Non-objectivity of the time derivative of Cauchy stress Objective stress rate

figure 1. undeformed , deformed material element, , elemental cube cut out deformed element.

for physical understanding of above, consider situation shown in figure 1. in figure components of cauchy (or true) stress tensor denoted symbols




s

i
j




{\displaystyle s_{ij}}

. tensor, describes forces on small material element imagined cut out material deformed, not objective @ large deformations because varies rigid body rotations of material. material points must characterized initial lagrangian coordinates




x

i




{\displaystyle x_{i}}

. consequently, necessary introduce so-called objective stress rate






s
∘



i
j




{\displaystyle {\overset {\circ }{s}}_{ij}}

, or corresponding increment



Δ

s

i
j


=



s
∘



i
j


Δ
t


{\displaystyle \delta s_{ij}={\overset {\circ }{s}}_{ij}\delta t}

. objectivity necessary






s
∘



i
j




{\displaystyle {\overset {\circ }{s}}_{ij}}

functionally related element deformation. means






s
∘



i
j




{\displaystyle {\overset {\circ }{s}}_{ij}}

must invariant respect coordinate transformations, particularly rigid-body rotations, , must characterize state of same material element deforms.

the objective stress rate can derived in 2 ways:

by tensorial coordinate transformations, standard way in finite element textbooks
variationally, strain energy density in material expressed in terms of strain tensor (which objective definition)

while former way instructive , provides useful geometric insight, latter way mathematically shorter , has additional advantage of automatically ensuring energy conservation, i.e., guaranteeing second-order work of stress increment tensor on strain increment tensor correct (work conjugacy requirement).