Skip to main content

Recent Studies on the Multiscale Models for Predicting Fracture Toughness of Polymer Nanocomposites

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:

$$G_{{\text{Ic,nc}}}^{{}} = G_{{\text{Ic,m}}}^{{}} + 2 \times \int_{0}^{\rho *(\varphi = \pi /2)} {wd\rho } ,$$
(1)

where the GIc,nc and the GIc,m are the critical SERR of the PNC and matrix, respectively.

Fig. 1
figure 1

Overview of the multiscale strategy forpredictingfracture toughness ofPNC near thetip of macroscopic crack

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, GIc,comp, as follows:

$$G_{{\text{Ic,comp}}}^{{}} = G_{{\text{Ic,mat}}}^{{}} + \sum\limits_{i}^{{}} {\Delta G_{i}^{{}} } ,$$
(2)

where ΔGi 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:

$$\Delta G_{{{\text{db}}}} = f_{{\text{p}}}^{{}} \times \psi_{{{\text{db}}}} \times G_{{\text{Ic,comp}}}^{{}} ,$$
(3)

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

$$\Delta G_{{{\text{py}}}}^{{}} = f_{{\text{p}}} \times \psi_{{{\text{py}}}} \times G_{{\text{Ic,comp}}}^{{}} ,$$
(4)

where the parameters ψdb and ψpy 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:

$$G_{{\text{Ic,comp}}}^{{}} = \frac{{G_{{\text{Ic,mat}}}^{{}} }}{{1 - f_{{\text{p}}} (\psi_{{{\text{db}}}} + \psi_{{{\text{py}}}} )}}.$$
(5)

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

$$K_{{\text{Ic,mat}}} = \sqrt {\frac{{G_{{\text{Ic,mat}}} \times E_{{{\text{mat}}}} }}{{1 - v_{{{\text{mat}}}}^{2} }}} , \, K_{{\text{Ic,comp}}} = \sqrt {\frac{{G_{{\text{Ic,comp}}} \times E_{{{\text{comp}}}} }}{{1 - v_{{{\text{comp}}}}^{2} }}} .$$
(6)

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.

Fig. 2
figure 2

The proposed multiscale modelframeworkto characterize elastoplastic constitutive law of interphase.Reprinted from Ref. [34]by kind permission of Elsevier

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, Uinteraction, was obtained from Uinteraction = UcompUmatUpar, 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 (KIc,comp/KIc,mat) is increasing with decreasing nanoparticle radius. Meanwhile, the critical SIF ratio (KIc,comp/KIc,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.

Fig. 3
figure 3

a Interfacial fracture energy of epoxy/SiC nanocompositesfrom MD simulationand b predicted fracture toughness improvement of epoxy/SiC nanocomposites.Reprinted from Ref. [34]by kind permission of Elsevier

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.

Fig. 4
figure 4

The proposed multiscale approach for predictingthe fracture toughness improvement of the thermoplastic/epoxy blends. Reprinted from Ref. [37]by kind permission of Elsevier

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

$$\Delta G_{p}^{{}} = 2 \times \int_{0}^{{\rho_{{}}^{*} (\phi = \pi /2)}} {u_{p}^{{}} d\rho } = 2 \times \sum\limits_{k = 1}^{N} {u_{p}^{{}} (\rho_{k}^{{}} )\Delta \rho } ,$$
(7)

where N is the number of integration points, ρk = k × Δρ and Δρ = ρ*/N. ΔGp, can be obtained as follows:

$$\Delta G_{p}^{{}} = 2 \times \sum\limits_{k = 1}^{N} {f_{p}^{{}} \widetilde{u}_{p}^{{}} (\rho_{k}^{{}} )\left\{ {\frac{{(1 + \nu_{comp}^{{}} )_{{}}^{2} K_{I}^{2} }}{{9\pi S_{{}}^{*2} }} \cdot \frac{1}{N}} \right\}} = f_{p}^{{}} \times G_{Ic}^{{}} \times \psi_{p}^{{}} = G_{{\text{Im}}}^{{}} \times \frac{{f_{p}^{{}} \times \psi_{p}^{{}} }}{{1 - f_{p}^{{}} \times \psi_{p}^{{}} }},$$
(8)

by using the definition of the fracture toughness as GIc = KIc2 × (1-νcomp2)/Ecomp. Here, the contribution of the thermoplastic-particle yield mechanism, ψp, is

$$\psi_{p}^{{}} = 2 \times E_{{{\text{comp}}}}^{{}} \times \frac{{(1 + \nu_{{{\text{comp}}}}^{{}} )_{{}}^{2} }}{{(1 - \nu_{{{\text{comp}}}}^{{}} )_{{}}^{2} }} \times \frac{1}{{9\pi S_{{}}^{*2} }} \times \left\{ {\frac{1}{N}\sum\limits_{k = 1}^{N} {\widetilde{u}_{p}^{{}} (\rho_{k}^{{}} )} } \right\}.$$
(9)

The hydrostatic tension at kth integration point, Sk, can be obtained as follows:

$$S_{k}^{{}} = \frac{{\left( {1 + \nu_{{{\text{comp}}}}^{{}} } \right)K_{I}^{{}} }}{{3\sqrt {\pi \rho_{k}^{{}} } }} = \frac{{S_{{}}^{*} }}{{\sqrt {{{\rho_{k}^{{}} } \mathord{\left/ {\vphantom {{\rho_{k}^{{}} } {\rho_{{}}^{*} }}} \right. \kern-\nulldelimiterspace} {\rho_{{}}^{*} }}} }} = \frac{{S_{{}}^{*} }}{{\sqrt {{k \mathord{\left/ {\vphantom {k N}} \right. \kern-\nulldelimiterspace} N}} }}.$$
(10)

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

$$\Delta G_{t}^{{}} \approx 2w_{0}^{{}} r_{p}^{{}} f_{p}^{{}} = 2\left\{ {\left( {\varepsilon_{l,p}^{{}} - \frac{{\sigma_{Y,p}^{{}} }}{{E_{p}^{{}} }}} \right)\sigma_{Y,p}^{{}} } \right\}r_{p}^{{}} f_{p}^{{}} ,$$
(11)

where rp is the radius of thermoplastic particles. The total toughness improvement can be obtained by ΔGIc = ΔGp + ΔGt.

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

Fig. 5
figure 5

Prediction of multiscalemodel compared to experimental data [61, 62]. Reprinted from Ref. [37]by kind permission of Elsevier

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.

Fig. 6
figure 6

Proposedmultiscale model ofthe CNT/polymer nanocomposites. Reprinted from Ref. [35]by kind permission of Elsevier

The critical SERR of the CNT/epoxy NC, GIc,nc, can be written as follows [35]:

$$G_{{\text{Ic,nc}}}^{{}} = G_{{\text{Ic,m}}}^{{}} + \sum\limits_{{i \in \{ {\text{db,py,po}}\} }}^{{}} {\Delta G_{i}^{{}} } ,$$
(12)

where ΔGi 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:

$$\Delta G_{i}^{{}} = 2 \times \int_{0}^{\rho *(\varphi = \pi /2)} {w_{i}^{{}} d\rho } ,$$
(13)

where wi 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:

$$\Delta G_{i}^{{}} = G_{{\text{Ic,m}}}^{{}} \times \frac{{\lambda V_{f}^{{}} \Delta U_{i}^{{}} }}{{1 - \lambda V_{f}^{{}} \Delta U_{i}^{{}} }},$$
(14)

where Vf is volume fraction of CNT; λ can be determined as:

$$\lambda = \frac{{8H_{{}}^{2} E_{{{\text{nc}}}}^{{}} }}{{9\pi_{{}}^{2} \sigma_{{{\text{cr}}}}^{{2}} r_{{\text{p}}}^{{2}} l_{{\text{p}}}^{{}} }} \times \frac{{\overline{f}}}{{1 - \nu_{{{\text{nc}}}}^{{2}} }},$$
(15)

where Enc and νnc are the elastic modulus and Poisson’s ratio of NC; rp and lp are the radius and length of CNTs; the details for the computation of the critical stress (σcr), 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 (ΔUi) can be obtained as follows:

$$\Delta U_{{{\text{db}}}}^{{}} = 2\pi r_{{\text{p}}}^{{}} l_{{\text{p}}}^{{}} \gamma_{{{\text{db}}}}^{{}} ,$$
(16)
$$\Delta U_{{{\text{py}}}}^{{}} \approx \pi l_{{\text{p}}}^{{}} \times \frac{{\sigma_{{{\text{cr}}}}^{{}} }}{H} \times \frac{{\sigma_{{{\text{Ym}}}}^{{}} }}{{\sqrt 3 G_{{\text{m}}}^{{}} }} \times r_{{\text{i}}}^{{2}} \times \left\{ {\frac{{\frac{\sqrt 3 }{H}\frac{{\sigma_{{{\text{cr}}}}^{{}} }}{{\sigma_{{{\text{Ym}}}}^{{}} }} - \left( {1 - \frac{1}{{n_{{\text{m}}}^{{}} }}} \right)}}{{\frac{1}{{n_{{\text{m}}}^{{}} }} \times \frac{{\sigma_{{{\text{Yi}}}}^{{}} }}{{\sigma_{{{\text{Ym}}}}^{{}} }} \times \left( {\frac{{\varepsilon_{{{\text{Ym}}}}^{{}} }}{{\varepsilon_{{{\text{Yi}}}}^{{}} }}} \right)_{{}}^{{n_{{\text{m}}}^{{}} }} \left[ {\left( {\frac{{r_{{\text{i}}}^{{}} }}{{r_{{\text{p}}}^{{}} }}} \right)_{{}}^{{2n_{{\text{m}}}^{{}} }} - 1} \right] + \frac{1}{{n_{{\text{m}}}^{{}} }}}}} \right\}_{{}}^{{1/n_{{\text{m}}}^{{}} }} ,$$
(17)

where γdb is the interfacial fracture energy; σYm and σYi are the yield strength of matrix and interphase, respectively; εYm and εYi the yield strain of matrix and interphase, respectively; Gm and nm 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]:

$$\Delta G_{{{\text{po}}}}^{{}} = \left\{ {\begin{array}{*{20}l} {\frac{{V_{{\text{f}}}^{{}} l_{{\text{p}}}^{{2}} \tau_{{\text{i}}}^{{}} }}{{12r_{{\text{p}}}^{{}} }}{\text{ if }}l_{{\text{p}}}^{{}} < l_{{\text{c}}}^{{}} } \hfill \\ {\frac{{V_{{\text{f}}}^{{}} l_{{\text{c}}}^{{2}} \tau_{{\text{i}}}^{{}} }}{{12r_{{\text{p}}}^{{}} }}{\text{ if }}l_{{\text{p}}}^{{}} > l_{{\text{c}}}^{{}} } \hfill \\ \end{array} } \right.,$$
(18)

where the τi is the interfacial shear strength; the critical length of CNTs can be determined as [43]:

$$l_{{\text{c}}}^{{}} = \frac{{r_{{\text{p}}}^{{}} \sigma_{{{\text{up}}}}^{{}} }}{{\tau_{{\text{i}}}^{{}} }},$$
(19)

where σup 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 Eint = χEmat and σYi = χσYm. 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.

Fig. 7
figure 7

Experimental confirmation: a Case I [19], b Case II [64], c Case III [65], and d Case IV [63].Reprinted from Ref. [35]by kind permission of Elsevier

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 FE2 approach, the systematic reduction techniques, and the fracture mechanics theory should be merged for the multiscale fracture mechanics analysis.

References

  1. 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)

    Article  Google Scholar 

  2. 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)

    Article  Google Scholar 

  3. 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)

    Article  Google Scholar 

  4. 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)

    Article  Google Scholar 

  5. 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)

    Article  Google Scholar 

  6. 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)

    Article  Google Scholar 

  7. 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)

    Article  Google Scholar 

  8. 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)

    Google Scholar 

  9. 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)

    Article  Google Scholar 

  10. 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)

    Article  Google Scholar 

  11. 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)

    Article  Google Scholar 

  12. 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)

    Article  Google Scholar 

  13. 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)

    Article  Google Scholar 

  14. 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)

    Article  Google Scholar 

  15. 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)

    Article  Google Scholar 

  16. 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)

    Article  Google Scholar 

  17. 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)

    Article  Google Scholar 

  18. 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)

    Article  Google Scholar 

  19. 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)

    Article  Google Scholar 

  20. 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)

    Article  Google Scholar 

  21. 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)

    Article  Google Scholar 

  22. 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)

    Article  Google Scholar 

  23. 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)

    Article  Google Scholar 

  24. 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)

    Article  Google Scholar 

  25. 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)

    Article  Google Scholar 

  26. 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)

    Article  Google Scholar 

  27. 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).

  28. B. Wetzel, P. Rosso, F. Haupert, K. Friedrich, Epoxy nanocomposites–fracture and toughening mechanisms. Eng. Fract. Mech. 73(16), 2375–2398 (2006)

    Article  Google Scholar 

  29. Y.L. Liang, R. Pearson, The toughening mechanism in hybrid epoxy-silica-rubber nanocomposites (HESRNs). Polymer 51(21), 4880–4890 (2010)

    Article  Google Scholar 

  30. 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)

    Article  Google Scholar 

  31. 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)

    Article  Google Scholar 

  32. 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)

    Article  Google Scholar 

  33. 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)

    Article  Google Scholar 

  34. 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)

    Article  Google Scholar 

  35. H. Shin, Multiscale model to predict fracture toughness of CNT/epoxy nanocomposites. Compos. Struct. 272, 114236 (2021)

    Article  Google Scholar 

  36. 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)

    Article  Google Scholar 

  37. 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)

    Article  Google Scholar 

  38. M.M. Shokrieh, A. Zeinedini, Effect of CNTs debonding on mode I fracture toughness of polymeric nanocomposites. Mater. Design 101, 56–65 (2016)

    Article  Google Scholar 

  39. 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)

    Article  Google Scholar 

  40. 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)

    Article  Google Scholar 

  41. 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)

    Article  Google Scholar 

  42. M. Quaresimin, K. Schulte, M. Zappalorto, S. Chandrasekaran, Toughening mechanisms in polymer nanocomposites: from experiments to modelling. Compos. Sci. Technol. 123, 187–204 (2016)

    Article  Google Scholar 

  43. H.D. Wagner, P.M. Ajayan, K. Schulte, Nanocomposite toughness from a pull-out mechanism. Compos. Sci. Technol. 83, 27–31 (2013)

    Article  Google Scholar 

  44. B. Lauke, On the effect of particle size on fracture toughness of polymer composites. Compos. Sci. Technol. 68(15–16), 3365–3372 (2008)

    Article  Google Scholar 

  45. 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)

    Article  Google Scholar 

  46. A.G. Evans, S. Williams, P.W.R. Beaumont, On the toughness of particulate filled polymers. J. Mater. Sci. 20(10), 3668–3674 (1985)

    Article  Google Scholar 

  47. 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)

    Article  Google Scholar 

  48. S. Yang, M. Cho, Scale bridging method to characterize mechanical properties of nanoparticle/polymer nanocomposites. Appl. Phys. Lett. 93, 043111 (2008)

    Article  Google Scholar 

  49. S. Yu, S. Yang, M. Cho, Multi-scale modeling of cross-linked epoxy nanocomposites. Polymer 50, 945–952 (2009)

    Article  Google Scholar 

  50. 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)

    Article  Google Scholar 

  51. G.M. Odegard, T.C. Clancy, T.S. Gates, Modeling of the mechanical properties of nanoparticle/polymer composites. Polymer 46, 553–562 (2005)

    Article  Google Scholar 

  52. 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)

    Article  Google Scholar 

  53. 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)

    Article  Google Scholar 

  54. 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)

    MATH  Article  Google Scholar 

  55. 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)

    Article  Google Scholar 

  56. 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)

    Article  Google Scholar 

  57. 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)

    Article  Google Scholar 

  58. 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)

    Article  Google Scholar 

  59. 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)

    Article  Google Scholar 

  60. 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)

    Article  Google Scholar 

  61. 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)

    Article  Google Scholar 

  62. 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).

  63. 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)

    Article  Google Scholar 

  64. 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)

    Article  Google Scholar 

  65. 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)

    Article  Google Scholar 

  66. H. Shin, K. Baek, J.-G. Han, M. Cho, Homogenization analysis of polymeric nanocomposites containing nanoparticulate clusters. Compos. Sci. Technol. 138, 217–224 (2017)

    Article  Google Scholar 

  67. 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)

    Article  Google Scholar 

  68. 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)

    Article  Google Scholar 

  69. 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)

    Article  Google Scholar 

  70. 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)

    Article  Google Scholar 

  71. 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)

    Article  Google Scholar 

Download references

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

Authors

Corresponding author

Correspondence to Hyunseong Shin.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

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

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

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

Keywords

  • Multiscale analysis
  • Fracture toughness
  • Micromechanics
  • Molecular dynamics