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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)
Problems with the traditional formulation
Rigid rotating frames:
tˆ  t ,
xˆ  h(t )  Q(t )x
Noll (1958)
Four-transformations, four-Jacobian:
cˆ  Jˆ c ,
a
a b
b
where
a
ˆ
 1 0
x
a
ˆ

J b  b  


x
 h  Qx Q 
0
0  c 0  
 cˆ 0   1

c
   
 



0
 cˆ  h  Q

 x Q  c   (h  Q

x
)
c

Qc
  
  

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=R3R?
M=EI?
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)
t 
 t 



x(t )  
4-vectors:  x  , world lines:
 
 x(t ) 
(4-coordinates w.r.t. an inertial coordinate system K)
0
The spacelike 4-vectors  
x
Euclidean vector space.
are absolute: form a 3-dimensional
Xa=(t0,Xi)
~
x ( X)
xa=(t,xi)
World lines
of a body –
material
manifold?
0j   1 0j 
~a ~ ~a  t  ~ ~  1
(t ,  j )   ~ ~ i ~ ~ i    ~ i ~ i 
Y b  b x   ~ i
 x (t , X i ) 
 t x  j x   v F j 
a, b {0,1,2,3},
i, j {1,2,3}
deformation gradient
Objective derivative of a vector
d 0

c
~ b  c0  ~ ~ 
dt
 a cˆ   ~ 1 j k ( t , i )  
d
~
1 j
k
F
c

k


 F kc
 dt
~ 0
0


c
c


~
 F 1 c  v  c  F 1  c  F  c 



 





~ ~ 1 j k 
i F k c 

 
~ 0
i c



c  c  c0 v
Upper convected
Tensorial property (order, co-) determines the form of the derivative.
Kinematics
Spacetime-allowed quantities:
1
•
the velocity field v [the 4-velocity field   ],
 v
 
•
its integral curves (world lines of material points),
1
1
j
i
 i v , ( i v   j v ), ( i v j   j v i )
2
2
The problem of reference time: is the continuum completely relaxed,
undisturbed, totally free at t0, at every material point?
•
j
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.
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 be
determined by ∂ivj
Cauchy
E
  R u 
Example: instead of
from now on:
S
 Cauchy   v S+ initial condition
E
R
(e.g. knowing the initial stress and a linear elastic const. rel.)
Remark: the natural strain measure rate equation is
   v A  A v T generalized left Cauchy-Green
A
R
R
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:
~ i
 :  0 e  i w  0,
~ ij
i
i


v


T

0
,
i :
0
j
~ i
j
i

 i : F j   j v  0i j.
~ material space derivative

Tij first Piola-Kirchhoff stress
Constitutive space:
~ i ~ i i ~ i
e, i e, v ,  j v , F j , k F j
Constitutive functions:
wi , T ij , s, J i
Remark:
0kj 
~ a ~ ~ a  1 0 j  ~ ~ a ~ ~ a  0k
Y b  b x   ~ i ~ i , kY b  kc x   ~ ~ i ~ ~ i 
v F j 
 k v k F j 
Second Law:
~ i
~ i
~ ij
~ j
~
j
i
i i

0 s  i J   ( 0e  i w0)si (i J0v 0, jT )   i ( F j  i v )  0,
Consequences:
Function restrictions:
wi (e, v i , F i j ), T ij (e, v i , F i j ), s (e, v i , F i j ),
J i   e swi   v j sT ji  K (e, v i , F i j )
v2 i
s (e  , F j )
2
- Internal energy
Dissipation inequality:
 


1 1 i
i 0f F i j  0
 Fii jj 
f : F
 i ( e s )w qv T   i K Te0sT
0 F j
T T
i
ij
j R
Constitutive theory of a third grade elastic solid
Balances:
~ i
 :  0 e   i w  0,
~ ij
i
i


v


T

0
,
Constitutive functions:
i :
0
j
i
ij
i
~
~
i
ik
i
j
w
,
T
,
s
,
J
ˆi :  0  j v   jkT  0 j ,
~ i
j
i

 i : F j   j v  0i j ,
~ i
i

ˆ jk i : ~
 k F j   kj v  0i jk
~ i ~ i ~ i i ~ i ~ i
e, i e, v ,  j v ,  jk v , F j , k F j , kl F j
Constitutive space:
Second Law:
~ i
~ i
~ ij
~ j
i
j
i

0 s  i J   ( 0e  i w )~ ii ( 0v   jT )   i ( F j  i v )


s
 i J~  0, ~
0
~
~
j
ˆ jk i  F i j   v i  0,
 ˆi 0  j vi   jkT ik  
k
kj




Consequences:
Function restrictions:
~ i i ~ i
i ~ i
i ~
T (e, v ,  j v , F j ,  k F j ), s(e, v ,  j v , F j , k F i j ),
i
i
ji
i ~ i
i ~
J   e sw   v j sT  ....  K (e, v ,  j v , F j ,  k F i j )
ij
i
v2 
s(e   Tr (F  F )  ..., F i j )
2 2
- Internal energy
Dissipation inequality:


Double stress
 i ( e s) wi  v jT ij 


   
 1 i1
 i
0
~i j
~
es kRT Rk T   0FFi ij jsfk 0~ RF i j s ~ FFi j fj : 0F  0
q  R e sT j T
k
k
es
 T T

Waves- 1d
Quadratic energy – without isotropy
 i j ˆ ~ i ~ k j
~ i
f ( F j ,  k F j )  F j F i   k F j  F i  ...
2
2
i
T    T   
0 Fi j


~
f  0 k  ~ F i j f : F  0
k
~
~
~
T  F    xxT  ˆ  xx F   t F  0,
~
~
 0 tt F   xxT  0.
ˆ
 0
c2
2.5
2.0
- Stable
- Dispersion relation
1.5

0
1.0
   k   ˆk
c2    
2
 k  0 1  k
2
2
2
0.5
k
0
1
2
3
4
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
Thermodinamics - Mechanics
Impossible world –
Second Law is violated
Real world –
Second Law is valid
Ideal world –
there is no dissipation