AFRL

Background on Composite Property Estimation and Measurement

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• 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

Effective Properties of Particulate Composites

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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

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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

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TRANSVERSELY ISOTROPIC MATERIALS

TL 321 ,

TEEE

TEEE

TEE

TTT

TT

L

LT

TT

TT

TL

LT

LT

TL

L

--

--

--

ssse

ssse

ssse

3213

32

12

321

1 )(

LL

LL

TT

GG

GG

GG

2,

2,

2,

1212

1212

1313

1313

2323

2323

e

e

e

x2-x3 plane is isotropic – all

properties transverse to

x1 axis are same

21 = 31 = TL

12 = 13 = LT

23 = 32 = TT

E1 = EL

E2 = E3 = ET

G12 = G13 = GL

G23 = GT

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ALSO, AS WITH GENERAL ORTHOTROPIC MATERIALS, T

TL

L

LT

EE

7 INDEPENDENT THERMOELASTIC CONSTANTS:

2 E’S, 1or2 G’S, 2or1 ’s, 2 ’s

AN APPROXIMATION: MOST TRANSVERSELY-ISOTROPIC COMPOSITES HAVE GL~GT

SINCE T.I. PROPERTIES ARE NOT

DIRECTIONALLY DEPENDENT, )1(2 TTT GE

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TRANSVERSELY ISOTROPIC MATERIALS

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Anisotropic material properties: calculation

using phase (particulate and matrix) properties

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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:

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• 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

Effective Composite Properties

sij = Cijkl*ekl

eij = Sijkl*skl

Cijkl* = effective elastic stiffnesses

Sijkl* = effective elastic compliances

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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

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• 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*

)43(5)2(61

*

mmm

mmm

mi

im

GKGGKv

GG

vGG

-

Particulate Reinforced Composite Moduli

11vi = volume fraction of inclusion; vm = volume fraction of matrix

AFRLAFRLResults of Bounding Theorems for Particulate Reinforced Composite Materials

Best lower bound is from Arbitrary Phase Geometry (APG) boundBest upper bound is from Composite Spheres Assemblage (CSA) bound

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G* = effective shear modulus of particulate reinforced composite