低速冲击下损伤层合/功能梯度板壳的非线性动力学研究
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摘要
本论文以层合复合材料/功能梯度板壳为研究对象,综合考虑损伤效应、物理非线性、几何非线性、环境温度等因素,系统地研究了低速冲击下层合复合材料/功能梯度板壳的动力响应、基体损伤、界面损伤等问题,揭示了层合复合材料/功能梯度板壳在低速冲击下的非线性动力学特性和损伤机理。其研究工作不仅丰富了接触力学和冲击损伤理论,同时,具有重要的工程应用价值。本论文的主要研究内容如下。
基于中厚壳几何非线性理论和损伤理论,采用应变描述的失效准则,导出了层合中厚浅球壳含基体损伤和纤维-基体剪切损伤的损伤本构关系,并假设每一单层沿厚度方向有相同的损伤程度,建立了低速冲击下用位移分量表示的轴对称正交对称铺设层合中厚浅球壳的非线性运动控制方程。将未知位移函数在空间域采用正交配置法离散,时间域采用Newmark-β方法离散,对整个问题进行迭代求解。具体讨论了基体损伤、冲击速度以及结构的几何尺寸等对低速冲击下结构所受的冲击载荷和结构的非线性动力响应的影响。
根据Reddy的高阶剪切变形理论,建立复合材料层合中厚板的位移场和Von Karman型几何关系,并采用Talreja张量内变量损伤模型,建立了具基体损伤复合材料层合中厚板的弹性损伤本构关系;然后基于Hertz接触理论,得到考虑法向接触力和切向接触力同时变化时的接触力和接触变形关系,并据此建立了弹性小球低速斜冲击下复合材料层合中厚板的冲击动力学模型。综合运用有限差分法、Newmark-β和迭代法,求解了低速冲击下具损伤复合材料层合中厚板的非线性动力响应问题,讨论了冲击角度、冲击速度和结构几何尺寸等对斜冲击作用下复合材料层合中厚板的非线性动力响应和结构损伤的影响。
基于弹塑性理论,建立了一个与应力球张量有关的正交各向异性材料的混合硬化屈服准则,并根据几何非线性理论和损伤理论,采用基于应变描述的Hashin失效准则,导出了层合中厚浅球壳含层内损伤的增量型弹塑性损伤本构关系,并运用改进的弹塑性接触模型建立了系统的冲击动力学模型,在此基础上建立了低速冲击下轴对称正交对称铺设弹塑性层合中厚浅球壳的增量型非线性运动控制方程。在数值分析中,讨论了低速冲击下结构内部的损伤发展、弹塑性变形以及结构的几何尺寸等对结构的非线性动力响应和接触力的影响。
采用适用于功能梯度浅球壳的接触模型,建立了在刚性小球低速冲击下具功能梯度涂层浅球壳的冲击动力学模型;采用基于连续体损伤理论的粘结域损伤模型建立界面损伤本构关系,并由此得到低速冲击下用位移分量表示的具功能梯度涂层弹性浅球壳的非线性运动控制方程。采用配点法和Newmark-β方法对整个问题进行迭代求解。数值算例中,分析了低速冲击下功能梯度材料指数、涂层厚度等对界面损伤的影响,并讨论了界面损伤对低速冲击下具功能梯度涂层弹性浅球壳的非线性动力响应和接触力的影响。
根据一维热传导方程,求解了受沿厚度变化温度荷载作用时功能梯度浅球壳内的温度场,其中考虑了功能梯度材料参数与温度相关的特性;然后基于功能梯度结构的二维轴对称接触模型,推导出弹性小球与功能梯度浅球壳的接触模型,模型中考虑了接触过程中弹性小球的变形,并由此得到功能梯度浅球壳在弹性小球低速冲击下的冲击动力学模型;通过建立功能梯度中厚浅球壳的几何关系,以及与温度相关弹性常数沿厚度呈指数分布形式时功能梯度材料的本构关系,从而建立了低速冲击下用位移分量表示的轴对称功能梯度中厚浅球壳的非线性运动控制方程。对未知位移函数在空间域采用正交配点法离散,时间域采用Newmark-β方法离散,整个问题采用迭代法求解。数值算例中,讨论了弹性小球低速冲击下功能梯度中厚浅球壳的非线性动力响应和接触力变化,以及功能梯度材料参数、温度环境以及结构几何参数等对冲击接触力和结构非线性动力响应的影响。
基于连续体介质损伤理论,建立了各向异性材料的损伤本构关系,并据此将功能梯度板分成N层,且假设在每一单层内功能梯度材料常数和损伤沿厚度不变化,建立了弹性模量沿厚度呈指数分布形式时功能梯度材料含面内损伤的本构关系;通过ABAQUS/Explicit模块中的子程序VUMAT嵌入功能梯度材料的损伤本构关系,并基于此进行了低速冲击下功能梯度板的非线性动力响应和损伤研究。数值算例中,讨论了低速冲击下功能梯度板的非线性动力响应和损伤的发展,并研究了功能梯度材料弹性模量变化形式、冲击速度等对功能梯度板的非线性动力响应、接触力以及损伤的影响。
In this dissertation, considering the effect of nonlinear factor and environment conditions synthetically, the dynamic response, impact resistance, matrix damage, interfacial damage of the laminated composite/functionally graded plates and shells under low velocity impact are studied, and the essential characters of the mechanical and damage property are illustrated precisely. The research results are not only contribute to enrichment and development of contact theory and impact resistance research, but also have an important meaning in the practical engineering. The main results contain as follows.
Based on the geometric nonlinear theory of the moderately thick shallow spherical shells and the damage theory,the constitutive relation for the laminated shallow spherical shells with matrix damage and matrix-fiber shear damage was established by employing a strain-based failure criterion.And a set of nonlinear equations of motion for the cross-ply laminated moderately thick shallow spherical shells under the low velocity impact were derived.By using the orthogonal collocation point method and the Newmark method to discrete the unknown variable functions in space domain and in time domain respectively,the whole problem is solved by the iterative method synthetically.The numerical results show that the damage,the initial velocity of the striking object and the shell’s geometrical parameters all affect the contact force and the dynamic response of the structure under the low velocity impact to some extent.
Based on the Reddy's higher-order shear theory and Talreja’s damage model with tensor internal variables, the displacement field and Von Karman geometrical relations as well as the damage constitutive relations of the composite plate with matrix damage are established. When considering the variation of the normal contact force as well as the tangential contact force, the relations between contact force and contact deformation are founded by using Hertz contact model, and subsequently oblique impacting dynamic model of the composite plate are established. The nonlinear motion equations are solved by applying finite difference method, Newmark-βand iterative methods synthetically. In numerical examples, the effects of the impacting angle, initial velocity of the impactor and geometrical size on the nonlinear dynamic response and damage evolution of the composite laminated plate are investigated.
Based on the elasto-plastic mechanics, the damage analysis and dynamic response of an elasto-plastic laminated composite shallow spherical shell under low velocity impact are carried out. A yielding criterion related to spherical tensor of stress is proposed to model the mixed hardening orthotropic material, and accordingly an incremental elasto-plastic damage constitutive relation for the laminated shallow spherical shell is founded when a strain-based Hashin failure criterion is applied to assess the damage initiation and propagation. By using a modified elasto-plastic contact law to establish the dynamic impacting model, and applying the presented constitutive relations and the classical nonlinear shell theory, a set of incremental nonlinear equations of motion for the axisymmetric laminated cross-ply moderately thick shallow spherical shell are obtained. In numerical examples, the effect of damage, elasto-plastic deformation and geometrical parameters on the contact force and the dynamic response of the structure under low velocity impact are investigated.
The contact model of A.E.Giannakopoulos’s 2-D functionally graded material (FGM) contact model is applied to predict contact force of shallow spherical shell with FGM coating under low velocity impact. And the interfacial damage analytical model is established based on the damage model of cohesive domain of the continuum damage theory. The nonlinear governing equations of motion for the shallow spherical shell substrate and FGM coating are obtained by Reissner variation. The questions are solved by using the orthogonal collocation point method, the Newmark method and the iterative method synthetically. In numerical examples, the dynamic response of shallow spherical shell with FGM coating and contact force are obtained and the effects of the functionally graded material index, the coating thickness, and geometrical parameters of FGM coating on interfacial damage and contact force are discussed.
By applying steady-state heat conducting equation, the temperature field in FGM shallow spherical subjected to a uniform temperature load on the surface of shell and the temperature varies along the thickness direction are obtained. Then based on the contact model of Giannakopoulos’s 2-D functionally graded material (FGM), a modified contact model is put forward to deal with impact problem of the functionally graded shallow spherical shell in thermal environment,in which the deformation of impacted sphere is considered. The displacement field and geometrical relations of the FGM shallow spherical shell are established on the basis of Timoshenko–Midlin theory. And the nonlinear equations of motion for the FGM shallow spherical shell under low velocity impact and thermal environment in terms of displacement variable functions are founded. Using the orthogonal collocation point method and the Newmark method to discretize the unknown variable functions in space and in time domain, the whole problem is solved by the iterative method. In numerical examples, the contact force and nonlinear dynamic response of the FGM shallow spherical shell under low velocity impact are investigated. The effects of temperature field, material and geometrical parameters on contact force and dynamic response of the FGM shallow spherical shell are discussed.
The damage constitutive relations of the anisotropic materials are established by applying the continuum damage theory. Dividing the FGM plate to N sub-plies, and assuming the material property and damage are constant along the thickness of every single layer, the damage constitutive relations of the FGM plate with the elastic modulus varying along the thickness of the plate as the power law function are founded. By using the subroutine VUMAT of ABAQUS software, the constitutive relations of the FGM are embedded, and consequently the nonlinear dynamic property and damage of the FGM plate under low velocity impact are investigated by using the finite element software ABAQUS. The effects of the functionally graded material index, the impact velocity and the geometrical size on the nonlinear dynamic response and the damage of structure are discussed.
基于中厚壳几何非线性理论和损伤理论,采用应变描述的失效准则,导出了层合中厚浅球壳含基体损伤和纤维-基体剪切损伤的损伤本构关系,并假设每一单层沿厚度方向有相同的损伤程度,建立了低速冲击下用位移分量表示的轴对称正交对称铺设层合中厚浅球壳的非线性运动控制方程。将未知位移函数在空间域采用正交配置法离散,时间域采用Newmark-β方法离散,对整个问题进行迭代求解。具体讨论了基体损伤、冲击速度以及结构的几何尺寸等对低速冲击下结构所受的冲击载荷和结构的非线性动力响应的影响。
根据Reddy的高阶剪切变形理论,建立复合材料层合中厚板的位移场和Von Karman型几何关系,并采用Talreja张量内变量损伤模型,建立了具基体损伤复合材料层合中厚板的弹性损伤本构关系;然后基于Hertz接触理论,得到考虑法向接触力和切向接触力同时变化时的接触力和接触变形关系,并据此建立了弹性小球低速斜冲击下复合材料层合中厚板的冲击动力学模型。综合运用有限差分法、Newmark-β和迭代法,求解了低速冲击下具损伤复合材料层合中厚板的非线性动力响应问题,讨论了冲击角度、冲击速度和结构几何尺寸等对斜冲击作用下复合材料层合中厚板的非线性动力响应和结构损伤的影响。
基于弹塑性理论,建立了一个与应力球张量有关的正交各向异性材料的混合硬化屈服准则,并根据几何非线性理论和损伤理论,采用基于应变描述的Hashin失效准则,导出了层合中厚浅球壳含层内损伤的增量型弹塑性损伤本构关系,并运用改进的弹塑性接触模型建立了系统的冲击动力学模型,在此基础上建立了低速冲击下轴对称正交对称铺设弹塑性层合中厚浅球壳的增量型非线性运动控制方程。在数值分析中,讨论了低速冲击下结构内部的损伤发展、弹塑性变形以及结构的几何尺寸等对结构的非线性动力响应和接触力的影响。
采用适用于功能梯度浅球壳的接触模型,建立了在刚性小球低速冲击下具功能梯度涂层浅球壳的冲击动力学模型;采用基于连续体损伤理论的粘结域损伤模型建立界面损伤本构关系,并由此得到低速冲击下用位移分量表示的具功能梯度涂层弹性浅球壳的非线性运动控制方程。采用配点法和Newmark-β方法对整个问题进行迭代求解。数值算例中,分析了低速冲击下功能梯度材料指数、涂层厚度等对界面损伤的影响,并讨论了界面损伤对低速冲击下具功能梯度涂层弹性浅球壳的非线性动力响应和接触力的影响。
根据一维热传导方程,求解了受沿厚度变化温度荷载作用时功能梯度浅球壳内的温度场,其中考虑了功能梯度材料参数与温度相关的特性;然后基于功能梯度结构的二维轴对称接触模型,推导出弹性小球与功能梯度浅球壳的接触模型,模型中考虑了接触过程中弹性小球的变形,并由此得到功能梯度浅球壳在弹性小球低速冲击下的冲击动力学模型;通过建立功能梯度中厚浅球壳的几何关系,以及与温度相关弹性常数沿厚度呈指数分布形式时功能梯度材料的本构关系,从而建立了低速冲击下用位移分量表示的轴对称功能梯度中厚浅球壳的非线性运动控制方程。对未知位移函数在空间域采用正交配点法离散,时间域采用Newmark-β方法离散,整个问题采用迭代法求解。数值算例中,讨论了弹性小球低速冲击下功能梯度中厚浅球壳的非线性动力响应和接触力变化,以及功能梯度材料参数、温度环境以及结构几何参数等对冲击接触力和结构非线性动力响应的影响。
基于连续体介质损伤理论,建立了各向异性材料的损伤本构关系,并据此将功能梯度板分成N层,且假设在每一单层内功能梯度材料常数和损伤沿厚度不变化,建立了弹性模量沿厚度呈指数分布形式时功能梯度材料含面内损伤的本构关系;通过ABAQUS/Explicit模块中的子程序VUMAT嵌入功能梯度材料的损伤本构关系,并基于此进行了低速冲击下功能梯度板的非线性动力响应和损伤研究。数值算例中,讨论了低速冲击下功能梯度板的非线性动力响应和损伤的发展,并研究了功能梯度材料弹性模量变化形式、冲击速度等对功能梯度板的非线性动力响应、接触力以及损伤的影响。
In this dissertation, considering the effect of nonlinear factor and environment conditions synthetically, the dynamic response, impact resistance, matrix damage, interfacial damage of the laminated composite/functionally graded plates and shells under low velocity impact are studied, and the essential characters of the mechanical and damage property are illustrated precisely. The research results are not only contribute to enrichment and development of contact theory and impact resistance research, but also have an important meaning in the practical engineering. The main results contain as follows.
Based on the geometric nonlinear theory of the moderately thick shallow spherical shells and the damage theory,the constitutive relation for the laminated shallow spherical shells with matrix damage and matrix-fiber shear damage was established by employing a strain-based failure criterion.And a set of nonlinear equations of motion for the cross-ply laminated moderately thick shallow spherical shells under the low velocity impact were derived.By using the orthogonal collocation point method and the Newmark method to discrete the unknown variable functions in space domain and in time domain respectively,the whole problem is solved by the iterative method synthetically.The numerical results show that the damage,the initial velocity of the striking object and the shell’s geometrical parameters all affect the contact force and the dynamic response of the structure under the low velocity impact to some extent.
Based on the Reddy's higher-order shear theory and Talreja’s damage model with tensor internal variables, the displacement field and Von Karman geometrical relations as well as the damage constitutive relations of the composite plate with matrix damage are established. When considering the variation of the normal contact force as well as the tangential contact force, the relations between contact force and contact deformation are founded by using Hertz contact model, and subsequently oblique impacting dynamic model of the composite plate are established. The nonlinear motion equations are solved by applying finite difference method, Newmark-βand iterative methods synthetically. In numerical examples, the effects of the impacting angle, initial velocity of the impactor and geometrical size on the nonlinear dynamic response and damage evolution of the composite laminated plate are investigated.
Based on the elasto-plastic mechanics, the damage analysis and dynamic response of an elasto-plastic laminated composite shallow spherical shell under low velocity impact are carried out. A yielding criterion related to spherical tensor of stress is proposed to model the mixed hardening orthotropic material, and accordingly an incremental elasto-plastic damage constitutive relation for the laminated shallow spherical shell is founded when a strain-based Hashin failure criterion is applied to assess the damage initiation and propagation. By using a modified elasto-plastic contact law to establish the dynamic impacting model, and applying the presented constitutive relations and the classical nonlinear shell theory, a set of incremental nonlinear equations of motion for the axisymmetric laminated cross-ply moderately thick shallow spherical shell are obtained. In numerical examples, the effect of damage, elasto-plastic deformation and geometrical parameters on the contact force and the dynamic response of the structure under low velocity impact are investigated.
The contact model of A.E.Giannakopoulos’s 2-D functionally graded material (FGM) contact model is applied to predict contact force of shallow spherical shell with FGM coating under low velocity impact. And the interfacial damage analytical model is established based on the damage model of cohesive domain of the continuum damage theory. The nonlinear governing equations of motion for the shallow spherical shell substrate and FGM coating are obtained by Reissner variation. The questions are solved by using the orthogonal collocation point method, the Newmark method and the iterative method synthetically. In numerical examples, the dynamic response of shallow spherical shell with FGM coating and contact force are obtained and the effects of the functionally graded material index, the coating thickness, and geometrical parameters of FGM coating on interfacial damage and contact force are discussed.
By applying steady-state heat conducting equation, the temperature field in FGM shallow spherical subjected to a uniform temperature load on the surface of shell and the temperature varies along the thickness direction are obtained. Then based on the contact model of Giannakopoulos’s 2-D functionally graded material (FGM), a modified contact model is put forward to deal with impact problem of the functionally graded shallow spherical shell in thermal environment,in which the deformation of impacted sphere is considered. The displacement field and geometrical relations of the FGM shallow spherical shell are established on the basis of Timoshenko–Midlin theory. And the nonlinear equations of motion for the FGM shallow spherical shell under low velocity impact and thermal environment in terms of displacement variable functions are founded. Using the orthogonal collocation point method and the Newmark method to discretize the unknown variable functions in space and in time domain, the whole problem is solved by the iterative method. In numerical examples, the contact force and nonlinear dynamic response of the FGM shallow spherical shell under low velocity impact are investigated. The effects of temperature field, material and geometrical parameters on contact force and dynamic response of the FGM shallow spherical shell are discussed.
The damage constitutive relations of the anisotropic materials are established by applying the continuum damage theory. Dividing the FGM plate to N sub-plies, and assuming the material property and damage are constant along the thickness of every single layer, the damage constitutive relations of the FGM plate with the elastic modulus varying along the thickness of the plate as the power law function are founded. By using the subroutine VUMAT of ABAQUS software, the constitutive relations of the FGM are embedded, and consequently the nonlinear dynamic property and damage of the FGM plate under low velocity impact are investigated by using the finite element software ABAQUS. The effects of the functionally graded material index, the impact velocity and the geometrical size on the nonlinear dynamic response and the damage of structure are discussed.
引文
[1] Johnson K L. Contact Mechanics. 2nd Edition, Cambridge University Press, New York. 1985
[2]徐秉业,罗学富,刘信声,宋国华,孙学伟.接触力学,高等教育出版社,1987
[3] Bradley R S. The cohesive force between solid surfaces and the surface energy of solids. Philosophical Magazine Series, 1932, 7(13): 853-862
[4] Derjaguin B. Untersuchungenüber die Reibung und Adhion. Theorie des Anhaftens kleiner Teilchen. 1934, 69:155-164
[5] Johnson K L, Kendall K, Roberts A D. Surface energy and the contact of elastic solids. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 1971, 324:301-313
[6] Johnson K L. A note on the adhesion of elastic solids. British Journal of Applied Physics 1958, 9:199–200
[7] Ding H J, Chen W, Zhang L. The adhesion of particles. Journal of Colloid and Interface Science, 2006, 53:314-326
[8] Tabor D. Surface forces and surface interactions. Journal of Colloid and Interface Science, 1977, 58:2-13
[9] Greenwood J A, Johnson K L. The mechanics of adhesion of viscoelastic solids. Philosophical Magazine A , 1981, 43:697-711
[10] Muller V M , Yushchenko V S, Derjaguin B V. On the influence of molecular forces on the deformation of an elastic sphere and its sticking to a rigid plane. Journal of Colloid and Interface Science,1980,77:91-101
[11] Maugis D. Adhesion of spheres: the JKR-DMT transition using a dugdale model. Journal of Colloid and Interface Science, 1992,150:243-269
[12] Green A E, Zerna W. Theoretical Elasticity, Oxford, 1968
[13] Lekhnitskii S G. Theory of Elasticity of an Anisotropic Elastic Body: Gostekhizdat, Moscow (in Russian) 1950; English translation, Holden-Day, San Francisco, 1963
[14] Willis J R. Hertzian contact of anisotropic bodies. Journal of Mechanics and Physics of Solids, 1966,14:163–176
[15] Stephen R S. Contact deformation and stress in orthotropic plates. Composites: Part A,2005,36:1421-1429
[16] Jiang D Z, Shen W, Wang X Y. Dynamic contact algorithm of an isotropic projectile contacting an orthotropic target. Computer Methods in Applied Mechanics and Engineering, 2000,189:575-585
[17] Loktev A V. The dynamic contact of an impactor and an elastic orthotropic plate when there are propagating thermoelastic waves. Journal of Applied Mathematics and Mechanics, 2008,72:475-480
[18] Fukumasu N K, Souza R M. Numerical analysis of the contact stresses developed during the indentation of coated systems with substrates with orthotropic properties. Surface & Coatings Technology,2006,201:4294-4299
[19] Swanson S R. Hertzian contact of orthotropic materials. International Journal of Solids and Structures,2004,41:1945-1959
[20] Pedersen P. On the influence of clearance in orthotropic disc-pin contacts. Composite Structures,2007,79:554-561
[21] Cattaneo C. Sul contatto di due corpi elastici: distribuzione locale degli sforzi. Academia dei Lincei, Rendicotti,1938,27(6):342-348
[22] Cattaneo C. Sul contatto di due corpi elastici: distribuzione locale degli sforzi. Academia dei Lincei, Rendicotti,1938,27(6),434-436
[23] Cattaneo C. Sul contatto di due corpi elastici: distribuzione locale degli sforzi. Academia dei Lincei, Rendicotti, 1938.27(6):474-478
[24] Mindlin R D. Compliance of elastic bodies in contact. ASME Journal of Applied Mechanics, 1949,16:259-268
[25] Mindlin R D, Deresiewicz H. Elastic spheres in contact under varying oblique forces. ASME Journal of Applied Mechanics, 1953,20:327-344
[26] Chandrasekaran N, Haisler W, Goforth R. Finite element analysis of hertz contact problem with friction. Finite Elements in Analysis and Design, 1987,3:39-56
[27] Cuttiono J, Dow T. Contact between elastic bodies with an elliptic contact interface in torsion. Journal of Applied Mechanics, 1997,64:144-148
[28] Brock L M. Frictionless indentation by a rigid wedge: the effect of tangential displacements in the contact zone. International Journal of Engineering Science, 1979, 17:365-372
[29] Georgiadis L M. Tangential-displacement effects in the wedge indentation of an elastic half-space d an integral-equation approach. Computational Mechanics, 1998,21:347-352
[30] Argatov I I. Approximate solution of an axisymmetric contact problem withallowance for tangential displacements on the contact surface. Journal of Applied Mechanics and Technical Physics, 2004(a),45:118-123
[31] Argatov I I. Axisymmetric Hertz problem with tangential displacements on the contact surface. Doklady Physics, 2004(b),49:222-225
[32] Sch?fer J, Dippel S, Wolf D E. Force schemes in simulations of granular materials. Journal of Physics I France,1996,6:5-20
[33] Tijskens E, Ramon H, De Baerdemaeker J. Discrete element modeling for process simulation in agriculture. Journal of Sound Vibration, 2003,266: 493-514
[34] Zeebroeck M V, Dintwa E, Tijskens E, et al. Determining tangential contact force model parameters for viscoelastic materials (apples) using a rheometer Postharvest. Biology and Technology,2004, 33:111-125
[35] Argatov I I, Mishuris G S. Axisymmetric contact problem for a biphasic cartilage layer with allowance for tangential displacements on the contact surface. European Journal of Mechanics A/Solids, 2010,29:1051-1064
[36] Chen W W, Wang Q J. A numerical model for the point contact of dissimilar materials considering tangential tractions. Mechanics of Materials, 2008,40:936 -948
[37] Sevostianov I, Kachanov M. Normal and tangential compliances of interface of rough surfaces with contacts of elliptic shape. International Journal of Solids and Structures,2008,45:2723-2736
[38] Allen M, A Ellis J. Stress and strain distribution in the vicinity of interference fit fasteners. AFFDL-TR, 1973,72-153
[39] Crews J H, An elastic-plastic analysis of a uniaxially loaded sheet with an interference fit bolt. NASA-TN-D 1974, 7748
[40] Wanlin G. Elastic-plastic analysis of a finite sheet with a cold worked hole. Engineering Fracture Mechancis, 1993,45:857-864
[41] Christensen P W. A semi-smooth Newton method for elasto-plastic contact problems. International Journal of Solids and Structures, 2002,39:2323-2341
[42] Jackson R L, Green I. A finite element study of elasto-plastic hemispherical contact. ASME Journal of Tribology,2005,127(3):43-54
[43] Robert L, Itzhak G. A statistical model of elasto-plastic asperity contact between rough surfaces. Tribology International, 2006,39:906-914
[44] Grunsweig J, Longman I M, Pci N J. Calculations and measurements of wedge indentations. Journal of the Mechanics and Physics of Solids, 1954,2:81
[45] Haddow J B. On a plane strain wedge indentation paradox. International Journal of Mechanical Science, 1967,9:159
[46] Bhasin Y P, Oxley P L B, Roth R V. An experimentally-determined slip-line field for plain strain wedge indentation of a strain-hardening material. Journal of the Mechanics and Physics of Solids, 1980,2:43
[47] Lockett F J. Indentation of a rigid plastic material by a conical indenter. Journal of the Mechanics and Physics of Solids, 1963,11:345
[48] Pei A L, Hyunb S, Molinaria J F, et al. Finite element modeling of elasto-plastic contact between rough surfaces. Journal of the Mechanics and Physics of Solids, 2005,53:2385-2409
[49] Dumas G, Baronet C N. Elasto-plastic indentation of a half-space by a long, rigid cylinder. International Journal of Mechanical Science,1971,13:519
[50] Skalski K. Contact problems for elasto-plaslic body. (in polish Prace Naukowe Politechniki Warszawskiej),seria "Mechanika", 1986
[51] Follansbee P S, Sinclair GB. Qua-static normal indentation of an elasto-plastic half-space by a rigid sphere. International Journal of Solids and Structures, 1984,20:81
[52] Hardy C, Baronet C N, Trdion G V. Elasto-plastic indentation of a half-space by a rigid sphere. International Journal for Numerical Methods in Engineering, 1971,3:451-462
[53] Krupp H, Sperling G. Theory of adhesion of small particles. Journal of Applied Physics, 1966,37:4176-4180
[54] Wang L G, Sun X P, Huang Y. Friction analysis of microcosmic elastic-plastic contact for extrusion forming. Journal of Materials Processing Technology, 2007,187-188:631-634
[55] Sztok S. Elasto-plastic solution of fictional contact problem by finite element method. Proceeding of International Conference on Barcelona, 1987
[56] Gun H. Elasto-plastic static stress analysis of 3D contact problems with friction by using the boundary element method. Engineering Analysis with Boundary Elements, 2004,28:779-790
[57] Zolotarevskiy V, Kligerman Y, Etsion I. Elastic-plastic spherical contact under cyclic tangential loading in pre-sliding G. Model WEA-99746 7
[58] Chen W M, Li M, Cheng Y T. Analysis on elasto-plastic spherical contact and its deformation regimes, the one parameter regime and two parameter regime by finite element simulation. Vacuum, 2011,85:898-903
[59] Kadin Y, Kligerman Y, Etsion I. Multiple loading of an elastic-plastic spherical contact. International Journal of Solids and Structures, 2006,7:19-127
[60] Bellemare S, Dao M, Suresh S. The frictional sliding response of elasto-plastic materials in contact with a conical indenter. International Journal of Solids and Structures,2007,44:1970-1989
[61] Chang W R, Etsion I, Bogy D B. An elastic-plastic model for the contact of rough surfaces ASME Journal of Tribology, 1987, 109:300-319
[62] Vu-Quoc L, Lesburg L. Contact model of contact law for spherical particles accounting for plastic deformation. International Journal of Solids and Strucutres, 2000,44:170–189
[63] Vu-Quoc L, Zhang X. An Elasto-Plastic contact force-displacement model in the normal direction: Displacement-Driven version, Proceeding of Royal Society in London, Series A, 1991,455:4013-4044
[64] Vu-Quoc L, Zhang X, Lesburg L. A normal force-displacement model for contacting spheres accounting for plastic deformation: force-Driven formation. ASME Journal of Applied Mechanics,2000, 67(2):363-371
[65] Thorton C. Coefficient of restitution for nonlinear collisions of elastic perfectly plastic spheres. ASME Journal of Applied Mechanics, 1997, 64:383-386
[66] Xu H M, Yao X F, Feng X Q, et al. Fundamental solution of a power-law orthotropic and half-space functionally graded material under line loads, Composites Science and Technology, 2008,68:27-34
[67] Kassir M K. Boussinesq problems for non-homogeneous materials. ASCE Journal of Engineering Mechanics Division, 1972,98:457-470
[68] Booker J R, Balaam N P, Davis E H. The behavior of an elastic non-homogen- eous half-space. Part–Circular and strip footings. International Journal of Analytical Methods in Geomechanics,1985,9:369-381
[69] Giannakopoulos A E, Suresh S. Indentation of solids with gradients in elastic properties: Part I. Point Force. International Journal of Solids and Structures,1997,34(19): 2357-2392
[70] Giannakopoulos A E, Pallot P. Two-dimensional contact analysis of elastic graded materials. Journal of the Mechanics and Physics of Solids,2000, 48(8): 1597-1631
[71] Jorgensen O, Giannakopoulos A E, Suresh S. Spherical indentation of composite laminates with controlled gradients in elastic anisotropy. International Journal of Solids and Structures,1998,35(36):5097-5113
[72]温卫东,徐颖,崔海坡.低速冲击下复合材料层合板损伤分析材料工程, 2007,7
[73] Richaardson M O W, Wisheart M J. Review of low velocity impact properties. Composites Part A , 1996,27(A): 1123-1131
[74] Joshi S P, Sun C T. Impact induced fracture in a laminated composite. Journal of Composite Materials,1985,19:51-66
[75] Choi H Y, Wu H Y T, Chang F K. A new approach toward understanding damage mechanisms and mechanics of laminated composites due to low-velocity impact Part II Analysis. Journal of Composite Materials, 1991, 25:1012-1038
[76] Cantwell W J, Morton J. Geometrical effects in the low velocity impact response of CFRP. Composite Structures,1989,12:39-59
[77] Liu, D. and Malvem, L.E. Matrix cracking in impacted glass/epoxy plates. Journal of Composite Materials,1987,21:594-609
[78] Wu, H Y T, Springer G S. Measurements of matrix cracking and delamination caused by impact on composite plates. Journal of Composite Materials, 1988,22: 518-532
[79] Chang F K, Choi H Y, Jeng S T. Study on impact damage in laminated composites. Mechanics of Materials, 1990,10:83-95
[80] Choi H Y, Downs R J,Chang F K. A new approach toward understanding damage mechanisms and mechanics of laminated composites due to low-velocity impact: Part I Experiments. Journal of Composite Maters,1991,25:992-1011
[81] Choi H Y, Chang F Y. Impact damage threshold of laminated composites. In‘Failure Criteria and Analysis in Dynamic Response’, AMD Vol. 107, ASME Applied Mechanics Division,Dallas,TX,1990,31-35
[82] Choi H Y, Wang H S, Chang F K. Effect of laminate configuration and impactor’s mass on the initial impact damage of graphite/epoxy composite plates due to line loading impact. Journal of composite materials, 1992,26(6): 804-827
[83] Chang F K, Chang K Y. A progressive damage model for laminated composites containing stress concentrations. Journal of composite materials, 1987,21: 834 -855
[84] Li S, Soden P D, Reid S R, et al. Indentation of laminated filament-wound composite tubes. Composites,1993,24:407-21
[85] Li S, Soden P D, Reid S R. A finite strip analysis of cracked laminates. Mechanics of Mater erial,1994,18:289-311
[86] Li S, Reid S R, Soden P D. A continuum damage model for transverse matrix cracking in laminated fibre-reinforced composites. Philosophical Transactions of the Royal Society in Lond A, 1998, 356:379-412
[87] Camanho P P, Davila C G, Moura M F D. Numerical simulation of mixed-mode progressive delamination in composite materials, Journal of Composites Materials, 2003, 37:1415-1424
[88] Li CF, Hu N, Yin YJ, et al. Low-velocity impact induced damage of continuous fiber-reinforced composite laminates. Part I. An FEM numerical model. Composite Part A 2002;33(8):1053–1060
[89] Collombet F, Lalbin X, Lataillade J L. Impact behavior of laminated composites: physical basis finite element analysis. Composite Science and Technology, 1998,58:463–478
[90] Choi H Y, Chang F K. A model for predicting damage in graphite/epoxy laminated composites resulting from low-velocity point impact. Journal of Composite Materials, 1992,26:2134–2169
[91] Zou Y, Tong L, Steven G P. Vibration-based model-dependent damage (delamination) identification and health monitoring for composite structures-a review. Journal of sound and vibration, 2000,230(2):357-378
[92] Lee S, Zahuta P. Instrumented impact and static indentation of composites. Journal of Composite Materials, 1991,25:204-222
[93] El-Habak A M. Effect of impact perforation load on GFRP composites. Composites 1993,24(4):341-345
[94] Dorey G. Impact damage in composites-development, consequences and prevention. The 6th International Conference on Composite Materials and 2nd European Conference on Composite Materials, Imperial College, London, 1988, 3:3.1-3.26
[95] Potti S V, Sun C T. Prediction of impact induced penetration and delamination in thick composite laminates. International Journal of Impact Engineering, 1997,19(1):31-48
[96] Gama B A, Gillespie J W. Punch shear based penetration model of ballistic impact of thick-section composites. Composite Structures, 2008,86:356-369
[97] Jiang W G, Hallett S R, Green B G, et al. A concise interface constitutive law for analysis of delamination and splitting in composite materials and its application to scaled notched tensile specimens. International Journal for Numerical Methods in Engineering, 2007, 69:1982–1995
[98] Wardle M W, Tokarsky E W. Drop weight impact testing of laminates reinforced with Kevlar aramid fibres, E-glass and graphite. Composites Technology Research 1983,5:4-10
[99] Hsieh CY, Mount A, Jang B Z, et al. Response of polymer composites to high and low velocity impact. International Proceeding 22nd International SAMPE Technology Conference 199,(6-8):14-27
[100] Caprino G, Lopresto V. The significance of indentation in the inspection of CFRP panels damaged by low-velocity impact. Composite Science and Technology, in press
[101] Cantwell W J, Morton J. Impact perforation of carbon fiber reinforced plastics. Composite Science and Technology,1990,38:119-141
[102] Caprino G, Lopresto V. Factors affecting the penetration energy of glass reinforced plastics subjected to a concentrated transverse load. Accepted for presentation at ECCM 9
[103] Babic L, Dunn C, Hogg P J. Damage development and its significance in GRP subjected to impact. Plastic Rubber Process Application,1989,12:199-207
[104] Bibo G, Leicy D, Hogg P J, Kemp M. High-temperature damage tolerance of carbon reinforced plastics. Part 1: Impact characteristics. Composites, 1994, 25: 414-424
[105] Bibo G A, Hogg P J. Influence of reinforcement architecture on damage mechanisms and residual strength of glass /epoxy composite systems. Composi- te Science and Technology,1998,58:803-813
[106] Delfosse D, Poursartip A. Experimental parameter study of static and dynamic out-of-plane loading of CFRP laminates. International Proceeding,10th, Internat- ional Conference on Composite Materials (ICCM1O), Whistler, Canada, 1985, V:583-590
[107] Caprino G, Lopresto V. On the penetration energy for reinforced plastics under low-velocity impact conditions, Composites Science and Technology, 2001, 61: 65-73
[108] Gama B A, Gillespie J W. Finite element modeling of impact, damage evolution and penetration of thick-section composites, International Journal of Impact Engineering,2011,38(4),181-197
[109] Richaardson M O W, Wisheart M J. Review of low velocity impact properties. Composites Part A, 1996,27(A): 1123-1131
[110] Krawczuk M, Ostachowicsw O, Zak A. Dynamic of cracked composite materialstructures. Computational Mechanics,1997,20:79-83
[111] Crisfield M A, Alfano G. Finite element interface models for the delamination analysis of laminated composites: mechanical and computational issues, International Journal for Numerical Methods in Engineering, 2000, 50:1701– 1736
[112] Borg R, Nilsson L, Simonsson K. Modelling of delamination using a discreized cohesive zone and damage formulation, Composite Science and Technology, 2002, 62:1299–1314
[113] Kinloch A J, Wang Y, Williams J G, et al. The mixed-mode delamination of fibre composite materials, Composite Science and Technology, 1993,47: 225–237
[114] Sankar B V. A finite element for modeling delaminations in composite beams. Computers and Structures, 1991,38:239-246
[115] Barbero E J, ReddyJ N. Modeling of delamination in composite laminates using a layer wise plate theory. nternational Journal of Solids and Structure, 1991,28: 373-388
[116] Harper P W, Hallett S R. Cohesive zone length in numerical simulations of composite delamination. Engineering Fracture Mechanics, 2008,75(16):4774- 4792
[117] Robinson P, Galvanetto U, Tumino D, et al. Numerical simulation of fatigue-driven delamination using interface elements, International Journal for Numerical Methods in Engineering, 2004, 63:1824–1848
[118] Turon A, Davila C G, Camanho P P, et al. An engineering solution for mesh size effects in the simulation of delamination using cohesive zone models, Engine- ering Fracture Mechanics, 2007,74:1665–1682
[119] Turon A, Costa J, Camanho P P. Simulation of delamination in composites under high-cycle fatigue, Compos A:Applied Science Manufacture, 2007,38: 2270- 2282
[120] Dorey G. Impact damage tolerance and assessment in advanced composite materials. Seminar on Advanced Composites, Cranfield Institute of Technology, Cranfield, UK, 1986
[121] Dorey G, Sigety P, Stellbrink K, Hart W G J. Impact damage tolerance of carbon fibre and hybrid laminates. RAE Technical Report 87 057, Royal Aerospace Establishment, Farnborough, UK, 1987
[122] Asp L E, Sjogren A,Greenhalgh E S. Delamination growth and thresholds in acarbon/epoxy composite under fatigue loading, Journal of Composites Technol- ogy and Research,2001,23:55-68
[123] Dugdale D S. Yielding of steel sheets containing slits. Journal of the Mechanics and Physics of Solids 1960,8:100-104
[124] Barenblatt G I. The mathematical theory of equilibrium cracks in brittle fracture. Advances in Applied Mechanics ,1962,7:55-129
[125] Hillerborg A, Mode M, Petersson P E. Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements. Cement and Concrete Research,1976,6:773-782
[126] Blackman B R K, Hadavinia H, Kinloch A J, et al. The use of a cohesive zone model to study the fracture of fibre composites and adhesively-bonded joints, International Journal of Fracture, 2003,19: 25–46
[127] Allix O, Ladeveze P, Corigliano A. Damage analyses of interlaminar fracture specimens, Composite Structures , 1995,31:66–74
[128] Reedy E D, Mello F J, Guess T R. Modeling the initiation and growth of delamination in composite structures, Journal of Composite Materials, 1997,31 (8):812–831
[129] Wisheart M, Richardson M. The finite element analysis of impact induced delamination in composite materials using a novel interface element, Composite Part A,1998, 29(A):301–303
[130] Johnson A, Holzpapfel M. Modeling of soft body impact on a composite structure, Composite Structures, 2003, 61:103–113
[131] Touron A, Camanho P P, Costa J, et al. A damage model for the simulation of delamination in advanced composite under variable model loading, Mechanics of Materials, 2006,38:1072–1089
[132] Whittingham B, Marshall I H, Mitrevski T, et al. The response of composite structures with pre-stress subject to low velocity impact damage. Composite Structures,2004,66:685-698
[133] Robb M D, Arnold W S, Marshall I H. The damage tolerance of GRP laminates under biaxial pre-stress. Composite Structures, 1995,32:141-149
[134] Cantwell W, Morton J. The impact resistance of composite materials-a review. Composites 1991,22:347-362
[135] Fuoss E, Straznicky P, Poon C. Effects of stacking sequence on the impact resistance in composite laminates-part 1: parametric study. Composite Structures, 1998,41:67-77
[136] Fuoss E, Straznicky P, Poon C. Effects of stacking sequence on the impact resistance in composite laminates-part 2: prediction method. Composite Structures,1998,41:177-186
[137] Dost E, Ilcewiz L, Avery W, Coxon B. Effects of stacking sequence on impact damage resistance and residual strength for quasi-isotropic laminates. In: Brien T O, Editor, Composites materials: fatigue and fracture, American Society for Testing and Materials 1991,3:476-500 .
[138] Strait L, Karasek M, Amateau M. Effects of stacking sequence on the impact resistance of carbon fiber reinforced thermoplastic toughened epoxy laminates. Journal of Composite Materials, 1992,26(12):1725-1740
[139] Lopes C S, Seresta O, Coquet Y, et al. Low-velocity impact damage on dispersed stacking sequence laminates. Part I: Experiments, Composites Science and Technology, 2009,69:926-936
[140] Choi I H, Kim I G, Ahn S M, et al. Analytical and experimental studies on the low-velocity impact response and damage of composite laminates under in-plane loads with structural damping effects. Composites Science and Technology, 2010,70 :1513-1522
[141] Nakatani H, Kosaka T, Osaka K, et al. Damage characterization of titanium/GFRP hybrid laminates subjected to low velocity Impact. Composites: Part A (In press)
[142] Zhao Y Y, Xiao T Y, Xue S X, et al. Experiments and simulated calculations on the resistance to low-velocity impact of layered plates with a sandwiched ERM. Journal of Sound and Vibration, 2004,271:615-633
[143] Anderson T, Madenci E. Experimental investigation of low-velocity impact characteristics of sandwich composites. Composite Structures, 2000,50:239-247
[144] Hoppmann W H J. Implusive loads on Beams. Proceeding of Society Experiments, Stress Analysis, 1952, 10.(1) 157
[145] Hosseinzadeh R, Shokrieh M M, Lessard L B. Parametric study of automotive composite bumper beams subjected to low-velocity impacts. Composite Structures, 2005,68:419-427
[146] Sankar B V, Zhu H S. The effect of stitching on the low-velocity impact response of delaminated composite beams. Composites Science and Technology, 2000,60:2681-2691
[147] Lam K Y, Sathiyamoorthy T S. Response of composite beam under low-velocity impact of multiple masses. Composite Structures, 1999,44:2055-220
[148] Karas K. Platten Unter Seitlichem Stoss. Ing Arch, 1939,10:237
[149] Chun L, lam K Y. Dynamic response of fully-clamped laminated composite plates subjected to low-velocity impact of a mass. International Journal of Soli- ds and Structures, 1998,35(11): 963-979
[150] Palazotto A N, Herup E J, Gummadi L N B. Finite element analysis of low-velocity impact on composite sandwich Plates. Composite Structures, 2000, 49:209-227
[151] Besant T, Davis G A O, Hitchings D. Finite element modeling of low velocity impact of composite sandwich panels. Composite, Part A,2001,32:1189-1196
[152] Chun K S,Kassegne S K. Low Velocity Impact Dynamic Behavior of Laminated Composite Nonprismatic Folded Plate Structures. ASME Journal of Engineering Mechanics 2005, 131(7):678
[153] Wu Y, Wu Y D, Wang Y X, Zhong W F. Study on the response to low-velocity impact of a composite plate improved by shape memory alloy. Acta Mechanica Solida Sinica, 2007,20(4):357-365
[154] Choi I-K H. Contact force history analysis of composite sandwich plates subjected to low-velocity impact. Composite Structures, 2006,75:582-586
[155] Yigit A S, Christoforou A P. Limits of asymptotic solutions in low-velocity impact of composite plates. Composite Structures, 2007,81:568-574
[156] Giovanni B, Roberto V. Low velocity impact tests of laminate glass-fiber-epoxy matrix composite material plates. International Journal of Impact Engineering, 2002,27:213-229
[157] Tiberkak R, Bachene M, Rechak S, et al. Damage prediction in composite plates subjected to low velocity impact. Composite Structures , 2008,83:73-82
[158] Setoodeh A R, Malekzadeh P, Nikbin K. Low velocity impact analysis of laminated composite plates using a 3D elasticity based layerwise FEM. Materials and Design, 2009,30:3795-3801
[159] Zhang Y, Zhu P, Lai X M. Finite element analysis of low-velocity impact damage in composite laminated plates. Materials and Design, 2006,27:513-519
[160] Roh J H, Kim J H. Hybrid smart composite plate under low velocity impact. Composite Structures, 2002,56:175-182
[161] Mehmet A,Ramazan K K, Yusuf A. Compression-after impact behavior of laminated composite plates subjected to low velocity impact in high temp- eratures. Composite Structures, 2009,89:77-82
[162] Meo M, Antonucci E, Duclaux P, et al. Finite element simulation of low velocityimpact on shape memory alloy composite plates. Composite Structures, 2005, 71:337-342
[163] Shokuhfar A, Khalili S M R, Ashenai G F, et al. Analysis and optimization of smart hybrid composite plates subjected to low-velocity impact using the response surface methodology (RSM). Thin-Walled Structures, 2008,46: 1204-1212
[164] Ganapathy S, Rao K P. Failure analysis of laminated composite cylindrical/sphe- rical shell panels subjected to low-velocity impact. Computers and Structures, 1998,68:627-641
[165] Khalili S M R, Soroush M, Davar A, et al. Finite element modeling of low- velocity impact on laminated composite plates and cylindrical shells. Composite Structures, 2011,9:1363-1375
[166] Ganapathy S, Rao K I. Interlaminar stresses in laminated composite plates, cylindrical/spherical shell panels damaged by low-velocity impact. Composite Structures, 1997,38(1-4):157-168
[167] Her S C, Liang Y C. The finite element analysis of composite laminates and shell structures subjected to low velocity impact. Composite Structures, 2004,66:277 -285
[168] Toh S L, Gong S W, Shim W. Transient stresses generated by low velocity impact on orthotropic laminated cylindrical shells. Composite Structures, 1995, 31:213-228
[169] Malekzadeh K, Khalili M R, Olsson R, et al. Higher-order dynamic response of composite sandwich panels with flexible core under simultaneous low-velocity impacts of multiple small masses. International Journal of Solids and Structures, 2006, 43:6667–6687
[170] Yu J L, Wang E H, Li J R, et al. Static and low-velocity impact behavior of sandwich beams with closed-cell aluminum-foam core in three-point bending. International Journal of Impact Engineering, 2008,35:885-894
[171] Mannana M N, Ansaria R, Abbas L H. Failure of aluminum beams under low velocity impact. International Journal of Impact Engineering, 2008,35: 1201- 1212
[172] Harsoor R, Ramachandra L S. Influence of notch on the elastic-plastic response of clamped beams subjected to low velocity impact. International Journal of Impact Engineering, 2009,36:1058-1069
[173] Wu C L, Sun C T. Low velocity impact damage in composite sandwich beams.Composite Structures, 1996,34:21-27
[174] Aslan Z H, Karakuzu R Z, Okutan B. The response of laminated composite plates under low-velocity impact loading. Composite Structures, 2003,59: 119 -127
[175] Mili F, Necib B. Impact behavior of cross-ply laminated composite plates under low Velocities. Composite Structures, 2001,51:237-244
[176] Etemadi E, Khatibi A A , Takaffoli M. 3D finite element simulation of sandwich panels with a functionally graded core subjected to low velocity impact. Composite Structures, 2009, 89(1):28-34
[177] Apetre N A, Sankar B V, Ambur D R. Low velocity impact response of sandwich beams with functionally graded core. International Journal of Solids and Structures, 2006,43(9):2479-2496
[178] Gong S W, Lam K Y, Reddy J N. The elastic response of functionally graded cylindrical shells to low-velocity impact. International Journal of Impact Engineering, 1999, 22(4):397-417
[179] Pantele C L, Librescu L V. Low velocity impact on a functionally graded circular thin plate. Proceeding ESDA,2006
[180] Reid L T, Larson A. Anthony P. Low velocity impact analysis of functionally graded circular plates. Proceeding IMECE,2006
[181] Recep R, Aydin M, Apalak M K, et al. The elasto-plastic impact analysis of functionally graded circular plates under low-velocities. Composite Structures, 2011,93:860-869
[182] Kubair D V, Lakshmana B K. Cohesive modeling of low-velocity impact damage in layered functionally graded beams. Mechanics Research Communications, 2008, 35: 104-114
[183] Collombet F,Bonini J,Lataillade J L.A three-dimensional modeling of low velocity impactdamage in composite laminates. International Journal for Numerical Methods in Engineering,1996,39:1491-1516
[184] Choi H Y,Chang F K. A model for predicting damage in graphite/epoxy laminated composites resulting from low-velocity point impact. Journal of Composite Materials,1992,26(14):2134-2169
[185]张彦,来新民等.复合材料铺层板低速冲击作用下损伤的有限元分析,复合材料学报,2006,23(2):150-157
[186] Krishnamurthy K S,Mahajan P,Mittal R K. A parametric study of the impact response and damage of laminated cylindrical composite shells. CompositesScience and Technology,2001,61:1655-1669
[187]邓梁波,叶天麒.圆板在物体撞击下的非线性动力响应.力学学报,1990,22(4):420-428
[188] Hou P J, Petrinic N, Ruiz C,et al.Prediction of impact damage in composite plates. Composites Science and Technology,2000,60:273-281
[189] Talreja R,Stiffness properties of composite laminates with matrix cracking and internal delaminations. Engineering Fracture Mechanics,1986, 25(6):751-762
[190]傅衣铭.结构非线性动力学分析.广州:暨南大学出版社,1997
[191] Liu Z S,Somasak S P.Response of plate and shell structures due to low velocity impact. Journal of Engineering Mechanics,1997,123(12):1230-1237
[192] Chung Y W,Ching H Y.Impact damage in composite laminates.Computers Structures,1990,37(6):967-982.
[193] Reddy J N. A simple higher-order theory for laminated composite plates. Journal of Applied Mechanics,1984,51:745-752
[194]杨光松.损伤力学与复合材料损伤.北京,国防工业出版社,1994
[195] Mitchell J A, Reddy J N. A refined hybrid plate theory for composite laminates with piezoelectric laminate. International Journal of Solids and Structures,1996,32(16) :2345-2367
[196] Mindlin R D, Deresiewicz H. Elastic spheres in contact under varying oblique forces. ASME Journal of Applied Mechanics, 1953,20:327-344
[197] Mindlin R D. Compliance of elastic bodies in cotact. ASME Journal of Applied Mechanics, 1949,16:259-268
[198] Vu-Quoc L, Zhang X, Lesburg L. Normal and tangential force-displacement relations for frictional elasto-plastic contact of spheres. International Journal of Solids and Strucutures,2001,38:6455-6489
[199] Chang W R, Etsion I, Bogy D B. An elastic-plastic model for the contact of rough surfaces. ASME Journal of Tribology, 1987,109,:300-319
[200] Vu-Quoc L, Zhang X. An Elasto-Plastic contact force-displacement model in the normal direction: Displacement-Driven version, Proceeding of Review on Society in London, Ser. A, 1999,455:4013-4044
[201] Vu-Quoc L, Zhang X, Lesburg L. A normal force-displacement model for contacting spheres accounting for plastic deformation: force-Driven formation. ASME Journal of Applied Mechanics, 2000,67(2):363-371
[202] Thorton C. Coefficient of restitution for nonlinear collisions of elastic perfectly plastic spheres. ASME Journal of Applied Mechanics,1997, 64:383-386.
[203] Mi Y, Crisfiled M A, Davies G A O. Progressive delamination using interface elements. Journal of Composite Materials, 1998,32:1246-1272
[204] Alfano G, Crisfieled M A. Finite element interface models for delamination analysis of laminated composites: mechanical and computational issues. International Journal of Numerical Methods and Engineering,2001,50:1701- 1736
[205] Zou Z, Reid S R, Li S, Soden P D. Modelling interlaminar and interlaminar damage in filament wound pipes under quasi-static indentation. Journal of Composite Materials, 2002,36:477-499
[206] Zou Z, Reid S R, Soden P D, Li S. Mode separation of energy release rate for delamination in composite laminates using sublaminates. International Journal of Solids and Structures, 2001,38:2597-2613
[207] Choi I-k H. Contact force history analysis of composite sandwich plates subjected to low-velocity impact. Composite Structures, 2006, 75:582-586
[208] Greszczuk L B. Damage in composite materials due to low velocity impact, in Zukas, J A, et al. editors, Impact Dynamics, John Wiley &Sons, New York,1982
[209] Shivakumar K N, Elber W, Illg W. Prediction of impact force and duration due to low-velocity impact on circular composite laminates, Journal of Applied Mechanics, 1985,52:674-680
[210] Marur P R, Tippur H V. Numerical analysis of crack-tip fields in functionally graded materials with a crack normal to the elastic gradient. International Journal of Solids and Structures, 2000,37:5353-5370
[211] Ueda S. Crack in functionally graded piezoelectric strip bonded to elastic surface layers under electromechanical loading. Theoretical and Applied Fracture Mechanics, 2003,40:225-236
[212] Jin B, Zhong Z. A moving mode-III crack in functionally graded piezoelectric material: permeable problem. Mechanics Research Communications, 2002, 29:217-224
[213] Chen J, Liu Z X, Zou Z Z. Electromechanical impact of a crack in a functionally graded piezoelectric medium. Theoretical and applied fracture mechanics, 2003, 39:47-60
[214]吴鸿遥.损伤力学.北京,国防工业出版社,1990
[215] Wang S X, Wu L Z, Ma L. Low-velocity impact and residual tensile strength analysis to carbon fiber composite laminates. Materials and Design, 2010, 31: 118-125
[216]徐芝纶,弹性力学,高等教育出版社,1998
[2]徐秉业,罗学富,刘信声,宋国华,孙学伟.接触力学,高等教育出版社,1987
[3] Bradley R S. The cohesive force between solid surfaces and the surface energy of solids. Philosophical Magazine Series, 1932, 7(13): 853-862
[4] Derjaguin B. Untersuchungenüber die Reibung und Adhion. Theorie des Anhaftens kleiner Teilchen. 1934, 69:155-164
[5] Johnson K L, Kendall K, Roberts A D. Surface energy and the contact of elastic solids. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 1971, 324:301-313
[6] Johnson K L. A note on the adhesion of elastic solids. British Journal of Applied Physics 1958, 9:199–200
[7] Ding H J, Chen W, Zhang L. The adhesion of particles. Journal of Colloid and Interface Science, 2006, 53:314-326
[8] Tabor D. Surface forces and surface interactions. Journal of Colloid and Interface Science, 1977, 58:2-13
[9] Greenwood J A, Johnson K L. The mechanics of adhesion of viscoelastic solids. Philosophical Magazine A , 1981, 43:697-711
[10] Muller V M , Yushchenko V S, Derjaguin B V. On the influence of molecular forces on the deformation of an elastic sphere and its sticking to a rigid plane. Journal of Colloid and Interface Science,1980,77:91-101
[11] Maugis D. Adhesion of spheres: the JKR-DMT transition using a dugdale model. Journal of Colloid and Interface Science, 1992,150:243-269
[12] Green A E, Zerna W. Theoretical Elasticity, Oxford, 1968
[13] Lekhnitskii S G. Theory of Elasticity of an Anisotropic Elastic Body: Gostekhizdat, Moscow (in Russian) 1950; English translation, Holden-Day, San Francisco, 1963
[14] Willis J R. Hertzian contact of anisotropic bodies. Journal of Mechanics and Physics of Solids, 1966,14:163–176
[15] Stephen R S. Contact deformation and stress in orthotropic plates. Composites: Part A,2005,36:1421-1429
[16] Jiang D Z, Shen W, Wang X Y. Dynamic contact algorithm of an isotropic projectile contacting an orthotropic target. Computer Methods in Applied Mechanics and Engineering, 2000,189:575-585
[17] Loktev A V. The dynamic contact of an impactor and an elastic orthotropic plate when there are propagating thermoelastic waves. Journal of Applied Mathematics and Mechanics, 2008,72:475-480
[18] Fukumasu N K, Souza R M. Numerical analysis of the contact stresses developed during the indentation of coated systems with substrates with orthotropic properties. Surface & Coatings Technology,2006,201:4294-4299
[19] Swanson S R. Hertzian contact of orthotropic materials. International Journal of Solids and Structures,2004,41:1945-1959
[20] Pedersen P. On the influence of clearance in orthotropic disc-pin contacts. Composite Structures,2007,79:554-561
[21] Cattaneo C. Sul contatto di due corpi elastici: distribuzione locale degli sforzi. Academia dei Lincei, Rendicotti,1938,27(6):342-348
[22] Cattaneo C. Sul contatto di due corpi elastici: distribuzione locale degli sforzi. Academia dei Lincei, Rendicotti,1938,27(6),434-436
[23] Cattaneo C. Sul contatto di due corpi elastici: distribuzione locale degli sforzi. Academia dei Lincei, Rendicotti, 1938.27(6):474-478
[24] Mindlin R D. Compliance of elastic bodies in contact. ASME Journal of Applied Mechanics, 1949,16:259-268
[25] Mindlin R D, Deresiewicz H. Elastic spheres in contact under varying oblique forces. ASME Journal of Applied Mechanics, 1953,20:327-344
[26] Chandrasekaran N, Haisler W, Goforth R. Finite element analysis of hertz contact problem with friction. Finite Elements in Analysis and Design, 1987,3:39-56
[27] Cuttiono J, Dow T. Contact between elastic bodies with an elliptic contact interface in torsion. Journal of Applied Mechanics, 1997,64:144-148
[28] Brock L M. Frictionless indentation by a rigid wedge: the effect of tangential displacements in the contact zone. International Journal of Engineering Science, 1979, 17:365-372
[29] Georgiadis L M. Tangential-displacement effects in the wedge indentation of an elastic half-space d an integral-equation approach. Computational Mechanics, 1998,21:347-352
[30] Argatov I I. Approximate solution of an axisymmetric contact problem withallowance for tangential displacements on the contact surface. Journal of Applied Mechanics and Technical Physics, 2004(a),45:118-123
[31] Argatov I I. Axisymmetric Hertz problem with tangential displacements on the contact surface. Doklady Physics, 2004(b),49:222-225
[32] Sch?fer J, Dippel S, Wolf D E. Force schemes in simulations of granular materials. Journal of Physics I France,1996,6:5-20
[33] Tijskens E, Ramon H, De Baerdemaeker J. Discrete element modeling for process simulation in agriculture. Journal of Sound Vibration, 2003,266: 493-514
[34] Zeebroeck M V, Dintwa E, Tijskens E, et al. Determining tangential contact force model parameters for viscoelastic materials (apples) using a rheometer Postharvest. Biology and Technology,2004, 33:111-125
[35] Argatov I I, Mishuris G S. Axisymmetric contact problem for a biphasic cartilage layer with allowance for tangential displacements on the contact surface. European Journal of Mechanics A/Solids, 2010,29:1051-1064
[36] Chen W W, Wang Q J. A numerical model for the point contact of dissimilar materials considering tangential tractions. Mechanics of Materials, 2008,40:936 -948
[37] Sevostianov I, Kachanov M. Normal and tangential compliances of interface of rough surfaces with contacts of elliptic shape. International Journal of Solids and Structures,2008,45:2723-2736
[38] Allen M, A Ellis J. Stress and strain distribution in the vicinity of interference fit fasteners. AFFDL-TR, 1973,72-153
[39] Crews J H, An elastic-plastic analysis of a uniaxially loaded sheet with an interference fit bolt. NASA-TN-D 1974, 7748
[40] Wanlin G. Elastic-plastic analysis of a finite sheet with a cold worked hole. Engineering Fracture Mechancis, 1993,45:857-864
[41] Christensen P W. A semi-smooth Newton method for elasto-plastic contact problems. International Journal of Solids and Structures, 2002,39:2323-2341
[42] Jackson R L, Green I. A finite element study of elasto-plastic hemispherical contact. ASME Journal of Tribology,2005,127(3):43-54
[43] Robert L, Itzhak G. A statistical model of elasto-plastic asperity contact between rough surfaces. Tribology International, 2006,39:906-914
[44] Grunsweig J, Longman I M, Pci N J. Calculations and measurements of wedge indentations. Journal of the Mechanics and Physics of Solids, 1954,2:81
[45] Haddow J B. On a plane strain wedge indentation paradox. International Journal of Mechanical Science, 1967,9:159
[46] Bhasin Y P, Oxley P L B, Roth R V. An experimentally-determined slip-line field for plain strain wedge indentation of a strain-hardening material. Journal of the Mechanics and Physics of Solids, 1980,2:43
[47] Lockett F J. Indentation of a rigid plastic material by a conical indenter. Journal of the Mechanics and Physics of Solids, 1963,11:345
[48] Pei A L, Hyunb S, Molinaria J F, et al. Finite element modeling of elasto-plastic contact between rough surfaces. Journal of the Mechanics and Physics of Solids, 2005,53:2385-2409
[49] Dumas G, Baronet C N. Elasto-plastic indentation of a half-space by a long, rigid cylinder. International Journal of Mechanical Science,1971,13:519
[50] Skalski K. Contact problems for elasto-plaslic body. (in polish Prace Naukowe Politechniki Warszawskiej),seria "Mechanika", 1986
[51] Follansbee P S, Sinclair GB. Qua-static normal indentation of an elasto-plastic half-space by a rigid sphere. International Journal of Solids and Structures, 1984,20:81
[52] Hardy C, Baronet C N, Trdion G V. Elasto-plastic indentation of a half-space by a rigid sphere. International Journal for Numerical Methods in Engineering, 1971,3:451-462
[53] Krupp H, Sperling G. Theory of adhesion of small particles. Journal of Applied Physics, 1966,37:4176-4180
[54] Wang L G, Sun X P, Huang Y. Friction analysis of microcosmic elastic-plastic contact for extrusion forming. Journal of Materials Processing Technology, 2007,187-188:631-634
[55] Sztok S. Elasto-plastic solution of fictional contact problem by finite element method. Proceeding of International Conference on Barcelona, 1987
[56] Gun H. Elasto-plastic static stress analysis of 3D contact problems with friction by using the boundary element method. Engineering Analysis with Boundary Elements, 2004,28:779-790
[57] Zolotarevskiy V, Kligerman Y, Etsion I. Elastic-plastic spherical contact under cyclic tangential loading in pre-sliding G. Model WEA-99746 7
[58] Chen W M, Li M, Cheng Y T. Analysis on elasto-plastic spherical contact and its deformation regimes, the one parameter regime and two parameter regime by finite element simulation. Vacuum, 2011,85:898-903
[59] Kadin Y, Kligerman Y, Etsion I. Multiple loading of an elastic-plastic spherical contact. International Journal of Solids and Structures, 2006,7:19-127
[60] Bellemare S, Dao M, Suresh S. The frictional sliding response of elasto-plastic materials in contact with a conical indenter. International Journal of Solids and Structures,2007,44:1970-1989
[61] Chang W R, Etsion I, Bogy D B. An elastic-plastic model for the contact of rough surfaces ASME Journal of Tribology, 1987, 109:300-319
[62] Vu-Quoc L, Lesburg L. Contact model of contact law for spherical particles accounting for plastic deformation. International Journal of Solids and Strucutres, 2000,44:170–189
[63] Vu-Quoc L, Zhang X. An Elasto-Plastic contact force-displacement model in the normal direction: Displacement-Driven version, Proceeding of Royal Society in London, Series A, 1991,455:4013-4044
[64] Vu-Quoc L, Zhang X, Lesburg L. A normal force-displacement model for contacting spheres accounting for plastic deformation: force-Driven formation. ASME Journal of Applied Mechanics,2000, 67(2):363-371
[65] Thorton C. Coefficient of restitution for nonlinear collisions of elastic perfectly plastic spheres. ASME Journal of Applied Mechanics, 1997, 64:383-386
[66] Xu H M, Yao X F, Feng X Q, et al. Fundamental solution of a power-law orthotropic and half-space functionally graded material under line loads, Composites Science and Technology, 2008,68:27-34
[67] Kassir M K. Boussinesq problems for non-homogeneous materials. ASCE Journal of Engineering Mechanics Division, 1972,98:457-470
[68] Booker J R, Balaam N P, Davis E H. The behavior of an elastic non-homogen- eous half-space. Part–Circular and strip footings. International Journal of Analytical Methods in Geomechanics,1985,9:369-381
[69] Giannakopoulos A E, Suresh S. Indentation of solids with gradients in elastic properties: Part I. Point Force. International Journal of Solids and Structures,1997,34(19): 2357-2392
[70] Giannakopoulos A E, Pallot P. Two-dimensional contact analysis of elastic graded materials. Journal of the Mechanics and Physics of Solids,2000, 48(8): 1597-1631
[71] Jorgensen O, Giannakopoulos A E, Suresh S. Spherical indentation of composite laminates with controlled gradients in elastic anisotropy. International Journal of Solids and Structures,1998,35(36):5097-5113
[72]温卫东,徐颖,崔海坡.低速冲击下复合材料层合板损伤分析材料工程, 2007,7
[73] Richaardson M O W, Wisheart M J. Review of low velocity impact properties. Composites Part A , 1996,27(A): 1123-1131
[74] Joshi S P, Sun C T. Impact induced fracture in a laminated composite. Journal of Composite Materials,1985,19:51-66
[75] Choi H Y, Wu H Y T, Chang F K. A new approach toward understanding damage mechanisms and mechanics of laminated composites due to low-velocity impact Part II Analysis. Journal of Composite Materials, 1991, 25:1012-1038
[76] Cantwell W J, Morton J. Geometrical effects in the low velocity impact response of CFRP. Composite Structures,1989,12:39-59
[77] Liu, D. and Malvem, L.E. Matrix cracking in impacted glass/epoxy plates. Journal of Composite Materials,1987,21:594-609
[78] Wu, H Y T, Springer G S. Measurements of matrix cracking and delamination caused by impact on composite plates. Journal of Composite Materials, 1988,22: 518-532
[79] Chang F K, Choi H Y, Jeng S T. Study on impact damage in laminated composites. Mechanics of Materials, 1990,10:83-95
[80] Choi H Y, Downs R J,Chang F K. A new approach toward understanding damage mechanisms and mechanics of laminated composites due to low-velocity impact: Part I Experiments. Journal of Composite Maters,1991,25:992-1011
[81] Choi H Y, Chang F Y. Impact damage threshold of laminated composites. In‘Failure Criteria and Analysis in Dynamic Response’, AMD Vol. 107, ASME Applied Mechanics Division,Dallas,TX,1990,31-35
[82] Choi H Y, Wang H S, Chang F K. Effect of laminate configuration and impactor’s mass on the initial impact damage of graphite/epoxy composite plates due to line loading impact. Journal of composite materials, 1992,26(6): 804-827
[83] Chang F K, Chang K Y. A progressive damage model for laminated composites containing stress concentrations. Journal of composite materials, 1987,21: 834 -855
[84] Li S, Soden P D, Reid S R, et al. Indentation of laminated filament-wound composite tubes. Composites,1993,24:407-21
[85] Li S, Soden P D, Reid S R. A finite strip analysis of cracked laminates. Mechanics of Mater erial,1994,18:289-311
[86] Li S, Reid S R, Soden P D. A continuum damage model for transverse matrix cracking in laminated fibre-reinforced composites. Philosophical Transactions of the Royal Society in Lond A, 1998, 356:379-412
[87] Camanho P P, Davila C G, Moura M F D. Numerical simulation of mixed-mode progressive delamination in composite materials, Journal of Composites Materials, 2003, 37:1415-1424
[88] Li CF, Hu N, Yin YJ, et al. Low-velocity impact induced damage of continuous fiber-reinforced composite laminates. Part I. An FEM numerical model. Composite Part A 2002;33(8):1053–1060
[89] Collombet F, Lalbin X, Lataillade J L. Impact behavior of laminated composites: physical basis finite element analysis. Composite Science and Technology, 1998,58:463–478
[90] Choi H Y, Chang F K. A model for predicting damage in graphite/epoxy laminated composites resulting from low-velocity point impact. Journal of Composite Materials, 1992,26:2134–2169
[91] Zou Y, Tong L, Steven G P. Vibration-based model-dependent damage (delamination) identification and health monitoring for composite structures-a review. Journal of sound and vibration, 2000,230(2):357-378
[92] Lee S, Zahuta P. Instrumented impact and static indentation of composites. Journal of Composite Materials, 1991,25:204-222
[93] El-Habak A M. Effect of impact perforation load on GFRP composites. Composites 1993,24(4):341-345
[94] Dorey G. Impact damage in composites-development, consequences and prevention. The 6th International Conference on Composite Materials and 2nd European Conference on Composite Materials, Imperial College, London, 1988, 3:3.1-3.26
[95] Potti S V, Sun C T. Prediction of impact induced penetration and delamination in thick composite laminates. International Journal of Impact Engineering, 1997,19(1):31-48
[96] Gama B A, Gillespie J W. Punch shear based penetration model of ballistic impact of thick-section composites. Composite Structures, 2008,86:356-369
[97] Jiang W G, Hallett S R, Green B G, et al. A concise interface constitutive law for analysis of delamination and splitting in composite materials and its application to scaled notched tensile specimens. International Journal for Numerical Methods in Engineering, 2007, 69:1982–1995
[98] Wardle M W, Tokarsky E W. Drop weight impact testing of laminates reinforced with Kevlar aramid fibres, E-glass and graphite. Composites Technology Research 1983,5:4-10
[99] Hsieh CY, Mount A, Jang B Z, et al. Response of polymer composites to high and low velocity impact. International Proceeding 22nd International SAMPE Technology Conference 199,(6-8):14-27
[100] Caprino G, Lopresto V. The significance of indentation in the inspection of CFRP panels damaged by low-velocity impact. Composite Science and Technology, in press
[101] Cantwell W J, Morton J. Impact perforation of carbon fiber reinforced plastics. Composite Science and Technology,1990,38:119-141
[102] Caprino G, Lopresto V. Factors affecting the penetration energy of glass reinforced plastics subjected to a concentrated transverse load. Accepted for presentation at ECCM 9
[103] Babic L, Dunn C, Hogg P J. Damage development and its significance in GRP subjected to impact. Plastic Rubber Process Application,1989,12:199-207
[104] Bibo G, Leicy D, Hogg P J, Kemp M. High-temperature damage tolerance of carbon reinforced plastics. Part 1: Impact characteristics. Composites, 1994, 25: 414-424
[105] Bibo G A, Hogg P J. Influence of reinforcement architecture on damage mechanisms and residual strength of glass /epoxy composite systems. Composi- te Science and Technology,1998,58:803-813
[106] Delfosse D, Poursartip A. Experimental parameter study of static and dynamic out-of-plane loading of CFRP laminates. International Proceeding,10th, Internat- ional Conference on Composite Materials (ICCM1O), Whistler, Canada, 1985, V:583-590
[107] Caprino G, Lopresto V. On the penetration energy for reinforced plastics under low-velocity impact conditions, Composites Science and Technology, 2001, 61: 65-73
[108] Gama B A, Gillespie J W. Finite element modeling of impact, damage evolution and penetration of thick-section composites, International Journal of Impact Engineering,2011,38(4),181-197
[109] Richaardson M O W, Wisheart M J. Review of low velocity impact properties. Composites Part A, 1996,27(A): 1123-1131
[110] Krawczuk M, Ostachowicsw O, Zak A. Dynamic of cracked composite materialstructures. Computational Mechanics,1997,20:79-83
[111] Crisfield M A, Alfano G. Finite element interface models for the delamination analysis of laminated composites: mechanical and computational issues, International Journal for Numerical Methods in Engineering, 2000, 50:1701– 1736
[112] Borg R, Nilsson L, Simonsson K. Modelling of delamination using a discreized cohesive zone and damage formulation, Composite Science and Technology, 2002, 62:1299–1314
[113] Kinloch A J, Wang Y, Williams J G, et al. The mixed-mode delamination of fibre composite materials, Composite Science and Technology, 1993,47: 225–237
[114] Sankar B V. A finite element for modeling delaminations in composite beams. Computers and Structures, 1991,38:239-246
[115] Barbero E J, ReddyJ N. Modeling of delamination in composite laminates using a layer wise plate theory. nternational Journal of Solids and Structure, 1991,28: 373-388
[116] Harper P W, Hallett S R. Cohesive zone length in numerical simulations of composite delamination. Engineering Fracture Mechanics, 2008,75(16):4774- 4792
[117] Robinson P, Galvanetto U, Tumino D, et al. Numerical simulation of fatigue-driven delamination using interface elements, International Journal for Numerical Methods in Engineering, 2004, 63:1824–1848
[118] Turon A, Davila C G, Camanho P P, et al. An engineering solution for mesh size effects in the simulation of delamination using cohesive zone models, Engine- ering Fracture Mechanics, 2007,74:1665–1682
[119] Turon A, Costa J, Camanho P P. Simulation of delamination in composites under high-cycle fatigue, Compos A:Applied Science Manufacture, 2007,38: 2270- 2282
[120] Dorey G. Impact damage tolerance and assessment in advanced composite materials. Seminar on Advanced Composites, Cranfield Institute of Technology, Cranfield, UK, 1986
[121] Dorey G, Sigety P, Stellbrink K, Hart W G J. Impact damage tolerance of carbon fibre and hybrid laminates. RAE Technical Report 87 057, Royal Aerospace Establishment, Farnborough, UK, 1987
[122] Asp L E, Sjogren A,Greenhalgh E S. Delamination growth and thresholds in acarbon/epoxy composite under fatigue loading, Journal of Composites Technol- ogy and Research,2001,23:55-68
[123] Dugdale D S. Yielding of steel sheets containing slits. Journal of the Mechanics and Physics of Solids 1960,8:100-104
[124] Barenblatt G I. The mathematical theory of equilibrium cracks in brittle fracture. Advances in Applied Mechanics ,1962,7:55-129
[125] Hillerborg A, Mode M, Petersson P E. Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements. Cement and Concrete Research,1976,6:773-782
[126] Blackman B R K, Hadavinia H, Kinloch A J, et al. The use of a cohesive zone model to study the fracture of fibre composites and adhesively-bonded joints, International Journal of Fracture, 2003,19: 25–46
[127] Allix O, Ladeveze P, Corigliano A. Damage analyses of interlaminar fracture specimens, Composite Structures , 1995,31:66–74
[128] Reedy E D, Mello F J, Guess T R. Modeling the initiation and growth of delamination in composite structures, Journal of Composite Materials, 1997,31 (8):812–831
[129] Wisheart M, Richardson M. The finite element analysis of impact induced delamination in composite materials using a novel interface element, Composite Part A,1998, 29(A):301–303
[130] Johnson A, Holzpapfel M. Modeling of soft body impact on a composite structure, Composite Structures, 2003, 61:103–113
[131] Touron A, Camanho P P, Costa J, et al. A damage model for the simulation of delamination in advanced composite under variable model loading, Mechanics of Materials, 2006,38:1072–1089
[132] Whittingham B, Marshall I H, Mitrevski T, et al. The response of composite structures with pre-stress subject to low velocity impact damage. Composite Structures,2004,66:685-698
[133] Robb M D, Arnold W S, Marshall I H. The damage tolerance of GRP laminates under biaxial pre-stress. Composite Structures, 1995,32:141-149
[134] Cantwell W, Morton J. The impact resistance of composite materials-a review. Composites 1991,22:347-362
[135] Fuoss E, Straznicky P, Poon C. Effects of stacking sequence on the impact resistance in composite laminates-part 1: parametric study. Composite Structures, 1998,41:67-77
[136] Fuoss E, Straznicky P, Poon C. Effects of stacking sequence on the impact resistance in composite laminates-part 2: prediction method. Composite Structures,1998,41:177-186
[137] Dost E, Ilcewiz L, Avery W, Coxon B. Effects of stacking sequence on impact damage resistance and residual strength for quasi-isotropic laminates. In: Brien T O, Editor, Composites materials: fatigue and fracture, American Society for Testing and Materials 1991,3:476-500 .
[138] Strait L, Karasek M, Amateau M. Effects of stacking sequence on the impact resistance of carbon fiber reinforced thermoplastic toughened epoxy laminates. Journal of Composite Materials, 1992,26(12):1725-1740
[139] Lopes C S, Seresta O, Coquet Y, et al. Low-velocity impact damage on dispersed stacking sequence laminates. Part I: Experiments, Composites Science and Technology, 2009,69:926-936
[140] Choi I H, Kim I G, Ahn S M, et al. Analytical and experimental studies on the low-velocity impact response and damage of composite laminates under in-plane loads with structural damping effects. Composites Science and Technology, 2010,70 :1513-1522
[141] Nakatani H, Kosaka T, Osaka K, et al. Damage characterization of titanium/GFRP hybrid laminates subjected to low velocity Impact. Composites: Part A (In press)
[142] Zhao Y Y, Xiao T Y, Xue S X, et al. Experiments and simulated calculations on the resistance to low-velocity impact of layered plates with a sandwiched ERM. Journal of Sound and Vibration, 2004,271:615-633
[143] Anderson T, Madenci E. Experimental investigation of low-velocity impact characteristics of sandwich composites. Composite Structures, 2000,50:239-247
[144] Hoppmann W H J. Implusive loads on Beams. Proceeding of Society Experiments, Stress Analysis, 1952, 10.(1) 157
[145] Hosseinzadeh R, Shokrieh M M, Lessard L B. Parametric study of automotive composite bumper beams subjected to low-velocity impacts. Composite Structures, 2005,68:419-427
[146] Sankar B V, Zhu H S. The effect of stitching on the low-velocity impact response of delaminated composite beams. Composites Science and Technology, 2000,60:2681-2691
[147] Lam K Y, Sathiyamoorthy T S. Response of composite beam under low-velocity impact of multiple masses. Composite Structures, 1999,44:2055-220
[148] Karas K. Platten Unter Seitlichem Stoss. Ing Arch, 1939,10:237
[149] Chun L, lam K Y. Dynamic response of fully-clamped laminated composite plates subjected to low-velocity impact of a mass. International Journal of Soli- ds and Structures, 1998,35(11): 963-979
[150] Palazotto A N, Herup E J, Gummadi L N B. Finite element analysis of low-velocity impact on composite sandwich Plates. Composite Structures, 2000, 49:209-227
[151] Besant T, Davis G A O, Hitchings D. Finite element modeling of low velocity impact of composite sandwich panels. Composite, Part A,2001,32:1189-1196
[152] Chun K S,Kassegne S K. Low Velocity Impact Dynamic Behavior of Laminated Composite Nonprismatic Folded Plate Structures. ASME Journal of Engineering Mechanics 2005, 131(7):678
[153] Wu Y, Wu Y D, Wang Y X, Zhong W F. Study on the response to low-velocity impact of a composite plate improved by shape memory alloy. Acta Mechanica Solida Sinica, 2007,20(4):357-365
[154] Choi I-K H. Contact force history analysis of composite sandwich plates subjected to low-velocity impact. Composite Structures, 2006,75:582-586
[155] Yigit A S, Christoforou A P. Limits of asymptotic solutions in low-velocity impact of composite plates. Composite Structures, 2007,81:568-574
[156] Giovanni B, Roberto V. Low velocity impact tests of laminate glass-fiber-epoxy matrix composite material plates. International Journal of Impact Engineering, 2002,27:213-229
[157] Tiberkak R, Bachene M, Rechak S, et al. Damage prediction in composite plates subjected to low velocity impact. Composite Structures , 2008,83:73-82
[158] Setoodeh A R, Malekzadeh P, Nikbin K. Low velocity impact analysis of laminated composite plates using a 3D elasticity based layerwise FEM. Materials and Design, 2009,30:3795-3801
[159] Zhang Y, Zhu P, Lai X M. Finite element analysis of low-velocity impact damage in composite laminated plates. Materials and Design, 2006,27:513-519
[160] Roh J H, Kim J H. Hybrid smart composite plate under low velocity impact. Composite Structures, 2002,56:175-182
[161] Mehmet A,Ramazan K K, Yusuf A. Compression-after impact behavior of laminated composite plates subjected to low velocity impact in high temp- eratures. Composite Structures, 2009,89:77-82
[162] Meo M, Antonucci E, Duclaux P, et al. Finite element simulation of low velocityimpact on shape memory alloy composite plates. Composite Structures, 2005, 71:337-342
[163] Shokuhfar A, Khalili S M R, Ashenai G F, et al. Analysis and optimization of smart hybrid composite plates subjected to low-velocity impact using the response surface methodology (RSM). Thin-Walled Structures, 2008,46: 1204-1212
[164] Ganapathy S, Rao K P. Failure analysis of laminated composite cylindrical/sphe- rical shell panels subjected to low-velocity impact. Computers and Structures, 1998,68:627-641
[165] Khalili S M R, Soroush M, Davar A, et al. Finite element modeling of low- velocity impact on laminated composite plates and cylindrical shells. Composite Structures, 2011,9:1363-1375
[166] Ganapathy S, Rao K I. Interlaminar stresses in laminated composite plates, cylindrical/spherical shell panels damaged by low-velocity impact. Composite Structures, 1997,38(1-4):157-168
[167] Her S C, Liang Y C. The finite element analysis of composite laminates and shell structures subjected to low velocity impact. Composite Structures, 2004,66:277 -285
[168] Toh S L, Gong S W, Shim W. Transient stresses generated by low velocity impact on orthotropic laminated cylindrical shells. Composite Structures, 1995, 31:213-228
[169] Malekzadeh K, Khalili M R, Olsson R, et al. Higher-order dynamic response of composite sandwich panels with flexible core under simultaneous low-velocity impacts of multiple small masses. International Journal of Solids and Structures, 2006, 43:6667–6687
[170] Yu J L, Wang E H, Li J R, et al. Static and low-velocity impact behavior of sandwich beams with closed-cell aluminum-foam core in three-point bending. International Journal of Impact Engineering, 2008,35:885-894
[171] Mannana M N, Ansaria R, Abbas L H. Failure of aluminum beams under low velocity impact. International Journal of Impact Engineering, 2008,35: 1201- 1212
[172] Harsoor R, Ramachandra L S. Influence of notch on the elastic-plastic response of clamped beams subjected to low velocity impact. International Journal of Impact Engineering, 2009,36:1058-1069
[173] Wu C L, Sun C T. Low velocity impact damage in composite sandwich beams.Composite Structures, 1996,34:21-27
[174] Aslan Z H, Karakuzu R Z, Okutan B. The response of laminated composite plates under low-velocity impact loading. Composite Structures, 2003,59: 119 -127
[175] Mili F, Necib B. Impact behavior of cross-ply laminated composite plates under low Velocities. Composite Structures, 2001,51:237-244
[176] Etemadi E, Khatibi A A , Takaffoli M. 3D finite element simulation of sandwich panels with a functionally graded core subjected to low velocity impact. Composite Structures, 2009, 89(1):28-34
[177] Apetre N A, Sankar B V, Ambur D R. Low velocity impact response of sandwich beams with functionally graded core. International Journal of Solids and Structures, 2006,43(9):2479-2496
[178] Gong S W, Lam K Y, Reddy J N. The elastic response of functionally graded cylindrical shells to low-velocity impact. International Journal of Impact Engineering, 1999, 22(4):397-417
[179] Pantele C L, Librescu L V. Low velocity impact on a functionally graded circular thin plate. Proceeding ESDA,2006
[180] Reid L T, Larson A. Anthony P. Low velocity impact analysis of functionally graded circular plates. Proceeding IMECE,2006
[181] Recep R, Aydin M, Apalak M K, et al. The elasto-plastic impact analysis of functionally graded circular plates under low-velocities. Composite Structures, 2011,93:860-869
[182] Kubair D V, Lakshmana B K. Cohesive modeling of low-velocity impact damage in layered functionally graded beams. Mechanics Research Communications, 2008, 35: 104-114
[183] Collombet F,Bonini J,Lataillade J L.A three-dimensional modeling of low velocity impactdamage in composite laminates. International Journal for Numerical Methods in Engineering,1996,39:1491-1516
[184] Choi H Y,Chang F K. A model for predicting damage in graphite/epoxy laminated composites resulting from low-velocity point impact. Journal of Composite Materials,1992,26(14):2134-2169
[185]张彦,来新民等.复合材料铺层板低速冲击作用下损伤的有限元分析,复合材料学报,2006,23(2):150-157
[186] Krishnamurthy K S,Mahajan P,Mittal R K. A parametric study of the impact response and damage of laminated cylindrical composite shells. CompositesScience and Technology,2001,61:1655-1669
[187]邓梁波,叶天麒.圆板在物体撞击下的非线性动力响应.力学学报,1990,22(4):420-428
[188] Hou P J, Petrinic N, Ruiz C,et al.Prediction of impact damage in composite plates. Composites Science and Technology,2000,60:273-281
[189] Talreja R,Stiffness properties of composite laminates with matrix cracking and internal delaminations. Engineering Fracture Mechanics,1986, 25(6):751-762
[190]傅衣铭.结构非线性动力学分析.广州:暨南大学出版社,1997
[191] Liu Z S,Somasak S P.Response of plate and shell structures due to low velocity impact. Journal of Engineering Mechanics,1997,123(12):1230-1237
[192] Chung Y W,Ching H Y.Impact damage in composite laminates.Computers Structures,1990,37(6):967-982.
[193] Reddy J N. A simple higher-order theory for laminated composite plates. Journal of Applied Mechanics,1984,51:745-752
[194]杨光松.损伤力学与复合材料损伤.北京,国防工业出版社,1994
[195] Mitchell J A, Reddy J N. A refined hybrid plate theory for composite laminates with piezoelectric laminate. International Journal of Solids and Structures,1996,32(16) :2345-2367
[196] Mindlin R D, Deresiewicz H. Elastic spheres in contact under varying oblique forces. ASME Journal of Applied Mechanics, 1953,20:327-344
[197] Mindlin R D. Compliance of elastic bodies in cotact. ASME Journal of Applied Mechanics, 1949,16:259-268
[198] Vu-Quoc L, Zhang X, Lesburg L. Normal and tangential force-displacement relations for frictional elasto-plastic contact of spheres. International Journal of Solids and Strucutures,2001,38:6455-6489
[199] Chang W R, Etsion I, Bogy D B. An elastic-plastic model for the contact of rough surfaces. ASME Journal of Tribology, 1987,109,:300-319
[200] Vu-Quoc L, Zhang X. An Elasto-Plastic contact force-displacement model in the normal direction: Displacement-Driven version, Proceeding of Review on Society in London, Ser. A, 1999,455:4013-4044
[201] Vu-Quoc L, Zhang X, Lesburg L. A normal force-displacement model for contacting spheres accounting for plastic deformation: force-Driven formation. ASME Journal of Applied Mechanics, 2000,67(2):363-371
[202] Thorton C. Coefficient of restitution for nonlinear collisions of elastic perfectly plastic spheres. ASME Journal of Applied Mechanics,1997, 64:383-386.
[203] Mi Y, Crisfiled M A, Davies G A O. Progressive delamination using interface elements. Journal of Composite Materials, 1998,32:1246-1272
[204] Alfano G, Crisfieled M A. Finite element interface models for delamination analysis of laminated composites: mechanical and computational issues. International Journal of Numerical Methods and Engineering,2001,50:1701- 1736
[205] Zou Z, Reid S R, Li S, Soden P D. Modelling interlaminar and interlaminar damage in filament wound pipes under quasi-static indentation. Journal of Composite Materials, 2002,36:477-499
[206] Zou Z, Reid S R, Soden P D, Li S. Mode separation of energy release rate for delamination in composite laminates using sublaminates. International Journal of Solids and Structures, 2001,38:2597-2613
[207] Choi I-k H. Contact force history analysis of composite sandwich plates subjected to low-velocity impact. Composite Structures, 2006, 75:582-586
[208] Greszczuk L B. Damage in composite materials due to low velocity impact, in Zukas, J A, et al. editors, Impact Dynamics, John Wiley &Sons, New York,1982
[209] Shivakumar K N, Elber W, Illg W. Prediction of impact force and duration due to low-velocity impact on circular composite laminates, Journal of Applied Mechanics, 1985,52:674-680
[210] Marur P R, Tippur H V. Numerical analysis of crack-tip fields in functionally graded materials with a crack normal to the elastic gradient. International Journal of Solids and Structures, 2000,37:5353-5370
[211] Ueda S. Crack in functionally graded piezoelectric strip bonded to elastic surface layers under electromechanical loading. Theoretical and Applied Fracture Mechanics, 2003,40:225-236
[212] Jin B, Zhong Z. A moving mode-III crack in functionally graded piezoelectric material: permeable problem. Mechanics Research Communications, 2002, 29:217-224
[213] Chen J, Liu Z X, Zou Z Z. Electromechanical impact of a crack in a functionally graded piezoelectric medium. Theoretical and applied fracture mechanics, 2003, 39:47-60
[214]吴鸿遥.损伤力学.北京,国防工业出版社,1990
[215] Wang S X, Wu L Z, Ma L. Low-velocity impact and residual tensile strength analysis to carbon fiber composite laminates. Materials and Design, 2010, 31: 118-125
[216]徐芝纶,弹性力学,高等教育出版社,1998