A novel multiscale model for mixed-mode fatigue crack growth in laminated composites

Highlights

• Novel multi-scale mixed-mode fatigue crack growth model for laminated composites.

• Connecting scales through introducing an effective critical distance parameter.

• Considering microscopic and mesoscopic phenomena in a novel integrated framework.

• A computationally efficient model capable of reasonably predictions of experiments.

• Remarkable effect of fiber bridging toughening on fatigue crack growth rate.

Abstract

A multiscale model is developed to predict mixed-mode fatigue crack growth (FCG) behavior in laminated composites by considering the micromechanical effects of fiber bridging. The proposed model consists of a micromechanical part dedicated to the fiber bridging mechanism near the crack tip, a macroscopic FCG model based on the maximum principal strain criterion, and an effective crack tip processing zone (i.e., an equivalent critical distance) connecting the two parts of the model. The core novelty of the presented model is to connect two scales (micro and meso) through a measurable physical quantity – called equivalent critical distance – accounting for various micromechanical as well as macroscopic phenomena occurring at both scales. The suggested model is validated against FCG test data available in the literature for laminated glass fiber composites under mode I, mode II, and mixed mode I/II conditions. It is found that the proposed framework is superior to its conventional (pure macroscopic) version, which eliminates fiber bridging effects, in the prediction of the cyclic mixed-mode crack propagation behavior in laminated composites. The multiscale nature of the proposed FCG model offers less complexity than fully micromechanical approaches while providing higher accuracy as compared to pure macroscopic models that ignore micromechanical phenomena occurring at the fiber scale.

Introduction

Fatigue crack growth (FCG) in laminated fiber composites has been studied extensively during the past few decades [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18] yet has remained one of the least understood topics in applied mechanics due to the simultaneous acting of various complex phenomena. The cycling delamination usually involves different mechanisms including fiber bridging, matrix failure, fiber failure, fiber/matrix interfacial debonding, and frictional sliding that may occur concurrently near the crack tip in laminated fiber composites (e.g., glass fiber-reinforced and carbon fiber-reinforced). The toughening mechanism resulting from the fiber bridging detracts crack growth rate per cycle (da/dN) as the delamination extends. While the shielding contribution of the fiber bridging is affected by many factors including the loading profile (i.e., the higher applied load is usually accompanied by the lower degrees of fiber bridging toughening and vice versa [19]), it is well understood that such micro-mechanisms must be included in predictive FCG models for being applicable to a wide range of operational conditions.

Most available studies on FCG of laminated composites utilize the so-called Paris power law [20] which essentially relates crack growth rate per cycle (da/dN) to cyclic stress intensity factor (SIF) range (ΔK = K_max − K_min), or alternatively, cyclic energy release rate (ΔG = G_max − G_min):

$$\frac{da}{dN}=A{\left(\Delta \Psi \right)}^b,\Psi \equiv KorG,$$

where the constants A and b depend on material properties, temperature, stress ratio, and frequency [21]. However, while being straightforward and having a bright history of application in different materials and specimens (e.g., [13,14,[22], [23], [24], [25], [26], [27]]), the traditional version of the Paris power law seems to be insufficient for laminated fiber composite materials for several reasons. The Paris power law was originally proposed for pure mode I conditions where cracks propagate along their initial lines (although it has been extended to handle pure mode II cases later [28], [29], [30], [31]) and is ill-suited for mixed-mode conditions (i.e., combined opening/sliding crack tip displacements) which exist in most real-world applications as it ignores the mean stress effect on the crack growth [32,33]. Thus, many researchers advanced the field by proposing more sophisticated models to be applicable to FCG under mixed-mode loading conditions. Most notable advancements are performed by extending the mixed mode fracture criteria to the mixed-mode FCG models (e.g., the strain energy density [32,33], minimum potential energy [34], etc.). For example, studies showed that the FCG models developed based on the strain energy density can be properly applied to isotropic and orthotropic material systems under mixed-mode conditions [32,33]. However, when it comes to laminated composite materials, such models are insufficient since they do not include the effects of fiber/matrix interactions in their formulations. The macroscopic form of Eq. (1), in essence, seems not to allow us to discern the role of fiber bridging toughening in the FCG behavior of laminated fiber composites.

Different FCG modeling approaches have been proposed for fiber composite materials during the past few decades including macroscopic-based [35], [36], [37], [38], micromechanical-based [19,[39], [40], [41], [42]], and combined macro/micromechanical-based [43], [44], [45], [46], [47] models. Early models are mostly pure macroscopic relying only on the macroscopic material constants without paying attention to the micromechanical behavior at the fiber scale. Examples are the works on glass fiber-reinforced polymer composites by Shindo et al. [36] (woven laminates) and Kenane and Benzeggagh [37] (unidirectional laminates). Such models are developed by extending the previous isotropic FCG models to their anisotropic versions. However, while being straightforward, the major disadvantage of such models is their incapacitation in providing information about fiber bridging toughening which is crucial in the design of composite materials. Micromechanical-based models, on the other hand, are mainly focused on details of micromechanical phenomena occurring around the propagating crack tip. Bridging of fibers – as one of the most important events taking place near the delamination zone – causes an increase in crack propagation resistance which can be measured macroscopically by the resistance curve (i.e., R-curve), or microscopically through a change in the local strain energy release rate [48,49]. Developing a sophisticated micromechanical-based FCG model requires an accurate understanding of various phenomena (e.g., fiber breaking, matrix failure, fiber/matrix interactions, etc.) and including several geometrical and material parameters at the microscale (e.g., dimensions, moduli, tensile strength, and distribution of fibers, material properties of the matrix, crack tip opening/sliding displacements, etc.) into the model. Examples are the efforts by Sorensen and co-workers [39,40] on the prediction of the cyclic crack propagation behavior of glass-epoxy composites by considering progressive fiber failure and fiber/matrix frictional sliding. However, the complex nature of most micromechanical-based models makes them inapplicable to tracking large-scale FCG behavior which is critical when it comes to practical applications. To bridge the gap between the micromechanical and macroscopic models, several multiscale models have been proposed during the past few years [43], [44], [45], [46], [47] by combining both viewpoints into a single framework. For example, Tevatia and Srivastava [47] have presented a multiscale FCG model for discontinuous fiber-reinforced metal matrix composites by merging the delamination micromechanics and an energy balance equation. Nevertheless, almost all the existing multiscale FCG models for fiber composites do not connect the scales through an easy-to-measure physical quantity – a major deficiency making them impractical for industrial applications.

In the present work, a multiscale FCG model is proposed for laminated fiber composites under mixed-mode I/II loading conditions by combining the fiber/matrix micromechanics and a macroscopic strain-based criterion. The key parameter relating to both perspectives is an equivalent critical distance defined to insert additional absorbed strain energy resulting from fiber bridging toughening into a macroscopic FCG model. The macroscopic portion of the model is formed based on the maximum principal strain criterion for anisotropic solid domains. Among the major advantages of the proposed model are i) presenting a smooth connection between the scales involved through defining a measurable physical quantity (i.e., an equivalent critical distance), and ii) being more computationally efficient and more accurate–compared with pure micromechanical and traditional macroscopic models, respectively–while providing satisfactory information about microscale fiber/matrix interactions. The following sections consist of theoretical developments (Section 2), results and discussion (Section 3) where the developed framework will be verified by experimental data available in the literature, and finally, the conclusion of this study which is presented in Section 4.