A micromechanical model for prediction of mixed mode I/II delamination of laminated composites considering fiber bridging effects

Highlights

• A model was proposed based on delamination bridging zone energy absorption concept.

• Effects of fiber bridging and related micro-mechanisms were investigated.

• The traction-separation behaviors of the crack plane were simulated by the model.

• The mixed mode I/II delamination of laminated composite was extracted by the model.

Abstract

In unidirectional laminated composites, fiber bridging as a toughening mechanism has a significant effect on the behavior of mixed mode I/II delamination. In the present paper, effects of fiber bridging and related micro-mechanisms were investigated quantitatively. To this end, a novel micromechanical model called “mixed mode I/II micromechanical bridging model” was proposed based on the calculation of the delamination crack bridging zone energy absorption. Firstly, different failure micro-mechanisms, occurring during the fiber bridging process, were identified. Then, different loading conditions on the bridged fibers were applied. In the next step, the absorbed energy of each failure micro-mechanisms was calculated. Finally, the energy absorbed by the fiber bridging zone was obtained by summation of the absorbed energy of each failure micro-mechanisms. The traction-separation behaviors in both the normal and tangential directions of the crack plane are the outcome of the proposed model. Moreover, the mixed mode I/II delamination failure response of the laminated composite was extracted by plotting G_I versus G_II and compared with the available experimental data. The results show that the proposed model is able to predict the mixed mode I/II delamination behavior of laminated composites considering fiber bridging effects.

Introduction

The widespread applications of laminated composites in different industries make the necessity of the prediction of probable failure modes in such materials for the improvement of their performance. During the service life of composite specimens, delamination as one of the most important failure modes in laminated composites often occurs under mixed mode I/II loading. Due to the creation of the fracture process zone (FPZ) around a delamination crack tip, the initiation and propagation of delamination in laminated composites is a complex phenomenon [1], [2]. This zone delays the fracture of the specimen by absorption of strain energy and prevents the catastrophic failure. The FPZ in unidirectional reinforced laminates includes the fiber bridging zone consisting of non-fractured fibers across the delamination crack surfaces. From a practical point of view, crack bridging is useful, since it leads to increase the fracture resistance of composite materials via propagation of delamination crack [3], [4], [5], [6]. The bridging laws are defined as the relationship between bridging stresses and crack opening displacements. In investigation of the crack bridging in composites, bridging laws are used for calculation of the energy absorption of the fiber bridging toughening mechanism in the fracture process zone [7], [8], [9], [10]. Bridging laws can be extracted via experiments or micromechanical models. A number of micromechanical models have been developed for the mode I delamination with considering fiber bridging effects [5], [11], [12]. Spearing and Evans [5] predicted a relationship between the normal stress and the normal opening by modeling of the bridged fibers as short rectangular cross section beams which overcome the fracture resistance through peeling away from the crack faces. Their model was based on the classical beam theory in which the fiber sliding with friction (observed experimentally [13]) has been ignored. Kaute et al. [11] presented a model based on in situ observation of the fiber pull-out from the matrix during testing of ceramic matrix composites. Their model presented a relation for crack closure force as a function of the crack opening and predicted a decrease of bridging stress due to the fiber failure using the Weibull statistics. Ivens et al. [12] developed a model based on the Wells model [14] to investigate the influence of the interface strength on the fiber bridging. In their model, a double cantilever beam (DCB) with a single fiber bridging a crack was analyzed based on energy considerations. They assumed that the fibers are not pulled out from an increasing depth, and then the elastic stored energy in the bridged fiber only includes the energy used for the fiber peel-off.

Many researchers have qualitatively investigated the fiber bridging effects on the delamination failure phenomenon of unidirectional composites under the mixed mode I/II loading through experimental observations [15], [16], [17]. However, due to the complexity, these effects are rarely studied quantitatively. Sørensen et al. [18] proposed a micromechanical model for prediction of the mixed mode I/II bridging laws based on in situ observations of the mixed-mode crack growth in a unidirectional carbon/epoxy composite. Their model was extended from mode I model of Spearing and Evanse [5] to the mixed mode I/II by considering a tangential crack opening displacement. In their model, the only failure of fiber detachment across a cracking plane (involving fiber/matrix interface and matrix cracking) has been considered and fiber failure has been ignored. They also assumed that the number of bridged fibers per unit crack area is constant and independent of the actual opening path. This assumption is not consistent with the fact that the number of bridged fibers decreases with fiber failure due to increasing the crack opening displacement. During the crack bridging phenomenon, there are several micro-mechanisms responsible for energy absorption across the bridged delamination cracks. However, due to complexity, only a few of these micro-mechanisms effects have been considered in the available bridging models.

In the present research, a novel micromechanical model called “mixed mode I/II micromechanical bridging (MMMB) model” has been developed for prediction of the mixed mode I/II delamination growth in laminated composites. In the present model, fiber bridging effects have been considered by a breakdown of the failure micro-mechanisms involved during the fiber bridging phenomenon. Firstly, these micro-mechanisms are identified by studying the experimental observations of different researchers [19], [20], [21], [22], [23] using the fractography of the mode I and mode II interlaminar fracture surfaces, reviewing their assumptions and results; and observing the fiber bridging zone and comparing it under pure modes I and II and mixed mode I/II loading. Then, the bridged fiber is modeled as a beam and by applying different loading conditions on the bridged fiber; the absorbed energy of each of these failure micro-mechanisms is calculated. Finally, the absorbed energy by the fiber bridging zone is obtained from the accumulation of the energy terms associated with each of these micro-mechanisms. The traction-separation behaviors in both the normal and tangential directions of the crack plane are the outcome of the MMMB model. Afterward, the mixed-mode I/II delamination failure response of the laminated composite is extracted by plotting G_I versus G_II and compared with the available experimental data. The proposed model is easy to employ and only needs the fiber and matrix properties, the fiber–matrix interfacial zone properties and also the statistical distribution of the fiber strength. A comparison of results with the available experimental data indicates that the proposed model can predict the initiation and propagation of the mixed mode I/II delamination in laminated composites correctly with a reasonable accuracy. The model can be used for arbitrary values of the mode mixities.