Background on Composite Property Estimation and Measurement
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Transcript of Background on Composite Property Estimation and Measurement
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