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