A hybrid embedded cohesive element method for predicting matrix cracking in composites
Introduction
Safety–critical fibre-reinforced polymer composite structures are designed so that no damage growth, such as matrix cracking from the resin-rich region at the end of a dropped ply, will occur at the design limit load [1], [2]. Furthermore, any matrix cracks emanating from manufacturing flaws or impact damage by foreign objects cannot grow to exceed critical size limits during service. Therefore matrix-dominated cracking is a major concern for the design and operation of safety–critical fibre reinforced composite structures. Many low cost out-of-autoclave manufacturing techniques, such as closed moulding processes (e.g. Resin Transfer Moulding (RTM)), vacuum-assisted resin infusion, and pultrusion, are capable of producing high-quality composite components with properties comparable to those produced by autoclave using prepregs. However, these processes can result in more complex resin-rich regions and reinforcement architectures than laminates made of prepregs. Computational analyses of these composite structures require the time intensive and laborious development of a very fine finite element mesh to accurately represent the geometrical features of resin regions and reinforcement fabrics or fibre tows. In automotive and maritime applications, the fibre composite components can be even more geometrically complex and thicker than the aerospace structures that are typically thin-gauge structures reinforced with stringers or honeycomb cores. Determination of the critical loads that may cause matrix cracking in thick composite structures requires accurate analysis of the 3D stress state through the thickness of the component. Currently accurate prediction of the 3D stress state typically involves modelling the structure with a fine mesh of three-dimensional (3D) elements. Generating this contiguous 3D solid mesh for a composite structure that has complex geometry and resin-rich regions pertinent to ply-drops and inserted fibre tows require significant effort, making this conventional approach inefficient.
The embedded element technique [3], [4], [5], [6], [7] offers an alternative approach, allowing the polymer matrix and fibre reinforcement to be meshed separately. Specifically, the entire volume of the composite component is meshed using hexahedral or tetrahedral elements to form the host domain. The embedded domain comprises all the reinforcement layers. Each layer is initially meshed as a 2D surface, which is extruded to form the 3D solid mesh. These two sets of meshes are fused together by tying the nodal displacements of the embedded domain with the interpolated displacements of the host domain. The resulting model accurately represents the stiffness of the composite structure. The embedded element technique has shown its potential as a computationally efficient modelling technique in a broad range of applications from the analysis of reinforced concrete [4], reinforced rubber tyres [3] and fibre reinforced composite structures [5], [6], [7], [8], [9], [10].
Yang and Cox [9], [10] used the embedded element technique (which is also called the “Binary Model”) to predict the strength of un-notched and notched (open hole) composite specimens, and demonstrated that this method offered a significant increase in computational efficiency over a homogenised finite element (FE) model. The binary model, in its original formulation for application to textile composites, represents the contribution of the axial stiffness of fibre tows by 1D “tow elements” embedded in a 3D “effective medium” that accounts for all other stiffness contributions. Yang and Cox demonstrated the importance of gauge-averaging methods used in conjunction with the binary model formulation [9] and also generalised the formulation by using strings of 1D elements, rather than a single string of elements, to represent a single tow [9]. When averaged over a gauge length comparable to the tow width, the predicted local strain variations became independent of the number of strings, N, for N = 4 or 16. Yang and Cox remarked that in the limit N → ∞, the binary model becomes equivalent to a representation in which the fibre tows appear as continuous 3D bodies embedded in an effective medium, as in the formulation of Fish [11], but with more limited degrees of freedom available to describe local elasticity. The 3D formulation originated by Fish [11], [12] was further developed by Hallett and colleagues under the name “domain superposition technique” [8] and more recently by Tabatabaei et al. [7], [13].
So far the embedded element technique has been used primarily to determine the stiffness of and failure onset in composite materials [7], [8], [9], [10], [13], [14], [15]. However, modelling the progression of damage using the embedded element approach has not been attempted to the authors’ knowledge. Two key challenges are (1) the appropriate method of combining the stress fields of the host and embedded domains to enable strength prediction and (2) the introduction of damage progression criteria into the embedded element modelling framework to simulate growth of in-service damage or manufacturing defects.
The principal objective of this study is to advance the embedded element technique by introducing a damage progression modelling technique to predict delamination cracking in composite structures containing resin-rich regions (e.g. near ply-drops). Introducing cohesive elements requires prior knowledge of the crack path and this process is relatively straightforward in monolithic laminates fabricated with continuous plies, as the cracking is generally along ply interfaces. Resin-rich regions due to ply-drops and fibre tow, which are more commonly encountered in low-cost manufacturing techniques such as RTM, can cause crack deflection when the matrix crack tip reaches the resin-rich region. If the crack deflection path is known, cohesive elements or cohesive surface can be employed to simulate the propogation. Although the issue of pre-seeding the crack path could be avoided using mesh insensitive techniques [16], [17], [18], the cohesive modelling approach is adopted in this study because of its relative maturity for implementation by the composites industry. In addition, a cohesive model can be used as a benchmark to evaluate other modelling approaches in terms of accuracy and computational costs. The ability to model interlaminar failure within the embedded element framework represents a major advancement towards the development of an efficient modelling technique for complex structures. This new approach is validated by predicting the evolution of matrix cracking in a modified Double Cantilever Beam (DCB) sample of a composite material that contains two discontinuous plies.
Section snippets
Existing embedded element approach
To facilitate the description of the damage progression model, it is advantageous to briefly summarise the essential features of the embedded element approach. When modelling fibre-reinforced composites, referring to Fig. 1(a), the host domain encloses the volume of a structure or component as shown in Fig. 1(b). The fibre reinforcement domain is located within the matrix and may fill all or part of the matrix domain as shown in Fig. 1(c). The two domains are coupled together by tying the
Experimental details
To validate the proposed modelling technique, further details are given in Section 4, experimental tests were carried out using carbon fibre-epoxy composites based on the geometry of the DCB specimen as described in ASTM D5528 [20]. The composite lay-up at the mid-plane of the laminate contained two discontinuous plies, as shown in Fig. 4. These two discontinuous plies created a resin-rich region along the primary delamination path. The purpose of this feature is to introduce a resin-rich
Predicting the evolution of matrix cracking with the embedded element technique
Following the description of the hybridised embedded cohesive element technique in Section 2.2 the identification of appropriate cohesive properties used to define the initiation and propagation of matrix cracking is described.
Comparison with experimental results
The predicted load–displacement responses of the three models are compared with experimental test data in Fig. 14. Crack initiation was predicted at the same load level as observed in the experiments. Following crack initiation, the load level was approximately constant as fibre bridging took effect. At approximately 5.6 mm of applied displacement (point A) the crack tip had reached the end of the ply termination and crack growth was arrested and the load was observed to increase. As the crack
Discussion
Previous research [7], [9], [10], [14] has focussed on the use of the embedded element technique for static stiffness and strength calculation. The work reported herein extends the embedded element approach to predicting delamination growth in fibre reinforced composite structures. By employing a coupled cohesive element/embedded element model, accurate predictions were achieved for the crack path and fracture loads of matrix crack growing across a resin-rich region. In the current study
Conclusion
A new approach to analyse interlaminar crack growth has been presented. In this method the embedded element technique is hybridised with a cohesive damage model to predict the onset and propagation of matrix cracking. A toughening mechanism was experimentally observed as the crack propagated through the resin-rich region. Crack deflection was due to the mismatched stiffness at the bi-material interface and the higher strength of the bulk epoxy. It has been shown that with appropriate
Acknowledgements
This research was conducted as part of a collaborative research program involving RMIT University, Deakin University, the University of Miami, Teledyne Scientific co. and industry partner Carbon Revolution. This research was supported under Australian Research Council’s Linkage Projects funding scheme (project LP120200046). Financial support from the ARC and Carbon Revolution is gratefully acknowledged. The authors would like to acknowledge the support of R. Ryan and P. Tkatchyk (RMIT) for
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2019, Composite StructuresCitation Excerpt :The two techniques have been previously combined to model matrix cracking [8] or delamination between plies [12], but the cohesive elements served there as ‘a host domain’ together with the matrix. The most important difference between the current method and those in the Refs. [8,12] is that the current method allows for modelling of interface damage in the framework of superimposed meshes where a physical object such as interface does not even exist. The interface is an elusive entity that lies between the embedded domain (reinforcements) and the host domain (matrix).