Vectors_Tensors_09_Cartesian_Tensors

Vectors_Tensors_09_Cartesian_Tensors - Section 1.9 1.9...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
Section 1.9 Solid Mechanics Part III Kelly 75 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, c b a ) ( , with 3 2 1 2 e e e a + = , 3 2 1 2 e e e b + + = and 3 2 1 e e e c + + = , is 3 2 1 2 2 4 ) ( ) ( e e e c b a c b a + = = 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 i i e e e e e e e e + + 3 3 2 2 1 1 (1.9.1) This follows from () ( ) ( ) ( ) ()()() u e e e u e e u e e u e e u e e u e e u e e u e e e e e e = + + = + + = + + = + + 3 3 2 2 1 1 3 3 2 2 1 1 3 3 2 2 1 1 3 3 2 2 1 1 u u u Note also that the identity tensor can be written as ( ) j i ij e e I = δ , 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 dyadic involving the Cartesian base vectors e i 1 . Consider an arbitrary second-order tensor T which operates on a to produce b , b a T = ) ( , or b e T = ) ( i i a . 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 §1.14)
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Section 1.9 Solid Mechanics Part III Kelly 76 b e T e T e T = + + ) ( ) ( ) ( 3 3 2 2 1 1 a a a Just as T transforms a into b , it transforms the base vectors e i into some other vectors; suppose that w e T v e T u e T = = = ) ( , ) ( , ) ( 3 2 1 , then () () () () () ( ) [] a e w e v e u a e w a e v a e u w e a v e a u e a w v u b 3 2 1 3 2 1 3 2 1 3 2 1 + + = + + = + + = + + = a a a and so 3 2 1 e w e v e u T + + = (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 () ( ) ( ) L L L + + + = + + + + = 1 3 3 1 2 2 1 1 1 3 2 1 3 3 2 2 1 1 e e e e e e e e e e e e T u u u u u u Introduce nine scalars ij T by letting 3 2 1 , , i i i i i i T w T v T u = = = , so that 3 3 33 2 3 32 1 3 31 3 2 23 2 2 22 1 2 21 3 1 13 2 1 12 1 1 11 e e e e e e e e e e e e e e e e e e T + + + + + + + + = T T T T T T T T T Second-order Cartesian Tensor (1.9.3) These nine scalars ij T are the components of the second order tensor T in the Cartesian coordinate system.
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 9

Vectors_Tensors_09_Cartesian_Tensors - Section 1.9 1.9...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online