convected_coordinates1

7/30/2019 Convected_Coordinates1

1/17

Section 2.10

Solid Mechanics Part III Kelly279

2.10 Convected Coordinates

In this section, the deformation and strain tensors described in 2.2-3 are now describedusing convected coordinates (see 1.16). Note that all the tensor relations expressed in

symbolic notation already discussed, such as CU = , iii nNF = , lFF =& , are independent

of coordinate system, and hold also for convected coordinates.

2.10.1 Convected Coordinates

Introduce the curvilinear coordinates i . The material coordinates can then be written as

),,(321

= XX (2.10.1)

so iiX EX = and

i

i

i

i ddXd GEX == , (2.10.2)

where iG are the covariant base vectors in the reference configuration, with corresponding

contravariant base vectorsi

G , Fig. 2.10.1, with

i

jj

i= GG (2.10.3)

Figure 2.10.1: Curvilinear Coordinates

The coordinate curves, curves of constant i , form a net in the undeformed configuration.One says that the curvilinear coordinates are convected orembedded, that is, the coordinate

curves are attached to material particles and deform with the body, so that each material

11,xX

22 , xX

33 , xX

X

11,eE

22 ,eE

1g

2g

current

configurationreference

configuration

1G

2G

x


2/17

Section 2.10


particle has the same values of the coordinates i in both the reference and currentconfigurations.

In the current configuration, the spatial coordinates can be expressed in terms of a new,

current, set of curvilinear coordinates

),,,( 321 t= xx , (2.10.4)

with corresponding covariant base vectors ig and contravariant base vectorsi

g , with

i

i

i

iddxd gex == , (2.10.5)

Example

Consider a motion whereby a cube of material, with sides of length 0L , is transformed into a

cylinder of radius R and height H, Fig. 2.10.2.

Figure 2.10.2: a cube deformed into a cylinder

A plane view of one quarter of the cube and cylinder are shown in Fig. 2.10.3.

Figure 2.10.3: a cube deformed into a cylinder

0L

R

0LH

1X

2X

1x

2x

0L

R

X x P p


3/17

Section 2.10


The motion and inverse motion are given by

)(Xx = ,

( )( ) ( )

( ) ( )3

0

3

2221

21

0

2

2221

21

0

1

2

2

XL

Hx

XX

XX

L

Rx

XX

X

L

Rx

=

+=

+=

(basis: ie )

and

)(1 xX = ,

( ) ( )

( ) ( )

303

2221

1

2

02

222101

2

2

xH

LX

xxx

x

R

LX

xxR

LX

=

+=

+=

(basis: iE )

Introducing a set of convected coordinates, Fig. 2.10.4, the material and spatial coordinates

are

),,( 321 = XX ,

303

2102

101

tan2

2

=

=

=

H

LX

R

LX

R

LX

and (these are simply cylindrical coordinates)

),,( 321 = xx ,33

212

211

sin

cos

=

=

=

x

x

x

A typical material particle (denoted byp) is shown in Fig. 2.10.4. Note that the position

vectors forphave the same i values, since they represent the same material particle.


4/17

Section 2.10


Figure 2.10.4: curvilinear coordinate curves

2.10.2 The Deformation Gradient

With convected curvilinear coordinates, the deformation gradient is

i

i GgF = , (2.10.6)

which is consistent with

( ) XFGGggx dddd jiijjj === (2.10.7)

The deformation gradient F, the transpose TF and the inversesT1 , FF , map the base

vectors in one configuration onto the base vectors in the other configuration:

i

i

i

i

i

i

i

i

gGF

GgF

gGF

GgF

==

=

=

T

T

1

ii

ii

ii

ii

GgFgGF

GgF

gFG

==

=

=

T

T

1

Deformation Gradient (2.10.8)

Thus the tensors F and 1F map the covariant base vectors into each other, whereas the

tensors TF and TF map the contravariant base vectors into each other, as illustrated in Fig.2.10.5.

1X

2X

1x

2x

1

2

1

2

R=1

4

2 =

p p


5/17

Section 2.10


Figure 2.10.5: the deformation gradient, its transpose and the inverses

Components of F

F has different components with respect to the different bases:

j

i

i

jj

ij

iji

ijji

ij

j

i

i

jj

ij

iji

ijji

ij

ffff

FFFF

gggggggg

GGGGGGGGF

====

====

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) jii

jj

ij

iji

ijji

ij

j

i

i

jj

ij

iji

ijji

ij

ffff

FFFF

gggggggg

GGGGGGGGF

====

====

1111

11111

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ji

i

jj

ij

iji

ijji

ij

j

i

i

jj

ij

iji

ijji

ij

ffff

FFFF

gggggggg

GGGGGGGGF

====

====

TTTT

TTTTT

( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ji

i

jj

ij

iji

ijji

ij

j

i

i

jj

ij

iji

ijji

ij

ffff

FFFF

gggggggg

GGGGGGGGF

====

====

TTTT

TTTTT

(2.10.9)

The components ofF with respect to the reference bases { } { }ii GG , are

1G

2G

1G

2G

1g

2g

1g

2g

covariant basis

contravariant basisTF

F

1F

TF


6/17

Section 2.10


j

m

m

i

j

i

j

ii

j

ki

jkj

i

j

i

k

ijkjiij

j

m

i

m

jijiij

x

XF

GF

GF

xXF

===

====

===

gGFGG

gGFGG

gGFGG

gGFGG

(2.10.10)

and similarly for the components with respect to the current bases.

Components of the Base Vectors in different Bases

Now

( ) ( )

m

m

i

m

mi

j

im

m

j

j

i

m

mj

i

j

m

m

ji

jm

mjii

FF

FF

FF

GG

GG

GGGGGGFGg

==

==

===

(2.10.11)

showing that some of the components of the deformation gradient can be viewed also as

components of the base vectors. Similarly,

( ) ( ) mm

i

m

miiiff gggFG

=== 111 (2.10.12)

For the contravariant base vectors, one has

( ) ( ) ( ) ( )

( ) ( )

( ) ( ) mimm

mi

i

j

mj

m

i

jm

mj

i

j

mj

m

i

jm

mjii

FF

FF

FF

GG

GG

GGGGGGGFg

==

==

===

TT

TT

TTT

(2.10.13)

and

( ) ( ) mi

mm

miii ff gggFG

=== TTT (2.10.14)

2.10.3 Reduction to Material and Spatial Coordinates

Material Coordinates

Suppose that the material coordinates iX with Cartesian basis are used (rather than the

convected coordinates with curvilinear basis iG ), Fig. 2.10.6. Then


7/17

Section 2.10


jj

i

jj

i

i

ji

j

ji

j

i

ijj

i

jj

i

i

iji

j

ji

j

iii

xX

x

X

xx

XX

X

X

XX

X

eeg

eeg

EEEG

EEEG

=

=

=

=

=

=

=

=

=

=

,, (2.10.15)

and

XeEgEgGF

xEeEgGgF

grad

Grad

1 =

===

=

===

jij

ii

i

i

i

i

ji

ji

i

i

i

x

X

X

x

(2.10.16)

which are Eqns. 2.2.2, 2.2.4. Thus xGrad is the notation forF to be used when the materialcoordinates iX are used to describe the deformation.

Figure 2.10.6: Material coordinates and deformed basis

Spatial Coordinates

Similarly, when the spatial coordinates

i

x are to be used as independent variables, then

ij

j

ij

j

ii

iji

j

ji

j

i

j

j

ij

j

ii

ji

j

ji

j

iii

x

x

x

x

xx

X

x

X

x

XX

x

eeeg

eeeg

EEG

EEG

=

=

=

=

=

=

=

=

=

=

,, (2.10.17)

and

1X

2X

3X

X1E

2E

1g

2g

current

configuration

reference

configuration


8/17

Section 2.10


XeEeGgGF

xEeGeGgF

grad

Grad

1 =

===

=

===

iji

ji

i

i

i

j

ij

ii

i

i

i

xX

X

x

(2.10.18)

The descriptions are illustrated in Fig. 2.10.7. Note that the base vectors iG , ig are not the

same in each of these cases (curvilinear, material and spatial).

Figure 2.10.7: deformation described using different independent variables

1X

2X

1x

2x

1X

2X

1x

2x

1X

2X

1x

2x

1G2G

1g2g

1E

2E

1g

2g

2G

1G1e

2e

i

i GgF =

xEeF Grad=

= jij

i

X

x

i

i gGF =1

XeEF grad1 =

= jij

i

x

X


9/17

Section 2.10


2.10.4 Strain Tensors

The Cauchy-Green tensors

The right Cauchy-Green tensorC and the left Cauchy-Green tensorb are defined by Eqns.

2.2.10, 2.2.13,

( )( )

( )( ) ( )( )( )( )( ) ( ) ji

ij

ji

ij

j

ji

i

ji

ij

ji

ij

j

ji

i

ji

ij

ji

ij

j

ji

i

ji

ij

ji

ij

j

ji

i

bG

bG

Cg

Cg

gggggGGgFFb

gggggGGgFFb

GGGGGggGFFC

GGGGGggGFFC

===

===

===

===

11T1

T

1T11

T

(2.10.19)

Thus the covariant components of the right Cauchy-Green tensor are the metric coefficients

ijg , the covariant components of the identity tensor with respect to the convected bases in the

current configuration, jiijg gggI = . It is possible to evaluate other components ofC,

e.g.ij

C , and also its components with respect to the current basis through 2.10.14, but only

the components ijC with respect to the reference basis will be used in the analysis. Similarly,

for 1b , the components ( )ijb 1 with respect to the current configuration will be used.

The Stretch

Now, analogous to 2.2.9, 2.2.12,

xxbXX

XXCxx

dddddS

ddddds

12

2

==

==(2.10.20)

so that the stretches are, analogous to 2.2.17,

( ) jiji

j

ij

i

xdbxddddd

dd

dsdS

XdCXdddd

d

d

d

dS

ds

1

1112

2

2

2

22

===

===

xbxxxb

xx

XCXX

XC

X

X

(2.10.21)

The Green-Lagrange and Euler-Almansi Tensors

The Green-Lagrange strain tensorE and the Euler-Almansi strain tensore are definedthrough 2.2.22, 2.2.24,


10/17

Section 2.10


( )

( ) xxexbIx

XXEXICX

dddddSds

dddddSds

=

=

122

22

21

2

2

1

2(2.10.22)

The components ofE and e can be evaluated through (writing IG , the identity tensorexpressed in terms of the base vectors in the reference configuration, and Ig , the identity

tensor expressed in terms of the base vectors in the current configuration)

( ) ( ) ( )

( ) ( ) ( ) jiijjiijijjiijjiij

ji

ij

ji

ijij

ji

ij

ji

ij

eGgGg

EGgGg

ggggggggbge

GGGGGGGGGCE

===

===

2

1

2

1

2

1

2

1

2

1

2

1

1

(2.10.23)

Note that the components ofE and e with respect to their bases are equal, ijij eE = (although

this is not true regarding their other components, e.g. ijij eE ).

2.10.5 Intermediate Configurations

Stretch and Rotation Tensors

The polar decompositions vRRUF == have been described in 2.2.5. The decompositionsare illustrated in Fig. 2.10.8. In the material decomposition, the material is first stretched by

U and then rotated by R. Let the base vectors in the associated intermediate configuration be

{ }ig . Similarly, in the spatial decomposition, the material is first rotated by Rand thenstretched by v. Let the base vectors in the associated intermediate configuration in this case

be { }iG . Then, analogous to Eqn. 2.10.8, {Problem 1}

i

i

ii

i

i

i

i

gGU

GgU

gGU

GgU

T

T

1

==

=

=

ii

ii

ii

ii

GgU

gGU

GgU

gUG

==

=

=

T

T

1

(2.10.24)

i

i

i

i

i

i

i

i

gGv

Ggv

gGv

Ggv

=

=

=

=

T

T

1

ii

ii

ii

ii

Ggv

gGv

Ggv

gGv

T

T

1

=

=

=

=

(2.10.25)


11/17

Section 2.10


Note that U and v symmetric, TUU = , Tvv = , so

i

ii

i

i

ii

i

GggGU

gGGgU

==

==

1

ii

ii

ii

ii

gGUGgU

GgUgUG

,

,

11 ==

==

(2.10.26)

i

ii

i

i

ii

i

GggGv

gGGgv

1 ==

==

ii

ii

ii

ii

gGvGgv

GvggGv

==

==

,

,

11 (2.10.27)

Similarly, for the rotation tensor, with Rorthogonal,T1

RR = ,

i

ii

i

i

ii

i

GGGGR

GGGGR

T ==

== ii

ii

ii

ii

GGRGGR

GRGGRG

==

==

,

,

TT (2.10.28)

i

ii

i

i

ii

i

ggggR

ggggR

==

==

T

ii

ii

ii

ii

ggRggR

ggRggR

,

,

TT ==

==(2.10.29)

The above relations can be checked using Eqns. 2.10.8 and RUF = , vRF = , 11 = RFv ,etc.

Figure 2.10.8: the material and spatial polar decompositions

Various relations between the base vectors can be derived, for example,

( ) ( )

j

i

j

i

j

i

j

i

jiji

jijijiji

gGgG

gGgG

gGgG

gGgRRGgRRGgG

T

==

==

==

===

L

L

L(2.10.30)

{ }iG

{ }igU R

{ }ig

i

GR v


12/17

Section 2.10


Deformation Gradient Relationship between Bases

The various base vectors are related above through the stretch and rotation tensors. The

intermediate bases are related directly through the deformation gradient. For example, from2.10.26a, 2.10.28b,

iiii GFGURUGg TT === (2.10.31)

In the same way,

ii

ii

ii

ii

gFG

gFG

GFg

GFg

T

1

T

==

=

=

(2.10.32)

Tensor Components

The stretch and rotation tensors can be decomposed along any of the bases. ForU the most

natural bases would be { }iG and { }iG , for example,

j

i

j

i

j

ij

ij

i

j

i

j

ii

j

j

i

i

j

m

jimjiij

ji

ij

jijiij

ji

ij

UU

UUGUU

UU

GgUGGGGU

gGUGGGGUgGUGGGGU

gGUGGGGU

===

======

===

,

,,

,

(2.10.33)

with jij

i

i

j

i

j

jiij

jiij UUUUUUUU

==== ,,, . One also has

j

i

j

i

j

ij

ij

i

j

i

j

ii

j

j

i

i

j

m

jimjiij

ji

ij

jijiij

ji

ij

vv

vv

Gvv

vv

GgGvGGGv

gGGvGGGv

gGGvGGGv

gGGvGGGv

,,

,

,

======

===

===

(2.10.34)

with similar symmetry. Also,


13/17

Section 2.10


( ) ( )

( ) ( )

( ) ( )

( ) ( ) jijij

ij

ij

i

j

i

j

ii

j

j

i

i

j

j

m

imjiij

ji

ij

jijiij

ji

ij

UU

UU

gUU

UU

gGgUgggU

GggUgggU

gGgUgggU

gGgUgggU

,

,

,

,

1111

1111

1111

1111

===

===

===

===

(2.10.35)

and

( ) ( )

( ) ( )

( ) ( )( ) ( ) jijij

ij

ij

i

j

i

j

ii

j

j

i

i

j

i

m

mjjiij

ji

ij

jijiij

ji

ij

vv

vv

gvv

vv

gGgvgggv

Gggvgggv

gGgvgggv

gGgvgggv

===

===

===

===

,

,

,

,

1111

1111

1111

1111

(2.10.36)

with similar symmetry. Note that, comparing 2.10.33a, 2.10.34a, 2.10.35a, 2.10.36a and

using 2.10.30,

( )

( )

( ) ( )ijijijij

jiij

ji

ij

ji

ij

ji

ij

vvUU

v

U

v

U

11

11

11

===

=

=

=

=

ggv

ggU

GGv

GGU

(2.10.37)

Now note that rotations preserve vectors lengths and, in particular, preserve the metric, i.e.,

jiijjiij

jiijjiij

gg

GG

gggg

GGGG

===

===(2.10.38)

Thus, again using 2.10.30, and 2.10.33-2.10.36, the contravariant components of the above

tensors are also equal, ( ) ( )ijijijij vvUU 11 === .

As mentioned, the tensors can be decomposed along other bases, for example,

jijiij

ji

ij vv gGvggggv === , (2.10.39)

2.10.6 Eigenvectors and Eigenvalues

Analogous to 2.2.5, the eigenvalues ofC are determined from the eigenvalue problem


14/17

Section 2.10


( ) 0det = IC C (2.10.40)

leading to the characteristic equation 1.11.5

0IIIIII 23 =+ CCCCCC (2.10.41)

with principal scalar invariants 1.11.6-7

[ ] ( )

321321

1332212122

21

321

detIII

)tr()(trII

trI

CCCC

CCCCCCC

CCCC

C

CC

C

===

++===

++===

kji

ijk

j

i

i

j

j

j

i

i

i

i

CCC

CCCC

A

(2.10.42)

The eigenvectors are the principal material directions iN , with

( ) 0NIC = ii (2.10.43)

The spectral decomposition is then

=

=3

1

2

i

iii NNC (2.10.44)

where2

ii =C and the i are the stretches. The remaining spectral decompositions in

2.2.37 hold also. Note also that the rotation tensor in terms of principal directions is (see2.2.35)

i

ii

i NnNnR == (2.10.45)

where in are the spatial principal directions.

2.10.7 Displacement and Displacement Gradients

Consider the displacement u of a material particle. This can be written in terms of covariant

components iU and iu :

i

i

i

i uU gGXxu == . (2.10.46)

The covariant derivative ofu can be expressed as

m

im

m

imiuU gG

u==

(2.10.47)


15/17

Section 2.10


The single line refers to covariant differentiation with respect to the undeformed basis, i.e.

the Christoffel symbols to use are functions of the ijG . The double line refers to covariant

differentiation with respect to the deformed basis, i.e. the Christoffel symbols to use are

functions of the ijg .

Alternatively, the covariant derivative can be expressed as

iiiiiGg

Xxu=

=

(2.10.48)

and so

( ) ( ) mm

imi

mm

i

m

imii

m

m

imi

mm

i

m

imii

fuu

FUU

ggggG

GGGGg

===

=+=+=

1

(2.10.49)

The last equalities following from 2.10.11-12.

The components of the Green-Lagrange and Euler-Almansi strain tensors 2.10.23 can be

written in terms of displacements using relations 2.10.49 {Problem 2}:

( ) ( )

( ) ( )j

n

inijjiijijij

j

n

i

n

i

j

j

iijijij

uuuuGge

UUUUGgE

+==

++==

2

1

2

12

1

2

1

(2.10.50)

In terms of spatial coordinates, ( ) jijiiiii XxX egEG === /,, , jiji XUU = / , thecomponents of the Euler-Lagrange strain tensor are

( )

+

+

=

==

j

k

i

k

i

j

j

i

ijmnj

n

i

m

ijijijX

U

X

U

X

U

X

U

X

x

X

xGgE

2

1

2

1

2

1 (2.10.51)

which is 2.2.46.

2.10.8 The Deformation of Area and Volume Elements

Differential Volume Element

Consider a differential volume element formed by the elements ii

d G in the undeformed

configuration, Eqn. 1.16.36:


16/17

Section 2.10


321 = dddGdV (2.10.52)

where, Eqn. 1.16.13,

[ ] jiijij GGG GG == ,det (2.10.53)

The same volume element in the deformed configuration is determined by the elements

i

id g :

321 = dddgdv (2.10.54)

where

[ ] jiijij ggg gg == ,det (2.8.55)

From 1.16.22 et seq., 2.10.11,

F

GGG

ggg

det

321

321

321

G

GFFF

FFF

g

ijk

kji

kji

kji

=

=

=

=

(2.10.56)

where ijk is the Cartesian permutation symbol, and so the Jacobian determinant is (see

2.2.53)

Fdet===G

g

dV

dvJ (2.10.57)

and Fdet is the determinant of the matrix with components ijF .

Differential Area Element

Consider a differential surface (parallelogram) element in the undeformed configuration,

bounded by two vector elements )1(Xd and )2(Xd , and with unit normal N . Then the vector

normal to the surface element and with magnitude equal to the area of the surface is, using1.16.23, given by

( ) kjiijkj

j

i

i ddedddddS GGGXXN G)2()1()2()1()2()1( === (2.10.58)


17/17

Section 2.10


where ( )Gijke is the permutation symbol associated with the basis iG , i.e.

( ) Ge ijkkjiijkijk == GGGG

. (2.10.59)

Using kk gFG T= , one has

kji

ijk ddGdS gFNT)2()1( = (2.10.60)

Similarly, the surface vector in the deformed configuration with unit normal n is

( ) kjiijkj

j

i

i ddeddddds gggxxng )2()1()2()1()2()1( === (2.10.61)

where ( )gijke is the permutation symbol associated with the basis ig , i.e.

( )ge ijkkjiijkijk == ggg

g . (2.10.62)

Comparing the two expressions for the areas in the undeformed and deformed configurations,

2.10.60-61, one finds that

( ) dSdSG

gds NFFNFn

TT det == (2.10.63)

which is Nansons relation, Eqn. 2.2.59.

2.10.9 Problems

1. Derive the relations 2.10.24.2. Use relations 2.10.49, with jiijg gg = and jiijG GG = , to derive 2.10.50

( ) ( )

( ) ( )jn

inijjiijijij

j

n

inijjiijijij

uuuuGge

UUUUGgE

+==

++==

21

21

2

1

2

1

convected_coordinates1

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