Lecture6_mechanics-rom-particles

Lecture6_mechanics-rom-particles - Rule of Mixtures –...

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Unformatted text preview: Rule of Mixtures – Particle Reinforcement Reinforcement Dr. Anthony Brennan Materials Science & Engineering University of Florida Mechanics - ROM - Particles Agenda Agenda • Introduction – Terminology • Rule of Mixtures – Einstein’s Theory - Equation of State – Mooney-Rivlin Theory – Packing Density – Kerner Equation • Summary Mechanics - ROM - Particles Einstein’s Theory – Equation of State Einstein’s η =η 1 (1+ k Eφ2 ) Assumes a dilute phase where concentration Assumes approaches zero and Where: η is the viscosity of the suspension is η 1 is the viscosity of the suspending liquid medium (fluid or matrix) kE is the Einstein coefficient φ 2 is volume fraction of suspended/dispersed phase phase Mechanics - ROM - Particles Mooney Rivlin Theory – Particle Reinforcement Mooney cont’d cont’d k Eφ2 η = ln η 1 1 − φ2 φ m Assumes concentration approaches 1 and where: Assumes η is the viscosity of the suspension is η 1 is the viscosity of the suspending liquid medium (fluid or matrix) kE is the Einstein coefficient φ 2 is volume fraction of suspended/dispersed phase is φ m is the maximum volume fraction of dispersed phase due to packing order restrictions phase Mechanics - ROM - Particles Mechanics Mooney Rivlin Theory – Particle Reinforcement cont’d cont’d φm = where: where: Vf is true volume of filler Vf Ve Ve is effective or apparent volume occupied by the filler Particles Structure Density HCP Hexagonal Close Pack 0.7405 FCC Face Centered Cubic 0.7405 RCP Random Close Packed 0.637 BCC Body Centered Cubic 0.60 SC Simple Cubic Mechanics - ROM - Particles 0.5236 Einstein Coefficient – Effect of Particle Aggregation, φ < φ m 2.5 2.5V f s + Vl kE = = φa V fs • ke Einstein coefficient ∀ φ a = volume fraction agglomerates • Vfs = volume fraction spheres • Vl = volume faction of matrix (liquid) Mechanics - ROM - Particles Effect of Particle Aspect Ratio Effect Adapted from Nielson and Landel, Marcel Dekker, NY, 1994 Mechanics - ROM - Particles Mooney: Effect of Particle Aggregation, φ > φ m η = k Eφ2 / φa ln η 1 1 − φ2 φφ m a • ke Einstein coefficient ∀ φ a = volume fraction agglomerates • Vfs = volume fraction spheres • Vl = volume faction of matrix (liquid) Mechanics - ROM - Particles Effect of Aggregation Effect Mechanics - ROM - Particles Correlation between Viscosity and Shear Modulus and η −1 = 4 − 5ν 1 ( G − 1) η 3 − 3ν 1 G1 1 ∀ η = viscosity viscosity • G = Shear modulus ∀ ν = Poisson’s ratio, where 1 is the matrix, 2 is the filler/reinforcing phase. matrix, Mechanics - ROM - Particles Over predicts Shear Modulus Over η −1 = 4 − 5ν 1 ( G − 1) η 3 − 3ν 1 G1 1 • Possible reasons: – Density – effective volume fraction – Thermal stresses – Interfacial bonding – Poisson’s ratio < 0.5 Mechanics - ROM - Particles Kerner Equation Kerner G =1+ 15(1 −ν 1 ) (φ2 ) G 1 (8 − 10ν 1 ) φ1 ∀ η = viscosity viscosity • G = Shear modulus ∀ ν = Poisson’s ratio, where c designates composite, m is matrix and f is filler. Alternatively, 1 is the matrix, 2 is the filler/ Alternatively, reinforcing phase. Mechanics - ROM - Particles Kerner Equation for Foams and Elastomers/Rubbers Elastomers/Rubbers G1 =1+ 15(1 −ν 1 ) (φ2 ) G (7 − 5ν 1 ) φ1 ∀ η = viscosity viscosity • G = Shear modulus ∀ ν = Poisson’s ratio, where c designates composite, m is matrix and f is filler. Alternatively, 1 is the matrix, 2 is the filler/ Alternatively, reinforcing phase. Mechanics - ROM - Particles Generalized Kerner Equation Generalized M 1+ ABφ2 M = 1− Bψφ 1 2 • M – modulus (E, G, K, B) of composite modulus M2 −1 (particle) (particle) A = ke −1 − φ2 ψφ =1− exp 1− φ2 φ M Mechanics - ROM - Particles M1 B= M2 − A M1 1 + φm ψ =1+ 2 φ2 φ m Effective Volume Fraction Function Function Mechanics - ROM - Particles Vollenberg Vollenberg Vollenberg, PHT and Heikens, D, Polymer, 30, 1656 (1989) Mechanics - ROM - Particles Vollenberg Vollenberg Vollenberg, PHT and Heikens, D, Polymer, 30, 1656 (1989) Mechanics - ROM - Particles Summary Silanes Summary • Organofunctional to provide covalent linkage Organofunctional between reinforcement and matrix polymer between • Chemistry – Step Growth combination of hydrolysis Chemistry and condensation and • Acid Catalyzed generates more linear species, rapid Acid condensation condensation • Base Catalyzed generates more particulate species, Base rapid hydrolysis, negative surface charge rapid • Dual stage process of acid hydrolysis followed by Dual base condensation or vice versa alternates base • Hybrid compounds that provide oligomers with Hybrid terminal alkoxides. terminal Mechanics - ROM - Particles ...
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