# Tensors grads 20Mar02 by DgjB6vi

VIEWS: 5 PAGES: 47

• pg 1
```									  1

Objective

Vectors

Tensors

Outer-
products    27-750, Advanced Characterization
Eigen-       and Microstructural Analysis,
analysis
Spring 2002, A. D. Rollett
Polar
Decomp.
2

Objective
Objective   • The objective of this lecture is to
Vectors      introduce the student to the concept of
Dyads        tensors and to review some basic
Tensors
concepts relevant to tensors, including
Outer-
products    • Many of the concepts reviewed in this
Eigen-       lecture are useful or essential in
analysis
discussions of elasticity and plasticity.
Polar
Decomp.
3

Tensors
Objective   • Tensors are extremely useful for
Vectors      describing anisotropic properties in
Dyads        materials. They permit complicated
Tensors
behaviors to be described in a compact
fashion that can be easily translated
Outer-
products      into numerical form (i.e.
Eigen-       programming).
analysis

Polar
Decomp.
4

Objective   • We are familiar with constructing
Vectors      vectors as triples of coefficients
Dyads        multiplying the unit vectors: we call
these tensors of first order.
Tensors                                       3
Outer-
products
v  v1e1  v2 e2  v3e3 
ˆ       ˆ      ˆ            viei  vi ei
ˆ       ˆ
i 1
Eigen-
analysis
• In order to work with higher order
Polar        tensors, it is very useful to construct
Decomp.
dyads from the unit vectors.
5

Objective   • Define the dyadic product of two
Vectors      vectors. Note coordinate free.
Dyads        Properties are the following:
Tensors     u  v  (ui ei )  (v je j )  uiv j ei  e j
ˆ          ˆ             ˆ    ˆ
Outer-
products    (u )  v  u  (v )   u  v 
Eigen-
analysis    u  (v  w )  u  v  u  w
scalar
(u  v )  w  u  w  v  w
Polar
Decomp.
6

Objective   • Transformation (l) of the dyadic product,
Vectors
from one coordinate system to another,
leaves it invariant. This is demonstrated in
the following construction:
Tensors     v  u  ui ˆi e   lirurl js vs (litet )  ( l jueu )
vj e ˆ j                          ˆ            ˆ
Outer-
products
 lir l jslit l juurvs et  eu
ˆ    ˆ
Eigen-               rtsuur vset  eu
ˆ ˆ
analysis
 ut vu et  eu
ˆ    ˆ
Polar
Decomp.              uiv jei  e j
ˆ    ˆ
7

Inner products from Dyadics
Objective    • The dyadic product is similar to the
Vectors       vector product: it is not commutative.
Dyads       • Inner products can be combined with
Tensors        the dyadic product:
Outer-
products           (u  v ) w  u(v w ) ,
Eigen-
analysis           w  (u  v )  (w u )v
Polar
Decomp.
8

Objective   • We can construct unit dyads from the
Vectors      unit vectors:
ei  e j
ˆ    ˆ
Tensors

Outer-
products      For now we will leave these as they
Eigen-       are and not introduce any new
analysis      symbols.
Polar
Decomp.
9

Dyad example: dislocation slip
Objective    • We commonly form a dyad for the
Vectors       strain, m, produced on a slip system (or
Dyads         twinning system) by combining unit
Tensors
vectors that represent slip (twin shear
direction) direction, b, and slip plane
Outer-
products       [normal] (twin plane), n.
Eigen-
analysis
ˆ
mij  bi  n j
ˆ
Polar
Decomp.
10

Second Order Tensors
Objective   • Unit dyads form the basis for second
Vectors      order (rank) tensors, just as the unit
Dyads        vectors do for vectors, where the Tij are
Tensors
the (nine) coefficients of the tensor.
coefficient
T  Tijei  e j
ˆ    ˆ           Unit tensor
Outer-
products

Eigen-
analysis      Example = stress:
Polar
Decomp.
   ijei  e j
ˆ    ˆ
11    Second Order Tensors example:
strain from slip
Objective   • The dyad for crystallographic slip
Vectors      forms the basis for a second order
Dyads        (rank) strain tensor, eslip, where the
Tensors
magnitude of the tensor is given by the
amount of shear strain, ∆g, on the
Outer-
products      given system.
Eigen-
analysis           e   slip        ˆ n
 g bi ˆ j
ij
Polar
Decomp.
12

Unit (spherical) tensors
Objective   • The unit tensor, I, is formed from the
Vectors

Tensors
I   ijei  e j
ˆ ˆ
Outer-       Note that this tensor is invariant under
products      transformations. An extension of this
Eigen-       idea is the isotropic tensor, where C is
analysis      a constant (scalar),
Polar
Decomp.                Cij ei  e j
ˆ    ˆ
13

Symmetric, skew-symmetric tensors
Objective   • A (second order) tensor is said to be
Vectors      symmetric (e.g. stress, strain tensors) if
Dyads                          Tij = Tji
Tensors     • Similarly a tensor is said to be skew-
symmetric or antisymmetric (as in small
Outer-
products
rotations) if
Eigen-
Tij = -Tji
analysis    • Any tensor can be decomposed into a
Polar
symmetric and a skew-symmetric part.
Decomp.
14

Tensor: transformations
Objective   • Transformation of tensors follows the
Vectors      rules set up for vectors and the unit
vectors:
Dyads                                     ˆ  ˆ 
T  Tijei  e j  Trser  es
ˆ    ˆ
Tensors

Outer-                   Tij (lrier )  (lsjes )
ˆ        ˆ 
products

Eigen-                   lri lsj Tijer  es
ˆ ˆ 
analysis      thus:
Polar
Decomp.
Trs  lrilsjTij

15

Right, left inner products
Objective   • Right and left inner-products of the second-
order tensor, T, with a vector:
Vectors

Tensors

Outer-
Tv  Tij v jei , v T  viTij e j
ˆ                ˆ
products

Eigen-       Note the order of the indices. Note also that
analysis
we can speak of a tensor acting on a vector
Polar        to send it onto another vector.
Decomp.
16

Inner products of tensors
Objective
• The composition of, or dot product between
two second-order tensors in the dyadic
Vectors      notation:
Dyads       S  Sij ei  e j  (U ir ei  er )  (Tsj es  e j )
ˆ    ˆ           ˆ    ˆ           ˆ    ˆ
Tensors
 U ir Tsj (er  es )ei  e j
ˆ ˆ ˆ         ˆ
Outer-
products         U ir Trj ei  e j
ˆ    ˆ
Eigen-
analysis         U T
Polar
• Notice that the dot product between two
Decomp.       tensors involves a contraction of the inner
indices, r & s. This is also called an inner
product.
17

Outer products of tensors
Objective   • Consider the outer product of a tensor
Vectors      of second-order with a vector to
Dyads        produce a tensor of third-order:
Tensors           T  v  (Tijei  e j )  (v k ek )
ˆ    ˆ            ˆ
Outer-
products                   Tijvk ei  e j  ek
ˆ ˆ        ˆ
Eigen-
analysis
• Fourth-order tensor is similar:
Polar
Decomp.        T  U  TijUrsei  e j  er  es
ˆ ˆ        ˆ    ˆ
18

General Cartesian tensors
Objective   • More generally, Cartesian tensors of
Vectors
order n are defined by components
Ti1i2 ...in by the expression:

Tensors

Outer-       T  Ti1i2 ...in ei1  ei2  ...  ein
ˆ     ˆ           ˆ
products

Eigen-     • The nth order polydyadics form a
analysis
complete orthogonal basis for tensors
Polar        of order n.
Decomp.
19

n th   order tensor transformations
Objective     • Changes in the coordinate frame
Vectors        change the components of the nth order
Dyads          tensor according to a simple extension:
Tensors

Outer-
Ti12 ...in 
i
products

Eigen-
analysis
li j li j ...li j T j
1 1       2 2       n n    1 j 2 ... j n
Polar
Decomp.
20        Inner products of higher order
tensors
Objective    • Inner-products on tensors of higher
Vectors       order are defined by contracting over
Dyads         one or more indices. For example,
Tensors
contracting the last n-p indices of
tensor T (of order n) with the first n-p
Outer-
products       indices of a tensor U (of order m) gives
Eigen-        a new tensor S (of order 2p+m-n)
analysis       according to the following.
Polar
Decomp.
21

Higher order inner products
Objective
S  TU  (Ti i ...i ei  ei  ... ei )(U j j ... j e j  e j  ... e j )
ˆ    ˆ         ˆ                ˆ     ˆ          ˆ
12 n 1         2         n     12 m 1            2          m
Vectors
 Ti i ...i U j j ... j (ei
ˆ        e j )(ei
ˆ     ˆ       e j )...
ˆ
12 n 1 2 m              p 1      1     p2      2
(ein  e jn p )ei1  ... ei p  e jn p 1  ... e jm
ˆ ˆ            ˆ          ˆ      ˆ                 ˆ
Tensors
 Ti1...in Uip 1 ...in jn p1 ...jm ei1  ... eip  e jn p1  ... e jm
ˆ          ˆ     ˆ                ˆ
Outer-
 Si ...i j           e  ... ei  e j
ˆ       ˆ    ˆ           ... e j
ˆ
products          1 p n  p1 ... jm i1        p     n  p1            m

Eigen-     Here, 0<p<n. From (2.30) it should be evident that the
analysis    order of each of the tensors S, T and U (as specified by m, n
Polar
and p) must be known in order to correctly form the
Decomp.     product. The order of the contraction is n-p (sometimes
denoted by the number of dots between the symbols).
22

Higher order outer products
Objective   • A natural generalization of the outer
Vectors      product to higher-order tensors is
Dyads        obvious. The outer product of two
Tensors
tensors T and U (of order n and m,
respectively) is a new tensor S of order
Outer-
products      n+m according to the expression
S T U
Eigen-
analysis     (Ti1i2 ...in ei1  ei2  ... ein )  (U j1 j2 ... jm e j1  e j2  ... e jm )
ˆ     ˆ          ˆ                       ˆ      ˆ           ˆ
 Ti1i2 ...in U j1 j2 ... jm ei1  ... ein  e j1  ... e jm
ˆ          ˆ     ˆ           ˆ
Polar
Decomp.          Si ...i j ...j ei  ... ei  e j  ... e j
1   n1    m 1
ˆ
n
ˆ
1
ˆ          ˆ
m
23

Eigenanalysis of tensors
Objective   • It is very useful to perform
Vectors      eigenanalysis on tensors of all kinds,
Dyads        whether rotations, physical quantities
Tensors
or properties.
Outer-
• We look for solutions to this equation,
products      where µ is a scalar:
Eigen-
analysis                 T v  v
Polar
Decomp.
or,      (T  I)  v  0
24

Characteristic equation
Objective   • The necessary condition for the
Vectors      relation above to have non-trivial
Dyads        solutions is given by:
Tensors                det(T  I)  0
Outer-
products      When the (cubic) characteristic
Eigen-       equation is solved, three roots, µi, are
analysis      obtained which are the eigenvalues of
Polar        the tensor T. They are also called the
Decomp.
principal values of the tensor.
25

Eigenvectors
Objective   • Assume that the three eigenvalues are
Vectors      distinct. The ith eigenvalue, µi, can be
Dyads        reintroduced into the previous relation
Tensors
in order to solve for the eigenvectors,
v(i):
Outer-
(i)
products
(T  i I)  v         0
Eigen-
analysis

Polar
Decomp.
26

Real, Symmetric Tensors
Objective   • Consider the special case where the
Vectors      components of T are real and
Dyads        symmetric, e.g. stress, strain tensors.
Tensors
Now let’s evaluate the effect on the
eigenvalues and             (i)         (i)
Outer-
products      eigenvectors:        T v  iv
Eigen-       ,which the symmetric
analysis      nature of the tensor
(i)              (i)
Polar
Decomp.
allows it to be      v  T  iv
re-written as:
27        Eigenvalues of real, symmetric
tensors
Objective   • Now take the complex conjugate of the
Vectors      components of each element in the
above, keeping in
Tensors
mind that T is real: v          T  i * v
Outer-
• Next, take the left inner product of the
(i)*
products      previous relation with   v and subtract
Eigen-       it from the right inner product of the
analysis                              (i)
above relation with v :
Polar                                        (i)* (i)
Decomp.                     ( i *  i )v        v  0
28

real eigenvalues
(i)*        (i)
Objective     v          v         0
Vectors
• Given this consequence of non-trivial
solutions for the eigenvectors, we see
Tensors
that the eigenvalues of a real-
Outer-
products
symmetric matrix must be themselves
be real valued in order for the previous
Eigen-
analysis      relation to be satisfied
Polar
Decomp.
29

eigenvectors
Objective   • Next, take the left inner product of the
(i)           (i)
Vectors      previous relation, T v  iv
( j)
Dyads        with  v       to obtain  v ( j )  T  v (i)  v ( j )  iv (i)
and subtract it from the right inner product
Tensors
v  T   jv
( j)           ( j)             (i)
of                       with v :
Outer-
products                                     (i )        ( j)
Eigen-
(j    
      i )v          v          0
analysis
       • If the eigenvectors are distinct, the inner
Polar
Decomp.
product of the associated eigenvectors must
be zero.
30

Eigenvectors are orthogonal
Objective    • If inner (scalar) products of the
Vectors       eigenvectors are zero, then they are
Tensors      • The eigenvectors of a real-symmetric
Outer-
tensor, associated with distinct
products       eigenvalues, are orthogonal.
Eigen-      • In general the eigenvectors can be
analysis
normalized by an appropriate selection
Polar
Decomp.        of scalar multiplier to have unit length.
31

Orthonormal eigenvectors
Objective   • Convenient to select the set of eigenvectors
in a right-handed manner such that:
Vectors

(1)    (2)      (3)
ˆ       v
ˆ     v
ˆ       1
v
Tensors
• The axes of the coordinate system defined
Outer-       by this orthonormal set of eigenvectors are
products
often called the principal axes of a tensor, T,
Eigen-       and their directions are called principal
analysis      directions.
Polar
Decomp.
32

Diagonalizing the tensor
Objective   • Consider the right and left inner
Vectors      product of tensor T with the
Dyads        eigenvectors according to:
(i)       ( j)        (i ) ( j )
Tensors
v T  v   j v  v   j ij
ˆ         ˆ          ˆ      ˆ
Outer-
products      The left hand side of this relation can
Eigen-       be expressed in the dyadic notation as:
analysis      ˆ(i) (Trser  es )  v( j )
v         ˆ    ˆ ˆ
Polar
Decomp.           (v (i )  er )(v ( j)  es )Trs  l irl jsTrs
ˆ        ˆ ˆ           ˆ
33

Transformation to Diagonal form
(i)
Objective   l ir  v
ˆ      er are the direction
ˆ
Vectors
cosines linking the orthonormal set of
eigenvectors to the original coordinate
system for T. Combining the
Tensors
equations above, we get the following,
Outer-
products
where superscript “d” denotes the
Eigen-
diagonal form of the tensor:
d
 lirl jsTrs   j ij
analysis

Polar
Tij
Decomp.
34

Principal values, diagonal matrix
d
Objective   • Tij are components of the real-
Vectors
symmetric tensor T in the coordinate
frame of its eigenvectors. It is evident
that the matrix of components of Tij d
Tensors       is diagonal, with the eigenvalues
Outer-       appearing along the diagonal of the
products      matrix. (The superscript d highlights
Eigen-       the “diagonal” nature of the
analysis      components in the frame of the
Polar        eigenvectors.)
Decomp.
35

Invariants of         2 nd   order tensor
Objective    • The product of eigenvalues, µ1µ2µ3, is
Vectors       equal to the determinant of tensor T.
ˆ (i)  T  v(j)  det  j ij  12 3
det v           ˆ
Tensors
 det l irl jsTrs  det T T
Outer-
T
products
 (det  )(detT)(det)
Eigen-
analysis                        detT
Polar                          detT
Decomp.
see slide 28, and recall that a transformation has unit det.
36

Invariants 2
Objective   • Other combinations of components
Vectors
which form (three) invariants of
second-order tensors include, where
T2=T•T (inner, or dot product):
Tensors

Outer-            I1  trT  Tii  1  2  3
products

Eigen-
analysis
1
2
    2      2

I2  trT  trT  2 3  13  12
Polar               I3 = det T
Decomp.
37

Deviatoric tensors
Objective   • Another very useful concept in
Vectors      elasticity and plasticity problems is
Dyads        that of deviatoric tensors.
Tensors
A’ = A - 1/3I trA
Outer-
• The tensor A’ has the property that its
products      trace is zero. If A is symmetric then
Eigen-       A’ is also symmetric with only five
analysis
independent components (e.g. the
Polar        strain tensor, e).
Decomp.
38

Deviatoric tensors: 2
• Frequently we decompose a tensor into
Objective
its deviatoric and spherical parts (e.g.
Vectors
stress):
A = A’ + 1/3I trA
Tensors       e.g.       = s + 1/3I tr= s + m
Outer-
products
• Non-zero invariants of A’ :
I’2=-1/2{(tr A’)2- tr A’2}
Eigen-
analysis      I’3= det A’ = 1/3 tr A’3
Polar      • Re-arrange: I’2=-1/3I12+I2.
Decomp.
I’3=I3-(I1I2)/3+2/27I13
39

Positive definite tensors
Objective     u T u  0
Vectors    • The tensor T is said to be positive
Dyads        definite if the above relation holds for
Tensors
any non-zero values of the vector u. A
Outer-
necessary and sufficient condition for
products      T to be positive definite is that the
Eigen-       eigenvalues of T are all positive.
analysis

Polar
Decomp.
40

Polar Decomposition
Objective   • Polar decomposition is defined as the
Vectors      unique representation of an arbitrary
Dyads        second-order tensor*, T, as the product of
Tensors
an orthogonal tensor, R, and a positive-
definite symmetric tensor, either U or V,
Outer-
products      according to:
Eigen-
analysis
T  R U  V  R
Why do this? For finite deformations, this
Polar
Decomp.      allows us to separate the rotation from the
“stretch” expressed as a positive definite matrix.
* T must have a strictly positive determinant
41

Polar Decomposition: 2
Objective   • Define a new second-order tensor, A = T-1T.
A is clearly symmetric, and that it is positive
Vectors      definite is clear from considering the
T
v  A v  v  T  T  v  v  v 
Tensors

Outer-
products      The right-hand side ofthis equation is
Eigen-       positive for any non-zero vector v, and
analysis      hence vAv is positive for all non-zero v.
Polar
Decomp.
42

Polar Decomposition: 3
Objective   • Having shown that A is symmetric,
Vectors      positive-definite, we are assured that A
Dyads        has positive eigenvalues. We shall
Tensors
denote these by µ12, µ22, µ32, where,
without loss of generality, µ1, µ2, µ3,
Outer-
products      are taken to be positive. It is easily
Eigen-       verified that the same eigenvectors
analysis      which are obtained for T are also
Polar        eigenvectors for A; thus
(i)        ( j)      2
Decomp.
v  A v   j  ij
ˆ         ˆ
43

Polar Decomposition: 4
Objective   • Next we define a new tensor, U, with a
Vectors      diagonal (principal values) matrix, D,
Dyads        and a rotation, R, according to:
T
Tensors

Outer-
U  R * D R *
products
D   jijei  e j
ˆ    ˆ
Eigen-
analysis
( j)
Polar       R*  (v ˆ         ei )ei  e j  lijei  e j
ˆ ˆ      ˆ        ˆ    ˆ
Decomp.
44

Polar Decomposition: 5
Objective   • Thus, D is a diagonal tensor whose
Vectors      elements are the eigenvalues of T, and
Dyads        R* is the rotation that takes the base
ˆi into the eigenvectors v (i)
vectors e                         ˆ
Tensors
associated with T. U is symmetric and
Outer-
products      positive definite, and since R* is
Eigen-       orthogonal
analysis
2            2      T
Polar             U  R *D  R *  A
Decomp.
45

Polar Decomposition: 6
Objective   • The (rotation) tensor R associated with
Vectors      the decomposition is defined by:
1
R  T U
Tensors
That R has the required orthogonality
Outer-
products      is clear from the following:
Eigen-           T
R R U       1 T
T  T U   1  
analysis

Polar
U 1  A U 1  U 1 U 2 U 1  I
Decomp.
46

Polar Decomposition: 7
Objective   • Thus the (right) U-decomposition of
Vectors      tensor T is defined by relations (2.66)
Dyads        and (2.69). If the (left) V-
Tensors
decomposition is preferred then the
following applies:
Outer-
T
products

Eigen-
V  R U  R
analysis

Polar
Decomp.
47

Summary
Objective   • The important properties and relationships
for tensors have been reviewed.
Vectors
• Second order tensors can written as
Dyads        combination sof coefficients and unit dyads.
Tensors     • Orthogonal tensors can be used to represent
rotations.
Outer-
products    • Tensors can be diagonalized using
eigenanalysis.
Eigen-
analysis    • Tensors can decomposed into a combination
of a rotation (orthogonal tensor) and a
Polar
Decomp.       stretch (positive-definite symmetric tensor).

```
To top