Page 32 - Mechanics of Asphalt Microstructure and Micromechanics

Introduction and Fundamentals for Mathematics and Continuum Mechanics 25

The density is a scalar function of position and time and thus may vary from point
to point within a given body.
ρ = ρ(, )
xt
i
or
ρ = ρ(, )
xt
1.6.4.3 Mass Conservation (Material Derivative)
For any domain, the mass cannot be destroyed or produced. Therefore, the mass flow-
ing in or out of a specific volume and its local mass change (within the volume) will
sum up to zero.
∫∫ ρvndS + ∂ t ∂ ∫∫∫ ρdV = 0 (1-123)
•
S V
Considering that: ∂( ρv )
∫∫ ρvndS = ∫∫∫ x ∂ i dV
•
S V i
If the above equations hold for any domain, one has the following equation:
∂(ρv ) ∂
i + ρ = 0
∂x ∂t
i
∂(ρv ) ∂ρ ∂v
i = v + ρ i
∂x ∂x i ∂x
i i i
Considering the material derivative in Section 3.2.5, one has:
dρ ρ ∂ ρ ∂
= + v
dt t ∂ x ∂ i
i
Therefore, the mass conservation or the continuity equation can be represented as:
dρ v ∂
+ ρ i = 0 (1-124)
dt x ∂
i
1.6.4.4 Linear Momentum Equilibrium Equation
For any finite domain (body), the force equilibrium equation can be represented as:
∫∫ tdS + ∫∫∫ ρ bdV = 0 (1-125)
n ()
S V
∫∫ i ij ∫∫∫ b dV = 0
Its indicial format: nσ dS + ρ
j
S V
By applying the divergence theorem, one has:
∂σ
∫∫∫ ( ij + ρbdV = 0
)
j
V ∂x i
If the above equation holds for any of the domain, the following will also hold:
∂σ
ij
+ ρb = 0 (1-126)
∂x j
i