Modelling of fibre pull-out in cracked fibre reinforced composites with linear elastic fracture mechanics

A method of finite element modelling the fracture mechanics of a fibre reinforced cement composite is presented. It embodies the use of beam elements with negative extensional stiffness. The method compares favourably with experimental work. Whilst fibre reinforced cement composites are used in this work, the techniques used are applicable to any type of fibre composite material.     

Shakedown of unidirectional composites

The shakedown problem for a composite lamina made of an elastic-plastic matrix and elastic cylindrical fibers is studied. The plastic deformation modes of the lamina are reviewed, and it is concluded that significant shakedown effects can be caused only by the I₁ = 1/2(T₁₁ + T₂₂) and I₂ = T₃₃ components of the remotely applied stress field which are symmetric about the axis x₃ of the fiber; T₁₁ and T₂₂ are the normal composite stresses in the transverse plane. It is shown that the I₁I₂ stress system is needed also to represent thermal loads caused by a uniform change of temperature in the composite.Two methods for evaluation of shakedown limits in the I₁I₂-plane are described. First, the classical approach involving the determination of parametric families of self-stress fields and the solution of mathematical programming problems is used. Results are presented for selected B-Al, Be-Al, B-Ti and B-Mg composites.In the second method, the shakedown problem is related to the recently developed kinematic hardening rules for fibrous composites. It is shown that the composite will shake down for any loading program within a prescribed domain in the I₁I₂-plane, providing that the domain can be contained within a translated initial yield surface. This approach leads to a closed-form evaluation of shakedown limits for any arbitrary combination of mechanical and thermal variable cyclic loads in fibrous composites with temperaturedependent matrix yield strengths.The relationship between shakedown and fatigue in metal matrix composites is discussed.     

Strength probability of unidirectional hybrid composites

A probability strength analysis of an unidirectional three-component hybrid composite (HC) is carried out for the cases of high modulus (HM) fibres and low modulus (LM) elastic fibers regularly embedded in a low elastic modulus matrix. Both single layer intraply and multilayer HC are considered. The fiber strength is assumed to be a random variable with a Weibull distribution. Breaking of the HM fibers are accumulated initially while probability of LM fiber failure is low. Failure modes tend to be covered by the two extreme cases of alternative failure of HM and LM fibers only. These modes can be categorized by using graph technique. Developed are the algorithm for finding the most probable pattern of fiber breaking and method for estimating the strength and fiber damage of a HC. The stress level at which the LM fibers are found to break represents a lower bound of the HC strength. Damage of HM fibers in a three-component HC is much higher than in a two-component HM fibre composite. Negative ‘hybrid effect’ for strength is obtained.     

Compression failure modes in composites

The relationship between the tendency for growth of a delamination in a layered material and the presence of out-of-plane displacements due to Euler instability of the delaminated plies, is examined. By considering the work done by an external load, an improved expression for the energy release rate, G associated with a layered isotropic plate containing a through width delamination and subject to in-plane compressive loading, is derived and compared with finite element and experimental results.     

Fundamental solutions of the theory of unidirectional composites

Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, No. 3, pp. 120–127, May–June, 1992.     

Compressive strength of fibre composites with random fibre waviness

The compressive strength of unidirectional long fibre composites is predicted for plastic microbuckling from a random two-dimensional distribution of fibre waviness. The effect of the physical size of waviness is addressed by using couple stress theory, with the fibre bending resistance scaling with the fibre diameter d. The predicted statistical distribution of compressive strength is found using a Monte Carlo method. An ensemble of fibre waviness profiles is generated from an assumed spectral density of waviness and the compressive strength for each such realisation is calculated directly by the finite element method. The average predicted strength agrees reasonably with practical values, confirming the hypothesis that microbuckles can be initiated by fibre misalignment. It is found that the probability distribution of strength is well matched by a Weibull fit, and the dependence of the Weibull parameters upon the spectral density of waviness is determined. For the practical range of fibre distributions considered, it is concluded that the strength depends mainly upon the root mean square amplitude of fibre misalignment, with the shape of the power spectral density function playing only a minor role. An engineering model for predicting the compressive strength is proposed, akin to weakest link theory for materials containing flaws. A specimen containing randomly distributed waviness is examined to locate regions of high-fibre misalignment. The strength of each of these weak regions is estimated from a look-up table derived from calculations with idealised circular or elliptical patches of waviness. The strength of the composite is given by the failure stress associated with the weakest such patch. For random distributions of waviness, the predictions using this engineering approach are in good agreement with the direct calculations of strength using the finite element method.     

Micromechanics-based constitutive modeling for unidirectional laminated composites

A micromechanics-based constitutive model is developed to predict the effective mechanical behavior of unidirectional laminated composites. A newly developed Eshelby’s tensor for an infinite circular cylindrical inclusion [Cheng, Z.Q., Batra, R.C., 1999. Exact Eshelby tensor for a dynamic circular cylindrical inclusion. J. Appl. Mech. 66, 563–565] is adopted to model the unidirectional fibers and is incorporated into the micromechanical framework. The progressive loss of strength resulting from the partial fiber debonding and the nucleation of microcracks is incorporated into the constitutive model. To validate the proposed model, the predicted effective stiffness of transversely isotropic composites under far field loading conditions is compared with analytical solutions. The constitutive model incorporating the damage models is then implemented into a finite element code to numerically characterize the elastic behavior of laminated composites. Finally, the present predictions on the stress–strain behavior of laminated composite plate containing an open hole is compared with experimental data to verify the predictive capability of the model.     

Effects of voids on postbuckling delamination growth in unidirectional composites

This work examines the effects of manufacturing induced voids on the postbuckling behavior of delaminated unidirectional composites. In the finite element model developed, a through-width delamination is introduced close to one surface of a flat panel, and a void is placed in the delamination plane ahead of each delamination front. The panel is subjected to compression in the fiber direction. The postbuckling delamination growth is studied by calculating the strain energy release rate (SERR) using the virtual crack closure technique. Local stress analyses of the region near the delamination front are also performed to further investigate the void effects. It is found that although the presence of void does not significantly alter the postbuckling transverse displacement of the delaminated panel, the induced stress perturbation by the void affects the SERR. The Mode II SERR as well as the total SERR increase depending on the size of the void and its distance from the delamination front. Since the Mode I SERR shows non-monotonic behavior with the applied load, the effects of voids are studied on its maximum value.     

Global load-sharing model for unidirectional hybrid fibre-reinforced composites

A promising strategy to increase the tensile failure strain of carbon fibre-reinforced composites is to hybridise carbon fibres with other, higher-elongation fibres. The resulting increase in failure strain is known as the hybrid effect. In the present article, a global load-sharing model for hybrid composites is developed and used to carry out a parametric study for carbon/glass hybrids. Hybrid effects of up to 15% increase in failure strain are predicted, corresponding reasonably well to literature data. Scatter in the carbon fibre strength is shown to be crucial for the hybrid effect, while the scatter in glass fibre strength is much less important. In contrast to reports in earlier literature, the ratio of failure strains of the two fibres has only a small influence on the hybrid effect. The results provide guidelines for designing optimal hybrid composites.     

Locally-exact homogenization theory for transversely isotropic unidirectional composites

The locally-exact homogenization theory for unidirectional composites with square periodicity and isotropic phases proposed by Drago and Pindera [18] is extended to architectures with hexagonal symmetry and transversely isotropic phases. The theory employs Fourier series representation for the displacement fields in the fiber and matrix phases in the cylindrical coordinate system that satisfies exactly the equilibrium equations and continuity conditions in the unit cell's interior. The inseparable exterior problem involves satisfaction of periodicity conditions for the hexagonal unit cell geometry demonstrated herein to be readily achievable using the previously introduced balanced variational principle for square geometries. This variational principle plays a key role in the employed unit cell solution, ensuring rapid convergence of the Fourier series coefficients with relatively few harmonic terms, yielding converged homogenized moduli and local stress fields with little computational effort. The solution's stability is illustrated using the dilute case which is shown to reduce to the Eshelby solution regardless of the employed number of harmonic terms. Comparison with published results and predictions of a finite-volume based homogenization in a wide fiber volume range and different fiber/matrix modulus contrast validates the approach's accuracy, and its utility is demonstrated through rapid local stress recovery in a multi-scale application. This extension completes the development of the theory for three important classes of unidirectional reinforcement arrays, thereby providing an efficient alternative to finite-element based homogenization techniques or approximate micromechanical schemes, as well as an efficient standard against which other methods may be compared.     

Cyclic behavior of unidirectional and cross-ply titanium matrix composites

Relatively simple and efficient micromechanical models are used to obtain the uniaxial response of SCS-6/Timetal 21S with [0]₄ and [0/90]_s laminates when subjected to isothermal and thermomechanical fatigue (TMF) loadings. Features of the modeling that are required to obtain the accurate deformation behavior for this class of materials under these loadings are highlighted. To this end, a comparison is made between the concentric cylinder model and the uniaxial stress model for representing the [0] laminate. The axial stresses from the two models are very similar under mechanical loading. The greatest differences appear under thermal loading alone. The differences on the composite response between a time-independent elastic-plastic and a viscoplastic matrix constitutive model are also examined. The latter is based on the Bodner-Partom unified constitutive model. The [0/90] laminate is treated by adding a parallel element with smeared [90] ply properties to the [0] model and invoking axial strain compatibility as well as stress equilibrium. The proposed constitutive law for the [90] ply includes both matrix viscoplasticity and fiber/matrix separation damage and is based on damage mechanics concepts. The effect of cyclic frequency on TMF behavior is examined. The in-phase TMF life is shown to be very sensitive to frequency due to the relaxation of matrix stress and the attendant increase in fiber stress.     

Modeling long-term creep rupture by debonding in unidirectional fibre-reinforced composites

Delayed fracture due to debonding can be observed in many unidirectional fibre-reinforced composites when the fibre/matrix interface experiences creep. The aim of this work is to describe such a phenomenon within the recently proposed modeling framework of transverse isotropy that allows for a neat decomposition of the mechanical behavior into fibre-directional, transverse, and pure shear parts. Specifically, debonding is here chosen to be governed by the tension transverse to the fibres. One can then speak of a mode-I debonding if use is made of the terminology adopted in fracture mechanics. On another hand, the time-dependent response is attributed to the matrix constituent. As the role of this latter is to deform and support stresses primarily in shear, a viscoelastic behavior is introduced that affects solely the pure shear part of the behavior. We show that both characteristics can be easily embedded into the aforementioned formulation. Among others, the occurrence of tertiary creep is made possible to predict. It is otherwise found that the predicted debonding path always propagates along the direction of the fibres in agreement with many experimental observations found in the literature. On the numerical side, the algorithmic treatment of debonding is independent of the one for viscoelasticity. This renders the implementation within the context of the finite element method very easy.     

单向增强玻璃钢复合材料静/动态拉伸实验研究

本文针对单向增强玻璃钢复合材料,进行了一系列静/动态拉伸试验,利用高速摄影与DIC相结合的方法,获得了材料不同方向、不同应变率的应力-应变曲线以及材料在不同方向上的动态失效应变,精确地描述了材料的静/动态拉伸及失效行为。实验结果表明,纤维增强方向在不同应变率(10^-3、10、10^2 s^-1)拉伸应力-应变曲线均存在一个刚度减小的刚度变化点N,变化后的Echanged分别为初始弹性模量Einitial的67.5%、39.0%、21.4%。此材料在不同应变率(10^-3、10、10^2 s^-1)拉伸情况下,纤维增强的方向1上强度最高(分别为608、967、1 123 MPa),方向2强度最低(分别为75、67、58 MPa),方向3强度较低(分别为90、151、221 MPa)。利用高速摄影与DIC相结合的方法,获得了100 s^-1应变率下,不同铺层方向破坏时刻的动态失效参数(方向1~3的动态失效应变分别为0.267、0.078、0.099),可以更加精确地描述此单向增强玻璃钢复合材料的动态失效行为。     

Accurate evaluation of the transverse shear modulus in unidirectional fibrous composites

A new relation for the prediction of the transverse shear modulus in unidirectional fiber composites has been derived. The theoretical results of this relationship are in better agreement with the experiments than those of other relations, existing in the literature. The discrepancies, which are observed among the theoretical predictions and the experimental values, are explained by the consideration of the boundary layers existing between the matrix and the fibers of the composite. A new model, which includes the intermediate phase between the matrix and the fiber, called the mesophase, is considered in order to take into account the above-mentioned layers.     

Nonlinear creep of unidirectional fibrous composites tensioned along the reinforcement

Creep strains in unidirectional organic-fiber-and organic-glass-fiber-reinforced plastics subject to tension in the reinforcement direction are predicted. Based on homogeneity condition and statistical criteria, it is shown that the creep of the composites is essentially nonlinear. To predict creep strains, use is made of a nonlinear creep model based on a modified Rabotnov’s similarity hypothesis for isochrones and of material constants determined from tests on specimens made of a composite as a whole and specimens made of its separate components, the mixing rule applied in the latter case. The calculated results and the experimental data are in satisfactory agreement __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 5, pp. 20–34, May 2007.