Nonlinear Continua Part 4 pps

2.13.2 The Eulerian strain rate tensor and the spin vorticity tensor We can decompose the velocity gradient tensor into its symmetric and skew-symmetric components: where, wd = wdW = 1 2

2.13 Strain rates 51

It is important to remember that the functional dependence is

w

[˙dD = w[˙dD({E> w) = (2.109b) Using the chain rule in Eq (2.109a),

w

[˙dD = wyd|ow[oD= (2.109c)

We define in the spatial configuration the velocity gradient tensor,

wl = wyd|o wgdwgo > (2.110a)

we can write the above as

Hence, we can write Eq.(2.109c) as

w

[˙dD = wodo w[oD (2.111a) and therefore,

w

It is important to realize that the above is the material time derivate of the deformation gradient tensor,wX˙ =G

w

 X

Gw 2.13.2 The Eulerian strain rate tensor and the spin (vorticity) tensor

We can decompose the velocity gradient tensor into its symmetric and skew-symmetric components:

where,

wd = wdW = 1

2 (

is the Eulerian strain rate tensor (defined in the spatial configuration) and,

w$ = w$W = 1

2 (

is the spin or vorticity tensor , also defined in the spatial configuration Let us assume a deformation process referred to a fixed Cartesian system The principal directions ofwUform, in the reference configuration, a Cartesian system known as Lagrangian system The principal directions ofw

Vform, in the spatial configuration, a Cartesian system known as a Eulerian system (Hill 1978)

Fig 2.5 Rotations

We can go from one of the above-defined coordinate systems to another one using the rotation tensors sketched in Fig 2.5

From Fig 2.5, we get

For two consecutive rotations,

w+w

and therefore,

w

R˙ = limw $ 0

"w+w

w R wg

w

#

· wR= (2.114b)

We can define a rotation rate

w

U= limw $ 0

"w+w

w R wg

w

#

(2.114c) and using it in Eq (2.114b) (Hill 1978), we get

w

R˙ =w

in the same way,

w

R˙

O =w

O·wR

wR˙

H =w

H·wR

2.13 Strain rates 53

Since the rotation tensors are orthogonal we can write

w

taking the time derivative of the above equation and using Eq (2.115a), we have

w

U + w W

in the same way,

w

 O + w W

w

H + w W

The above equations indicate that w U , w O and w H are skew-symmetric tensors

2.13.3 Relations between dierent rate tensors

The time derivative of Eq (2.113) leads to

w

H · wR

H = w

U · wR · wR

O + wR · w O · wRO > (2.117a) and therefore,

w

RW · (w H  w U) · wR = w O = (2.117b) Using Eqs (2.111b) and (2.40),

wl = wR˙ · wRW + wR · wU˙ · wU1 · wRW > (2.118a) splitting the above equation into its symmetric and skew-symmetric compo-nents, we get

w

RW ·wd · wR = 1

2 (

w

U˙ · wU1 + wU1 · wU)˙ (2.118b) and

w

RW · (w$  w U) · wR = 1

2 (

w

U˙ · wU1  wU1 · wU)˙ = (2.118c)

It is very important to recognize that (Hill 1978):

w

U = g=, w$ = w

An example of the above situation is the beginning of the deformation process (w = 0)

In the deformation process depicted in Fig 2.6 (Truesdell & Noll 1965, Malvern 1969) we can write, using Eqs (2.30a-2.30b):

Fig 2.6 Relative deformation gradients

For a fixed w-configuration, we can write

d d

X = d

d

Using Eq (2.111b) and the polar decomposition in the above equation, we get

l · X = (d

d

wR · wU + wR · dd wU) · wX> (2.119c) for  = w it is obvious that 

wU = wR = g , and since the above equation has to hold for any wX,

d d

wR| =w + d

d

wU| =w = wl = (2.119d)

It is easy to show that the first tensor on the l.h.s of the above equation

is skew-symmetric and the second one is symmetric; hence,

d d

wR| =w = 1

2(

wl  wlW) = w$ (2.120a) d

d

wU| =w = 1

2(

wl + wlW) = wd= (2.120b)

We can obtain an interesting picture of the deformation process by refer-ring the Lagrangian tensors to the Lagrangian coordinate system (principal directions ofw

U) and the Eulerian tensors to the Eulerian coordinate system (principal directions ofw

V) Hence (Hill 1978):

2.13 Strain rates 55

• In the Lagrangian system the components ofw

Uarewu(we assume them

to be dierent); the components of w

U˙ are w˙uv and the components of

w

 O arew uvO

• In the Eulerian system the components of wVare of course also wu; the components ofwdarewguv; the components ofw$ arew$uv; the components

ofw H arew uvH and the components ofw U arew uvU

From Eq (2.117b) we have

w H

uv  w uvU = w Ouv> (2.121a) from Eq (2.118b) we get

wguv = w˙uv

wu + wv

and from Eq (2.118c) we get

w$uv  w U

uv = w˙uv

wu  wv

2wuwv

(in Eqs (2.121b) and (2.121c) we do not use the summation convention)

In the fixed Cartesian system the components ofwUform the matrix [wX ]; hence,

[wX] = [wUO] [w] [wUO]W > (2.122a) where,

[w] =

5 7

w1 0 0

0 w2 0

0 0 w3

6

Taking the time derivative of Eq (2.122a), we obtain

[wX] = [˙ w

UO] [w] [˙ w

UO]W + [w O] [wUO] [w] [w

UO]W

 [wUO] [w] [wUO]W [w O] (2.122c) hence,

[wUO]W [wX] [˙ wUO] = [w] + [˙ wUO]W [w O] [wUO] [w]

 [w] [wUO]W [w O][wUO]= (2.122d) The above equation shows that:

w˙uv = (wv wu)w

 uvO (u 6= v) = (2.123) From Eq (2.121b) for the caseu = v (diagonal components), we get

wguu =

w˙u

wu

dw (ln

Example 2.16 JJJJJ Using Eqs (2.65), (2.35), (2.111b) and (2.112b) we can show that:

w

% =˙ 1 2

w

C˙ =w

XW · wd· wX=

JJJJJ

The Hencky strain tensor components in the fixed Cartesian system are,

[wK] = [wUO] ln [w] [wUO]W >

hence,

[wK] = [˙ w

UO] [w]1 [w] [˙ w

UO]W + [w O] [wUO] ln [w] [w

UO]W

 [w

UO] ln [w] [w

UO]W [w

JJJJJ

2.14 The Lie derivative

In the deformation process represented in Fig 2.2 we can define, for a Eulerian tensorwt, its Lie derivative associated to the flow of the spatial configuration (Simo 1988, Marsden & Hughes 1983):

Lw v(wt) = w!

½ d dw

£w

!(wt)¤¾

As we already know (see Sect 2.9) the operation of pull-back is not a tensor operation since it operates on components Hence, for calculating a Lie derivative using Eq (2.125) it is important to identify the components ofwt that we are using

The Lie derivative of a scalar is

Lw v = d

dw =

C

Cw +

C

Cw{d

The covariant components of the Lie derivative of a spatial vector ww are

¡

Lw v ww¢

l = (w[1)Dl

½ d

dw[(

w

[)mD wzm]

¾

after some algebra,

2.14 The Lie derivative 57

¡

Lw v ww¢

l = Cwzl

Cwzl

Cw{d

wyd + Cwyd

Cw{l

Since,

we can write

¡

Lw v ww¢l w

zl+wzl ¡

Lw v ww¢

l =

µ d dw

wzl

¶

wzl+wzl

µ d dw

wzl

¶ (2.128b)

and from the above we get the contravariant components of the Lie derivative

of a spatial vectorww,

¡

Lw v ww¢l

= Cwzl

Cwzl

Cw{d

wyd  wzd C

wyl

Cw{d = (2.128c) Following the above procedure we can show that the mixed components

of the Lie derivative of a general Eulerian tensorwtare

¡

Lw v(wt)¢d===e

f===g = Cwwd===e

f===g

C wwd===e

f===g

Cw{s

C

wyd

Cw{s

wws===ef===g  · · · C

wye

Cw{s

wwd===s f===g

+C wys

Cw{f

wwd===e s===g + · · · + C

wys

Cw{g

wwd===e f===s

To calculate the Lie derivative of the spatial metric tensor wg we can directly use Eq (2.125)

³

Lw v wg´

lm = w!

 d dw

h

w!(wg)i

LM

¸

>

using now Eq (2.93a), we get

³

Lw v wg´

lm = h

w!(wC)˙ i

lm = Using the result in Example 2.16, we get

³

Lw v wg´

lm = ¡

2wd¢

lm =

JJJJJ

Example 2.19 JJJJJ

To calculate the Lie derivative of the Almansi deformation tensor we use

Eq (2.125) and get

¡

Lw v we¢

lm = w!

 d dw

£w

!(we)¤

LM

¸

and resorting to Eq.(2.94a),

¡

Lw v we¢

lm = £w

!(w%)˙ ¤

lm = Taking into account the result obtained in Example 2.16 we can finally

Lw v we¢

lm = ¡w

d¢

lm =

JJJJJ

The Lie derivative of the Finger deformation tensor is

¡

Lw v wb¢lm

= w!

 d dw

£w

!(wb)¤LM¸

= Using Eq (2.97b) we get

£w

!(wb)¤LM

= h

wB`iLM

= jLM

and since

˙g = 0

we get

¡

Lw v wb¢lm

= 0 =

JJJJJ

2.14.1 Objective rates and Lie derivatives

In this Section we will show that the Lie derivative is the adequate mathe-matical tool for deriving covariant (objective) rates from covariant (objective) Eulerian tensors

Let us consider the deformation processes schematized in Fig 2.7 It is obvious that

2.14 The Lie derivative 59

Fig 2.7 Deformation processes between three configurations

ˆ w

For a covariant Eulerian tensor wt, that without losing generality we take

as a second-order tensor:

ˆ

wwˆ ~ˆ  = ˆww[ˆ ~

d ˆ w

w[ˆe wwde> (2.130b)

¯

¯wˆ! ˆwwˆ~ˆ¯

¯DE = (wˆ[1)Dˆ~(wˆ[1)Eˆˆww[ˆ~d ˆww[ˆe wwde= (2.130c) Using Eq (2.130a) in the above, we get

¯

¯ˆw! ˆwwˆ ~ˆ ¯

¯DE = (w[1)Do (w[1)En wwon > (2.130d)

¯

¯ˆw! ˆwwˆ~ˆ¯

¯DE = ¯w

! wwon¯DE

From the above and from Eq (2.125) it follows that:

Lw ˆ(ˆwt)ˆdˆe = wwˆ[d ˆ

o ˆ w

w[ˆep h

Lw v (wt)iop

=

½

ˆ w

w! h

Lw v (wt)iop¾ˆ dˆ e

= (2.130f)

The above equality shows that the Lie derivative of a covariant Eulerian tensor is also a covariant Eulerian tensor

Example 2.21 JJJJJ Considering again the case of a moving Cartesian frame and a fixed one from Example 2.15, we get

wy = ˙f + ˙T

 (w}) + T

 (wy) >

therefore taking into account that c > Q(w) and ˙Q(w), for the case under consideration, are constant in space,

wo = T˙ C (

w})

Cw} + T (wo) C (w})

C w}

hence,

wo = T˙ (TW) + T (wo)(TW) = Comparing with Eq (2.101c) it is obvious thatwlis not an objective tensor Since the velocity gradient tensor is not objective in the classical sense we

From Example 2.19, we know thatwdis the result of a Lie derivative; hence

we can assess that the Eulerian strain rate tensor is a covariant (objective)

For a Eulerian tensorwt, we define in the reference configuration the tensor:

wT` = ¡w

[1¢D d

¡w

[1¢E e

wwde g

D

g

E

which can be written as

wT` = wX1 · wt · wXW =

Since, wX1 · wX = g we can derive that,

d dw

¡w

X1¢

=  wX1 · wl and

d dw

³

w

XW´

= wlW · wXW

we can write

d

dw

³

wT`´

= wX1 · w˙t · w

XW  wX1 · wl ·wt · wXW

wX1 · wt ·wlW · wXW =

2.15 Compatibility 61

Also, from Eqs (2.110c)

wlW = uwv = wys|q wgq wgs =

Considering that the time derivative of the reference configuration base vectors

is zero and using the above together with the Lie derivative definition in Eq (2.125), we get

£

Ow v (wt)¤de

= w˙wde  wwqe wyd|q  wwdo wye|o = The above equation is going to be used in Sect 3.4 for deriving objective

2.15 Compatibility

In our previous description of the kinematics of continuous media we went through the following path:

Assume the existence of a regular mappingw!

Calculate the tensorial components of dierent deformation measures

If, instead of the above, we want to start by defining the tensorial compo-nents of a given deformation measure, our freedom to define them is limited by the fact that they should guarantee the existence of a regular mapping from which they could be derived The conditions that the tensorial components

of a deformation measure should fulfill in order to assure the existence of a regular mapping are called their compatibility conditions

In what follows we will derive the compatibility conditions for the Green deformation tensor

From Eqs (2.61), (2.80c) and (2.93a) we have,

Eulerian tensor Spatial configuration Pull-back space

Coordinate dierential wd{d [(wd{d)`]D = d{D

Metric tensor wjde [(wjde)^]DE = w

FDE

Hence,

wd{d wjdewd{e = d{D wFDE d{E = (2.131) From the above equation it is obvious that the covariant components of the Green deformation tensor are the covariant components of the metric tensor

of the pull-back space Note that the pull-back space is by no means coinci-dent with the reference configuration, whose metric tensor has the covariant components jDE

Since we restrict our study of the kinematics of continuous media to the Euclidean space, we can assess that the Riemann-Christoel tensor is zero in the spatial configuration (McConnell 1957) Hence,

The above equation represents 81 compatibility conditions to be fulfilled in the spatial configuration However, the covariant components of the Riemann-Christoel tensor satisfy the following relations (Aris 1962)7:

wUsuvt =  wUusvt > (2.133a)

wUsuvt =  wUsutv> (2.133b)

We must also consider that wUllll = 0 ; wUlllm = 0 ; wUllmm = 0 can

be easily transformed into a trivial identity of the form 0 = 0 Hence we are left with only 6 significant equations, namely:

wU1212 = 0 ; wU1213 = 0 ; wU1223 = 0 ; (2.134)

wU1313 = 0 ; wU1323 = 0 ; wU2323 = 0=

From Eqs (A.79a-A.79e),

wUsuvt = 1

2

µ

C2 wjst

Cw{uCw{v + C2 wjuv

Cw{s Cw{t  C

2 wjsv

Cw{u Cw{t  C

2 wjut

Cw{s Cw{v

¶

+wjpq ¡w

uvp wstq  wutp wsvq¢

(2.135a) where the wlmn are the Christoel symbols of the first kind corresponding

to the coordinate system {w{d}

Hence, using Eq (A.79b)

wUsuvt = 1

2

µ C2wj

st

Cw{uCw{v+ C2 wjuv

Cw{sCw{t  C

2 wjsv

Cw{u Cw{t  C

2 wjut

Cw{s Cw{v

¶ (2.135b) +wjpq



1

4

µC wj

vp

Cw{u +Cwjpu

Cw{v C

wjuv

Cw{p

¶ µC wj

tq

Cw{s +C wjqs

Cw{t C

wjst

Cw{q

¶

1

4

µC wj

tp

Cw{u +Cwjpu

Cw{t C

wjut

Cw{p

¶ µC wj

vq

Cw{s +C wjqs

Cw{v C

wjsv

Cw{q

¶¸

= 0=

7 See Appendix.

2.15 Compatibility 63

Doing a pull-back operation on Eq (2.132) we obtain,

êw

!âw

Usuvtđô

S UVT = w[sS w[uUw[vV w[tT wUsuvt = 0= (2.136a)

Equation (2.135b) represents the components of the tensorial equation

wR = 0 =

If in the spatial configuration we change from the {w{l} coordinate system to the {w{˜l} system, we write Eq (2.135b) using,

wj˜st = wjop C

w{o

Cw{˜s

Cw{p

Cw{˜t

w˜jst = wjop C

w{˜s

Cw{o

Cw{˜t

Cw{p

and the equation would look like

wU˜suvt = 1

2

Ế C2w

˜

jst

Cw{˜uCw{˜v + · · ·

ả + w˜jpq

 1 4

ẾCw

˜

jvp

Cw{˜u + · · ·

ảÌ

= 0 =

If we now want to do a pull-back of Eq.(2.135b) the algebra can get quite lengthy, but we can use an analogy with the above tensor transformations:

êw

!(wjop)S T ô

= wjop C

w{o

C{S

Cw{p

C{T = wFS T

êw

!(wjop)ôS T

= wjop C

{S

Cw{o

C{T

Cw{p = âw

F1đS T

= Using this formal analogy we can easily write:

êw

! (wUsuvt)ô

S UVT = 1

2



C2 w

FS T

C{UC{V + C2 w

FUV

C{SC{T  C

2 w

FS V

C{UC{T

 C

2 w

FUT

C{SC{V

Ì + âw

F1đPQ



1

4

ẾC w

FVP

C{U +C w

FPU

C{V C

w

FUV

C{P

ả ẾC w

FTQ

C{S +C w

FQS

C{T C

w

FS T

C{Q

ả

1

4

ẾCw

FTP

C{U +C w

FPU

C{T C

w

FUT

C{P

ả ẾCw

FVQ

C{S +C w

FQ S

C{V C

w

FS V

C{Q

ảÌ JJJJJ

1

2



C2 w

FS T

C{UC{V + C2 w

FUV

C{SC{T  C

2 w

FS V

C{UC{T  C

2 w

FUT

C{SC{V

¸ + ¡w

F1¢PQ

(2.136b)



1

4

µ

Cw

FVP

C{U +C w

FPU

C{V C

w

FUV

C{P

¶ µ

Cw

FTQ

C{S +C w

FQS

C{T C

w

FS T

C{Q

¶

1

4

µC w

FTP

C{U +C w

FPU

C{T C

w

FUT

C{P

¶ µC w

FVQ

C{S +C w

FQ S

C{V C

w

FS V

C{Q

¶¸

= 0=

We can now define in the pull-back space, with metric wFDE, the Christof-fel symbols of the first kind:

UVP = 1

2

µ Cw

FVP

C{U + Cw

FPU

C{V  C

w

FUV

C{P

¶

(2.136c)

therefore we obtain the following 6 compatibility conditions for the covariant components of the Green deformation tensor:

!(wUsuvt) = 1

2

µ

C2 w

FS T

C{UC{V +

C2 w

FUV

C{SC{T 

C2 w

FS V

C{UC{T 

C2 w

FUT

C{SC{V

¶

+ ¡w

F1¢PQ

(UVP S TQ  UTP S VQ ) = 0= (2.136d) The above equation indicates thatwFDEis a metric of the Euclidean space Taking into account the Bianchi identities in Eq (A.82d) (Synge & Schild 1949) and Eqs.(2.135a), we get

wU1212|3 + wU1213|1 + wU1213|2 = 0>

wU1313|2 + wU1323|1  wU1213|3 = 0> (2.137)

wU2323 |1  wU1323 |2 + wU1223 |3 = 0>

and we reduce the number of independent compatibility conditions to 3

Using as a metric the Green tensor, we can define an analog to Eq.(A.59):

GRADw

 C (wA^) =



CwD^ L

C{D 

wD^GLDG

¸

gL gD

where for the Eulerian vectorwa,

³

wD^´

L = w[lL dl >