Vectors_Tensors_09_Cartesian_Tensors

# Vectors_Tensors_09_Cartesian_Tensors - Section 1.9 1.9...

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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)

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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.
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Vectors_Tensors_09_Cartesian_Tensors - Section 1.9 1.9...

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