On predicting the deformation and fracture properties of ABS / PMMA / Nano -SiO 2 composites

: The damage mechanism of ABS/PMMA/nano-SiO 2 composites was studied by both experiments and finite element analysis in this paper. A microstructure-based the Homogenization theory and the surface-based cohesive method was developed to predict the deformation and fracture behavior of ABS/PMMA/nano-SiO 2 composite. The constitutive behavior of the elastoplastic-damage in the matrix, the fracture for the nano-SiO 2 reinforement , and the traction-separation for interfaces, were simulated in this model. The validity of the modeling results were validated by the agreement of the experiment and the morphology of fracture section with those predicted by the simulation. The numerical results were used to provide insight into the damage mechamisms of ABS/PMMA/nano-SiO 2 composites, and the effects of nano-SiO 2 strength on ABS/PMMA blends mechanical properties were also discussed in detail.

Acrylonitrile-butadiene styrene (ABS), as a type of excellent rubber-toughened thermoplastic, has been widely used in many fields for many years such as automotive and household appliance industries due to its good processability, low water absorption, and high impact strength. However, neat ABS materials usually have low toughness and surface glossiness. Polymer blends have recently drawn considerable interest because these materials present outstanding properties at low cost. ABS/poly (methyl methacrylate) (PMMA) blends are characterized by a remarkable combination of physical properties, such as excellent surface glossiness, easy processability, and high mechanical strength. At the same time, the blends toughness is declined because of the inherent brittleness of PMMA [1][2][3][4][5][6]. It is difficult to toughen and stiffen polymeric matrix simultaneously through physical blending as well as know.
In recent years, using inorganic nanoparticles as modifiers to refine the morphology of polymer blends has received considerable attention. Compared with traditional microcomposites, the massive surface to volume ratio of the nanoparticles results in a big interfacial area and a high surface energy of nanoparticle fillers, which lead to a strong interfacial adhesion between polymeric matrix and the fillers and hence affects the properties of the overall composites [7]. There are generally two methods that have been used to study the mechanical behavior of the polymer modified with nanoparticles: experimental approach and numerical modeling. Many experimental studies describe the effects of nanoparticles in polymer blends [8][9][10][11][12][13][14].Mu et al. [15] investigated the effects of SiO2 on the morphology of micro PMMA and found that nano-SiO2 particles could grow on PMMA microspheres. Haghtalab and Rahimi [16] studied the effects of ABS and nanosilica with different nanoparticle sizes and various loadings. Zhang et al. [8] investigated of nano-CaCO3 on the rheological and mechanical properties of ABS/PMMA blends, the results showed that the tensile yield strength decreased slightly with the increase in nano-CaCO3 content, and all nano-CaCO3 particles significantly decrease the Izod impact strength. In our previous experimental study, we have investigated the influence of nano-SiO2 particle modification on the morphology and properties of ABS/PMMA polymer blends.When the contents of the nano-SiO2 fillers were below 2wt%, the toughness of the ternary blends was significantly improved [17]. These researches proved that nanoparticles and high filler loadings enhanced the nanoparticle surface area and the viscoelastic properties were intensified by increasing the polymer chain adsorption on the nanoparticles. However, the experimental approach can not characterize the actual interactions that occur between the polymer matrix, the modifier, and the interfaces during deformation and fracture processes.
In the numerical modeling of polymer nanoparticles composites, there exist two popular methods: discrete element method (DEM) and finite element method(FEM).The DEM model has a faster convergence rate than FEM, however ,it neglects some important factors that are necessary for the study of deformation and fracture mechanisms in polymer nanoparticles composites, such as the geometry of the particles, the brittle fracture between the matrix and the particles. FEM is often used to fully reconstruct the microstructure of the composite materials, including the irregular morphology of the particles, the inhomogeneous spatial distribution of particles and the anisotropy of properties in the particles. In the literatures there have been a few effective numerical models based on FEM proposed to study the mechanical properties of polymer composites. For example, based on sensitivity analysis (SA) methods, N.vuBac et al. [18] proposed a stochastic framework to quantify the key-input parameters influencing the Yong's modulus of polymer (epoxy) clay nanocomposites (PCNs). All stochastic methods predict that the key parameters for Yong's modulus are the epoxy stiffness followed by the clay volume fraction.
Pontefisso et al. [12] developed a new algorithm for the generation of 3D Representative Volume Elements(RVEs),which is easy to be meshed and imported in a FE code. The interphase thickness and properties on the elastic properties of nanocomposites was studied with a computational analysis .Yuan [13]  Although there are previous studies explicitly considering the fracture behavior of composite using FEM method, they are not able to account for the effect of nanoparticle morphology on the properties of ABS/PMMA/nano-SiO2 polymer blends.
Since these models are established based on spherical particle assumption, meanwhile, the multiphase failure of particles was not taken into account in these models. The remainder of the paper is organized as follows. In Section 2, the Homogenization theory and the surface-based cohesive method are described briefly. Section 3 presents the materials and the experimental procedures. Section 4 presents a microstructure-based finite element model to predict the interfacial debonding of ABS/PMMA/nano-SiO2 polymer blends. We validate the model by challenging it with experimental data in Section 5. Finally, some conclusion, and potential extension are discussed in Section 6.

Homogenization theory
The Homogenization theory is widely used in converting inhomogeneous composite into homogeneous material. It often relies on the finite element method to simulate the response of the samples discretely with four-node quadrangle elements and three-node triangular elements. Four edges of RVE are defined as periodic boundary conditions.
To minimize the effect of the mesh density, it was kept very fine.
The equivalent elastic constant tensor of composite ( "#$% ) as follows： where ' "# is the mean stress, and ̅ "# is the mean strain of a unit cell , "# , , "# and ̅ "# are calculated as follows：   In this way, the mean stress and strain of a whole RVE can be calculated by Eqs.

Surface-based cohesive method
Cohesive technology is widely used in simulation the damage between filler particles and matrix [19]. In this paper, a similar method named surface-based cohesive behavior is introduced to simulate the particles. To simulate the degradation and eventual failure of the bond between two surfaces, a damage model that consists of two ingredients a damage initiation criterion and a damage evolution law is established. In this paper , a typical traction-separation response which is used to simulate the damage between the nano-SiO2 particles and matrix, which is shown in where t consists of two components K and ˆ, which represent the normal and the shear traction, respectively. The corresponding separations and denoted by K and ˆ.
Matrix K is a constant before damage initiates and it will decrease according to the damage evolution law when damage initiates.
The maximum stress criterion is used as the damage initiation criterion in this paper.
It is assumed that damage initiates when the maximum contact stress is equal to the maximum stress. This criterion can be written as where the subscript "n" represents the normal or the shear direction and K ;E˜ refers to the maximum value of the effective separation attained in the loading history.

Matericals
The

Preparation of blends
All of the materials were dried at 80 °C for 12 h in an air-circulating oven prior to use. ABS/PMMA/nano-SiO2 was prepared by using a two-screw extruder ( °C from the hopper to the nozzle, and the holding pressure was 50 MPa. The mass ratio of ABS to PMMA was fixed at 80/20, and the content of nano-SiO2 is 2wt%.

Characterization
The tensile properties were conducted on a CMT 5305 electrical testing machine at room temperature. The crosshead speed was maintained at 4 mm/min.
Scanning Electron Microscope (SEM). The SEM samples for the morphology and fracture mechanism analyses were directly collected from the broken pieces after the mechanical tests. The fracture surface was coated with a thin layer of gold before being observed under a FEI-NOVA NANOSEM 450 SEM.
Transmission Electron Microscope (TEM). The morphological structure was examined by using a TEM (JEM-2100) at an accelerating voltage of 200 KV. The specimens were dyed with a mixture of sulfur, zinc stearate, and promoter (content ratio: 90/5/5) in an oven at 120 °C for 24 h. Then, 100 nm thick samples were prepared through ultra-cryomicrotomy at −100 °C by using a Leica UCT microtome.
A representative volume element was used to represent the selected composite microstructure in this paper. The morphology of ABS/PMMA and ABS/PMMA/nano-SiO2 blends was shown in Fig.2, which was clearly observed using TEM. The rubber particle morphologies are regular and have circle shapes, however, the particle morphologies of nano-SiO2 are irregular because of aggregation.
The structure modeling of ABS/PMMA/nano-SiO2 composite was created based on this information. The determination of the minimum accurate RVE size is an extremely laborious work so we just verify that the RVE size used in this paper is reasonable.The size of RVE model was 1mm × 1.2mm, and the average size of the ruber particles is 0.2mm. The size and shape of nano-SiO2 were simplified because of its aggregation. The RVE model of ABS/PMMA/nano-SiO2 was shown in Fig.3.
Note that the model generated here is a special one, in which no particles cross the surface. This is the limitation for surface-based cohesive element in Abaqus to simulate the ABS/PMMA/nano-SiO2 composite. In terms of the finite element discretization, the general linear solid element C3D8R in Abaqus was used to mesh the matrix and the nano-SiO2 particles. The surface-based cohesive element was adopted to account for the interface behavior between matrix and particles .The meshing of representative volume unit was shown in Fig. 4.
The parameters of Cohesive behavior and Damage should be selected before using Surface-based Cohesive to define the boundary condition. In this paper, the above parameters were given based on the relevant literatures [20][21][22][23],because it is difficult to define the parameters through our experience. They were listed in table1.
Meanwhile，we only considered the elastic stage of nano-SiO2 because of its high yield strength and elastic modulus relative to the PMMA and ABS. PMMA and ABS are input corresponding elastic-plastic properties respectively.

Numerical results and discussion
A visible stress and strain evolution, as well as the crack generation and propagation inside the composite during tensile loading we shown in Fig.5 and Fig.6, respectively. Analyzing the numerical model, one could find that the first fracture behavior happened between nano-SiO2 particles and matrix when ε was 0.05, and the damage area was becoming larger with increasing the value of ε.
In this simulation, We can identify that the initial stage of high stress area occurs at the interface between nano-SiO2 particles and matrix. During the composite deformation, as shown in Fig.5, the aggregation nano-SiO2 particles act as stress concentration points. When ε was 0.075，the high stress area occurred in the equator of rubber particles . One can see that the high stress area of rubber particles is gradually expanding from the rubber equator to the poles, with the increase of extension area, rubber particles caused further interface debonding behavior between the matrix. One can find that the aggregation of nano-SiO2 and weak interface bonding with matrix promote easier interface debonding, which decreases the stiffness and strength of ABS/PMMA/nano-SiO2 composite. The SEM micrograph in Fig.7 shows the morphology of an actual fracture section , the fracture direction is about 45°，which is consistent with the direction observed in the numerical model.