1

Objective

Vectors

Dyads

Tensors

Outer-products

Eigen-analysis

Polar Decomp.

Tensors, Dyads

27-750, Advanced Characterization and Microstructural Analysis,

Spring 2002, A. D. Rollett

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Objective

Vectors

Dyads

Tensors

Outer-products

Eigen-analysis

Polar Decomp.

Objective

• The objective of this lecture is to introduce the student to the concept of tensors and to review some basic concepts relevant to tensors, including dyads.

• Many of the concepts reviewed in this lecture are useful or essential in discussions of elasticity and plasticity.

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Vectors

Dyads

Tensors

Outer-products

Eigen-analysis

Polar Decomp.

Tensors

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

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Vectors

Dyads

Tensors

Outer-products

Eigen-analysis

Polar Decomp.

Dyads: 1

• We are familiar with constructing vectors as triples of coefficients multiplying the unit vectors: we call these tensors of first order.

• In order to work with higher order tensors, it is very useful to construct dyads from the unit vectors.

v v1ˆ e 1 v2 ˆ e 2 v3 ˆ e 3 vi ˆ e i

i1

3

vi ˆ e i

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Vectors

Dyads

Tensors

Outer-products

Eigen-analysis

Polar Decomp.

Dyads: 2

• Define the dyadic product of two vectors. Note coordinate free. Properties are the following:

u

v (ui ˆ e i ) (v j ˆ e j ) uivj ˆ e i ˆ e j

(u )

v

u (

v )

u

v

u (

v

w )

u

v

u

w

(u

v ) w

u w

v w

scalar

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Objective

Vectors

Dyads

Tensors

Outer-products

Eigen-analysis

Polar Decomp.

Dyads: 3

• Transformation () of the dyadic product, from one coordinate system to another, leaves it invariant. This is demonstrated in the following construction:

v

u u i v j ˆ e i ˆ e j irur jsvs(it ˆ e t ) ( juˆ e u)

ir jsit juurvs ˆ e t ˆ e u

rtsuurvs ˆ e t ˆ e u utvu ˆ e t ˆ e u uiv jˆ e i ˆ e j

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Vectors

Dyads

Tensors

Outer-products

Eigen-analysis

Polar Decomp.

Inner products from Dyadics

• The dyadic product is similar to the vector product: it is not commutative.

• Inner products can be combined with the dyadic product:

(u

v )w

u (v

w ) ,

w (u

v ) (

w

u )

v

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Objective

Vectors

Dyads

Tensors

Outer-products

Eigen-analysis

Polar Decomp.

Unit Dyads

• We can construct unit dyads from the unit vectors:

For now we will leave these as they are and not introduce any new symbols.

ˆ e i ˆ e j

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Objective

Vectors

Dyads

Tensors

Outer-products

Eigen-analysis

Polar Decomp.

Dyad example: dislocation slip

• We commonly form a dyad for the strain, m, produced on a slip system (or twinning system) by combining unit vectors that represent slip (twin shear direction) direction, b, and slip plane [normal] (twin plane), n.

mij ˆ b i ˆ n j

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Vectors

Dyads

Tensors

Outer-products

Eigen-analysis

Polar Decomp.

Second Order Tensors

• Unit dyads form the basis for second order (rank) tensors, just as the unit vectors do for vectors, where the Tij are the (nine) coefficients of the tensor.

T Tij ˆ e i ˆ e jExample = stress:

ij ˆ e i ˆ e j

Unit tensorcoefficient

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Objective

Vectors

Dyads

Tensors

Outer-products

Eigen-analysis

Polar Decomp.

Second Order Tensors example: strain from slip

• The dyad for crystallographic slip forms the basis for a second order (rank) strain tensor, eslip, where the magnitude of the tensor is given by the amount of shear strain, ∆ on the given system.

eijslip ˆ b i ˆ n j

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Objective

Vectors

Dyads

Tensors

Outer-products

Eigen-analysis

Polar Decomp.

Unit (spherical) tensors

• The unit tensor, I, is formed from the unit dyad thus:

Note that this tensor is invariant under transformations. An extension of this idea is the isotropic tensor, where C is a constant (scalar),

I ij ˆ e i ˆ e j

Cijˆ e i ˆ e j

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Dyads

Tensors

Outer-products

Eigen-analysis

Polar Decomp.

Symmetric, skew-symmetric tensors

• A (second order) tensor is said to be symmetric (e.g. stress, strain tensors) if

Tij = Tji

• Similarly a tensor is said to be skew-symmetric or antisymmetric (as in small rotations) if

Tij = -Tji• Any tensor can be decomposed into a symmetric

and a skew-symmetric part.

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Dyads

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Outer-products

Eigen-analysis

Polar Decomp.

Tensor: transformations

• Transformation of tensors follows the rules set up for vectors and the unit vectors:

thus:

T Tij ˆ e i ˆ e j T rsˆ e r ˆ e s

Tij (ri ˆ e r) (sj ˆ e s)

risjTij ˆ e r ˆ e s

T rs risjTij

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Objective

Vectors

Dyads

Tensors

Outer-products

Eigen-analysis

Polar Decomp.

Right, left inner products

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

left: right:

Note the order of the indices. Note also that we can speak of a tensor acting on a vector to send it onto another vector.

Tv Tijvj ˆ e i ,

v T viTij ˆ e j

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Vectors

Dyads

Tensors

Outer-products

Eigen-analysis

Polar Decomp.

Inner products of tensors• The composition of, or dot product between two

second-order tensors in the dyadic notation:

• Notice that the dot product between two tensors involves a contraction of the inner indices, r & s. This is also called an inner product.

SSijˆ e i ˆ e j (Uir

ˆ e i ˆ e r )(Tsjˆ e s ˆ e j )

UirTsj ( ˆ e r ˆ e s) ˆ e i ˆ e j UirTrj

ˆ e i ˆ e j UT

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Vectors

Dyads

Tensors

Outer-products

Eigen-analysis

Polar Decomp.

Outer products of tensors

• Consider the outer product of a tensor of second-order with a vector to produce a tensor of third-order:

• Fourth-order tensor is similar:

Tv (Tij ˆ e i ˆ e j ) (vk ˆ e k )

Tijvk ˆ e i ˆ e j ˆ e k

TU TijUrs ˆ e i ˆ e j ˆ e r ˆ e s

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Dyads

Tensors

Outer-products

Eigen-analysis

Polar Decomp.

General Cartesian tensors

• More generally, Cartesian tensors of order n are defined by components by the expression:

• The nth order polydyadics form a complete orthogonal basis for tensors of order n.

T Ti1i2 ...inˆ e i1 ˆ e i2 ... ˆ e in

Ti1i2...in

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Eigen-analysis

Polar Decomp.

nth order tensor transformations

• Changes in the coordinate frame change the components of the nth order tensor according to a simple extension:

T i1i2 ...in

i1 j1i2 j2

...in jnT j1 j2 ... jn

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Dyads

Tensors

Outer-products

Eigen-analysis

Polar Decomp.

Inner products of higher order tensors

• Inner-products on tensors of higher order are defined by contracting over one or more indices. For example, contracting the last n-p indices of tensor T (of order n) with the first n-p indices of a tensor U (of order m) gives a new tensor S (of order 2p+m-n) according to the following.

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Eigen-analysis

Polar Decomp.

Higher order inner products

S TU (Ti1i2 ...inˆ e i1 ˆ e i2 ... ˆ e in )(U j1j2...jm

ˆ e j1 ˆ e j2 ... ˆ e jm )

Ti1i2 ...inU j1j2 ...jm

( ˆ e ip 1ˆ e j1 )( ˆ e i p2

ˆ e j2 )...

(ˆ e in ˆ e jn p)ˆ e i1 ... ˆ e i p

ˆ e jn p 1 ... ˆ e jm

Ti1...in Uip 1...in jn p1...jmˆ e i1 ... ˆ e ip ˆ e jn p1

... ˆ e jm

Si1...ip jn p1...jmˆ e i1 ... ˆ e ip ˆ e jn p1

... ˆ e jm

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

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Vectors

Dyads

Tensors

Outer-products

Eigen-analysis

Polar Decomp.

Higher order outer products

• A natural generalization of the outer product to higher-order tensors is obvious. The outer product of two tensors T and U (of order n and m, respectively) is a new tensor S of order n+m according to the expressionSTU(Ti1i2...in

ˆ e i1 ˆ e i2 ... ˆ e in ) (U j1 j2...jmˆ e j1 ˆ e j2 ... ˆ e jm )

Ti1i2 ...inU j1j2 ...jm

ˆ e i1 ... ˆ e in ˆ e j1 ... ˆ e jm Si1...in j1...jm

ˆ e i1 ... ˆ e in ˆ e j1 ... ˆ e jm

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Vectors

Dyads

Tensors

Outer-products

Eigen-analysis

Polar Decomp.

Eigenanalysis of tensors

• It is very useful to perform eigenanalysis on tensors of all kinds, whether rotations, physical quantities or properties.

• We look for solutions to this equation, where µ is a scalar:

T v

v

(T I) v

0 or,

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Eigen-analysis

Polar Decomp.

Characteristic equation

• The necessary condition for the relation above to have non-trivial solutions is given by:

When the (cubic) characteristic equation is solved, three roots, µi, are obtained which are the eigenvalues of the tensor T. They are also called the principal values of the tensor.

det(T I) 0

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Dyads

Tensors

Outer-products

Eigen-analysis

Polar Decomp.

Eigenvectors

• Assume that the three eigenvalues are distinct. The ith eigenvalue, µi, can be reintroduced into the previous relation in order to solve for the eigenvectors, v(i):

(T i I) v (i)

0

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Dyads

Tensors

Outer-products

Eigen-analysis

Polar Decomp.

Real, Symmetric Tensors

• Consider the special case where the components of T are real and symmetric, e.g. stress, strain tensors. Now let’s evaluate the effect on the eigenvalues andeigenvectors:,which the symmetric nature of the tensor allows it to be re-written as:

T v (i) i

v (i)

v (i) T i

v (i)

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Objective

Vectors

Dyads

Tensors

Outer-products

Eigen-analysis

Polar Decomp.

Eigenvalues of real, symmetric tensors

• Now take the complex conjugate of the components of each element in the above, keeping in mind that T is real:

• Next, take the left inner product of the previous relation with and subtract it from the right inner product of the above relation with :

v (i)* T i *

v (i)*

v (i)

v (i)*

(i * i )v (i)*

v (i) 0

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Vectors

Dyads

Tensors

Outer-products

Eigen-analysis

Polar Decomp.

real eigenvalues

• Given this consequence of non-trivial solutions for the eigenvectors, we see that the eigenvalues of a real-symmetric matrix must be themselves be real valued in order for the previous relation to be satisfied

v (i)*

v (i) 0

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Dyads

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Eigen-analysis

Polar Decomp.

eigenvectors

• Next, take the left inner product of the previous relation, with to obtain and subtract it from the right inner product of with :

( j i )v (i )

v ( j) 0

v (i)

v ( j) T

v (i) i

v (i)

v ( j ) T j

v ( j )

• If the eigenvectors are distinct, the inner product of the associated eigenvectors must be zero.

v ( j ) T

v (i)

v ( j ) i

v (i)

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Dyads

Tensors

Outer-products

Eigen-analysis

Polar Decomp.

Eigenvectors are orthogonal

• If inner (scalar) products of the eigenvectors are zero, then they are orthogonal.

• The eigenvectors of a real-symmetric tensor, associated with distinct eigenvalues, are orthogonal.

• In general the eigenvectors can be normalized by an appropriate selection of scalar multiplier to have unit length.

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Outer-products

Eigen-analysis

Polar Decomp.

Orthonormal eigenvectors

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

• The axes of the coordinate system defined by this orthonormal set of eigenvectors are often called the principal axes of a tensor, T, and their directions are called principal directions.

ˆ v (1) ˆ v (2) ˆ v (3) 1

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Dyads

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Outer-products

Eigen-analysis

Polar Decomp.

Diagonalizing the tensor

• Consider the right and left inner product of tensor T with the eigenvectors according to:

The left hand side of this relation can be expressed in the dyadic notation as:

ˆ v (i) T ˆ v ( j ) j ˆ v (i ) ˆ v ( j ) j ij

ˆ v (i) (Trsˆ e r ˆ e s) ˆ v ( j )

(ˆ v (i ) ˆ e r )(ˆ v ( j) ˆ e s)Trs ir jsTrs

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Polar Decomp.

Transformation to Diagonal form

are the direction cosines linking the orthonormal set of eigenvectors to the original coordinate system for T. Combining the equations above, we get the following, where superscript “d” denotes the diagonal form of the tensor:

ir ˆ v (i) ˆ e r

Tijd ir jsTrs j ij

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Eigen-analysis

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Principal values, diagonal matrix

• are components of the real-symmetric tensor T in the coordinate frame of its eigenvectors. It is evident that the matrix of components of is diagonal, with the eigenvalues appearing along the diagonal of the matrix. (The superscript d highlights the “diagonal” nature of the components in the frame of the eigenvectors.)

Tijd

Tijd

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Invariants of 2nd order tensor

• The product of eigenvalues, µ1µ2µ3, is equal to the determinant of tensor T.det ˆ v (i) T ˆ v (j) det j ij 123

detir jsTrs det T T

(det T )(det T)(det)

det T

det Tsee slide 28, and recall that a transformation has unit det.

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Eigen-analysis

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Invariants 2

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

I3 = det T

I1 trT Tii 1 2 3

I2 1

2trT 2 trT2 23 13 12

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Eigen-analysis

Polar Decomp.

Deviatoric tensors

• Another very useful concept in elasticity and plasticity problems is that of deviatoric tensors.

A’ = A - 1/3I trA• The tensor A’ has the property that its

trace is zero. If A is symmetric then A’ is also symmetric with only five independent components (e.g. the strain tensor, ).

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Eigen-analysis

Polar Decomp.

Deviatoric tensors: 2• Frequently we decompose a tensor into its

deviatoric and spherical parts (e.g. stress):A = A’ + 1/3I trA

e.g. = s + 1/3I tr= s + m

• Non-zero invariants of A’ :I’2=-1/2{(tr A’)2- tr A’2}I’3= det A’ = 1/3 tr A’3

• Re-arrange: I’2=-1/3I12+I2.

I’3=I3-(I1I2)/3+2/27I13

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Positive definite tensors

• The tensor T is said to be positive definite if the above relation holds for any non-zero values of the vector u. A necessary and sufficient condition for T to be positive definite is that the eigenvalues of T are all positive.

u T

u 0

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Polar Decomp.

Polar Decomposition

• Polar decomposition is defined as the unique representation of an arbitrary second-order tensor*, T, as the product of an orthogonal tensor, R, and a positive-definite symmetric tensor, either U or V, according to:

T RU V RWhy do this? For finite deformations, this allows us to separate the rotation from the “stretch” expressed as a positive definite matrix.

* T must have a strictly positive determinant

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Polar Decomposition: 2

• Define a new second-order tensor, A = T-1T. A is clearly symmetric, and that it is positive definite is clear from considering the following:

The right-hand side ofthis equation is positive for any non-zero vector v, and hence vAv is positive for all non-zero v.

v A

v

v TT T

v

v

v

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Polar Decomp.

Polar Decomposition: 3

• Having shown that A is symmetric, positive-definite, we are assured that A has positive eigenvalues. We shall denote these by µ1

2, µ22, µ3

2, where, without loss of generality, µ1, µ2, µ3, are taken to be positive. It is easily verified that the same eigenvectors which are obtained for T are also eigenvectors for A; thus

ˆ v (i) Aˆ v ( j) j2 ij

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Polar Decomp.

Polar Decomposition: 4

• Next we define a new tensor, U, with a diagonal (principal values) matrix, D, and a rotation, R, according to:

U R * DR *T

D jij ˆ e i ˆ e j

R* (ˆ v ( j) ˆ e i )ˆ e i ˆ e j ij ˆ e i ˆ e j

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Polar Decomposition: 5

• Thus, D is a diagonal tensor whose elements are the eigenvalues of T, and R* is the rotation that takes the base vectors into the eigenvectors associated with T. U is symmetric and positive definite, and since R* is orthogonal

ˆ e i ˆ v (i)

U2 R *D2 R *T A

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Polar Decomp.

Polar Decomposition: 6

• The (rotation) tensor R associated with the decomposition is defined by:

That R has the required orthogonality is clear from the following:

R T U 1

RT R U 1 TT T U 1

U 1 AU 1 U 1 U2 U 1 I

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Polar Decomp.

Polar Decomposition: 7

• Thus the (right) U-decomposition of tensor T is defined by relations (2.66) and (2.69). If the (left) V-decomposition is preferred then the following applies:

V RU RT

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Summary

• The important properties and relationships for tensors have been reviewed.

• Second order tensors can written as combination sof coefficients and unit dyads.

• Orthogonal tensors can be used to represent rotations.

• Tensors can be diagonalized using eigenanalysis.• Tensors can decomposed into a combination of a

rotation (orthogonal tensor) and a stretch (positive-definite symmetric tensor).