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.     

Boundary-layer hygrothermal stresses in laminated, composite, circular, cylindrical shell panels

Free-edge effects in laminated, circular, cylindrical shell panels subjected to hygrothermal loading are studied by utilizing displacement-based technical theories. Starting from the most general displacement field of elasticity for long, circular, cylindrical shells, appropriate reduced displacement fields are determined for laminated composite shell panels with cross-ply and antisymmetric angle-ply layups. An equivalent single-layer shell theory is used to analytically determine the constant parameters appearing in the reduced displacement fields. A layerwise shell theory is then employed to analytically determine the local displacement functions and the boundary-layer interlaminar stresses in cross-ply and antisymmetric angle-ply shell panels under hygroscopic and/or thermal changes. Several numerical examples for the distributions of transverse shear and normal stresses in various shell panels under a uniform temperature change are presented and discussed.     

Residual stresses and warpage in woven-glass/epoxy laminates

Residual stresses are introduced in composite laminates during curing as a result of differential thermal expansion of the various plies. Residual stresses coupled with asymmetries in the laminate produce warpage. To study these phenomena, symmetric and asymmetric glass-fabric-reinforced laminates were fabricated. The laminate material was fully characterized by determining its physical and mechanical properties at room and elevated temperatures. Thermal strains during curing and subsequent thermal cycling were measured by means of embedded strain gages. Residual stresses were then calculated using the mechanical properties determined before. Warpage for known types of asymmetry was calculated by means of lamination theory and compared with experimental measurements using a projection moire technique. The residual stresses in the studied laminates were very low, owing to the balancing effect of the woven-fabric reinforcement. A crossplied antisymmetric laminate showed saddle-shaped warpage in agreement with the analytical prediction. Unexpected warpage found in symmetric laminates may be due to imperfections in fiber orientations and/or temperature nonuniformities during laminations. Paper was presented at the 1985 SEM Fall Conference on Experimental Mechanics held in Grenelefe, FL on Nov. 17–20.     

Steady-state aeroelasticity of fluid flow over a laminated composite plate

A stress-function-based variational approach is used to determine the interlaminar stresses in a multilayered strip of laminate subject to arbitrary combinations of axial extension, bending, and the steady-state aerodynamic loading of fluid flow over the upper surface of laminated composite plate. Symmetric four-layer, cross-ply and angle-ply laminates are considered in details. Some numerical solutions by using a personal computer are obtained. The present results for four-layer laminates show that the aerodynamic loading has significant effects on the interlaminar stresses near the free-edge regions.     

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

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Effect of laminate construction on residual stresses in graphite/polyimide composites

The influence of ply-stacking sequence and ply orientation on the magnitude of lamination residual stresses in graphite/polyimide angle-ply laminates was investigated. The effect of stacking sequence was investigated with laminates of [0₂/±45]_s, [±45/0₂]_s, [0/+45/0/?45]_s and [+45/0₂/?45]_s layup. The effect of ply orientation was evaluated with additional specimens of [0₂/±15]_s and [0₂/90₂]_s layup. Thermal strains were measured using embedded-gage techniques. Residual strains were determined by comparing thermal strains in the angle-ply laminates with those of a unidirectional laminate. The ply-stacking sequence did not have an influence on the magnitude of residual strains. The highest residual strains occur in the [0₂/90₂]_s laminate and the lowest, approximately one-fourth in magnitude, occur in the [0₂/±15]_s laminate. The maximum residual strains in the [0₂/±45]_s group are slightly lower than those in the [0₂/90₂]_s laminate. Residual stress computations show that, at room temperature, the transverse-to-the-fibers stresses in all plies, except those of the [0₂/±15]_s laminate, exceed the transverse tensile strength of the unidirectional material.     

A two-dimensional continuum theory for angle-ply laminates

A two-dimensional continuum theory of microstructure is developed for stress analysis of angle-ply laminates under in-plane loading. An example problem is used to evaluate the results of the theory against a reference solution obtained by the finite element method. The results are in satisfactory agreement; they also show that the in-plane stresses reach somewhat higher peak values than reported in previous literature.The theory is also presented in a simplified version, which is found to be adequate for predicting interlaminar stresses and in-plane stress resultants, but does not give acceptable results for the variation of in-plane stresses through the thickness of the laminations.     

复合材料层板分层伴以横向裂纹扩展的三维有限元分析

本文采用三维有限元模型,对典型铺设的[0_2/±45_2/90_2]_s碳/环氧复合材料层板中分层伴以横向裂纹的产生和扩展导致的层间应力分布进行了分析。计算结果表明层间裂纹首先在90°层中部出现并开裂至相邻界面处而产生横向裂纹,横向裂纹的出现引起局部分层按三角形状扩展;并指出分层损伤过程是一个主导性的稳定扩展过程,这是导致刚度下降的主要因素。最后,数值计算结果与实验结果比较,两者是吻合的。     

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.     

基体开裂的正交复合材料对称层合板的粘弹性力学行为

本文讨论了含有均匀基体裂纹的正交复合材料对称层合板的线性粘弹性力学行为.采用二维剪切滞后模型并对其层间剪应力在厚度方向进行线性假设分布,求得层合板的平均应力应变的线弹性解,利用等效约束模型和经典层合理论可得到层合板因为含有基体裂纹而所引起的刚度退化现象.在弹性-粘弹性对应原理的基础上对其层合板的线粘弹性进行了讨论和研究.结果表明:层合板的松弛模量和蠕变泊松比随着时间的增加而减少,到达稳态后其值基本上是恒值.并跟Zocher的解析解和有限元数值解作了比较,发现结果非常吻合.     

Assessment of improved zigzag and smeared theories for smart cross-ply composite cylindrical shells including transverse normal extensibility under thermoelectric loading

The improved zigzag theory recently developed by the authors for smart, piezoelectric, and laminated cylindrical shells is assessed for the response of finite-length cross-ply shells and shell panels under mechanical, potential, and thermal loading, in direct comparison with the exact three-dimensional (3D) piezothermoelasticity solution. This theory has the unique features of including the transverse normal strain due to thermoelectric loading without introducing additional deflection variables, capturing the nonlinear potential field and actual temperature profile across laminate thickness, accounting for the layerwise (zigzag) variation of inplane displacements, and satisfying the conditions on transverse shear stresses at the layer interfaces and at the inner and outer surfaces. For the assessment, new results are obtained for the 3D exact solution for smart cylindrical shells having a test laminate with widely different material properties across layers, a piezo-composite laminate and a piezo-sandwich laminate. To ascertain the contribution of the layerwise terms in the inplane displacements, the theory is compared with its smeared counterpart with the same number of primary variables. The effect of inclusion of transverse normal extensibility in these theories is established by comparing with their conventional counterparts that assume constant deflection across the thickness. The effect of span angle (for shell panels), length, and thickness parameters on the error of the 2D theories is illustrated.     

Shear deformation in thermal bending analysis of laminated plates by the GDQ method

The interlaminar stresses and deflections in a laminated rectangular plate under thermal bending were determined by using the generalized differential quadrature method involving the effect of shear deformation. The approximate stress and deflection solutions are obtained under the bending of sinusoidal temperature of thermal load for layer in cross-ply laminates and angle-ply laminates. Numerical results show that the shear deformation has significant effects on the dominant interlaminar stresses and deflections in the laminated plate of thermal bending analysis.     

Non-linear design and control optimization of composite laminated plates with buckling and postbuckling objectives

Multiobjective design and control optimization of composite laminated plates is presented to minimize the postbuckling dynamic response and maximize the buckling load. The control objective aims at dissipating the postbuckling elastic energy of the laminate with the minimum possible expenditure of control energy using a closed-loop distributed force. The layer thicknesses and fiber orientations are taken as design variables. The objectives of the optimization problem are formulated based on a shear deformation theory including the von-Karman non-linear effect for various cases of boundary conditions. The non-linear control problem is solved iteratively until an appropriate convergence criterion is satisfied based on Liapunov–Bellman theory. Liapunov function is taken as a sum of positive definite functions with different degrees. Comparative examples for three-layer symmetric and four-layer antisymmetric laminates are given for various cases of edges conditions. Graphical study is carried out to assess the accuracy of results obtained due to the successive iterations. The influences of the boundary conditions, orthotropy ratio, shear deformation, aspect ratio on the laminate optimal design are elucidated.     

Wave propagation in elastic laminates using a second order microstructure theory

A generalization of the simpler microstructure theory developed earlier for elastic laminates by Sun, Achenbach and Herrmann is used to analyze steady state plane wave propagation. This new version incorporates higher-order thickness variations in the displacement functions and includes restrictions on both displacement and stress at the laminate interfaces.To assess the potential of a second-order microstructure theory for accurate modeling of mechanical processes in laminates, dispersion results and especially mode shape data for both displacements and stresses are obtained and compared to corresponding solutions obtained by the theory of elasticity. The comparisons indicate that while dispersion results may be nearly identical, extremely significant differences may be observed in the mode shapes.     

Properties degradation induced by transverse cracks in general symmetric laminates

This paper presents the details of a methodology for predicting the thermoelastic properties degradation in general symmetric laminates with uniform ply cracks in some or all of the 90° layers. First, a stress transfer method is derived by using the concept of state space equation. The laminate can be subjected to any combination of in-plane biaxial and shear loading, and the uniform thermal loading is also taken into account. The method takes into account all independent material constants and guarantees continuous fields of all interlaminar stresses across interfaces between material layers. By this method, a laminate may be composed of an arbitrary number of monoclinic layers and each layer may have different material property and thickness. Second, the concept of the effective thermoelastic properties of a cracked laminate is introduced. Based on the numerical solutions of specially designed loading cases, the effective thermoelastic constants of a cracked laminate can be obtained. Finally, the applications of the methodology are shown by numerical examples and compared with numerical results from other models and experiment data in the literature. It is found that the theory provides good predictions of the thermoelastic properties degradation in general symmetric laminates.     

Further assessment of a generalized plate model: stress analysis of angle-ply laminates

This study is a continuation of the investigation presented in reference (International Journal of Mechanical Sciences 44 (2002) 287) and deals with an assessment of the stress analysis performance of the generalised shear deformable theory presented in (Acta Mechanica 123 (1997c) 163) when dealing with angle-ply laminated beams. One of the main conclusions is that the existing elasticity solutions for simply supported laminates cannot anymore be considered as safe means in checking and testing the effectiveness of other, conventional shear deformable theories, at least as far as angle-ply laminates are concerned.     

Effect of transverse cracks on the effective thermal expansion coefficient of aged angle-ply composites laminates

A modified shear-lag analysis, taking into account the concept of stress perturbation function, is developed and applied to evaluate the effect of transverse cracks on the effective thermal expansion coefficient of aged angle-ply composites laminates. Effects of number of 90° layers and number of θ° layers in the outer angle-ply laminates on the reduction of the effective axial coefficient of thermal expansion have also been studied. The results of this paper represent well the dependence of the reduction of the effective axial coefficient of thermal expansion on the hygrothermal conditions, the fibre orientation angle of the outer layers, the number of cracked cross-ply layers and the number of un-cracked outer θ° layers in laminate.     

Micro-stress prediction in composite laminates with high stress gradients

The objective of this research is to develop a macroscopic theory, which can provide the connection between macro-mechanics and micro-mechanics in characterizing the micro-stress of composite laminates in regions of high macroscopic stress gradients. The micro-polar theory, a class of higher-order elasticity theory, of composite laminate mechanics is implemented in a well-known Pipes–Pagano free edge boundary problem. The micro-polar homogenization method to determine the micro-polar anisotropic effective elastic moduli is presented. A displacement-based finite element method based on micro-polar theory in anisotropic solids is developed in analyzing composite laminates. The effects of fiber volume fraction and cell size on the normal stress along the artificial interface resulting from ply homogenization of the composite laminate are also investigated. The stress response based on micro-polar theory is compared with those deduced from the micro-mechanics and classical elasticity theory. Special attention of the investigation focuses on the stress fields near the free edge where the high macro-stress gradient occurs. The normal stresses along the artificial interface and especially, the micro-stress along the fiber/matrix interface on the critical cell near the free edge where the high macro-stress gradient detected are the focus of this investigation. These micro-stresses are expected to dominate the failure initiation process in composite laminate. A micro-stress recovery scheme based on micro-polar analysis for the prediction of interface micro-stresses in the critical cell near the free edge is found to be in very good agreement with “exact” micro-stress solutions. It is demonstrated that the micro-polar theory is able to capture the micro-stress accurately from the homogenized solutions.     

A unified periodical boundary conditions for representative volume elements of composites and applications

An explicit unified form of boundary conditions for a periodic representative volume element (RVE) is presented which satisfies the periodicity conditions, and is suitable for any combination of multiaxial loads. Starting from a simple 2-D example, we demonstrate that the “homogeneous boundary conditions” are not only over-constrained but they may also violate the boundary traction periodicity conditions. Subsequently, the proposed method is applied to: (a) the simultaneous prediction of nine elastic constants of a unidirectional laminate by applying multiaxial loads to a cubic unit cell model; (b) the prediction of in-plane elastic moduli for [±θ]_n angle-ply laminates. To facilitate the analysis, a meso/micro rhombohedral RVE model has been developed for the [±θ]_n angle-ply laminates. The results obtained are in good agreement with the available theoretical and experimental results.     

Residual Stress Analysis of Orthotropic Materials by the Through-hole Drilling Method

The present study deals with the development and the application of the through-hole drilling method for the residual stress analysis in orthotropic materials. Through a systematic theoretical study of the stress field present on orthotropic plates with a circular hole, the relationships between the relaxed strains measured by a rectangular strain gauge rosette and the Cartesian components of the unknown residual stresses are obtained. The theoretical formulas of each influence coefficient allow the user an easy application of the method to the analysis of uniform-residual stresses on a generic homogeneous orthotropic material. Furthermore, to extend the method to the analysis of the residual stresses on orthotropic laminates, caused by initial in-plane loadings, an alternative formulation is implemented. The accuracy of the proposed method has been assessed through 3D numerical simulations and experimental tests carried out on unidirectional, cross-ply and angle-ply laminates.