Tensors grads 20Mar02 by DgjB6vi

VIEWS: 5 PAGES: 47

									  1




Objective

 Vectors

 Dyads
                  Tensors, Dyads
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
              dyads.
 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

                          Dyads: 1
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

                        Dyads: 2
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

                            Dyads: 3
Objective   • Transformation (l) of the dyadic product,
 Vectors
              from one coordinate system to another,
              leaves it invariant. This is demonstrated in
 Dyads
              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

                      Unit Dyads
Objective   • We can construct unit dyads from the
 Vectors      unit vectors:
 Dyads
                          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
              unit dyad thus:
 Dyads

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

 Dyads                  left:             right:
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:
 Dyads

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
 Dyads
                (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
 Dyads                               (i)*                 (i)*
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
 Dyads
              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
 Dyads         orthogonal.
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
 Dyads
                             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
 Dyads
            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
 Dyads
              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.
 Dyads
                    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
 Dyads
              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):
 Dyads
                        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
 Dyads        following:
                                      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
 Dyads
                          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