Page 147 - Mechanics of Asphalt Microstructure and Micromechanics

Mixture T heor y and Micromechanics Applications 139

There are a few interesting features for the Green function, for example, symmetry
for locations x, y, that is the displacement at point y produced by a unit force at point x
will be equal to the displacement at point x produced by a unit force at point y. For
isotropic materials, the Green function can be further expressed as:
(
)
∞
i
j j
j
Gx y = 1 {(34 v)δ + ( x − y x − y ) } (5-58)
−
i
)
(,
−
ij ij 2
16πμ (1 − vx y xy
)
−

The above solution refers to the unbound (infinite) domain problem. It can include
both the stress and displacement boundary conditions.
5.2.1.2 Eigenstrains
Eigenstrains refers to inelastic strains such as thermal strains, plastic strains, etc. It is
*
typically represented as e ij while e ij and e ij often represent total strain and elastic strain.
Hooke’s law can be presented as:
σ = L ( ε − ε ) or e = M σ (5-59a, b)
*
ij ijkl kl kl ij ijkl kl
By replacing σ = L ( ε − ε ) into the equilibrium equation, it can be proven that
*
ijkl
kl
kl
ij
the eigenstrain problem is equivalent to the problem where the body force is equal to
f =− L ε . Therefore, the eigenstrain problem can be related to the Green function. It
*
i ijkl kl j '
can be shown that the resulted displacement field will be:
∞
x y)
ux() = ∫ ∞ L ε * y ( ) ∂ G (, dy (5-60)
mi
i −∞ mjkl kl y ∂
j
From this displacement, the strain field and stress field can be obtained.
A typical problem involves the strain and stress field due to an ellipsoid inclusion
with uniform eigenstrain in the inclusion. The solution to this problem is the Eshelby
solution, which has significant meaning in micromechanics. The derivation can be
found in several references such as Qu and Cherkaoui (2006). The problem can be stated
as follows:
For an unbounded domain with an ellipsoid inclusion of size a 1 , a 2 , and a 3 , embed-
x x x
) +
) +
) ≤
ded at location (0,0,0) [domain, ( 1 2 ( 2 2 ( 3 2 1], if there is a uniform eigen-
a 1 a 2 a 3
*
strain e ij within the inclusion, what is the total strain and the stress in the inclusion?
Through the use of the Green’s function, the solution can be obtained as:
ε x() = S ε * (5-61)
ij ijkl kl
Where S ijkl is the Eshelby inclusion tensor. S ijkl poses certain symmetry such that
S ijkl = S jikl = S ijlk but S ijkl ≠ S klij . It is nonsingular and independent of the eigenstrain. With
the Eshelby inclusion tensor, the stress can be calculated in the inclusion,
σ = L ( S ε − ε ) . The traction continuity will result in surface tractions. The ele-
*
*
ij ijkl ijkl kl kl
ments for the Eshelby inclusion tensor of inclusions of different shapes are different. It
is related to the shape of the inclusions. The elements of the Eshelby tensor for special
cases such as ellipsoids and spheres are documented in Appendix B. These Eshelby ten-
sor elements for special cases can be used for computing the effective modulus pre-
sented in the following sections.