复合材料弹性力学特性数值预测研究
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摘要
复合材料由于其优异的性能而备受关注,本文主要探讨其宏观弹性性质的预测。本文采用直接建模有限元方法,研究复合材料的有效弹性性质,并出于验证和比较的目的回顾和分析了经典的理论上下界限和解析方法。由于针对传统的夹杂-基体型复合材料的理论方法和数值建模方法较为成熟,本文研究的复合材料主要是指微观结构中无法清楚区分基体相和夹杂相的非夹杂-基体型复合材料。本文主要的工作如下:
1.分析了经典的复合材料弹性性质预测的理论上下界限和解析方法,比较了不同组分材料弹性模量比的情形下,Hashin-Shtrikman界限和Voigt-Reuss界限所确定的有效模量的理论界限范围;针对两相复合材料,研究了不同基体夹杂组合时,广义自洽方法、Mori-Tanaka方法和Halpin-Tsai方法的预测结果及其与Hashin-Shtrikman界限的关系。
2.提出了直接建模有限元方法的建模思路并介绍了其具体实现方法:通过将初始样本模型离散成大小形状相同的初始材料域,按照一定概率分布规则赋予材料域相应组分材料属性并细分材料域为若干个有限元单元,实现了对复合材料微观结构的模拟,建立了复合材料的平面和三维模型。
3.运用直接建模有限元方法,建立了两相随机复合材料的有限元模型,并通过后续的有限元分析得到了平面和三维情况下,样本复合材料有效模量的预测值。通过将预测值与Hashin-Shtrikman理论上下界限的比较验了本文提出方法的有效性;通过与广义自洽方法预测结果的比较,证明了经典解析方法对非夹杂-基体复合材料有效弹性模量的预测存在不足,而本文提出的方法可以有效的预测这种复合材料的有效弹性性质。
4.研究了直接建模有限元方法中初始材料域大小及其单元密度这两个建模参数对有效模量预测结果的影响,为选择合适的建模参数以减少计算量且满足精度要求提供了依据;在选择适当的建模参数基础上,研究了组分材料体积分数对复合材料弹性力学性能的影响。
本文提出了一种新的复合材料有效弹性模量数值预测方法,即直接建模有限元方法。研究结果表明该方法能有效预测非夹杂-基体型复合材料的弹性力学性质,特别是在只知道组分材料性质和体积分数的情况下,该方法是一种简洁、快速预测复合材料弹性模量的有效手段。
Composite materials have attracted dramatic attention in the last few decades for their outstanding properties and the main focus of this thesis concerns the predictions of their macroscopically elastic moduli. To predict the effective moduli of composite materials, a direct modeling strategy based on the finite element method is investigated and the well-known Hashin-Shtrikman bounds and the general self-consistent method are reintroduced for the validation and comparison purposes. Considering the extensive and relatively ripe study on the traditional inclusion-matrix composite materials, this thesis addresses the non-particulate composites, in which the matrix and particulate (or fiber) phases can not be distinguished clearly. The specific works finished are listed as below:
1. The classical bounds and analytical models on the elastic moduli of composite materials are reviewed and analyzed. The well-known Hashin-Shtrikman bounds and Voigt-Reuss bounds are compared under the different scenarios of the elastic property contrast of constituent phases. The connections between the Hashin-Shtrikman bounds and the analytical models (the general consistent method, the Mori-Tanaka method and the Halpin-Tsai method) are investigated numerically by applying to bi-continuous composite materials.
2. The direct modeling strategy based on the finite element method is proposed. Simply stated, the model is constructed from equal-size domains, which are then assigned with material labels to distinguish constituent phases and further divided into several finite elements to mimic the microstructure of the composite materials. Based on such method, the two-dimension and three-dimension sample models have been constructed.
3. The direct modeling strategy is applied to build up bi-continuous composite materials and the successive finite element analysis (FEA) gives out the predictions of the effective elastic moduli of the constructed samples under the planar and three-dimensional settings. The validity of the prediction is endorsed by the agreement of the FEA results to the corresponding Hashin-Shtrikman bounds. The comparison of the predictions from the direct modeling strategy and the general consistent method shows that classical analytical models for inclusion-matrix composites can not be simply extended to non-inclusion composites.
4. The effects of the two parameters introduced by the modeling process, the size of the initial domains and their further refinement (element density) on the predictions of the effective elastic moduli are discussed to provide indications for selecting the parameters properly to reduce the computation time while the accuracy issue is not affected. With the proper selection of the two modeling parameters, the effect of the constituent phase’s volume fraction on the elastic property of the composites is investigated.
In short, the direct modeling strategy based the finite element method proposed in this dissertation can provide valuable information on the elastic properties of non-particulate composites with random constituent phase distribution when only the phase properties and volume fractions of each constituent phase are available.
1.分析了经典的复合材料弹性性质预测的理论上下界限和解析方法,比较了不同组分材料弹性模量比的情形下,Hashin-Shtrikman界限和Voigt-Reuss界限所确定的有效模量的理论界限范围;针对两相复合材料,研究了不同基体夹杂组合时,广义自洽方法、Mori-Tanaka方法和Halpin-Tsai方法的预测结果及其与Hashin-Shtrikman界限的关系。
2.提出了直接建模有限元方法的建模思路并介绍了其具体实现方法:通过将初始样本模型离散成大小形状相同的初始材料域,按照一定概率分布规则赋予材料域相应组分材料属性并细分材料域为若干个有限元单元,实现了对复合材料微观结构的模拟,建立了复合材料的平面和三维模型。
3.运用直接建模有限元方法,建立了两相随机复合材料的有限元模型,并通过后续的有限元分析得到了平面和三维情况下,样本复合材料有效模量的预测值。通过将预测值与Hashin-Shtrikman理论上下界限的比较验了本文提出方法的有效性;通过与广义自洽方法预测结果的比较,证明了经典解析方法对非夹杂-基体复合材料有效弹性模量的预测存在不足,而本文提出的方法可以有效的预测这种复合材料的有效弹性性质。
4.研究了直接建模有限元方法中初始材料域大小及其单元密度这两个建模参数对有效模量预测结果的影响,为选择合适的建模参数以减少计算量且满足精度要求提供了依据;在选择适当的建模参数基础上,研究了组分材料体积分数对复合材料弹性力学性能的影响。
本文提出了一种新的复合材料有效弹性模量数值预测方法,即直接建模有限元方法。研究结果表明该方法能有效预测非夹杂-基体型复合材料的弹性力学性质,特别是在只知道组分材料性质和体积分数的情况下,该方法是一种简洁、快速预测复合材料弹性模量的有效手段。
Composite materials have attracted dramatic attention in the last few decades for their outstanding properties and the main focus of this thesis concerns the predictions of their macroscopically elastic moduli. To predict the effective moduli of composite materials, a direct modeling strategy based on the finite element method is investigated and the well-known Hashin-Shtrikman bounds and the general self-consistent method are reintroduced for the validation and comparison purposes. Considering the extensive and relatively ripe study on the traditional inclusion-matrix composite materials, this thesis addresses the non-particulate composites, in which the matrix and particulate (or fiber) phases can not be distinguished clearly. The specific works finished are listed as below:
1. The classical bounds and analytical models on the elastic moduli of composite materials are reviewed and analyzed. The well-known Hashin-Shtrikman bounds and Voigt-Reuss bounds are compared under the different scenarios of the elastic property contrast of constituent phases. The connections between the Hashin-Shtrikman bounds and the analytical models (the general consistent method, the Mori-Tanaka method and the Halpin-Tsai method) are investigated numerically by applying to bi-continuous composite materials.
2. The direct modeling strategy based on the finite element method is proposed. Simply stated, the model is constructed from equal-size domains, which are then assigned with material labels to distinguish constituent phases and further divided into several finite elements to mimic the microstructure of the composite materials. Based on such method, the two-dimension and three-dimension sample models have been constructed.
3. The direct modeling strategy is applied to build up bi-continuous composite materials and the successive finite element analysis (FEA) gives out the predictions of the effective elastic moduli of the constructed samples under the planar and three-dimensional settings. The validity of the prediction is endorsed by the agreement of the FEA results to the corresponding Hashin-Shtrikman bounds. The comparison of the predictions from the direct modeling strategy and the general consistent method shows that classical analytical models for inclusion-matrix composites can not be simply extended to non-inclusion composites.
4. The effects of the two parameters introduced by the modeling process, the size of the initial domains and their further refinement (element density) on the predictions of the effective elastic moduli are discussed to provide indications for selecting the parameters properly to reduce the computation time while the accuracy issue is not affected. With the proper selection of the two modeling parameters, the effect of the constituent phase’s volume fraction on the elastic property of the composites is investigated.
In short, the direct modeling strategy based the finite element method proposed in this dissertation can provide valuable information on the elastic properties of non-particulate composites with random constituent phase distribution when only the phase properties and volume fractions of each constituent phase are available.
引文
[1]周曦亚.复合材料.北京:化学工业出版社,2005,1-10
[2]沈观林,胡更开.复合材料力学.北京:清华大学出版社,2006,287-304
[3]张双寅,刘济庆,于晓霞,等.复合材料结构的力学性能.北京:北京理工大学出版社,1992,1-9
[4]张锦,张乃恭.新型复合材料力学机理及其应用.北京:北京航空航天大学出版社,1992,1-13
[5] P. A. Buchan, J. F. Chen. Blast resistance of FPR composites and polymer strengthened concrete and masonry structures– A state-of-the-art review. Composites: Part B, 2007, 38:509-522
[6] M. Martina, G. Subramanyam, J. C. Weaver, et al. Developing macroporous bicontinuous materials as scaffolds for tissue engineering. Biomaterials, 2005, 26:5609-5616
[7] X. Q. Feng, Z. Tian, Y. H. Liu, et al. Effective elastic and plastic properties of interpenetrating multiphase composites. Applied Composite Materials, 2004,11:33-55
[8]陈建桥.复合材料力学概论.北京:科学出版社,2006,1-5
[9] G. S. Daehn, B. Starck, L. Xu, et al. Elastic and plastic behavior of a co-continuous alumina/ aluminum composite.Acta. Mater., 1996, 44(1):249-261
[10] H. X. Peng, Z. Fan, J. R. G. Evans. Bi-continuous metal matrix composites. Mat. Sci. Eng. A, 2001, 303:37-45
[11] J. D. Eshelby. The determination of the elastic field of an ellipsoidal inclusion and relateid problems. Proc. Roy. Soc. London, 1957, A241:376-396
[12] Z. Hashin. The elastic moduli of heterogeneous materials. J. Appl. Mech., 1962, 29:143-150
[13] R. Hill. Continuum micromechanics of elastoplastic polycrystals. J. Mech. Phys. Solids, 1965, 13: 89-101
[14] B. Budiansky. On the elastic moduli of some heterogeneous materials. J. Mech. Phys. Solids, 1965, 13: 223-227
[15] R. M. Christensen, K. H. Lo. Solution for the effective shear properties in three phase sphere and cylinder models. J. Mech. Phys. Solids, 1979, 27:315-330
[16] Y. Benveniste. A new approach to the application of Mori-Tanaka’s theory in composite materials. Mech. Mater., 1987, 6:147-157
[17] R. M. Christensen, A critical evaluation for a class of micromechanics models. J. Mech. Phys. Solids, 1990, 38(3):379-404
[18] J. C. Halpin, J. L. Kardos.The Halpin-Tsai equations: A review. Polymer Engineering and Science, 1976, 16(5):344-352
[19] S. Torquato. Effective stiffness tensor of composite media I: Exact series expansions. J. Mech. Phys. Solids, 1997, 45(9):1421-1448
[20] S.Torquato. Effective stiffness tensor of composite media II: Applications to isotropic dispersions. J.Mech.Phys.Solids, 1998, 46(8):1411-1440
[21] A. Bhattacharyya, Edmonton, Alberta, et al. Effective elastic moduli of two-phase transversely isotropic composites with aligned clustered fibers. Acta. Mechanica., 2000, 145:65-93
[22]杜丹旭.多相材料有效性质的理论研究:[博士学位论文],北京:清华大学,2000,2-10
[23] H. F. Zhao, G. K. Hu, T. J. Lu. Cross-property relations for two-phase planar composites. Comput. Mat. Sci., 2006, 35:408-415
[24] E. J. Garboczi, A. R. Day. An algorithm for computing the effective linear elastic properties of heterogeneous materials: Three-dimensional results for composites with equal phase Poisson ratios. J. Mech. Phys. Solids, 1995, 43:1349-1362
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[26] S. Torquato. Characterization of the microstructure of disordered media: A unified approach. Phys. Rev.B, 1987, 35(10):5385-5387
[27] B. L. Lu, S. Torquato.General formalism to characterize the microstructure of polydispersed random media. Phys. Rev. A, 1991, 43(4):2078-2080
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[30] S. Torquato, B. L. Lu. Chord-length distribution function for two-phase random media. Phys. Rev. E, 1993, 47(4): 2950-2953
[31] L. V. Gibiansky, S. Torquato. Geometrical-Parameter bounds on the effective moduli of composites. J. Mech. Phys. Solid, 1995, 43(10):1587-1613
[32] J. Quintanilla. Microstructures and properties of random heterogeneous materials: a review of theoretical results. Polymer Engineering and Science, 1999, 39(3):559-585
[33] Z. Hashin, Shtrikman. A variational approach to the theory of elastic behaviour of multiphase materials. J. Mech. Phys. Solids, 1963, 11:127-140
[34] Z. Hashin. On elastic behaviour of fibre reinforced materials of arbitrary transverse phase geometry. J. Mech. Phys.Solids, 1965, 13:119-134
[35] J. R. Brockenborough, S. Suresh, H. A. Wienecke. Deformation of metal-matrix composites with continuous fibers: Geometrical effects of fiber distribution and shape. Acta. Metall. Mat., 1991, 39(5):735-752
[36] Y. L. Shen, M. Finot, A. Needleman, et al. Effective elastic response of two-phase composites. Acta. Metall. Mat., 1994, 42(1):77-97
[37] M. Dong, S. Schmauder. Transverse mechanical behaviour of reinforced composites– FE modeling with embedded cell models. Comput. Mat. Sci., 1996, 5: 53-66
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[40] P. A. Crossley, L.M. Schwartz, J. R. Banavar. Image-based models of porous-media--application to vycor glass and carbonate rocks. Appl. Phys. Lett, 1991, 59:3553-3555
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[58] J. Chen, L. M. Xu, H. Li. Investigation on a direct modeling strategy for the effective elastic moduli prediction of composite material. Mat. Sci. Eng. A, 2008, doi: 10. 1016/ j. msea. 2008. 02. 024
[2]沈观林,胡更开.复合材料力学.北京:清华大学出版社,2006,287-304
[3]张双寅,刘济庆,于晓霞,等.复合材料结构的力学性能.北京:北京理工大学出版社,1992,1-9
[4]张锦,张乃恭.新型复合材料力学机理及其应用.北京:北京航空航天大学出版社,1992,1-13
[5] P. A. Buchan, J. F. Chen. Blast resistance of FPR composites and polymer strengthened concrete and masonry structures– A state-of-the-art review. Composites: Part B, 2007, 38:509-522
[6] M. Martina, G. Subramanyam, J. C. Weaver, et al. Developing macroporous bicontinuous materials as scaffolds for tissue engineering. Biomaterials, 2005, 26:5609-5616
[7] X. Q. Feng, Z. Tian, Y. H. Liu, et al. Effective elastic and plastic properties of interpenetrating multiphase composites. Applied Composite Materials, 2004,11:33-55
[8]陈建桥.复合材料力学概论.北京:科学出版社,2006,1-5
[9] G. S. Daehn, B. Starck, L. Xu, et al. Elastic and plastic behavior of a co-continuous alumina/ aluminum composite.Acta. Mater., 1996, 44(1):249-261
[10] H. X. Peng, Z. Fan, J. R. G. Evans. Bi-continuous metal matrix composites. Mat. Sci. Eng. A, 2001, 303:37-45
[11] J. D. Eshelby. The determination of the elastic field of an ellipsoidal inclusion and relateid problems. Proc. Roy. Soc. London, 1957, A241:376-396
[12] Z. Hashin. The elastic moduli of heterogeneous materials. J. Appl. Mech., 1962, 29:143-150
[13] R. Hill. Continuum micromechanics of elastoplastic polycrystals. J. Mech. Phys. Solids, 1965, 13: 89-101
[14] B. Budiansky. On the elastic moduli of some heterogeneous materials. J. Mech. Phys. Solids, 1965, 13: 223-227
[15] R. M. Christensen, K. H. Lo. Solution for the effective shear properties in three phase sphere and cylinder models. J. Mech. Phys. Solids, 1979, 27:315-330
[16] Y. Benveniste. A new approach to the application of Mori-Tanaka’s theory in composite materials. Mech. Mater., 1987, 6:147-157
[17] R. M. Christensen, A critical evaluation for a class of micromechanics models. J. Mech. Phys. Solids, 1990, 38(3):379-404
[18] J. C. Halpin, J. L. Kardos.The Halpin-Tsai equations: A review. Polymer Engineering and Science, 1976, 16(5):344-352
[19] S. Torquato. Effective stiffness tensor of composite media I: Exact series expansions. J. Mech. Phys. Solids, 1997, 45(9):1421-1448
[20] S.Torquato. Effective stiffness tensor of composite media II: Applications to isotropic dispersions. J.Mech.Phys.Solids, 1998, 46(8):1411-1440
[21] A. Bhattacharyya, Edmonton, Alberta, et al. Effective elastic moduli of two-phase transversely isotropic composites with aligned clustered fibers. Acta. Mechanica., 2000, 145:65-93
[22]杜丹旭.多相材料有效性质的理论研究:[博士学位论文],北京:清华大学,2000,2-10
[23] H. F. Zhao, G. K. Hu, T. J. Lu. Cross-property relations for two-phase planar composites. Comput. Mat. Sci., 2006, 35:408-415
[24] E. J. Garboczi, A. R. Day. An algorithm for computing the effective linear elastic properties of heterogeneous materials: Three-dimensional results for composites with equal phase Poisson ratios. J. Mech. Phys. Solids, 1995, 43:1349-1362
[25] I. Sevostianov, N. Yilmaz, V. Kushch, et al. Effective elastic properties of matrix composites with transversely-isotropic phase. International Journal of Solids and Structures, 2005, 42: 455-476
[26] S. Torquato. Characterization of the microstructure of disordered media: A unified approach. Phys. Rev.B, 1987, 35(10):5385-5387
[27] B. L. Lu, S. Torquato.General formalism to characterize the microstructure of polydispersed random media. Phys. Rev. A, 1991, 43(4):2078-2080
[28] B. L. Lu, S. Torquato. Lineal-path function for random heterogeneous materials. Phys. Rev. A, 1992, 45(2):922-929
[29] B. L. Lu, S. Torquato. Lineal-path function for random heterogeneous materials II: Effect of polydispersivity. Phys. Rev. A, 1992, 45(10):7292-7301
[30] S. Torquato, B. L. Lu. Chord-length distribution function for two-phase random media. Phys. Rev. E, 1993, 47(4): 2950-2953
[31] L. V. Gibiansky, S. Torquato. Geometrical-Parameter bounds on the effective moduli of composites. J. Mech. Phys. Solid, 1995, 43(10):1587-1613
[32] J. Quintanilla. Microstructures and properties of random heterogeneous materials: a review of theoretical results. Polymer Engineering and Science, 1999, 39(3):559-585
[33] Z. Hashin, Shtrikman. A variational approach to the theory of elastic behaviour of multiphase materials. J. Mech. Phys. Solids, 1963, 11:127-140
[34] Z. Hashin. On elastic behaviour of fibre reinforced materials of arbitrary transverse phase geometry. J. Mech. Phys.Solids, 1965, 13:119-134
[35] J. R. Brockenborough, S. Suresh, H. A. Wienecke. Deformation of metal-matrix composites with continuous fibers: Geometrical effects of fiber distribution and shape. Acta. Metall. Mat., 1991, 39(5):735-752
[36] Y. L. Shen, M. Finot, A. Needleman, et al. Effective elastic response of two-phase composites. Acta. Metall. Mat., 1994, 42(1):77-97
[37] M. Dong, S. Schmauder. Transverse mechanical behaviour of reinforced composites– FE modeling with embedded cell models. Comput. Mat. Sci., 1996, 5: 53-66
[38] D. Fang, H. Qi, S. Tu. Elastic an plastic properties of metal-matrix composites: Geometrical effects of particles. Comput. Mat. Sci., 1996, 6: 303-309
[39] A. P. Roberts. Statistical reconstruction of three-dimensional porous media from two-dimensional images. Phys. Rev. E, 1997, 56(3):3203-3212
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