3-D numerical study on the bending of symmetric composite laminates

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Structure and properties of the solution space of general anisotropic laminates

The algebraic structure of the solution space of all types of anisotropic laminates is determined. The full space is shown to be the direct sum of a number of orthogonal eigenspaces, one for each simple or multiple eigenvalue, whose dimension equals the multiplicity. There are eight different types of eigenvalues, which combine to yield eleven distinct types of laminates with peculiar representations of the general solution. All such representations are explicitly obtained, along with the pseudo-metrics based on the binary product of the eigenvectors. This leads to the projection operators in the solution space, spectral sums and intrinsic tensors analogous to the Stroh–Barnett–Lothe tensors in 2-D elasticity. The present theoretical results are obtained by adopting a mixed formulation involving the deflection function and Airy’s stress function, and by using new laminate elasticity matrices different from the conventional stiffness matrices A, B and D. The new formulation also discloses an isomorphism relating each anisotropic laminate to an image laminate, such that every equilibrium solution of the former directly yields an image solution of the latter by interchanging the kinematical and kinetic variables and the in-plane and out-of-plane variables. This implies, in particular, that the classical bending theory of homogeneous plates and symmetric laminates is not a distinct subject, despite its historical development and pedagogical recognition, but is mathematically identical to the plane stress problem of anisotropic elasticity.     

纤维曲线铺放的复合材料层合板的自由振动分析

假设曲线纤维的方向角沿板的长度方向按照线性规律变化,导出了纤维曲线铺设时的参考路径,将参考路径沿板的宽度方向平移可得一种曲线纤维增强复合材料单层板,当这种纤维曲线铺设的单层板对称铺放时即可得相应的曲线纤维增强复合材料层合板．基于弹性薄板的小挠度理论,建立了曲线纤维增强复合材料层合板自由振动问题的基本方程,采用微分求积法进行数值求解,得到了层合板的自振频率及相应的振型．与已有文献计算结果的比较,验证了本文计算结果的正确性．通过数值算例分析了微分求积法求解本问题时的收敛性,研究了纤维铺放路径和边界条件的不同对曲线纤维增强复合材料层合板频率及振型的影响,研究结果可为该种结构的设计提供一定的参考．     

Three-dimensional microscopic interlaminar analysis of cross-ply laminates based on a homogenization theory

In this study, a method for three-dimensional microscopic interlaminar analysis of cross-ply laminates is developed based on a homogenization theory to analyze microscopic interactions between unidirectional long fiber-reinforced laminae. For this, a unit cell of a cross-ply laminate, which includes interlaminar areas, is defined under the assumption that each lamina in the laminate has a transversely square fiber array. Then, showing that the laminate has a point-symmetric internal structure, the symmetry is utilized to introduce half of the unit cell as the domain of analysis. Moreover, the domain of analysis is divided into substructures using a substructure method combined with the homogenization theory, significantly reducing the computational costs. The present method is then applied to the analysis of interlaminar stress distributions in a carbon fiber/epoxy cross-ply laminate subjected to in-plane uniaxial tension. It is shown that microscopic shear stress noticeably occurs at the interface between the 0°- and 90°-plies. It is also shown that the microscopic interaction between the two plies is observed only in the vicinity of the interface.     

Variational principle of hybrid energy and the fundamentals of 3-D laminate theory—A new approach for the analysis of interlaminar stresses in composite laminates

This paper discusses the discontinuity of stresses and strains at interlaminar surfaces of the composue laminate and presents a 3-D laminate theory for composite materials. This paper also presents a new type of elastic energy based on the globally continuous variables in laminates, different from the traditional potential energy and complementary energy. Then, a variational principle corresponding to the 3-D laminate theory is developed. The theory and the principle could be a basis of verifying the 2-D laminate theory and determining the interlaminar stresses near the free edges.     

Analytical solutions to edge effect of composite laminates based on symplectic dual system

In the symplectic space composed of the original variables, displacements, and their dual variables, stresses, the symplectic solution for the composite laminates based on the Pipes-Pagano model is established in this paper. In contrast to the traditional technique using only one kind of variables, the symplectic dual variables include displacement components as well as stress components. Therefore, the compatibility conditions of displacement and stress at interfaces can be formulated simultaneously. After being introduced into the symplectic dual system, the uniform schemes, such as the separation of variables and symplectic eigenfunction expansion method, can be implemented conveniently to analyze composite laminate problems. An analytical solution for the free edge effect of composite laminates is obtained, showing the effectiveness of the symplectic dual method in analyzing composite laminates.     

基于近场动力学理论的层压板损伤分析方法

提出了一种基于近场动力学理论 的纤维增强复合材料层压板的渐进损伤分析方法.在弹性力学和复合材料力学的基础上, 推导了适用于近场动力学建模的微模量和临界伸长率等基本参量, 结合经典层压板理论中的偏轴模量, 构建了适用于各向异性材料的对点力函数, 可分析3种形式的损伤:纤维断裂, 基体开裂和分层破坏.分析了含圆孔层压板在拉伸载荷作用下的破坏过程, 预测结果与试验结果吻合良好.     

多椭圆孔有限大复合材料层板的应力研究

基于经典层板理论，将复合材料层板的弹性问题化归为均匀各向异性板求解，采用各向异性体平面弹性理论中的复势方法，以Ｆａｂｅｒ级数、保角映射及最小二乘边界配置技术为工具，提出了多椭圆孔有限大层板在任意外载作用下的级数解，详细探讨了各参数对孔边应力分布的影响规律，得到了许多有益结论。     

含椭圆孔的变角度复合材料层合板的屈曲性能研究

本文利用变角度复合材料的纤维方向角可沿平面位置任意连续变化的特点,提出在孔附近采用与孔同心的椭圆曲线作为纤维铺设路径的层合板铺层方案,以改善含椭圆孔的层合板的孔边应力集中,进而提高层合板的抗屈曲性能．主要研究内容有：利用ABAQUS软件分析本文提出的孔边特殊铺层方式下变角度复合材料层合板的面内应力分布及屈曲性能,通过与传统直线铺层方式以及线性变角度铺层方式进行比较,说明了本文提出的新铺层方式的优越性,并详细分析了椭圆孔的离心率、开孔尺寸及开孔方位对层合板的屈曲临界荷载的影响．研究结果可为含椭圆孔的变角度复合材料层合板的结构设计和优化提供一定的参考．     

Singular full-field stresses in composite laminates with open holes

A method of the superposition of a hybrid and displacement approximation was developed to provide the accurate stress fields in a multilayered composite laminate, including the singular neighborhood of the ply interface and the hole edge. Asymptotic analysis was used to derive the hybrid stress functions. The displacement approximation is based on the polynomial B-spline functions. The method provides the determination of the coefficient of the singular term along with convergent stress components including the singular regions. Reissner’s variational principle was employed. Simple [45/−45]_s and quasi-isotropic IM7/5250 [45/90/−45/0]_s laminates were analyzed. Uniaxial loading and residual stress calculation (quasi-isotropic laminate) were considered. A convergence study showed that accurate values of the coefficient of the singular term of the asymptotic stress expansion could be obtained with coarse out-of-plane and in-plane subdivisions. The interaction between the singular terms on the neighboring interfaces was found to be important for the convergence with coarse subdivisions. Converged transverse interlaminar stress components as a function of the distance from the hole edge, were shown for all examples.     

含椭圆刚性核有限大复合材料层板弯曲应力分析Bending problem of a finite composite laminated plate with elliptical rigid inclusions

基于经典的复合材料层板理论,将有限大复合材料层板等效成各向异性弹性平板。采用复变函数理论中的Faber级数分析方法,使用最小二乘边界配点法,对含多椭圆刚性核有限大各向异性板弯曲问题进行应力分析,得到了该问题的级数解形式,分析了含椭圆刚性核层板在弯曲载荷下的应力分布,并讨论了形状和结构参数对应力分布的影响。结果表明,本文方法对于分析含多个椭圆形刚性核有限大薄板弯曲应力问题非常有效,该方法具有精度高及计算方便等优点。     

Elliptical inclusion in an anisotropic plane: non-uniform interface effects

We study the plane deformation of an elastic composite system made up of an anisotropic elliptical inclusion and an anisotropic foreign matrix surrounding the inclusion. In order to capture the influence of interface energy on the local elastic field as the size of the inclusion approaches the nanoscale, we refer to the Gurtin-Murdoch model of interface elasticity to describe the inclusion-matrix interface as an imaginary and extremely stiff but zero-thickness layer of a finite stretching modulus. As opposed to isotropic cases in which the effects of interface elasticity are usually assumed to be uniform (described by a constant interface stretching modulus for the entire interface), the anisotropic case considered here necessitates non-uniform effects of interface elasticity (described by a non-constant interface stretching modulus), because the bulk surrounding the interface is anisotropic. To this end, we treat the interface stretching modulus of the anisotropic composite system as a variable on the interface curve depending on the specific tangential direction of the interface. We then devise a unified analytic procedure to determine the full stress field in the inclusion and matrix, which is applicable to the arbitrary orientation and aspect ratio of the inclusion, an arbitrarily variable interface modulus, and an arbitrary uniform external loading applied remotely. The non-uniform interface effects on the external loading-induced stress distribution near the interface are explored via a group of numerical examples. It is demonstrated that whether the nonuniformity of the interface effects has a significant effect on the stress field around the inclusion mainly depends on the direction of the external loading and the aspect ratio of the inclusion.     

材料非线性对复合材料层合板热自由边界效应的影响

本文用准三维有限元法研究了材料非线性对复合材料层合板热自山边界效应的影响,给出了修正型Hahn-Tsai非线性应力-应变关系的三维形式。由本文非线性分析方法得到的层间应力与以往由线性分析方法得到的层间应力做了比较,结果表明:材料非线性能显著降低层间剪应力的集中程度,但对层间正应力影响不太明显。     

大开口复合材料层合板强度破坏研究

复合材料层合板的各向异性及非均质,使得复合材料层合板内部的破坏形式非常复杂.在复合材料结构的设计中,为满足制造及使用功能上的需求,在复合材料层合板承力结构件上不可避免地需要设计各种开口.然而,含大开口复合材料层合板的强度破坏问题变得更为复杂,使得现有的强度理论面临新的挑战.针对碳纤维增强复合材料大开口层合板受单向拉伸载荷作用下的强度破坏问题进行了数值分析和实验研究.首先,根据Hashin准则和刚度退化模型,对含不同圆形开口尺寸的[0]_(10)单向铺层、[0/90]_5和[±45]_5正交铺层的层合板,进行了单向拉伸载荷作用下渐进失效的数值模拟分析,获得了对应结构的极限载荷和破坏模式.在此基础上,采用数字图像相关方法,进行复合材料大开口层合板强度破坏的实验研究.研究结果表明,大开口复合材料层合板在单向拉伸加载下主要呈现脆性破坏形式,破坏起始位置处于应力集中区.此外,破坏强度和失效模式与复合材料铺层方式和开口尺寸大小密切相关.其中[±45]_5铺层的开口层合板承载能力最弱,分层破坏最严重.开口尺寸越大,结构的极限载荷值越低.同实验测试结果相比,数值模拟对复合材料层合板的损伤失效分析略显不足,往往很难全面分析复合材料层合板破坏失效过程中的各种因素的影响.     

Nonlocal stress field of interface dislocations

Summary The basic theory of nonlocal elasticity is stated with emphasis on the difference between the nonlocal theory and classical continuum mechanics. The concept of Nonlocal Interface Residual (NIR) is introduced in nonlocal theory. With the concept of NIR and the nonlocal constitutive equation, we calculate nonlocal stresses due to an edge dislocation on the interface of bi-materials. The nonlocal stress distribution along an interface is quite different from the classical one. Instead of the singularity in the dislocation core, nonlocal stress gives a finite value in the core. A maximum of the stress is also found near the dislocation core. Received 27 May 1997; accepted for publication 1 July 1997     

Interlaminar hygrothermal stresses in laminated plates

Within the elasticity formulation the most general displacement field for hygrothermal problems of long laminated composite plates is presented. The equivalent single-layer theories are then employed to determine the global deformation parameters appearing in the displacement fields of general cross-ply, symmetric, and antisymmetric angle-ply laminates under thermal and hygroscopic loadings. Reddy’s layerwise theory is subsequently used to determine the local deformation parameters of various displacement fields. An elasticity solution is also developed in order to validate the efficiency and accuracy of the layerwise theory in predicting the interlaminar normal and shear stress distributions. Finally, various numerical results are presented for edge-effect problems of several cross-ply, symmetric, and antisymmetric angle-ply laminates subjected to uniform hygrothermal loads. All results indicate high stress gradients of interlaminar normal and shear stresses near the edges of laminates.     

含多椭圆孔(核)复合材料加筋层板的应力集中研究

基于各向异性体平面弹性理论中的复势方法,应用杂交变分原理建立了一种与常规有限元相协调的含任意椭圆核各向异性板杂交应力有限元,采用该杂交应力有限元来描述层板的椭圆核区域,采用杆单元来描述加强筋(杆单元的刚度取为层板沿筋条方向的刚度),其余区域采用常规8节点等参单元进行模拟,建立起分析含多椭圆核复合材料加筋壁板问题的力学分析方法,详细讨论了椭圆核大小、位置、筋条尺寸、相对位置、铺层比例等诸参数的影响规律,得到了一些有益的结论。     

A state space approach for exact analysis of composite laminates and functionally graded materials

A class of problems of composite laminates and functionally graded materials (FGM) under extension, twisting, and bending is formulated in the state space setting. A solution approach for exact analysis of the deformation and stress fields in the media is developed. Exact solutions for torsion of cross-ply laminates and certain FGM are derived, which satisfy exactly the equations of anisotropic elasticity, the end conditions, the traction-free boundary conditions on the bounding planes of the rectangular section, and the interfacial continuity conditions in multilayered composite laminates, regardless of the number of layers. The solutions serve as useful benchmarks for numerical modeling and material characterization of composite laminates and FGM.     

层状复合材料的弹塑性模型

通过将宏微观应力及应变分成面内及面外两部分,利用理想界面上的面外应力与面内应变连续条件,应用微观应力及应变均匀化方法得到了两层复合材料的弹塑性分析模型,并在此基础上构造了多层复合材料的全量型三维显式本构模型.各向同性层组成的复合材料算例表明了该模型的可靠性,同时该模型完全适用于各向异性层组成的复合材料弹塑性分析.     

CONTACT ANALYSIS FOR FIBER-REINFORCED, DELAMINATED LAMINATES WITH KINEMATIC NONLINEARITY

A macro-micro analytical approach for the anti-penetrating contact problem at the interfaces of the delamination in symmetrically cross-plied,fiber-reinforced rectangular laminates is presented in this paper.The laminate is simply supported and subjected to a uniform transverse load with a through-width delamination buried at the center position.A contact factor is defined to characterize the contact efect and determined using the micro-mechanics of composite material.By analyzing the kinematics of nonlinear deformation at the interfaces of the delamination,the contact force is derived.Asymptotic solutions from perturbation analysis are presented.It is found that the deformation of the laminate involves a global deflection and a local buckling.The antipenetrating contact efects are characterized by the local buckling and are intrinsic properties of the laminates,relying only on the geometries of the delamination and the material properties.Parametric analyses show that the location and size of the contact areas and the distribution of the contact force are hardly afected by the aspect ratio.