非均匀材料有效力学性能和破坏过程的数值模拟
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摘要
由于微结构的存在和相互作用,多数材料在本质上都是非均匀的。通常的数值分析通过将非线性本构关系赋给每个单元来模拟材料的宏观非线性。这种确定性模型无法反映材料特性和作用载荷的随机性。需要使用统计方法才能对结构系统的输出进行分析和定量化,以便给出可靠的结构安全设计。本文从细观非均匀性的本质出发,以空间随机分布的、具有不同材料特性的线性本构关系的单元来模拟非均匀脆性材料的宏观非线性。
非均匀材料试件离散为几何尺寸相同的有限元格子模型。格子单元的弹性模量和破坏强度由按照统计分布生成的相应离散序列随机地确定,以模拟材料性质的初始非均匀分布。在破坏模拟中,每个载荷步中破坏单元的数目用自适应加载步长来实现控制。引入“空气模量”单元代替破坏单元,使损伤试件的几何连续性得以保持,从而可通过不断更新结构刚度矩阵和迭代求解来实现对非均匀脆性材料三维破坏过程的模拟。借助于自适应载荷步长加载方法和“空气模量”单元,准静态加载条件下非均匀脆性材料的破坏过程可以更容易地得到模拟。
对于两相复合材料,诸如颗粒增强复合材料、层叠复合材料和加筋复合材料,建立了相应的格子模型,以反映材料性质的非均匀分布。对于非均匀性对以上复合材料的等效刚度和破坏过程的影响进行了分析。得到的结论可以为复合材料的生产、加工工艺提供有用的指导。
对于非均匀脆性材料的一维情况,假设弹性模量和破坏强度随机分布且相互独立,推导了描述其破坏过程的一维应力应变关系解析表达式。这些解析解对于帮助理解材料的宏观非线性与非均匀分布参数之间的本质关系,以及快速算得非均匀材料结构弹性场的有关统计量,都是很有帮助的。
采用子域边界元法对具有多种材料性质的结构进行了应力分析。针对非均匀脆性材料格子模型的特点,采用重复子域边界元法模拟了二维非均匀脆性材料的破坏过程。由于此法只需要进行一次子域系数矩阵的积分,因而大大提高了分析效率。
Most materials are heterogeneous in nature due to the presence and interaction of the microstructures within. In conventional numerical analyses the macro non-linearity of the materials is simulated through assigning the nonlinear constitutive relation of homogeneous material to each element. Such deterministic model cannot reflect the randomness of the material properties. The statistical methods are needed to analyze and quantify the output of structural system in order to give reliable structural safety design. Based on the nature of meso-heterogeneity, the macro non-linearity of heterogeneous brittle material is numerically simulated by spatial-randomly distributed elements with linear constitutive relation of different material properties in this paper.The heterogeneous material specimen is discretized into lattice elements with the same size. The elastic modulus and failure strength of the lattice elements are randomly determined by corresponding discrete sequences produced by statistical distribution to represent the initial heterogeneity of material properties. In the failure simulation, the number of failed elements in each time step is well controlled by self-adaptive load-step size. The element with "air modulus" is introduced to replace the failed element so that the geometric continuity keeps unchanged for the damaged specimen and the 3-D failure process of the heterogeneous brittle material can be simulated through continual update of the structural stiffness matrix and iterative solution. By means of the self-adaptive load-step and "air modulus" element, the failure process of heterogeneous brittle material under quasi-static loading can be simulated more easily.The two-phase composites, such as granular reinforced composite, the laminated composite and the reinforced composite are modeled with the lattice approach to reflect the heterogeneous distribution of material component. The effects of heterogeneity on equivalent modulus and failure process are analyzed. The results obtained can provide useful instructions for the producing technique and manufacturing procedure of the composite materials.
For the one-dimensional case of the heterogeneous brittle material, assumed that the distributions of the elastic modulus and the failure strength are random and independent, the one-dimensional analytical stress-strain relations with the heterogeneity parameters can be derived. These analytical expressions are helpful to understand the essential relations between macro non-linearity and the parameters of heterogeneous distribution, and to quickly calculate the statistical constants of the elastic field in the 1-D heterogeneous material structures.The Multi-sub-region Boundary Element Method is employed to analyze the structures consists of components with different material properties. The Reduplicate Sub-region Boundary Element Method is used to simulate the full failure process of heterogeneous brittle material since the lattice model with the same geometric size is used to discretize the specimen. The coefficient matrices are only generated just once, so the computational efficiency is increased significantly.
非均匀材料试件离散为几何尺寸相同的有限元格子模型。格子单元的弹性模量和破坏强度由按照统计分布生成的相应离散序列随机地确定,以模拟材料性质的初始非均匀分布。在破坏模拟中,每个载荷步中破坏单元的数目用自适应加载步长来实现控制。引入“空气模量”单元代替破坏单元,使损伤试件的几何连续性得以保持,从而可通过不断更新结构刚度矩阵和迭代求解来实现对非均匀脆性材料三维破坏过程的模拟。借助于自适应载荷步长加载方法和“空气模量”单元,准静态加载条件下非均匀脆性材料的破坏过程可以更容易地得到模拟。
对于两相复合材料,诸如颗粒增强复合材料、层叠复合材料和加筋复合材料,建立了相应的格子模型,以反映材料性质的非均匀分布。对于非均匀性对以上复合材料的等效刚度和破坏过程的影响进行了分析。得到的结论可以为复合材料的生产、加工工艺提供有用的指导。
对于非均匀脆性材料的一维情况,假设弹性模量和破坏强度随机分布且相互独立,推导了描述其破坏过程的一维应力应变关系解析表达式。这些解析解对于帮助理解材料的宏观非线性与非均匀分布参数之间的本质关系,以及快速算得非均匀材料结构弹性场的有关统计量,都是很有帮助的。
采用子域边界元法对具有多种材料性质的结构进行了应力分析。针对非均匀脆性材料格子模型的特点,采用重复子域边界元法模拟了二维非均匀脆性材料的破坏过程。由于此法只需要进行一次子域系数矩阵的积分,因而大大提高了分析效率。
Most materials are heterogeneous in nature due to the presence and interaction of the microstructures within. In conventional numerical analyses the macro non-linearity of the materials is simulated through assigning the nonlinear constitutive relation of homogeneous material to each element. Such deterministic model cannot reflect the randomness of the material properties. The statistical methods are needed to analyze and quantify the output of structural system in order to give reliable structural safety design. Based on the nature of meso-heterogeneity, the macro non-linearity of heterogeneous brittle material is numerically simulated by spatial-randomly distributed elements with linear constitutive relation of different material properties in this paper.The heterogeneous material specimen is discretized into lattice elements with the same size. The elastic modulus and failure strength of the lattice elements are randomly determined by corresponding discrete sequences produced by statistical distribution to represent the initial heterogeneity of material properties. In the failure simulation, the number of failed elements in each time step is well controlled by self-adaptive load-step size. The element with "air modulus" is introduced to replace the failed element so that the geometric continuity keeps unchanged for the damaged specimen and the 3-D failure process of the heterogeneous brittle material can be simulated through continual update of the structural stiffness matrix and iterative solution. By means of the self-adaptive load-step and "air modulus" element, the failure process of heterogeneous brittle material under quasi-static loading can be simulated more easily.The two-phase composites, such as granular reinforced composite, the laminated composite and the reinforced composite are modeled with the lattice approach to reflect the heterogeneous distribution of material component. The effects of heterogeneity on equivalent modulus and failure process are analyzed. The results obtained can provide useful instructions for the producing technique and manufacturing procedure of the composite materials.
For the one-dimensional case of the heterogeneous brittle material, assumed that the distributions of the elastic modulus and the failure strength are random and independent, the one-dimensional analytical stress-strain relations with the heterogeneity parameters can be derived. These analytical expressions are helpful to understand the essential relations between macro non-linearity and the parameters of heterogeneous distribution, and to quickly calculate the statistical constants of the elastic field in the 1-D heterogeneous material structures.The Multi-sub-region Boundary Element Method is employed to analyze the structures consists of components with different material properties. The Reduplicate Sub-region Boundary Element Method is used to simulate the full failure process of heterogeneous brittle material since the lattice model with the same geometric size is used to discretize the specimen. The coefficient matrices are only generated just once, so the computational efficiency is increased significantly.
引文
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[2] Renaudin P., Breysse D., Frantziskonis G. Heterogeneous solids-Part Ⅱ:numerical 2-D results on boundary and other relevant phenomena. European Journal of Mechanics, A/Solids, 1997, 16(3):425～443
[3] S.C. Blair, N. G. Cook. Analysis of compressive fracture in rock using statistical techniques:Part Ⅰ. a non-linear rule-based model. International Journal of Rock Mechanics and Mining Sciences, 1998, 35(7):837～848
[4] Blair S. C., Cook N. G. Analysis of compressive fracture in rock using statistical techniques:Part Ⅱ. Effect of microscale heterogeneity on macroscopic deformation. International Journal of Rock Mechanics and Mining Sciences, 1998, 35(7):849～861
[5] Kumar R., Ray R. Modeling microstructure heterogeneity in materials by using Q-mode factor analysis. Materials Characterization, 1998, 40:7～13
[6] Frantziskonis G, Tucson, Arizona. Heterogeneity and implicated surface effects:statistical, fractal formulation and relevant analytical solution. Acta Mechanica, 1995, 108:157～178
[7] Muneo hori, Junochiro Kubo. Analysis of probability and range of average stress in each phase of heterogeneity materials. Journal of the Mechanics and Physics of Solids, 1998, 46(3):537～556
[8] Mishnaevsky L., Schmauder S. Damage evolution and heterogeneity of materials:model based on fuzzy set theory. Engineering Fracture Mechanics, 1997, 57(6):625～636
[9] Tang F. F., Desai C. S., Frantziskonis G. Heterogeneity and degradation in brittle materials. Engineering Fracture Mechanics, 1992, 43(5):779～796
[10] Ala Tabiei, Sun Jianmin. Statistical aspects of strength size effect of laminated composites materials. Composites Structures, 1999, 46:209～216
[11] Ostoja-Starzewski M. Random field models of heterogeneous materials. International Journal of Solids and Structures, 1998, 35(19):2429～2455
[12] Zohdi T., Wriggers P. A domain decompostion method for bodies with heterogeneous microstructures based on material regularization. International Journal of Solids and Structures, 1999, 36:2507～2525
[13] Kolednik O. The yield stress gradient effect in inhomogeneous materials. International Journal of Solids and Structures, 2000, 37:781～808
[14] Dai H., Frantziskonis G. Heterogeneity, spatial correlations, size effects and dissipated energy in brittle materials. Mechanics of Materials, 1994, 18:103～118.
[15] Grigoriu Mircea. Stochastic mechanics. International Journal of Solids and Structures, 2000, 37(1):197-214
[16] Zdenek, Bazant P. Size effect. International Journal of Solids and Structures, 2000, 37:69～80
[17] Read H., Hegemier G. Softening of rock, soil and concrete-a review article. Mechanics of Materials, 1984, 3:271～294
[18] 唐春安.岩石损伤参量与本构关系的统计理论解及试验确定.岩石、混凝土断裂与强度,1988,1:80～86
[19] 唐春安.岩石破裂过程中的灾变.北京:煤炭工业出版社,1993
[20] Tang Chun'an. Numerical simulation of progressive rock failure and associated seismicity. International Journal of Rock Mechanics and Mining Sciences, 1997, 34(n):249～261
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