A multi-scale and multi-mechanism approach for the fracture toughness assessment of polymer nanocomposites M. Quaresimin ⇑ , M. Salviato, M. Zappalorto Department of Management and Engineering, University of Padova, Stradella San Nicola 3, 36100 Vicenza, Italy article info Article history: Received 8 August 2013 Received in revised form 13 November 2013 Accepted 15 November 2013 Available online 28 November 2013 Keywords: A. Nanoparticles B. Fracture toughness B. Interphase abstract In the present work, a multi-scale modelling strategy to assess the fracture toughness of nanoparticle filled thermosetting polymers is presented. The model accounts for the main damaging mechanisms arising in this kind of materials, i.e. nanoparticle debonding, plastic yielding of nanovoids and plastic shear banding of the polymer. Further, the proposed analytical framework considers the influence of an interphase around nanoparticles, a particular feature of nanocomposites. Comparison of the theory to a bulk of experimental data from the literature shows a very good agreement. Ó 2013 Elsevier Ltd. All rights reserved. 1. Introduction Nanotechnology has recently emerged as a suitable tool to optimise properties of materials by designing their internal structure at the very nanoscale thus assisting in the achievement of desirable combinations of physical and mechanical properties [1–3]. However, to fully exploit the potential benefits of nanomo- dification, appropriate models able to soundly predict the macro- scale mechanical properties from material structure need to be developed. With the aim to explain the significant improvements of poly- mer toughness achievable with low nanofiller contents and consid- ering the importance of the several damaging mechanisms that might take place at the nanoscale, some authors have recently sug- gested to use a ‘‘multi-mechanism’’ modelling strategy [4–8]. However, modelling the effects of nanoscale damaging mecha- nisms on macroscale properties is far from easy, essentially be- cause at that length scale classical micromechanics is no longer valid. Instead, the adoption of a multi-scale strategy is necessary in order to describe the nanocomposite material behaviour, physi- cally and mathematically, in each individual scale of interest. In the recent literature several authors dealt with the analysis of toughening mechanisms in nanocomposites. Chen et al. [9] carried out a theoretical study on the amount of energy dissipated by interfacial debonding of nanoparticles and provided a close form solution for the critical detachment stress. The size distribution of particles and the debonding probability were included into the analytical formulation using a logarithmic normal distribution and the Weibull distribution function, respectively. Some years later, the present authors refined the analysis car- ried out in [9] studying the effects of a small interphase zone embedding the nanoparticle [10] and of surface elastic constants [11] on the critical debonding stress. In both cases, the range of the nanoparticle radii where those effects are significant was proved to be limited to the nanoscale [10,11]. The energy dissipation phenomena due to particle debonding, voiding and subsequent yielding of the polymer have been ana- lysed by Lauke [4] who used a simple geometrical model of parti- cle–particle interaction in a regular particle arrangement. By further applying a critical stress criterion, Lauke found a dissipa- tion zone which was independent of the particle diameter and jus- tified the increase of crack resistance with decreasing particle size by the increase in the specific debonding energy [4]. Williams [5] re-analysed in detail the toughening of particle filled polymers assuming that plastic void growth around debond- ed or cavitated particles is the dominant mechanism for energy dissipation. He assumed a tri-axial state of stress around the spher- ical particle and supposed the debonding and cavitation conditions to be controlled by either surface energy or the cohesive energy of the particle. Williams further noted that, even if the debonding process is generally considered to absorb little energy, it is essen- tial to reduce the constraint at the crack tip and, in turn, to allow the epoxy polymer to deform plastically via a void-growth mecha- nism. A similar result was found also by the present authors [12]. Hsieh et al. [6,7] studied the fracture toughness improvements resulting from nanomodification of epoxy resins with silica nanoparticles. Based on experimental observations, they identified 0266-3538/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compscitech.2013.11.015 ⇑ Corresponding author. Tel.: +39 0444 998723; fax: +39 0444 998888. E-mail address: marino.quaresimin@unipd.it (M. Quaresimin). Composites Science and Technology 91 (2014) 16–21 Contents lists available at ScienceDirect Composites Science and Technology journal homepage: www.elsevier.com/locate/compscitech