Section 1.9 1.9 Cartesian Tensors As with the vector, a (higher order) tensor is a mathematical object which represents many physical phenomena and which exists independently of any coordinate system. In what follows, a Cartesian coordinate system is used to describe tensors. 1.9.1 Cartesian Tensors A second order tensor and the vector it operates on can be described in terms of Cartesian components. For example, (a b)c , with a 2e1 e2 e3 , b e1 2e2 e3 and c e1 e 2 e3 , is (a b)c a(b c) 4e1 2e 2 2e3 Example (The Unit Dyadic or Identity Tensor) The identity tensor, or unit tensor, I, which maps every vector onto itself, has been introduced in the previous section. The Cartesian representation of I is e1 e1 e 2 e 2 e3 e3 ei ei (1.9.1) This follows from e1 e1 e 2 e 2 e3 e3 u e1 e1 u e 2 e 2 u e3 e3 u e1 e1 u e 2 e 2 u e3 e3 u u1e1 u2e 2 u3e3 u Note also that the identity tensor can be written as I ij ei e j , in other words the Kronecker delta gives the components of the identity tensor in a Cartesian coordinate system. ■ Second Order Tensor as a Dyadic In what follows, it will be shown that a second order tensor can always be written as a 1 dyadic involving the Cartesian base vectors ei . Consider an arbitrary second-order tensor T which operates on a to produce b, T(a) b , or T(aiei ) b . From the linearity of T, 1 this can be generalised to the case of non-Cartesian base vectors, which might not be orthogonal nor of unit magnitude (see §1.16) Solid Mechanics Part III 75 Kelly Section 1.9 a T(e ) a T(e ) a T(e ) b 1 1 2 2 3 3 Just as T transforms a into b, it transforms the base vectors ei into some other vectors; suppose that T(e1 ) u, T(e 2 ) v, T(e3 ) w , then b a1u a2 v a3w a e1 u a e 2 v a e3 w u e1 a v e 2 a w e3 a u e1 v e 2 w e3 a and so T u e1 v e 2 w e3 (1.9.2) which is indeed a dyadic. Cartesian components of a Second Order Tensor The second order tensor T can be written in terms of components and base vectors as follows: write the vectors u, v and w in (1.9.2) in component form, so that T u1e1 u2e 2 u3e3 e1 e 2 e3 u1e1 e1 u2e 2 e1 u3e3 e1 Introduce nine scalars Tij by letting ui Ti1 , vi Ti2 , wi Ti3 , so that T T11e1 e1 T12e1 e 2 T13e1 e3 T21e 2 e1 T22e 2 e 2 T23e 2 e3 Second-order Cartesian Tensor (1.9.3) T31e3 e1 T32e3 e 2 T33e3 e3 These nine scalars Tij are the components of the second order tensor T in the Cartesian coordinate system. In index notation, T Tij ei e j Thus whereas a vector has three components, a second order tensor has nine components. Similarly, whereas the three vectors ei form a basis for the space of vectors, the nine dyads ei e j form a basis for the space of tensors, i.e. all second order tensors can be expressed as a linear combination of these basis tensors. It can be shown that the components of a second-order tensor can be obtained directly from {▲Problem 1} Tij ei Te j Components of a Tensor (1.9.4) Solid Mechanics Part III 76 Kelly Section 1.9 which is the tensor expression analogous to the vector expression ui ei u . Note that, in Eqn. 1.9.4, the components can be written simply as ei Te j (without a “dot”), since ei Te j ei T e j . Example (The Stress Tensor) Define the traction vector t acting on a surface element within a material to be the force acting on that element2 divided by the area of the element, Fig. 1.9.1. Let n be a vector normal to the surface. The stress σ is defined to be that second order tensor which maps n onto t, according to t σn The Stress Tensor (1.9.5) x2 n 31 11 t t 21 x1 e1 x 3 Figure 1.9.1: stress acting on a plane If one now considers a coordinate system with base vectors ei , then σ ij ei e j and, for example, σe1 11e1 21e 2 31e3 Thus the components 11 , 21 and 31 of the stress tensor are the three components of the traction vector which acts on the plane with normal e1 . Augustin-Louis Cauchy was the first to regard stress as a linear map of the normal vector onto the traction vector; hence the name “tensor”, from the French for stress, tension. ■ 2 this force would be due, for example, to intermolecular forces within the material: the particles on one side of the surface element exert a force on the particles on the other side Solid Mechanics Part III 77 Kelly Section 1.9 Higher Order Tensors The above can be generalised to tensors of order three and higher. The following notation will be used: , , … 0th-order tensors (“scalars”) a, b, c … 1st-order tensors (“vectors”) A, B, C … 2nd-order tensors (“dyadics”) A, B, C … 3rd-order tensors (“triadics”) A, B, C … 4th-order tensors (“tetradics”) An important third-order tensor is the permutation tensor, defined by E ijk ei e j e k (1.9.6) whose components are those of the permutation symbol, Eqns. 1.3.10-1.3.13. A fourth-order tensor can be written as A Aijkl ei e j e k el (1.9.7) It can be seen that a zeroth-order tensor (scalar) has 30 1 component, a first-order tensor has 31 3 components, a second-order tensor has 32 9 components, so A has 33 27 components and A has 81 components. 1.9.2 Simple Contraction Tensor/vector operations can be written in component form, for example, Ta Tij ei e j ak e k T a e e e ij k i j k (1.9.8) Tij ak jk ei Tij a j ei This operation is called simple contraction, because the order of the tensors is contracted – to begin there was a tensor of order 2 and a tensor of order 1, and to end there is a tensor of order 1 (it is called “simple” to distinguish it from “double” contraction – see below). This is always the case – when a tensor operates on another in this way, the order of the result will be two less than the sum of the original orders. An example of simple contraction of two second order tensors has already been seen in Eqn. 1.8.4a; the tensors there were simple tensors (dyads). Here is another example: Solid Mechanics Part III 78 Kelly Section 1.9 TS Tij ei e j S kl e k el T S e e e e ij kl i j k l (1.9.9) Tij Skl jk ei el Tij S jl ei el From the above, the simple contraction of two second order tensors results in another second order tensor. If one writes A TS , then the components of the new tensor are related to those of the original tensors through Aij Tik S kj . Note that, in general, AB BA AB C ABC … associative (1.9.10) AB C AB AC … distributive The associative and distributive properties follow from the fact that a tensor is by definition a linear operator, §1.8.2; they apply to tensors of any order, for example, ABv ABv (1.9.11) To deal with tensors of any order, all one has to remember is how simple tensors operate on each other – the two vectors which are beside each other are the ones which are “dotted” together: a bc b ca a b c d b c a d (1.9.12) a b c d e b c a d e a b c d e f c d a b e f An example involving a higher order tensor is A E Aijkl Emn ei e j ek el e m en Aijkl Eln ei e j ek en and u v AB C Au v Ab C AB C Note the relation (analogous to the vector relation ab cd ab cd , which follows directly from the dyad definition 1.8.3) {▲Problem 10} Solid Mechanics Part III 79 Kelly Section 1.9 AB CD AB CD (1.9.13) Powers of Tensors Integral powers of tensors are defined inductively by T0 I , Tn Tn1T, so, for example, T2 TT The Square of a Tensor (1.9.14) T3 TTT , etc. 1.9.3 Double Contraction Double contraction, as the name implies, contracts the tensors twice as much a simple contraction. Thus, where the sum of the orders of two tensors is reduced by two in the simple contraction, the sum of the orders is reduced by four in double contraction. The double contraction is denoted by a colon (:), e.g. T : S . First, define the double contraction of simple tensors (dyads) through a b: c d a cb d (1.9.15) So in double contraction, one takes the scalar product of four vectors which are adjacent to each other, according to the following rule: a b c : d e f b dcea f For example, T : S Tij ei e j : Skl ek el Tij Skl ei ek e j el (1.9.16) Tij Sij which is, as expected, a scalar.