18 TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS ANALYTICAL MODELLING OF ELASTIC PROPERTIES OF NANOCOMPOSITES WITH A NEW TYPE OF REINFORCEMENT I.A. Guz 1 *, J.J. Rushchitsky 2 , A.N. Guz 2 1 Centre for Micro- and Nanomechanics (CEMINACS), University of Aberdeen, Aberdeen, UK 2 Timoshenko Institute of Mechanics, Kiev, Ukraine * Corresponding author ( i.guz@abdn.ac.uk ) Keywords : composite materials, nanowires, analytical method, effective properties cell) consists of the matrix and a coated fibre, Fig. 2. 1 Introduction At that, the coating itself has several sub- In their paper, Wang et al. [1] reported a new brush- components, which is a new feature of the model: like nano-composition: the CdTe nanowires coated the coated fibre consists of three different parts: a with SiO 2 nanowires, nearly parallel bristles growing solid core, a solid coating (homogeneous shell), and perpendicular to the surface. The structure “CdTe a “bristled coating” (composite shell). The fourth nanowire - SiO 2 nanowires” consists of three component in the model is the matrix. A segment of components: CdTe wire forms the solid core, which the model representing the core fibre with solid is jointed continuously with coating from SiO 2 coating and bristles attached to the solid coating is nanowires in the form of some solid shell, and then shown in Fig. 3. Subsequently, the following the shell is coated periodically by bristled and notations are used to distinguish the four smooth zones, Fig. 1. components of composite: (1) for the fibre core, (2) Bristled nanowires can be used for the ultimate for the fibre solid coating, (3) for the fibre bristled purpose of fabricating composite materials with coating and (4) for the matrix. improved fibre-matrix adhesion and hence the Three out of four components are homogeneous increased shear strength. In order to study the materials, e.g., EPON828 epoxy matrix, SiO 2 solid effective properties of the entire composite and the coating and CdTe solid fibre core, with known effect of reinforcement with bristled nanowires on properties (Young’s modulus, shear modulus, the overall performance of the material, a four- Poisson’s ratio, density etc.). However, the bristled component model is required - the matrix being the coating is itself a composite consisting of, e.g., the fourth component, in addition to the three EPON828 matrix reinforced by SiO 2 nanowires, components mentioned above. Figs. 2 and 3. The effective properties of this This paper gives the details of the method for component are evaluated separately beforehand. The deriving the explicit formulas for effective elastic easiest way to do it is by using the classical Voigt constants of such materials. and Reuss bounds. For this purpose, we would need to know the diameter and the length of bristles, and their number per unit surface area of microfibres or 2 Methodology nanowires. Also, in order to use the known formulas This work concerns the development of a new four- of the rule of mixture, we assume here that all the component model for predicting the mechanical bristles are parallel to each other, i.e. that the properties of nanocomposites reinforced with properties of the bristled coating (3), Fig. 2, do not bristled nanowires. It generalises the approaches change with the radius. This assumption, being, of presented in [2-4]. course, a certain simplification, seems reasonable, The model assumes that fibres are arranged in the since the length of SiO 2 nanowires used for matrix periodically as a quadratic or hexagonal reinforcement is rather small. lattice. Then the representative volume element (unit
The mathematical formulation of the model is based ϕ = ( z ) on considering the four simple states of plane elastic ( 4 ) equilibrium of the unit cell (a square with the side ⎛ + ⎞ ∞ 2 n 1 1 z ∑ + ( 4 ) − λ 2 ( 4 ) ⎜ − α ⎟ a z a ⎜ ⎟ l unit , Fig. 4) - i.e. longitudinal tension, transverse 0 2 n , 0 + z 2 n 1 (4) ⎝ ⎠ = n 1 tension, longitudinal shear and transverse shear - and + λ + ∞ ∞ z 2 n 1 ∞ a ( 4 ) 2 k 2 using the Muskhelishvili complex potentials [5] for ∑∑ ∑ + + ( 4 ) λ 2 k 2 α − + a 2 k 2 + 2 k 2 n , k + + + each domain occupied by a separate component. The 2 k 1 2 n 1 ( 2 k 1 ) z = = = k 1 n 1 k 1 model yields the explicit formulas for five effective where α n,k are the constants used in the theory of elastic constants. ( 1 ) ( 2 ) Weierstrass functions, λ = 2 r (3) / l unit , and a , a , The possible case of imperfect adhesion between the 2 k 2 k fibre core and the matrix can be taken into account ( 3 ) ( 4 ) a , a are the yet unknown coefficients in the 2 k 2 k by considering one of the four components, i.e. the series given by Eqs. (1)-(4). coating layer, with the appropriately reduced Then the averaged stresses and strains for each of properties. the domains are calculated using the contour integrals, Fig 5. This procedure results in the 3 Deriving the explicit expressions for effective following expression for the effective longitudinal constants shear modulus of entire four-component composition, G * : The procedure of deriving the explicit expressions for effective elastic constants of the suggested four- = = + + − G * G * [ G c ( 4 ) 2 c ( 3 ) c ( 4 ) G G 1 12 ( 4 ) ( 4 ) ( 3 ) component structural model is by no means a trivial ( ) − 1 + ( 2 ) + − 1 4 c 1 G G mathematical exercise. For the lack of space, here it ( 3 ) ( 2 ) (5) ( ) ( ) can be given only in outline for one of the constants, − − 1 1 + + − + − 8 c ( 1 ) 1 G G 1 1 G G 1 ] namely the shear modulus, G . Hereafter, z is a ( 3 ) ( 2 ) ( 2 ) ( 1 ) [ ] − 1 complex co-ordinate in the transverse cross section, − × ( 4 ) + − ( 4 ) 1 c ( 2 c ) G G ( 4 ) ( 3 ) and i is imaginary unit) where c (1) , c (2) , c (3) and c (4) are the volume fractions The cornerstone of the analytical procedure is the of the fibre core, the fibre solid coating, the fibre representation of the Muskhelishvili potentials by: bristled coating and the matrix, respectively. a harmonic complex function, which is regular in the domain of fibre core (circle A (1) in Fig. 4): Eq. (5) yields the well-known formulas for two- component and three-component models as the + ∞ 2 k 1 z ∑ particular cases, see [3,4]. The latter will follow ϕ = ( 1 ) ( z ) a (1) ( 1 ) 2 k + 2 k 1 from Eq. (5) if the volume fraction of bristled = k 0 coating c (3) = 0 and the shear moduli of bristled, G (3) , a function in the form of Laurent series, which is and solid, G (2) , coatings are the same; the former will regular in the domain of fibre solid coating (ring A (2) follow from Eq. (5) if, additionally, the volume in Fig. 4): fraction of solid coating c (2) = 0 and the shear moduli of solid coating, G (2) , and fibre core, G (1) , are the ∞ ∑ ϕ = + ( z ) a ( k z 2 ) 2 k 1 (2) same. ( 2 ) 2 = −∞ k Similarly, the explicit expressions for other four effective constants for the entire four-component a function in the form of Laurent series, which is composition are deduced. regular in the domain of fibre bristle coating (ring A (3) in Fig. 4): 4 Results ∞ ∑ ϕ = + ( 3 ) 2 k 1 ( z ) a k z (3) ( 3 ) 2 The values of all five effective elastic constants k = −∞ representing the transversely-isotropic response of a doubly-periodic function constructed utilising the the entire composite were computed by the method Weierstrass functions in the domain of the matrix outlined in the previous section. The results for ( A (4) in Fig. 4): composites reinforced by the bristled nanowires
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