Page 34 - Mechanics of Asphalt Microstructure and Micromechanics

Introduction and Fundamentals for Mathematics and Continuum Mechanics 27

The rate of work by body force: ∫ V ρbvdV (1-131)
ii

The power of mechanical force is: P(t) = V ∫ ρb v dV + ∂ V ∫ t v dS (1-132)
ii
ii
The rate of thermal energy includes heat supply r (heat supply per unit mass per
unit time) and heat flux q (heat flux per unit surface area per unit time). It can be repre-
sented as:
Qt () ∫ ρ rdV − ∫ q n dS (1-133)
V ∂ V ii
The rate of internal energy U accounts for all the other energies stored in the medi-
um. It is abstract at this stage. The specific internal energy u can be introduced so that:
.
.
Ut () ∫ ρ udV Ut () = ∫ ρ udV (1-134a, b)
V V
Using the above terms, the energy conservation law can be represented as:
. .
Ut () + K t () = P t () + Q t () (1-135)

. d 1 .
Kt() = V ∫ ρ v v dV = V ∫ ρ v v dV
dt 2 ii i i
By using the divergence theorem and the stress-surface traction relationship, one
can convert the surface integral ∫ t i v i dS into a volume integral:
V
∂ σ ( v ) ∂σ v ∂
∫ ∂ t v dS = ∫ ∂ σ ij n v dS = ∫ x ∂ ij i dv = ∫ V ∂ ij j v +σ ij x ∂ i ) dV
(
i
ii
j i
V
V
V
j x j j
V ∫ qn dS = V ∫ q ∂ i dV = V ∫ div q dV
()
∂ ii x ∂
i
.
.
∫ ρudV + ∫ ρv vdV ∫ ρb vdV + V ∫ ( ∂ σ ij v + σ i ij v ∂ x ∂ i ) dV + ∫ ρ rdV − ∫ div q dV
( )
x ∂
i
ii
i
i
V
V
V
j j V V
. . ∂ σ v ∂
∫ ρudV ∫ (− ρv + ρb + x ∂ ij ) vdV + V ∫ σ ij i dV + ∫ ρ rdV − ∫ div q dV
()
i
i
i
i
x
V
V
j ∂x j V V
⎛ . ∂σ ⎞
⎜
By using the momentum equation, −ρv i + ρb i + ij ⎟ 0, and
⎝ ∂x j ⎠
∫ σ ij x ∂ v ∂ i dV = V ∫ σ ( D + W dV = V ∫ σ ij D dV
)
ij
ij
ij
ij
V
j
The energy equation will reduce to:
.
+
ρu− σ D − ρr div q) = 0 (1-136)
(
: