A modified peridynamic method to model the fracture behaviour of nanocomposites
Introduction
Epoxy resin, used as a matrix of composite materials, has received great attention due to its wide application in the aerospace industry as it allows to enhance the lightweight and anti-corrosion features of composite materials. However, epoxy resin is also the possible weak point of composite materials, considering its mechanical features, i.e., low strength, poor fracture resistance and brittleness. It behaves as the dominant damage location inside composites under some in-service loading conditions like impact [1] and fatigue [2]. As a result, improving the fracture resistance of epoxy resin is important, and adding particle materials is one of the promising solutions [3]. According to existing investigations, introducing polylactide [4], graphene oxides [5], nanoparticles [6] and hyperbranched polyester (HBP) [7] to epoxy resin can greatly increase its fracture toughness. In particular, neat epoxy or composites, which contain HBP, show not only outstanding fracture properties [7], [8] but also excellent thermal properties along with an increased glass transition temperature. Additionally, HBP particles are also a low-cost option for industrial applications.
According to the work of Verrey et al. [9], by introducing HBP to the matrix of carbon-fibre-reinforced polymer (CFRP), double cantilever beam tests presented a higher mode-I fracture toughness compared with neat epoxy. In addition, mode-II fracture toughness can also be improved by adding HBP to the matrix due to the interaction between the epoxy resin and the HBP particles [10], where a perfect interphase can be witnessed between both components [11]. In order to further investigate the effect of HBP on the matrix of composites, attention should focus on the structure and properties of the pure resin with HBP particles [7], [8], [12]. It has been reported that HBP increases the fracture toughness of epoxy by 60–90% with a HBP weight fraction of 1–5 wt% [7]. Such an improvement is also affected by the type of HBP and the chemical functions carried by the particles [8]. More so, potential enhancements on the neat epoxy could also be achieved by mixing HBP with other nanoparticles [13], [14].
Considering the cost of experimental activities and the various factors (e.g., chemical functions, weight fraction, etc.) that can affect the strengthening effect of HBP on the epoxy resin, the development of virtual tests to investigate the fracture properties is of great interest. For neat resin, numerical methods are capable to provide reliable results of the quasi-static fracture tests for both compact-tension (CT) [15] and single-edge-notched bending (SENB) [16] specimens. Nevertheless, it is easier to conduct such investigations on neat and typical composite materials than on nanocomposites. The effect of nanoparticles like HBP, even under simple loading configurations, including flexure [10] and tension [12], leads to the limitation of applying related mechanical properties in fracture simulations, unlike the neat material [17]. Furthermore, due to the complex geometries and uncertainty distributions of HBP, the modelling process remains challenging and, thus, limited numerical simulations are presented in existing studies. However, as far as the interface is concerned, a perfect bond between the HBP and epoxy can be assumed in the model [18], and the small size of the HBP allows for the simplification of numerical modelling [19].
Generally, replicating the fracture behaviour in a classical numerical environment is not straightforward, especially the replication of the random crack path and the generated free surfaces. With the recent developments in computational mechanics, new methods have been proposed to address the aforementioned numerical issues in fracture simulations. For instance, extended finite element modelling is able to produce a crack path in any position of the numerical model without being limited by the mesh morphology; however, the analysis requires many parameters [20]. In addition, atomistic molecular dynamics (MD) simulation is one of the competitive techniques [21], which builds the model based on the molecular structures and bonds between them. MD simulations can endow the free surface energy on a molecular scale, and the breaking of the bonds can introduce more freedom for the crack path. However, most of the studies employing MD simulations consider unit cells under simple loading conditions instead of fracture of a real component [22]. This choice can mainly be attributed to the time-consuming calculations, which hinder its application to large structures, including SENB specimens. Compared with MD simulations, the cutting-edge method proposed by Silling [23], which employs a peridynamic model, is a better choice to model the fracture behaviour of structures. The peridynamic theory is based on integral–differential equations. The governing equations are always valid, regardless of whether or not there is any discontinuity in the structure. The method is further based on basic equations of motion and forces (in Greek, ‘peri’ means ‘horizon’ and ‘dynamics’ means ‘force’). Therefore, it can mimic random cracks, and the mechanical behaviour, by setting bonds between elements, which helps to easily generate free fracture surfaces. It is capable of providing reliable results with only a limited number of required parameters [24], [25], especially for linear-elastic/brittle materials [26]. Additionally, the peridynamic method has been modified frequently to replicate the mechanical behaviour of nanocomposites due to its inherent non-local features [27], [28]. However, for these materials, an enhanced material model development and a reconstructed stress calculation are required based on a strong mathematical basis.
In the present work, a phenomenological modified peridynamic method is proposed to mimic the fracture behaviour of nanocomposites reinforced with nanoparticles. The objective of the present work is to check the capability of the proposed method to capture experimental data [29] from fracture tests [7], [8] using SENB samples of HBP-reinforced epoxy resin. The proposed method uses peridynamics to simulate the fracture behaviour with randomly constrained nodes mimicking the effect of HBP particles. Additionally, the Monte Carlo method has been employed to achieve the random distributions of the HBP particles. In comparison with a typical modelling methodology on nanocomposites, the proposed modified peridynamic method is not only able to characterise the behaviour of nanocomposites at a macroscale, but it also greatly simplifies the modelling process.
Section snippets
Background of the peridynamic method
Unlike typical mechanical methods employing spatial differential terms to present deformation, the peridynamic method uses non-local nodal integral terms to build the load transformation among the material points, which are the basic elements for the peridynamic method. Actually, peridynamics can be regarded as an update of MD simulation at a macroscale, which does not consider too many parameters of the microstructures [28] and which is also suitable for the non-local replication of
Validation for the neat epoxy
Prior to the application on nanocomposites, the numerical model and the related parameters were validated for the neat case. Fig. 6 presents the results from the numerical model and its comparison with experimental data. The formation of a centre crack from the notch can be witnessed in the red-marked region presenting the failure location from the numerical model (see Fig. 6(a)), owing to the breaking of the bonds among the elements at the crack tip. Additionally, a crack map (see Fig. 6(b))
Conclusions and future development
A phenomenological modified numerical method in a peridynamic environment was proposed. By randomly constraining the nodes, with a certain number based on the weight fraction, and involving statistical calculations using Monte Carlo simulations to cover sufficient possible cases, the experimental results from the fracture tests can be successfully captured. Following the validation, the present method was utilised to analyse the effect of the HBP distribution on the fracture properties. With
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgement
The author Dayou Ma would like to thank Prof. F. Han, from Dalian University of Technology, for the discussion about the application of peridynamics. Support by the Italian Ministry of Education, University and Research, through the project Department of Excellence LIS4.0 (Integrated Laboratory for Lightweight e Smart Structures), is acknowledged.
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