Lecture #4 Objective stress rate (Rong Tian, posted on iMechanica)
The Cauchy stress is related to PK2 stress as
σ = J −1φ* [S ]
(43)
The time derivative of PK2 stress Sɺ is objective, whereas that of Cauchy stress is not.
4.1 Truesdell rate
Truesdell stress rate is defined by
σ ∇ = J −1φ* Sɺ 
(44)
where Sɺ is the material time derivative of PK2 stress.
Using σ ∇T to denote the Truesdell rate, we derive it as
σ ∇T = J −1φ* Sɺ 
d

= J −1F  ( JF −1σ F -T )  F T
 dt

-T
−1
−1
ɺ
ɺ
= J F  JF σ F + JF −1σ F -T + JF −1σɺ F -T + JF −1σ Fɺ -T  F T
Note that
d
ɺ −1 + FFɺ −1 = 0 , we have
( FF −1 ) = FF
dt
ɺ −1 = −F −1L
Fɺ −1 = −F −1FF
Consider
Jɺ = J trace ( D ) = J trace ( L )
(45)
(46)
(47)
We obtain
σ ∇T = J −1F  JɺF −1σ F -T − JF −1Lσ F -T + JF −1σɺ F -T + JF −1σ Fɺ -T  F T
= J −1  Jtrace ( L ) σ − JLσ + J σɺ − J σ LT 
(48)
= σɺ − Lσ − σ LT + trace ( L ) σ
This is the Truesdell stress rate.
4.2 Green-Naghdi rate
Ignore the stretch component of deformation and assume
F = RU ≈ R
then we obtain
ɺ −1 = RR
ɺ T , trace ( RR
ɺ T ) = trace ( Ω ) = 0
L = FF
(49)
(50)
where Ω is the angular velocity matrix, Ωii = 0 . Substitute (50) into the Truedell rate
(48), we obtain Green-Naghdi rate as
ɺ Tσ + σ RR
ɺ T
σ ∇G = σɺ − RR
(51)
4.3 Jaumann rate
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Using polar decomposition F = RU and spin tensor of W =
1
L − LT ) , we can obtain
(
2
ɺ T − 1 R ( UU
ɺ -1 + U -T U
ɺ ) RT
W = RR
2
(52)
If we assume
ɺ T
W ≈ RR
and substitute into the Green-Naghdi rate (51), then we obtain Jaumann rate
σ ∇G = σɺ − Wσ + σ W
(53)
(54)
The relationship among Truesdell, Green-Naghdi, and Jaumann rates are summarized in
Figure 3.1. It is noted that the Jaumann and Green-Naghdi stress rates are the
approximate of the Truesdell rate; they should not be expected to be the same accurate
as the Truesdell rate for a general non-rigid-body motion. This might explain the
difference of the shear stress computed by the three different stress rates for the same
shear deformation.
σ
∇T
Truesdell stress rate
= σɺ − Lσ − σ LT + trace ( L ) σ
Truesdell stress rate (Kirchhoff)
J σ ∇T = τ ∇J
τ ∇J = τɺ − Lτ − τ LT
Assume F ≈ R , ignoring the stretch component of F
Simplification 1
Green-Naghdi stress rate
ɺ T − RR
ɺ Tσ
σ ∇G = σɺ + σ RR
ɺ T − 1 R ( UU
ɺ -1 + U -T U
ɺ ) R T ≈ RR
ɺ T
Assume W = RR
2
Simplification 2
Jaumann stress rate
σ ∇J = σɺ + σ W − Wσ
Figure 3.1. Relationship among Truesdell, Green-Naghdi, and Jaumann rates.
Quick question: why are we often using Jaumann rate instead of Truesdell rate?
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