## Abstract

In this review, we introduce a series of the recent studies that attempted to develop the multiscale models for predicting the fracture toughness of polymer nanocomposites (PNC). Firstly, the overview of the multiscale schematics for predicting the fracture toughness of PNC. Secondly, according to the multiscale schematics, the multiscale models for predicting the fracture toughness of PNC are described: (i) epoxy nanocomposites (NC) including rigid spherical nanoparticles, (ii) thermoplastic/epoxy blends, and (iii) epoxy NC including carbon nanotubes. Finally, we summarize the discussion and provide our perspective on future challenging issues.

## Introduction

Epoxy have received many attention from researchers because of their excellent properties (such as thermal stability, high chemical resistance and mechanical properties [1,2,3,4,5]). Meanwhile, epoxy have excellent strength and high rigidity benefit from the high degree of cross-linking properties [6,7,8]. However, its highly cross-linked structure can make the epoxy exhibit serious problems such as high brittleness, low fracture toughness and low resistance to crack propagation [9,10,11]. This greatly limits the application of epoxy materials in fields such as aerospace and automotive components. Therefore, research to enhance the fracture toughness of epoxy materials has received increasing attention. Conventionally, in order to achieve this goal, the researchers by adding diverse organic and inorganic nanoparticles [12, 13], polymer nanoparticles [14, 15], thermos-plastics and rubber materials [1, 8, 16]. In recent years, many research have shown that the researchers focus on organic and inorganic nanoparticles, such as graphene and its derivatives [17,18,19,20,21,22], silica [23,24,25], alumina [26], and clay nanoparticles [27] to make epoxy composites effectively improves the fracture toughness [28, 29]. This effective strategy is due to the fact that epoxy composites combine the advantages of fillers and epoxy [30]. Besides, there are many factors affecting the fracture toughness, mechanical properties and thermo-mechanical properties of the epoxy composites, such as the nanoparticles size, nanoparticles shape, degree of dispersion of the nanoparticles and the interphase zone effect [31,32,33]. Therefore, the proposed of a theoretical model for predicting the fracture toughness of PNC is a very necessary work.

According to the related literatures [34,35,36,37,38,39,40,41,42,43,44,45,46], the toughening mechanisms are mainly induced by the nanoscale energy dissipations due to the nanoscale damaging mechanisms (such as the nanoparticle debonding, the plastic nanovoid growth, the pull-out, the plastic shear band, etc.). Actually, the macroscopic crack extension problem of the PNC is the multiscale problem. The nanoscale energy dissipations near the tip of macroscopic cracks contribute to resist the macroscopic crack extension of the PNC. Meanwhile, the nanoscale energy dissipations near the macroscopic crack tip are determined by the macroscopic stress fields near the macroscopic crack tip, which can be quantified by the fracture mechanics theory. Even though the macroscopic crack propagation problem of the PNC is the multiscale problem, some researchers have developed the multiscale models for predicting the fracture toughness of the PNC [34,35,36,37,38,39,40,41].

The multiscale approach has been firstly adopted by Quaresimin et al. [39] by reflecting the toughening mechanisms of the PNC including the rigid spherical nanoparticles. As we all know three different damage mechanisms (the nanoparticle debonding, the plastic nanovoid growth, and the plastic shear banding) are considered by using the multiscale approach. The predictive model shows good agreement with the experimental data. Shin et al. [36] extends the aforementioned approach to the multiscale models for predicting fatigue crack growth of the PNC. According to the literature [36], the proposed model shows the sufficient agreement predictions by comparison with the experimental data. Shin et al. [37] extends the aforementioned approach to the multiscale model for predicting the fracture toughness of the thermoplastic/epoxy blends. In the mechanical viewpoint, the thermoplastic/epoxy blends is the composite materials including the spherical shape of the elasto-plastic inclusions. In this approach, the molecular dynamics simulation is used to characterize the elasto-plastic behavior of the thermoplastic particle and the nanoscale energy dissipations. After the characterization of the elasto-plastic behavior of the thermoplastic particle, the two main damage mechanisms (the plastic yielding of the thermoplastic particle, and the particle bridging of the crack wake) are considered. In the Ref. 37, the multiscale framework is explained in detail. In conclusion, the proposed model is validated by comparison to the results of the experiment. Shin [35] develops the multiscale model for predicting the CNT/Epoxy NC focused on explain the three toughening mechanisms (nanoparticle debonding, plastic nanovoid growth, and pull-out of CNT). A detailed explanation of the multiscale formulation is given in Ref. [35]. Finally, the proposed model is also validated by the experimental data [35].

In this paper, we present the recent studies on the multiscale models for predicting fracture toughness of PNC. In Sect. 2, we provide a brief overview on the concept of the multiscale framework for predicting the fracture toughness of PNC. In Sect. 3, we introduce the multiscale models for predicting the fracture toughness of the PNC: (i) epoxy NC including rigid spherical nanoparticles, (ii) thermoplastic/epoxy blends, and (iii) epoxy NC including carbon nanotubes. In the last section, we summarize the discussion and provide our perspective on future challenging issues.

## Overview of Multiscale Strategy for Predicting the Fracture Toughness of PNC

In this section, the overview of multiscale strategy for predicting the fracture toughness of PNC is described, as shown in Fig. 1. From the fracture mechanics theory, the macroscopic stress fields can be obtained. By adopting the micromechanics theory, the microscopic boundary value problem (BVP) of the representative volume element can be defined. Then, by solving the microscopic BVP, the energy dissipations of the representative volume element can be quantified. By taking the J-integral near the tip of the macroscopic crack, the mode I critical strain energy release rate (SERR) of the PNC as follow:

where the *G*_{Ic,nc} and the *G*_{Ic,m} are the critical SERR of the PNC and matrix, respectively.

## Review of the Multiscale Models for Predicting Fracture Toughness of PNC

### Epoxy Nanocomposites Including Rigid Spherical Nanoparticles

The rigid spherical nanoparticles can effectively enhance the fracture toughness of PNC. Many papers [39,40,41,42,43,44,45,46] have shown that the three different damage mechanisms (nanoparticle debonding, plastic nanovoid growth, and the plastic shear banding) are closely related to the fracture toughness improvement. The multiscale models to predict fracture toughness of polymer nanocomposites including the rigid spherical nanoparticles were firstly developed by the Quaresimin et al. [39] with consideration of the three main toughening mechanisms (interfacial debonding, plastic nanovoid growth, and plastic shear banding).

The critical SERR of epoxy/SiC NC, *G*_{Ic,comp}, as follows:

where Δ*G*_{i} is the enhancement of the SERR, the subscripts *i* represent by each toughening mechanism nanoparticle debonding (*i* = db) and plastic nanovoids growth (*i* = py). The SERR enhancement due to nanoparticle debonding can be obtained as follows:

where the plastic nanovoids growth the critical SERR enhancement, as follows:

where the parameters *ψ*_{db} and *ψ*_{py} quantify the stress intensity factors (SIF) caused by nanoparticle debonding and plastic nanovoids growth, respectively. The overall critical SERR of the epoxy/SiC NC can be obtained as follows:

The relationship between the SIF and SERR for the epoxy/SiC NC and neat matrix can be obtained as follows:

The predictive results show good agreement with the experimental data for the low weight fraction of nanoparticles (up to 8wt%) [39].

According to the developed models by Quaresimin et al. [39], the interfacial fracture energy and the interphase elasto-plastic constitutive law is very critical in the prediction of the fracture toughness of polymer nanocomposites. Although many researchers have developed predictive models for fracture toughness and considered the existence of interphase zone effect [39,40,41], no models reflect the interphase constitutive law and interfacial energy on the nanoparticle size. There are many studies show [46,47,48,49,50,51,52,53,54,55,56,57,58,59,60] that NC high performance is closely related to the interphase zone, with the contribution of the interphase region to the NC increasing as the radius of the nanoparticles decreases. Therefore, the effect of the size of the rigid spherical nanoparticles on the toughening mechanism (i.e. nanoparticle debonding and plastic nanovoid growth) should be reflected together in the theoretical model.

Wang and Shin [34] proposed a multiscale framework for predicting fracture toughness improvement of epoxy/SiC NC focused on explain nanoparticle debonding and plastic nanovoid growth mechanisms by using molecular dynamics (MD) simulations and the multiscale fracture toughness model of epoxy/SiC NC, as shown in Fig. 2. Here, for molecular modelling, the Material Studio 2021 were used and the uniaxial tensile loading simulations were performed by employing the LAMMPS. Then, the stress–strain curves obtained from the MD simulations were compared with the mean-field (MF) homogenization stress–strain curves of the three-phase continuum model, to reverse calculation the interphase elastoplastic constitutive parameters.

In this present model, it is important to point out that the rigid spherical nanoparticles are assumed to be uniformly dispersed. The interfacial fracture energy, *U*_{interaction}, was obtained from *U*_{interaction} = *U*_{comp}—*U*_{mat}—*U*_{par}, as shown in Fig. 3a. The MD predicted results show that as the nanoparticle size increased, the interfacial fracture energy decreased. As shown in Fig. 3b, the fracture toughness improvement of epoxy/SiC NC due to nanoparticle debonding and plastic nanovoids growth was obtained from the proposed fracture toughness multiscale model. The critical SIF ratio (*K*_{Ic,comp}/*K*_{Ic,mat}) is increasing with decreasing nanoparticle radius. Meanwhile, the critical SIF ratio (*K*_{Ic,comp}/*K*_{Ic,mat}) increases with the increase of the nanoparticle volume fraction. There are two remarkable points: (i) this is the first attempt to obtain the fracture toughness improvements of PNC with characterization of interphase plastic behavior, (ii) as the nanoparticulate radius increased, the fracture toughness is clearly decreased.

### Thermoplastic/Epoxy Blends

Some studies have shown that the plastic deformation near the macroscopic crack tip and the particle bridging in the crack wake are two central elements in the toughening mechanism of thermoplastic/epoxy NC. Shin et al. [37] developed a multiscale model for predicting the fracture toughness improvement of thermoplastic/epoxy blends by plastic yielding of toughening agents, as shown in Fig. 4.

The numerical integration is employed to compute the contour integral can be obtained as follows:

where *N* is the number of integration points, *ρ*_{k} = *k* × Δ*ρ* and Δ*ρ* = *ρ*^{*}/*N*. Δ*G*_{p}, can be obtained as follows:

by using the definition of the fracture toughness as *G*_{Ic} = *K*_{Ic}^{2} × (1-*ν*_{comp}^{2})/*E*_{comp}. Here, the contribution of the thermoplastic-particle yield mechanism, *ψ*_{p}, is

The hydrostatic tension at *k*th integration point, *S*_{k}, can be obtained as follows:

Toughness improvement due to rubbery particle bridging mechanism can be determined as:

where *r*_{p} is the radius of thermoplastic particles. The total toughness improvement can be obtained by Δ*G*_{Ic} = Δ*G*_{p} + Δ*G*_{t}.

As shown in Fig. 5, the multiscale model predictions show high agreement with the experimental data.

### Epoxy Nanocomposites Including Carbon Nanotubes

Many experiments show that the carbon nanotubes (CNTs) can effectively improve the fracture toughening of epoxy NC. Interfacial debonding mechanisms, plastic nanovoids growth mechanisms, and pull-out of CNTs mechanisms are considered to be crucial for improving the fracture toughness of CNT/polymer NC [35, 42].

According to the literature [35, 38, 43], many theoretical models for predicting the fracture toughness of CNT/polymer NC have been developed. Shokrieh and Zeinedini [38] focused on the consideration of interfacial debonding mechanisms and developed multiscale models for predicting the toughness improvement of CNT/polymer NC. However, this multiscale model only considers the interfacial debonding mechanism, so the experimental data are much bigger than prediction results. Also, the other literature [43, 63] focused on the fracture toughness improvement of CNT/polymer NC due to the pull-out of CNTs mechanism. However, the aforementioned models are not the integrated models including the three main toughening mechanisms. Shin et al. [35] considered that the three toughening mechanisms should be reflected together in the theoretical model, so proposed a new multiscale model for predicting the fracture toughness of CNT/epoxy NC, as shown in Fig. 6.

The critical SERR of the CNT/epoxy NC, *G*_{Ic,nc}, can be written as follows [35]:

where Δ*G*_{i} is the enhancement of the SERR, the subscripts *i* represent by each toughening mechanism, interfacial debonding (*i* = db), plastic nanovoids growth (*i* = py), and the pull-out of CNTs (*i* = po). The SERR enhancement can be determined as:

where *w*_{i} is the dissipated energy density caused by each toughening mechanism. From the analytic formulations [35], the enhancement of the SERR of the CNT/epoxy NC can be determined as:

where *V*_{f} is volume fraction of CNT; *λ* can be determined as:

where *E*_{nc} and *ν*_{nc} are the elastic modulus and Poisson’s ratio of NC; *r*_{p} and *l*_{p} are the radius and length of CNTs; the details for the computation of the critical stress (*σ*_{cr}), the radial part of the stress concentration tensor (*H*), and the orientation average of the function *f* (\(\overline{f}\)) are explained in Ref. 38. For the interfacial debonding mechanisms and the plastic nanovoid growth mechanisms, the dissipated energy (Δ*U*_{i}) can be obtained as follows:

where *γ*_{db} is the interfacial fracture energy; *σ*_{Ym} and *σ*_{Yi} are the yield strength of matrix and interphase, respectively; *ε*_{Ym} and *ε*_{Yi} the yield strain of matrix and interphase, respectively; *G*_{m} and *n*_{m} are the shear modulus and hardening exponent of matrix. For the pull-out of CNT mechanism, the SERR enhancements due to pull-out of CNT mechanisms can be determined as [43]:

where the *τ*_{i} is the interfacial shear strength; the critical length of CNTs can be determined as [43]:

where *σ*_{up} is the ultimate tensile strength of CNT.

In the present model, it is necessary to point out that the CNTs are assumed to be uniformly dispersed and randomly oriented. Therefore, the filler agglomeration (such as CNT bundle) and waviness of CNTs are not considered.

The Eq. (12) is validated by the experimental comparison, as shown in Fig. 7. The predictive results show satisfactory agreement with the experimental data. Here, the non-dimensional parameter, *χ*, is introduced to consider the influences of the interphase elasto-plastic behaviors, as *E*_{int} = *χE*_{mat} and *σ*_{Yi} = *χσ*_{Ym}. There are two core points: (i) the harder interphase zone contribute to the fracture toughness improvement of the CNT/epoxy NC, (ii) for the CNT/epoxy NC, the contribution of interphase effect is more prominent the smaller the diameter of CNT.

## Summary and Perspective

In this review, we introduce a series of the recent studies that attempted to develop the multiscale models for predicting the fracture toughness of PNC. Firstly, the overview of the multiscale schematics for predicting the fracture toughness of PNC. Secondly, based on the multiscale schematics, the multiscale models for predicting the fracture toughness of PNC are described: (i) epoxy NC including rigid spherical nanoparticles, (ii) thermoplastic/epoxy blends, and (iii) epoxy NC including carbon nanotubes.

Even though some recent studies on the multiscale models for predicting fracture toughness of PNC are reviewed, the aforementioned multiscale approaches are only applicable to the PNC including the well-dispersed nanofillers. For the efficient representative volume element analysis, the well-established analytic micromechanics methods (such as Mori–Tanaka model) are employed. This approach needs the assumption that the embedded nanoparticles should be well dispersed. However, according to the many related literatures [66,67,68,69], the influences of the filler agglomeration on the interphase constitutive law are not negligible. Some experimental evidence also shows that the influences of filler agglomeration on the fracture toughness are critical [39, 70]. Meanwhile, in some cases, the influences of filler agglomeration on the fracture toughness could be negligible [71]. Therefore, the numerical homogenization approaches based on the finite element method will be more applicable to the consideration of the filler agglomeration during the representative volume element analysis. During the J-integral near the macroscopic crack tip, the nanoscale energy dissipations should be obtained in the representative volume elements at each integration points. Therefore, the well-established FE^{2} approach, the systematic reduction techniques, and the fracture mechanics theory should be merged for the multiscale fracture mechanics analysis.

## References

Z. Liu, J. Li, X. Liu, Novel functionalized BN nanosheets/epoxy composites with advanced thermal conductivity and mechanical properties. ACS Appl. Mater. Interfaces

**12**, 6503–6515 (2020)X. Zhang, Z. Liu, Y. Li, C. Wang, Y. Zhu, H. Wang, J. Wang, Robust superhydrophobic epoxy composite coating prepared by dual interfacial enhancement. Chem. Eng. J.

**371**, 276–285 (2019)Q. Wu, J. He, F. Wang, X. Yang, J. Zhu, Comparative study on effects of covalentcovalent, covalent-ionic and ionic-ionic bonding of carbon fibers with polyether amine/GO on the interfacial adhesion of epoxy composites. Appl. Surf. Sci.

**532**, 147359 (2020)S.Y. Mun, J. Ha, S. Lee, Y. Ju, H.M. Lim, D. Lee, Prediction of enhanced interfacial bonding strength for basalt fiber/epoxy composites by micromechanical and thermomechanical analyses. Compos. A Appl. Sci. Manuf.

**142**, 106208 (2020)J. He, H. Wang, Q. Qu, Z. Su, T. Qin, X. Tian, Three-dimensional network constructed by vertically oriented multilayer graphene and SiC nanowires for improving thermal conductivity and operating safety of epoxy composites with ultralow loading. Compos. A Appl. Sci. Manuf.

**139**, 106062 (2020)X. Han, T. Wang, P.S. Owuor, S.H. Hwang, C. Wang, J. Sha et al., Ultra-stiff graphene foams as three-dimensional conductive fillers for epoxy resin. ACS Nano

**12**(11), 11219–11228 (2018)V.N. Mochalin, I. Neitzel, B.J. Etzold, A. Peterson, G. Palmese, Y. Gogotsi, Covalent incorporation of aminated nanodiamond into an epoxy polymer network. ACS Nano

**5**(9), 7494–7502 (2011)M.F. DiBerardino, R.A. Pearson, The effect of particle size on synergistic toughening of boron nitride-rubber hybrid epoxy composites. ACS Symp. Ser. Am. Chem. Soc.

**759**, 213–229 (2000)C. Zhou, Z. Li, J. Li, T. Yuan, B. Chen, X. Ma et al., Epoxy composite coating with excellent anticorrosion and self-healing performances based on multifunctional zeolitic imidazolate framework derived nanocontainers. Chem. Eng. J.

**385**, 123835 (2020)J. Sun, C. Wang, J.C.C. Yeo, D. Yuan, H. Li, L.P. Stubbs, C. He, Lignin epoxy composites: preparation, morphology, and mechanical properties. Macromol Mater. Eng.

**301**(3), 328–336 (2016)Y. Zeng, L. Ci, B.J. Carey, R. Vajtai, P.M. Ajayan, Design and reinforcement: vertically aligned carbon nanotube-based sandwich composites. ACS Nano

**4**(11), 6798–6804 (2010)L. Chen, S. Chai, K. Liu, N. Ning, J. Gao, Q. Liu et al., Enhanced epoxy/silica composites mechanical properties by introducing graphene oxide to the interface. ACS Appl. Mater. Interfaces

**4**(8), 4398–4404 (2012)L. Zhu, C. Feng, Y. Cao, Corrosion behavior of epoxy composite coatings reinforced with reduced graphene oxide nanosheets in the high salinity environments. Appl. Surf. Sci.

**493**, 889–896 (2019)L.-C. Tang, Y.-J. Wan, K. Peng, Y.-B. Pei, L.-B. Wu, L.-M. Chen et al., Fracture toughness and electrical conductivity of epoxy composites filled with carbon nanotubes and spherical particles. Compos. A Appl. Sci. Manuf.

**45**, 95–101 (2013)X. Huang, T. Iizuka, P. Jiang, Y. Ohki, T. Tanaka, Role of interface on the thermal conductivity of highly filled dielectric epoxy/AlN composites. J. Phys. Chem. C

**116**(25), 13629–13639 (2012)L.-X. Gong, L. Zhao, L.-C. Tang, H.-Y. Liu, Y.-W. Mai, Balanced electrical, thermal and mechanical properties of epoxy composites filled with chemically reduced graphene oxide and rubber nanoparticles. Compos. Sci. Technol.

**121**, 104–114 (2015)L.-C. Tang, Y.-J. Wan, D. Yan, Y.-B. Pei, L. Zhao, Y.-B. Li et al., The effect of graphene dispersion on the mechanical properties of graphene/epoxy composites. Carbon

**60**, 16–27 (2013)J. Jia, X. Sun, X. Lin, X. Shen, Y.-W. Mai, J.-K. Kim, Exceptional electrical conductivity and fracture resistance of 3D interconnected graphene foam/epoxy composites. ACS Nano

**8**(6), 5774–5783 (2014)S. Chandrasekaran, N. Sato, F. Tolle, R. Mülhaupt, B. Fiedler, K. Schulte, Fracture toughness and failure mechanism of graphene based epoxy composites. Compos. Sci. Technol.

**97**, 90–99 (2014)Y.-J. Wan, L.-C. Tang, L.-X. Gong, D. Yan, Y.-B. Li, L.-B. Wu et al., Grafting of epoxy chains onto graphene oxide for epoxy composites with improved mechanical and thermal properties. Carbon

**69**, 467–480 (2014)Y.-J. Wan, L.-X. Gong, L.-C. Tang, L.-B. Wu, J.-X. Jiang, Mechanical properties of epoxy composites filled with silane-functionalized graphene oxide. Compos. A Appl. Sci. Manuf.

**64**, 79–89 (2014)Y.T. Park, Y. Qian, C. Chan, T. Suh, M.G. Nejhad, C.W. Macosko et al., Epoxy toughening with low graphene loading. Adv. Funct. Mater.

**25**(4), 575–585 (2015)L.-C. Tang, H. Zhang, S. Sprenger, L. Ye, Z. Zhang, Fracture mechanisms of epoxybasedternary composites filled with rigid-soft particles. Compos. Sci. Technol.

**72**(5), 558–565 (2012)M. Kucharek, W. MacRae, L. Yang, Investigation of the effects of silica aerogel particles on thermal and mechanical properties of epoxy composites. Compos. A Appl. Sci. Manuf.

**139**, 106108 (2020)Y. Ma, H. Di, Z. Yu, L. Liang, L. Lv, Y. Pan et al., Fabrication of silica-decorated graphene oxide nanohybrids and the properties of composite epoxy coatings research. Appl. Surf. Sci.

**360**, 936–945 (2016)J. Ligoda-Chmiel, R.E. Sliwa, M. Potoczek, Flammability and acoustic absorption of alumina foam/tri-functional epoxy resin composites manufactured by the infiltration process. Compos. Part B-Eng.

**112**, 196–202 (2017)D.V. A. Ceretti, L.C. Escobar da Silva, M. do Carmo Gonçalves, D.J. Carastan, The role of dispersion technique and type of clay on the mechanical properties of clay/ epoxy composites. Macromolecular Symposia: Wiley Online Library 1800055 (2019).

B. Wetzel, P. Rosso, F. Haupert, K. Friedrich, Epoxy nanocomposites–fracture and toughening mechanisms. Eng. Fract. Mech.

**73**(16), 2375–2398 (2006)Y.L. Liang, R. Pearson, The toughening mechanism in hybrid epoxy-silica-rubber nanocomposites (HESRNs). Polymer

**51**(21), 4880–4890 (2010)J. Fu, M. Zhang, L. Jin, L. Liu, N. Li, L. Shang et al., Enhancing interfacial properties of carbon fibers reinforced epoxy composites via Layer-by-Layer self assembly GO/SiO2 multilayers films on carbon fibers surface. Appl. Surf. Sci.

**470**, 543–554 (2019)O. Zabihi, M. Ahmadi, S. Nikafshar, K.C. Preyeswary, M. Naebe, A technical review on epoxy-clay nanocomposites: Structure, properties, and their applications in fiber reinforced composites. Compos. Part B-Eng.

**135**, 1–24 (2018)X. Xu, B. Zhang, K. Liu, D. Liu, M. Bai, Y. Li, Finite element simulation and analysis of the dielectric properties of unidirectional aramid/epoxy composites. Polym. Compos.

**39**(S4), 2226–2233 (2018)L.-C. Hao, Z.-X. Li, F. Sun, K. Ding, X.-N. Zhou, Z.-X. Song et al., High-performance epoxy composites reinforced with three-dimensional Al2O3 ceramic framework. Compos. A Appl. Sci. Manuf.

**127**, 105648 (2019)H. Wang, H. Shin, Influence of nanoparticulate diameter on fracture toughness improvement of polymer nanocomposites by a nanoparticle debonding mechanism: a multiscale study. Eng. Fract. Mech.

**261**, 108261 (2022)H. Shin, Multiscale model to predict fracture toughness of CNT/epoxy nanocomposites. Compos. Struct.

**272**, 114236 (2021)H. Shin, M. Cho, Multiscale model to predict fatigue crack propagation behavior of thermoset polymeric nanocomposites. Compos. A Appl. Sci. Manuf.

**99**, 23–31 (2017)H. Shin, B. Kim, J.-G. Han, M.Y. Lee, J.K. Park, M. Cho, Fracture toughness enhancement of thermoplastic/epoxy blends by the plastic yield of toughening agents: a multiscale analysis. Compos. Sci. Technol.

**145**, 173–180 (2017)M.M. Shokrieh, A. Zeinedini, Effect of CNTs debonding on mode I fracture toughness of polymeric nanocomposites. Mater. Design

**101**, 56–65 (2016)M. Quaresimin, M. Salviato, M. Zappalorto, A multi-scale and multi-mechanism approach for the fracture toughness assessment of polymer nanocomposites. Compos. Sci. Technol.

**91**, 16–21 (2014)M. Zappalorto, M. Salviato, M. Quaresimin, A multiscale model to describe nanocomposite fracture toughness enhancement by the plastic yielding of nanovoids. Compos. Sci. Technol.

**72**(14), 1683–1691 (2012)M. Salviato, M. Zappalorto, M. Quaresimin, Plastic shear bands and fracture toughness improvements of nanoparticle filled polymers: a multiscale analytical model. Compos. Part A Appl. Sci. Manuf.

**48**, 144–152 (2013)M. Quaresimin, K. Schulte, M. Zappalorto, S. Chandrasekaran, Toughening mechanisms in polymer nanocomposites: from experiments to modelling. Compos. Sci. Technol.

**123**, 187–204 (2016)H.D. Wagner, P.M. Ajayan, K. Schulte, Nanocomposite toughness from a pull-out mechanism. Compos. Sci. Technol.

**83**, 27–31 (2013)B. Lauke, On the effect of particle size on fracture toughness of polymer composites. Compos. Sci. Technol.

**68**(15–16), 3365–3372 (2008)Y. Huang, A. Kinloch, Modeling of the toughening mechanisms in rubber-modified epoxy polymers part II: a quantitative description of the microstructure fracture property relationships. J. Mater. Sci.

**27**, 2763–2769 (1992)A.G. Evans, S. Williams, P.W.R. Beaumont, On the toughness of particulate filled polymers. J. Mater. Sci.

**20**(10), 3668–3674 (1985)J. Choi, S. Yu, S. Yang, M. Cho, The glass transition and thermoelastic behavior of epoxy based nanocomposites: a molecular dynamics study. Polymer

**52**, 5197–5203 (2011)S. Yang, M. Cho, Scale bridging method to characterize mechanical properties of nanoparticle/polymer nanocomposites. Appl. Phys. Lett.

**93**, 043111 (2008)S. Yu, S. Yang, M. Cho, Multi-scale modeling of cross-linked epoxy nanocomposites. Polymer

**50**, 945–952 (2009)H. Shin, J. Choi, M. Cho, An efficient multiscale homogenization modeling approach to describe hyperelastic behavior of polymer nanocomposites. Compos. Sci. Technol.

**175**, 128–134 (2019)G.M. Odegard, T.C. Clancy, T.S. Gates, Modeling of the mechanical properties of nanoparticle/polymer composites. Polymer

**46**, 553–562 (2005)S. Yu, S. Yang, M. Cho, Multiscale modeling of cross-linked epoxy nanocomposites to characterize the effect of particle size on thermal conductivity. J. Appl. Phys.

**110**, 124302 (2011)H. Shin, S. Yang, S. Chang, S. Yu, M. Cho, Multiscale homogenization modeling for thermal transport properties of polymer nanocomposites with Kapitza thermal resistance. Polymer

**54**, 1543–1554 (2013)M. Cho, S. Yang, S. Chang, S. Yu, A study on the prediction of the mechanical properties of nanoparticulate composites using the homogenization method with the effective interface concept. Int. J. Numer. Meth Eng.

**85**, 1564–1583 (2011)B. Kim, J. Choi, S. Yang, S. Yu, M. Cho, Influence of crosslink density on the interfacial characteristics of epoxy nanocomposites. Polymer

**60**, 186–197 (2015)S. Yang, S. Yu, W. Kyoung, D.-S. Han, M. Cho, Multiscale modeling of size-dependent elastic properties of carbon nanotube/polymer nanocomposites with interfacial imperfections. Polymer

**53**, 623–633 (2012)J. Choi, S. Yang, S. Yu, H. Shin, M. Cho, Method of scale bridging for thermoelasticity of cross-linked epoxy/SiC nanocomposites at a wide range of temperatures. Polymer

**53**, 5178–5189 (2012)S. Yang, J. Choi, M. Cho, Elastic stiffness and filler size effect of covalently grafted nanosilica polyimide composites: molecular dynamics study. ACS Appl. Mater. Interfaces

**4**, 4792–4799 (2012)H. Shin, S. Chang, S. Yang, B.D. Youn, M. Cho, Statistical multiscale homogenization approach for analyzing polymer nanocomposites that include model inherent uncertainties of molecular dynamics simulations. Compos. Part B Eng.

**87**, 120–131 (2016)H. Shin, S. Yang, J. Choi, S. Chang, M. Cho, Effect of interphase percolation on mechanical behavior of nanoparticle-reinforced polymer nanocomposite with filler agglomeration: a multiscale approach. Chem. Phys. Lett.

**635**, 80–85 (2015)S.J. Park, K. Li, S.K. Hong, Thermal stabilities and mechanical interfacial properties of polyethresulfone-modified epoxy resin. Solid State Phenom.

**111**, 159–162 (2006)J. Stein, A. Wilkilson, The influence of pes and triblock copolymer on the processing and properties of highly crosslinked epoxy matrices 15th European Conference of Composite Materials, Venice, Italy (2012).

N. Lachman, H.W. Daniel, Correlation between interfacial molecular structure and mechanics in CNT/epoxy nano-composites. Compos. Part A Appl. Sci. Manuf.

**41**(9), 1093–1098 (2010)F.H. Gojny, M.H.G. Wichmann, B. Fiedlerf, K. Schultes, Influence of different carbon nanotubes on the mechanical properties of epoxy matrix composites—a comparative study. Compos. Sci. Technol.

**65**(15–16), 2300–2313 (2005)M.R. Ayatollahi, S. Shadlou, M.M. Shokrieh, Fracture toughness of epoxy/multiwalled carbon nanotube nano-composites under bending and shear loading conditions. Mater. Des.

**32**(4), 2115–2124 (2011)H. Shin, K. Baek, J.-G. Han, M. Cho, Homogenization analysis of polymeric nanocomposites containing nanoparticulate clusters. Compos. Sci. Technol.

**138**, 217–224 (2017)K. Baek, H. Shin, T. Yoo, M. Cho, Two-step multiscale homogenization for mechanical behaviour of polymeric nanocomposites with nanoparticulate agglomerations. Compos. Sci. Technol.

**179**, 97–105 (2019)K. Baek, H. Shin, M. Cho, Multiscale modeling of mechanical behaviors of nano-SiC/epoxy nanocomposites with modified interphase model: effect of nanoparticle clustering. Compos. Sci. Technol.

**203**, 108572 (2021)K. Baek, H. Park, H. Shin, S. Yang, M. Cho, Multiscale modeling to evaluate combined effect of covalent grafting and clustering of silica nanoparticles on mechanical behaviors of polyimide matrix composites. Compos. Sci. Technol.

**206**, 108673 (2021)Y.-S. Kim, J.-H. Lee, S.-J. Park, Effect of ambient plasma treatment on single-walled carbon nanotubes-based epoxy/fabrics for improving fracture toughness and electromagnetic shielding effectiveness. Compos. Part A Appl. Sci. Manuf.

**148**, 106456 (2021)N. Domun, H. Hadavinia, T. Zhang, T. Sainsbury, G.H. Liaghat, S. Vahid, Improving the fracture toughness and the strength of epoxy using nanomaterials—a review of the current status. Nanoscale

**7**, 10294–10329 (2015)

## Acknowledgements

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2021R1C1C1004353).

## Author information

### Authors and Affiliations

### Corresponding author

## Additional information

### Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Rights and permissions

## About this article

### Cite this article

Wang, H., Shin, H. Recent Studies on the Multiscale Models for Predicting Fracture Toughness of Polymer Nanocomposites.
*Multiscale Sci. Eng.* **4, **1–9 (2022). https://doi.org/10.1007/s42493-022-00075-y

Received:

Revised:

Accepted:

Published:

Issue Date:

DOI: https://doi.org/10.1007/s42493-022-00075-y

### Keywords

- Multiscale analysis
- Fracture toughness
- Micromechanics
- Molecular dynamics