Abstract
The fracture energy is a substantial material property that measures the ability of materials to resist crack growth. The reinforcement of the epoxy polymers 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 phantom 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 parameters, a polynomial chaos expansion surrogate model is constructed followed by a sensitivity analysis. The variance in the fracture energy was mostly influenced by the maximum allowable principal stress and Young’s modulus of the epoxy matrix.
Similar content being viewed by others
References
Anderson TL (2005) Fracture mechanics: fundamentals and applications, 3rd edn. CRC Press, Boca Raton
Arash B, Park HS, Rabczuk T (2015) Tensile fracture behavior of short carbon nanotube reinforced polymer composites: a coarse-grained model. Compos Struct 134:981–988
Arash B, Park HS, Rabczuk T (2016) Coarse-grained model of the J-integral of carbon nanotube reinforced polymer composites. Carbon 96:1084–1092
Areias P, Rabczuk T (2008) Quasi-static crack propagation in plane and plate structures using set-valued traction-separation laws. Int J Numer Method Eng 74(3):475–505
Berveiller M, Sudret B, Lemaire M (2006) Stochastic finite element: a non intrusive approach by regression. Eur J Comput Mech/Revue Européenne de Mécanique Numérique 15(1–3):81–92
Bhuiyan MA, Pucha RV, Worthy J, Karevan M, Kalaitzidou K (2013) Understanding the effect of CNT characteristics on the tensile modulus of CNT reinforced polypropylene using finite element analysis. Comput Mater Sci 79:368–376
Bondioli F, Cannillo V, Fabbri E, Messori M (2005) Epoxy-silica nanocomposites: preparation, experimental characterization, and modeling. J Appl Polym Sci 97(6):2382–2386
Boutaleb S, Zaïri F, Mesbah A, Naït-Abdelaziz M, Gloaguen JM, Boukharouba T, Lefebvre JM (2009) Micromechanics-based modelling of stiffness and yield stress for silica/polymer nanocomposites. Int J Solids Struct 46(7):1716–1726
Chau-Dinh T, Zi G, Lee PS, Rabczuk T, Song JH (2012) Phantom-node method for shell models with arbitrary cracks. Comput Struct 92:242–256
Chen J, Huang Z, Zhu J (2007) Size effect of particles on the damage dissipation in nanocomposites. Compos Sci Technol 67(14):2990–2996
Choi SK, Grandhi RV, Canfield RA, Pettit CL (2004) Polynomial chaos expansion with latin hypercube sampling for estimating response variability. AIAA J 42(6):1191–1198
Crestaux T, Le Maıtre O, Martinez JM (2009) Polynomial chaos expansion for sensitivity analysis. Reliab Eng Syst Saf 94(7):1161–1172
Dittanet P, Pearson RA (2012) Effect of silica nanoparticle size on toughening mechanisms of filled epoxy. Polymer 53(9):1890–1905
Dominkovics Z, Hári J, Kovács J, Fekete E, Pukánszky B (2011) Estimation of interphase thickness and properties in pp/layered silicate nanocomposites. Eur Polymer J 47(9):1765–1774
Garcia-Cabrejo O, Valocchi A (2014) Global sensitivity analysis for multivariate output using polynomial chaos expansion. Reliab Eng Syst Saf 126:25–36
Ghanem RG, Spanos PD (1991) Stochastic finite elements: a spectral approach. Springer, New York
Griffith AA (1921) The phenomena of rupture and flow in solids. Philos Trans R Soc Lond Ser A Contain Pap Math Phys Charact 221:163–198
Grigoriu M (2010) Probabilistic models for stochastic elliptic partial differential equations. J Comput Phys 229(22):8406–8429
Guilleminot J, Soize C (2013) On the statistical dependence for the components of random elasticity tensors exhibiting material symmetry properties. J Elast 111(2):109–130
Hamdia KM, Msekh MA, Silani M, Vu-Bac N, Zhuang X, Nguyen-Thoi T, Rabczuk T (2015) Uncertainty quantification of the fracture properties of polymeric nanocomposites based on phase field modeling. Compos Struct 133:1177–1190
Hamdia KM, Zhuang X, He P, Rabczuk T (2016) Fracture toughness of polymeric particle nanocomposites: evaluation of models performance using Bayesian method. Compos Sci Technol 126:122–129
Hansbo A, Hansbo P (2004) A finite element method for the simulation of strong and weak discontinuities in solid mechanics. Comput Methods Appl Mech Eng 193(33):3523–3540
Hbaieb K, Wang Q, Chia Y, Cotterell B (2007) Modelling stiffness of polymer/clay nanocomposites. Polymer 48(3):901–909
Huang S, Mahadevan S, Rebba R (2007) Collocation-based stochastic finite element analysis for random field problems. Probab Eng Mech 22(2):194–205
Huang Y, Kinloch A (1992) Modelling of the toughening mechanisms in rubber-modified epoxy polymers. part II a quantitative description of the microstructure-fracture property relationships. J Mater Sci 27(10):2763–2769
Iman RL, Conover W (1982) A distribution-free approach to inducing rank correlation among input variables. Commun. Stat Sim Comput 11(3):311–334
Irwin G (1957) Analysis of stresses and strains near the end of a crack traversing a plate. J Appl Mech 24:361–364
Isukapalli SS (1999) Uncertainty analysis of transport-transformation models. PhD thesis, The State University of New Jersey
Le TT, Guilleminot J, Soize C (2016) Stochastic continuum modeling of random interphases from atomistic simulations. application to a polymer nanocomposite. Comput Methods Appl Mech Eng 303:430–449
Li Y, Waas AM, Arruda EM (2011) A closed-form, hierarchical, multi-interphase model for composites-derivation, verification and application to nanocomposites. J Mech Phys Solids 59(1):43–63
Liang Y, Pearson R (2009) Toughening mechanisms in epoxy-silica nanocomposites (ESNs). Polymer 50(20):4895–4905
Matthies HG, Keese A (2005) Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations. Comput Methods Appl Mech Eng 194(12):1295–1331
Mortazavi B, Bardon J, Ahzi S (2013) Interphase effect on the elastic and thermal conductivity response of polymer nanocomposite materials: 3D finite element study. Comput Mater Sci 69:100–106
Msekh MA, Silani M, Jamshidian M, Areias P, Zhuang X, Zi G, He P, Rabczuk T (2016) Predictions of J integral and tensile strength of clay/epoxy nanocomposites material using phase field model. Compos Part B Eng 93:97
Odegard G, Clancy T, Gates T (2005) Modeling of the mechanical properties of nanoparticle/polymer composites. Polymer 46(2):553–562
Pontefisso A, Zappalorto M, Quaresimin M (2015) An efficient RVE formulation for the analysis of the elastic properties of spherical nanoparticle reinforced polymers. Comput Mater Sci 96:319–326
Qiao R, Brinson LC (2009) Simulation of interphase percolation and gradients in polymer nanocomposites. Compos Sci Technol 69(3):491–499
Quaresimin M, Salviato M, Zappalorto M (2014) A multi-scale and multi-mechanism approach for the fracture toughness assessment of polymer nanocomposites. Compos Sci Technol 91:16–21
Rabczuk T, Zi G, Gerstenberger A, Wall WA (2008) A new crack tip element for the phantom-node method with arbitrary cohesive cracks. Int J Numer Method Eng 75:577–599
Saltelli A (2002) Making best use of model evaluations to compute sensitivity indices. Comput Phys Commun 145(2):280–297
Saltelli A, Ratto M, Andres T, Campolongo F, Cariboni J, Gatelli D, Saisana M, Tarantola S (2008) Global sensitivity analysis. The primer. Wiley, Hoboken
Scocchi G, Posocco P, Danani A, Pricl S, Fermeglia M (2007) To the nanoscale, and beyond!: multiscale molecular modeling of polymer-clay nanocomposites. Fluid Phase Equilib 261(1):366–374
Shokrieh MM, Rafiee R (2010) Stochastic multi-scale modeling of CNT/polymer composites. Comput Mater Sci 50(2):437–446
Silani M, Ziaei-Rad S, Esfahanian M, Tan V (2012) On the experimental and numerical investigation of clay/epoxy nanocomposites. Compos Struct 94(11):3142–3148
Silani M, Talebi H, Ziaei-Rad S, Kerfriden P, Bordas SP, Rabczuk T (2014) Stochastic modelling of clay/epoxy nanocomposites. Compos Struct 118:241–249
Sobol’ IM (1990) On sensitivity estimation for nonlinear mathematical models. Matematicheskoe Modelirovanie 2(1):112–118
Song JH, Areias P, Belytschko T (2006) A method for dynamic crack and shear band propagation with phantom nodes. Int J Numer Meth Eng 67(6):868–893
Sudret B (2008) Global sensitivity analysis using polynomial chaos expansions. Reliab Eng Syst Saf 93(7):964–979
Thostenson ET, Li C, Chou TW (2005) Nanocomposites in context. Compos Sci Technol 65(3):491–516
Tserpes K, Papanikos P, Labeas G, Pantelakis SG (2008) Multi-scale modeling of tensile behavior of carbon nanotube-reinforced composites. Theor Appl Fract Mech 49(1):51–60
Vu-Bac N, Nguyen-Xuan H, Chen L, Lee CK, Zi G, Zhuang X, Liu GR, Rabczuk T (2013) A phantom-node method with edge-based strain smoothing for linear elastic fracture mechanics. J Appl Math 2013:12. doi:10.1155/2013/978026
Vu-Bac N, Lahmer T, Zhang Y, Zhuang X, Rabczuk T (2014) Stochastic predictions of interfacial characteristic of polymeric nanocomposites (PNCs). Compos B Eng 59:80–95
Vu-Bac N, Rafiee R, Zhuang X, Lahmer T, Rabczuk T (2015a) Uncertainty quantification for multiscale modeling of polymer nanocomposites with correlated parameters. Compos B Eng 68:446–464
Vu-Bac N, Silani M, Lahmer T, Zhuang X, Rabczuk T (2015b) A unified framework for stochastic predictions of mechanical properties of polymeric nanocomposites. Comput Mater Sci 96:520–535
Wang H, Zhou H, Peng R, Mishnaevsky L (2011) Nanoreinforced polymer composites: 3D FEM modeling with effective interface concept. Compos Sci Technol 71(7):980–988
Williams J (2010) Particle toughening of polymers by plastic void growth. Compos Sci Technol 70(6):885–891
Xiu D, Karniadakis GE (2002) The wiener–askey polynomial chaos for stochastic differential equations. SIAM J Sci Comput 24(2):619–644
Xiu D, Karniadakis GE (2003) Modeling uncertainty in flow simulations via generalized polynomial chaos. J Comput Phys 187(1):137–167
Yu S, Yang S, Cho M (2009) Multi-scale modeling of cross-linked epoxy nanocomposites. Polymer 50(3):945–952
Zamanian M, Mortezaei M, Salehnia B, Jam J (2013) Fracture toughness of epoxy polymer modified with nanosilica particles: Particle size effect. Eng Fract Mech 97:193–206
Zappalorto M, Salviato M, Quaresimin M (2011) Influence of the interphase zone on the nanoparticle debonding stress. Compos Sci Technol 72(1):49–55
Zhao J, Jiang JW, Jia Y, Guo W, Rabczuk T (2013) A theoretical analysis of cohesive energy between carbon nanotubes, graphene and substrates. Carbon 57:108–119
Acknowledgements
The authors gratefully acknowledge the support for this research provided by the Deutsche Forschungsgemeinschaft (DFG).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Hamdia, K.M., Silani, M., Zhuang, X. et al. Stochastic analysis of the fracture toughness of polymeric nanoparticle composites using polynomial chaos expansions. Int J Fract 206, 215–227 (2017). https://doi.org/10.1007/s10704-017-0210-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10704-017-0210-6