CONTINUUM MECHANICS for ENGINEERS

With the use of Eqs 5.10-29 and 5.10-30 it is evident that
+
n = Q n
k kj j
(5.10-31)
All that remains is to determine how the stress components transform
under superposed rigid body motion. All the results are now in place to find
this transformation. In the superposed rigid body motion frame, the stress
+ vector ti can be written as
+ + + +
t = σ n = σ Q n
i ij j ij jk k
(5.10-32)
where Eq 5.10-31 has been used. The assumed transformation for the stress
vector yields
+
ti = Qijtj = Qijσn jk k
(5.10-33)
where, as in Eq 5.10-32, Cauchy’s stress formula has been used. Equating
the stress vector in the superposed rigid body reference frame leads to
+ ( σijQ − Q σ
jk ij n jk) = 0 k
which holds for all n k. Thus, the terms in parentheses must equal zero.
Multiplying the remaining terms by Q mk results in the following expression
for stress component transformation under superposed rigid body motion:
+
σ = Q Q σ
ij im jn mn
(5.10-34)
In plasticity, as well as explicit finite element formulation, the stress constitutive
response is usually formulated in an incremental form. This means
that the stress rate is used. The stress rate must be objective, meaning
˙ +
σ = Q˙ Q σ + Q Q˙ σ + Q Q σ˙
ij im jn mn im jn mn im jn mn
(5.10-35)
It is clear from this equation that the stress rate is not objective even though
the stress is objective. This result is serious since the stress rate as shown in
Eq 5.10-35 could not be used as a response function or in the independent
variable list of a response function. Luckily, there are several ways to express
a form of the stress rate in an invariant manner.
One way to obtain an objective stress rate is found from using the spin
tensor Wij. Using Eqs 5.10-25 and 5.10-15 to solve for in terms of the spin
˙ +
Q = W Q −Q
W
ip ij jp im mp
˙Q ij
(5.10-36)