Composite laminate failure analysis using multicontinuum theory

Damage in a composite material typically begins at the constituent level and may, in fact, be limited to only one constituent in some situations. An accurate prediction of constituent damage at sampling points throughout a laminate provides a genesis for progressively analyzing failure of a composite structure from start to finish. Multicontinuum Theory is a micromechanics based theory and associated numerical algorithm for extracting, virtually without a time penalty, the stress and strain fields for a composites’ constituents during a routine finite element analysis. A constituent stress-based failure criterion is used to construct a nonlinear progressive failure algorithm for investigating the material failure strengths of composite laminates. The proposed failure analysis methodology was used to simulate the nonlinear laminate behavior and progressive damage of selected laminates under both uniaxial and biaxial load conditions up to their ultimate strength. This effort was part of a broader project to compare the predictive capability of current composite failure criteria.     

A finite strip analysis of cracked laminates

As a step towards the construction of a comprehensive model for damaged composites, the deformation of transversely cracked laminates is analysed. The approach is based on a generalised plane strain approach and uses the finite strip method which allows general lay-ups of laminates rather than just cross-ply to be considered. General loading is treated. In addition, the method permits a more flexible control to be exercised between accuracy and the extensiveness of the calculations than can be achieved using existing analyses. This method could be integrated with the finite element analysis of structures to allow for changes in the effective material properties of locally damaged regions. Results produced by the approach are compared with those available in the literature and good agreement is demonstrated.     

Curvature effects in the buckling of symmetrically-laminated rectangular plates with transverse shear deformation

It is shown that the effect of stress discontinuities from ply-to-ply must be taken into account when curvature terms are included along with shear deformation in the buckling analysis of rectangular, symmetrically-laminated plates. Such ply-stress discontinuities lead to curvature terms in the governing equations which differ considerably from those derived for homogeneous plates. Critical buckling loads are determined for orthotropic laminates subjected to biaxial inplane loading and for cylindrical bending of anisotropic plates subjected to uniaxial compression loading. Simply-supported boundary conditions are considered in conjunction with the rectangular, orthotropic laminate, while simply-supported and clamped boundaries are considered for the case of cylindrical bending of anisotropic plates. Numerical results indicate that the curvature terms have little effect on critical buckling loads for the laminates investigated. The effect of transverse shear deformation is shown to depend on the degree of boundary constraint, laminate stacking geometry, and inplane load ratio.     

Analysis of bolted–bonded composite single-lap joints under combined in-plane and transverse loading

A semi-analytical solution method is developed for stress analysis of single-lap hybrid (bolted/bonded) joints of composite laminates under in-plane as well as lateral loading. The laminate and bolt displacements are based on the Mindlin and Timoshenko beam theories, respectively. For the adhesive, the displacement field is expressed in terms of those of laminates by using the shear-lag model. The derivation of the governing equations of equilibrium of the joint is based on the virtual work principle, where the kinematics of each laminate are approximated by local and global functions and the bolt kinematics is assumed in terms of cubic Hermitian polynomials. The capability of the present approach is justified by validation and demonstration problems, including the analysis of bolted and bonded joints and hybrid joints with and without considering a disbond between the adhesive and laminates.     

Analysis of interlaminar stresses in general cross-ply laminates with distributed piezoelectric actuators

This paper represents an analytical solution to determine the interlaminar stresses of general cross-ply laminates with piezoelectric layers as actuators under transverse mechanical loads. The three-dimensional constitutive equations of piezoelectricity are considered and the governing equations are derived within the framework of second-order shear-deformation plate theory as a set of partial differential equations. These equations are solved for two kinds of boundary conditions and the unknown assumed functions of displacement field are found. Numerical results show that this approach can generally predict the behavior of interlaminar stresses. Also, they clearly indicate the singular behavior of interlaminar normal and shear stresses in the boundary region near the edges of the laminate.     

Interlaminar stresses around hole boundaries of composite laminates subjected to in-plane loading

Due to the complex stress state near the curvilinear edge of composite laminates, three-dimensional (3D) finite element analysis is usually applied even it is time consuming. In this paper, in order to save computer time and storage, an asymptotic analysis is employed to separate the 3D problem into two plane problems. One is an interior problem, the other is a boundary layer problem. The former is treated by classical lamination theory and is solved by a special boundary element, the latter is then solved by the finite element developed for the generalized plane deformation problems. When the ratio of laminate thickness to hole diameter is small, the solutions for the laminates containing a circular hole are in good agreement with those obtained by the 3D finite element method.     

Non‐linear dynamic stability analysis of composite laminates under periodic in‐plane compressive loads

This work deals with the investigation of the non‐linear instability behaviour of the composite laminates subjected to periodic in‐plane/axial load, through the finite element formulation with dynamic response analysis. Here, C¹ eight‐noded shear‐flexible plate element, based on a new kind of kinematics which allows to exactly ensure the continuity conditions for displacements and stresses at the interfaces between the layers of the laminate, and also the boundary conditions at the top and bottom surfaces of the laminate, is employed. The non‐linear governing equations obtained are solved using the Newmark direct integration method coupled with a modified Newton–Raphson iteration procedure. The analysis brings out various characteristic features of the dynamic stability such as existence of beats, their dependency on the forcing frequency, and the typical character of vibrations in the different regions. Numerical results are also presented to highlight the influence of ply‐angle and lay‐up of the laminate on dynamic stability behaviour of the composite laminates. Copyright © 1999 John Wiley & Sons, Ltd.     

Gradient piezoelectricity for cracks under an impact load

The flexoelectric effect on elastic waves is investigated in nano-sized cracked structures. The strain gradients are considered in the constitutive equations of a piezoelectric solid for electric displacements and the higher-order stress tensor. The governing equations with the corresponding boundary conditions are derived from the variational principle. The finite element method (FEM) is developed from the principle of virtual work. It is equivalent to the weak-form of derived governing equations in gradient elasticity. The computational method can be applied to analyze general 2D boundary value problems in size-dependent piezoelectric elastic solids with cracks under a dynamic load. The FEM formulation is implemented for strain-gradient piezoelectricity under a dynamic load.     

Nonlinear static analysis of composite laminated plates with evolving damage

Considering geometric nonlinearity and damage evolution, the static response characteristics of laminated composite plates subjected to uniformly distributed loading are investigated using finite element approach based on the first-order shear deformation theory. The damage evolution is modeled employing generalized macroscopic continuum theory within the framework of irreversible thermodynamics. The governing nonlinear equations are solved using Newton–Raphson iterative technique. The resulting finite element-based continuum damage model enables to predict the progressive damage and failure load. A detailed parametric study is carried out to investigate the influences of damage evolution, boundary conditions, span-to-thickness ratio, and lamination scheme on the static response of laminated plates undergoing moderately large deformation. It is revealed that the in-plane stretching forces owing to geometric nonlinearity significantly influence the failure load, damage, and stress distribution for immovable thin laminates.     

Antisymmetric bending analysis of symmetric laminated plates including transverse shear and normal effects

Antisymmetric bending analysis of symmetric laminated plates is presented here. The transverse shear and normal strain and stress effects on bending of such laminates are considered. The displacement fields and the transverse shear and normal stress fields are assumed to preserve the displacement and traction continuity conditions at the interface between layers. A set of twelfth-order governing equations and consistent boundary conditions are given from a mixed variational theorem. Solutions for simply-supported cross-ply plates and a strip are discussed. The numerical results are compared with elasticity solutions and results given from other theories. The present theory is found to agree closely with three-dimensional elasticity solutions.     

Stress intensity factors as the fracture parameters for delamination crack growth in composite laminates

A “mutual integral” approach is used to calculate the mixed-mode stress intensity factors for a free-edge delamination crack in a laminate under tensile loading conditions. This “mutual integral” approach, for generalized plane strain conditions, is based on the application of the path-independent J integral to a linear combination of three solutions: one, the problem of the laminate to be solved using the quasi 3-D finite element method, the second, an “auxiliary” solution with a known asymptotic singular solution, and the third, the particular solution due to the out-of-plane loading. A comparison with the exact solutions is made to determine the accuracy and efficiency of this numerical method. With this “mutual integral” approach, it was found that the calculated mixed-mode stress intensity factors of the free-edge delamination crack remain relatively constant as the crack propagates into the laminate. It was also found that the fracture criterion based on the mixed-mode stress intensity factors is more consistent with the experimental observations than the criterion based on the total energy release rate, and hence demonstrates the importance of the ability to calculate each individual component of the stress intensity factors. Furthermore, it was found that the fracture toughness measurements from double cantilever beam specimens can be used directly to predict the onset of delamination crack growth between two dissimilar laminae. Using these fracture toughness measurements from the double cantilever beam specimens, some examples are given to show that the fracture criterion based on the mixed-mode stress intensity factors can accurately predict the failure load for various laminates under tensile loading conditions.     

Variational approach development in analysis of matrix cracking and induced delamination of cross-ply composite laminates subjected to in-plane shear loading

Matrix microcracking and induced delamination propagating from the edge of microcracks in cross-ply composite laminates with [0_n/90_m]_s and [90_m/0_n]_s layups under in-plane static shear loading are investigated. An admissible stress field, which satisfies all of equilibrium equations, boundary conditions, and continuity of interfaces, is approximated. Then using the principle of the minimum complementary energy, the stress state is obtained from calculations of variation. The calculated stress state gives the stiffness reduction and the total strain energy of the laminated composite structure. Finally, the strain energy release rate of a general cross-ply laminate due to initiation and propagation of matrix cracking and induced delamination can be deduced. Results of the developed approach are in good agreement with experimental observations and finite element analyses, which confirms its accuracy.     

Multilayer hybrid-stress finite element analysis of composite laminate with through-thickness cracks

An assumed hybrid-stress finite element model using a simple composite multilayer element is developed to analyze generally thin or moderately thick composite laminates with through-thickness cracks. The assumed stress field satisfies: (i) equilibrium conditions within each layer, (ii) the traction reciprocity conditions at interlaminar boundaries, and (iii) the traction-free boundary conditions at the top and bottom faces of the laminate. Since the number of nodes and assumed stress parameters are independent of the number of layers, the multilayer element devised is quite effective especially for the laminate with a large number of layers. Several selected composite laminates with a through-thickness edge crack are solved. Many fewer degrees of freedom and only a one-step solution are necessary for the present technique. The variations of mixed-mode stress intensity factors across the thickness of the composite laminate are also computed. Excellent agreements between the present results and referenced solutions are drawn. The technique developed is also applicable to analyze the structural behaviors of the cracked laminate with arbitrary fiber orientation and stacking sequence for which the stress singularity has not yet been found.     

Study of layup influences on the nonlinear behavior of composites by evaluation of ply stiffness reduction

The influence of the stacking sequence on the nonlinear response of composite laminates is investigated. It is shown that a layup dependency solely emerges from damage evolution mechanisms, whereas damage initiation and viscoelastic and viscoplastic strain accumulation are not affected by the layup. This is a result of a proposed procedure that enables the evaluation of the stiffness reduction on lamina level. The residual ply stiffness components can be determined at large deformations and for various laminates under in-plane loading conditions. A finite element study is utilized to characterize the properties of a ply containing discrete cracks. The relationship between transverse and shear stiffness reduction is derived from the FE results. This allows the combined determination of the residual lamina moduli from an axial laminate stiffness. The analysis approach is validated by angle-ply specimens with different layups.     

Initial strain effects in multilayer composite laminates

A boundary integral formulation for the analysis of stress fields induced in composite laminates by initial strains, such as may be due to temperature changes and moisture absorption is presented. The study is formulated on the basis of the theory of generalized orthotropic thermo-elasticity and the governing integral equations are directly deduced through the generalized reciprocity theorem. A suitable expression of the problem fundamental solutions is given for use in computations. The resulting linear system of algebraic equations is obtained by the boundary element method and stress interlaminar distributions in the boundary-layer are calculated by using a boundary only discretization. The approach is general and it does not require a priori assumptions. Numerical results are presented to show the potential of the proposed approach.     

Generalized theoretical analysis method for free-edge delaminations in composite laminates

A simplified method for determining the individual mode components of the strain energy release rate of free-edge delaminations in composite laminates is proposed. Interlaminar stresses are evaluated as an interface moment and interface shear forces obtained from equilibrium equations of stress resultants at the interface between the adjacent layers. The deformation of edge-delaminated laminate is calculated by using a generalized quasi-three dimensional classical laminated plate theory developed by the authors. The analysis provides closed-form expressions for the Mode-I, Mode-II and Mode-III component of the strain energy release rate by combining the deformation of the edge-delaminated laminate with the interface moment and the interface shear forces. The presented method is compared with existing method suggested by Li for the asymmetry laminate. Comparison of the results with a finite element analysis using the virtual crack closure technique shows good agreement.     

层合板在压剪载荷下屈曲分析的三角有限差分法

本文利用J.M.Housner和M.Stein采用过的、具有以较少的自由度获得较精确结果优点的三角有限差分方法,对复合材料加筋层合板的总位能表达式进行单元划分后的差分变换,再利用能量变分原理导出以差分节点挠度为未知特征向量、外加均布载荷为未知特征值的矩阵形式的屈曲控制方程。对在简支、固支或用挠曲弹簧和弯曲弹簧模拟的弹性支持情况下,承受面内单向压力、剪切或压、剪联合载荷作用的复合材料加筋层合板进行了有效的屈曲分析。文中对以上几种情况进行了数值计算,计算结果同实验结果吻合良好。     

Transverse matrix cracks in cross-ply laminates: Stress transfer, stiffness reduction and crack opening profiles

Summary Cross-ply laminates with transverse plies containing through-width matrix cracks across the thickness of the transverse plies are studied using an energy-based approach,complementary to that of Hashin. An upper bound on the effective axial stiffness of the cracked laminates with a uniform distribution of transverse cracks is derived. The equations governing the field variables in a typical RVE are derived using thelayerwise laminate theory of Reddy and are solved using the finite element method. The predicted reduction in the effective axial modulus is in good agreement with experimental results, and it approaches a fixed value with increase in crack density for laminates with bothstaggered andnon-staggered cracking. Laminates with staggered cracks showed a greater reduction in effective modulus at lower crack densities. The stress distribution and mechanics of load transfer is examined in detail, at two crack densities including the characteristic damage state. The crack opening profile has been normalized in a special way in terms of the crack density, layup parameters and material properties.     

Simulation of inelastic response of multidirectional laminates based on stress failure criteria

Abstract

The primary purpose of this paper is to simulate the non-linear stress–strain curve of a multidirectional laminate subjected to an arbitrary in plane load using constituent material data and laminate geometrical parameters. The simulation is performed at a ply level. The classical laminated plate theory is employed to determine the load shared by each lamina in the laminate, while internal stresses in the constituent fibre and matrix of the lamina are obtained using a recently developed bridging micromechanics model. Thus, various failure criteria can be incorporated to detect the failure of a lamina in the laminate, and a progressive failure process is assumed by stiffness discount. Another objective of this paper is to investigate the influence of three typical failure criteria, i.e. the maximum normal stress criterion, the Tsai–Wu criterion, and the Hashin–Rotem criterion, on the simulation. Prediction has been made for T300/5208 graphite–epoxy laminates of a number of layups subjected to uniaxial tension. For the considered laminates, the predicted curves agree well with available experimental data. It is found that the predic tions based on the maximum normal stress criterion are comparable with those based on the other two criteria. As the maximum normal stress criterion is the simplest in application, it is recommended as the first candidate for laminate non-linear and failure analysis.     

Viscoplastic BEM fracture analysis of creeping metallic cracked structures in plane stress using complex variable techniques

This paper extends a boundary element approach developed earlier by the present author to the solution of viscoplastic fracture problems arising in creeping cracked metallic structural components undergoing plane stress deformation conditions. By using a plane stress viscoplastic formulation this work adopts the procedure of Lyness and Moler to express the derivative of displacement rate as the ratio of the imaginary part of the displacement rate to the imaginary part of a complex variable representing the coordinates of a collocation point. In this manner the evaluation of creep stresses at internal points is greatly simplified. The results thus obtained are highly accurate in comparison with conventional BEM. The application of the present BEM approach is illustrated by obtaining creep stress and strain distribution for center cracked and standard CT fracture specimens subjected to remote tensile and shear stress loading under plane stress deformation conditions.