Free-edge and ply cracking effect in cross-ply laminated composites under uniform extension and thermal loading

The interlaminar stresses and displacements near the free-edges and ply cracks are investigated by using the state space equation method for general cross-ply laminates subjected to extension and/or thermal loading. By this approach, a laminated plate may be composed of an arbitrary number of orthotropic layers, each of which may have different material properties and thickness. The method takes into account all independent material constants and guarantees continuous fields of all interlaminar stresses across interfaces between material layers. Numerical solutions are compared with results obtained from other methods. It is found that the theory provides a satisfactory approximation to the stress singularity occurring in the vicinity of the free-edges and ply cracks.     

Thermal expansion coefficients of cracked laminates

Effective thermal expansion coefficients of cross-ply laminates containing many intralaminar ply cracks are evaluated as functions of ply thermomechanical properties and crack density. The method employed is applicable to any cracked laminate if internal stress fields due to mechanical loads are known. Results obtained demonstrate strong reduction of expansion coefficients due to cracks. It is shown that because of opening of cracks by tension and closing by compression the laminate has several different expansion coefficients, each valid for a different load-temperature combination.     

Analysis of stiffness loss in cross-ply composite laminates

The behaviour of laminated composite plates beyond first-ply failure has been the subject of much research work. It is well known that generally, the load-bearing capability of laminated composite plates can remain significant despite the presence of some damage in the plies. Traditionally, the ply-discount method has been used among analysts and designers, although the approach is generally regarded as too conservative. It is therefore desirable to develop models for the prediction of the mechanical properties of damaged composite laminates at various applied loads, and to be able to correlate the changes in properties with the amount of damage and cracking within each constituent ply. Generally, if the models are to be useful as predictive tools, they must be capable of not only sufficiently describing the damage state but also the nature of the damage evolution with loading. This ‘evolution law’ is often obtained through fracture analysis, although it should be noted that the diffused nature of cracks and the multiplicity of failure modes in composites in general greatly complicates the analysis. The problem of transverse matrix cracking in cross-ply laminates under uniaxial tension is considerably simpler because it is essentially dominated by mode I fracture. Thus it is necessarily the first step for any model aiming to predict stiffness losses in composite laminates. In this paper, a constitutive model of the damage state for composite laminates, first proposed by Allen et al., is used with a damage evolution criterion based on strain energy to predict the stiffness loss due to matrix cracking in cross-ply laminated composite plates. Although the constitutive model does not require the determination of many constants, the state of damage is described by a vector of internal state variables (ISV), which contains information on the crack geometry and fracture modes. A series of parametric finite element analyses was performed to determine the effects of relative ply thicknesses, crack density and crack opening profile on the vector of ISVs. A computer algorithm was written for the analysis of cross-ply laminates based on the damage evolution criterion proposed in this work. The results of the analysis compare favourably with experimental measurements of progressive stiffness loss in damaged cross-ply graphite-epoxy laminates obtained from other researchers.     

Stiffness and fracture analysis of laminated composites with off-axis ply matrix cracking

Matrix cracks parallel to the fibres in the off-axis plies is the first intralaminar damage mode observed in laminated composites subjected to static or fatigue in-plane tensile loading. They reduce laminate stiffness and strength and trigger development of other damage modes, such as delaminations. This paper is concerned with theoretical modelling of unbalanced symmetric laminates with off-axis ply cracks. Closed-form analytical expressions are derived for Mode I, Mode II and the total strain energy release rates associated with off-axis ply cracking in [0/θ]_s laminates. Stiffness reduction due to matrix cracking is also predicted analytically using the Equivalent Constraint Model (ECM) of the damaged laminate. Dependence of the degraded stiffness properties and strain energy release rates on the crack density and ply orientation angle is examined for glass/epoxy and carbon/epoxy laminates. Suitability of a mixed mode fracture criterion to predict the cracking onset strain is also discussed.     

A Higher Order Synergistic Damage Model for Prediction of Stiffness Changes due to Ply Cracking in Composite Laminates

A non-linear damage model is developed for the prediction of stiffness degradation in composite laminates due to transverse matrix cracking. The model follows the framework of a recently developed synergistic damage mechanics (SDM) approach which combines the strengths of micro-damage mechanics and continuum damage mechanics (CDM) through the so-called constraint parameters. A common limitation of the current CDM and SDM models has been the tendency to over-predict stiffness changes at high crack densities due to linearity inherent in their stiffness-damage relationships. The present paper extends this SDM approach by including higher order damage terms in the characterization of ply cracking damage inside the material. Following the SDM procedure, predictions are aided by suitable micromechanical computations of crack opening displacements. A nonlinear SDM model is developed and applied for multiple classes of composite laminate layups. Stiffness predictions for damaged laminates using the developed model are compared with the experimental data for cross-ply ([0m/90n]s), angle-ply ([±θm/90n]s), off-axis ([0/±θ4/01/2]s) and quasi-isotropic ([0/90/±45]s) laminates. A comparison with current linear damage models showcases the usefulness of the proposed nonlinear SDM approach.     

Thermal shock behaviour of angle-ply and woven dense ceramic-matrix composites

The behaviour of two Nicalon/calcium aluminosilicate ceramic composite laminates (a (±45°)_3s and a plain-weave woven) under conditions of thermal shock has been studied. Test specimens heated at various temperatures were quenched into room-temperature water. This was followed by detailed damage characterisation. In addition, post-shock mechanical properties were assessed by tensile tests (for the woven laminate) and flexural tests (for both laminates). Both materials were found to have comparable thermal shock resistance. Crack morphologies comprised matrix cracks of various orientations that exhibited similar characteristics to those described for thermally shocked cross-ply laminates with the same constituents, but cracking was found to be less widespread in the woven laminate. Fibre breaks were also detected on the woven material when high-temperature degradation of the fibre–matrix interface was present. A gradual reduction in properties (stiffness, proportional limit stress, fracture strength) of thermally shocked specimens was identified, which began at larger shocks than those at which thermal shock damage initiated. This was attributed to the extension of some matrix cracks into the bulk of material.     

Residual Stiffness of Cracked Cross-Ply Composite Laminates Under Multi-Axial In-plane Loading

In this paper, the Equivalent Constraint Model (ECM) together with a 2-D shear lag stress analysis approach is applied to predict residual stiffness properties of polymer and ceramic matrix [0/90 _n /0] cross-ply laminates subjected to in-plane biaxial loading and damaged by transverse and longitudinal matrix cracks. It is found that the longitudinal Young’s modulus, shear modulus and major Poisson’s ratio undergo large degradation as the matrix crack density increases, with Poisson’s ratio appearing to be the most affected by transverse cracking. In cross-ply laminates with thick 90° layer strip-shaped delaminations begin to initiate and grow from the tips of matrix cracks at the 0°/90° interface. These delaminations contribute to further stiffness degradation of such laminates, and hence have to be taken into account in failure analysis models. The thickness of the 90° layer plays an important role; the thicker the 90° layer, the bigger stiffness reduction suggesting a size (volume) effect at ply level. In SiC/CAS cross-ply laminates reduction in the longitudinal modulus occurs mainly due to transverse cracks, while the shear modulus appears to be the most affected by the presence of longitudinal cracks. The shear modulus reduction ratio predicted previously by a semi-empirical formula is, in the most of cases, within 10% of the current ECM/2-D shear lag approach value. In some cases, though, the error of the semi-empirical finite element expression can be as big as 20% since it fails to capture damage mode interaction.     

Statistical model of the transverse ply cracking in cross-ply laminates by strength and fracture toughness based failure criteria

Cross-ply laminate subjected to tensile loading provides a relatively well understood and widely used model system for studying progressive cracking of the transverse ply. This test allows to identify material strength and/or toughness characteristics as well as to establish relation between damage level and the composite stiffness reduction. The transverse ply cracking is an inherently stochastic process due to the random variability of local material properties of the plies. The variability affects both crack initiation (governed by the local strength) and propagation (governed by the local fracture toughness). The primary aim of the present study is elucidation of the relative importance of these phenomena in the fragmentation process at different transverse and longitudinal ply thickness ratios. The effect of the random crack distribution on the mechanical properties reduction of the laminate is also considered. Transverse ply cracking in glass fiber/epoxy cross-ply laminates of the lay-ups [0₂/90₂]_s, [0/90₂]_s, and [0/90₄]_s is studied. Several specimens of each lay-up were subjected to uniaxial quasistatic tension to obtain crack density as a function of applied strain. Crack spacing distributions at the edge of the specimen also were determined at a predefined applied strain. Statistical model of the cracking process is derived, calibrated using crack density vs. strain data, and verified against the measured crack spacing distributions.     

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.     

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.     

Damage-induced property changes in composites subjected to cyclic thermal loading

Damage in the form of transverse cracks resulting from thermal loading is studied as it relates to the dimensional stability of flat laminates and stiffness changes in cylindrical tubes. Graphite-epoxy specimens were subjected to cyclic thermal loading in the temperature range −250 to +250°F. It is shown that transverse cracking is the dominant damage mechanism in both types of structural elements. Fiber splitting is also quite common at the low test temperatures. Experimental results indicate that damage significantly reduces the inplane coefficients of thermal expansion of flat laminates and the torsional stiffness of the tubes. Theoretical predictions for coefficients of thermal expansion as a function of crack spacing in flat laminates followed the same trend as experimental results.     

A study of transverse ply cracking using a discrete element method

We study the transverse cracking of the 90° ply in [0/90]_S cross-ply laminates by means of a discrete element method. To model the 90° ply a two-dimensional triangular lattice of springs is constructed where nodes of the lattice model fibers, and springs with random breaking thresholds represent the disordered matrix material in between. The spring-lattice is coupled by interface springs to two rigid bars which represent the two 0° plies in the model, which could be sublaminate as well. Molecular dynamics simulation is used to follow the time evolution of the model system. It was found that under gradual loading of the specimen, after some distributed cracking, segmentation cracks occur in the 90° ply which then develop into a saturated state where the ply cannot support additional load. The stress distribution between two neighboring segmentation cracks was determined, furthermore, the dependence of the microstructure of damage on the ply thickness was also studied. To give a quantitative characterization of stiffness degradation, the Young modulus of the system is monitored as a function of the density of segmentation cracks. The results of the simulations are in satisfactory agreement with experimental findings and with the results of analytic calculations.     

The behaviour of Kevlar fibre-epoxy laminates under static and fatigue loadings. Part I—experimental

Kevlar 49 fibre and unidirectional Kevlar fibre reinforced plastic (KFRP) laminates both show an increase in stiffness under monotonic tensile loading. This stiffening effect is time-dependent and is reversible once the load is removed. In contrast, the modulus of a cross-ply KFRP laminate is affected primarily by matrix cracking of the transverse (90°) ply, and is sensitive to strain-rate and temperature. In cyclic (tensile) loading, however, the modulus of the cross-ply laminate depends on a combination of the fibre stiffening effect and transverse matrix cracking.     

Effect of Off – Axis Matrix Cracking on Stiffness of Symmetric Angle-Ply Composite Laminates

Matrix cracking models developed for cross-ply composite laminates cannot easily be applied to more complicated geometries. In this paper a detailed analysis of the effect of matrix cracking on the longitudinal Young’s Modulus of a [0/45]s plate under uniaxial tension is attempted. The theoretical approach, based on a semi-analytical generalized plane strain model, is compared to experimental data obtained by microscopic strain measurements on a fiber sensor using the technique of laser Raman Spectroscopy. The experimental results are in a good agreement with theoretical stiffness degradation predictions obtained using the semi-analytical model.     

Predicting evolution of ply cracks in composite laminates subjected to biaxial loading

An energy-based model is developed to predict the evolution of sub-critical matrix crack density in symmetric multidirectional composite laminates for the case of multiaxial loading. A finite element-based numerical scheme is also developed to evaluate the critical strain energy release rate, G_Ic, associated with matrix micro-cracking, a parameter that previously required fitting with experimental data. Furthermore, the prediction scheme is improved to account for the statistical variation of G_Ic within the material volume by using a two-parameter Weibull distribution. The variation of G_Ic with increasing crack density is also accounted for based on reported experimental evidence. The simulated results for carbon/epoxy and glass/epoxy cross-ply laminates demonstrate the ability of the improved model to predict the evolution of multidirectional ply cracking. By integrating this damage evolution model with the synergistic damage mechanics approach for stiffness degradation, the stress-strain response of the studied laminates is predicted. Finally, biaxial stress envelopes for ply crack initiation and pre-determined stiffness degradation levels are predicted to serve as representative examples of stiffness-based design and failure criterion.     

A probabilistic approach for transverse crack evolution in a composite laminate under variable amplitude cyclic loading

This paper presents a theoretical approach for predicting transverse cracking behavior in a cross-ply laminate with a thick transverse ply under variable amplitude loads for which the cracks grow instantaneously, or very quickly, across the specimen width. The transverse crack density was derived on the basis of the slow crack growth (SCG) concept using the Paris law in conjunction with the Weibull distribution for a brittle material subjected to multi-stage cyclic loading. A fracture criterion obtained was related with the empirical rules by Miner and Broutman & Sahu. Next, the probabilistic SCG model was applied to transverse cracking in a cross-ply laminate under multi-stage cyclic loading. The two-stage fatigue tests with various loading sequences and amplitudes were conducted for carbon fibre reinforced plastic (CFRP) cross-ply laminates in addition to single-stage fatigue tests for various maximum stresses. The experiment results were compared with the predictions to verify the validity of the model.     

Failure mechanisms in laminates of silicon carbide/calcium-aluminosilicate ceramic composite

A systematic investigation was conducted of failure mechanisms and their quantitative effects on the overall mechanical behaviour in laminates of a fibre-reinforced ceramic-matrix composite (silicon carbide/calcium-aluminosilicate) under static tensile loading. Three laminates were investigated: quasi-isotropic, cross-ply and angle-ply. Matrix cracking initiated in all laminates well below the proportional limit of the stress/strain relationship. This cracking grew in a random manner with increasing load; however, no continuous crack was observed across the whole width of the specimen. The measured Young's modulus and Poisson's ratio agreed well with the corresponding values from classical laminated plate theory, but this was not the case for the first ply failure and ultimate strengths.     

The role of carbon nanofibers on thermo-mechanical properties of polymer matrix composites and their effect on reduction of residual stresses

In this research, the effects of carbon nanofibers (CNFs) on thermo-elastic properties of carbon fiber (CF)/epoxy composite for the reduction of thermal residual stresses (TRS) using micromechanical relations were studied. In the first step, micromechanical models to calculate the coefficient of thermal expansion (CTE) and Young's modulus of CNF/epoxy and CNF/CF/epoxy nanocomposites were developed and compared with experimental results of the other researchers. The obtained results of the CTE and Young's modulus of modified Schapery and Halpin-Tsai theories have good agreement with the experimental results. In the second step, the classical lamination theory (CLT) was employed to determine the TRS for CNF/CF/epoxy laminated nanocomposites. Also, the theoretical results of the CLT were compared with experimental results. Finally, reduction of the TRS using the CLT for different lay-ups such as cross ply, angle ply, and quasi-isotropic laminates were obtained. The results demonstrated that the addition of 1% weight fraction of CNF can reduce the TRS that the most reduction occurred in the unsymmetric cross-ply laminate by up to 27%.     

Interlaminar stresses in composite laminates: Thermoelastic deformation

An approximate elasticity solution for prediction of the displacement, stress and strain fields within the m-layer, symmetric and balanced angle-ply composite laminate of finite-width and subjected to uniform axial extension was developed earlier [4]. In the present paper, the authors have extended that solution to treat thermal stresses and deformations induced by a uniform change in laminate temperature. The results have revealed not only the complex fields within the laminate, but also inter-relationships between the lamina axial and shearing coefficients of thermal expansion and the effective laminate coefficients of thermal expansion. Further, the solution is shown to recover laminated plate theory predictions for thermally induced fields at interior regions of the laminate, thereby confirming the boundary layer nature of the interlaminar phenomena for the thermoelastic case. Finally, the results exhibit the anticipated response in congruence with the mechanical solution of Ref. [4] and the thermoelastic results satisfy the conditions of self-equilibration necessary for the finite-width laminate subjected to free thermal deformation. Integration of the stress σ_x over the laminate cross-section in the y–z plane is shown to converge to zero as the number of Fourier terms is increased. While the exact solution for mechanical loading is known to exhibit singular behavior, non-convergence of the interlaminar shearing strain is also seen to occur at the intersection of the free edge and planes between lamina of +θ and −θ orientation under thermal loading. The analytical results show excellent agreement with the finite-element predictions for the same boundary-value problem.