Page 148 - Mechanics of Asphalt Microstructure and Micromechanics

140 Ch a p t e r Fiv e

5.3 Effective Properties of Mixture
The Eshelby solution is very important in various topics in micromechanics. One ma-
jor application of the solution is in the estimation of the effective properties, mainly
the effective modulus of elasticity of the mixture from the properties, of components,
and their volume fractions. The effective modulus is defined on the average stress and
strain relationship.

5.3.1 Average Stress and Average Strain Theorem

Average stress and stress rate (Nemat-Nasser and Hori, 1999)
It is understood that even though the surface traction follows the distribution resulted
from a uniform stress condition (when the material is homogenous), due to the inhomo-
geneities, the stress distribution is not uniform. It can be proved that as long as the
prescribed surface traction is self-equilibrating on ∂V, the volume average of the stress
field s (x) is related only to the prescribed boundary tractions (regardless of how the
inhomogeneities are distributed in the volume).
If the volume average of a field integrable variable is denoted as:
1
< T >≡ ∫ Tx dV (5-62)
()
V V
The unweighted volume average stress is:
σ ≡< σ > (5-63)

The average stress is:
1 1
σ = V ∫ σ dV = V ∫ xt dS (5-64)
ij V ij V ∂ ij
σ = 1 ∫ xn σ dS
ij ∂ i k kj
V V
Using the divergence theorem, the above surface integral can be converted into a
volume integral.
1 ∫ ∂( xσ kj ) dV 1 ∫ (δσ + x ∂σ kj ) dV 1 1 ∫ (σ + x ∂ σ kj ) dV (5-65)
i
V V x ∂ V V ik kj i x ∂ V V ij i x ∂
k k k
∂σ
Considering the equilibrium condition, kj = 0 the theorem is proved.
∂x
It can also be proven that: k
1
σ ij ≡< σ >= V ∫ xt dS (5-66)
ij V ∂ i j

Average strain and strain rate (Nemat-Nasser and Hori, 1999)
It can also be proven that the average strain is completely determined by the surface
displacements regardless of the distribution and volume fractions of the heterogene-
ities. The theorems on average stress, strain, and stress rate and strain rate serve as the
foundation of the representative volume element (RVE) methods.
1
u =< u >= V ∫ nu dS (5-67)
ji , ji , V ∂ i j