VIEWS: 5 PAGES: 47 POSTED ON: 2/16/2012 Public Domain
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- rtsuur 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. Cij 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- Ti12 ...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 p2 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 p1 ...jm ei1 ... eip e jn p1 ... e jm ˆ ˆ ˆ ˆ Outer- Si ...i j e ... ei e j ˆ ˆ ˆ ... e j ˆ products 1 p n p1 ... jm i1 p n p1 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 12 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 13 12 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 jijei 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).