Page 150 - Mechanics of Asphalt Microstructure and Micromechanics

142 Ch a p t e r Fiv e

The corresponding stress and strain fields:
d
⎧ ⎪ L :( ε + ε ( )) in M in M
o
x
σ() = : L ε ( () − ε ()) = ⎨ (5-76)
*
x
x
x
ε
d
*
o
⎩ ⎪ L :( ε + ε x() − ε x()) in Ω
Integral operator:
ε x() ≡ S x; ε ) (5-77)
d
*
(
ij ij
Similarly, the concept of eigenstress can be introduced:
⎧0 in M
*
σ () = ⎨ * (5-78)
x
⎩ ⎪ σ in Ω
So that elasticity tensor is uniform everywhere, including Ω, so that the strains and
stress can be represented in the following equations.
ε() = ε + ε ()
o
d
x
x
d
⎧ ⎪ L :( ε + ε ( )) in M
o
x
σ() = L ε : () + σ () = ⎨ (5-79)
*
x
x
x
σ
o
*
d
x
⎩ ⎪ L :( ε + ε ( )) +σ ()x in Ω
−
−
N
N
ε ij = 1 V ∫ ε dV = 1 ∑ ∫ ε dV = ∑ c ε = ε (5-80)
V ij V Ω i ij i i
i=0 i=0
Where c i is the volume fraction of the ith heterogeineity and c 0 is the volume fraction
of the matrix. Strain concentration tensor is so defined that (Qu and Ckerkaoui, 2006):
ε = A ε (5-81)
i i
–
e is the average strain for the entire composite material. Equation 5-80 can be further
written in the following format:
N
N
−
−
−
−
ε
c ε =− ∑ c ε =− ∑ c A ε − (5-82)
ε
0 0 i i i i
=
=
i 1 i 1
For stresses, the following similar representations can be used:
−
−
N
σ = 1 V ∫ σ dV = σ = ∑ c σ (5-83)
ij V ij i=0 i i
− −
σ = L ε
i i i i ( 0 , N)
Making use of Equation 5-82:
− − N −
c L
σ = cL ε + ∑ i i ε i (5-84)
0
00
= i 1
− N −
σ = [L 0 + ∑ i ( c L i − L 0 )A i ε ] (5-85)
= i 1