Page 175 - Mechanics of Asphalt Microstructure and Micromechanics

Fundamentals of Phenomenological Models 167

∂ε
Denoting C = ij , the above equation can be represented as:
ijkl ∂σ
kl
ε = C σ (6-3)
ij ijkl kl
∂ε
Generally, C = ij is a function of s kl , which may include a constant term, a lin-
ijkl ∂σ
kl
ear term, and a high order term. The simplest case is that it has only the constant term. In
that case, it represents the linear elasticity in the most well established elasticity, plasticity,
4
viscoelasticity, and viscoplasticity. In theory, C ijkl should have 3 = 81 constants. Due to the
symmetry of the stress tensor and the strain tensor, C ijkl = C jikl = C ijlk , therefor this will re-
duce the number of combinations to 6 6 = 36 (the nine combinations of ij or kl are re-
duced to six independent combinations and therefore will have 36 independent coef-
1
ficients). If the strain energy exists, then u = σε = 1 σ C σ , which means if one s ij
2 ij ij 2 ij ijkl kl
with s kl , u will not change and therefore, C ijkl = C klij . This symmetry will reduce the inde-
⎛ 36 6 ⎞
−
⎟
pendent constants to ⎜ ⎝ 2 + 6 = 21. This can be represented as a matrix C PQ , where
⎠
P and Q have six independent variations. So if C PQ is symmetric, it has only 21 indepen-
dent variables. Through the transformation of stress and strain tensors and making use
of the symmetry properties of the tensors, it can be proved that there are only five inde-
pendent parameters for the cross-anisotropic case and there will be only two indepen-
dent parameters for the isotropic case. A very good textbook on elasticity is by Green
and Zerna (1954).
For isotropic elasticity, there are several methods to express the stress-strain rela-
tionship in terms of the two independent parameters. Typical ones include the Lame’s
constant, Young’s modulus (E), shear modulus (G), Lame’s constant (l), Poisson’s ratio
(n), and bulk modulus (K). Their interrelationship is presented in Table 6.1.
The Hooke’s Law represented by these constants is respectively:

Lame’s Shear Young’s Poisson’s Bulk
Modulus, λ Modulus, G Modulus, E Ratio, ν Modulus, K
λ ,G G (3λ + 2 ) λ 3λ + 2G
G
λ + G 2(λ + G ) 3
, λν λ (1 2 ) λ (1 ν )(1 2 ) λ (1 ν )
ν
−
ν
−
+
+
2ν ν 3ν
−
−
λ , K 3(k λ ) 9(KK λ ) λ
−
−
2 3K λ 3K λ
+
+
, G ν 2Gν 2(1 ν ) 2(1 ν )
G
G
−
12ν 3(1 2 )
ν
−
, GK 3K − 2G 9KG 3K − 2G
+
+
3 3KG 2(3KG )
, E ν ν E E E
ν
−
+
(1 ν )(1 2 ) ν 2(1 ν ) 3(1 2 )
+
−
, EK 3(3KK − E ) 3EK 3K − E
(9K − E ) 9K − E 6K
TABLE 6.1 Interrelationship among elastic constants.