1 TENSORS/ 3-D STRESS STATE

2 Tensors Tensors are specified in the following manner: –A zero-rank tensor is specified by a sole component, independent of the system of reference (e.g., mass, density). –A first-rank tensor is specified by three (3) components, each associated with one reference axis (e.g., force). –A second-rank tensor is specified by nine (9) components, each associated simultaneously with two reference axes (e.g., stress, strain). –A fourth-rank tensor is specified by 81 components, each associated simultaneously with four reference axes (e.g., elastic stiffness, compliance).

3 The Number of Components (N) required for the description of a TENSOR of the n th Rank in a k-dimensional space is: N = k n EXAMPLES (a)For a 2-D space, only four components are required to describe a second rank tensor. (b)For a 3-D space, the number of components N = 3 n Scalar quantities  3 0  Rank Zero Vector quantities  3 1  Rank One Stress, Strain  3 2  Rank Two Elastic Moduli  3 4  Rank Four (4-1)

4 The indicial (also called dummy suffix) notation will be used. The number of indices (subscripts) associated with a tensor is equal to its rank. It is noted that: –density (  ) does not have a subscript –force has one (F 1, F 2, etc.) –stress has two (  12,  22, etc.) The easiest way of representing the components of a second- rank tensor is as a matrix For the tensor T, we have:

5 The collection of stresses on an elemental volume of a body is called stress tensor, designated as  ij. In tensor notation, this is expressed as: where i and j are iterated over x, y, and z, respectively. (4-2)

6 Here, two identical subscripts (e.g.,  xx ) indicate a normal stress, while a differing pair (e.g.,  xy ) indicate a shear stress. It is also possible to simplify the notation with normal stress designated by a single subscript and shear stresses denoted by , so:  x   xx  xy   xy (4-3)

7 In general, a property T that relates two vectors p = [p 1, p 2, p 3 ] and q = [q 1, q 2, q 3 ] in such a way that where T 11, T 12, ……. T 33 are constants in a second rank tensor. (4-4)

8 (Eqn. 4-4) can be expressed matricially as: Equation 4-5 can be expressed in indicial notation, where The symbol is usually omitted, and the Einstein’s summation rule used. (4-5) (4-6) (4-7) Free Subscript “dummy” Subscript (appears twice)

9 Transformations Transformation of vector p [p 1, p 2, p 3 ] from reference system x 1, x 2, x 3 to reference x ’ 1, x ’ 2, x ’ 3 can be carried out as follows where = X1X1 X’1X’1 X2X2 X3X3 X’2X’2 X’3X’3 Angle between X ’ i X j New Old p

10 In vector notation: p = p 1 i 1 + p 2 i 2 + p 3 i 3 where i 1, i 2, and i 3 are unit vectors p ’ = p 1 cos(X ’ 1 X 1 ) + p 2 cos(X ’ 1 X 1 ) + p 3 cos(X ’ 1 X 1 ) = a 11 p 1 + a 12 p 2 + a 13 p 3 where a ij = cos (X ’ i X j ) is the direction cosine between X ’ i and X j. New Old (4-8) (4-9)

11 The nine angles that the two systems form are as follows: = This is known as the TRANSFORMATION Matrix Old System New System (4-10)

12 What is the transformation matrix for a simple rotation of 30 o about the z-direction? 30 o

13 For any Transformation from p to p ’, determine the Transformation Matrix and use as follows: This can be written as: It is also possible to perform the opposite operation, i.e., new to old (4-11) (4-12) (4-13)

14 The Transformation of a second rank Tensor [T kl ] from one reference frame to another is given as: OR, for stress Eqn. 4-14(a) is the transformation law for tensors and the letters and subscripts are immaterial. Transformation from new to old system is given as: (4-14a) (4-15) (4-14b)

15 NOTES on Transformation Transformation does not change the physical integrity of the tensor, only the components are transformed. Stress/strain Transformation results in nine components. Each component of the transformed 2 nd rank tensor has nine terms. l ij and T ij are completely different, although both have nine components. –L ij is the relationship between two systems of reference. –T ij is a physical entity related to a specific system of reference.

16 Transformation of the stress tensor  ij from the system of axes to the We use eqn. 4-14: –First sum over j = 1, 2, 3 –Then sum over i = 1, 2, 3 (4-16)

17 For each value of k and l there will be an equation similar to eqn To find the equation for the normal stress in the x ’ 1 direction, let m = 1 and n = 1. Let us determine the shear stress on the x ’ plane and the z ’ direction, that is  ’ 13 or  x ’ z ’ for which m = 1 and n = 3

18 The General definition of the Transformation of an n th -rank tensor from one reference system to another (i.e., T  T ’ ) is given by: T ’ mno……. = l mi l nj l ok …………….T ijk……….. Note that a ij = l mi (the letters are immaterial) The transformation does not change the physical integrity of the tensor, only the components are transformed (4-16)