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 2

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 3

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 4

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 5

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 6

7. Anisotropic material properties: calculation using phase (particulate and matrix) properties7

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: 8

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

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 10

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 11

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 12