Page 33 - Mechanics of Asphalt Microstructure and Micromechanics

26 Ch a p t e r O n e

1.6.4.5 Angular Momentum
For any finite domain (body), the angular momentum equilibrium equation can be rep-
resented as:
∫∫ x t dS + ∫∫∫ ρ x bdV = 0 (1-127)
×
×
n ()
S V
The indicial format can be represented as:
×=
x t ex t e = ex n σ e
pqr p q r pqr p j jq r
xb×= e x b e
pqr p q r
Using the divergence theorem for surface integral one obtains:
⎧ ∂ ⎫ ⎪
⎪
∫∫∫ e pqr ⎨ x ∂ x ( σ jq ) + ρ x b dV = 0
p q ⎬
p
V ⎩ ⎪ j ⎭ ⎪
⎧ ∂( x σ ) ⎫
⎪
⎪
e pqr ⎨ p jq + ρ xb ⎬ = 0
pq
⎩ ⎪ x ∂ j ⎭ ⎪
⎛ ∂σ ⎞
e { x ⎜ jq + ρ b ⎟ +σ } = 0
pqr p x ∂ q pq
⎝ j ⎠
Since:
∂σ
jq
+ ρb = 0
∂x q
j
e σ = 0
pqr pq
or
σ = σ (1-128)
pq qp
This actually means that stress tensor is symmetric. The condition is that there is no
distributed angular momentum.
1.6.4.6 Energy Conservation
The energy conservation principle can be stated as: for any domain of interest, the rate
of change of the kinetic and internal energy is equal to the rate of work done by the
surface tractions and body forces, plus other rates of energies entering or leaving the
surface. Other energies may include all types of energies but are limited to thermal and
mechanical energies in this book. While a small section may focus on some polar me-
dium, the majority of the book deals with non-polar media.

1
The kinetic energy in any domain: K(t) = ∫ ρvv dV (1-129)
2 V ii

The rate of work by surface traction: V ∫ tv dS (1-130)
∂ ii