epoxy blends by the plastic yield of toughening agents: A multiscale analysis

Accepted Manuscript Fracture toughness enhancement of thermoplastic/epoxy blends by the plastic yield of toughening agents: A multiscale analysis Hyunseong Shin, Byungjo Kim, Jin-Gyu Han, Man Young Lee, Jong Kyoo Park, Maenghyo Cho PII:

S0266-3538(16)30877-6

DOI:

10.1016/j.compscitech.2017.03.028

Reference:

CSTE 6713

To appear in:

Composites Science and Technology

Received Date: 3 August 2016 Revised Date:

18 February 2017

Accepted Date: 17 March 2017

Please cite this article as: Shin H, Kim B, Han J-G, Lee MY, Park JK, Cho M, Fracture toughness enhancement of thermoplastic/epoxy blends by the plastic yield of toughening agents: A multiscale analysis, Composites Science and Technology (2017), doi: 10.1016/j.compscitech.2017.03.028. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT

Fracture Toughness Enhancement of Thermoplastic/Epoxy Blends by the Plastic Yield of Toughening Agents: A Multiscale Analysis

and Maenghyo Choa,* a

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Hyunseong Shina, Byungjo Kima, Jin-Gyu Hana, Man Young Leeb, Jong Kyoo Parkb,

Department of Mechanical and Aerospace Engineering, Seoul National University, San

56-1, Shillim-Dong, Kwanak-Ku, Seoul 151-742, South Korea

The 4th R&D Institute-4, Agency for Defense Development, Daejeon 305-600, South

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b

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Korea

Abstract

We propose a strategy of multiscale analysis to predict enhancements in fracture toughness of thermoplastic/epoxy blends by the plastic yield of toughening agents. As

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main mechanisms of toughness enhancement of thermoplastic/epoxy blends, we considered plastic deformation in the material near the macroscopic crack tip, and particle bridging in the crack wake. The proposed multiscale model is validated through

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the reported experimental data. Using the proposed multiscale model, some useful

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guidelines for the proper selection of toughening agent were devised.

Keywords:

Polymer-matrix composites (PMCs); Fracture toughness; Multiscale modeling

* Corresponding author, [email protected] 1

ACCEPTED MANUSCRIPT 1. Introduction A highly crosslinked epoxy system has excellent properties such as good thermal stability, creep resistance, excellent adhesion properties, and relatively high modulus.

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However, the high crosslink density leads to low fracture toughness, and inferior impact strength, which limit their application in high performance areas such as the automotive, aerospace, and defense industries. Therefore, it has been a challenging issue to improve

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the fracture toughness of an epoxy by modifying the resin with a secondary phase such as rubber [1,2], thermoplastic [3-7], and inorganic particles [8-10]. In particular,

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thermoplastic/epoxy blends are widely employed, because the high toughness enhancement through modification by the thermoplastic was not accompanied by critical reduction in the elastic modulus [3,5]. Despite the importance of thermoplastictoughened epoxy systems in various sectors, there have been insufficient studies to

[6].

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predict the toughness enhancement by thermoplastic polymers as the secondary phase

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EP

Based on the experimental observations, it was reported that the ductile thermoplastic

Fig. 1. The proposed multiscale approach to predict the fracture toughness enhancements induced by toughening mechanisms (plastic yield of thermoplastic particle and particle bridging) of the thermoplastic/epoxy blends 2

ACCEPTED MANUSCRIPT particle plays two key roles in toughening mechanisms of thermoplastic/epoxy blends (the plastic deformation in the material near the macroscopic crack tip and the particle bridging in the crack wake [11,12]), as shown in Fig. 1. For the first mechanism,

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multiscale approach [8-10] is useful when developing an analytical model with an infinite number of particles embedded in the matrix domain. The density of energy dissipated by the damage mechanism of the representative volume element (RVE) near

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the crack tip can be described by micromechanics models. The fracture toughness enhancement of the composites can be obtained quantitatively through the J-integral

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near the macroscopic crack tip [13]. However, multiscale models for the toughness enhancements of thermoplastic/epoxy blends have not been reported, unlike rigid nanoparticles [8-10]. For this reason, we focused on the development of a model to describe the fracture toughness improvement by thermoplastic particle yield near the

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crack tip. The main objective of the present paper is to quantify the toughness enhancement due to the plastic yield of the thermoplastic particle. With this motivation, we conducted the formulation to quantify the plastic dissipation

EP

energy of thermoplastic particles near the macroscopic crack tip. To quantify the plastic dissipation energy of thermoplastic particles, molecular dynamics (MD) simulation was

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then conducted to obtain the curve of nonlinear hydrostatic stress versus volumetric strain. The proposed multiscale model is validated through the reported experimental data. Based on results obtained using the multiscale model, design guidelines regarding the proper selection of toughening agents are proposed for enhancing the fracture toughness of thermoplastic/epoxy blends.

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ACCEPTED MANUSCRIPT 2. Review of multiscale strategy to describe fracture toughness enhancement by microscopic damage mechanisms near the macroscopic crack tip A multiscale strategy to calculate the dissipated energy by considering the damage

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mechanism on the nanoscale near the crack tip of macroscopic structures has been developed [2,10]. Through the theory of linear fracture mechanics under the assumption of plane strain conditions, hydrostatic components of macroscopic stress fields, Σh, are

2(1 + ν comp ) K I 3 2πρ

cos

φ

2

(1)

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Σh =

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described as follows:

where KI and νcomp are the stress intensity factor of the macroscopic stress field and the Poisson’s ratio of the composites, respectively [10]. ρ and ϕ are geometric quantities that can be found in Fig. 1. Using the multiscale approach, the microscale RVE is subjected to the hydrostatic tension, S, that satisfies force equilibrium with Σh near the

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crack tip, as shown in Fig. 1. By taking J-integral near the crack tip, the enhancement in fracture toughness due to the ith mechanism, ∆Gi, can be determined as follows [10,13]:

∆Gi = 2 × ∫

ρ * (φ =π 2)

ui dρ

(2)

EP

0

where ui is the microscopic dissipation energy produced by ith damage mechanisms in

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the RVE. ρ* is the critical radius of the yield domain such that ui is zero in the range of ρ>ρ*.

The link between the macroscopic quantities and the microscopic quantities is made by considering the volume average of microscopic quantities defined in the RVE:

{Σ, E} =

1 Y 4

∫ {σ, ε} dV Y

(3)

ACCEPTED MANUSCRIPT where Y is domain of the RVE. By means of the micromechanics model, a RVE that is subjected to hydrostatic tension, S, can be approximated as an equivalent inclusion model. The Mori-Tanaka model [14] assumes that the only single inhomogeneity is

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embedded in the uniform matrix domain by neglecting interactions between inhomogeneities. Due to this reason, it has been reported that the Mori-Tanaka model is more applicable to the low volume fraction systems (<6%) than high volume fraction

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systems [8-10]. When we consider the cubic array at this low volume fraction ranges, the allowable interparticulate distances are nearly "2.12×rp (radius of particle)" [9]. In

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the following section, we will propose the formulation used to calculate the energy produced by the thermoplastic particle yield, up, and its corresponding fracture toughness enhancement, ∆Gp.

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3. Thermoplastic particle yield-induced toughness enhancement 3.1. Description of the plastic energy from thermoplastic particles 3.1.1. Mechanical description of the thermoplastic/epoxy blends system

EP

It is assumed that the thermoplastic particles are well phase-separated from the epoxy domain as a continuous and homogeneous phase, as shown in the SEM image in Fig. 1

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[6]. Reportedly, the spherical particulate diameter of the thermoplastic polyethersulfone (PES) phase, which is employed as toughening agent in this study, of the epoxy-rich composites is set as 0.1-1.5µm [6].

3.1.2. General solutions of displacement and stress fields of the particle and matrix The hardening behavior of the matrix was simplified as the following linear elasto-

5

ACCEPTED MANUSCRIPT power law [15,16]:  ε (el),m = E − 1 σ (el),m if σ (el),m ≤ σ m Ym   (pl),m (pl),m n m σ  ε (pl),m  if σ = ≥ σ Ym   σ Ym   ε Ym  

(4a, b)

is the equivalent operator, and Em, nm, εYm and σYm are Young’s modulus,

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where

hardening exponent, yield strain, and yield stress, respectively, of the matrix. (·)(el),* and (·)(pl),* are quantities during the elastic and the plastic deformation, respectively, and

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(·)(*),m and (·)(*),p are those of matrix and particle, respectively. The stress values and

{σ

(pl),m rr

, σ

(pl),m

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displacement fields in the plastic zone (r < Rp) of the matrix can be expressed as: nm  2 1 ε Ym  C1    , u r(pl),m =  C 2 − C1 n m r −3/ nm , C1 r − 3/ nm ,    3 2 r 2  σ Ym    

}

(5)

where r is the distance from the center of particle (Fig. 1). Outside the plastic core (r >

expressed as:

{σ

( el ), m rr

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Rp), the stress components and displacement fields of linear elastic matrix zone can be

 C3   C3  C C C 4   , σ θθ( el ), m , u r( el ), m } =  E m  −2 3 4 + 3 4  , Em   , C3 r + 2  r (1 + ν m )  r    1 − 2ν m  1 − 2ν m r (1 + ν m ) 

(6)

EP

where νm is Poisson’s ratio of the matrix [17], and C1~C4 are constants. To avoid the singular solution of the equivalent stress at the center of particle (r = 0), we introduced

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the curve of nonlinear hydrostatic stress versus volumetric strain as a constitutive model. We defined the radial component of strain, ε rr( pl ), p , as one of the unknown variables, C5, as follows:

u r( pl ), p = ε rr( pl ), p r = C5 r

6

(7)

ACCEPTED MANUSCRIPT 3.1.3. Determination of coefficients (C1 to C5) and radius of the plastic zone (Rp) 3.1.3.1. The partially yielded domain (rp < Rp < rm) In this domain, yielding occurs in the entire particle and a part of the matrix. The

σ rr( el ), m

r = Rp

= σ rr( pl ), m

r = Rp

, σ rr( pl ), m

r = rp

= σ rr( pl ), p

r = rp

, u r( el ), m

r = Rp

= u r( pl ), m

r = Rp

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continuity in displacement and stress is satisfied as follows: , u r( pl ), m

r = rp

= u r( pl ), p

r = rp

(8)

From the definition of Rp and traction boundary condition, the additional boundary

σ rr( pl ), m

r = Rp

= σ Ym , σ rr( el ), m

= σ Ym R 3/p nm =

(9)

 2 3 ε Ym 3  Rp  σ Ym R 3/p nm nm rp−3/ nm + σ h( ,plMD), p  ε v =   3 2 rp3  

(

)

1 ε Ym 3 Rp 2 rp3

(10a-e)

3/ n m     3 ε Ym 3   R p   ( pl ), p  σ ε R − S − 1 + =       h , MD  v p  3   r 2 r p   p     

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 S  C = +2 3 4  (1 − 2ν m ) rm (1 + ν m )   Em −1   2 E  1 1    2 m σ Ym nm =  3 − 3    (1 + ν m )  rm R p    3  =

=S

EP

 C1   C 2   C  3    C 4   C  5 

r = rm

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The solutions of the coefficients (C1-C5) are then

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conditions are defined as:

where Rp can be obtained from the following equation using the bisection method

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withint the interval (rp, rm):

f ( R p ) = R 3p −

2C 4 ( R p )

ε Ym − 2 C 3 ( C 4 ( R p ))

= 0, R p ∈ ( rp , rm )

(11)

3.1.3.2. The fully yielded domain (Rp = rm) In this domain, yielding occurs in the entire particle and the entire matrix. The continuity in displacement and stress and the traction boundary condition are satisfied 7

ACCEPTED MANUSCRIPT by following equations: σ rr( pl ), m

r = rp

= σ rr( pl ), p

r = rp

, u r( pl ), m

r = rp

= u r( pl ), p

r = rp

, σ rr( el ), m

r = rm

=S

(12)

The solutions C1-C5 are   

 2  ), p C1 n m rp− 3/ nm + σ h( ,plMD ε v = 3C 5 )  ( 3 

1 nm

,

where the solutions of C5 (C5,1 < C5,2) can be obtained from

(14a-c)

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)

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C 5,1 ( S < S C = C )  5 5,double root C5 =  C 5,2 (S > S C5 = C5,double root ) 1 nm    2C5  2  {C 5,1 , C5,2 } = C 5 3 σ Ym ⋅  ε  nm 1 − f p1 nm = S − σ h( ,plMD), p (ε v = 3C 5 )  Ym   

(

(13)

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  2 r 3C {C1 , C 2 } =  εp 5   Ym

where S has double roots at C5 = C5,double root, and fp is volume fraction of particle.

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3.1.4. Quantification of the density of dissipated plastic energy

The density of plastic energy dissipated by the particle yield in the composites, up, can be expressed as the following form:

EP

up = U p ×

3 fp

4π rp3

= f p u% p (ε v )

(15) ε v = 3 C5

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where Up and u% p are the dissipated plastic energy and the dissipated plastic energy density in the particle, respectively.

3.2. Modeling of toughness enhancement induced by thermoplastic particle yield 3.2.1. Determination of domain in which yielding occurs in a particle The critical radii of the partially yield domain, ρ*, and the fully yielded domain, ρ**, can be described by the following equation because it is preferable for the contour line 8

ACCEPTED MANUSCRIPT for the J-integral to be parallel to the y-axis (φ=π/2) [13]: ρ* =

2(1 + ν comp ) 2 K I2 9π S

*2

⋅ cos 2

ϕ 2

(1 + ν comp ) 2 K I2

=

9π S

ϕ =π 2

*2

, ρ ** =

(1 + ν comp ) 2 K I2 9π S **2

(16)

where S* and S** are the critical surface traction forces such that Rp is converges to rp

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and rm, respectively, which can be obtained by equations in section 3.1.3.1.

3.2.2. Numerical analysis of toughness enhancement

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To compute the contour integral (Eq. (2)) of up, a numerical integration is employed

∆G p = 2 × ∫

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to compute the contour integral as follows: ρ * (φ =π / 2 )

0

N

u p d ρ = 2 × ∑ u p ( ρ k )∆ρ

(17)

k =1

where N is the number of integration points, ρk=k×∆ρ and ∆ρ=ρ*/N. By introducing Eqs. (15) and (16) into Eq. (17), the toughness enhancement, ∆Gp, can be represented by (18)

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N  (1 + ν comp ) 2 K I2 1  f p ×ψ p ∆ G p = 2 × ∑ f p u% p ( ρ k )  ⋅  = f p × G Ic ×ψ p = G Im × *2 9π S N  1 − f p ×ψ p k =1 

by using the definition of the fracture toughness as GIc=KIc2×(1-νcomp2)/Ecomp. Here, the

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contribution of the thermoplastic-particle yield mechanism, ψp, is ψ p = 2 × E comp ×

(1 + ν comp ) 2 (1 − ν comp )

2

×

1 1 × *2 9π S N

N

∑ u% k =1

p

 (ρk ) 

(19)

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The elastic moduli (Ecomp and νcomp) of the composites is determined by the MoriTanaka model [14], while u% p ( ρ k ) can be determined by the proposed formulation in section 6.1. The hydrostatic tension at kth integration point, Sk, can be determined as Sk =

(1 + ν

comp

)K

3 πρ k

I

=

9

S*

ρk ρ

*

=

S* k N

(20)

ACCEPTED MANUSCRIPT 4. Thermoplastic particle bridging-induced toughness enhancement Douglass et al. [2] proposed the toughness enhancement due to rubbery particle bridging mechanism. We quantified the thermoplastic particle bridging-induced

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toughness enhancement in a similar way. Even though the fracture toughness of PES has not been reported, the elongation limit (εl,p) and tensile strength (σY,p) were reported as 6-80% and 67.6-95.2 MPa, respectively [18]. Here, the Young’s modulus (Ep) is

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determined by MD simulation results (Table 1). Using the linear elasto-perfect plastic model, the dissipated energy density of PES due to failure, w0, can be estimated as 2.33-

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72.78 MPa. Then, toughness enhancement due to this mechanism can be obtained as the following equation [2]:

σY,p   ∆ Gt ≈ 2 w0 rp f p = 2   ε l , p −  Ep  

   σ Y , p  rp f p  

(21)

where rp is the radius of thermoplastic particles. We used the diameter of thermoplastic

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particles (2×rp) as 0.1-1.5µm [6] due to the lack of information to validate. The total toughness enhancement can be obtained by ∆GIc = ∆Gp + ∆Gt.

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Table 1. Comparison of linear elastic properties and glass transition temperature (Tg) of PES obtained from molecular dynamics simulation with experimental literature. Present study

Experiment

G (GPa)

ν

ρ (g/cm3)

Tg (K)

E (GPa) [19]

ρ (g/cm3) [18]

Tg (K) [19]

2.68

0.99

0.36

1.32

499

2.8±0.1

1.37-1.46

498.19

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E (GPa)

5. Molecular modeling and molecular dynamics simulation of bulk PES 5.1. Molecular modeling of bulk PES For the molecular modeling and relaxation simulation, commercial molecular 10

ACCEPTED MANUSCRIPT simulation software Material Studio 5.5 was used [20], and the polymer consistent force-field (PCFF) [21] was employed to describe both the inter- and intra-atomic interactions. The chemical structure of PES is given in Fig. 2. In this study, a PES chain

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containing 60 repeating units was used as an end group. The initial target density of the unit cell containing five PES chains was set to 0.01 g/cm3 for modeling of a dilute condition and periodic boundary conditions were applied in all directions. The initial

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molecular structure of PES was gradually compressed and equilibrated, using the NVT ensemble (with constant number of particles, volume, and temperature) at 300K

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followed by the NPT ensemble (with constant number of particles, pressure, and temperature) at 300K and 1atm. The simulation details to predict the elastic properties

EP

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of unit cell can be found elsewhere [22,23].

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Fig. 2. Molecular modeling of PES: (a) chemical structure of PES, (b) monomer of PES molecule, and (c) configuration of amorphous unit cell with five PES chains.

5.2. Simulation of triaxial tensile loading and unloading in bulk PES MD simulation was employed to obtain the triaxial tensile loading and unloading simulation response of PES. A small displacement was applied to the atoms on the 11

ACCEPTED MANUSCRIPT surface of the unit cell according to the predefined strain rate of 108/s along every axis, and the system was relaxed via the NVT ensemble simulation. After the relaxation process, the atoms on the surface of the unit cell were displaced again, followed by the

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same relaxation process. In each relaxation process, the time-averaged stress tensors calculated from the virial theorem were stored as the stress components corresponding to the strain state. This process was repeated until the volumetric strain of the unit cell

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reached 120%. After the loading simulation, unloading simulations followed at strains of 15%, 45%, 90%, and 120%. During the unloading simulation, the atoms on the

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surface of the unit cell were displaced according to the predefined strain rate of -108/s along every axis. We used the averaged stress values over three-times MD simulations for different initial atomistic velocities distributions to reduce the model uncertainties that stem from the initial atomistic velocities distributions. For all the ensemble

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simulations, an open source code for large-scale atomic and molecular simulation tool (LAMMPS) [24] was used with the same force field (PCFF).

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6. Simulation results and discussion

6.1. Molecular dynamics simulation results of bulk PES

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To validate the unit cell model of PES, we compared the Young’s modulus (E), density (ρ), and glass-transition temperature (Tg) of PES obtained by MD simulation with experimental data reported in the literature [19], as shown in Table 1. Here, the Young’s modulus and the glass transition temperature are obtained from the “Unreinforced PES” of Table 2 and “PES + 0% CNF” of Table 6 in the Ref. 19. The obtained nonlinear hydrostatic stress-volumetric strain curve is as shown in Fig. 3,

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ACCEPTED MANUSCRIPT which shows trends similar to those reported in the literature [25]. MD data were fitted as the well-fitting functional form by using the least square method:    ε exp  − 0.5 ×  ln  v σ h , MD (ε v ) = εv    a 2 

 1 ×  a3

  

2

   

(22)

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a1

where the coefficients (a1, a2, and a3) are listed in Table 2 for each strain rate. As shown in Fig. 3, the simulation results indicate that the secant modulus on the unloading path is

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almost constantly held and the volumetric strain recovers near the zero at the free hydrostatic stress state. Therefore, the dissipated plastic energy density of particle can

∫

εv 0

σ h , M D (ε ) d ε −

1 σ h , M D (ε v ) × ε v 2

(23)

EP

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u% p ( ε v ) =

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be simply described as follows:

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Fig. 3. Stress-strain response of triaxial tensile simulation of PES: (a) loading and unloading curves at 108/s of strain rate, and (b) effects of strain rates.

6.2. Experimental validation of the proposed multiscale model To validate the proposed multiscale model, we compared the normalized toughness of thermoplastic/epoxy blends with experimental data as shown in Fig. 4. The values of GIm, Em, νm, σYm, and nm were set as 150 J/m2, 3.36 GPa, 0.38, 68 MPa, and 3.4, 13

ACCEPTED MANUSCRIPT respectively [8,26]. A satisfactory agreement with experimental data is evident as shown in Fig. 4. The error bar, which is determined by the lower bound and the upper bound of predictions, of multiscale model is propagated from the uncertainties of mechanical

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properties of PES (described in section 4) in the prediction of thermoplastic particle bridging-induced toughness enhancement mechanisms. As shown in Fig. 4, the magnitudes of error bars are negligibly small (as 1.60% and 2.87% of the predicted

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fracture toughness at 5% and 10% volume fraction systems, respectively), even though the experimental values (described in section 4) cover a large range due to the low

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sensitivity on the prediction of the fracture toughness enhancements.

Table 2. The coefficients of the fitted curves for different strain rate and their corresponding toughness enhancements due to the plastic yield mechanisms of thermoplastic particles Fitted curve Strain rate 109/sec

a0 [MPa]

a1 0.527

56.694

0.523

7

54.186

0.520

10 /sec 10 /sec 0

10 /sec (extrapolated)

R

1.276

∆Gp/GIm

Vol.f=5%

Vol.f=10%

0.995

0.109 (-8.4%)

0.207 (-9.2%)

1.229

0.992

0.119 (0)

0.228 (0)

1.216

0.993

0.121 (1.7%)

0.233 (2.2%)

0.164

0.327

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63.511

8

a2

2

-

EP

Here, each reference data are determined by Fig. 2 of Ref. 27 and RHMW data in Fig. 6 of Ref. 28. In the specimen of both cases, the phase separation between PES and

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epoxy is investigated. Especially, in the Ref. 27, authors discussed that the blended PES plays role as a stress concentrator. In the case of GIc of Ref. 28, the experimental data (KIc) of Ref. 28 and the simple rule, GIc=KIc2×(1-νcomp2)/Ecomp, are used. Here, Ecomp and νcomp are obtained from the Mori-Tanaka model, and the material properties of each phase can be found in Table 1 (particle) and section 6.2 (matrix). The error bar of Ref. 28 can be found in the corresponding paper directly. It is noteworthy that the 14

ACCEPTED MANUSCRIPT configuration is not related to the prediction of fracture toughness enhancements

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because GIc of the PES/epoxy blends are material property.

Fig. 4. Validation of multiscale model compared to experimental data [27,28]. Here, the "Multiscale model (*)" is based on our MD simulation results with 108/s strain rate. The "Multiscale model (**)" is the expected upper limit value (at the 100/s strain rate) influenced by the sensitivity on the strain rate of MD simulations. (The dotted line is just a power law fitted curve of experimental reference data through the GIc/GIm=1 at 0% volume fraction.)

The fracture toughness enhancements for different strain rates are listed in Table 2.

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Even though the deviation by changing the one order of magnitude (from 108/s to 109/s) is near the 10% deviations, the sensitivity on the strain rate of the fracture toughness enhancements is reduced as the order of magnitude of the strain rate decreases (near the

EP

2% deviations from 108/s to 107/s). We listed the predicted toughness enhancements

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within the actual experimental range (~100/s) in Table 2, using the linear regression on the “log10(dε/dt)~∆Gp/GIm”. Even though it is expected that the sensitivity of “∆Gp/GIm” on the “log10(dε/dt)” decreases as the “log10(dε/dt)” decreases, we apply the linear regression method to validate the prediction by the most extremely influenced case. The predicted toughness enhancements exist within the satisfactory range of experimental data of Fig. 4.

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ACCEPTED MANUSCRIPT 6.3. Dependence of the toughness enhancement on mechanical properties of thermoplastic particles The effect of the nonlinear hydrostatic stress-volumetric strain response of

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thermoplastic particles on the toughness enhancement of the thermoplastic/epoxy composites was investigated to determine which materials are appropriate for the particles. Two parameters, mx and my, were introduced to simplify the description of

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various hydrostatic stress-volumetric strain responses of the particles. The stress-strain curves are magnified by mx and my along strain and stress direction, respectively, as

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shown in Fig. 5 (a). For fixed fracture energy of thermoplastic particle (mx×my=1), parametric studies are conducted to provide design guideline. The bulk modulus of the thermoplastic particle is determined by Kpar(mx, my)=Kpar,MD×(my/mx), and Kpar,MD is determined as 3.19GPa by Table 1 (e.g., 2.22GPa and 4.98GPa at mx=1.2 and mx =0.8,

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EP

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respectively, when the fracture energy of the thermoplastic particle is fixed).

Fig. 5. An influence of mechanical properties of thermoplastic particles: (a) curve of hydrostatic stress versus volumetric strain of thermoplastic particles magnified view along strain direction (constant fracture energy) and (b) sensitivities of predicted fracture toughness enhancements for varied representative quantities (mx and my) at 5% of volume fraction system. Here, the representative quantities imply mx (for fixed my case), my (for fixed mx case), and mx (for mx ×my=1).

As shown in Fig. 5 (b), it can be concluded that optimal selection of particles to 16

ACCEPTED MANUSCRIPT enhance the fracture toughness through the plastic yield of the particles can be achieved at high mx and low my. Even though the fracture energy of particle is fixed, the toughness enhancement can be improved dramatically by increasing mx and decreasing

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my. As the ultimate stress (represented by my) of the thermoplastic particles increases, the critical radius of yielded domain decreases (Table 3). This is the origin of increasing the fracture toughness enhancements as the ultimate stress of the thermoplastic particles

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decreases in spite of the fixed fracture energy of the employed thermoplastic particle.

5.00

Magnitude

∆Gp/GIm

Critical radius of yielded domain Partially Fully yielded yielded

Density of dissipated plastic energy density [MPa] Partially Fully yielded yielded

mx

my

0.80

1.25

0.096

0.664

0.447

0.360

143.1

0.90

1.11

0.107

0.820

0.441

8.773

152.1

1.00

1.00

0.119

1.000

0.438

12.36

162.8

1.10

0.91

0.127

1.205

0.437

12.66

166.1

0.83

0.131

1.436

0.436

12.26

165.4

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1.20

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Volume fraction (%)

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Table 3. Effect of stress-strain characteristics on the improvement in fracture toughness of a thermoplastic/epoxy composites at fixed fracture energy. (The critical radius of the yielded domain is normalized by the critical radius of the partially yielded domain at mx=my=1.)

6.4. An influences of the number of integration points

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To compute Eq. (17), we used the numerical integration using the number of integration points. Here, the error analysis using the parametric studies on the number of integration points, where the error is defined as follows: Error =

G Ic( N ) − G Ic(10,000 ) where N is the number of integration points G Ic(10,000)

(24)

As shown in Table 4, When we consider the “sufficiently accurate” as 0.1% error, we 17

ACCEPTED MANUSCRIPT can determine a lower limit as the 500 number of integration points. Even though it needs a large number of integration points, the computational time was less than 10 seconds when we used Intel Core i5 processor.

The number of integration points (N) 10,000

5,000

1,000

500

100

10%

0.000%

0.000%

0.011%

0.008%

0.040%

20%

0.000%

0.017%

0.082%

0.090%

0.112%

50

10

0.235%

1.552%

1.581%

5.295%

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Volume fractions

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Table 4. Error computations for different number of integration points

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6.5. Discussions on the bonding conditions

When we consider the plastic yield of thermoplastic particle as a main toughening mechanism, the prediction with perfect bonding condition can be regarded as the overestimated value of the predicted toughness enhancements. Meanwhile, the

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interfacial debonding with the elastic deformation of thermoplastic particle can be regarded as the underestimated value of the predicted toughness enhancements. For the elastic particle, the model to predict toughness enhancement due to interfacial

EP

debonding-induced void growth mechanism was developed [8], which can be computed by GIc=fp×(ψdb+ψpy)/{1-fp×(ψdb+ψpy)}, and nm  2 γ 1 + ν comp E comp 1 + ν comp E comp   σ cr 8  −1 db ψ = × × × × × × − + , , 3 C n 1 }    db py h m 2 2 π C h 1 − ν comp σ Ym n m Gm    3π rp 1 − ν comp σ cr × (C h )

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{ψ

   

(25)

Here, the detail computations of Ch and σcr can be found in Ref. 8. We predict the fracture toughness enhancement due to the interfacial debonding-induced toughness enhancements by conducting the parametric studies (γdb=0.01J/m2 and γdb=0.05J/m2). The ranges of γdb are similar values to the acknowledged surface energy of polymeric materials. The particulate diameter is simply determined as 0.8µm by the mean value of 18

ACCEPTED MANUSCRIPT experimentally observed ranges of particulate diameter, 0.1-1.5µm [6]. As a result, the interfacial debonding with the elastic deformation of thermoplastic particle cases show the similar order of magnitudes predictions within 45% deviations from the perfect

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bonding condition. Table 5. An influence of bonding conditions on the prediction of fracture toughness enhancements GIc/GIm Perfect bonding condition (proposed approach

Interfacial debonding with elastic deformation of particle [8] γdb=0.01J/m2

1.138

1.078

Vol.f = 10%

1.264

1.167

1.085

1.182

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Vol.f = 5%

γdb=0.05J/m2

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System

7. Conclusion

A multiscale model was developed to describe the fracture toughness enhancement of

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thermoplastic/epoxy blends by the plastic yield of toughening agents. As the main mechanisms of toughness enhancement of thermoplastic/epoxy blends, we considered plastic deformation in the material near the macroscopic crack tip, and particle bridging

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in the crack wake. The proposed multiscale model shows a satisfactory agreement with

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experimental data. From the viewpoint of the plastic yield of toughening agents, the proposed multiscale model can provide design guidelines to enhance the fracture toughness of thermoplastic/epoxy blends. In future works, we will investigate the influences of the mechanical interactions between the microcracks inside the continuum phase and the established toughening mechanisms on the overall fracture toughness, which has not been considered in the current multiscale-multimechanism approaches [810]. 19

ACCEPTED MANUSCRIPT Acknowledgements This work was supported by the Defense Acquisition Program Administration and Agency for Defense Development under the contract UE135112GD, and the National

(No. 2012R1A3A2048841).

References

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Research Foundation of Korea (NRF) Grant funded by the Korea government (MSIP)

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