Validation of Numerical Modeling for the Prediction of Elastic and Failure Behavior of Diamond Powder Filled Woven Composites

A numerical investigation was carried out to examine the role of micro-sized diamond powder filler on the on-axis tensile stiffness properties of the standard modulus T300 and the high modulus YS90A woven fabric composite plates by progressive damage modeling. Finite element modeling (FEM) results for the T300 composite with and without diamond powder predicted a specific case of fiber failure in all the plies showing the characteristics of brittle failure. Static tensile tests were carried out on the YS90A composite coupons containing no diamond powder (DP) and filled with 6% and 12% volume fractions of DP. A higher content of diamond powder in the coupons led to agglomeration. This induced stress concentrations and subsequently reduced the mechanical properties. FEM was carried out considering specimens with and without an induced stress concentration geometry in the YS90A coupons filled with DP. The results of the on-axis tensile tests indicated a delamination type of failure in both cases with additional fiber fracture in the Open Hole Tensile (OHT) coupons.


Introduction
Textile structural composites are a widely used class of composites finding their applications in aerospace, automotive and manufacturing industries. They possess relatively high ratios of strain to failure in tension, compression or impact load as compared to traditional unidirectional prepreg composites [1].
The research and developments reported in this paper have been conducted within the framework of the experiments for particle physics, such as those carried out at the Large Hadron Collider (LHC) at the European Organization for Nuclear Research (CERN). Here light-weight but very stable support structures for high precision sensors are required with the additional constraint that they need to be able to remove the heat produced from the read-out electronics and the sensors themselves. Carbon fiber reinforced composites are used in this case to provide mechanical support and protection from thermal runaway of the detector [2]. This study has been carried out to investigate the potential improvements in the elastic and failure properties of the composite materials used, with the intention of improving the mechanical behavior of the support structure of the pixel detector of the ATLAS experiment at the LHC.
Woven fabric composite laminates offer a number of attractive properties compared to their unidirectional tape counterparts. Some of these properties are, lower production costs, better drapability, good resistance to fracture and transverse rupture due to weaving resistance and high impact strength [3]. Although the mechanical properties are not as good as those of their non-woven counterparts, they still offer reasonable specific stiffness and strength with particularly good impact and energy absorption characteristics [4]. Typical failure modes of woven composites include fiber fracture, matrix fracture, fiber-matrix interface debonding and interlaminar delamination [5] [6]. The cohesive matrix fracture and adhesive fracture of the fiber-matrix interface are known as inter-fiber fracture [6]. Inter-fiber fracture is a macro type damage which starts by the initiation of matrix micro-cracks at the fiber-matrix interface during matrix curing or due to transverse stress as a result of loading, thermal stresses, or fatigue loading.
The micro-cracks subsequently propagate through the matrix to form macro-cracks. In addition to the inter-fiber fracture of composites, the low failure strain and ductility are other drawbacks in the mechanical response when composites in general are loaded in fiber direction (on-axis loading) [7]. It is not uncommon that composite parts are subjected to large-deflection bending and multiple impacts in service conditions and that these quasi-static and dynamic loads generate high local stresses and strains leading to complex damage modes due to heterogeneity and anisotropy of composite laminates. In a bending scenario, a laminate experiences transverse shear and normal stresses resulting in interlaminar delamination damage, because of their low through-thickness strength and stiffness [8]. Damage evolution results in significant reduction of in-service mechanical properties and leads to a loss of structural integrity of the composite.
One of the common techniques to overcome the premature failure of composites due to delamination/debonding is to reinforce the composites with fillers [3].
Carbon nanotubes (CNTs), nano-clay, graphene nanoparticles (GNPs), and nano-silica are common non-metallic nanoparticles used to fabricate nanocomposites [9] [10]. Discovered by Ijima in 1991 [11], CNTs have attracted the attention of many scientists worldwide, because of their exceptional mechanical properties compared with conventional structural materials. Hossain et al. [12] reported a 49% and 31% increase in the flexural strength and modulus of on-axis woven E-glass/polyester composites reinforced by 0.1 -0.4 wt% carbon nano fibers (CNF). Qiu et al. [13] examined the tensile and shear behavior of (GFRP) composites, with 1.0% by weight functionalised multi-walled carbon nanotubes (MWCNTs). It was reported that a 14% and 5% increase in on-axis tensile and shear strengths respectively was achieved. The tensile strength and modulus of the composites filled with carbon black (CB) were shown to increase with increasing filler contents but impact strength and elongation at break were reduced. It was also reported that the composites with more than 30% CB were very brittle and were difficult to mix.
Few studies have examined the use of fillers to improve thermal, tensile and failure behavior of woven fabric composites. In previous research carried out by authors, an improvement in the out of plane thermal conductivity by a factor of 2.3 and 2.8 was achieved for the standard modulus T300 and the high modulus YS90A composites filled with 14% and 12% volume fractions of DP respectively [2]. This investigation is an extension of the previous research and it has been intended to predict the mechanical and failure behavior of T300 and YS90A woven composites filled with diamond powder under tensile loading conditions. The objectives of this research were: 1) To predict the elastic properties of epoxy-diamond powder matrix through a homogenization method.
2) To develop a micromechanical unit cell model to predict the elastic properties of the diamond powder filled standard modulus T300 and the high modulus YS90A woven composites.
3) To carry out the possible experiments and the numerical models to evaluate the on-axis tensile behavior of the diamond powder filled T300 and YS90A woven composites through progressive damage modeling.

Homogenization of Heterogeneous Matrix Microstructure
The macro response of the epoxy-diamond powder matrix was calculated from the micro response through numerical homogenization. The mechanical properties of a micro heterogeneous matrix material were characterized by a spatially variable elasticity tensor C. Generally, in order to demonstrate the homogenized effective macroscopic response of such materials, the relation between averages are described as shown in the following: The homogenized effective elastic modulus of diamond powder filled polymer matrix composite was obtained by finite element analysis with a 2D RVE consisting of randomly distributed spherical diamond powder fillers. An algorithm as reported in the author's previous research [2] [14] was used to generate the fillers content up to 25% volume fraction ( Figure 1). The effective elastic properties of the composite were predicted from the properties of their constituents.
Both the diamond powder filler and the epoxy matrix were considered to be iso- repeating nature. They can be used to simulate a bulk material by modeling a finite RVE [15]. For the RVE shown in Figure 1, the PBC's were applied to the RVE to ensure a macroscopically uniform stress or displacement field for tension and shear load (Equations (3) and (4) [16]).
where u is the displacement vector of any node on the boundary. The subscripts A, B, C and D correspond to the vertices. AB, BC, AD and DC correspond to the edges between the two vertices ( Figure 1). The PBC's were implemented in the form of equations using the constraint option in ABAQUS® through a python script.

PBC as Constraint Equations in ABAQUS
A general linear homogeneous equation is defined as follows where R is the node, k is the degree of freedom and AN is a constant coefficient that defines the relative motion of the nodes [15]. In order to apply the PBC using the constraint equations as described in Equation (5), a dummy node was introduced in ABAQUS®. In Equation (5), zero on the right side of the equation would be replaced by a nonzero value 1 2P where û is the prescribed displacement value. In ABAQUS®, the prescribed value û was applied through a dummy node, I, which was not attached to any other part in the model. A reference point ( Figure 1) with arbitrary coordinates was defined to represent the dummy node. This node was specified as a boundary condition with the value û in a certain direction, as shown in Equation where n is the direction. A load step was defined in order to apply the prescribed displacement û as a boundary condition.
The homogenization scheme was implemented with the volume averaged stress and strain calculated in each element to predict the effective elastic properties of the diamond powder filled polymer matrix composites. As expected, the diamond powder epoxy matrix became stiffer as the tensile and shear moduli increased with the increase in content of the diamond powder (Table 1).

Prediction of Elastic Properties of DP Filled Woven Composites
The prediction of the macromechanical properties of T300 and YS90A woven composites filled with diamond powder were evaluated with the periodic unit cell structure through the micromechanical method. The unit cell was modelled using TexGen [17] by considering that the warp and weft yarns posses geometric and material similarities ( Figure 2). The developed geometric models together with the information of textile data (Table 2), fiber material data (Table 3) and mesh details were imported into ABAQUS® through an input file containing voxel based mesh details using 8 node 3D linear brick elements. The elastic properties of diamond powder filled polymer matrix which were predicted through numerical homogenization in the previous section (    of woven composites. The contents of the diamond powder were limited to volume fractions of 14%, 25% in epoxy corresponding to 6% and 12% by volume fraction in the composite. The orthotropic behavior of the yarns were defined by a 3D stiffness matrix consisting of nine independent constants and are shown in Equation (10). The displacement boundary conditions for the unit cell have been defined following the procedures defined by Li et al [20]. As specified by Li et al. [20] the unit cells were treated by using the translational symmetry transformation. The macros- γ as shown in Equation (8), were treated as six extra degrees of freedom through which loads to the unit cell have been prescribed.
Concentrated forces were applied to these degrees of freedom and effectively, macroscopic stresses were applied to the unit cell [20]. Periodic boundary conditions were applied in the form of equations for all the unit cells that can be implemented using an equation option in ABAQUS®. Uniaxial loads (F x , F y , F z , F yz , F xz , F xy and ΔT) were applied to the unit cell at any point in the cell defined as constraint driven point ( Figure 3) assigned as x = 0, y = 1, z = 2, xy = 3, xz = 4 and yz = 5 in the analysis to obtain the elastic properties [20].
With the macroscopic stresses being expressed in terms of forces applied to the unit cell, the effective properties of the material represented by the unit cell were obtained in terms of the independent degrees of freedom ( x ε , y ε , z ε , yz γ , xz γ , xy γ ) and the applied loads (F x , F y , F z , F yz , F xz , F xy and ΔT) [20]. The predicted elastic properties of T300 and YS90A composites with and without diamond powder through this micromechanical method are listed in Table 4.

Experimental Tensile Testing
The experimental testing was only carried out for the high modulus YS90A composite filled with diamond powder in order to characterize the elastic properties of the composite filled with diamond powder under tensile loading.  Twenty six layers of the plain weave fabric (On-axis 0˚ -90˚) with an areal weight of 125 g•m −2 relating to a fiber volume fraction of 50% were used for the fabrication of specimens. The first sample was prepared without DP and the re-

Finite Element Modeling
Numerical modeling was carried out for both the standard modulus T300 and the high modulus YS90A composites under tensile loading. Mesomechanical damage models were developed for damage detection, evolution and propagation through a progressive failure analysis methodology. Material strength, in-terlaminar strength and fracture toughness properties of T300 woven composite ( cracking and fiber failure) and cohesive elements were used to capture delamination at ply interfaces. The intralaminar and interlaminar damage models implemented are briefly described below.

Interlaminar Damage
The onset of damage is attained when the total stress acting on the interface reaches a critical value max N as given by: where in the Equation (10) where η is the Benzeggaagh-Kenane material parameter, G IC and G IIC are the critical fracture toughness values for pure Mode I and Mode II fracture

Intralaminar Damage
The constitutive stress-strain relationship are formulated in a local Cartesian coordinate system considering the base vectors aligned with the fiber directions.
The fabric reinforced ply was modelled as a homogeneous orthotropic elastic material with the potential to sustain progressive stiffness degradation due to fiber/matrix cracking and plastic deformation under shear loading [23]. It has been assumed that the elastic stress-strain relationship is given by orthotropic damage elasticity. Four failure modes including fiber tension, fiber compression, matrix tension, and matrix compression are considered. Initiation of damage, which refers to the onset of damage at a material point is based on the Hashin damage model and four different failure modes based on the Hashin Criteria [24] are described in Equations (12) to (15):  Once the damage initiation function has been satisfied, the associated damage variable would be different from zero and further loading would cause degradation of the material stiffness coefficients [23]. The stiffness matrix of a damaged ply could therefore be defined as where D as shown in Equation (18)    been increased, the element will experience a larger displacement before it reaches ultimate failure [23].

Model Set-Up
Progressive failure analysis for a specimen was performed with the on-axis laminates (0˚ -90˚) of 117 mm in length, 15 mm in width and 3 mm in thickness respectively ( Figure 6). A 5-ply laminate with a ply thickness of 0.6 mm along with a cohesive layer of nearly zero thickness between each ply was modeled ( Figure 6) for the tensile loading.
Each ply was modelled in ABAQUS® using quadrilateral continuum shell elements, type SC8R which is an 8-node, quadrilateral, and uses first-order inter-  Table 4. The strength and the toughness properties of the T300 composite (Table 5) were taken from [21] [22].
As suggested by by H. Ullah et al. in [22], normal and shear strengths of the YS90A composite were examined from a FEM model of an undamaged coupon under the same boundary conditions. In the numerical analysis the interlaminar Figure 6. Finite element model representing the laminate and cohesive zone modeling for the on-axis tensile and bending.
shear stress was assumed as the interlaminar shear strength and normal stress at the ultimate load was taken as the normal strength of the laminate (Table 5). For the on-axial tensile models, fully clamped boundary conditions were applied at one end of the specimen and an uniform axial displacement was prescribed at the opposite end ( Figure 6). Three different configurations of T300 and YS90A filled composite under static tension corresponding to coupons without DP and coupons with 6% and 12% DP by volume fraction in the composite, were simulated by means of an implicit numerical scheme. On average, each on-axis tensile simulation took about five hours to run on a single 3.4 GHz Intel® Core TM i7 processor.

T300 Composite
The FEM results of the on-axis tensile stress-strain behavior of the standard modulus T300 woven composite with and without DP are presented in Figure 7.    drastically (Table 6, Figure 7) as the diamond powder enhanced epoxy matrix became stiffer and more brittle. As a result of enhancing the epoxy matrix with diamond powder, the polymer viscosity, tensile modulus and tensile strength of the composite increases while the elongation at break reduces substantially, which means that high strength fillers like DP change the matrix behavior into a more rigid-like material.

YS90A Composite 1) Experimental tensile results
The experimental tensile testing results showed a brittle fiber fracture along with delamination. From the tensile testing, a drastic reduction in the elastic modulus and strength of the YS90A composite filled with diamond powder was observed ( Figure 11, Table 7). The average ultimate tensile strength of the composite reduced to 320 MPa with 6% volume fraction of DP in the composite, while with 12% volume fraction of DP in the composite, the ultimate tensile strength reduced to 286 MPa ( Figure 11). The failure strain of the composite increased to 0.56% with 12% volume fraction of DP in the composite. Elastic moduli reduced significantly to 157.7 GPa and 129.6 GPa with DP volume fractions of 6% and 12% in the composite. One of the reasons for the severe reduction in the strength and stiffness are due to the non-homogeneous dispersion ( Figure 10) of the DP in the matrix.
Ultrasonication techniques were not used to achieve uniform dispersion and the manufacturing procedure was carried out by hand-layup, which led to a reduction in the production quality of the testing coupons. A higher content of diamond powder led to agglomerations and also the sharp edges of the DP itself can contribute to stress concentrations through geometric defects. A good dispersion and the interfacial properties between the epoxy resin and DP are crucial for determining the properties of the composite as the interface between the two is the region where the stresses are transferred from the epoxy resin to diamond   [27]. Sobia et al. reported [28] that the mechanical properties of epoxy-diamond composites initially increased at lower contents of DP but with the higher content the mechanical properties decreased due to agglomeration. High modulus YS90A being a pitch based carbon fiber was extremely fragile to handle in terms of fabrication of the samples and the interface bonding between diamond powder-epoxy resin was not perfectly formed which can be seen from the SEM image shown in Figure 10.
From the SEM image, it can be noted that, as the content of DP increased in the matrix, the DP filler network could have trapped some portion of epoxy resin preventing it from infiltrating the reinforcing fibers properly. The above hypothesis could be one of the reasons for the reduction in the tensile properties ( Figure 11, Table 7) of the DP filled composite.
To understand the above mentioned behavior further, finite element modeling was performed with and without stress concentrations in the present composite specimen.

2) FEM without stress concentrations
The FEM results of the on-axis tensile behavior of the YS90A woven compo-site (without stress concentrations) exhibited a delamination type failure. Delamination triggered a substantial level of load drop due to dissipation of energy ( Figure 11). At the ultimate tensile strength of the composite (Figure 11) just before the occurrence of massive delamination, it is worth noting that fiber failure was not detected in any of the plies (Figure 12(a)) and that matrix damage was also not present. After this point, the propagation of delamination ( Figure   12(b), Figure 13) at the interface developed further. Figure 12(a), Figure 12(b) represents the damage mode occuring in the plies and demonstrates that fibers in the on-axis plies have not failed even as the load drops significantly to a level corresponding to the failure stress.
The FEM analysis of laminates without DP showed a tensile strength and Figure 11. Comparison between the experimental tensile testing results and FEM tensile behavior without considering stress concentrations in the laminate.    Figure 11, Table 8) which are in close agreement with the experimental results ( Figure 11). The analysis of laminates with 6 vol % and 12 vol % of DP shows an increasing trend in the elastic modulus whereas the experimental results shows a decreasing trend due to the presence of DP induced stress concentration in the samples.

3) FEM with stress concentrations
To study the behavior of stress concentrations in the YS90A composite filled with diamond powder, Open Hole Tension (OHT) coupons were modeled with diameters of 6 mm (laminate filled with 6 vol % DP, Figure 14(a)) and 10 mm (laminate filled with 12 vol % DP, Figure 14 Table 9. The FEM results with the OHT coupons predicted a reduction in the elastic modulus for the composite filled with DP (6% and 12% volume fraction) with the predictions showing reasonable accuracy when compared to the experimental results (Table   9). From the data given above, it is evident that the reduction in the elastic modulus is due to inhomogeneities in the dispersion of the DP and the presence of possible stress concentrations in the composite. The FEM results also show that a higher content of DP causes weak adhesion in the fiber-matrix interface which reduces the the strength of the composite and also reduces the ability of the composite to resist the damage (Figure 15(a), Figure 15(b), Table 9). The FEM results also show that the presence of a rigid filler reduces the elongation at the    (Table 9). In general, the elastic and the failure properties reduce significantly with the presence of stress concentrations. The drastic increase in the failure strain (Table 9, Figure 11) as shown by the experimental results for the composite filled with 12% volume fraction of DP requires further investigation.

Conclusions
In this study, the elastic and failure characteristics of standard modulus T300 and high modulus YS90A woven composites filled with diamond powder were A detailed investigation of the mesh sensitivity towards the predicted damage behavior must also be carried out. Different constitutive laws considering through the thickness mechanical properties as well as strain rate effects in the diamond powder filled composite are required and need to be developed which could be implemented through an explicit rather than implicit FEA modeling scheme.