Effective stiffness concept in bending modeling of laminates with damage in surface 90-layers

Simple approach based on Classical Laminate Theory (CLT) and effective stiffness of damaged layer is suggested for bending stiffness determination of laminate with intralaminar cracks in surface 90-layers and delaminations initiated from intralaminar cracks. The effective stiffness of a layer with damage is back-calculated comparing the in-plane stiffness of a symmetric reference cross-ply laminate with and without damage. The in-plane stiffness of the damaged reference cross-ply laminate was calculated in two ways: (1) using FEM model of representative volume element (RVE) and (2) using the analytical GLOB-LOC model. The obtained effective stiffness of a layer at varying crack density and delamination length was used to calculate the A, B and D matrices in the unsymmetrically damaged laminate. The applicability of the effective stiffness in CLT to solve bending problems was validated analyzing bending of the damaged laminate in 4-point bending test which was also simulated with 3-D FEM.     

Regularity of the transverse crack spacing in a damaged laminate

In evaluating elastic constants for a damaged laminate, the assumption is often made that transverse matrix cracks are equally spaced. To investigate the plausibility of this assumption, an analytical expression for the probability density function of the next failure location in a region of the cross-ply laminate bounded by two existing transverse matrix cracks is derived. The derivation employs the method developed by Oh and Finney for modeling the characteristics of the location of failure in conjunction with a shear-lag solution for the stress state in a transversely cracked layer of a multi-ply laminate. In exercising this model, it is found that the assumption of regularly spaced transverse matrix cracks becomes more plausible as the crack density increases (crack spacing decreases).     

EVOLUTION OF TRANSVERSE MATRIX CRACKING IN CROSS-PLY LAMINATES

The objective of this paper is to develop a micromechanics model for the onset and subsequent multiplication of transverse cracks in the 90° layers of a cross-ply laminate under uniaxial tension. The micromechanics study of transverse matrix cracking is performed by a combination of finite element and analytical analyses. Based on the Griffith fracture criterion, the evolution of transverse matrix crack density is predicted. The prediction compares well with existing experimental data in the literature. Analytical expressions of the non-linear effective stress–strain curves are also obtained for a wide range of material systems. In addition, an analytical expression of the effective stiffness is derived based on the differential self-consistent method. The analytical results agree very well with the finite element 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.     

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.     

Characterization and evolution of matrix and interface related damage in [0/90]S laminates under tension. Part I: Numerical predictions

This paper considers damage development mechanisms in cross-ply laminates using an accurate numerical method that assumes a Generalized Plane Strain (GPS) state. A 2D Boundary Element Method (BEM) model is generated to investigate the two types of damage progression in a [0/90]_S laminate: transverse cracks in the 90° lamina and delamination between both laminae. The model permits the contact between the surfaces of the cracks. The study is carried out in terms of the dependence of the Energy Release Rates (ERR) of the two types of crack on their respective lengths. A special emphasis is put on the mechanisms of the joining of the two aforementioned types of crack, including the study of the distribution of the stresses along the interface between the two plies when the transverse crack is approaching this interface.     

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.     

An investigation of non-linear elastic behavior of CFRP laminates and strain measurement using Lamb waves

This paper investigates the non-linear elastic behavior of unidirectional and cross-ply CFRP laminates and proposes a new method to measure tensile strain using Lamb waves. Young’s modulus was measured as a function of strain in situ using Lamb wave velocity during a tensile test. The stiffening effect of the carbon fibers on [0]₈ specimens and the softening effect of the epoxy matrix on [90]₈ specimens were accurately evaluated. Young’s modulus of the 0° ply was obtained as a quadratic function of strain. Using the function and the rule of mixture, the dependence of Young’s modulus on strain was accurately predicted for cross-ply laminates. Based on the results, the tensile strain was quantitatively correlated with the corresponding arrival time of the Lamb waves. The strains obtained from the proposed method agreed well with those from the strain gauge. Finally, the effect of transverse cracks on the in situ Young’s modulus of the cross-ply laminate under a tensile load was investigated. This method clearly detected even a small decrease in the Young’s modulus due to the transverse cracks in stiffening cross-ply laminate.     

Acoustic structural health monitoring of composite materials : Damage identification and evaluation in cross ply laminates using acoustic emission and ultrasonics

The characterisation of the damage state of composite structures is often performed using the acoustic behaviour of the composite system. This behaviour is expected to change significantly as the damage is accumulating in the composite. It is indisputable that different damage mechanisms are activated within the composite laminate during loading scenario. These “damage entities” are acting in different space and time scales within the service life of the structure and may be interdependent. It has been argued that different damage mechanisms attribute distinct acoustic behaviour to the composite system. Loading of cross-ply laminates in particular leads to the accumulation of distinct damage mechanisms, such as matrix cracking, delamination between successive plies and fibre rupture at the final stage of loading. As highlighted in this work, the acoustic emission activity is directly linked to the structural health state of the laminate. At the same time, significant changes on the wave propagation characteristics are reported and correlated to damage accumulation in the composite laminate. In the case of cross ply laminates, experimental tests and numerical simulations indicate that, typical to the presence of transverse cracking and/or delamination, is the increase of the pulse velocity and the transmission efficiency of a propagated ultrasonic wave, an indication that the intact longitudinal plies act as wave guides, as the transverse ply deteriorates. Further to transverse cracking and delamination, the accumulation of longitudinal fibre breaks becomes dominant causing the catastrophic failure of the composite and is expected to be directly linked to the acoustic behaviour of the composite, as the stiffness loss results to the velocity decrease of the propagated wave. In view of the above, the scope of the current work is to assess the efficiency of acoustic emission and ultrasonic transmission as a combined methodology for the assessment of the introduced damage and furthermore as a structural health monitoring tool.     

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.     

Determination of damage micromechanisms and fracture resistance of glass fiber/epoxy cross-ply laminate by means of X-ray computed microtomography

The onset and evolution of the damage in three dimensions was studied by X-ray computed micro-tomography (XCT) in a notched glass fiber/epoxy cross-ply laminate subjected to three-point bending. It was found that damage began by formation of intraply cracks in the 90° plies followed by intraply cracking the 0° plies. Fiber fracture in front of the notch tip occurred at 65% of the maximum load and finally fiber kinking and interply delamination took place under the loading point. Finite element (FE) simulations were carried out to understand crack initiation and the redistribution of stresses upon crack propagation. The crack area corresponding to each damage mechanism was quantified from the XCT images, and this information was used to determine the effective fracture resistance curve of the cross-ply laminate.     

Tensile behaviour of stainless steel fibre/epoxy composites with modified adhesion

In this study we investigate the tensile behaviour of unidirectional and cross-ply composites reinforced with ductile stainless steel fibres and modified adhesion to the epoxy matrix. Results show that annealed stainless steel fibres have a potential in designing tough polymer composites for structural applications. The stiffness of the UD composites made from these fibres is 77GPa combined with the strain-to-failure between 15% and 18% depending on the level of adhesion. Silane treatments were used to modify the adhesion. By treating the stainless steel fibres with different silane coupling agents, an increase of 50% in the transverse 3-point-bending strength was realised. Increasing the adhesion by 50% leads to a higher tensile strength and strain-to-failure in both UD and cross-ply laminates and a higher in-situ strength of the 90° plies. It also delays formation of matrix cracks and hinders growth of debonding.     

Fatigue damage mechanics of composite materials Part IV: Prediction of post-fatigue stiffness

A model has been developed for predicting the stiffness of cross-ply carbon fibre composite laminates containing a notch from which damage, in the form of matrix cracks, splits and delaminations, has grown. A combination of experimental and theoretical results have been used to deduce appropriate degraded stiffness properties for the damaged regions of the laminate. These degraded stiffnesses have then been incorporated into a finite element representation of the notched laminate to predict the overall stiffness. Agreement with experimental data is satisfactory.     

Matrix cracking in transverse layers of cross-ply beams subjected to bending and its effect on vibration frequencies

The solution for the matrix crack distribution in transverse layers of a cross-ply beam subjected to bending is developed based on the model of Han and Hahn (Compos Sci Technol 35 (1989) 377). This solution accounts for a bimodulus-type of the material response. As a result, it is possible to predict the extent and distribution of matrix cracks as well as the stiffness distribution in the beam subjected to bending, accounting for residual thermal stresses. The results of the solution of the bending problem are employed in the analysis of free and forced vibrations of a cross-ply beam with matrix cracks in transverse layers. The solution differs from the case of vibrations of an intact beam due to a difference in response of damaged layers under tension and compression. The results are shown for the natural frequency of a representative ceramic matrix composite (CMC) beam.     

Evaluation of long-term durability in high temperature resistant CFRP laminates under thermal fatigue loading

Thermal fatigue tests were conducted on high temperature resistant carbon fiber reinforced plastics cross-ply laminates to evaluate microscopic damage progress which affects macroscopic mechanical behavior of the laminates. Materials system used were thermoplastic polyetheretherketone based, AS4/PEEK and thermoset bismaleimide based, G40-800/5260. Several types of laminate configuration were used to clarify the effect of ply thickness on microscopic damage progress. Microscopic damages were observed using optical microscopy and soft X-ray radiography. Energy release rate associated with transverse cracking was calculated using variational analysis. The modified Paris law was used to predict transverse cracking. From comparison to mechanical fatigue test results, it is clarified that transverse crack accumulation rate was larger under thermal fatigue loading at same energy release rate range due to the dependence of the fracture toughness on temperature.     

On transverse matrix cracking in cross-ply laminates loaded in simple bending

A one-dimensional analysis of a cross-ply laminate, containing cracked transverse plies, loaded in flexure is presented. Simple bending theory is used in conjunction with a shear-lag analysis, to calculate the degraded longitudinal modulus of a cracked transverse ply, enabling the flexural modulus of the laminate to be determined. The solution is shown to agree well with a more sophisticated stress transfer model in the literature. The analysis is then extended to calculate the applied bending moment at transverse crack onset under flexural loading using a fracture mechanics approach. The results suggest that the in situ transverse ply stress at which matrix cracking commences for the beam loaded in flexure is very close to the stress level at which the same ply would crack if the laminate were loaded in tension.     

Ply cracking and stiffness degradation in cross-ply laminates under biaxial extension, bending and thermal loading

Transverse ply cracking often leads to the loss of stiffness and reduction in thermal expansion coefficients. This paper presents the thermoelastic degradation of general cross-ply laminates, containing transverse ply cracks, subjected to biaxial extension, bending and thermal loading. The stress and displacement fields are calculated by using the state space equation method [Zhang D, Ye JQ, Sheng HY. Free-edge and ply cracking effect in cross-ply laminated composites under uniform extension and thermal loading. Compos Struct [in press].]. 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. After introducing the concept of the effective thermoelastic properties of a laminate, the degradations of axial elastic moduli, Poisson’s ratios, thermal expansion coefficients and flexural moduli are predicted and compared with numerical results from other methods or available test results. It is found that the theory provides good predictions of the stiffness degradation in both symmetric and antisymmetric cross-ply laminates. The predictions of stiffness reduction in nonsymmetric cross-ply laminates can be used as benchmark test for other methods.     

Prediction for progression of transverse cracking in CFRP cross-ply laminates using Monte Carlo method

This study predicted transverse cracking progression in laminates including 90° plies. The refined stress field (RSF) model, which takes into account thermal residual strain for plies including transverse cracks is formulated, and the energy release rate associated with transverse cracking is calculated using this model. For comparison, the energy release rate based on the continuum damage mechanics (CDM) model is formulated. Next, transverse cracking progression in CFRP cross-ply laminates including 90° plies is predicted based on both stress and energy criteria using the Monte Carlo method. The results indicated that the RSF model and the CDM model proposed in this study can predict the experiment results for the relationship between transverse crack density and ply strain in 90° ply. The models presented in this paper can be applied to an arbitrary laminate including 90° plies.