Stiffness Reduction Of Laminates Research Articles

Effective shear modulus of a damaged ply in laminate stiffness analysis: Determination and validation

The concept of the “effective stiffness” for plies in laminates containing intralaminar cracks is revisited presenting rather accurate fitting expressions for the effective stiffness dependence on crack density in the ply. In this article, the effective stiffness at certain crack density is back-calculated from the stiffness difference between the undamaged and damaged laminate. Earlier finite element method analysis of laminates with cracked 90-plies showed that the effective longitudinal modulus and Poisson’s ratio of the ply do not change during cracking, whereas the transverse modulus reduction can be described by a simple crack density dependent function. In this article, focus is on the remaining effective constant: in-plane shear modulus. Finite element method parametric analysis shows that the dependence on crack density is exponential and the fitting function is almost independent of geometrical and elastic parameters of the surrounding plies. The above independence justifies using the effective ply stiffness in expressions of the classical laminate theory to predict the intralaminar cracking caused stiffness reduction in laminates with off-axis plies. Results are in a very good agreement with (a) finite element method calculations; (b) experimental data, and (c) with the GLOB-LOC model, which gives a very accurate solution in cases where the crack face opening and sliding displacements are accurately described.

Model to accurately predict out-of-plane shear stiffness reduction in general cracked laminates

The practically important problem of the modification of laminate out-of-plane shear stiffness by ply cracks is hardly investigated in the literature. In this paper, out-of-plane shear stiffness reduction of laminates containing uniform and non-uniform distributions of ply cracks is studied. A novel variational model is developed to determine accurately stress transfer mechanisms and consequently out-of-plane shear stiffness reduction of general cracked laminates under applied out-of-plane shear loads. It is shown that the presence of ply cracks in a laminate under out-of-plane shear loads, perturbs the uniform distribution of shear stresses and induces high gradients of in-plane stresses leading to large shear stiffness reductions. The results are compared with those of the finite element method (FEM) implementing periodic boundary conditions. It is shown that there is excellent accordance between the results obtained from these approaches. The outcome of the paper provides necessary information for determination of damage-based constitutive laws for composites.

Intralaminar Damage Model for Laminates Subjected to Membrane and Flexural Deformations

Prediction of transverse damage initiation and evolution for not necessarily symmetric laminates under membrane and/or bending loads is the subject of this work. The laminate stiffness reduction is computed via crack opening displacement (COD) methods and the generalization to multiple cracking laminas is made via continuum damage mechanics (CDM) concepts. Using available COD solutions combined with homogenization techniques leads to an analytical constitutive model capable of predicting the initiation and evolution of crack density versus applied strain, as well as laminate modulus degradation, not only for symmetric laminates subjected to membrane deformation but also for general laminates subjected to flexural deformations as well. To adjust the model parameters, experimental data is required in the form of crack density, or modulus reduction, versus strain for two laminates of the same material system. Then, the model is capable of predicting crack density and modulus degradation for other laminate stacking sequences. The model takes into account crack closure, which is important under flexure, as well as the case of the center lamina straddling the neutral axis. The effect of thermal stresses is incorporated in the formulation.

Damage Characterization in Glass Fiber/Epoxy Laminates Using Electronic Speckle Pattern Interferometry

The degradation of the elastic properties of composite laminates with intralaminar cracks is caused by reduced stress in the damaged layer which is mainly due to two parameters: the crack opening displacement (COD) and the crack sliding displacement (CSD). In this paper these parameters are measured experimentally, providing laminate stiffness reduction models with valuable information for validation of used assumptions and for defining limits of their application. In particular, the displacement field on the edges of a [0/ +704/ −704]s glass fiber/epoxy laminate specimens with multiple intralaminar cracks is studied, and the COD and CSD dependence on the applied mechanical load is measured. The specimen full-field displacement measurement is carried out using ESPI (electronic speckle pattern interferometry). By studying the displacement discontinuities, the crack face displacements were measured. A comparison between the COD and the CSD (for the same crack) is performed.

Characterization of damaged composite laminates by an optical measurement of the displacement field

The degradation of the elastic properties of composite laminates with intralaminar cracks is caused by reduced stress in the damaged layer which is mainly due to two parameters: the crack opening displacement (COD) and the crack sliding displacement (CSD). In this paper these parameters are measured experimentally providing laminate stiffness reduction models with valuable information for validation of used assumptions and for defining limits of their application. In particular, the displacement field on the edges of a [0/ +704/ −704]s glass fiber/epoxy laminate specimens with multiple intralaminar cracks is studied and the COD and CSD dependence on the applied mechanical load is measured. The specimen full-field displacement measurement is carried out using ESPI (Electronic Speckle Pattern Interferometry). By studying the displacement discontinuities, the crack face displacements were measured. A comparison between the COD and the CSD (for the same crack) is performed.

A Two-Parameter Model of FRP Laminates Stiffness Reduction

Matrix crack and fibre breakage are the main damage models of the fibre reinforced polymer (FRP) laminates under cyclic loading. In this paper, meso-mechanical analysis is used and a two-parameter model is developed to describe the stiffness reduction. Based on the probability distribution function of fiber strength, the evolution of fibre breakage is deduced. Then with the help of the damage evolution, the stiffness reduction of laminates can be predicted. As an example, the stiffness reduction of grass fibre reinforced polymer (GFRP) laminate is made and the simulation results show that the proposed model has good capacity to describe the stiffness reduction of FRP laminates resulted in the combination of matrix crack and fibre breakage.

Stiffness Reduction in Laminates at High Intralaminar Crack Density: Effect of Crack Interaction

The previously developed closed form expressions for thermo-elastic properties of laminates with intralaminar cracks (Lundmark and Varna, 2005) contain crack surface opening and sliding as main local parameters. The dependence of these parameters on various material and geometrical characteristics was in (Lundmark and Varna, 2005) described by power functions valid only for noninteractive cracks in a given layer (low crack density). In this article the 90-layer crack interaction in terms of its effect on the crack opening displacement (COD) is discussed. The effect on COD is described by the introduced ‘interaction function’ which is determined fitting results of finite element (FE) analysis for cross-ply laminates. To simplify the application in stiffness predictions, the numerically found weak dependence of the interaction on geometrical and material parameters is neglected and the interaction function is presented as a function of crack density only. Using the interaction function to determine the COD, the previously developed calculation scheme (Lundmark and Varna, 2005) has been used to predict stiffness reduction in the entire crack density region. The results are validated comparing with tests and FE simulations.

Estimation of laminate stiffness reduction due to cracking of a transverse ply by employing crack initiation-and propagation-based master curves

The applicability range of toughness-and strength-based criteria for progressive cracking of a transverse layer in a cross-ply composite laminate subjected to tensile loading is considered. Using a deterministic cracking model, approximate relations for the crack density as a function of stress are derived for initiation-and propagation-controlled types of cracking. The master-curve approach is applied to progressive cracking in glass/epoxy laminates. The accuracy of estimation of laminate stiffness reduction by using crack density master curves is evaluated.

Energy criterion for modelling damage evolution in cross-ply composite laminates

The energy dissipated in cross-ply laminates during loading–unloading loops is obtained from stress–strain curves for cross-ply laminates and used in an energy based approach to predict the development of matrix cracking. The dissipated energy is correlated to the crack density growth data recorded for a reference laminate. The critical strain energy release rate, G c obtained in this way is increasing with the applied strain. This phenomenon reflects the statistical nature of G c distribution in the 90-layer: the first cracks (lower strain) develop in positions with lower fracture toughness. The obtained G c data are in a good agreement with fracture toughness data obtained using LEFM based “compliance calibration” model in which the stiffness change with increasing strain is used. Finally, the matrix cracking development is successfully simulated using in the LEFM model, the data for critical strain energy release rate and an earlier derived stiffness–crack density relationship. It has been demonstrated that knowing the laminates geometry and measuring the laminate stiffness reduction with strain or (alternatively measuring the dissipated energy) the damage evolution may be simulated, thus reducing the necessity for optical observations to validation only.

Damage Mechanics Formulation of Cracking Evolution in Composite Laminates. 2nd Report. Formulation of Laminate Constitutive Equation.

In Part I of this study, the constitutive equation of a cracking layer was formulated introducing the new concept that a cracked lamina was replaced by an equivalent uniform work-softening layer. Part II formulates laminate constitutive equation in damage process using the results obtained in Part I and the laminate theory. In this formulation, influence of lamina stacking sequence is discussed through the magnification of an initial damage surface. It is shown that the damage evolution law in Part I can be used for general laminates even if the initial damage surface is enlarged, as long as cracking layer energy release rate is adopted as a cracking criterion. Present model is shown to reproduce experimental results with good accuracy. Finally, discussions about laminate stiffness reduction and stress path in damage process are given.

Damping in partially delaminated composites

A simple model was developed to predict the material damping in partially delaminated composites. First, we evaluated the damping loss factors experimentally in three kinds of specimens corresponding to various partial delamination areas. Second, the stiffness loss with delamination growth was assumed to result directly in the loss of energy from the oscillatory system because the delamination due to interlaminar stress is accompanied with stiffness loss in numerous laminated composites. By correlating the laminate stiffness reduction and the corresponding delamination area, a model for their basic material damping properties was formulated using the elastic-viscoelastic principle, the rule-of-mixtures law and modified Hashin’s Model. We predicted the damping of any partially delaminated composites with different stacking sequences based on Adams and Ni’s work and their basic damping loss factors. Numerical and experimental results demonstrate that damping is significantly influenced by the size of delamination area in laminated composites. In addition, experimental improvements in making accurate damping measurements are discussed as well.

Statistical Fatigue Life Prediction of Cross-Ply Composite Laminates

Within the framework of statistical models of residual strength and fatigue life established in previous work, a new model for determining the stress redistribution function is proposed to predict fatigue life of cross-ply laminates in terms of the principal fatigue mechanisms, i.e., progressive matrix cracking in 90° plies. The simple shear-lag analysis is applied to describe the cycle-number-dependent stress redistribution process and the stiffness reduction of laminates resulting from the progressive matrix cracking in transverse plies. From the strength degradation behaviour and the static strength distribution of 0° plies as well as the fatigue behaviour of 90° plies, the fatigue life of cross-ply laminates with various lay-ups can be calculated from the model. Predictions of fatigue life from the model are compared with experimental data for [0/902] s, [02/902] s and [0/904] s, graphite/epoxy cross-ply laminates, and good agreement indicates success of the approach.

Effect of matrix cracking on stiffness of composite laminates

ABSTRACT Effects of matrix cracking on the sitiffness of composite laminates were investigated. Laminate stiffness reduction caused by matrix cracking was predicted by a finite element model, and the prediction was compared with some existing analytical models and available experimental data. The effective moduli of a cracked layer in a composite laminate were studied. It was found that normalized crack density (thickness of the cracked layer/crack spacing) is a more critical parameter to characterize matrix cracking damage; location of the cracked layer in a laminate has negligible effect on its effective moduli; and orientation of the constraining sublaminate has considerable influence on the effective moduli of the cracked layer. An experimental method to investigate effective shear modulus of a cracked layer was also presented, and the test results were compared with predictions by the finite element model and other existing models.

Matrix cracking in crossply ceramic matrix composites under quasi-static and cyclic loading

Experimental data are presented for the development of 90° ply and 0° ply cracking in two crossply silicon carbide fibre/calcium aluminosilicate matrix laminates under quasi-static loading. under mechanical fatigue loading it is found that there is an increase in ply crack densities and a corresponding laminate stiffness reduction with cycling. Possible mechanisms to account for these observations are proposed. A model is presented which describes the stress/strain behaviour as a function of crack densities based on assumptions of frictional load transfer between fibre and matrix in the longitudinal plies and elastic bonding between the longitudinal and transverse plies.

Interaction between matrix cracking and edge delamination in composite laminates

Matrix cracking and edge delamination are two main damage modes in continuous-fibre composite laminates. They are often investigated separately, and so the interaction between two damage modes has not yet been revealed. In this paper, a simple parallel-spring model is introduced to model the longitudinal stiffness reduction due to matrix cracking and edge delamination together. The energy release rate of edge delamination eliminating the matrix crack effect and the energy release rate of matrix cracking in the presence of edge delamination are then obtained. Experimental materials include carbon- and glass-fibre-reinforced bismaleimide composite laminates under static tension. The growth of matrix cracks and edge delaminations was recorded by means of NDT techniques. Results show that matrix cracks may initiate before or after edge lamination. This depends on the laminate layup, and especially on the thickness of the 90° plies. Edge delamination may also induce matrix cracking. Matrix cracking has a significant effect on the stiffness reduction in GRP laminates. The present model can predict the stiffness reduction in a laminate containing both matrix cracks and edge delaminations. The mixed-mode delamination fracture toughness obtained from the present model shows up to 50% differences compared with O'Brien's model for GRP laminates. However, matrix cracking has a small effect on the mixed-mode interlaminar fracture toughness of the CFRP laminates.

Prediction of Transverse Cracking and Stiffness Reduction in Cross-Ply Laminated Composites

A modified shear lag analysis, taking into account the concept of in terlaminar shear layer, is employed to predict the onset of a transverse crack and multiple transverse cracking. In this analysis, the boundary conditions are satisfied for any trans verse crack spacing. The crack multiplication is simulated by adjusting the crack spacing. Based on this analysis the laminate stiffness reduction due to the multiple transverse crack ing is also evaluated in cross-ply laminated composites. The energy concept is utilized to assess the effect of 90° layer thickness and the constraining effect of 0° layer on the trans verse cracking behavior of cross-ply laminated composites. Predictions of the onset of a transverse crack and stiffness reduction due to transverse cracks are compared with those of previous analyses and existing experimental data. The present analysis is simple, yet its results show reasonable agreement with experimental results.