Mathematical & Mechanical
Method in Mechanical Engineering
Dr. Wang Xingbo
Fall，2005
Mathematical & Mechanical
Method in Mechanical Engineering
Introduction to Tensors
1. Concept of Tensors
2. Tensor Algebra
3. Tensor Calculus
4. Application of Tensors
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Concept of Tensors
A coordinate transformation in an ndimensional space
 i u j
i j
i
u

u


u
,
det

j
j  0

j
u

i
u i  u u j   i u j , (  i )  ( i ) 1
j
j
j
j

u
i
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Concept of Tensors
T is a quantity with ns components
represented by one of the following three
forms
T i1i2 ...is (i1 , i2 ,..., is  1, 2,..., n)
Ti1i2 ...is (i1 , i2 ,..., is  1, 2,..., n)
j1 j2 ... jm
i1i2 ...il
T
(l  m  s, i1 , i2 ,..., il , j1, j2 ,..., jm  1, 2,..n)
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Concept of Tensors
Components of T are represented by
the first one and transformed by
T
j1 j2 ... js
   ... T
j1
k1
j2
k2
js
ks
k1k2 ...ks
u j1 u j2 u js k1k2 ...ks
 k1
... ks T
k2
u u
u
T is called a contravariant tensor
of order s.
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Concept of Tensors
Components of T are represented by
the second one and transformed by
Ti1i2 ...is    ... T j1 j2 ... js
j1
k1
j2
k2
js
ks
u j1 u j2 u js
 k1
... ks T j1 j2 ... js
k3
u u
u
T is called a covariant tensor of
order s.
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Concept of Tensors
Components of T are represented by
the third one and transformed by
j1 j2 ... jm
jm h1 h2
hl k1k2 ...km
j1
j2
Ti1i2 ...il   k1  k2 ... km i1 i2 ...il Th1h2 ...hl
u u
u u u
u k1k2 ...km
 k1
... km
... il Th1h2 ...hl
k2
i1
i2
u u
u u u
u
j1
j2
jm
h1
h2
hl
T is called a mix-variant tensor of
order s.
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Concept of Tensors
Tensor product
Let U, V be two vector spaces of dimension m, n,
Tensor product of U and V is an mn–dimensional vector
space W denoted by W=UV.
Symbol  is used to denote a tensor product
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Concept of Tensors
Symbol  is used to denote a tensor product
(u  v)  w  u  w  v  w
s  (u  v)  s  u  s  v
 (u  v)  ( u )  v  u  ( v)
0u  u 0  0
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Concept of Tensors
Since UV is an mn-dimensional space, it has mn basis
vectors.
All pairs (i,j) produce exactly mn pairs of (ui,vj)
It often uses symbol ui  v j to denote the basis of
W=UV.
ij
The elements of W=UV are w ui  v j
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Concept of Tensors
Tensor basis
Covariant tensor basis
Ei1i2 ...is (i1 , i2 ,..., is  1, 2,..., n)
are defined by tensor product of covariant vector basis
Ei1i2 ...is  ei1  ei2  ...  eis
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Concept of Tensors
Tensor basis
Contravariant tensor basis
E i1i2 ...is (i1 , i2 ,..., is  1, 2,..., n)
are defined by tensor product of contravariant vector
basis
E
i1i2 ...is
 e  e  ...  e
i1
i2
is
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Concept of Tensors
Tensor basis
Mix-variant tensor basis
i1i2 ...il
***** j1 j2 ... jm
E
(l  m  s, i1 , i2 ,..., il , j1, j2 ,..., jm  1, 2,..n)
are defined by tensor product of covariant and
contravariant vector basis
i1i2 ...il
il
i1
i2
E*****

e

e

...

e
 e j1  e j2  ...  e jm
j1 j2 ... jm
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Concept of Tensors
A vector is a first-order tensor
Take s =1, the two forms of components
i1
T Ti1
The transformations
T
j1
u
k1
  T  k1 T
u
j1
k1
k1
j1
u
Ti1   T j1  k1 T j1
u
j1
k1
j1
This is what a contravariant vector or a covariant vector is!
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Concept of Tensors
Second-order tensor
ij
T

T
Eij
contravariant tensor T can be represented by
The transformations
i
j

u

u
kl
T ij   ki  l jT kl  k
T
u u l
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Concept of Tensors
Second-order tensor
covariant vector T can be represented by
T  Tij E
The transformations
u u
Tij    T  i
T
j kl
u u
k
k
i
l
j kj
l
ij
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Concept of Tensors
Second-order tensor
Mix covariant vector T can be represented by T
The transformations
i
l

u

u
k
T. ji   ki  ljT.lk  k
T
j .l
u u
 T. ij Ei. j
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Concept of Tensors
Second-order tensor
Matrix Form
 T 11 T 12 T 13 
ij  21
22
23 
(T )  T
T
T 
 T 31 T 32 T 33 


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Second order tensor
Sample
Aij x x  1
i
( Aij )
j
is a covariant tensor of order 2
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Second order tensor
Quantity to illustrate strain in an elastic material is a covariant
tensor of order 2
Let A, B be two points in an elastic body and let .
A  r , B  r  r
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Second order tensor
After deformation A =
r + d,B = r + r + u(r + r )
r = AB = r + u(r + r ) - u(r )
A
r
r
B
u(r)
r+r
r+u(r)
u(r+r)
A
O
r+r+u(r+r)
B
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Second order tensor
u  (u1 ( x1 , x2 , x3 ), u2 ( x1 , x2 , x3 ), u3 ( x1 , x2 , x3 ))
xi  xi  ui ( x1  x1 , x2  x2 , x3  x3 )  ui ( x1 , x2 , x3 )
ui
xi  xi 
xk
xk
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Second order tensor
l   xi x j   xk xl
2
ij
kl
ui
ui ui
2
xi xk 
xk xl
xk
xk xl
 ui uk ul ul 



 xi xk
 xk xi xi xk 
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Second order tensor
1  ui uk ul ul 
uik  



2  xk xi xi xk 
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Second order tensor
Let us change the Cartesian coordinate transformation O to O
Assume
j
j
xi  i x j  x0i ,det(i )  0
1  ui uk ul ul 
uik  



2  xk xi xi xk 
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Second order tensor
xi  i x j  x0i ,
xi  i j x j  x0i ,(i j )  (i j )1
j
xi
k
 i
xk
xi
 ik
xk
   i
k
i
k
j
 
k
i
j
i
k
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Strain tensor
1   imum xn  kt ut xm  lr ur xm  ls us xn 
uik  



2  xn xk
xm xi
xm xi xn xk 
1  m um k
n un
i
r ur
i
s u s
k 
  i
n   k
m  l
 m l
n 
2
xn
xm
xm
xn

1 m n  um un ur ur 
 i  k 



2
 xn xm xm xn 
  im knumn
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Tensor algebra
Addition and Subtract of Two Tensors
Ai1i2 ...is  Bi1i2 ...is  Ci1i2 ...is
Contraction of Tensors :
Forcing one upper index equal to a lower index
and invoking the summation convention
A special operation on mix-variant tensors
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Geometric Meanings of Cross Product
Contraction of Tensors
u
u u
u k1k2 ...km h2
 k2 ... km
... il Th1h2 ...hl  k1
i1
u
u u
u
u j2 u jm u h1 u hl k1k2 ...km
 k2 ... km
... il Th1k1 ...hl
i1
u
u u
u
 Ti1ij32......iljm
i2 j2 ... jm
i1i2 ...il
T
jm
j2
h1
hl
A  A  A  A  ...  A
i
ik
1
1k
2
2k
3
3k
n
nk
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Out Product of Two Tensors
The product of two tensors is a
tensor whose order is the sum of
the orders of the two tensors,
and whose components are products
of a component of one tensor with
any component of the other tensor.
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Out Product of Two Tensors
A=AikEik , B=BlmElm,
Ciklm = AikBlm
A  Aipq Eipq , B  Brj E rj
C
ij
pqr
A B
i
pq
j
r
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Inner product of Two Tensors
Multiplying two tensors and then contracting the product
with respect to indices belonging to different factors
A A E
ij
k
k
ij
l
B  Bmnt
Elmnt
contraction by i  k
l
ijl
jl
Akij Bmnt
 Ckmnt

 Cmnt
ij
k
l
mnt
A B
C
ijl
kmnt
contraction by i  m
 C
jl
knt
contraction by l  k
l
ijl
ij
Akij Bmnt
 Ckmnt

 Cmnt
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Quotient Law
Assume Brqs
IF
C sp are two arbitrary tensors.
A B C
r
pq
Then A is a tensor
qs
r
s
p
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Some Useful and Important Tensors
•
Metric tensors
gij  ei  e j , g  e  e , gi  ei  e
ij
G  gij E
 g11
G 
 g21
ij
g12 

g22 
i
j
, G  g Eij
ij
j
j
G  gij E ij
2
det G  g11 g 22  g122  g11 g 22  g 21
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Metric tensors
In 3-dimensional space
det( gij )  (e1  (e2  e3 ))2
 g e1  e2  e3

2
 g e  e3  e1

3
g
e

e

e
1
2

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The Alternating Tensor of Third Order
 jkl  (e j  ek )  el
εjkl = 1, if j, k, l cyclic permutation of 1, 2, 3
εjkl = -1, if j, k, l cyclic permutation of 2, 1, 3
εjkl = 0, otherwise.
( A  B)i   ijk Aj Bk
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Absolute Derivatives & Differential
A  A( r ) be a vector field
in covariant frame-vectors {ei }
A  A je j
dA  d ( A e j )
j
dA is called absolute differential or
covariant differential of vector field A
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Absolute Derivatives & Differential
dA  d ( A j e j )  e j dA j  A j de j
Absolute differential of A is composed of two parts
j

A
j
k
dA  k dx
x
reflects the relationship of the contravariant
components changing with spatial position
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Absolute Derivatives & Differential
j

e
j
k
de  k dx
x
reflects that of frame-vectors components changing
with spatial position
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Absolute Derivatives & Differential
e j
A
j
k
dA  ( k e j  A
)dx
k
x
x
j
e j
A A
j
 k ej  A
k
x
x
x k
A k
dA  k dx
x
j
Let
then
A
k
x
absolute derivative represented by contravariant
components
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Absolute Derivatives & Differential
We also can derive absolute derivative
represented by covariant components
 A A j j
e j
 k  k e  A j k
x
x
x

A k

dA  k dx

x
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Absolute Derivatives & Differential
the absolute derivative of a vector field
can be represented by either
contravariant components or covariant
components.
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Derivatives of Frame-vectors and Christoffel Symbols
e j e j
x k x k
are vectors , be linear
combination of frame-vectors
ei
k
  ijk e
j
x
ei
k


e
ij
k
j
x
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Derivatives of Frame-vectors and Christoffel Symbols
ijk

k
ij
the first kind of Christoffel symbols
the second kind of Christoffel symbols
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Derivatives of Frame-vectors and Christoffel Symbols
e j
ei
 r
2r
2r
 r
 j ( i) j i  i j  i ( j) i
j
x
x x
x x x x
x x
x
 
k
ij
k
ji
ijk   jik
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Derivatives of Frame-vectors and Christoffel Symbols
,j

Partial Derivative Operator
j
x

 ,2
2
x

 ,k
k
x
,j
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Derivatives of Frame-vectors and Christoffel Symbols
ei , j  el  ijk e  el     ijl
k
k
ijl l
ei , j  e   e  e     
l
k
ij k
l
l l
ij k
l
ij
ijl = ei , j  el   e  el   gkl
k
ij k
k
ij
  ei , j  e  ijk e  e  ijk g
l
ij
l
k
l
kl
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Derivatives of Frame-vectors and Christoffel Symbols
Relationships between ijk and the metric tensor
gij ,k  ei ,k  e j  ei  e j ,k  ikj   jki
gik , j  ijk   kij
g jk ,i   jik   kij
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Derivatives of Frame-vectors and Christoffel Symbols
ijk
1
 ( g jk ,i  g ki , j  gij ,k )
2
1 lk
  g ( g jk ,i  g ki , j  gij ,k )
2
l
ij
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Derivatives of Frame-vectors and Christoffel Symbols
The Christoffel symbols are only symbols but not
components of any tensor though they look like the
form of tensor-components
 x x
x x x k
  i j i  i
ij
j
k
x x x x x x
2 i
k
i
j
k
k
ij
ijk  g mn
 2 x m x n xi x j x k
 i
ijk
i
j
k
j
k
x x x
x x x
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Derivatives of Frame-vectors and Christoffel Symbols
The derivatives of the contravariant frame-vectors
e ej  
i
i
j
(e  e j ),k  e  e j  e  e j ,k  0
i
i
,k
i
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Derivatives of Frame-vectors and Christoffel Symbols
e  e j  
i
,k
i
jk
e
i
i
k
= e, j   jk e
j
x
i
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Derivatives of Frame-vectors and Christoffel Symbols
Recalling the gradient of a scalar field 
grad    
 j
e

j
where   i ei x
x

j j ... j
i i ...i



E
i i ...i
j j ... j
Let the symbol  i denote
i
x
 i
d  i dx   i dxi
1 2
12
x
absolute differential
m
s
12
1 2
l
m
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Derivatives of Frame-vectors and Christoffel Symbols
Since
ei  e j   ij
it yields
d  (e  s )  (ei dx )
s
Let
  e s  s ,one can see
d  dr 
i
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Derivatives of Frame-vectors and Christoffel Symbols
Let us take  to be a tensor of order 2 as an example .
Suppose
j i
  i E j
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Derivatives of Frame-vectors and Christoffel Symbols
  e s  s  e s  s  ij e i  e j )]
 e s  s  ij   ij ( s e i )  e j   ij e i  ( s e j )]
 e s  s  ij   ij ( isk e k )  e j   ij e i  ( kjs ek )]
 e s [ s  ij  e i  ( ij  kjs ek )  ( ij  isk e k )  e j ]
 e s [ s  ij  e i  (1i (11s e1  12s e2 )   i2 (12 s e1   22 s e2 ))
 (1j (1s1e1  1s 2 e 2 )   2j ( 2s1e1   2s 2 e 2 ))  e j ]
Mathematical & Mechanical
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Derivatives of Frame-vectors and Christoffel Symbols
 e s { s  ij  e1  (12 (11s e1  12s e2 )  12 (12 s e1   22 s e2 ))
e 2  ( 22 (11s e1  12s e2 )   22 (12 s e1   22 s e2 ))
 (11 (1s1e1  1s 2 e 2 )  12 ( 2s1e1   2s 2 e 2 ))  e1
 (12 (1s1e1  1s 2 e 2 )   22 ( 2s1e1   2s 2 e 2 ))  e2 }
 e s [ s  ij  (12 11s + 12 12 s - 111s1  12  2s1 )e1  e1 
 (12 12s + 12  22 s - 12 1s1 -  22  2s1 )e1  e2
 ( 22 11s +  22 12 s - 111s 2 - 12  2s 2 )e 2  e1
 (( 22 12s   22  22 s  12 1s 2   22  2s 2 )e 2  e2
Mathematical & Mechanical
Method in Mechanical Engineering
Derivatives of Frame-vectors and Christoffel Symbols
  e s  s  e s  s  ij e i  e j )]
 e s [ s  ij  (12 11s + 1212 s - 111s1  12  2s1 )e1  e1 
 (12 12s + 12  22 s - 12 1s1 -  22  2s1 )e1  e2
 ( 22 11s +  22 12 s - 111s 2 - 12  2s 2 )e 2  e1
 (( 2212s   22 22 s  121s 2   22 2s 2 )e 2  e2
 e s [ s  ij  ( ih  hsj   hj ish )e i  e j ]
Mathematical & Mechanical
Method in Mechanical Engineering
Derivatives of Frame-vectors and Christoffel Symbols
Let 
sι   s       
j
j
i
h
i
j
hs
j
h
h
is
then
  sι e  e j ]e
j i
We call
 sι
j
covariant derivative of 
s
Mathematical & Mechanical
Method in Mechanical Engineering
Derivatives of Frame-vectors and Christoffel Symbols
Similarly, we have
sij   s ij  ih  hj 
h
js
 s   s       
ij
ij
ih
j
hs
hj
i
sh
h
is
Mathematical & Mechanical
Method in Mechanical Engineering
Derivatives of Frame-vectors and Christoffel Symbols
Therefore,let
A  A( r )
then
A
j l
k A  k e j  A  jk el
x
j
Mathematical & Mechanical
Method in Mechanical Engineering
Tensors of Order 2
Three Representations
T  Tij E , T  T Eij , T  Ti E
ij
ij
x x
Tij  i
T
j mn
x x
m
n
j
i
j
i
j

x

x
T ij  m n T mn
x x
 T11 T12 T13 


(Tij )   T21 T22 T23 
T T T 
 32 32 33 
Mathematical & Mechanical
Method in Mechanical Engineering
Tensors of Order 2
Transpose, Symmetry, Skew-symmetry and Unit
 D11

DT   D12
D
 13
Transpose of a Tensor:DT
( DT )T  D
Symmetric Tensors
skew-symmetric tensor
D D
T
D21
D22
D23
D31 

D32 
D33 
Dij  D ji
Dij   D ji , Dii  0
Mathematical & Mechanical
Method in Mechanical Engineering
Circulation
Unit
I   ij ei  e j
1 0 0


I  0 1 0
0 0 1


Mathematical & Mechanical
Method in Mechanical Engineering
Tensors of Order 2
Inner Product of Two Tensors with Lower Order
Du  v
A linear transformation
Let D, E be two tensor of order two
D E  H
Dik Ekj  H ij
Mathematical & Mechanical
Method in Mechanical Engineering
Tensors of Order 2
Principal Value, Direction and Invariants
u
Let n 
|u|
T
u
=

u
then
T | u | n =  | u | n
(T   I )  n  0
Mathematical & Mechanical
Method in Mechanical Engineering
Tensors of Order 2
Principal Value, Direction and Invariants
T  Tij e  e , n  ni e
i
j
i
(Tij e i  e j   ij e i  e j )  (n j e j )
 (Tij n j (e i  e j )  e j   ij n j (e i  e j )  e j )
 (Tij n j   ij n j )e i  0
Mathematical & Mechanical
Method in Mechanical Engineering
Tensors of Order 2
Principal Value, Direction and Invariants
T21
T31  n1 
 T11  

 
T22  
T32  n2   0
 T12
 T
 n 
T
T


23
33
 13
 3 
shows that, the principal value and direction of tensor T can
be found by the eigenvalue and vector of the matrix [Tij]
Mathematical & Mechanical
Method in Mechanical Engineering
Tensors of Order 2
Principal Value, Direction and Invariants
T11  
T21
T31
T12
T22  
T32
T13
T23
T33  
0
  IT   IIT   IIIT  0
3
2
Mathematical & Mechanical
Method in Mechanical Engineering
Tensors of Order 2
Principal Value, Direction and Invariants
  IT   IIT   IIIT  0
3
2
IT  Tii  trT
1
IIT  (TiiT jj  TijT ji )
2
IIIT | Tij | det T
Mathematical & Mechanical
Method in Mechanical Engineering
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