A novel 3D mixed finite‐element model for statics of angle‐ply laminates

Analysis of angle‐ply laminates becomes critical and computationally involved because of the presence of extension–shear coupling. A refined three‐dimensional, mixed, 18‐node finite element (FE) model has been developed to analyse angle‐ply laminates under static loading. The minimum potential energy principle has been used for the development of the mixed FE model, where the transverse stress components (τ_xz, τ_yz and σ_z, where z is the thickness direction) have been incorporated as the nodal degrees of freedom, in addition to the three displacement fields. Further, continuity of transverse stress fields through the thickness of the plate and layerwise continuity of displacement fields have been enforced in the formulation. Because all the constitutive and the compatibility conditions have been ensured within the continuum, the present formulation is unique amongst the family of mixed FE models. Results have been obtained for various angle‐ply laminates and compared with analytical and finite‐element solutions, which have been found to be in good agreement with them. Some new results on angle‐ply with clamped–clamped support condition have also been presented to serve as benchmark results. Copyright © 2003 John Wiley & Sons, Ltd.     

Numerical investigation of matrix cracking propagation in cross‐ply laminated composites subjected to three‐point bending load using concurrent multiscale model

The purpose of the present study is to analyze fiber‐matrix debonding and induced matrix cracking formation as two major micromechanical damage modes in cross‐ply composite laminates using a two‐dimensional numerical approach. To this aim, the cross‐ply laminates containing 90‐degree layers are modeled, where the fibers are arranged randomly in transverse plies. Damage modes in this numerical model are simulated by the cohesive surface method. The performed analyses reveal that in the laminates with 90‐degree layers located in the outer positions, the primary micro damage mode is micro matrix cracking which is initiated from the fiber‐matrix debonding damage mode and will be followed by matrix cracking. The main benefit of the present study in comparison to other numerical methods is proposing a virtual test method for damage analysis of different cross‐ply laminates in which, the matrix cracking formation will emerge physically in a random and antisymmetric pattern similar to the experimental observations.     

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.     

Influence of intralaminar cracking on the apparent interlaminar mode I fracture toughness of cross‐ply laminates

One of the major difficulties in interlaminar fracture tests of multidirectional laminates is the high tendency for intralaminar cracking and the resulting wavy crack propagation. Experimental work showed that this occurred in double cantilever beam (DCB) tests of cross‐ply laminates having a starter crack on a 0°/90° interface. Moreover, under steady‐state propagation conditions, the apparent values of the critical strain energy release rate G_Ic were two times higher than those of 0°/0° specimens. In this paper, a finite‐element‐based progressive damage model was used to simulate crack propagation in cross‐ply specimens. The results showed that transverse cracking alone cannot be responsible for the above difference of G_Ic values. Therefore, the higher propagation G_Ic values for cross‐plies must be attributed to the more extensive fibre bridging observed and to plastic deformations of the 90° interfacial ply.     

Accurate calculation of transverse shear stresses for soft-core sandwich laminates

Accurate evaluation of transverse stresses in soft-core sandwich laminates using the existing 2D finite element (FE) models involves cumbersome post-processing techniques. In this paper, a simple and robust method is proposed for accurate evaluation of through-the-thickness distribution of transverse stresses in soft-core sandwich laminates by using a displacement-based C⁰ continuous 2D FE model derived from refined higher-order shear deformation theory (RHSDT) and a least square error (LSE) method. In this refined higher-order shear deformation theory (RHSDT), the in-plane displacement field for the face sheets and the core is obtained by superposing a global cubically varying displacement field on a zigzag linearly early varying displacement field. The transverse displacement is assumed to have a quadratic variation within the core, and it remains constant in the faces beyond the core. The proposed C⁰ FE model satisfies the condition of transverse shear stress continuity at the layer interfaces and the zero transverse shear stress condition at the top and bottom of the sandwich plate. The nodal field variables are chosen in an efficient manner to circumvent the problem of C¹ continuity requirement of the transverse displacements associated with the RHSDT. The LSE method is applied to the 3D equilibrium equations of the plate problem at the post-processing stage, after in-plane stresses are calculated by using the above FE model based on RHSDT. Thus, the proposed method is quite simple and elegant compared to the usual method of integrating the 3D equilibrium equations at the post-processing stage for the calculation of transverse stresses in a sandwich laminates. The accuracy of the proposed method is demonstrated in the numerical examples through the comparison of the present results with those obtained from different models based on HSDT and 3D elasticity solutions.     

TORSIONAL STRESS ANALYSIS OF CORD-RUBBER COMPOSITE LAMINATES

SUMMARY

The structural response of two-ply cord-rubber composite laminates subjected to torsional loading is investigated using non-linear three-dimensional finite element analysis. The present model treats a two-ply composite as an equivalent three/six-ply system with rubber and cord-rubber elements. The present finite element model is validated by comparing the results obtained to the existing experimental and analytical solutions. The effects of torque and bias angle on twist angle, axial displacement, coupled strain ratio, and interply and interface shear stresses are presented. Also, the effects of interply thickness and rubber elasticity were studied. The presented results illustrate the torsional behaviour of two-ply cord-rubber composites from three-dimensional analysis.     

Finite Element Computation of Woven Ply Laminated Composite Structures up to Rupture

For unidirectional ply laminates, the great diversity of the damage mechanisms and their patterns of evolution make it extremely difficult to estimate the strength margins. In the case of woven ply laminates, the number of damage mechanisms is fairly small (no transverse rupture occurs and the material has a greater resistance to delamination) and the behaviour of the material is fairly simple to model up to rupture. In this study, a numerical model for woven ply laminated composite structures up to rupture is developed. The implementation is performed in a Euler Backward scheme and the consistent tangent stiffness matrix is calculated. Comparison with some experiments on structures are made and the model predicts these experiments well.     

Buckling analysis of composite laminates under end shortening by higher‐order shear deformable finite strips

A higher‐order shear deformable finite strip is developed and employed in the buckling analysis of laminated composite plates when subjected to uniform end shortening. This enables the transverse shear deformation to be accurately incorporated. The permitted laminate material properties are quite general, encompassing anisotropy and full coupling between in‐plane and out‐of‐plane behaviour. Results with respect to the number of plies, thickness of laminate and ratios of E₁₁/E₂₂ are presented for unsymmetric cross‐ply and angle‐ply lay‐ups and for laminates with arbitrary lay‐up arrangements. Copyright © 2002 John Wiley & Sons, Ltd.     

A computational continuum damage mechanics model for predicting transverse cracking and splitting evolution in open hole cross‐ply composite laminates

The present study focuses on a computational constitutive model which predicts the matrix cracking evolution and fibre breakage in cross‐ply composite laminates with open hole under in‐plane loading. To consider the effects of matrix cracking on the nonlinear response of laminates, a simplified crack density based model is applied which evaluates the representative damage parameters of matrix cracking. Furthermore, a developed subroutine based on continuum damage mechanics concepts is applied in ANSYS code which is capable to consider the transverse cracking/splitting evolution and predict the final failure load of mentioned laminate under monotonic loading in a progressive damage analyses. It is shown that the obtained stress–strain behaviours and the damage evaluation of considered laminates are in good agreement with the available experimental results.     

Layerwise modeling of progressive damage in fiber-reinforced composite laminates

This paper investigates the effects of discrete layer transverse shear strain and discrete layer transverse normal strain on the predicted progressive damage response and global failure of fiber-reinforced composite laminates. These effects are isolated using a hierarchical, displacement-based 2-D finite element model that includes the first-order shear deformation model (FSD), type-I layerwise models (LW1) and type-II layerwise models (LW2) as special cases. Both the LW1 layerwise model and the more familiar FSD model use a reduced constitutive matrix that is based on the assumption of zero transverse normal stress; however, the LW1 model includes discrete layer transverse shear effects via in-plane displacement components that are C ⁰ continuous with respect to the thickness coordinate. The LW2 layerwise model utilizes a full 3-D constitutive matrix and includes both discrete layer transverse shear effects and discrete layer transverse normal effects by expanding all three displacement components as C ⁰ continuous functions of the thickness coordinate. The hierarchical finite element model incorporates a 3-D continuum damage mechanics (CDM) model that predicts local orthotropic damage evolution and local stiffness reduction at the geometric scale represented by the homogenized composite material ply. In modeling laminates that exhibit either widespread or localized transverse shear deformation, the results obtained in this study clearly show that the inclusion of discrete layer kinematics significantly increases the rate of local damage accumulation and significantly reduces the predicted global failure load compared to solutions obtained from first-order shear deformable models. The source of this effect can be traced to the improved resolution of local interlaminar shear stress concentrations, which results in faster local damage evolution and earlier cascading of localized failures into widespread global failure.     

A simple estimation method of stress intensity factors for through-cracks in angle-ply laminates

In the present paper a simple method of estimation of stress intensity factors for through-cracks in angle-ply laminates is developed. In this procedure, Savin's elasticity solution for an elliptical hole in two-dimensional infinite plate is used as a basic solution for the stress distribution in each ply of laminate. The present method is applied to the problems of through cracks in a $\text{(}\text{0°}\text{90}\text{)}$_s laminate and a $\text{(}\text{?45°}\text{+ 45°}\text{)}$_s laminate. Comparison with existing numerical solutions obtained by three-dimensional finite element analysis shows good agreement. The simplicity of the present method gives the design engineer a useful tool for estimating stress concentration due to the presence of a hole or a crack.     

Characterisation of Transverse Cracking in a Quasi-Isotropic GFRP Laminate under Flexural Loading

Transverse cracking behaviour in a quasi-isotropic glass/epoxy (GFRP) laminate loaded in flexure is studied experimentally and theoretically. A theory developed for cross-ply laminates is applied to a [0°/90°/–45°/45°] _S quasi-isotropic laminate. An equivalent laminate is introduced to derive the Young's modulus of a cracked transverse ply on the basis of a shear lag analysis. The model predicts the flexural stiffness, the neutral axis position and the residual curvature as a function of the transverse crack density and the in-situ ply stress at first ply failure. Experimental results are obtained with the use of the applied moment – strain data in four-point flexural tests and compared with predictions. Time-dependent behaviour of the residual curvature is also investigated.The theoretical predictions are in reasonably good agreement with the experimental results. It is found that the decrease in the residual curvature after unloading is mainly ascribed to viscoelasticity of the material.     

The Effect of Crack Spacing Distribution on Stiffness Reduction of Cross-ply Laminates

The effect of transverse crack distribution on the effective mechanical properties of cross-ply laminates is considered. Young’s modulus and Poisson’s ratio dependence on the transverse ply crack density is obtained experimentally for glass fiber/epoxy laminates of lay-ups [0₂/90₂]_s, [0/90₂]_s, and [0/90₄]_s subjected to uniaxial tensile loading. Crack spacing distributions at the edge of the specimen are also measured at a predefined applied strain. Mechanical property reduction is evaluated for two crack spacing distributions: uniform spacing routinely considered in theoretical derivations and the experimental crack spacing distribution; the results are compared with test data.     

The behaviour of cross-ply hybrid matrix composite laminates: Part 2: Modelling

In Part 1 of this work experimental data were presented for the initiation and propagation of damage in hybrid matrix and uniform matrix laminates. The data showed that during the extension of cross-ply laminates, either constrained (stable) matrix cracking or brittle (unstable) matrix cracking occurs in the transverse plies, with the transverse ply thickness and level of urethane in the transverse ply determining which type of cracking is observed. In the present paper the stable cracking behaviour is modelled using a shear-lag stress analysis combined with an energy balance and the unstable cracking behaviour is discussed in terms of a statistical distribution of transverse ply strengths.     

Influences of artificial pre-stressing on ply stresses and tensile properties of Vinylon fibre-reinforced aluminium laminates (VIRALL)

A simple micromechanical model based primarily upon the rule-of-mixtures is developed which allows prediction of the effects of fibre preload on the ply stresses (i.e., the initial residual stresses of fibre, adhesive and matrix) induced within VIRALL laminates, and the tensile stress-strain curves and mechanical properties of VIRALL laminates. The analysis of the ply stresses of VIRALL laminates indicates that pre-stressing will dramatically influence the ply stresses. The predicted tensile stress-strain curves of VIRALL laminates are in good agreement with the experimental curves and the results show that the stress-strain curves of VIRALL laminates move upwards when the prestress increases. The predicted tensile mechanical properties of VIRALL laminates at room temperature show good agreement with those obtained experimentally; both show that prestress can improve the tensile properties (i.e., elastic limit strength, 0.2% yield strength and failure strength) of VIRALL laminates.     

Behaviour of unidirectional and crossply ceramic matrix composites under quasi-static tensile loading

Experimental results are presented for the quasi-static tensile behaviour of unidirectional, (0/90)_s, (0₂/90₄)_s and (0/90)_3s silicon carbide fibre (Nicalon) reinforced calcium aluminosilicate glass-ceramic matrix laminates. The stress-strain behaviour and associated damage development is described in detail for each laminate. The damage development is quantified by counts of crack density (in both the longitudinal and transverse plies) and stiffness reduction as functions of applied strain. The damage initiation and growth (and its effect on residual properties) are discussed with reference to the Aveston-Cooper-Kelly (ACK) theory for unidirectional ply cracking and crossply laminate shear-lag (originally developed for polymer matrix composites) to describe the transverse ply cracking behaviour.     

Free vibration of orthotropic laminated plates according to partial hybrid plate element

Abstract

The free vibration analysis of orthotropic composite laminates is investigated by using the partial hybrid plate element. The Hellinger‐Reissner principle is modified by adding kinetic energy. The through thickness effect is properly predicted since the transverse shear stress fields are assumed in the hybrid stress version. The natural frequencies are therefore accurately predicted. Apparently, the present study is more accurate than the displacement‐based higher‐order plate element.     

Effects of Shear‐Transverse Coupling and Plasticity in the Formulation of an Elementary Ply Composites Damage Model,Part I: Model Formulation and Validation

Abstract: Continuum damage mechanics (CDM) models have been employed successfully in the literature to predict the response of laminated composite materials. Some sophisticated models can include the effects of non‐linear shear and transverse damage progression, plasticity and shear‐transverse damage coupling. However, these models require non‐standard test data for calibration that may not always be available to a modeller. In this two‐part study, we examine the effect of neglecting plasticity parameters, and also the effect of neglecting both plasticity and shear‐transverse coupling parameters in simplified CDM models for predicting monotonic tensile strength. In part I, we develop simplified versions of the CDM model and test their ability to accurately predict the failure response of angle‐ply laminates. In part II, we provide details of the experimental test series carried out to determine the input parameters for the models. It was found that neglecting plasticity requires some approximations in the damage development laws, but the resulting model can predict well the response of the angle‐ply laminates tested under monotonic loading to failure. Neglecting shear‐transverse coupling is acceptable for the some materials.     

Model for prediction of simultaneous time-dependent damage evolution in multiple plies of multidirectional polymer composite laminates and its influence on creep

A model to predict time-dependent evolution of simultaneous transverse cracking developed in multiple plies during creep loading and its effects on creep of multidirectional polymer matrix composite laminates is presented. The stress states in the intact regions of the plies are determined using the lamination theory during an incremental change in time. The stored elastic energy, determined using this stress state, is compared with a critical stored elastic energy value for damage to determine if a ply would fracture after the increment. If fracture is predicted, variational analysis is used to determine the perturbation in ply stresses due to cracking. This procedure is repeated to determine the crack evolution and creep strain. Model predictions compared well with experimental results for a [±θ_m/90_n]_s laminate.     

An improved isoparametric quadratic element based on refined zigzag theory to compute interlaminar stresses of multilayered anisotropic plates

An improved eight-noded isoparametric quadratic plate bending element based on refined higher-order zigzag theory (RHZT) has been developed in the present study to determine the interlaminar stresses of multilayered composite laminates. The C⁰ continuous element has been formulated by considering warping function in the displacement field based on the RHZT. Shear locking phenomenon is avoided by considering substitute shear strain field. The continuity of transverse shear stresses cannot be ensured by the proposed zigzag formulation directly, and hence, the continuity conditions of transverse shear stresses have been established by using the three-dimensional (3D) stress equilibrium equations in the present study. The transverse shear stresses are computed in a simplified manner using the differential equations of stress equilibrium. A finite element code is developed by using MATLAB software package. The performance of the present finite element model is validated by comparing the results with 3D elasticity solutions. The superiority of the proposed element in view of computational efficiency, simplicity, and accuracy has been examined by comparing the present solutions with those available in published literature using other elements.