非均质材料力学性能与失效分析的多尺度有限元法研究
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摘要
自然界中存在的、以及工程应用中的大部分材料都具有多尺度特征,且当考察尺度小到一定程度后,都将表现为非均质性。对这类材料而言,其微观非均质特征对宏观行为影响非常大,实际上,材料在宏观尺度所表现出来的大部分物理力学现象大多是由微观非均质结构决定的。以航空航天工业中广泛使用的复合材料为例,其力学性能与各组分相的材料属性、尺寸、形状、空间分布及界面连接特性等细观特征密切相关。另一方面,材料的宏观失效通常是由微观缺陷、微观裂纹等引起并发展所致。所以,研究微观现象和宏观行为的相互影响已成为工程应用中的一个迫切需要解决的问题。当采用传统数值方法(如有限元法)直接求解这类多尺度特征问题时,常常因为需要大量的计算机存储和计算时间而遇到求解困难。多尺度计算方法作为求解这类问题的一种有效方法,已成为近年来的一个研究热点。
本文首先提出了非均质材料弹性力学性能分析的扩展多尺度有限元法(EMsFEM)。基于多尺度有限元法(MsFEM)的思想,数值构造了能反映粗网格单元内部小尺度非均质信息的矢量场多尺度基函数。和标量场问题不同的是,在考虑固体变形的矢量场问题中材料内部会发生体积膨胀/收缩效应(泊松效应),从而导致材料各个方向的变形是相互耦合的。为了处理这种耦合关系,我们提出了一种新型的多尺度基函数构造方式,并在多尺度基函数中添加了耦合附加项。对不同构造方法下的多尺度基函数性能进行比较,结果表明,新型多尺度基函数能正确地反映粗网格单元内部的耦合变形,在相同的边界条件下可得到和常规精细有限元解完全一致的微观变形场,从而很大程度地提高了多尺度方法的计算精度。比较了不同边界条件下多尺度基函数的构造对多尺度方法精度的影响,并分析了尺度效应和误差产生的原因。所发展的方法实施方便,具有较好的数值精度。与传统均匀化方法相比,本文方法无尺度分离和周期性假设的限制,且易于进行降尺度计算以获得微观尺度的真实应力应变信息。
其次,基于扩展多尺度有限元法基本思想,提出了一类新的平面矩形单元和空间长方体单元(广义单元)。通过弹性力学伽辽金方程,理论推导出单元内部位移场和节点值之间的函数关系,并应用于新单元形函数的构造中。新单元能更合理地考虑单元内部的耦合变形,在不增加计算自由度的情况下得到比传统单元更精确的结果,为构造精度更高的新型有限单元提供新的思路。
进一步,发展了非均质材料小变形弹塑性分析的扩展多尺度有限元法。对于弹塑性材料的多尺度分析,随着微观物理变量的演化,微观尺度节点上将产生不平衡节点力。为了处理这些不平衡节点力产生的响应,我们提出了一种位移分解技术,将微观不平衡节点力响应等效成宏观等效力和微观扰动力的共同作用。其中宏观等效力用来驱动结构的宏观位移响应,而微观扰动力用来获得局部的微观位移扰动响应。在位移分解技术的基础上,进一步提出了两尺度协同迭代算法。在两尺度计算框架下,对宏微观非线性方程组进行协同增量迭代计算。最后,对方法的计算复杂度进行分析并与传统有限元法进行比较。数值结果表明,所发展的方法在模拟非均质材料弹塑性问题时具有很好的精度,且能显著地减少计算机存储需求和计算时间。
此外,发展了周期性桁架材料应变局部化分析的自适应多尺度方法(AMM)。局部化现象是引起工程结构破坏的主要原因之一,传统基于体积平均思想的均匀化方法由于受到局部周期性假设和尺度分离假设的限制而不能很好地用于模拟这类问题。在计算过程中,AMM自适应地将整体计算域分解为两个部分,即细网格计算域和粗网格计算域,EMsFEM提供了多尺度框架。这样,变形剧烈的局部化域可采用有限元法在细尺度网格上直接求解,保证了算法的精度；而对于其它变形较缓和的域,则仍采用EMsFEM在粗网格尺度上进行非线性计算。同时,尺度连接通过建立在尺度界面上的主从节点约束方程完成。提出一种基于宏观单元位移梯度的网格识别器来自动更新细网格计算域。数值结果表明AMM具有较高的计算效率,能准确预测局部化问题的剪切带位置,且避免了宏观单元网格依赖性问题。
最后,发展了一种模拟准静态裂纹扩展问题的自适应多尺度方法。采用扩展有限元法(XFEM)和水平集法(LSM)的组合对细尺度模型的不连续问题(包括强不连续和弱不连续问题)进行模拟,其中,XFEM用来计算不连续模型的应力应变场,而LSM用来对固定不连续界面(材料界面)和移动不连续界面(裂纹面)进行几何描述。这样,有限元网格可独立于不连续界面,模拟裂纹扩展问题时不需要对有限元网格重新剖分。而对于不需要精细分析的其它子域,采用EMsFEM在粗尺度网格上进行求解。提出一种基于距离参数的自适应网格识别技术来对求解域进行自适应分解,其距离参数可由水平集函数的演化方程自动获取。方法具有很好的精度,能准确地预测裂纹扩展路径,有效地减少裂纹扩展模拟所需的计算自由度和计算时间。
Almost all natural materials, as well as industrial and engineering materials, have multiple scale natures. At the same time, they are heterogeneous at a certain scale. For these materials, the heterogeneous nature can drastically impact on their macroscopic behaviors. In fact, most observed physical and mechanical properties of micro-heterogeneous materials depend on their heterogeneous micro-structures. Taking the composites which have been widely used in aerospace industry for example, the macroscopic mechanics properties of composites are closely related to the material properties, size, shape, and spatial distribution of the microstructural constituents and their respective interfaces. On the other hand, the macroscopic failures are usually induced by the microscopic defects and microscopic cracks. Studying the relations between the microscopic phenomena and macroscopic behavior has become an essentical problem in engineering applications. When the traditional numerical method (such as finite element method, FEM) is directly adopted for solving these multiscale problems, it will encounter difficulties due to the tremendous requirement of computer memory and computing time. Multiscale computational method servers as an effective approach for sovling these problems and has become a hot area of research in recent years.
Firstly, an extended multiscale finite element method (EMsFEM) is proposed for solving the mechanical problems of heterogeneous materials in elasticity. Based on the idea of multiscale finite element method (MsFEM), the multiscale base functions (MBFs) for vector field are constructed numerically which can reflect the small-scale heterogeneities within a coarse element. Unlike the scalar field problem, the bulk expansion/contraction phenomena (Poisson effect) have to be considered in the construction of MBFs in the vector field. In other words, the displacement fields in different directions in an element are coupled. To deal with this coupling relation, a new technique is proposed for the construction of MBFs, in which the additional coupling terms are considered. The performances of MBFs with different construction techniques are compared and the results show that the new type of MBFs can exactly reflect the coupling deformations within the coarse element and can obtain an identical micro-deformation field with the traditional FEM, thus, the EMsFEM can improve the computational accuracy dramatically. Several different kinds of boundary conditions are introduced for the construction of MBFs and their influences are investigated. Moreover, the reasons for the scale effect and error are discussed. The method proposed can be implemented conveniently and has a good precision. Comparing with the traditional homogenization method, the EMsFEM here does not need the assumptions of the scale separation and microstructural periodicity, and can perform the downscaling computations easily.
Secondly, a new kind of rectangular elements (i.e., Generalized Elements) is proposed based on the idea of the EMsFEM. By means of the Galerkin equation in elastic mechanics, the relations between the displacement field inside the element and the displacements at nodal points of the element are deduced theoretically. The new element can consider the coupling effect of the displacements more reasonable and obtain more accuracy results then the traditional one. Thus, our research can provide ideas for constructing new kinds of elements, which have better precision.
The EMsFEM is further extended for modeling the elasto-plastic behavior of heterogeneous materials under small deformation. When the material nonlinearity is considered in the multiscale analysis, it will induce unbalanced nodal forces at the small scales with the emergence of plastic deformation. To deal with the microscopic unbalanced nodal forces, a displacement decomposition technique is proposed. In this context, the microscopic unbalanced nodal forces can be treated as the combined effects of the macroscopic equivalent forces and microscopic perturbed forces, in which the macroscopic equivalent forces are used to solve the macroscopic displamcents and the microscopic perturbed forces are used to obtain the local perturebed displacement field in microscopic scale. Then, a two-scale concurrent computational modeling with successive iteration scheme is proposed. Lastly, we make an estimate of the computer memory and CPU time for the EMsFEM, and compare them with those of the traditional FEM. Extensive numerical experiments have shown that the method developed provides excellent precision of the nonlinear response for the heterogeneous materials and can reduce the comtational cost dramatically.
Also, an adaptive multiscale method (AMM) is developed for strain localization analysis of periodic lattice truss materials. In terms of practical applications, localization phenomena can be recognized as the main reasons for structure failure. Volume average based homogenization methods have some limitations in simulating localization phenomena, stemming from their basic assumptions, i.e., scale separation and microstructural periodicity. During the computional processes, the overall computional region in the AMM is adaptively decomposed into two parts:i.e., fine-scale region and coarse-scale region. The EMsFEM serves as a multiscale framework. Since the direct fine-scale simulation is used to model the localized zone, the motion of the truss elements which have severe deformations can be accurately modeled. On the other hand, the EMsFEM is adopted to capture the macroscopic nonlinear response on the coarse region whose strains are moderate. The degrees of freedom on the two different scales are then bridged at the scale-interface by virtue of the master-slave constraints. A displacement gradient based mesh indicator is proposed for the adaptive scheme. The results of the AMM show insensitivity to the coarse-scale discrtization and can capture well the localization phenomena. Moreover, the computational cost is reduced dramatically.
At last, a new multiscale method for simulating crack propagation is proposed based on the AMM. The combination of the extended finite element method (XFEM) and level set method (LSM) is employed to model the discontinuities (including strong and weak discontinuities), in which the XFEM is used for computing the stress and displacement fields necessary for determining the rate of crack growth, and the LSM is used for geometrical description of the discontinuities. In this way, the method allows the crack geometry to be represented independently of the finite element mesh, thus, remeshing is not needed. For the subregions which surround the refined zone, the EMsFEM is used and resolved on a coarse-scale mesh. A displacement parameters based mesh indicator is proposed to identify adaptively the fine-scale zone. The parameters can be obtained by the level set functions. The method developed can be well used for the simulation of evolving discontinuities and can reduce the computational cost dramatically.
本文首先提出了非均质材料弹性力学性能分析的扩展多尺度有限元法(EMsFEM)。基于多尺度有限元法(MsFEM)的思想,数值构造了能反映粗网格单元内部小尺度非均质信息的矢量场多尺度基函数。和标量场问题不同的是,在考虑固体变形的矢量场问题中材料内部会发生体积膨胀/收缩效应(泊松效应),从而导致材料各个方向的变形是相互耦合的。为了处理这种耦合关系,我们提出了一种新型的多尺度基函数构造方式,并在多尺度基函数中添加了耦合附加项。对不同构造方法下的多尺度基函数性能进行比较,结果表明,新型多尺度基函数能正确地反映粗网格单元内部的耦合变形,在相同的边界条件下可得到和常规精细有限元解完全一致的微观变形场,从而很大程度地提高了多尺度方法的计算精度。比较了不同边界条件下多尺度基函数的构造对多尺度方法精度的影响,并分析了尺度效应和误差产生的原因。所发展的方法实施方便,具有较好的数值精度。与传统均匀化方法相比,本文方法无尺度分离和周期性假设的限制,且易于进行降尺度计算以获得微观尺度的真实应力应变信息。
其次,基于扩展多尺度有限元法基本思想,提出了一类新的平面矩形单元和空间长方体单元(广义单元)。通过弹性力学伽辽金方程,理论推导出单元内部位移场和节点值之间的函数关系,并应用于新单元形函数的构造中。新单元能更合理地考虑单元内部的耦合变形,在不增加计算自由度的情况下得到比传统单元更精确的结果,为构造精度更高的新型有限单元提供新的思路。
进一步,发展了非均质材料小变形弹塑性分析的扩展多尺度有限元法。对于弹塑性材料的多尺度分析,随着微观物理变量的演化,微观尺度节点上将产生不平衡节点力。为了处理这些不平衡节点力产生的响应,我们提出了一种位移分解技术,将微观不平衡节点力响应等效成宏观等效力和微观扰动力的共同作用。其中宏观等效力用来驱动结构的宏观位移响应,而微观扰动力用来获得局部的微观位移扰动响应。在位移分解技术的基础上,进一步提出了两尺度协同迭代算法。在两尺度计算框架下,对宏微观非线性方程组进行协同增量迭代计算。最后,对方法的计算复杂度进行分析并与传统有限元法进行比较。数值结果表明,所发展的方法在模拟非均质材料弹塑性问题时具有很好的精度,且能显著地减少计算机存储需求和计算时间。
此外,发展了周期性桁架材料应变局部化分析的自适应多尺度方法(AMM)。局部化现象是引起工程结构破坏的主要原因之一,传统基于体积平均思想的均匀化方法由于受到局部周期性假设和尺度分离假设的限制而不能很好地用于模拟这类问题。在计算过程中,AMM自适应地将整体计算域分解为两个部分,即细网格计算域和粗网格计算域,EMsFEM提供了多尺度框架。这样,变形剧烈的局部化域可采用有限元法在细尺度网格上直接求解,保证了算法的精度；而对于其它变形较缓和的域,则仍采用EMsFEM在粗网格尺度上进行非线性计算。同时,尺度连接通过建立在尺度界面上的主从节点约束方程完成。提出一种基于宏观单元位移梯度的网格识别器来自动更新细网格计算域。数值结果表明AMM具有较高的计算效率,能准确预测局部化问题的剪切带位置,且避免了宏观单元网格依赖性问题。
最后,发展了一种模拟准静态裂纹扩展问题的自适应多尺度方法。采用扩展有限元法(XFEM)和水平集法(LSM)的组合对细尺度模型的不连续问题(包括强不连续和弱不连续问题)进行模拟,其中,XFEM用来计算不连续模型的应力应变场,而LSM用来对固定不连续界面(材料界面)和移动不连续界面(裂纹面)进行几何描述。这样,有限元网格可独立于不连续界面,模拟裂纹扩展问题时不需要对有限元网格重新剖分。而对于不需要精细分析的其它子域,采用EMsFEM在粗尺度网格上进行求解。提出一种基于距离参数的自适应网格识别技术来对求解域进行自适应分解,其距离参数可由水平集函数的演化方程自动获取。方法具有很好的精度,能准确地预测裂纹扩展路径,有效地减少裂纹扩展模拟所需的计算自由度和计算时间。
Almost all natural materials, as well as industrial and engineering materials, have multiple scale natures. At the same time, they are heterogeneous at a certain scale. For these materials, the heterogeneous nature can drastically impact on their macroscopic behaviors. In fact, most observed physical and mechanical properties of micro-heterogeneous materials depend on their heterogeneous micro-structures. Taking the composites which have been widely used in aerospace industry for example, the macroscopic mechanics properties of composites are closely related to the material properties, size, shape, and spatial distribution of the microstructural constituents and their respective interfaces. On the other hand, the macroscopic failures are usually induced by the microscopic defects and microscopic cracks. Studying the relations between the microscopic phenomena and macroscopic behavior has become an essentical problem in engineering applications. When the traditional numerical method (such as finite element method, FEM) is directly adopted for solving these multiscale problems, it will encounter difficulties due to the tremendous requirement of computer memory and computing time. Multiscale computational method servers as an effective approach for sovling these problems and has become a hot area of research in recent years.
Firstly, an extended multiscale finite element method (EMsFEM) is proposed for solving the mechanical problems of heterogeneous materials in elasticity. Based on the idea of multiscale finite element method (MsFEM), the multiscale base functions (MBFs) for vector field are constructed numerically which can reflect the small-scale heterogeneities within a coarse element. Unlike the scalar field problem, the bulk expansion/contraction phenomena (Poisson effect) have to be considered in the construction of MBFs in the vector field. In other words, the displacement fields in different directions in an element are coupled. To deal with this coupling relation, a new technique is proposed for the construction of MBFs, in which the additional coupling terms are considered. The performances of MBFs with different construction techniques are compared and the results show that the new type of MBFs can exactly reflect the coupling deformations within the coarse element and can obtain an identical micro-deformation field with the traditional FEM, thus, the EMsFEM can improve the computational accuracy dramatically. Several different kinds of boundary conditions are introduced for the construction of MBFs and their influences are investigated. Moreover, the reasons for the scale effect and error are discussed. The method proposed can be implemented conveniently and has a good precision. Comparing with the traditional homogenization method, the EMsFEM here does not need the assumptions of the scale separation and microstructural periodicity, and can perform the downscaling computations easily.
Secondly, a new kind of rectangular elements (i.e., Generalized Elements) is proposed based on the idea of the EMsFEM. By means of the Galerkin equation in elastic mechanics, the relations between the displacement field inside the element and the displacements at nodal points of the element are deduced theoretically. The new element can consider the coupling effect of the displacements more reasonable and obtain more accuracy results then the traditional one. Thus, our research can provide ideas for constructing new kinds of elements, which have better precision.
The EMsFEM is further extended for modeling the elasto-plastic behavior of heterogeneous materials under small deformation. When the material nonlinearity is considered in the multiscale analysis, it will induce unbalanced nodal forces at the small scales with the emergence of plastic deformation. To deal with the microscopic unbalanced nodal forces, a displacement decomposition technique is proposed. In this context, the microscopic unbalanced nodal forces can be treated as the combined effects of the macroscopic equivalent forces and microscopic perturbed forces, in which the macroscopic equivalent forces are used to solve the macroscopic displamcents and the microscopic perturbed forces are used to obtain the local perturebed displacement field in microscopic scale. Then, a two-scale concurrent computational modeling with successive iteration scheme is proposed. Lastly, we make an estimate of the computer memory and CPU time for the EMsFEM, and compare them with those of the traditional FEM. Extensive numerical experiments have shown that the method developed provides excellent precision of the nonlinear response for the heterogeneous materials and can reduce the comtational cost dramatically.
Also, an adaptive multiscale method (AMM) is developed for strain localization analysis of periodic lattice truss materials. In terms of practical applications, localization phenomena can be recognized as the main reasons for structure failure. Volume average based homogenization methods have some limitations in simulating localization phenomena, stemming from their basic assumptions, i.e., scale separation and microstructural periodicity. During the computional processes, the overall computional region in the AMM is adaptively decomposed into two parts:i.e., fine-scale region and coarse-scale region. The EMsFEM serves as a multiscale framework. Since the direct fine-scale simulation is used to model the localized zone, the motion of the truss elements which have severe deformations can be accurately modeled. On the other hand, the EMsFEM is adopted to capture the macroscopic nonlinear response on the coarse region whose strains are moderate. The degrees of freedom on the two different scales are then bridged at the scale-interface by virtue of the master-slave constraints. A displacement gradient based mesh indicator is proposed for the adaptive scheme. The results of the AMM show insensitivity to the coarse-scale discrtization and can capture well the localization phenomena. Moreover, the computational cost is reduced dramatically.
At last, a new multiscale method for simulating crack propagation is proposed based on the AMM. The combination of the extended finite element method (XFEM) and level set method (LSM) is employed to model the discontinuities (including strong and weak discontinuities), in which the XFEM is used for computing the stress and displacement fields necessary for determining the rate of crack growth, and the LSM is used for geometrical description of the discontinuities. In this way, the method allows the crack geometry to be represented independently of the finite element mesh, thus, remeshing is not needed. For the subregions which surround the refined zone, the EMsFEM is used and resolved on a coarse-scale mesh. A displacement parameters based mesh indicator is proposed to identify adaptively the fine-scale zone. The parameters can be obtained by the level set functions. The method developed can be well used for the simulation of evolving discontinuities and can reduce the computational cost dramatically.
引文
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