. The numerical study in theperformance of two types of Cosserat continuum finite elements, i.e. u8ω8(8-noded elementinterpolation approximations for both displacements u and microrotationω) and u8ω4(8-noded and 4-noded element interpolation approximations for u andωrespectively)elements is carried out. It is indicated that as compared with the u8ω8 element mesh, theu8ω4 element mesh possesses better performance in simulation of post-failure process. Bothtwo possess better performance in simulation of strain localization.
Then, based on the pressure-dependent elastoplastic Cosserat continuum model, progressive failure phenomena, which occured in the typical geotechnical engineeringproblems, such as the slope, excavation, retaining wall and soil foundation, characterized bystrain localization due to the material dilatancy, i.e. non-associated plasticity, or strainsoftening, are numerically simulated. Numerical results illustrate that as compared with theperformance of the finite element procedure for the classical continuum model, the finiteelement procedure based on Cosserat continuum model is capable of preserving thewell-posedness of the localization problems widely existing in geoteclanical engineering andsimulating the entire progressive failure phenomena characterized by strain localization due tostrain softening or the non-associated yield criterion adopted.
Besides, the Biot-Cosserat continuum model for coupled hydro-dynamic processes insaturated porous media is proposed by means of the combination of both Biot theory andCosserat continuum theory to simulate the strain localization phenomena due to the strainsoftening. In the present contribution, the Biot formulation of the skeleton material isextended to include the microrotation and correspondingly the coupled stresses defined in theCosserat model. The finite element formulations governing the coupled hydro-dynamicbehavior with the primary variables of the displacements and the microrotaion for the solid phase and the pressure for the fluid phase are derived on the basis of the Galerkin-weightedresidual method. The strain localization phenomena in saturated porous media due to thestrain softening are numerically simulated by using the developed model with correspondingfinite element method and the non-associated Drucker-Prager yield criterion particularlyconsidered for the pressure-dependent elasto-plastic Cosserat skeleton. Numerical results ofthe plane strain examples illustrate that the capability of the developed model in keeping thewell-posedness of the boundary value problems with strain softening behavior incorporated, and the availability of modeling the strain localization phenomena due to the strain softeningin saturated media.
In addition, an elastoplastic Cosserat continuum model for CAP constitutive model withnon-smooth multiplicative yield surfaces is presented. The consistent algorithm of the CAPelastoplastic model is derived in the framework of Cosserat continuum theory, i.e. the returnmapping algorithm for the integration of the rate constitutive equation and the closed form ofthe consistent elastoplastic tangent modulus matrix. The matrix inverse operation usuallyrequired in the calculation of elastoplastic tangent constitutive modulus matrix is avoided.The strain localization phenomena of the slope due to strain softening and the failure of thetunnel due to the excavation are numerically simulated using the developed model withcorresponding finite element method. Numerical results of the plane strain examples illustratethe capability and performance of the present model in keeping the well-posedness of theboundary value problems and ensuring the second order convergence rate and thecomputational efficiency of the model in numerical solution procedure. It also illustrates thepossibility of application of Cosserat continuum theory to various geotechnical constitutivemodels.
Finally, the two engineering examples, i.e. the failure of the two geotechnical structurescaused in the filling and the excavation processes respectively, are performed by using theproposed models and the developed algorithms. The non-associated perfect elastoplasticbehavior is particularly considered for the filling material and the strain softening behavior forthe excavation material. Numerical results indicate the capability and performance of Cosseratcontinuum model in keeping the well-posedness of the boundary value problems with strainsoftening behavior or non-associated perfect elastoplastic behavior incorporated and incompleting simulation of the whole failure progress.
采用引入了高阶连续结构的Cosserat连续体理论是引入正则化机制的主要途径之一。在Cosserat连续体中引入了旋转自由度和相应产生的微曲率，引入了与微曲率能量共轭的对偶应力、以及作为正则化机制在本构方程中具有“特征长度”意义的内尺度参数。本论文基于Cosserat连续体理论，在应变局部化有限元数值模拟与工程应用方面作了如下的工作：
提出了一个压力相关弹塑性Cosserat连续体模型。具体考虑非关联Drucker-Prager屈服准则。分离应力和应变速率向量的偏量和球量部分，在Cosserat连续体理论的框架下，推导出基于压力相关弹塑性Cosserat连续体本构模型的一致性算法：率本构方程积分的返回映射算法和压力相关弹塑性切线本构模量矩阵的闭合型显式表示。避免了计算切线本构模量矩阵时的矩阵求逆，保证了数值求解过程的收敛性与计算效率。利用所发展的模型和有限元法，数值模拟由应变软化引起的平面应变问题中的应变局部化现象。数值例题结果表明，所发展模型对保持应变局部化边值问题适定性、再现应变局部化问题特征：塑性应变局限于局部处急剧发生和发展以及随塑性变形发展的整体承载能力下降的有效性。
对Cosserat连续体模型中的本构参数Cosserat剪模、软化模量及内部长度参数对应变局部化数值模拟结果的影响进行了研究。指出在一定的取值范围内，Cosserat剪模对数值模拟结果几乎没有影响，并给出了具体数值计算时的取值范围；软化模量绝对值越大，后破坏段的荷载-位移曲线越陡，计算得到的剪切带宽度越窄；内部长度参数越大，后破坏段的荷载-位移曲线越平缓，计算得到的剪切带越宽。数值结果表明，u8ω8与u8ω4插值单元均具有模拟应变局部化的良好性能，但后者具有更好一点的模拟后破坏过程的能力。
由所发展的压力相关弹塑性Cosserat连续体模型，对一些典型的岩土工程如边坡、开挖、挡土墙及地基等结构中以应变局部化为特征的渐进破坏现象进行了有限元数值模拟。数值结果表明，对于岩土工程中广泛存在的非关联塑性流动或应变软化非弹性材料，Cosserat连续体模型具有保持应变局部化边值问题适定性、再现应变局部化问题特征的能力。
建立了适用于饱和多孔介质中应变局部化分析及动力渗流耦合分析的Biot-Cosserat连续体模型。基于饱和多孔介质动力渗流耦合分析的Biot理论，将固体骨架看作Cosserat连续体，并考虑与微极转角自由度相应的惯性，建立了饱和多孔介质动力渗流耦合分析的Biot-Cosserat连续体模型。基于Galerkin加权余量法，对所发展的模型推导了以固体骨架广义位移(包含旋转)及孔隙水压力为基本未知量的有限元公式。利用所发展的数值模型，对包含压力相关弹塑性固体骨架材料的饱和多孔介质进行了动力渗流耦合分析与应变局部化有限元模拟。结果表明，所发展的两相饱和多孔介质动力渗流耦合分析的Biot-Cosserat连续体模型能保持饱和两相介质应变局部化问题的适定性及模拟饱和多孔介质中由应变软化引起的应变局部化现象的有效性。
发展了基于Cosserat连续体的非光滑多重屈服面的CAP模型，考虑Cosserat连续体理论的特点，发展了非线性本构速率方程积分的一致性算法，且率本构方程积分的返回映射算法和一致性弹塑性切线模量矩阵均为显式表示，避免了计算切线本构模量矩阵时的矩阵求逆。利用所发展的模型和一致性算法，数值模拟了平面应变条件下的边坡稳定和隧道开挖问题，结果表明，所发展的模型和一致性算法能够保持应变局部化问题的适定性，保证数值求解过程的收敛性和计算效率。说明将Cosserat连续体理论推广应用于各种岩土体材料模型是可能的。
基于Cosserat连续体模型的有限元过程，对两个分别由填筑与开挖引起破坏的工程实例进行了数值模拟。填筑问题中的土体材料为服从非关联塑性流动的理想弹塑性材料，而开挖问题中的土体材料为应变软化弹塑性材料。数值结果表明，与基于经典连续体的有限元分析过程相比，所发展的基于Cosserat连续体模型的有限元过程具备保持应变局部化问题的适定性、模拟整个渐进破坏过程的能力。
As strain softening constitutive behavior is incorporated into a computational model inthe frame of classical plastic continuum theories, the initial and boundary value problem ofthe model will become ill-posed, resulting in pathologically mesh-dependent solutions.Furthermore, the energy dissipated at strain softening is incorrectly predicted to be zero, andthe finite element solutions converge to incorrect, physically meaningless ones as the elementmesh is refined. To correctly simulate strain localization phenomena characterized byoccurrence and severe development of the deformation localized into narrow bands of intenseirreversible straining caused by strain softening, it is required to introduce some type ofregularization mechanism into the classical continuum model to preserve the well-posednessof the localization problem.
One of the radical approaches to introduce the regularization mechanism into the modelis to utilize the Cosserat micro-polar continuum theory, in which high-order continuumstructures are introduced. As two dimensional problems are concerned, a rotational degree offreedom with the rotation axis orthogonal to the 2D plane, micro-curvatures as spatialderivatives of the rotational degree of freedom, coupled stresses energetically conjugate to themicro-curvatures and the material parameter defined as internal length scale are introduced inthe Cosserat continuum. In this paper, the Cosserat micro-polar continuum theory isintroduced into the FEM numerical model, which is used to simulate the strain localizationphenomena occuring in solid and saturated porous media.
First, a pressure-dependent elastoplastic Cosserat continuum model is presented. Thenon-associated Drucker-Prager yield criterion is particularly considered. Splitting the scalarproduct of the stress rate and the strain rate into their deviatoric and spherical parts, theconsistent algorithm of the pressure-dependent elastoplastic model is derived in theframework of Cosserat continuum theory, i.e. the return mapping algorithm for the integrationof the rate constitutive equation and the closed form of the consistent elastoplastic tangentmodulus matrix. The matrix inverse operation usually required in the calculation ofelastoplastic tangent constitutive modulus matrix is avoided, that ensures the second orderconvergence rate and the computational efficiency of the model in numerical solutionprocedure. The strain localization phenomena due to strain softening are numericallysimulated using the developed model with corresponding finite element method. Numerical results of the plane strain examples illustrate the capability and performance of the presentmodel in keeping the well-posedness of the boundary value problems with strain softeningbehavior incorporated and in reproducing the characteristics of strain localization problems, i.e. intense plastic straining development localized into the narrow band and a significantreduction of the load-carrying capacity of the structure in consideration.
Next, the effects of constitutive parameters, such as Cosserat shear module, the softeningmodule and the internal length scale for modelling the softening behaviour and adhering tothe Cosserat continuum, on the numerical results of the simulation of the strain localization, are studied. It is pointed out that the value of Cosserat shear module chosen within certainrange has no effect on the results; the greater the absolute value of the soften modules is, thesteeper the post-failure curve of load-displacement is and the narrower the width of shearband is; the greater the value of the internal scale is, the flatter the post-failure curve ofload-displacement is and the wider the width of shear band is. The numerical study in theperformance of two types of Cosserat continuum finite elements, i.e. u8ω8(8-noded elementinterpolation approximations for both displacements u and microrotationω) and u8ω4(8-noded and 4-noded element interpolation approximations for u andωrespectively)elements is carried out. It is indicated that as compared with the u8ω8 element mesh, theu8ω4 element mesh possesses better performance in simulation of post-failure process. Bothtwo possess better performance in simulation of strain localization.
Then, based on the pressure-dependent elastoplastic Cosserat continuum model, progressive failure phenomena, which occured in the typical geotechnical engineeringproblems, such as the slope, excavation, retaining wall and soil foundation, characterized bystrain localization due to the material dilatancy, i.e. non-associated plasticity, or strainsoftening, are numerically simulated. Numerical results illustrate that as compared with theperformance of the finite element procedure for the classical continuum model, the finiteelement procedure based on Cosserat continuum model is capable of preserving thewell-posedness of the localization problems widely existing in geoteclanical engineering andsimulating the entire progressive failure phenomena characterized by strain localization due tostrain softening or the non-associated yield criterion adopted.
Besides, the Biot-Cosserat continuum model for coupled hydro-dynamic processes insaturated porous media is proposed by means of the combination of both Biot theory andCosserat continuum theory to simulate the strain localization phenomena due to the strainsoftening. In the present contribution, the Biot formulation of the skeleton material isextended to include the microrotation and correspondingly the coupled stresses defined in theCosserat model. The finite element formulations governing the coupled hydro-dynamicbehavior with the primary variables of the displacements and the microrotaion for the solid phase and the pressure for the fluid phase are derived on the basis of the Galerkin-weightedresidual method. The strain localization phenomena in saturated porous media due to thestrain softening are numerically simulated by using the developed model with correspondingfinite element method and the non-associated Drucker-Prager yield criterion particularlyconsidered for the pressure-dependent elasto-plastic Cosserat skeleton. Numerical results ofthe plane strain examples illustrate that the capability of the developed model in keeping thewell-posedness of the boundary value problems with strain softening behavior incorporated, and the availability of modeling the strain localization phenomena due to the strain softeningin saturated media.
In addition, an elastoplastic Cosserat continuum model for CAP constitutive model withnon-smooth multiplicative yield surfaces is presented. The consistent algorithm of the CAPelastoplastic model is derived in the framework of Cosserat continuum theory, i.e. the returnmapping algorithm for the integration of the rate constitutive equation and the closed form ofthe consistent elastoplastic tangent modulus matrix. The matrix inverse operation usuallyrequired in the calculation of elastoplastic tangent constitutive modulus matrix is avoided.The strain localization phenomena of the slope due to strain softening and the failure of thetunnel due to the excavation are numerically simulated using the developed model withcorresponding finite element method. Numerical results of the plane strain examples illustratethe capability and performance of the present model in keeping the well-posedness of theboundary value problems and ensuring the second order convergence rate and thecomputational efficiency of the model in numerical solution procedure. It also illustrates thepossibility of application of Cosserat continuum theory to various geotechnical constitutivemodels.
Finally, the two engineering examples, i.e. the failure of the two geotechnical structurescaused in the filling and the excavation processes respectively, are performed by using theproposed models and the developed algorithms. The non-associated perfect elastoplasticbehavior is particularly considered for the filling material and the strain softening behavior forthe excavation material. Numerical results indicate the capability and performance of Cosseratcontinuum model in keeping the well-posedness of the boundary value problems with strainsoftening behavior or non-associated perfect elastoplastic behavior incorporated and incompleting simulation of the whole failure progress.
引文
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8．李锡夔，唐洪祥．压力相关弹塑性Cosserat连续体模型与应变局部化有限元模拟．岩石力学与工程学报，2005，24(9)：1497-1505．
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1. Li Xikui, Tang Hongxiang. A consistent return mapping algorithm for pressure-dependent elastoplastic Cosserat continua and modeling of strain localization. Computers & Structures, 2005, 83(1): 1-10.
2．李锡夔，唐洪祥．压力相关弹塑性Cosserat连续体模型与应变局部化有限元模拟．岩石力学与工程学报，2005，24(9)：1497-1505．
3．武文华．非饱和土中热-水力-力学-传质耦合过程模拟及土壤环境工程中的应用：博士学位论文．大连：大连理工大学，2002．
4．刘泽佳．多孔介质中化学-热-水力-力学耦合分析与混合元方法：博士学位论文．大连：大连理工大学，2005．
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7．李锡夔．多孔介质中非线性耦合问题的数值方法．大连理工大学学报，1999，39：02-06．
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1. Li Xikui, Tang Hongxiang. A consistent return mapping algorithm for pressure-dependent elastoplastic Cosserat continua and modeling of strain localization. Computers & Structures, 2005, 83(1): 1-10.
2．李锡夔，唐洪祥．压力相关弹塑性Cosserat连续体模型与应变局部化有限元模拟．岩石力学与工程学报，2005，24(9)：1497-1505．
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4．刘泽佳．多孔介质中化学-热-水力-力学耦合分析与混合元方法：博士学位论文．大连：大连理工大学，2005．
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