Page 221 - Intro to Tensor Calculus

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Figure 2.3-4. Stress vectors acting upon an element of volume

which is a summation of all the body forces and surface tractions acting upon our material. Here % is the
density of the material, dS is an element of surface area, and dτ is an element of volume.
~
The resultant moment M about the origin is similarly expressed as
ZZ ZZZ
~
~
M = r (n) dS + %(~ × b) dτ. (2.3.4)
r
~ × ~ t
S V
The global motion of the material is governed by the Euler equations of motion.
• The time rate of change of linear momentum equals the resultant force or
ZZZ ZZ ZZZ
d (n)
~
~
%~vdτ = F = ~ t dS + %bdτ. (2.3.5)
dt V S V
This is a statement concerning the conservation of linear momentum.
• The time rate of change of angular momentum equals the resultant moment or

ZZZ ZZ ZZZ
d (n)
~
~
r
r
%~ × ~vdτ = M = ~ × ~ t dS + %(~ × b) dτ. (2.3.6)
r
dt V S V
This is a statement concerning conservation of angular momentum.
The Stress Tensor
Define the stress vectors
1
~ t = σ 11 ˆ e 1 + σ 12 ˆ e 2 + σ 13 ˆ e 3
2
~ t = σ 21 ˆ e 1 + σ 22 ˆ e 2 + σ 23 ˆ e 3 (2.3.7)
3
~ t = σ 31 ˆ e 1 + σ 32 ˆ e 2 + σ 33 ˆ e 3 ,
ij
where σ ,i, j =1, 2, 3 is the stress tensor acting at each point of the material. The index i indicates the
i
i
coordinate surface x = a constant, upon which ~ t acts. The second index j denotes the direction associated
i
with the components of ~ t .