Position Vectors
Revision B
By Tom Irvine
Email: [email protected]
March 30, 2015
_____________________________________________________________________________________
Position Vectors
Let r  x i i i
x i
  ij
x j
r 


i i  x ji j 
x jij  x i,i  3
x i
x i
xr 


i i x x ji j  ijk
x j i k  ijk x j,i i k  0
x i
x i
 
 x i, i

r  x i, j i i  i j   x j,i
 x k ,i

x i, j
x j, j
x k, j
x i, k  1 0 0

x j, k   0 1 0
x k , k  0 0 1
1
APPENDIX A
Sample Problems
a)
v    R m  i i
 

R
R m  m R m1
ii
x i
x i
xi
x
R

1 2x i

xi xi 

 i
x i x i
2 xi xi
xi xi R
x
v  m R m 1 i i i  m R m  2 r
R
b)




x  x , j i j
x  x , j i j
x  
 

, j  ijk i k
x i
 
x  , ji  ijk i k  0
2
c)
 2     
 ij

x j

m R m  2 r 
 ij

x j

m R m  2 x i i i 
m R m  2 x i ij


x j


m R m2 xi
x i
 xi



x i

m R m  2  m R m  2 xxi
i
R
 x i mm  2 R m  3
 3m R m2
x i
x
 x i mm  2 R m  3 i  3 m R m  2
R
x x
 mm  2 R m  3 i i  3 m R m  2
R
 mm  2 R m  3
R2
 3m R m2
R
 mm  2 R m  3 R  3 m R m  2
 mm  2 R m  2  3 m R m  2
 m  2   3mR m  2
 m  1mR m  2
3
d)
v  


v  m R m  2  r  m R m  2 r
v   m R m  2 r
r  I
v  i i



m R m2  r  m R m2 I
x i

R 
v   mm  2 R m3
i i   r  m R m2 I
x i 

x 

v   mm  2 R m3 i i i   r  m R m2 I
R 

r

v   mm  2 R m3   r  m R m2 I
R

r r

v   mm  2 R m  2    m R m  2 I
R R

r r 

v  m R m  2 m  2   I 
R R 

4
APPENDIX B
Sphere Problem
a) Deformation map
R (r, t )  r  u (r, t )
R (r, t )  r  f (r, t ) r
R (r, t )  1  f (r, t )r
5
R (r, t )  (r, t ) r
b) Deformation gradient tensor
F   R (r, t )
F   (r, t ) r
F  r       r
F  r   1  f    I
F  r  f   I
F  r   f  r   I
r 
r
r
 r
F r  f  I
 r
F
f
rr I
r
c) Metric tensor
G  FT F
f 
 f 

G   r  r   I  r  r   I
r
 r

f
f 
 f 

G   r  r   r  r   2 r  r  2 I
r
r
 r

6
f 
G 
r
2

r  r  2 I
r
 r  r  r  r   2 f
d) Finite & infinitesimal strain tensors
u  f r 
 f r  r  f
r
 f I  f r
r
Finite
E* 
1
u  uT  uT u 


2
Infinitesmial
E
1
u  uT 

2 
7