Jaumann rate of the Cauchy stress Objective stress rate

the jaumann rate used in computations 2 reasons

recall spin tensor




w



{\displaystyle {\boldsymbol {w}}}

(the skew part of velocity gradient) can expressed as

w

=



r
˙



⋅


r


t


+


1
2


 

r

⋅
(



u
˙



⋅


u


−
1


−


u


−
1


⋅



u
˙



)
⋅


r


t




{\displaystyle {\boldsymbol {w}}={\dot {\boldsymbol {r}}}\cdot {\boldsymbol {r}}^{t}+{\frac {1}{2}}~{\boldsymbol {r}}\cdot ({\dot {\boldsymbol {u}}}\cdot {\boldsymbol {u}}^{-1}-{\boldsymbol {u}}^{-1}\cdot {\dot {\boldsymbol {u}}})\cdot {\boldsymbol {r}}^{t}}

thus pure rigid body motion

w

=



r
˙



⋅


r


t


=

Ω



{\displaystyle {\boldsymbol {w}}={\dot {\boldsymbol {r}}}\cdot {\boldsymbol {r}}^{t}={\boldsymbol {\omega }}}

alternatively, can consider case of proportional loading when principal directions of strain remain constant. example of situation axial loading of cylindrical bar. in situation, since

u

=

[





λ

x








λ

y









λ

z






]



{\displaystyle {\boldsymbol {u}}=\left[{\begin{array}{ccc}\lambda _{x}\\&\lambda _{y}\\&&\lambda _{z}\end{array}}\right]}

we have

u
˙



=

[








λ
˙




x











λ
˙




y












λ
˙




z






]



{\displaystyle {\dot {\boldsymbol {u}}}=\left[{\begin{array}{ccc}{\dot {\lambda }}_{x}\\&{\dot {\lambda }}_{y}\\&&{\dot {\lambda }}_{z}\end{array}}\right]}

also,

u


−
1


=

[




1

/


λ

x







1

/


λ

y








1

/


λ

z






]



{\displaystyle {\boldsymbol {u}}^{-1}=\left[{\begin{array}{ccc}1/\lambda _{x}\\&1/\lambda _{y}\\&&1/\lambda _{z}\end{array}}\right]}

of cauchy stress

therefore,

u
˙



⋅


u


−
1


=

[








λ
˙




x



/


λ

x











λ
˙




y



/


λ

y












λ
˙




z



/


λ

z






]

=

u

−
1





u
˙





{\displaystyle {\dot {\boldsymbol {u}}}\cdot {\boldsymbol {u}}^{-1}=\left[{\begin{array}{ccc}{\dot {\lambda }}_{x}/\lambda _{x}\\&{\dot {\lambda }}_{y}/\lambda _{y}\\&&{\dot {\lambda }}_{z}/\lambda _{z}\end{array}}\right]=u^{-1}{\dot {u}}}

this once again gives

w

=



r
˙



⋅


r


t


=

Ω



{\displaystyle {\boldsymbol {w}}={\dot {\boldsymbol {r}}}\cdot {\boldsymbol {r}}^{t}={\boldsymbol {\omega }}}

in general, if approximate

w

≈



r
˙



⋅


r


t




{\displaystyle {\boldsymbol {w}}\approx {\dot {\boldsymbol {r}}}\cdot {\boldsymbol {r}}^{t}}

the green–naghdi rate becomes jaumann rate of cauchy stress

σ
△


=



σ
˙



+

σ

⋅

w

−

w

⋅

σ



{\displaystyle {\overset {\triangle }{\boldsymbol {\sigma }}}={\dot {\boldsymbol {\sigma }}}+{\boldsymbol {\sigma }}\cdot {\boldsymbol {w}}-{\boldsymbol {w}}\cdot {\boldsymbol {\sigma }}}