复合材料层合板脱层分析界面元性能研究与脱层数值模拟
|
摘要
复合材料层合结构因其优异的力学性能而被越来越广泛地应用于不同的工程领域。虽然复合材料在纤维增强方向具有高比强度、高比模量等特点,但是它们铺层间的强度很低,尤其是在横向低速冲击等荷载下,层合结构内部会产生不可忽视的损伤,其力学性能将严重退化。脱层是复合材料层合板最为普遍的损伤形式之一,也是使层合结构的力学性能发生严重退化主要原因。随着复合材料层合结构的广泛应用,脱层破坏损伤分析也受到更多的关注。鉴于复合材料层合板脱层损伤主要是由界面层的破坏引起,对界面层材料的力学性能的分析以及合理的模拟与计算是分析层合结构脱层损伤的有效途径。
本文的主要研究工作有以下三个方面。
1首先,建立了一种基于粘接层的实际厚度和材料特性确定弹簧界面元等效初始刚度的计算模型。通过把所给弹簧界面等效刚度计算方法用于ANSYS中的非线性弹簧单元COMBIN39以及COMBIN14,对复合材料双悬臂梁(DCB)以及端边切口弯曲(ENF)实验进行了脱层扩展的有限元数值模拟;数值脱层预测结果与实验数据吻合。算例也表明,文中所给弹簧界面元等效刚度计算和长度确定方法不仅计算简单和准确,并且计算稳定。
2其次,采用了数值模拟的方法研究了材料模型和强度准则对ANSYS中实体壳单元模拟粘接层的脱层预测结果的影响。文中分别在两种不同的材料模型基础上建立了相应的强度准则,并利用其单元生死功能有效模拟了脱层的物理过程。数值计算结果表明:本文所选取的不同强度准则下的计算结果均与实验吻合的较好;而采取固体壳单元以及单元生死功能来模拟由粘接层破坏引起的脱层的方法行之有效。
3基于拟协调元理论的假设应变方法,从广义控制方程的弱形式出发,推导并构造了无各类自锁现象的三维实体壳单元。将所构造的单元应用于具有板壳类拓扑结构特征的问题中,计算结果表明该单元不仅精度高、收敛速度快,且具有计算量小的特点。
本论文所给出的新的弹簧界面元等效初始刚度的计算模型和粘接层材料模型的建模方法对复合材料层合板结构脱层损伤数值模拟有很大的应用意义,所推导的拟协调实体壳单元经过完善可用于层合板的高效数值分析。
Composite laminated structures have been more and more used in various industries because of their very outstanding mechanical properties. However, the transverse strength of laminated plates is very weak, especially at the adhesive layers which are the resin rich zones at the interlaminar interface. As a result, laminated plates would suffer damages, such as matrix cracking, debonding and inter-ply delaminations especially under the action of low-velocity transverse impact Among these different damages, the delamination at the interlaminar interfaces with different fiber orientations is the most dangerous and could be one main source that the mechanical capability of the structures is seriously degenerated. Therefore, it is very desireable and important to study the mechanical behavior of the interlaminar interfaces in order to obtain realistic failure prediction.
This thesis is composed of the followings three major parts.
1. A simple and natural model to evaluate the stiffness of the spring interface elements, which is based on the physics and the geometry of the adhesive layers, is proposed. The proposed model is applied to the nonlinear spring element COMBIN39 and COMBIN14 in ANSYS to simulate the delamination propagations of the standard DCB and ENF tests. The resulting delamination predictions agree well with the experimental results, and the numerical examples demonstrated that the proposed model is not only simple but also an accurate and robust way to evaluate the stiffness of the spring interface elements used for the delamination simulations of laminated plates.
2. The influence of the material modeling of the solid-shell interface element on the delaminations simulations of laminated plates is studied. The predicted results agree well with experimental results. It is shown that the solid-shell interface element is an efficient model for the adhesive layers of laminated plates.
3. Based on the assumed element strain fields of the Quasi-Conforming Element Techniques, a 3-D solid-shell finite element, designed for the 3-D analysis of plates is developed. The element stiffness matrix of the present solid-shell element is would be free of shear locking as well as numerical ill-conditioning. The presented solid-shell element is applied to the analysis of mechanical behavior of plates in the paper. The numerical results show that the element processes high accuracy as well as high computational efficiency. Therefore, the solid-shell element presented in this thesis provides a brand new element for the delaminations simulation of laminated plates.
The model of the stiffness evaluation of spring interface element and the material modeling approach for the solid-shell interface element presented in the thesis would be of great help in the delaminations simulating of composite laminated plates.
本文的主要研究工作有以下三个方面。
1首先,建立了一种基于粘接层的实际厚度和材料特性确定弹簧界面元等效初始刚度的计算模型。通过把所给弹簧界面等效刚度计算方法用于ANSYS中的非线性弹簧单元COMBIN39以及COMBIN14,对复合材料双悬臂梁(DCB)以及端边切口弯曲(ENF)实验进行了脱层扩展的有限元数值模拟;数值脱层预测结果与实验数据吻合。算例也表明,文中所给弹簧界面元等效刚度计算和长度确定方法不仅计算简单和准确,并且计算稳定。
2其次,采用了数值模拟的方法研究了材料模型和强度准则对ANSYS中实体壳单元模拟粘接层的脱层预测结果的影响。文中分别在两种不同的材料模型基础上建立了相应的强度准则,并利用其单元生死功能有效模拟了脱层的物理过程。数值计算结果表明:本文所选取的不同强度准则下的计算结果均与实验吻合的较好;而采取固体壳单元以及单元生死功能来模拟由粘接层破坏引起的脱层的方法行之有效。
3基于拟协调元理论的假设应变方法,从广义控制方程的弱形式出发,推导并构造了无各类自锁现象的三维实体壳单元。将所构造的单元应用于具有板壳类拓扑结构特征的问题中,计算结果表明该单元不仅精度高、收敛速度快,且具有计算量小的特点。
本论文所给出的新的弹簧界面元等效初始刚度的计算模型和粘接层材料模型的建模方法对复合材料层合板结构脱层损伤数值模拟有很大的应用意义,所推导的拟协调实体壳单元经过完善可用于层合板的高效数值分析。
Composite laminated structures have been more and more used in various industries because of their very outstanding mechanical properties. However, the transverse strength of laminated plates is very weak, especially at the adhesive layers which are the resin rich zones at the interlaminar interface. As a result, laminated plates would suffer damages, such as matrix cracking, debonding and inter-ply delaminations especially under the action of low-velocity transverse impact Among these different damages, the delamination at the interlaminar interfaces with different fiber orientations is the most dangerous and could be one main source that the mechanical capability of the structures is seriously degenerated. Therefore, it is very desireable and important to study the mechanical behavior of the interlaminar interfaces in order to obtain realistic failure prediction.
This thesis is composed of the followings three major parts.
1. A simple and natural model to evaluate the stiffness of the spring interface elements, which is based on the physics and the geometry of the adhesive layers, is proposed. The proposed model is applied to the nonlinear spring element COMBIN39 and COMBIN14 in ANSYS to simulate the delamination propagations of the standard DCB and ENF tests. The resulting delamination predictions agree well with the experimental results, and the numerical examples demonstrated that the proposed model is not only simple but also an accurate and robust way to evaluate the stiffness of the spring interface elements used for the delamination simulations of laminated plates.
2. The influence of the material modeling of the solid-shell interface element on the delaminations simulations of laminated plates is studied. The predicted results agree well with experimental results. It is shown that the solid-shell interface element is an efficient model for the adhesive layers of laminated plates.
3. Based on the assumed element strain fields of the Quasi-Conforming Element Techniques, a 3-D solid-shell finite element, designed for the 3-D analysis of plates is developed. The element stiffness matrix of the present solid-shell element is would be free of shear locking as well as numerical ill-conditioning. The presented solid-shell element is applied to the analysis of mechanical behavior of plates in the paper. The numerical results show that the element processes high accuracy as well as high computational efficiency. Therefore, the solid-shell element presented in this thesis provides a brand new element for the delaminations simulation of laminated plates.
The model of the stiffness evaluation of spring interface element and the material modeling approach for the solid-shell interface element presented in the thesis would be of great help in the delaminations simulating of composite laminated plates.
引文
[1]杜善义,先进复合材料与航空航天,复合材料学报, 2007, 24(1): 1~12.
[2] Robert I. M., Advanced composite structures research in Australia, Composite Structures, 2002, 57: 3~10.
[3]陈绍杰,浅谈空客A380的复合材料应用,高科技纤维与应用, 2008, 33(4):1~4, 24.
[4]杨光松,损伤力学与复合材料损伤,北京:国防工业出版社, 1995, 135~148.
[5]崔海坡,温卫东,崔海涛,复合材料层合板冲击损伤及剩余强度研究进展,材料科学与工程学报, 2005, 23(3): 466~472.
[6]冯世宁,陈浩然,含脱层损伤复合材料层合板非线性动力稳定性,复合材料学报, 2006, 23(1): 154~160.
[7] Reissner E., The effect of transverse shear deformation on the bending of elastic plates, ASME Journal of Applied Mechanics, 1945, 12: A69-A77.
[8] Mindlin R. D., Influence of rotator inertia and shear on flexural motions of isotropic elastic plates, ASME Journal of Applied Mechanics, 1951, 18: 31~38.
[9] Whitney J. M., Pagano N. J., Shear deformation in heterogeneous anisotropic plates, Journal of Applied Mechanics, 1970, 37(4): 1031~1036.
[10] Pagano N. J., Exact solutions for rectangular bidirectional and sandwich plates, Journal of Composite Material , 1970, 4(4): 20~34.
[11] Levinson M. A., An accurate simple theory of statics and dynamics of elastic plates. Mechanics Research Communications, 1980, 7(6): 343~350.
[12] Reddy J.N., A simple higher-order theory for laminated composite plates, ASME Journal of Applied Mechanics, 1984, 51: 745~752.
[13] Reddy J. N., Khdeir A. A. and Librescu L., The Levy type solutions for symmetric rectangular composite plates using the first-order shear deformation theory, Journal of Applied Mechanics, 1987, 54( 3) : 740~742.
[14] Khdeir A. A., Reddy J. N. and Librescu L., Analytical solution of refined deformation theory for rectangular composite plates, International Journal of Solids & Structures , 1987, 23(10): 1447~1463.
[15] Shi G.., A new simple third-order shear deformation theory of plates, International Journal of Solids & Structures, 2007, 44: 4399~4417.
[16] Beakou A., Touratier M., A rectangular finite element for analyzing composite multilayered shallow shells in statics, vibration and buckling, International Journal of Numerical Meth. Engineering, 1993, 36: 627~653.
[17] Touratier M., An efficient standard plate theory, International Journal of Engineering Science, 1991, 29(8): 901~916.
[18] Karama K. et al., Mechanics behavior of laminated composite beam by the new multi-layered laminated composite structures model with transverse shear stress continuity, International Journal of Solids & Structures, 2003, 40: 1525~1546.
[19] Ye Kaiyuan and Deng Liangbo, Free vibration of composite laminated plates clamped at four edges, Chinese Science Bulletin, 1989, 34(1): 25~30.
[20]夏传友,闻立洲,各种边界条件对称正交复合材料层板自由振动的解析解,复合材料学报, 1991, 8(4): 88~99.
[21] Shi G. and Voyiadjis, G. Z., A sixth-order theory of shear deformable beams with variational consistent boundary conditions, ASME. Journal of Applied Mechanics, 2011, 78(2): 021019-1~11.
[22] Bakis C. E., Stinchcomb H. R., and Reifsnider K. L., Damage Initiation and Growth in Notched Laminates Under Reserved Cyclic Loading, Composite Materials: Fatigue and Fracture, Second Volume, ASTM STP 1012. PaulA. Lagace, Ed. American Society of Testing and Materials, Philadelphi, 1989, 2: 66~83.
[23] Martin R. J., Sandhu R. S. and Palazotto A. N., Experimental and analytical comparisons of failure in thermo plastic composite laminates, Experimental Mechanics, 1994, March: 53~65.
[24] Herakovich C. T., Post D., Buczek M. B., et al, Free edge strain concentrations in real composite laminates: experimental theoretical correlation, Journal of Applied Mechanics, 1985, 52: 787~793.
[25]佟景伟,谢马岳,王世斌等,复合材料层合板的数值分析与实验,实验力学,2000, 15(2): 182~187.
[26]贺跃进,张恒,复合材料层合板脱层诊断的实验研究,材料导报,2008, 22(5): 140~142.
[27] Cui W., Wisnom M. R., A combined stress-based and fracture-mechanics-based model for predicting delamination in composites, Composites 1993, 24: 67~74.
[28] Lammerant L., Verpoestg, Modeling of the interaction between matrix cracks and delaminations during impact of composite plates, Composites Science and Technology, 1996, 56(10): 1171~1178.
[29] Iannucci L., Dynamic delamination modeling using interface elements, Computers and Structures, 2006, 2(84): 1029~1048.
[30] Li D. S., Winsnom M. R., Modeling damage in vitiation and progapation in composites using interface elements, Computer Aided Design in Composite Materials Technology, 1994: 213~220.
[31] Tvergaard V., Hutchinson J. W., The in?uence of plasticity on mixed mode interface toughness. J. Mech Phys Solids, 1993, 41: 1119~35.
[32] Adams D. F., A micromechanics analysis of the influence of the interface on the performance of polymer-matrix composites, J. Reinf. Plastics and Composites, 1987: 66~87.
[33] Zywicz E., Local stress and deformation due to fabrication and transverse loading in an ideal continuously reinforced Graphite/Aluminum metal matrix composites.. Sci .Thesis, Dept. of Mech. Eng. MIT, USA, 1986.
[34]赵鹏,石广玉,层合板界面层的弹簧界面元等效刚度计算模型,计算力学学报, 2011, 28, Sup.: 131~135,146.
[35]赵鹏,石广玉,简单而准确的脱层分析弹簧界面元等效刚度计算方法, 2009年度复合材料力学研讨会,长沙: 28~30.
[36] Papanicolaou G. C., Sravropoulos C. D., New approach for residual compre-strength prediction of impacted CFRP laminates, Composites, 1995, 26: 517~523.
[37] Olsson R., et al., Delamination threshold load for dynamic impact on plates, Int. J. Solids Structures, 2006, 43: 3124~3141.
[38] Zheng S., Sun C. T., A double-plate finite-element model for the impact-induced delamination problem, Compos. Sci. Tech., 1995, 53: 111~118.
[39] Choi H. Y., Chang F. K., A model for predicting damage in graphite/epoxy laminates composites resulting from low-velocity point impact, J. Compos. Mater, 1992, 26: 2134~2169.
[40] Hou J. P., Petrinic N., Ruiz C., Prediction of impact damage in composite plates. Composite Science and Technology, 2000, 60(2): 27~81.
[41]张彦,来新民,朱平,复合材料铺层板低速冲击作用下损伤的有限元分析,上海交通大学学报, 2006, 40(8): 1348~1353.
[42]关志东,郭渊,含缺陷的复合材料层合板低速冲击过程的有限元模拟,复合材料学报, 2006, 23(2): 181~184.
[43] Camando P. P., Davila C. G., Mixed-mode decohesion finite elements for the simulation of delamination in composite materials, NASA-TM_2002_211737, Hampton VA, 2002.
[44] Collombet F., A three dimensional modelling of low velocity impact damage in composite laminates, Int. J. Num Meth Eng., 1996, 39 (9): 1491~1516.
[45] Collombet F., Damage criteria for study impacted composite laminates, Composites Sci. and Tech., 1998, 58(5): 679~686.
[46] Melin L. G., A study of model cracks by high magnification more inter-geometry, Composite Sci. and Tech., 1998, 58 (34): 51~525.
[47] Melin L. G., Tensile impact delamination of a cross-ply interface studied by moiréphotography, Fatigue Fract Eng. Mater Struct., 1995, 18(10): 1101~1114.
[48] Reedy E. D., Mello F. J., Guess T. R., Modeling the initiation and growth of delaminations in composite structures, Journal of Composite Materials, 1997, 31(8): 812~831.
[49] Belingardi G., Gugliotta A., Vadori R., Numerical simulation of fragmentation of composite materials plates due to impact, Int. J. Impact Engng, 1998, 21(5): 335~347.
[50] Shi G., On the interface element for delamination simulation of laminated plates subjected to low-velocity impact, Computational Mechanics, Eds. Yao ZH, Yuan MW, Tsinghua University Press & Springer Press, 2007: 229.
[51] Argyris J., Tenck L., A natural triangular layered element for bending analysis of isotropic, sandwich, laminated composite and hybrid plates, Computer Meth. Appl. Mech. Eng., 1993, 109: 197~218.
[52] Iannucci L., Willows M. L., An energy based damage mechanics approach to modeling impact onto woven composite material Part-1: Numerical models, Composites, Part A: Applied Science and Manufacturing, 2006, 37: 2041~2056.
[53] Zou Z., Reid S. R., Li S., A continuum damage model for delaminations in laminated composites, Journal of the Mechanics and Physics of Solids, 2003, 51: 333~356.
[54] Robinson P., Song D. Q., A modified DCB specimen for mode I testing of multidirectional laminates, Composite Materials, 1992, 28: 1554~1557.
[55]张海波,新3D界面元在复合材料层合板分层损伤数值模拟中的应用,硕士学位论文,天津大学, 2009.
[56] ANSYS, Inc, ANSYS 12.0, Canonsburg, PA, USA.2007.
[57] Sze K. Y., Yao L., Pian T. H. H., An eighteen-node hybrid-stress shell element for homogeneous and laminated structures, Finite Element in Analysis and Design, 2002, 38(7): 353~374.
[58] Sze K. Y., Chan W. K., A six-node pentagonal assumed natural strain solid-shell element, Finite Elements in Analysis and Design, 2001, 37(8): 639~655.
[59] Kulikov G. M., Plotnikova S. V., Geometrically exact assumed stress-strain multilayered solid-shell elements based on the 3D analytical integration, Computer & Structures, 2006, 84: 1275~1287.
[60]郑世杰,余锦炎,几何非线性复合材料层合板固体壳单元,复合材料学报, 2003, 20(3): 7~12.
[61] Klinkel S. et al, A robust non-linear solid-shell element based on a mixed variational formulation, Computer Methods for Applied Mechanical Engineering, 2006, 195: 179~201.
[62] Shi G. and Voyiadjis G., Simple and efficient shear flexible two-node arch/beam and four-node cylindrical shell/plate finite element, International Journal for Numerical Methods in Engineering, 1990, 31: 759~766.
[63] Shi G. and Voyiadjis G., Efficient and accurate four-node quadrilateral C0 plate bending element based on assumed strain fields, International Journal for Numerical Methods in Engineering, 1990, 32: 1041~1055.
[64] Shi G. and Voyiadjis G., Geometrically nonlinear analysis of plates by assumed strain element with explicit tangent stiffness matrix, Computers & Structures, 1991, 41: 757~763.
[65]刘迎曦,石广玉,唐立民,拟协调厚薄通用梁、板单元,大连工学院学报,1984, 23: 79~85.
[66]陈万吉,刘迎曦,唐立民,拟协调元列式,大连工学院学报,1980, 19(2): 37~50.
[67]唐立民,陈万吉,刘迎曦,有限元分析中的拟协调元,大连工学院学报,1980, 19(2): 19~35.
[68] Shi G. and Voyiadjis G., A simple non-layered finite element for the elasto-plastic analysis of shear flexible plates, International Journal for Numerical Methods in Engineering, 1992, 33: 85~99.
[69] Shi G. and Voyiadjis G., A simple C0 quadrilateral thick/thin shell element based on the refined shell theory and the assumed strain fields, Int. J. Solids Structure, 1991, 27(3): 283~298.
[70] Shi G., Lam K. Y., and Tay T. E., Assumed strain quadrilateral C0 laminated plate element based on third-order shear deformation theory, Structural Engineering and Mechanics, 1999, 8(6): 623~627.
[71]王勖成,有限单元法,北京:清华大学出版社, 2003, 131~141.
[72]庄茁,曲小川,廖剑辉等,基于ABAQUS的有限元分析和应用,北京:清华大学出版社, 2009, 492~495.
[73] Timoshenko S., Woinowsky-Krieger S., Theory of Plates and Shells, McGRAW-HILL BOOK COMPANY, 1987, 197-198.
[2] Robert I. M., Advanced composite structures research in Australia, Composite Structures, 2002, 57: 3~10.
[3]陈绍杰,浅谈空客A380的复合材料应用,高科技纤维与应用, 2008, 33(4):1~4, 24.
[4]杨光松,损伤力学与复合材料损伤,北京:国防工业出版社, 1995, 135~148.
[5]崔海坡,温卫东,崔海涛,复合材料层合板冲击损伤及剩余强度研究进展,材料科学与工程学报, 2005, 23(3): 466~472.
[6]冯世宁,陈浩然,含脱层损伤复合材料层合板非线性动力稳定性,复合材料学报, 2006, 23(1): 154~160.
[7] Reissner E., The effect of transverse shear deformation on the bending of elastic plates, ASME Journal of Applied Mechanics, 1945, 12: A69-A77.
[8] Mindlin R. D., Influence of rotator inertia and shear on flexural motions of isotropic elastic plates, ASME Journal of Applied Mechanics, 1951, 18: 31~38.
[9] Whitney J. M., Pagano N. J., Shear deformation in heterogeneous anisotropic plates, Journal of Applied Mechanics, 1970, 37(4): 1031~1036.
[10] Pagano N. J., Exact solutions for rectangular bidirectional and sandwich plates, Journal of Composite Material , 1970, 4(4): 20~34.
[11] Levinson M. A., An accurate simple theory of statics and dynamics of elastic plates. Mechanics Research Communications, 1980, 7(6): 343~350.
[12] Reddy J.N., A simple higher-order theory for laminated composite plates, ASME Journal of Applied Mechanics, 1984, 51: 745~752.
[13] Reddy J. N., Khdeir A. A. and Librescu L., The Levy type solutions for symmetric rectangular composite plates using the first-order shear deformation theory, Journal of Applied Mechanics, 1987, 54( 3) : 740~742.
[14] Khdeir A. A., Reddy J. N. and Librescu L., Analytical solution of refined deformation theory for rectangular composite plates, International Journal of Solids & Structures , 1987, 23(10): 1447~1463.
[15] Shi G.., A new simple third-order shear deformation theory of plates, International Journal of Solids & Structures, 2007, 44: 4399~4417.
[16] Beakou A., Touratier M., A rectangular finite element for analyzing composite multilayered shallow shells in statics, vibration and buckling, International Journal of Numerical Meth. Engineering, 1993, 36: 627~653.
[17] Touratier M., An efficient standard plate theory, International Journal of Engineering Science, 1991, 29(8): 901~916.
[18] Karama K. et al., Mechanics behavior of laminated composite beam by the new multi-layered laminated composite structures model with transverse shear stress continuity, International Journal of Solids & Structures, 2003, 40: 1525~1546.
[19] Ye Kaiyuan and Deng Liangbo, Free vibration of composite laminated plates clamped at four edges, Chinese Science Bulletin, 1989, 34(1): 25~30.
[20]夏传友,闻立洲,各种边界条件对称正交复合材料层板自由振动的解析解,复合材料学报, 1991, 8(4): 88~99.
[21] Shi G. and Voyiadjis, G. Z., A sixth-order theory of shear deformable beams with variational consistent boundary conditions, ASME. Journal of Applied Mechanics, 2011, 78(2): 021019-1~11.
[22] Bakis C. E., Stinchcomb H. R., and Reifsnider K. L., Damage Initiation and Growth in Notched Laminates Under Reserved Cyclic Loading, Composite Materials: Fatigue and Fracture, Second Volume, ASTM STP 1012. PaulA. Lagace, Ed. American Society of Testing and Materials, Philadelphi, 1989, 2: 66~83.
[23] Martin R. J., Sandhu R. S. and Palazotto A. N., Experimental and analytical comparisons of failure in thermo plastic composite laminates, Experimental Mechanics, 1994, March: 53~65.
[24] Herakovich C. T., Post D., Buczek M. B., et al, Free edge strain concentrations in real composite laminates: experimental theoretical correlation, Journal of Applied Mechanics, 1985, 52: 787~793.
[25]佟景伟,谢马岳,王世斌等,复合材料层合板的数值分析与实验,实验力学,2000, 15(2): 182~187.
[26]贺跃进,张恒,复合材料层合板脱层诊断的实验研究,材料导报,2008, 22(5): 140~142.
[27] Cui W., Wisnom M. R., A combined stress-based and fracture-mechanics-based model for predicting delamination in composites, Composites 1993, 24: 67~74.
[28] Lammerant L., Verpoestg, Modeling of the interaction between matrix cracks and delaminations during impact of composite plates, Composites Science and Technology, 1996, 56(10): 1171~1178.
[29] Iannucci L., Dynamic delamination modeling using interface elements, Computers and Structures, 2006, 2(84): 1029~1048.
[30] Li D. S., Winsnom M. R., Modeling damage in vitiation and progapation in composites using interface elements, Computer Aided Design in Composite Materials Technology, 1994: 213~220.
[31] Tvergaard V., Hutchinson J. W., The in?uence of plasticity on mixed mode interface toughness. J. Mech Phys Solids, 1993, 41: 1119~35.
[32] Adams D. F., A micromechanics analysis of the influence of the interface on the performance of polymer-matrix composites, J. Reinf. Plastics and Composites, 1987: 66~87.
[33] Zywicz E., Local stress and deformation due to fabrication and transverse loading in an ideal continuously reinforced Graphite/Aluminum metal matrix composites.. Sci .Thesis, Dept. of Mech. Eng. MIT, USA, 1986.
[34]赵鹏,石广玉,层合板界面层的弹簧界面元等效刚度计算模型,计算力学学报, 2011, 28, Sup.: 131~135,146.
[35]赵鹏,石广玉,简单而准确的脱层分析弹簧界面元等效刚度计算方法, 2009年度复合材料力学研讨会,长沙: 28~30.
[36] Papanicolaou G. C., Sravropoulos C. D., New approach for residual compre-strength prediction of impacted CFRP laminates, Composites, 1995, 26: 517~523.
[37] Olsson R., et al., Delamination threshold load for dynamic impact on plates, Int. J. Solids Structures, 2006, 43: 3124~3141.
[38] Zheng S., Sun C. T., A double-plate finite-element model for the impact-induced delamination problem, Compos. Sci. Tech., 1995, 53: 111~118.
[39] Choi H. Y., Chang F. K., A model for predicting damage in graphite/epoxy laminates composites resulting from low-velocity point impact, J. Compos. Mater, 1992, 26: 2134~2169.
[40] Hou J. P., Petrinic N., Ruiz C., Prediction of impact damage in composite plates. Composite Science and Technology, 2000, 60(2): 27~81.
[41]张彦,来新民,朱平,复合材料铺层板低速冲击作用下损伤的有限元分析,上海交通大学学报, 2006, 40(8): 1348~1353.
[42]关志东,郭渊,含缺陷的复合材料层合板低速冲击过程的有限元模拟,复合材料学报, 2006, 23(2): 181~184.
[43] Camando P. P., Davila C. G., Mixed-mode decohesion finite elements for the simulation of delamination in composite materials, NASA-TM_2002_211737, Hampton VA, 2002.
[44] Collombet F., A three dimensional modelling of low velocity impact damage in composite laminates, Int. J. Num Meth Eng., 1996, 39 (9): 1491~1516.
[45] Collombet F., Damage criteria for study impacted composite laminates, Composites Sci. and Tech., 1998, 58(5): 679~686.
[46] Melin L. G., A study of model cracks by high magnification more inter-geometry, Composite Sci. and Tech., 1998, 58 (34): 51~525.
[47] Melin L. G., Tensile impact delamination of a cross-ply interface studied by moiréphotography, Fatigue Fract Eng. Mater Struct., 1995, 18(10): 1101~1114.
[48] Reedy E. D., Mello F. J., Guess T. R., Modeling the initiation and growth of delaminations in composite structures, Journal of Composite Materials, 1997, 31(8): 812~831.
[49] Belingardi G., Gugliotta A., Vadori R., Numerical simulation of fragmentation of composite materials plates due to impact, Int. J. Impact Engng, 1998, 21(5): 335~347.
[50] Shi G., On the interface element for delamination simulation of laminated plates subjected to low-velocity impact, Computational Mechanics, Eds. Yao ZH, Yuan MW, Tsinghua University Press & Springer Press, 2007: 229.
[51] Argyris J., Tenck L., A natural triangular layered element for bending analysis of isotropic, sandwich, laminated composite and hybrid plates, Computer Meth. Appl. Mech. Eng., 1993, 109: 197~218.
[52] Iannucci L., Willows M. L., An energy based damage mechanics approach to modeling impact onto woven composite material Part-1: Numerical models, Composites, Part A: Applied Science and Manufacturing, 2006, 37: 2041~2056.
[53] Zou Z., Reid S. R., Li S., A continuum damage model for delaminations in laminated composites, Journal of the Mechanics and Physics of Solids, 2003, 51: 333~356.
[54] Robinson P., Song D. Q., A modified DCB specimen for mode I testing of multidirectional laminates, Composite Materials, 1992, 28: 1554~1557.
[55]张海波,新3D界面元在复合材料层合板分层损伤数值模拟中的应用,硕士学位论文,天津大学, 2009.
[56] ANSYS, Inc, ANSYS 12.0, Canonsburg, PA, USA.2007.
[57] Sze K. Y., Yao L., Pian T. H. H., An eighteen-node hybrid-stress shell element for homogeneous and laminated structures, Finite Element in Analysis and Design, 2002, 38(7): 353~374.
[58] Sze K. Y., Chan W. K., A six-node pentagonal assumed natural strain solid-shell element, Finite Elements in Analysis and Design, 2001, 37(8): 639~655.
[59] Kulikov G. M., Plotnikova S. V., Geometrically exact assumed stress-strain multilayered solid-shell elements based on the 3D analytical integration, Computer & Structures, 2006, 84: 1275~1287.
[60]郑世杰,余锦炎,几何非线性复合材料层合板固体壳单元,复合材料学报, 2003, 20(3): 7~12.
[61] Klinkel S. et al, A robust non-linear solid-shell element based on a mixed variational formulation, Computer Methods for Applied Mechanical Engineering, 2006, 195: 179~201.
[62] Shi G. and Voyiadjis G., Simple and efficient shear flexible two-node arch/beam and four-node cylindrical shell/plate finite element, International Journal for Numerical Methods in Engineering, 1990, 31: 759~766.
[63] Shi G. and Voyiadjis G., Efficient and accurate four-node quadrilateral C0 plate bending element based on assumed strain fields, International Journal for Numerical Methods in Engineering, 1990, 32: 1041~1055.
[64] Shi G. and Voyiadjis G., Geometrically nonlinear analysis of plates by assumed strain element with explicit tangent stiffness matrix, Computers & Structures, 1991, 41: 757~763.
[65]刘迎曦,石广玉,唐立民,拟协调厚薄通用梁、板单元,大连工学院学报,1984, 23: 79~85.
[66]陈万吉,刘迎曦,唐立民,拟协调元列式,大连工学院学报,1980, 19(2): 37~50.
[67]唐立民,陈万吉,刘迎曦,有限元分析中的拟协调元,大连工学院学报,1980, 19(2): 19~35.
[68] Shi G. and Voyiadjis G., A simple non-layered finite element for the elasto-plastic analysis of shear flexible plates, International Journal for Numerical Methods in Engineering, 1992, 33: 85~99.
[69] Shi G. and Voyiadjis G., A simple C0 quadrilateral thick/thin shell element based on the refined shell theory and the assumed strain fields, Int. J. Solids Structure, 1991, 27(3): 283~298.
[70] Shi G., Lam K. Y., and Tay T. E., Assumed strain quadrilateral C0 laminated plate element based on third-order shear deformation theory, Structural Engineering and Mechanics, 1999, 8(6): 623~627.
[71]王勖成,有限单元法,北京:清华大学出版社, 2003, 131~141.
[72]庄茁,曲小川,廖剑辉等,基于ABAQUS的有限元分析和应用,北京:清华大学出版社, 2009, 492~495.
[73] Timoshenko S., Woinowsky-Krieger S., Theory of Plates and Shells, McGRAW-HILL BOOK COMPANY, 1987, 197-198.