CONTINUUM MECHANICS for ENGINEERS

Ε ABδ iAδ jB = e ij = 0(ε)
o
Examination of Eqs 5.5-21 and 5.5-22 relates stress measures σji, P , and Ai
s AB. To discuss this relationship in the linear case, we must find an expression
for the Jacobian as we let ε → 0. Starting with the definition of J in the form
J =
we substitute F iA = u i,A + δ iA, etc., to get
J =
Carrying out the algebra and after some manipulation of the indices
J =
1
6
1
6
1
ε ε 6 ijk ABCFF iA jBFkC ( ) ( + ) ( + )
ε ε u + δ u δ u δ
ijk ABC i,A iA j,B jB k,C kC
[ ]
ε ε δ δ δ δ δ ε
2
ijk ABC iA jB kC + 3uk,C iA jB + 0(
)
where terms on the order of ε 2 and higher have not been written out explicitly.
Since ε ijkε ijk = 6 and ε ijkε ijC = 2δ kC,
J = 1 + u k,k + 0( ) (5.5-24)
Now we can evaluate Eqs 5.5-21 and 5.5-22 as ε → 0, that is, for the case of
a linear theory. Since u k,k is 0(ε).
( )
With a similar argument for Eq 5.5-22, we find
σji + 0 ε = + (5.5-25)
PAi Aj 0 ε
σ ji = s ABδ Aiδ Bj as ε → 0 (5.5-26)
Eqs 5.5-25 and 5.5-26 demonstrate that in linear theory Cauchy, Piola-Kirchhoff,
and symmetric Piola-Kirchhoff stress measures are all equivalent.
5.6 Moment of Momentum (Angular Momentum) Principle
Moment of momentum is the phrase used to designate the moment of the
linear momentum with respect to some point. This vector quantity is also
frequently called the angular momentum of the body. The principle of angular
momentum states that the time rate of change of the moment of momentum
o δ
ε 2
( )