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Engineering Equations for Strength and Modulus of Particulate Reinforced Composite Materials M.E. 7501 – Reinforced Composite Materials Lecture 3 – Part 2 Particulate Reinforcement d Example: idealized cubic array of spherical particles s s s Experimental observations on effects of particulate reinforcement Experiments show that, for typical micron-sized particulate reinforcement, as the particle volume fraction increases, the modulus increases but strength and elongation decrease Flexural stress-strain curves for 30 µm glass bead-reinforced epoxy composites of various bead volume fractions. (From Sahu, S., and Broutman, L. J. 1972. Polymer Engineering and Science, 12(2), 91-100. With permission.) Yield strength of particulate composites Nicolais-Narkis semi-empirical equation for case with no bonding between particles and matrix S yc S ym (1 1.21v ) 2/3 p (6.65) where Syc is the yield strength of the composite Sym is the yield strength of the matrix material vp is the volume fraction of particles the coefficient 1.21 and the exponent 2/3 are selected so as to insure that Syc decreases with increasing vp, that Syc = Sym when vp=0, and that Syc=0 when vp=0.74 , the particle volume fraction corresponding to the maximum packing fraction for spherical particles of the same size in a hexagonal close packed arrangement Liang – Li equation includes particle – matrix interfacial adhesion S yc S ym (1 1.21sin2 v2/3 p ) where θ is the interfacial bonding angle, θ = 0o corresponds to good adhesion, and θ = 90o corresponds to poor adhesion (6.66) Finite element models for particulate composites (a) development of axisymmetric RVE (b) axisymmetric finite element models of RVE Finite element models for spherical particle reinforced composite. (From Cho, J., Joshi, M. S., and Sun, C. T. 2006. Composites Science and Technology, 66, 1941-1952. With permission) Modulus of particulate composites Katz -Milewski and Nielsen-Landel generalizations of the Halpin-Tsai equations Ec 1 ABv p Em 1 B v p where A kE 1 (6.67) E B E p p A / Em 1 / Em 1 v p max 1 2 v p max vp and where is the Young’s modulus of the composite Ec is the Young’s modulus of the particle E p is the Young’s modulus of the matrix Em is the Einstein coefficient is the particle volume fraction kE is the maximum particle packing fraction vp v p max Semi empirical Models Use empirical equations which have a theoretical basis in mechanics Halpin-Tsai Equations Where E2 1 v f Em 1 v f E E f f Em 1 Em (3.63) (3.64) And curve-fitting parameter 2 for E2 of square array of circular fibers 1 for G12 As As Rule of Mixtures 0 Inverse Rule of Mixtures 6 Young's Modulus (106 psi) 5 4 Experimental [62] 3 Eq. 3.27 Eq. 3.40 Eq. 6.67 2 1 0 0 0.1 0.2 0.3 0.4 0.5 Particle Volume Fraction Comparison of predicted and measured values of Young’s modulus for glass microsphere-reinforced polyester composites of various particle volume fractions. Hybrid multiscale reinforcements (a) (b) Improvement of mechanical properties of conventional unidirectional E-glass/epoxy composites by using silica nanoparticle-enhanced epoxy matrix. (a) off-axis compressive strength. (b) transverse tensile strength and transverse modulus. (From Uddin, M. F., and Sun, C. T. 2008. Composites Science and Technology, 68(7-8), 1637-1643. With permission.)