Page 149 - Mechanics of Asphalt Microstructure and Micromechanics

Mixture T heor y and Micromechanics Applications 141

Average strain and average strain rate:
1 1
ε ij ≡< ε >= V ∫ ε dV = V ∫ ( n u + u n dS (5-68)
)
ij V ij ∂ 2 i j i j
−
ε ≡< ε >= 1 ∫ 1 (nu j + u n j )dS (5-69)
ij
ij
i
i
V ∂V 2
An important relationship among the averages (and also the strain energy) is the
Hill’s Lemma. A concise proof is documented in Qu and Ckerkaoui (2006).
σε − σ ε ij = 1 S ∫ ( u − x ε ij)( σ n − σ ikn dS (5-70)
)
ij
ij ij V i j ik k k
1
Where σε = V ∫ σ ε dv
ij ij
V ik ij
Therefore, effective properties are defined as:
−
σ = L ijkl ε ij
ij
ε ij = M ijkl σ kl
Therefore:
−1
I
LM = ML = or L = M (5-71)
5.3.2 General Philosophy of Estimating Effective Properties
For a finite homogeneous linearly elastic solid with elasticity tensor L and compliance
tensor M, containing a linearly elastic and homogeneous inclusion Ω of arbitrary geom-
Ω
Ω
etry, with elasticity and compliance tensors L and M .
x
u =•ε on ∂ V
o
σ = L ε : o (5-72)
Hooke’s law:
⎪ ⎪ ε + ε x ( )) in M = V − Ω
o
d
⎧L :(
σ = ⎨ Ω (5-73)
o
d
⎩ ⎪ L :( ε + ε x ( )) in Ω
−
d
⎪ ⎪ σ + σ x ( )) in M = V Ω
o
⎧M :(
ε = ⎨ Ω o d (5-74)
⎩ ⎪ M :( σ + σ x ( )) in Ω
e (x) and s (x) are perturbed strain and stress.
d
d
The Eigenstrain theorem states that the heterogeneity can be treated by finding a
*
suitable strain field e (x) (eigenstrain) in Ω, such that the equivalent homogeneous solid
has the same strain and stress fields as the actual heterogeneous solid has under the
applied tractions or displacements.
⎧0in M
ε () = ⎨ (5-75)
*
x
⎩ ε ⎪ * in Ω