Objectivity and constitutive modelling Fülöp 2, T., Papenfuss 3, C. and Ván 12, P. 1 RMKI, Dep....
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Transcript of Objectivity and constitutive modelling Fülöp 2, T., Papenfuss 3, C. and Ván 12, P. 1 RMKI, Dep....
![Page 1: Objectivity and constitutive modelling Fülöp 2, T., Papenfuss 3, C. and Ván 12, P. 1 RMKI, Dep. Theor. Phys., Budapest, Hungary 2 Montavid Research Group,](https://reader035.fdocuments.in/reader035/viewer/2022081519/56649c745503460f949273dd/html5/thumbnails/1.jpg)
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)
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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
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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!
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What is non-relativistic space-time?
M=EI?
M=R3R?
E
I
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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
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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
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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
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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
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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
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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?
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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~~~
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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
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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
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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
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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
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Summary
- Noll is wrong.
- Aspects of non-relativistic spacetime cannot be avoided.
- Deformation and strain – wordline based rate definition.
- Gradient elasticity theories are valid.
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Thank you for your attention!
Continuum Physics and Engineering Applications 2010 June
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Real world –Second Law is valid
Impossible world –Second Law is violated
Ideal world –there is no dissipation
Thermodinamics - Mechanics