Background on Composite Property Estimation and Measurement

1. Background on Composite Property Estimation and Measurement

2. Effective Properties of Particulate Composites Concepts from Elasticity Theory Statistical Homogeneity, Representative Volume Element, Composite Material “Effective” Stress-Strain Relations Particulate composite effective moduli Unidirectional composite effective moduli Lamina constitutive relations Lamina off-axis constitutive relations

3. Orthotropic Material s-e Relations • Engineering materials having orthotropic properties are finding increased application in the design of structural systems. An orthotropic material is completely defined by nine independent elastic constants. The most common elastic constants are the following: • Elastic moduli E1, E2, E3, in three orthogonal directions • Poisson’s ratios, 12, for transverse strain in the j-directions due to stress in the i-direction • Shear moduli, G12, G23, G31 in the 1-2, 2-3, and 3-1 planes, respectively • The inverse of the elastic matrix which is called the compliance matrix, S, is then given by

4. Orthotropic Material s-e Relations The compliance matrix is symmetric so that the following symmetry relations must hold The nonzero stiffness coefficients, Cij, are found by inverting the compliance matrix (10.17) and are where

5. TRANSVERSELY ISOTROPIC MATERIALS x2-x3 plane is isotropic – all properties transverse to x1 axis are same E1 = EL E2 = E3 = ET G12 = G13 = GL G23 = GT n21 = n31 = nTL n12 = n13 = nLT n23 = n32 = nTT

6. TRANSVERSELY ISOTROPIC MATERIALS SINCE T.I. PROPERTIES ARE NOT DIRECTIONALLY DEPENDENT, ALSO, AS WITH GENERAL ORTHOTROPIC MATERIALS, 7 INDEPENDENT THERMOELASTIC CONSTANTS: 2 E’S, 1or2 G’S, 2or1 n’s, 2 a’s AN APPROXIMATION: MOST TRANSVERSELY-ISOTROPIC COMPOSITES HAVE GL~GT

7. Anisotropic material properties: calculation using phase (particulate and matrix) properties

8. Effective Composite Properties Statistical Homogeneity: to calculate effective properties, it is first necessary to introduce a representative volume element (RVE), which must be large compared to typical phase region dimensions (i.e., reinforcement diameters and spacing) RVE must be large enough so that average stress in RVE is unchanged as size increases:

9. Effective properties of a composite material define relations between averages of field variables s and e Effective Composite Properties Cijkl* and Sijkl* are reciprocals of one another Overbars denote RVE averages

10. Particulate Reinforced Composite Moduli • Provided dispersion of particulate reinforcement is uniform, and provided orientation of non-spherical particulates is random, stress-strain relations of such composite materials will be effectively isotropic • Two independent elastic moduli • For convenience, these are selected to be the bulk modulus (K) and the shear modulus (G) • All other elastic constants can be defined in terms of K and G

11. Particulate Reinforced Composite Moduli Effective elastic constants of particulate reinforced composites are obtained using multi-phase material solutions from elasticity theory Exact solutions are possible only in the case of spherical particles Approximate (bounding theory) results are used for other cases, such as non-spherical particles Example of a lower bound result is Arbitrary Phase Geometry (APG) lower bound on G* vi = volume fraction of inclusion; vm = volume fraction of matrix

12. Results of Bounding Theorems for Particulate Reinforced Composite Materials G* = effective shear modulus of particulate reinforced composite Best lower bound is from Arbitrary Phase Geometry (APG) bound Best upper bound is from Composite Spheres Assemblage (CSA) bound