Int J Fract DOI 10.1007/s10704-017-0210-6 ORIGINAL PAPER Stochastic analysis of the fracture toughness of polymeric nanoparticle composites using polynomial chaos expansions Khader M. Hamdia · Mohammad Silani · Xiaoying Zhuang · Pengfei He · Timon Rabczuk Received: 2 August 2016 / Accepted: 8 April 2017 © Springer Science+Business Media Dordrecht 2017 Abstract The fracture energy is a substantial material property that measures the ability of materials to resist crack growth. The reinforcement of the epoxy poly- mers by nanosize fillers improves significantly their toughness. The fracture mechanism of the produced polymeric nanocomposites is influenced by different parameters. This paper presents a methodology for stochastic modelling of the fracture in polymer/particle nanocomposites. For this purpose, we generated a 2D finite element model containing an epoxy matrix and rigid nanoparticles surrounded by an interphase zone. The crack propagation was modelled by the phan- tom node method. The stochastic model is based on six uncertain parameters: the volume fraction and the diameter of the nanoparticles, Young’s modulus and the maximum allowable principal stress of the epoxy matrix, the interphase zone thickness and its Young’s modulus. Considering the uncertainties in input param- eters, a polynomial chaos expansion surrogate model is constructed followed by a sensitivity analysis. The vari- ance in the fracture energy was mostly influenced by K. M. Hamdia · T. Rabczuk (B ) Duy Tan University, Institute of Research & Development, 3 Quang Trung, Danang, Vietnam e-mail: timon.rabczuk@uni-weimar.de M. Silani Department of Mechanical Engineering, Isfahan University of Technology, Isfahan 84156-83111, Iran X. Zhuang · P. He Tongji University, Shanghai, China the maximum allowable principal stress and Young’s modulus of the epoxy matrix. Keywords Polymeric nanoparticle composites · Interphase · Fracture toughness · Computational mechanics · Uncertainty quantification · Sensitivity analysis 1 Introduction Polymers are considered a promising material in sev- eral applications. Adding particles to the polymer matrix results in major improvements in the prop- erties of the created composite. Incorporating rigid fillers at the nano-scale has shown an improved fracture toughness without sacrificing other important thermo- mechanical properties. Three shapes of nanofillers are commonly used: spherical particles (e.g. silica, alumina and glass), layered (e.g. clay and graphite) and fibrous materials (nanotubes). Thanks to the low density and simple fabrication methods, polymeric nanocomposites (PNCs) have become a popular multi- functional material in numerous nanotechnology appli- cations (Thostenson et al. 2005). The remarkable improvement in the properties of the produced composite may be attributed to the large sur- face area-to-volume ratio of the nanofillers that creates an extreme interfacial zone between the nanofiller and the surrounding matrix. Due to the nanofillers, the adja- cent polymer chains are disordered forming interphase 123