7/28/2019 Convected Co Ordinates
1/1
ES 241 Notes on convected coordinates 1 2 3, , (JRR, 4/97 and
2/03)
Base vectors: Reference configuration: GX
i i=
. Current configuration: g
xi i=
.
Relation between them: Since d d i ix F X g F G= = , .
Reciprocal base vectors: giand Gi
satisfy g g G G
ij
ij j
i
if i j if i j = = = = 0 1, .Since g F G G G
ij
ij = ( ) , they are related by g F G
i i = .
Relations between stress and strain measures and rates:
E F F IG T
= ( / )( )1 2 and D F F D F E F= = sym( ) 1 1 1T G .
Thus, if we write E G GG iji j
= , where ij iG
j i j i j = G E G g g G G( / )( )1 2
= = = ( / )( ) [ ,1 2 g G g Gij ij ij i j ij i j g g G G are
metric tensors] , then D g g= iji j
.
= =F S F g g S G GPK T ij i jPK ij
i j2 2
,If = then .
* ( ) * = + = = + + and antisym F F D D F S F1 2PK T.
Thus * = + +D D g gij i j , so that ( * )ij i j = g D D g .
Rate form of constitutive relation: Suppose * : = L D , where L
is the 4th rank incrementalmodulus tensor. Then
( : )ij i j = g L D D D g
= [( ) : : ( ) ( )( ) ( )( )] g g L g g g g g g g g g gj i l k i
k j l i k j l kl
= [ ] L g gijkl ik jl ik jl kl where gij i j = g g and Lijkl are
the components
ofL when written L g g g g=Lijkl i j k l .
Prandtl-Reuss equations: * ( ) : + =tr PRD L D where, in
cartesian coordinates,
LE
h EijklPR
ik jl il jk ij klij kl
=
+
+ +
+
+ +
1
1
2 1 2
3
2 1 2 1 32
( )
[ ( ) / ].
Since = det(F), these relations are equivalent to * : = L D with
L F L= det( ) PR . To identify
Lijkl
, note that I e e e e g g g g= = = =i i ij i j j
jij
i jgcartesian
shows that ijcartesian
corresponds to gij
when components are relative to the base vectors gi.
Thus the Prandtl-Reuss equations are ij ijkl ik jl ik jl
klL g g= [ ]
with
LE
g g g g g gh E
ijkl ik jl il jk ij klij kl
=
+
+ +
+
+ +
det( ) ( )
[ ( ) / ]F
1
1
2 1 2
3
2 1 2 1 32
where = = ij i j ij ij klkl
g gg I I g[ ( / ) ( : )] 1 3 and 2 3 2= ( ) / g gik jlij kl
.
