1 Part B Tensors Or, inverting The graph above represents a transformation of coordinates when the system is rotated at an angle CCW
2 Let be two vectors in a Cartesian coordinate system. If is a transformation, which transforms any vector into some other vector, we can write 2B1 Tensors whereandare two different vectors. If For any arbitrary vectorandand scalar is called a LINEAR TRANSFORMATION and a Second Order Tensor. (1.1a) and (1.2a) can be written as,
3 2B1 Tensors Questions: Yes
4 2B1 Tensors Questions: Yes
5 2B2 Components of a Tensor Scalar has 1 component Vector has 3 component Tensor has ? component?
6 2B2 Components of a Tensor In general The components of T ij can be written as
7 2B2 Components of a Tensor (cont.) Example 2B2.3 Thus Q: Let R correspond to a righ-hand rotation of a rigid body about the x 3 axis by an angle . Find the matrix of R Q: Let R correspond to a 90° right-hand rigid body rotation about the x 3 axis. Find the matrix of R
8 2B3 Components of a transformed vector The transformationis linear if is then called a LINEAR TRANSFORMATION and is a second-order tensor.
9 2B3 Components of a transformed vector Now what is b? what are the components of b?
10 2B2 Components of a Tensor (cont.)
11 2B4 Sum of Tensors Definition:
12 2B5 Product of Two Tensors Components: or In general, the product of two tensors is not commutative:
13 Q: Let R correspond to a 90° right-hand rigid body rotation about the x 3 axis. Find the matrix of R 2B5 Product of Two Tensors Q: Let R correspond to a 90° right-hand rigid body rotation about the x 1 axis. Find the matrix of S Q: Let W correspond to a 90° right-hand rigid body rotation about the x 3 axis, then a 90° right-hand rigid body rotation about the x 1 axis. Find the matrix of W
14 2B6 Transpose of a Tensor Transpose of ~ T is denoted by ~ T T. If a and then Note
15 2B7 Dyadic Product Definition, dyadic product of two vectors is also a tensor. It transforms a vector into a vector parallel to
16 2B8 Trace of a Tensor
17 2B9 Identity Tensor A linear transformation which transforms every vector into itself is an identify tensor
18 2B9 Inverse Tensor Note that
19 2B10 Orthogonal Tensor Thus and
20 2B10 Orthogonal Tensor (cont.) Thus ~~~ QQI T Also, we have ~~ QQI 1, So Thus for an orthogonal matrix, the transpose is also its inverse. Example: A rigid body rotation is an ORTHOGONAL tensor,
21 2B11 Transformation Matrix Between Two Coordinate Systems We see that,
22 2B11 Transformation Matrix Between Two Coordinate Systems (cont.) Thus Q: Let be obtained by rotating the basis about the x 3 axis by an angle . Find the transformation matrix of R from to
23 2B12 Transformation laws for Vectors.
24 2B12 Transformation laws for Vectors. (cont.) Answer
25 2B13 Transformation law for tensor Let and Recall that the components of a tensor are:
26 2B13 Transformation law for tensor (cont.)
27 2B13 Transformation law for tensor (cont.) Find the matrix of T with respect to the basis
28 2B14 Tensors by Transformation laws
29 2B14 Tensors by Transformation laws (cont.) The components transform as follows:
30 2B14 Tensors by Transformation laws (cont.)