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Objectivity and constitutive modelling
Fülöp2, T., Papenfuss3, C. and Ván12, P.
1RMKI, Dep. Theor. Phys., Budapest, Hungary 2Montavid Research Group, Budapest, Hungary
3TU Berlin, Berlin, Germany
– Objectivity – contra Noll
– Kinematics – rate definition
– Frame independent Liu procedure– Second and third grade solids (waves)
Rigid rotating frames:
xQhx )()(ˆ,ˆ tttt Noll (1958)
,ˆˆ bab
a cJc
Four-transformations, four-Jacobian:
QxQh 01ˆˆ
b
aab x
xJwhere
QcxQhcQxQhc 0
000
)(
01
ˆ
ˆ
c
ccc
Problems with the traditional formulation
Consequences:
four-velocity is an objective vector, time cannot be avoided, transformation rules are not convenient, why rigid rotation?
arbitrary reference frames reference frame INDEPENDENT formulation!
What is non-relativistic space-time?
M=EI?
M=R3R?
E
I
The aspect of spacetime:
Special relativity: no absolute time, no absolute space, absolute spacetime; three-vectors → four-vectors, x(t) → world line (curve in the 4-dimensional spacetime).
The nonrelativistic case (Galileo, Weyl): absolute time, no absolute space, spacetime needed (t’ = t, x’ = x − vt)
4-vectors: , world lines:
(4-coordinates w.r.t. an inertial coordinate system K)
The spacelike 4-vectors are absolute: form a 3-dimensional
Euclidean vector space.
x
t
)()(
t
tt
xx
x
0
Xa=(t0,Xi)
xa=(t,xi)World lines of a body – material manifold?
)(~ Xx
jiij
ij
it
jjt
ii
abb
a
FvxxXtx
txY ~~
01~~~~
01)
~,
~(
),(~~~~
deformation gradient}3,2,1{,},3,2,1,0{, jiba
cFcFcvcF~
~
~~~
~
)~
,~
(~ˆ~
11
00
11
00
1
0
cc
cFcFdt
d
ccdt
d
cF
cc
kk
j
ik
kj
i
itkk
jb
a
Objective derivative of a vector
Upper convected
Tensorial property (order, co-) determines the form of the derivative.
vcc 0c
Spacetime-allowed quantities:• the velocity field v [the 4-velocity field ],
• its integral curves (world lines of material points),
•
The problem of reference time: is the continuum completely relaxed,undisturbed, totally free at t0, at every material point?
In general, not → incorrect for reference purpose.
The physical role of a strain tensor: to quantitatively characterize not a change but a state:
not to measure a change since a t0 but to express the difference of the current local geometric status from the status in a fully relaxed, ideally undisturbed state.
v
1
)(2
1),(
2
1, i
jj
ii
jj
ij
i vvvvv
Kinematics
What one can have in general for a strain tensor:a rate equation + an initial condition
(which is the typical scheme for field quantities, actually)
Spacetime requirements → the change rate of strain is to bedetermined by ∂ivj
Example: instead offrom now on:
+ initial condition
(e.g. knowing the initial stress and a linear elastic const. rel.)
Remark: the natural strain measure rate equation is
generalized left Cauchy-Green
SR
Cauchy uE
SR
Cauchy vE
TRR vAAvA
Frame independent constitutive theory:
variables: spacetime quantities (v!)
body related rate definition of the deformation
balances: frame independent spacetime relations
pull back to the body (reference time!) Entropy flux is constitutive!
Question: fluxes?
Constitutive theory of a second grade elastic solid
Balances:
material space derivative
Tij first Piola-Kirchhoff stress
Constitutive space:
Constitutive functions:
Remark:
.0~
,0~
,0~
0
0
jii
jji
iijj
i
ii
vF
Tv
we
~
ji
kjii
ji
i FFvvee ~
,,~
,,~
,
iiji JsTw ,,,
:
:
:
ij
i
ji
ki
k
kjkakcb
ak
jiija
bba
FvxY
FvxY ~~~~
00~~~~,~~
01~~~
0)( 0 ji
Fj
ie
ii
ijj
iei FfsTKTvws
ji
Second Law:
Consequences:
Function restrictions:
- Internal energy
Dissipation inequality:
),,(
),,,(),,,(),,,(
jiiji
v
ie
i
jii
jiiij
jiii
FveKsTswJ
FvesFveTFvew
j
),2
(2
jiF
ves
,0~
0 ii Js ,0)
~()
~()
~(
~000 j
iji
ijij
ji
ii
ii
i vFTvweJs
0:11
0 FTq fTT j
iFR
Constitutive theory of a third grade elastic solid
Balances:
Constitutive functions:
Constitutive space:Second Law:
jkii
kjji
k
jii
jji
jiik
jki
j
iijj
i
ii
vF
vF
Tv
Tv
we
0~~
,0~
,0~~
,0~
,0~
0
0
0
ji
klji
kjii
jki
ji
i FFFvvvee ~
,~
,,~
,~
,,~
,
iiji JsTw ,,,
,0~
0 ii Js
:ˆ
:
:ˆ
:
:
ijk
ij
ji
i
,0~~ˆ~~ˆ
)~
()~
()~
(~
0
000
i
kjji
kijkik
jki
jj
i
jij
ii
jijj
ii
ii
ii
vFTv
vFTvweJs
0~~
)(
~0
ji
FkFe
kj
ki
eji
e
ijj
iei
Fsss
TssT
Tvws
ji
kji
Consequences:
Function restrictions:
- Internal energy
Dissipation inequality:
)~
,,~
,,(....
),~
,,~
,,(),~
,,~
,,(
ji
kjii
jiji
v
ie
i
ji
kjii
ji
ji
kjii
jiij
FFvveKsTswJ
FFvvesFFvveT
j
)...,)(Tr22
(2
jiF
ves FF
0:11
~00
FTTq ffTT j
ikj
i FRFRRR
Double stress
0 1 2 3 4k
0 .5
1 .0
1 .5
2 .0
2 .5c2
Waves- 1dQuadratic energy – without isotropy
...~~
2
ˆ
2)
~,( i
jkj
iki
jj
ij
ikj
i FFFFFFf
.0~~
,0~~
ˆ~
0
TF
FFTFT
xxtt
txxxx
0:~
~00
FTT ffj
ikj
i FkF
- Stable- Dispersion relation
2
2
0
222
1
ˆ
k
kk
kc
0
0
ˆ
Summary
- Noll is wrong.
- Aspects of non-relativistic spacetime cannot be avoided.
- Deformation and strain – wordline based rate definition.
- Gradient elasticity theories are valid.
Thank you for your attention!
Continuum Physics and Engineering Applications 2010 June
Real world –Second Law is valid
Impossible world –Second Law is violated
Ideal world –there is no dissipation
Thermodinamics - Mechanics